Research ArticleDynamic Behaviour Analysis of Turbocharger Rotor-ShaftSystem in Thermal Environment Based on Finite ElementMethod
Zhihao Liu1 Renren Wang2 Fang Cao 1 and Pidong Shi1
1School of Mechanical and Automotive Engineering Qilu University of Technology (Shandong Academy of Sciences)Jinan 250353 China2School of Electrical Engineering and Automation Qilu University of Technology (Shandong Academy of Sciences)Jinan 250353 China
Correspondence should be addressed to Fang Cao caofangqlueducn
Received 17 March 2020 Revised 23 June 2020 Accepted 29 July 2020 Published 14 August 2020
Academic Editor oi Trung Nguyen
Copyright copy 2020 Zhihao Liu et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
e stable operation of a high-speed rotating rotor-bearing system is dependent on the internal damping of its materials In thisstudy the dynamic behaviours of a rotor-shaft system with internal damping composite materials under the action of atemperature field are analysed e temperature field will increase the tangential force generated by the internal damping of thecomposite material e tangential force will also increase with the rotor speed which can destabilise the rotor-shaft system Tobetter understand the dynamic behaviours of the system we introduced a finite element calculation model of a rotor-shaft systembased on a 3D high-order element (Solid186) to study the turbocharger rotor-bearing system in a temperature field e analysiswas done according to the modal damping coefficient stability limit speed and unbalance responsee results show that accurateprediction of internal damping energy dissipation in a temperature field is crucial for accurate prediction of rotor dynamicperformance is is an important step to understand dynamic rotor stress and rotor dynamic design
1 Introduction
Turbochargers are mechanical devices that can improve fuelefficiency and reduce greenhouse gas emissions e corecomponent of a turbocharger is a rotor composed mainly ofa turbine and a compressor e turbine is crucial because itcan recover energy from the exhaust gas and increase theintake air volume by driving the compressor [1] It ischaracterized by light weight and high speed To meet re-quirements for higher speeds greater power density and theability to adapt to harsh operating environments for a longtime structural designers use various materials to manu-facture the rotor parts is introduces high requirementsfor the stable rotation of the rotor erefore to analyse thestability of a rotor and its dynamic characteristics it isnecessary to accurately predict the damping effect Dampingis divided into external damping such as bearing dampingand internal damping such as material damping which inturn is mainly modelled by viscous damping and hysteretic
damping [2] In a dynamic composite rotor internaldamping is meaningful only when the matrix can damp [3]Also most single materials such as metals have vibrationdamping characteristics similar to those of hysteretic in-ternal damping but not viscous internal damping [4]
Many researchers have studied the effect of materialdamping on rotor dynamics and stability behaviour Sin [5]studied the instability phenomenon of a composite shaftwith internal damping and calculated the natural frequencyand instability threshold of the shaft by using the finiteelement model for beams ey found that the stackingsequence of layers of composite materials the arrangementdirections of material fibers and the transverse shear forcesof materials affected the natural frequency and instabilitythreshold of the shaft Wittgren [6] studied the stability offlexible rotors with flexible supports ey found that thecorrect combination of the asymmetry of the internaldamping of the shaft material and the anisotropy of bearingscould significantly reduce the amplitude at critical speed
HindawiShock and VibrationVolume 2020 Article ID 8888504 18 pageshttpsdoiorg10115520208888504
Montagnier [7] used EulerndashBernoulli shaft model to analysethe critical speed of a flexible internal resistance rotorsupported by elastic bearings to obtain the stable workingrange of the rotor ey found that most of the rotor in-stability was caused by the internal damping of the materialIncreasing the bearing damping could make the rotor runmore stably and they established a rotor instability analysismodel that included the internal resistance of the materialVitta [8] used linear and nonlinear evaluation methods toanalyse the effect of internal damping on the dynamic be-haviour of a shaft e results show that the critical sub-critical and stable limit speeds of a rotor can be obtained bylinear evaluation Newkirk [9] studied internal damping andfound that a rotor may vibrate violently at a speed higherthan the first critical speed Genta [10] explained that thehysteretic damping of rotating structures is stable in thesubcritical range but unstable in the supercritical range Allthose researchers proved that the instability speed is higherthan the first-order critical speed when considering thehysteretic damping of rotating structures Also many re-searchers have studied the combined effects of materialdamping and bearing damping e results show that in-creasing the bearing damping can improve the stability ofthe rotor but increasing the material damping can reducethe instability threshold [6]
e effect of the surrounding temperature on materialdamping of a turbocharger rotor cannot be disregarded inhigh-temperature environments It is very important tounderstand the dynamic behaviours of the structure invarious temperatures when designing a rotor to operate inthermal extremes San [11] established a nonlinear rotor-bearing finite element model e model considered thethermal effects of lubricating oil and the thermal expansionsof the rotor shaft and bearing and San Andres analysed thenatural frequency stability and unbalanced response of therotor e results show that the natural frequency andsynchronous speed amplitude obtained by the simulationwere completely consistent with the experimental valueswhich verified the feasibility of the finite element modelZych [12] used a finite element method to calculate thethermal stress of a radial axial microturbine in a high-temperature environment In the calculation they consid-ered the mass of the disc the rotation speed of the rotor andthe complex shape at the rear of the disc eir work showedthat numerical calculation helps to choose the best opti-mization method and they reduced the turbinersquos von Misesstress by approximately 45 Jeyaraj [13] uniformly heated athin plate that incorporated various internal dampingcomposite materials and did free vibration and forced vi-bration analyses e study found that with increasingtemperature the vibration amplitude (response) of the thinplate structure decreased but the modal loss factor increasedsignificantly with increasing temperature thereby reducingthe vibration amplitude Its response frequency also de-creased with increasing temperature Guo [14] did thermalvibration analysis of a rotating beam structure with con-strained layer damping e study found that as the tem-perature increased the modal frequency of the beamstructure decreased accordingly and the damping ratio
increased accordingly at study provides a basis for dy-namic analysis of high-speed rotating blades in variousthermal environments However the number of articles onthis subject is limited
erefore this research studied the dynamic characteristicsof internal material damping and an oil film force turbochargerrotor-bearing system under thermal environments (tempera-ture fields)We used the conjugate heat transfer (CHT)methodto simulate the temperature field of the solid part of a rotor-shaft system e temperature field was coupled with the finiteelement model of the rotor compared with the instancewithout considering the temperature field e rotor finiteelement is verified by experiment
2 Numerical Model and Research Methods
21 Fluid and Heat Transfer Sections Before the thermalmodal analysis the aerodynamic thermal analysis wasperformed e current industry standard modelling ap-proaches assume the turbine and compressor operate underadiabatic conditions [15] e CHT simulation method[16ndash22] is used to obtain the temperature distribution of therotor and the node temperature is provided for the modalsimulation part
211 Turbocharge and Compressor Geometry e turbo-charger had an impeller with 10 blades and the compressorhad an impeller with 6 blades and 6 splitters e turbo-charger and compressor parameters are shown in Table 1 Toimprove the accuracy of the calculations we modelled boththe turbocharger and the compressor using full passagesesolid parts and the air flow are shown in Figure 1
212 Grid Generation CHT involves the direct coupling offluids and solids ICEM grid discretisation software (AnsysInc USA) uses the same numerical principles and griddiscretisation for both regions is allows the non-interpolated exchange of heat flux between adjacent grids[23] e calculation accuracy of CHT is very sensitive to theresolution of the fluid boundary layer grid erefore thedimensionless distance y+ 1 or less (in this paper y+ 1)of the wall distance of the first layer of the grid can determinethe local heat flux with enough accuracy As shown inFigures 2 and 3 the fluid domain solid domain (rotorimpeller) and boundary layer grid were generated for CHTcalculations e total number of global grids was12329631 with 2067903 global grid nodes Following theshape of the outer contour of the rotor we used a triangulartetrahedral mesh with good adaptability to the outer con-tour and we used smoothing to optimize the mesh eimpeller grid and the fluid grid were connected in a generalgrid interface mode in Ansys CFX software
213 Boundary Conditions We determined the boundaryconditions for aerodynamic thermal analysis using experi-mental data provided by the turbocharger company theboundary conditions are shown in Table 2
2 Shock and Vibration
Table 1 Turbocharger and compressor parameters
Turbine side Compressor sideParameters Value and units Parameters Value and unitsBlades number 10 Blades number 6 + 6 (minus)Impeller inlet diameter 5505mm Impeller outlet diameter 565mmTip clearance 041mm Tip clearance 026mm
Turbineside
Solid casing Solid casingInlet
Inlet
Outlet
Impeller Impeller
Sha
Compressorside
Outlet
Figure 1 Centrifugal turbocharger and compressor with a solid casing
Figure 2 Global grid for conjugate heat transfer calculation
Turbinefluid
Turbineimpeller
Compressorfluid Compressor
impeller
Figure 3 Computational grids for conjugate heat transfer calculation
Table 2 e boundary conditions for aerothermal analysis
Turbine side Compressor sideMedium (intensity 5) Medium (intensity 5)Inlet mass flow 0074 kgs Inlet total pressure 999359 PaInlet total temperature 87297K Inlet total temperature 297201 KOutlet static pressure 968927 Pa Outlet static pressure 127578 Pa
Shock and Vibration 3
e heat transfer of the surrounding air was disregardedand the out-wall of the turbocharger was assumed to beadiabatic We applied ldquono-sliprdquo boundary conditions to allinner walls An interface was added between the rotatingdomain and the fixed domain and the interface was con-nected by a ldquofrozen rotorrdquo [24]
214 Numerical Methodology CHTrefers to a coupled heattransfer phenomenon in which the thermal properties of twomaterials occur through a medium or in direct contact eCHT method can calculate the heat transfer between fluidand solid and calculate the temperatures of fluids and solidsat the same time In this study we used the commercialsoftware Ansys CFX for numerical simulation CFX is acomputational fluid dynamics software package based on thecontrol volume method to solve NavierndashStokes equations Inthe fluid domain the mass conservation momentum andenergy transport equations are described as
where uf ρf p τ λf T htot and t represent the velocityvector density pressure stress tensor thermal conductivitytemperature total enthalpy and time of the fluidrespectively
In a solid domain the conservation of energy equationcan explain the heat transfer caused by solid motion con-duction and a volume heat source e energy equation is
zhtot ρs
zt+ nabla middot ρs us htot( 1113857 nabla middot λsnablaT( 1113857 + SE (4)
where us is the velocity vector of the solid us uf SE is theoptional volume heat source SE nabla middot (us middot τ) ρs is the densityof the solid and λs is the thermal conductivity of the solid
is study did not directly solve (1)ndash(4) Instead theywere converted to the steady-state Reynolds averageNavierndashStokes method to calculate turbulence e fluidmedium (exhaust gas and air) is an ideal gas and the shearstress transmission (SST) turbulence model was used be-cause the SST turbulence model has good accuracy for CHTcalculations [20] at model has both the accuracy of thek minus ω model in high-pressure gradient flow boundary layerprediction and the stability of the k minus e model in mainstreamprediction [18]
22 Modal Part
221 Rotor-Shaft System Geometry emodel of the rotor-shaft system was provided by the turbocharger company andagreed with the real onee rotor-shaft system comprised a
turbine wheel a compressor wheel a rotating shaft afloating ring bearing a thrust bearing a seal sleeve and anut as Figure 4 shows e turbine wheel and shaft wereconnected by friction welding whereas the compressor wasaxially and circumferentially fixed by a left-hand nut
e materials of the main parts of the rotor-shaft systemare shown in Table 3 based on the actual situation
is study investigated the effect of temperature andfound that the material properties changed as the temper-ature changed e function curves of the material prop-erties of each part as a function of temperature are shown inFigure 5 [25 26]
Publication [29] proposes a set of damping coefficientsand we verified the influence of damping coefficients on thestability limit displacement of the rotor through theoreticalanalysis and numerical simulation We selected this set ofdamping coefficients based on the theory of the publication
Table 4 shows what we assumed was the materialdamping coefficient of each part of the rotor
222 Floating Ring Bearing Parameters e rotor shaft wassupported by a floating ring bearing whose detailed pa-rameters are shown in Table 5
To aid the simulation we combined the stiffness anddamping coefficients of the inner and outer oil film into atotal impedance (equivalent stiffness and damping coeffi-cients) according to [27 28] Combining the stiffness anddamping coefficients of the inner and outer oil film providedby the turbocharger company with the parameters of thefloating ring bearing we calculated the total impedanceusing Dyrobes rotor dynamics software (Dyrobes USA)Figure 6 shows the total impedance (equivalent stiffness anddamping coefficients) of the floating ring bearing at variousspeeds
223 Finite Element Model of Rotor is study used a large-scale general software Ansys Workbench for simulation Toobtain the motion trace on the axis the rotor shaft wasequally divided into four parts along the axis direction asshown in Figure 7 e blue lines (solid and dotted) in thefigure represent the division positions
e solid model was discretised and an Ansys Solid186element was used e Solid186 element is a high-order 3D20-node element with quadratic displacement characteris-tics e element is defined by 20 nodes each node withthree degrees of freedom (translation of each node in the x-y- and z-directions) e element supports plasticitysuperelasticity creep stress hardening large deflection andlarge strain capacity In the shaft part the grid was generatedby a sweep as shown in Figure 8 depicting the ldquoithrdquo elementafter segmentation
We transformed (deformed) the original hexahedronelement to an approximately one-quarter cylinder elementIn Figure 8 the red dotted line represents the actual edge ofthe element and the red solid line represents the edge of theelement e advantage of a solid model is that it retains theproperties of the original element and describes the outercontour of the shaft model well
4 Shock and Vibration
To adapt to the more complicated outer contour of theblade tetrahedral pyramid and prismmethods were used togenerate the unit in the blade and the remaining partFigure 9 shows the (i + 1) (i + 2) and (i + 3) elementsgenerated by three methods
e effects of rotational inertia translational inertia gyromoment support stiffness support damping materialdamping rotational damping and thermal stress stiffnesswere considered in the model to establish the equation ofmotion e rotor model was discretised into Ne elementsand the total number of nodes became a e node dis-placement vector of the ith element is
qe xiI yiI ziI xiJ yiJ ziJ xiμ yiμ ziμ xiA1113966
yiA ziA xiB yiB ziB1113865T
(5)
where μ I J K L M N O P Q R S T U V W
X Y Z A B is the node number and the superscript T isthe matrixvector transpose ese writing formats are alsomentioned later in the text
After combining the governing equations of all elementsand combining the boundary conditions the equation ofmotion of the rotor-bearing system is
Meuroq (t) + C _q(t) + Kq (t) f (t) (6)
where M M trs + M rot C [minusΩG + Ccon + Cbrg] andK [Kbrg + Ksh + Ktem + Ktemσ]
HereM is a mass matrix that includes mass componentscaused by translation along the axis (Mtrs) and rotation alongthe axis (Mrot) C is a damping matrix (asymmetric matrixcoefficient of the velocity vector) that includes a skew-symmetric gyro matrix (G) a constant structural dampingmatrix (Ccon βKcon) with hysteresis characteristics causedby internal friction of the material and damping caused bythe floating ring bearing support matrix (Cbrg) K is thestiffness matrix (matrix coefficient of the displacementvector) that includes the stiffness matrix (Kbrg) caused by thesupport of the floating ring bearing the spin-softeningbending matrix (Ksh) generated by the rotor rotation thestiffness matrix (Ktem) of the elastic modulus the stiffnessmatrix (Ktemσ) caused by the nonuniform thermal stress andthermal deformation caused by temperature changes esize of all matrices is 3a times 3a Ω is the rotor speed in rpm βis the material damping (Rayleigh damping) stiffnessdamping coefficient
e total displacement and global force vectors q(t) andf(t) are expressed as
where matrices A and B state vector w(t) and force vectorF(t) are
A 0 M
M C1113890 1113891
6atimes6a
B M 0
0 minusK1113890 1113891
6atimes6a
w(t) _q (t)
q (t)1113896 1113897
6atimes1F(t)
0
f(t)1113896 1113897
6atimes1
(9)
where 0 is an empty matrix of size 3a times 3a and an emptyvector of size 3a times 1
Figure 10 shows a simplified finite element model of therotor-shaft system e grey-blue points in the figure areused to observe the modal shapes line trajectories andunbalanced harmonic response analysis
Figure 11 shows the rotor finite element global grid etotal number of grids was 3176485 with 546067 gridnodes In the geometry software each portion of the rotorwas treated as a part to avoid extra contact stiffness in thefinite element simulation and prevent affecting the simu-lation accuracy
To aid the comparison we considered two cases in thisstudy case 1 the rotor-shaft system without the influence oftemperature and case 2 the rotor-shaft system affected bytemperature In the following sections r-th forward and r-thbackward modes are represented as ldquor-Frdquo and ldquor-Brdquo modesrespectively
3 Results and Discussion
31 Heat Transfer Results and Experimental VerificationAfter the calculation we verified the numerical results of themass flow and the supercharger ratio (expansion ratio) enumerical results agreed with the experimental data and theerror control was approximately 7 as Figure 12 shows
e temperature distribution of the rotor was obtainedthrough postprocessing as Figure 13 shows the temperaturedistribution law was the same as that in the Bohn [19] studyresult and is shown in Figure 14
Figures 13 and 14 show that the highest temperature wasat the outer edge of the turbine wheel blades and that thelowest temperature was at the compressor wheel Overallthe rotor had a large temperature gradient
32 Modal Results
321 Campbell Diagram and Stability Diagram of NormalTemperature Environment Figure 15 shows the Campbelldiagram of the rotor-shaft system in case 1 that iswithout considering the temperature e figure was
Figure 5 Material properties of rotor-shaft system as a function of temperature (Youngrsquos modulus Poisson ratio specific heat capacity andthermal conductivity) (a) K418 (b) ZL105 (c) 42CrMo (d) 316 stainless steel [25 26]
Table 4 Damping coefficient of each partPart name Damping coefficientTurbo impeller 0002Compressor impeller and nut 0001Shaft 0005rust bearing and seal sleeve 0005
Table 5 Parameters of floating ring bearing that supported therotor shaftParameter name Parameter valueFloating ring mass 517 gInner length 514mmOuter length 796mmInner diameter 750mmOuter diameter 129mm
Shock and Vibration 7
Y
Divide
Z
X
14 cylinder
Figure 7 Rotor shaft divided into four parts and 14 cylinder
Figure 6 Total impedance of floating ring bearing (equivalent stiffness and damping coefficients) (a) Turbine side (b) Compressor side
8 Shock and Vibration
obtained from the whirl frequency (obtained from theimaginary part of the eigenvalue) and the rotational speedTo present a clear figure and express its basic
characteristics only the first four whirling directions aredescribed In the figure there are two forward whirls (thewhirl direction of the rotor agrees with the rotation
cyx cyx
cxy
cyy cyykyxkyx
kyy
kxxkxx kxycxx
cxy cxxkxy
kyy
1 2 3 4 5 6 7 8 9 10 11
Y
X Z
Figure 10 Finite element model of rotor-shaft system
N
Z
U
J
QM
Y
T L
B
P
IY
Z
X
V ON V
U A
RK X
S
M
YQ
I
O
W
P
B
LT
Z
JTransit (sweep)A
K
SX
R W
Figure 8 Solid186 element generated by the sweep method
N
Z
U
J
QM
Y
T L
B
P
I
V O
A
K
SX
R W
YZ
X
Transit (tetrahedral) (i + 1)
(i + 2)
(i + 3)
J K O N R AV Z
J K O N R A V Z
O N V
W U
P M XT
I
QY SK
BL
AR
P M X
W U
I
Q
Q
I
UM X
W
PB
L
SY
T
J Z
YS
T L B
Transit (pyramid)
Transit (prism)
Figure 9 Solid186 elements generated by tetrahedral pyramid and prism methods
Shock and Vibration 9
direction of the speed) which are marked as 1F and 2F inorder of increased frequency ere are also two backwardwhirls (the whirl direction of the rotor is opposite to therotation direction of the speed) that are marked as 3B and
4B respectively e synchronous whirl line is called thesynchronous excitation line In the figure the velocitiescorresponding to the intersection of the synchronouswhirl line and the frequency line of the forward whirl are
Temperature8177
Turbineside
Compressorside
(K)
7688
7199
6710
6222
5733
5244
4755
4266
3777
Figure 13 Postprocessing rotor temperature distribution
Figure 11 Global grid of rotor-shaft system
008
007
006
005
004
003
002
001
000
60000rpm
80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
12 13 14 15 16 17 18
ExperimentalCFD
(a)
014
012
010
008
006
004
002
000
60000rpm80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
10 11 12 13 14 1615 17
ExperimentalCFD
(b)
Figure 12 Numerical simulation verified by experiments (a) Turbine side (b) Compressor side
10 Shock and Vibration
Compressorside
Turbineside035 10
TToT times103 (K)
Figure 14 Rotor temperature distribution of Bohn [19]
Figure 16 Stability diagram of rotor-shaft system without considering temperature
Shock and Vibration 11
marked as Ωcr1 and Ωcr2 respectively and were calculatedas follows Ωcr1 5 0115 rpm and Ωcr2 43 622 rpmBecause the backward whirl could not be excited by theunbalanced force it is not marked or considered
When the speed was in the critical speed range be-tween Ωcr1 and Ωcr2 in the 1F mode the whirling fre-quency had a clear increasing trend at trend may havebeen caused by floating ring bearings because in thecritical speed range of Ωcr1 and Ωcr2 the stiffness anddamping of the oil film changed sharply When the criticalspeed of Ωcr2 was exceeded the stiffness and damping ofthe oil film changed smoothly and the eddy frequency inthe corresponding 1F mode also increased at a constantrate
Figure 16 shows that the curve of the damping ratio ofthe four modes in case 1 changed with the rotation speede modal damping ratio is the ratio of actual damping tothe critical damping of a certain mode and the dampingratio is the decisive parameter to describe the stability Apositive damping ratio means that the rotor-shaft systemwas in a stable region when the vibration energy wasdissipated A negative damping ratio means that therotation energy supported the rotor rotation by addingvibration energy resulting in the rotor-shaft systembecoming unstable e point where the damping ratiocurve intersects with the zero line is the stable limit speed(SLS) e figure shows that the rotor-shaft system be-came unstable when SLS 1 56 550 rpm in 1F mode andthen became unstable when SLS 2 97 946 rpm in 2Fmode
Figure 17 shows the four modes close to the SLS 1stability limit speed at 55000 rpm e black dotted linerepresents the axis the combination of the orange line andthe orange sphere represents the starting position of thetrajectory and the incomplete track curve represents thewhirl direction e figure shows that the 1F mode wascylindrical thus combining rigid body motion with rotorbending Furthermore the rotor was close to the unstablestate at this time In contrast the 2F mode was a conicalbending mode e figure shows that the 3B mode wasconical with both rigid body motion and rotor bendingLastly the 4B mode was cylindrical with both rigid bodymotion and rotor bending
Figure 18 shows the four modes close to the SLS 2stability limit speed at 97946 rpme rotor was close to theunstable state at this time e 1F and 3B modes were cy-lindrical and conical combining rigid body motion androtor bending In contrast the 2F mode was a conicalbending mode
322 Effect of Temperature on the Campbell Diagram andStability Diagram We exported the temperature data ofeach node in Figure 12 and inserted the temperature valueinto the corresponding rotor-shaft system node through the
Ansys software commands functionWe analysed the criticalspeed and stability of the rotor-shaft system in the thermalenvironment
Figure 19 shows that temperature affected the intrinsicfrequency (whirl frequency) Compared with case 1 thewhirl-frequency curves in the 1F 2F and 3B modes wereslightly lower whereas for the 4B mode the whirl-fre-quency curve was much lower In the figure the startingpoint at the left end (when the rotating speed was 0 rpm)was taken as the observation object In case 1 the frequencywas 23009 Hz whereas in case 2 the frequency decreasedto 21825 Hz Because the temperature changed the ma-terialrsquos elastic modulus and Poissonrsquos ratio it affected itsstructural stiffness For higher-order frequencies the effecton frequency became greater as the order increasedHowever the effect on the critical speed Ωcr1 did notchange much whereas Ωcr2 decreased from 43622 rpm to41815 rpm is decrease was also caused by the intrinsicfrequency drop
Figure 20 shows the stability diagram in case 2 and thatthe temperature had an important effect on the dampingratio and SLS Compared with case 1 SLS 1 of the 1F modedecreased from the original 56550 rpm to 55078 rpm andmade the 1F mode enter the unstable region ahead of timee 2F mode change was more obvious and the same SLS 2decreased from the original 97946 rpm to 95969 rpm ereason for this can be explained by the following formulaξr (α2ωhr) + (βωhr2) Here ξr is the damping ratiocoefficient of the rth mode α is the mass damping coeffi-cient β is the stiffness damping coefficient and ωhr is thenatural frequency of the rth mode In most cases α (α 0)can be disregarded that is ξr βωhr2 β did not changewhereas ωhr decreased because of the effect of temperatureresulting in a decrease in ξr In other words SLS 1 and SLS 2in the 1F and 2F modes were reduced erefore whenpredicting the stability limit reset and dynamic perfor-mance of a turbocharger rotor the effect of temperatureshould have a great influence on the dynamic design of therotor
323 Unbalance Response Analysis We applied a dummyunbalance mass of 001 kg middot mm at node 2 and the fre-quency range of the steady-state synchronous responseanalysis was 0 to 2500Hz (corresponding speed range0 to 150 000 rpm) We then measured the response am-plitude of the turbine end floating ring bearing centerposition node 5 In Figure 21 case 1 is represented by ablack curve (normal temperature) and case 2 by a redcurve (high temperature) In case 1 (case 2) the curvesrepresent 6F 6B and 20F modes e frequency at thewave crest corresponding to the frequency correspondingto the fast reverse rotation (the frequency of the rotationspeed within one minute) in the case 1 (case 2) Campbelldiagram as shown in Figures 22 and 23) is similar to the
12 Shock and Vibration
black (red) number in Figure 20 Because the steeringmode is dominant in the response diagram the criticalspeeds Ωcr1 and Ωcr2 in case 1 (case 2) 1F and 2F modesare not shown in the diagram and followed a phenom-enon similar to that reported by Chouksey and Roy[29 30] Because of the effect of temperature the steering
frequency of case 2 shifted to the left making the changeof response amplitude more obvious e responsevalue at the peak of the 6F and 6B modes decreased from0147mm to 0014 mm and that at the peak of the20F and 20B modes decreased from 0036 mm to00037 mm
324 Verification of Rotor FEM Natural frequency anal-ysis is very important as the basis of a dynamic analysisWe obtained the first two natural frequencies of the rotorby hammering experiments and verified them by nu-merical simulation Figure 24 shows the measured parts(rotor system) force hammer and piezoelectric acceler-ation sensor To reduce the measurement error rate we
adopted a sensor with a nonlinearity of less than 12 anda sensitivity of 10 mvg and placed the sensor in themiddle of the rotor shaft Table 6 compares the naturalfrequency experimental value with the simulated valuee experimental and simulated values of the first naturalfrequency were 1386Hz and 1453 Hz respectively esecond-order natural frequencies were 4100 Hz and
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
Montagnier [7] used EulerndashBernoulli shaft model to analysethe critical speed of a flexible internal resistance rotorsupported by elastic bearings to obtain the stable workingrange of the rotor ey found that most of the rotor in-stability was caused by the internal damping of the materialIncreasing the bearing damping could make the rotor runmore stably and they established a rotor instability analysismodel that included the internal resistance of the materialVitta [8] used linear and nonlinear evaluation methods toanalyse the effect of internal damping on the dynamic be-haviour of a shaft e results show that the critical sub-critical and stable limit speeds of a rotor can be obtained bylinear evaluation Newkirk [9] studied internal damping andfound that a rotor may vibrate violently at a speed higherthan the first critical speed Genta [10] explained that thehysteretic damping of rotating structures is stable in thesubcritical range but unstable in the supercritical range Allthose researchers proved that the instability speed is higherthan the first-order critical speed when considering thehysteretic damping of rotating structures Also many re-searchers have studied the combined effects of materialdamping and bearing damping e results show that in-creasing the bearing damping can improve the stability ofthe rotor but increasing the material damping can reducethe instability threshold [6]
e effect of the surrounding temperature on materialdamping of a turbocharger rotor cannot be disregarded inhigh-temperature environments It is very important tounderstand the dynamic behaviours of the structure invarious temperatures when designing a rotor to operate inthermal extremes San [11] established a nonlinear rotor-bearing finite element model e model considered thethermal effects of lubricating oil and the thermal expansionsof the rotor shaft and bearing and San Andres analysed thenatural frequency stability and unbalanced response of therotor e results show that the natural frequency andsynchronous speed amplitude obtained by the simulationwere completely consistent with the experimental valueswhich verified the feasibility of the finite element modelZych [12] used a finite element method to calculate thethermal stress of a radial axial microturbine in a high-temperature environment In the calculation they consid-ered the mass of the disc the rotation speed of the rotor andthe complex shape at the rear of the disc eir work showedthat numerical calculation helps to choose the best opti-mization method and they reduced the turbinersquos von Misesstress by approximately 45 Jeyaraj [13] uniformly heated athin plate that incorporated various internal dampingcomposite materials and did free vibration and forced vi-bration analyses e study found that with increasingtemperature the vibration amplitude (response) of the thinplate structure decreased but the modal loss factor increasedsignificantly with increasing temperature thereby reducingthe vibration amplitude Its response frequency also de-creased with increasing temperature Guo [14] did thermalvibration analysis of a rotating beam structure with con-strained layer damping e study found that as the tem-perature increased the modal frequency of the beamstructure decreased accordingly and the damping ratio
increased accordingly at study provides a basis for dy-namic analysis of high-speed rotating blades in variousthermal environments However the number of articles onthis subject is limited
erefore this research studied the dynamic characteristicsof internal material damping and an oil film force turbochargerrotor-bearing system under thermal environments (tempera-ture fields)We used the conjugate heat transfer (CHT)methodto simulate the temperature field of the solid part of a rotor-shaft system e temperature field was coupled with the finiteelement model of the rotor compared with the instancewithout considering the temperature field e rotor finiteelement is verified by experiment
2 Numerical Model and Research Methods
21 Fluid and Heat Transfer Sections Before the thermalmodal analysis the aerodynamic thermal analysis wasperformed e current industry standard modelling ap-proaches assume the turbine and compressor operate underadiabatic conditions [15] e CHT simulation method[16ndash22] is used to obtain the temperature distribution of therotor and the node temperature is provided for the modalsimulation part
211 Turbocharge and Compressor Geometry e turbo-charger had an impeller with 10 blades and the compressorhad an impeller with 6 blades and 6 splitters e turbo-charger and compressor parameters are shown in Table 1 Toimprove the accuracy of the calculations we modelled boththe turbocharger and the compressor using full passagesesolid parts and the air flow are shown in Figure 1
212 Grid Generation CHT involves the direct coupling offluids and solids ICEM grid discretisation software (AnsysInc USA) uses the same numerical principles and griddiscretisation for both regions is allows the non-interpolated exchange of heat flux between adjacent grids[23] e calculation accuracy of CHT is very sensitive to theresolution of the fluid boundary layer grid erefore thedimensionless distance y+ 1 or less (in this paper y+ 1)of the wall distance of the first layer of the grid can determinethe local heat flux with enough accuracy As shown inFigures 2 and 3 the fluid domain solid domain (rotorimpeller) and boundary layer grid were generated for CHTcalculations e total number of global grids was12329631 with 2067903 global grid nodes Following theshape of the outer contour of the rotor we used a triangulartetrahedral mesh with good adaptability to the outer con-tour and we used smoothing to optimize the mesh eimpeller grid and the fluid grid were connected in a generalgrid interface mode in Ansys CFX software
213 Boundary Conditions We determined the boundaryconditions for aerodynamic thermal analysis using experi-mental data provided by the turbocharger company theboundary conditions are shown in Table 2
2 Shock and Vibration
Table 1 Turbocharger and compressor parameters
Turbine side Compressor sideParameters Value and units Parameters Value and unitsBlades number 10 Blades number 6 + 6 (minus)Impeller inlet diameter 5505mm Impeller outlet diameter 565mmTip clearance 041mm Tip clearance 026mm
Turbineside
Solid casing Solid casingInlet
Inlet
Outlet
Impeller Impeller
Sha
Compressorside
Outlet
Figure 1 Centrifugal turbocharger and compressor with a solid casing
Figure 2 Global grid for conjugate heat transfer calculation
Turbinefluid
Turbineimpeller
Compressorfluid Compressor
impeller
Figure 3 Computational grids for conjugate heat transfer calculation
Table 2 e boundary conditions for aerothermal analysis
Turbine side Compressor sideMedium (intensity 5) Medium (intensity 5)Inlet mass flow 0074 kgs Inlet total pressure 999359 PaInlet total temperature 87297K Inlet total temperature 297201 KOutlet static pressure 968927 Pa Outlet static pressure 127578 Pa
Shock and Vibration 3
e heat transfer of the surrounding air was disregardedand the out-wall of the turbocharger was assumed to beadiabatic We applied ldquono-sliprdquo boundary conditions to allinner walls An interface was added between the rotatingdomain and the fixed domain and the interface was con-nected by a ldquofrozen rotorrdquo [24]
214 Numerical Methodology CHTrefers to a coupled heattransfer phenomenon in which the thermal properties of twomaterials occur through a medium or in direct contact eCHT method can calculate the heat transfer between fluidand solid and calculate the temperatures of fluids and solidsat the same time In this study we used the commercialsoftware Ansys CFX for numerical simulation CFX is acomputational fluid dynamics software package based on thecontrol volume method to solve NavierndashStokes equations Inthe fluid domain the mass conservation momentum andenergy transport equations are described as
where uf ρf p τ λf T htot and t represent the velocityvector density pressure stress tensor thermal conductivitytemperature total enthalpy and time of the fluidrespectively
In a solid domain the conservation of energy equationcan explain the heat transfer caused by solid motion con-duction and a volume heat source e energy equation is
zhtot ρs
zt+ nabla middot ρs us htot( 1113857 nabla middot λsnablaT( 1113857 + SE (4)
where us is the velocity vector of the solid us uf SE is theoptional volume heat source SE nabla middot (us middot τ) ρs is the densityof the solid and λs is the thermal conductivity of the solid
is study did not directly solve (1)ndash(4) Instead theywere converted to the steady-state Reynolds averageNavierndashStokes method to calculate turbulence e fluidmedium (exhaust gas and air) is an ideal gas and the shearstress transmission (SST) turbulence model was used be-cause the SST turbulence model has good accuracy for CHTcalculations [20] at model has both the accuracy of thek minus ω model in high-pressure gradient flow boundary layerprediction and the stability of the k minus e model in mainstreamprediction [18]
22 Modal Part
221 Rotor-Shaft System Geometry emodel of the rotor-shaft system was provided by the turbocharger company andagreed with the real onee rotor-shaft system comprised a
turbine wheel a compressor wheel a rotating shaft afloating ring bearing a thrust bearing a seal sleeve and anut as Figure 4 shows e turbine wheel and shaft wereconnected by friction welding whereas the compressor wasaxially and circumferentially fixed by a left-hand nut
e materials of the main parts of the rotor-shaft systemare shown in Table 3 based on the actual situation
is study investigated the effect of temperature andfound that the material properties changed as the temper-ature changed e function curves of the material prop-erties of each part as a function of temperature are shown inFigure 5 [25 26]
Publication [29] proposes a set of damping coefficientsand we verified the influence of damping coefficients on thestability limit displacement of the rotor through theoreticalanalysis and numerical simulation We selected this set ofdamping coefficients based on the theory of the publication
Table 4 shows what we assumed was the materialdamping coefficient of each part of the rotor
222 Floating Ring Bearing Parameters e rotor shaft wassupported by a floating ring bearing whose detailed pa-rameters are shown in Table 5
To aid the simulation we combined the stiffness anddamping coefficients of the inner and outer oil film into atotal impedance (equivalent stiffness and damping coeffi-cients) according to [27 28] Combining the stiffness anddamping coefficients of the inner and outer oil film providedby the turbocharger company with the parameters of thefloating ring bearing we calculated the total impedanceusing Dyrobes rotor dynamics software (Dyrobes USA)Figure 6 shows the total impedance (equivalent stiffness anddamping coefficients) of the floating ring bearing at variousspeeds
223 Finite Element Model of Rotor is study used a large-scale general software Ansys Workbench for simulation Toobtain the motion trace on the axis the rotor shaft wasequally divided into four parts along the axis direction asshown in Figure 7 e blue lines (solid and dotted) in thefigure represent the division positions
e solid model was discretised and an Ansys Solid186element was used e Solid186 element is a high-order 3D20-node element with quadratic displacement characteris-tics e element is defined by 20 nodes each node withthree degrees of freedom (translation of each node in the x-y- and z-directions) e element supports plasticitysuperelasticity creep stress hardening large deflection andlarge strain capacity In the shaft part the grid was generatedby a sweep as shown in Figure 8 depicting the ldquoithrdquo elementafter segmentation
We transformed (deformed) the original hexahedronelement to an approximately one-quarter cylinder elementIn Figure 8 the red dotted line represents the actual edge ofthe element and the red solid line represents the edge of theelement e advantage of a solid model is that it retains theproperties of the original element and describes the outercontour of the shaft model well
4 Shock and Vibration
To adapt to the more complicated outer contour of theblade tetrahedral pyramid and prismmethods were used togenerate the unit in the blade and the remaining partFigure 9 shows the (i + 1) (i + 2) and (i + 3) elementsgenerated by three methods
e effects of rotational inertia translational inertia gyromoment support stiffness support damping materialdamping rotational damping and thermal stress stiffnesswere considered in the model to establish the equation ofmotion e rotor model was discretised into Ne elementsand the total number of nodes became a e node dis-placement vector of the ith element is
qe xiI yiI ziI xiJ yiJ ziJ xiμ yiμ ziμ xiA1113966
yiA ziA xiB yiB ziB1113865T
(5)
where μ I J K L M N O P Q R S T U V W
X Y Z A B is the node number and the superscript T isthe matrixvector transpose ese writing formats are alsomentioned later in the text
After combining the governing equations of all elementsand combining the boundary conditions the equation ofmotion of the rotor-bearing system is
Meuroq (t) + C _q(t) + Kq (t) f (t) (6)
where M M trs + M rot C [minusΩG + Ccon + Cbrg] andK [Kbrg + Ksh + Ktem + Ktemσ]
HereM is a mass matrix that includes mass componentscaused by translation along the axis (Mtrs) and rotation alongthe axis (Mrot) C is a damping matrix (asymmetric matrixcoefficient of the velocity vector) that includes a skew-symmetric gyro matrix (G) a constant structural dampingmatrix (Ccon βKcon) with hysteresis characteristics causedby internal friction of the material and damping caused bythe floating ring bearing support matrix (Cbrg) K is thestiffness matrix (matrix coefficient of the displacementvector) that includes the stiffness matrix (Kbrg) caused by thesupport of the floating ring bearing the spin-softeningbending matrix (Ksh) generated by the rotor rotation thestiffness matrix (Ktem) of the elastic modulus the stiffnessmatrix (Ktemσ) caused by the nonuniform thermal stress andthermal deformation caused by temperature changes esize of all matrices is 3a times 3a Ω is the rotor speed in rpm βis the material damping (Rayleigh damping) stiffnessdamping coefficient
e total displacement and global force vectors q(t) andf(t) are expressed as
where matrices A and B state vector w(t) and force vectorF(t) are
A 0 M
M C1113890 1113891
6atimes6a
B M 0
0 minusK1113890 1113891
6atimes6a
w(t) _q (t)
q (t)1113896 1113897
6atimes1F(t)
0
f(t)1113896 1113897
6atimes1
(9)
where 0 is an empty matrix of size 3a times 3a and an emptyvector of size 3a times 1
Figure 10 shows a simplified finite element model of therotor-shaft system e grey-blue points in the figure areused to observe the modal shapes line trajectories andunbalanced harmonic response analysis
Figure 11 shows the rotor finite element global grid etotal number of grids was 3176485 with 546067 gridnodes In the geometry software each portion of the rotorwas treated as a part to avoid extra contact stiffness in thefinite element simulation and prevent affecting the simu-lation accuracy
To aid the comparison we considered two cases in thisstudy case 1 the rotor-shaft system without the influence oftemperature and case 2 the rotor-shaft system affected bytemperature In the following sections r-th forward and r-thbackward modes are represented as ldquor-Frdquo and ldquor-Brdquo modesrespectively
3 Results and Discussion
31 Heat Transfer Results and Experimental VerificationAfter the calculation we verified the numerical results of themass flow and the supercharger ratio (expansion ratio) enumerical results agreed with the experimental data and theerror control was approximately 7 as Figure 12 shows
e temperature distribution of the rotor was obtainedthrough postprocessing as Figure 13 shows the temperaturedistribution law was the same as that in the Bohn [19] studyresult and is shown in Figure 14
Figures 13 and 14 show that the highest temperature wasat the outer edge of the turbine wheel blades and that thelowest temperature was at the compressor wheel Overallthe rotor had a large temperature gradient
32 Modal Results
321 Campbell Diagram and Stability Diagram of NormalTemperature Environment Figure 15 shows the Campbelldiagram of the rotor-shaft system in case 1 that iswithout considering the temperature e figure was
Figure 5 Material properties of rotor-shaft system as a function of temperature (Youngrsquos modulus Poisson ratio specific heat capacity andthermal conductivity) (a) K418 (b) ZL105 (c) 42CrMo (d) 316 stainless steel [25 26]
Table 4 Damping coefficient of each partPart name Damping coefficientTurbo impeller 0002Compressor impeller and nut 0001Shaft 0005rust bearing and seal sleeve 0005
Table 5 Parameters of floating ring bearing that supported therotor shaftParameter name Parameter valueFloating ring mass 517 gInner length 514mmOuter length 796mmInner diameter 750mmOuter diameter 129mm
Shock and Vibration 7
Y
Divide
Z
X
14 cylinder
Figure 7 Rotor shaft divided into four parts and 14 cylinder
Figure 6 Total impedance of floating ring bearing (equivalent stiffness and damping coefficients) (a) Turbine side (b) Compressor side
8 Shock and Vibration
obtained from the whirl frequency (obtained from theimaginary part of the eigenvalue) and the rotational speedTo present a clear figure and express its basic
characteristics only the first four whirling directions aredescribed In the figure there are two forward whirls (thewhirl direction of the rotor agrees with the rotation
cyx cyx
cxy
cyy cyykyxkyx
kyy
kxxkxx kxycxx
cxy cxxkxy
kyy
1 2 3 4 5 6 7 8 9 10 11
Y
X Z
Figure 10 Finite element model of rotor-shaft system
N
Z
U
J
QM
Y
T L
B
P
IY
Z
X
V ON V
U A
RK X
S
M
YQ
I
O
W
P
B
LT
Z
JTransit (sweep)A
K
SX
R W
Figure 8 Solid186 element generated by the sweep method
N
Z
U
J
QM
Y
T L
B
P
I
V O
A
K
SX
R W
YZ
X
Transit (tetrahedral) (i + 1)
(i + 2)
(i + 3)
J K O N R AV Z
J K O N R A V Z
O N V
W U
P M XT
I
QY SK
BL
AR
P M X
W U
I
Q
Q
I
UM X
W
PB
L
SY
T
J Z
YS
T L B
Transit (pyramid)
Transit (prism)
Figure 9 Solid186 elements generated by tetrahedral pyramid and prism methods
Shock and Vibration 9
direction of the speed) which are marked as 1F and 2F inorder of increased frequency ere are also two backwardwhirls (the whirl direction of the rotor is opposite to therotation direction of the speed) that are marked as 3B and
4B respectively e synchronous whirl line is called thesynchronous excitation line In the figure the velocitiescorresponding to the intersection of the synchronouswhirl line and the frequency line of the forward whirl are
Temperature8177
Turbineside
Compressorside
(K)
7688
7199
6710
6222
5733
5244
4755
4266
3777
Figure 13 Postprocessing rotor temperature distribution
Figure 11 Global grid of rotor-shaft system
008
007
006
005
004
003
002
001
000
60000rpm
80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
12 13 14 15 16 17 18
ExperimentalCFD
(a)
014
012
010
008
006
004
002
000
60000rpm80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
10 11 12 13 14 1615 17
ExperimentalCFD
(b)
Figure 12 Numerical simulation verified by experiments (a) Turbine side (b) Compressor side
10 Shock and Vibration
Compressorside
Turbineside035 10
TToT times103 (K)
Figure 14 Rotor temperature distribution of Bohn [19]
Figure 16 Stability diagram of rotor-shaft system without considering temperature
Shock and Vibration 11
marked as Ωcr1 and Ωcr2 respectively and were calculatedas follows Ωcr1 5 0115 rpm and Ωcr2 43 622 rpmBecause the backward whirl could not be excited by theunbalanced force it is not marked or considered
When the speed was in the critical speed range be-tween Ωcr1 and Ωcr2 in the 1F mode the whirling fre-quency had a clear increasing trend at trend may havebeen caused by floating ring bearings because in thecritical speed range of Ωcr1 and Ωcr2 the stiffness anddamping of the oil film changed sharply When the criticalspeed of Ωcr2 was exceeded the stiffness and damping ofthe oil film changed smoothly and the eddy frequency inthe corresponding 1F mode also increased at a constantrate
Figure 16 shows that the curve of the damping ratio ofthe four modes in case 1 changed with the rotation speede modal damping ratio is the ratio of actual damping tothe critical damping of a certain mode and the dampingratio is the decisive parameter to describe the stability Apositive damping ratio means that the rotor-shaft systemwas in a stable region when the vibration energy wasdissipated A negative damping ratio means that therotation energy supported the rotor rotation by addingvibration energy resulting in the rotor-shaft systembecoming unstable e point where the damping ratiocurve intersects with the zero line is the stable limit speed(SLS) e figure shows that the rotor-shaft system be-came unstable when SLS 1 56 550 rpm in 1F mode andthen became unstable when SLS 2 97 946 rpm in 2Fmode
Figure 17 shows the four modes close to the SLS 1stability limit speed at 55000 rpm e black dotted linerepresents the axis the combination of the orange line andthe orange sphere represents the starting position of thetrajectory and the incomplete track curve represents thewhirl direction e figure shows that the 1F mode wascylindrical thus combining rigid body motion with rotorbending Furthermore the rotor was close to the unstablestate at this time In contrast the 2F mode was a conicalbending mode e figure shows that the 3B mode wasconical with both rigid body motion and rotor bendingLastly the 4B mode was cylindrical with both rigid bodymotion and rotor bending
Figure 18 shows the four modes close to the SLS 2stability limit speed at 97946 rpme rotor was close to theunstable state at this time e 1F and 3B modes were cy-lindrical and conical combining rigid body motion androtor bending In contrast the 2F mode was a conicalbending mode
322 Effect of Temperature on the Campbell Diagram andStability Diagram We exported the temperature data ofeach node in Figure 12 and inserted the temperature valueinto the corresponding rotor-shaft system node through the
Ansys software commands functionWe analysed the criticalspeed and stability of the rotor-shaft system in the thermalenvironment
Figure 19 shows that temperature affected the intrinsicfrequency (whirl frequency) Compared with case 1 thewhirl-frequency curves in the 1F 2F and 3B modes wereslightly lower whereas for the 4B mode the whirl-fre-quency curve was much lower In the figure the startingpoint at the left end (when the rotating speed was 0 rpm)was taken as the observation object In case 1 the frequencywas 23009 Hz whereas in case 2 the frequency decreasedto 21825 Hz Because the temperature changed the ma-terialrsquos elastic modulus and Poissonrsquos ratio it affected itsstructural stiffness For higher-order frequencies the effecton frequency became greater as the order increasedHowever the effect on the critical speed Ωcr1 did notchange much whereas Ωcr2 decreased from 43622 rpm to41815 rpm is decrease was also caused by the intrinsicfrequency drop
Figure 20 shows the stability diagram in case 2 and thatthe temperature had an important effect on the dampingratio and SLS Compared with case 1 SLS 1 of the 1F modedecreased from the original 56550 rpm to 55078 rpm andmade the 1F mode enter the unstable region ahead of timee 2F mode change was more obvious and the same SLS 2decreased from the original 97946 rpm to 95969 rpm ereason for this can be explained by the following formulaξr (α2ωhr) + (βωhr2) Here ξr is the damping ratiocoefficient of the rth mode α is the mass damping coeffi-cient β is the stiffness damping coefficient and ωhr is thenatural frequency of the rth mode In most cases α (α 0)can be disregarded that is ξr βωhr2 β did not changewhereas ωhr decreased because of the effect of temperatureresulting in a decrease in ξr In other words SLS 1 and SLS 2in the 1F and 2F modes were reduced erefore whenpredicting the stability limit reset and dynamic perfor-mance of a turbocharger rotor the effect of temperatureshould have a great influence on the dynamic design of therotor
323 Unbalance Response Analysis We applied a dummyunbalance mass of 001 kg middot mm at node 2 and the fre-quency range of the steady-state synchronous responseanalysis was 0 to 2500Hz (corresponding speed range0 to 150 000 rpm) We then measured the response am-plitude of the turbine end floating ring bearing centerposition node 5 In Figure 21 case 1 is represented by ablack curve (normal temperature) and case 2 by a redcurve (high temperature) In case 1 (case 2) the curvesrepresent 6F 6B and 20F modes e frequency at thewave crest corresponding to the frequency correspondingto the fast reverse rotation (the frequency of the rotationspeed within one minute) in the case 1 (case 2) Campbelldiagram as shown in Figures 22 and 23) is similar to the
12 Shock and Vibration
black (red) number in Figure 20 Because the steeringmode is dominant in the response diagram the criticalspeeds Ωcr1 and Ωcr2 in case 1 (case 2) 1F and 2F modesare not shown in the diagram and followed a phenom-enon similar to that reported by Chouksey and Roy[29 30] Because of the effect of temperature the steering
frequency of case 2 shifted to the left making the changeof response amplitude more obvious e responsevalue at the peak of the 6F and 6B modes decreased from0147mm to 0014 mm and that at the peak of the20F and 20B modes decreased from 0036 mm to00037 mm
324 Verification of Rotor FEM Natural frequency anal-ysis is very important as the basis of a dynamic analysisWe obtained the first two natural frequencies of the rotorby hammering experiments and verified them by nu-merical simulation Figure 24 shows the measured parts(rotor system) force hammer and piezoelectric acceler-ation sensor To reduce the measurement error rate we
adopted a sensor with a nonlinearity of less than 12 anda sensitivity of 10 mvg and placed the sensor in themiddle of the rotor shaft Table 6 compares the naturalfrequency experimental value with the simulated valuee experimental and simulated values of the first naturalfrequency were 1386Hz and 1453 Hz respectively esecond-order natural frequencies were 4100 Hz and
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
Table 1 Turbocharger and compressor parameters
Turbine side Compressor sideParameters Value and units Parameters Value and unitsBlades number 10 Blades number 6 + 6 (minus)Impeller inlet diameter 5505mm Impeller outlet diameter 565mmTip clearance 041mm Tip clearance 026mm
Turbineside
Solid casing Solid casingInlet
Inlet
Outlet
Impeller Impeller
Sha
Compressorside
Outlet
Figure 1 Centrifugal turbocharger and compressor with a solid casing
Figure 2 Global grid for conjugate heat transfer calculation
Turbinefluid
Turbineimpeller
Compressorfluid Compressor
impeller
Figure 3 Computational grids for conjugate heat transfer calculation
Table 2 e boundary conditions for aerothermal analysis
Turbine side Compressor sideMedium (intensity 5) Medium (intensity 5)Inlet mass flow 0074 kgs Inlet total pressure 999359 PaInlet total temperature 87297K Inlet total temperature 297201 KOutlet static pressure 968927 Pa Outlet static pressure 127578 Pa
Shock and Vibration 3
e heat transfer of the surrounding air was disregardedand the out-wall of the turbocharger was assumed to beadiabatic We applied ldquono-sliprdquo boundary conditions to allinner walls An interface was added between the rotatingdomain and the fixed domain and the interface was con-nected by a ldquofrozen rotorrdquo [24]
214 Numerical Methodology CHTrefers to a coupled heattransfer phenomenon in which the thermal properties of twomaterials occur through a medium or in direct contact eCHT method can calculate the heat transfer between fluidand solid and calculate the temperatures of fluids and solidsat the same time In this study we used the commercialsoftware Ansys CFX for numerical simulation CFX is acomputational fluid dynamics software package based on thecontrol volume method to solve NavierndashStokes equations Inthe fluid domain the mass conservation momentum andenergy transport equations are described as
where uf ρf p τ λf T htot and t represent the velocityvector density pressure stress tensor thermal conductivitytemperature total enthalpy and time of the fluidrespectively
In a solid domain the conservation of energy equationcan explain the heat transfer caused by solid motion con-duction and a volume heat source e energy equation is
zhtot ρs
zt+ nabla middot ρs us htot( 1113857 nabla middot λsnablaT( 1113857 + SE (4)
where us is the velocity vector of the solid us uf SE is theoptional volume heat source SE nabla middot (us middot τ) ρs is the densityof the solid and λs is the thermal conductivity of the solid
is study did not directly solve (1)ndash(4) Instead theywere converted to the steady-state Reynolds averageNavierndashStokes method to calculate turbulence e fluidmedium (exhaust gas and air) is an ideal gas and the shearstress transmission (SST) turbulence model was used be-cause the SST turbulence model has good accuracy for CHTcalculations [20] at model has both the accuracy of thek minus ω model in high-pressure gradient flow boundary layerprediction and the stability of the k minus e model in mainstreamprediction [18]
22 Modal Part
221 Rotor-Shaft System Geometry emodel of the rotor-shaft system was provided by the turbocharger company andagreed with the real onee rotor-shaft system comprised a
turbine wheel a compressor wheel a rotating shaft afloating ring bearing a thrust bearing a seal sleeve and anut as Figure 4 shows e turbine wheel and shaft wereconnected by friction welding whereas the compressor wasaxially and circumferentially fixed by a left-hand nut
e materials of the main parts of the rotor-shaft systemare shown in Table 3 based on the actual situation
is study investigated the effect of temperature andfound that the material properties changed as the temper-ature changed e function curves of the material prop-erties of each part as a function of temperature are shown inFigure 5 [25 26]
Publication [29] proposes a set of damping coefficientsand we verified the influence of damping coefficients on thestability limit displacement of the rotor through theoreticalanalysis and numerical simulation We selected this set ofdamping coefficients based on the theory of the publication
Table 4 shows what we assumed was the materialdamping coefficient of each part of the rotor
222 Floating Ring Bearing Parameters e rotor shaft wassupported by a floating ring bearing whose detailed pa-rameters are shown in Table 5
To aid the simulation we combined the stiffness anddamping coefficients of the inner and outer oil film into atotal impedance (equivalent stiffness and damping coeffi-cients) according to [27 28] Combining the stiffness anddamping coefficients of the inner and outer oil film providedby the turbocharger company with the parameters of thefloating ring bearing we calculated the total impedanceusing Dyrobes rotor dynamics software (Dyrobes USA)Figure 6 shows the total impedance (equivalent stiffness anddamping coefficients) of the floating ring bearing at variousspeeds
223 Finite Element Model of Rotor is study used a large-scale general software Ansys Workbench for simulation Toobtain the motion trace on the axis the rotor shaft wasequally divided into four parts along the axis direction asshown in Figure 7 e blue lines (solid and dotted) in thefigure represent the division positions
e solid model was discretised and an Ansys Solid186element was used e Solid186 element is a high-order 3D20-node element with quadratic displacement characteris-tics e element is defined by 20 nodes each node withthree degrees of freedom (translation of each node in the x-y- and z-directions) e element supports plasticitysuperelasticity creep stress hardening large deflection andlarge strain capacity In the shaft part the grid was generatedby a sweep as shown in Figure 8 depicting the ldquoithrdquo elementafter segmentation
We transformed (deformed) the original hexahedronelement to an approximately one-quarter cylinder elementIn Figure 8 the red dotted line represents the actual edge ofthe element and the red solid line represents the edge of theelement e advantage of a solid model is that it retains theproperties of the original element and describes the outercontour of the shaft model well
4 Shock and Vibration
To adapt to the more complicated outer contour of theblade tetrahedral pyramid and prismmethods were used togenerate the unit in the blade and the remaining partFigure 9 shows the (i + 1) (i + 2) and (i + 3) elementsgenerated by three methods
e effects of rotational inertia translational inertia gyromoment support stiffness support damping materialdamping rotational damping and thermal stress stiffnesswere considered in the model to establish the equation ofmotion e rotor model was discretised into Ne elementsand the total number of nodes became a e node dis-placement vector of the ith element is
qe xiI yiI ziI xiJ yiJ ziJ xiμ yiμ ziμ xiA1113966
yiA ziA xiB yiB ziB1113865T
(5)
where μ I J K L M N O P Q R S T U V W
X Y Z A B is the node number and the superscript T isthe matrixvector transpose ese writing formats are alsomentioned later in the text
After combining the governing equations of all elementsand combining the boundary conditions the equation ofmotion of the rotor-bearing system is
Meuroq (t) + C _q(t) + Kq (t) f (t) (6)
where M M trs + M rot C [minusΩG + Ccon + Cbrg] andK [Kbrg + Ksh + Ktem + Ktemσ]
HereM is a mass matrix that includes mass componentscaused by translation along the axis (Mtrs) and rotation alongthe axis (Mrot) C is a damping matrix (asymmetric matrixcoefficient of the velocity vector) that includes a skew-symmetric gyro matrix (G) a constant structural dampingmatrix (Ccon βKcon) with hysteresis characteristics causedby internal friction of the material and damping caused bythe floating ring bearing support matrix (Cbrg) K is thestiffness matrix (matrix coefficient of the displacementvector) that includes the stiffness matrix (Kbrg) caused by thesupport of the floating ring bearing the spin-softeningbending matrix (Ksh) generated by the rotor rotation thestiffness matrix (Ktem) of the elastic modulus the stiffnessmatrix (Ktemσ) caused by the nonuniform thermal stress andthermal deformation caused by temperature changes esize of all matrices is 3a times 3a Ω is the rotor speed in rpm βis the material damping (Rayleigh damping) stiffnessdamping coefficient
e total displacement and global force vectors q(t) andf(t) are expressed as
where matrices A and B state vector w(t) and force vectorF(t) are
A 0 M
M C1113890 1113891
6atimes6a
B M 0
0 minusK1113890 1113891
6atimes6a
w(t) _q (t)
q (t)1113896 1113897
6atimes1F(t)
0
f(t)1113896 1113897
6atimes1
(9)
where 0 is an empty matrix of size 3a times 3a and an emptyvector of size 3a times 1
Figure 10 shows a simplified finite element model of therotor-shaft system e grey-blue points in the figure areused to observe the modal shapes line trajectories andunbalanced harmonic response analysis
Figure 11 shows the rotor finite element global grid etotal number of grids was 3176485 with 546067 gridnodes In the geometry software each portion of the rotorwas treated as a part to avoid extra contact stiffness in thefinite element simulation and prevent affecting the simu-lation accuracy
To aid the comparison we considered two cases in thisstudy case 1 the rotor-shaft system without the influence oftemperature and case 2 the rotor-shaft system affected bytemperature In the following sections r-th forward and r-thbackward modes are represented as ldquor-Frdquo and ldquor-Brdquo modesrespectively
3 Results and Discussion
31 Heat Transfer Results and Experimental VerificationAfter the calculation we verified the numerical results of themass flow and the supercharger ratio (expansion ratio) enumerical results agreed with the experimental data and theerror control was approximately 7 as Figure 12 shows
e temperature distribution of the rotor was obtainedthrough postprocessing as Figure 13 shows the temperaturedistribution law was the same as that in the Bohn [19] studyresult and is shown in Figure 14
Figures 13 and 14 show that the highest temperature wasat the outer edge of the turbine wheel blades and that thelowest temperature was at the compressor wheel Overallthe rotor had a large temperature gradient
32 Modal Results
321 Campbell Diagram and Stability Diagram of NormalTemperature Environment Figure 15 shows the Campbelldiagram of the rotor-shaft system in case 1 that iswithout considering the temperature e figure was
Figure 5 Material properties of rotor-shaft system as a function of temperature (Youngrsquos modulus Poisson ratio specific heat capacity andthermal conductivity) (a) K418 (b) ZL105 (c) 42CrMo (d) 316 stainless steel [25 26]
Table 4 Damping coefficient of each partPart name Damping coefficientTurbo impeller 0002Compressor impeller and nut 0001Shaft 0005rust bearing and seal sleeve 0005
Table 5 Parameters of floating ring bearing that supported therotor shaftParameter name Parameter valueFloating ring mass 517 gInner length 514mmOuter length 796mmInner diameter 750mmOuter diameter 129mm
Shock and Vibration 7
Y
Divide
Z
X
14 cylinder
Figure 7 Rotor shaft divided into four parts and 14 cylinder
Figure 6 Total impedance of floating ring bearing (equivalent stiffness and damping coefficients) (a) Turbine side (b) Compressor side
8 Shock and Vibration
obtained from the whirl frequency (obtained from theimaginary part of the eigenvalue) and the rotational speedTo present a clear figure and express its basic
characteristics only the first four whirling directions aredescribed In the figure there are two forward whirls (thewhirl direction of the rotor agrees with the rotation
cyx cyx
cxy
cyy cyykyxkyx
kyy
kxxkxx kxycxx
cxy cxxkxy
kyy
1 2 3 4 5 6 7 8 9 10 11
Y
X Z
Figure 10 Finite element model of rotor-shaft system
N
Z
U
J
QM
Y
T L
B
P
IY
Z
X
V ON V
U A
RK X
S
M
YQ
I
O
W
P
B
LT
Z
JTransit (sweep)A
K
SX
R W
Figure 8 Solid186 element generated by the sweep method
N
Z
U
J
QM
Y
T L
B
P
I
V O
A
K
SX
R W
YZ
X
Transit (tetrahedral) (i + 1)
(i + 2)
(i + 3)
J K O N R AV Z
J K O N R A V Z
O N V
W U
P M XT
I
QY SK
BL
AR
P M X
W U
I
Q
Q
I
UM X
W
PB
L
SY
T
J Z
YS
T L B
Transit (pyramid)
Transit (prism)
Figure 9 Solid186 elements generated by tetrahedral pyramid and prism methods
Shock and Vibration 9
direction of the speed) which are marked as 1F and 2F inorder of increased frequency ere are also two backwardwhirls (the whirl direction of the rotor is opposite to therotation direction of the speed) that are marked as 3B and
4B respectively e synchronous whirl line is called thesynchronous excitation line In the figure the velocitiescorresponding to the intersection of the synchronouswhirl line and the frequency line of the forward whirl are
Temperature8177
Turbineside
Compressorside
(K)
7688
7199
6710
6222
5733
5244
4755
4266
3777
Figure 13 Postprocessing rotor temperature distribution
Figure 11 Global grid of rotor-shaft system
008
007
006
005
004
003
002
001
000
60000rpm
80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
12 13 14 15 16 17 18
ExperimentalCFD
(a)
014
012
010
008
006
004
002
000
60000rpm80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
10 11 12 13 14 1615 17
ExperimentalCFD
(b)
Figure 12 Numerical simulation verified by experiments (a) Turbine side (b) Compressor side
10 Shock and Vibration
Compressorside
Turbineside035 10
TToT times103 (K)
Figure 14 Rotor temperature distribution of Bohn [19]
Figure 16 Stability diagram of rotor-shaft system without considering temperature
Shock and Vibration 11
marked as Ωcr1 and Ωcr2 respectively and were calculatedas follows Ωcr1 5 0115 rpm and Ωcr2 43 622 rpmBecause the backward whirl could not be excited by theunbalanced force it is not marked or considered
When the speed was in the critical speed range be-tween Ωcr1 and Ωcr2 in the 1F mode the whirling fre-quency had a clear increasing trend at trend may havebeen caused by floating ring bearings because in thecritical speed range of Ωcr1 and Ωcr2 the stiffness anddamping of the oil film changed sharply When the criticalspeed of Ωcr2 was exceeded the stiffness and damping ofthe oil film changed smoothly and the eddy frequency inthe corresponding 1F mode also increased at a constantrate
Figure 16 shows that the curve of the damping ratio ofthe four modes in case 1 changed with the rotation speede modal damping ratio is the ratio of actual damping tothe critical damping of a certain mode and the dampingratio is the decisive parameter to describe the stability Apositive damping ratio means that the rotor-shaft systemwas in a stable region when the vibration energy wasdissipated A negative damping ratio means that therotation energy supported the rotor rotation by addingvibration energy resulting in the rotor-shaft systembecoming unstable e point where the damping ratiocurve intersects with the zero line is the stable limit speed(SLS) e figure shows that the rotor-shaft system be-came unstable when SLS 1 56 550 rpm in 1F mode andthen became unstable when SLS 2 97 946 rpm in 2Fmode
Figure 17 shows the four modes close to the SLS 1stability limit speed at 55000 rpm e black dotted linerepresents the axis the combination of the orange line andthe orange sphere represents the starting position of thetrajectory and the incomplete track curve represents thewhirl direction e figure shows that the 1F mode wascylindrical thus combining rigid body motion with rotorbending Furthermore the rotor was close to the unstablestate at this time In contrast the 2F mode was a conicalbending mode e figure shows that the 3B mode wasconical with both rigid body motion and rotor bendingLastly the 4B mode was cylindrical with both rigid bodymotion and rotor bending
Figure 18 shows the four modes close to the SLS 2stability limit speed at 97946 rpme rotor was close to theunstable state at this time e 1F and 3B modes were cy-lindrical and conical combining rigid body motion androtor bending In contrast the 2F mode was a conicalbending mode
322 Effect of Temperature on the Campbell Diagram andStability Diagram We exported the temperature data ofeach node in Figure 12 and inserted the temperature valueinto the corresponding rotor-shaft system node through the
Ansys software commands functionWe analysed the criticalspeed and stability of the rotor-shaft system in the thermalenvironment
Figure 19 shows that temperature affected the intrinsicfrequency (whirl frequency) Compared with case 1 thewhirl-frequency curves in the 1F 2F and 3B modes wereslightly lower whereas for the 4B mode the whirl-fre-quency curve was much lower In the figure the startingpoint at the left end (when the rotating speed was 0 rpm)was taken as the observation object In case 1 the frequencywas 23009 Hz whereas in case 2 the frequency decreasedto 21825 Hz Because the temperature changed the ma-terialrsquos elastic modulus and Poissonrsquos ratio it affected itsstructural stiffness For higher-order frequencies the effecton frequency became greater as the order increasedHowever the effect on the critical speed Ωcr1 did notchange much whereas Ωcr2 decreased from 43622 rpm to41815 rpm is decrease was also caused by the intrinsicfrequency drop
Figure 20 shows the stability diagram in case 2 and thatthe temperature had an important effect on the dampingratio and SLS Compared with case 1 SLS 1 of the 1F modedecreased from the original 56550 rpm to 55078 rpm andmade the 1F mode enter the unstable region ahead of timee 2F mode change was more obvious and the same SLS 2decreased from the original 97946 rpm to 95969 rpm ereason for this can be explained by the following formulaξr (α2ωhr) + (βωhr2) Here ξr is the damping ratiocoefficient of the rth mode α is the mass damping coeffi-cient β is the stiffness damping coefficient and ωhr is thenatural frequency of the rth mode In most cases α (α 0)can be disregarded that is ξr βωhr2 β did not changewhereas ωhr decreased because of the effect of temperatureresulting in a decrease in ξr In other words SLS 1 and SLS 2in the 1F and 2F modes were reduced erefore whenpredicting the stability limit reset and dynamic perfor-mance of a turbocharger rotor the effect of temperatureshould have a great influence on the dynamic design of therotor
323 Unbalance Response Analysis We applied a dummyunbalance mass of 001 kg middot mm at node 2 and the fre-quency range of the steady-state synchronous responseanalysis was 0 to 2500Hz (corresponding speed range0 to 150 000 rpm) We then measured the response am-plitude of the turbine end floating ring bearing centerposition node 5 In Figure 21 case 1 is represented by ablack curve (normal temperature) and case 2 by a redcurve (high temperature) In case 1 (case 2) the curvesrepresent 6F 6B and 20F modes e frequency at thewave crest corresponding to the frequency correspondingto the fast reverse rotation (the frequency of the rotationspeed within one minute) in the case 1 (case 2) Campbelldiagram as shown in Figures 22 and 23) is similar to the
12 Shock and Vibration
black (red) number in Figure 20 Because the steeringmode is dominant in the response diagram the criticalspeeds Ωcr1 and Ωcr2 in case 1 (case 2) 1F and 2F modesare not shown in the diagram and followed a phenom-enon similar to that reported by Chouksey and Roy[29 30] Because of the effect of temperature the steering
frequency of case 2 shifted to the left making the changeof response amplitude more obvious e responsevalue at the peak of the 6F and 6B modes decreased from0147mm to 0014 mm and that at the peak of the20F and 20B modes decreased from 0036 mm to00037 mm
324 Verification of Rotor FEM Natural frequency anal-ysis is very important as the basis of a dynamic analysisWe obtained the first two natural frequencies of the rotorby hammering experiments and verified them by nu-merical simulation Figure 24 shows the measured parts(rotor system) force hammer and piezoelectric acceler-ation sensor To reduce the measurement error rate we
adopted a sensor with a nonlinearity of less than 12 anda sensitivity of 10 mvg and placed the sensor in themiddle of the rotor shaft Table 6 compares the naturalfrequency experimental value with the simulated valuee experimental and simulated values of the first naturalfrequency were 1386Hz and 1453 Hz respectively esecond-order natural frequencies were 4100 Hz and
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
e heat transfer of the surrounding air was disregardedand the out-wall of the turbocharger was assumed to beadiabatic We applied ldquono-sliprdquo boundary conditions to allinner walls An interface was added between the rotatingdomain and the fixed domain and the interface was con-nected by a ldquofrozen rotorrdquo [24]
214 Numerical Methodology CHTrefers to a coupled heattransfer phenomenon in which the thermal properties of twomaterials occur through a medium or in direct contact eCHT method can calculate the heat transfer between fluidand solid and calculate the temperatures of fluids and solidsat the same time In this study we used the commercialsoftware Ansys CFX for numerical simulation CFX is acomputational fluid dynamics software package based on thecontrol volume method to solve NavierndashStokes equations Inthe fluid domain the mass conservation momentum andenergy transport equations are described as
where uf ρf p τ λf T htot and t represent the velocityvector density pressure stress tensor thermal conductivitytemperature total enthalpy and time of the fluidrespectively
In a solid domain the conservation of energy equationcan explain the heat transfer caused by solid motion con-duction and a volume heat source e energy equation is
zhtot ρs
zt+ nabla middot ρs us htot( 1113857 nabla middot λsnablaT( 1113857 + SE (4)
where us is the velocity vector of the solid us uf SE is theoptional volume heat source SE nabla middot (us middot τ) ρs is the densityof the solid and λs is the thermal conductivity of the solid
is study did not directly solve (1)ndash(4) Instead theywere converted to the steady-state Reynolds averageNavierndashStokes method to calculate turbulence e fluidmedium (exhaust gas and air) is an ideal gas and the shearstress transmission (SST) turbulence model was used be-cause the SST turbulence model has good accuracy for CHTcalculations [20] at model has both the accuracy of thek minus ω model in high-pressure gradient flow boundary layerprediction and the stability of the k minus e model in mainstreamprediction [18]
22 Modal Part
221 Rotor-Shaft System Geometry emodel of the rotor-shaft system was provided by the turbocharger company andagreed with the real onee rotor-shaft system comprised a
turbine wheel a compressor wheel a rotating shaft afloating ring bearing a thrust bearing a seal sleeve and anut as Figure 4 shows e turbine wheel and shaft wereconnected by friction welding whereas the compressor wasaxially and circumferentially fixed by a left-hand nut
e materials of the main parts of the rotor-shaft systemare shown in Table 3 based on the actual situation
is study investigated the effect of temperature andfound that the material properties changed as the temper-ature changed e function curves of the material prop-erties of each part as a function of temperature are shown inFigure 5 [25 26]
Publication [29] proposes a set of damping coefficientsand we verified the influence of damping coefficients on thestability limit displacement of the rotor through theoreticalanalysis and numerical simulation We selected this set ofdamping coefficients based on the theory of the publication
Table 4 shows what we assumed was the materialdamping coefficient of each part of the rotor
222 Floating Ring Bearing Parameters e rotor shaft wassupported by a floating ring bearing whose detailed pa-rameters are shown in Table 5
To aid the simulation we combined the stiffness anddamping coefficients of the inner and outer oil film into atotal impedance (equivalent stiffness and damping coeffi-cients) according to [27 28] Combining the stiffness anddamping coefficients of the inner and outer oil film providedby the turbocharger company with the parameters of thefloating ring bearing we calculated the total impedanceusing Dyrobes rotor dynamics software (Dyrobes USA)Figure 6 shows the total impedance (equivalent stiffness anddamping coefficients) of the floating ring bearing at variousspeeds
223 Finite Element Model of Rotor is study used a large-scale general software Ansys Workbench for simulation Toobtain the motion trace on the axis the rotor shaft wasequally divided into four parts along the axis direction asshown in Figure 7 e blue lines (solid and dotted) in thefigure represent the division positions
e solid model was discretised and an Ansys Solid186element was used e Solid186 element is a high-order 3D20-node element with quadratic displacement characteris-tics e element is defined by 20 nodes each node withthree degrees of freedom (translation of each node in the x-y- and z-directions) e element supports plasticitysuperelasticity creep stress hardening large deflection andlarge strain capacity In the shaft part the grid was generatedby a sweep as shown in Figure 8 depicting the ldquoithrdquo elementafter segmentation
We transformed (deformed) the original hexahedronelement to an approximately one-quarter cylinder elementIn Figure 8 the red dotted line represents the actual edge ofthe element and the red solid line represents the edge of theelement e advantage of a solid model is that it retains theproperties of the original element and describes the outercontour of the shaft model well
4 Shock and Vibration
To adapt to the more complicated outer contour of theblade tetrahedral pyramid and prismmethods were used togenerate the unit in the blade and the remaining partFigure 9 shows the (i + 1) (i + 2) and (i + 3) elementsgenerated by three methods
e effects of rotational inertia translational inertia gyromoment support stiffness support damping materialdamping rotational damping and thermal stress stiffnesswere considered in the model to establish the equation ofmotion e rotor model was discretised into Ne elementsand the total number of nodes became a e node dis-placement vector of the ith element is
qe xiI yiI ziI xiJ yiJ ziJ xiμ yiμ ziμ xiA1113966
yiA ziA xiB yiB ziB1113865T
(5)
where μ I J K L M N O P Q R S T U V W
X Y Z A B is the node number and the superscript T isthe matrixvector transpose ese writing formats are alsomentioned later in the text
After combining the governing equations of all elementsand combining the boundary conditions the equation ofmotion of the rotor-bearing system is
Meuroq (t) + C _q(t) + Kq (t) f (t) (6)
where M M trs + M rot C [minusΩG + Ccon + Cbrg] andK [Kbrg + Ksh + Ktem + Ktemσ]
HereM is a mass matrix that includes mass componentscaused by translation along the axis (Mtrs) and rotation alongthe axis (Mrot) C is a damping matrix (asymmetric matrixcoefficient of the velocity vector) that includes a skew-symmetric gyro matrix (G) a constant structural dampingmatrix (Ccon βKcon) with hysteresis characteristics causedby internal friction of the material and damping caused bythe floating ring bearing support matrix (Cbrg) K is thestiffness matrix (matrix coefficient of the displacementvector) that includes the stiffness matrix (Kbrg) caused by thesupport of the floating ring bearing the spin-softeningbending matrix (Ksh) generated by the rotor rotation thestiffness matrix (Ktem) of the elastic modulus the stiffnessmatrix (Ktemσ) caused by the nonuniform thermal stress andthermal deformation caused by temperature changes esize of all matrices is 3a times 3a Ω is the rotor speed in rpm βis the material damping (Rayleigh damping) stiffnessdamping coefficient
e total displacement and global force vectors q(t) andf(t) are expressed as
where matrices A and B state vector w(t) and force vectorF(t) are
A 0 M
M C1113890 1113891
6atimes6a
B M 0
0 minusK1113890 1113891
6atimes6a
w(t) _q (t)
q (t)1113896 1113897
6atimes1F(t)
0
f(t)1113896 1113897
6atimes1
(9)
where 0 is an empty matrix of size 3a times 3a and an emptyvector of size 3a times 1
Figure 10 shows a simplified finite element model of therotor-shaft system e grey-blue points in the figure areused to observe the modal shapes line trajectories andunbalanced harmonic response analysis
Figure 11 shows the rotor finite element global grid etotal number of grids was 3176485 with 546067 gridnodes In the geometry software each portion of the rotorwas treated as a part to avoid extra contact stiffness in thefinite element simulation and prevent affecting the simu-lation accuracy
To aid the comparison we considered two cases in thisstudy case 1 the rotor-shaft system without the influence oftemperature and case 2 the rotor-shaft system affected bytemperature In the following sections r-th forward and r-thbackward modes are represented as ldquor-Frdquo and ldquor-Brdquo modesrespectively
3 Results and Discussion
31 Heat Transfer Results and Experimental VerificationAfter the calculation we verified the numerical results of themass flow and the supercharger ratio (expansion ratio) enumerical results agreed with the experimental data and theerror control was approximately 7 as Figure 12 shows
e temperature distribution of the rotor was obtainedthrough postprocessing as Figure 13 shows the temperaturedistribution law was the same as that in the Bohn [19] studyresult and is shown in Figure 14
Figures 13 and 14 show that the highest temperature wasat the outer edge of the turbine wheel blades and that thelowest temperature was at the compressor wheel Overallthe rotor had a large temperature gradient
32 Modal Results
321 Campbell Diagram and Stability Diagram of NormalTemperature Environment Figure 15 shows the Campbelldiagram of the rotor-shaft system in case 1 that iswithout considering the temperature e figure was
Figure 5 Material properties of rotor-shaft system as a function of temperature (Youngrsquos modulus Poisson ratio specific heat capacity andthermal conductivity) (a) K418 (b) ZL105 (c) 42CrMo (d) 316 stainless steel [25 26]
Table 4 Damping coefficient of each partPart name Damping coefficientTurbo impeller 0002Compressor impeller and nut 0001Shaft 0005rust bearing and seal sleeve 0005
Table 5 Parameters of floating ring bearing that supported therotor shaftParameter name Parameter valueFloating ring mass 517 gInner length 514mmOuter length 796mmInner diameter 750mmOuter diameter 129mm
Shock and Vibration 7
Y
Divide
Z
X
14 cylinder
Figure 7 Rotor shaft divided into four parts and 14 cylinder
Figure 6 Total impedance of floating ring bearing (equivalent stiffness and damping coefficients) (a) Turbine side (b) Compressor side
8 Shock and Vibration
obtained from the whirl frequency (obtained from theimaginary part of the eigenvalue) and the rotational speedTo present a clear figure and express its basic
characteristics only the first four whirling directions aredescribed In the figure there are two forward whirls (thewhirl direction of the rotor agrees with the rotation
cyx cyx
cxy
cyy cyykyxkyx
kyy
kxxkxx kxycxx
cxy cxxkxy
kyy
1 2 3 4 5 6 7 8 9 10 11
Y
X Z
Figure 10 Finite element model of rotor-shaft system
N
Z
U
J
QM
Y
T L
B
P
IY
Z
X
V ON V
U A
RK X
S
M
YQ
I
O
W
P
B
LT
Z
JTransit (sweep)A
K
SX
R W
Figure 8 Solid186 element generated by the sweep method
N
Z
U
J
QM
Y
T L
B
P
I
V O
A
K
SX
R W
YZ
X
Transit (tetrahedral) (i + 1)
(i + 2)
(i + 3)
J K O N R AV Z
J K O N R A V Z
O N V
W U
P M XT
I
QY SK
BL
AR
P M X
W U
I
Q
Q
I
UM X
W
PB
L
SY
T
J Z
YS
T L B
Transit (pyramid)
Transit (prism)
Figure 9 Solid186 elements generated by tetrahedral pyramid and prism methods
Shock and Vibration 9
direction of the speed) which are marked as 1F and 2F inorder of increased frequency ere are also two backwardwhirls (the whirl direction of the rotor is opposite to therotation direction of the speed) that are marked as 3B and
4B respectively e synchronous whirl line is called thesynchronous excitation line In the figure the velocitiescorresponding to the intersection of the synchronouswhirl line and the frequency line of the forward whirl are
Temperature8177
Turbineside
Compressorside
(K)
7688
7199
6710
6222
5733
5244
4755
4266
3777
Figure 13 Postprocessing rotor temperature distribution
Figure 11 Global grid of rotor-shaft system
008
007
006
005
004
003
002
001
000
60000rpm
80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
12 13 14 15 16 17 18
ExperimentalCFD
(a)
014
012
010
008
006
004
002
000
60000rpm80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
10 11 12 13 14 1615 17
ExperimentalCFD
(b)
Figure 12 Numerical simulation verified by experiments (a) Turbine side (b) Compressor side
10 Shock and Vibration
Compressorside
Turbineside035 10
TToT times103 (K)
Figure 14 Rotor temperature distribution of Bohn [19]
Figure 16 Stability diagram of rotor-shaft system without considering temperature
Shock and Vibration 11
marked as Ωcr1 and Ωcr2 respectively and were calculatedas follows Ωcr1 5 0115 rpm and Ωcr2 43 622 rpmBecause the backward whirl could not be excited by theunbalanced force it is not marked or considered
When the speed was in the critical speed range be-tween Ωcr1 and Ωcr2 in the 1F mode the whirling fre-quency had a clear increasing trend at trend may havebeen caused by floating ring bearings because in thecritical speed range of Ωcr1 and Ωcr2 the stiffness anddamping of the oil film changed sharply When the criticalspeed of Ωcr2 was exceeded the stiffness and damping ofthe oil film changed smoothly and the eddy frequency inthe corresponding 1F mode also increased at a constantrate
Figure 16 shows that the curve of the damping ratio ofthe four modes in case 1 changed with the rotation speede modal damping ratio is the ratio of actual damping tothe critical damping of a certain mode and the dampingratio is the decisive parameter to describe the stability Apositive damping ratio means that the rotor-shaft systemwas in a stable region when the vibration energy wasdissipated A negative damping ratio means that therotation energy supported the rotor rotation by addingvibration energy resulting in the rotor-shaft systembecoming unstable e point where the damping ratiocurve intersects with the zero line is the stable limit speed(SLS) e figure shows that the rotor-shaft system be-came unstable when SLS 1 56 550 rpm in 1F mode andthen became unstable when SLS 2 97 946 rpm in 2Fmode
Figure 17 shows the four modes close to the SLS 1stability limit speed at 55000 rpm e black dotted linerepresents the axis the combination of the orange line andthe orange sphere represents the starting position of thetrajectory and the incomplete track curve represents thewhirl direction e figure shows that the 1F mode wascylindrical thus combining rigid body motion with rotorbending Furthermore the rotor was close to the unstablestate at this time In contrast the 2F mode was a conicalbending mode e figure shows that the 3B mode wasconical with both rigid body motion and rotor bendingLastly the 4B mode was cylindrical with both rigid bodymotion and rotor bending
Figure 18 shows the four modes close to the SLS 2stability limit speed at 97946 rpme rotor was close to theunstable state at this time e 1F and 3B modes were cy-lindrical and conical combining rigid body motion androtor bending In contrast the 2F mode was a conicalbending mode
322 Effect of Temperature on the Campbell Diagram andStability Diagram We exported the temperature data ofeach node in Figure 12 and inserted the temperature valueinto the corresponding rotor-shaft system node through the
Ansys software commands functionWe analysed the criticalspeed and stability of the rotor-shaft system in the thermalenvironment
Figure 19 shows that temperature affected the intrinsicfrequency (whirl frequency) Compared with case 1 thewhirl-frequency curves in the 1F 2F and 3B modes wereslightly lower whereas for the 4B mode the whirl-fre-quency curve was much lower In the figure the startingpoint at the left end (when the rotating speed was 0 rpm)was taken as the observation object In case 1 the frequencywas 23009 Hz whereas in case 2 the frequency decreasedto 21825 Hz Because the temperature changed the ma-terialrsquos elastic modulus and Poissonrsquos ratio it affected itsstructural stiffness For higher-order frequencies the effecton frequency became greater as the order increasedHowever the effect on the critical speed Ωcr1 did notchange much whereas Ωcr2 decreased from 43622 rpm to41815 rpm is decrease was also caused by the intrinsicfrequency drop
Figure 20 shows the stability diagram in case 2 and thatthe temperature had an important effect on the dampingratio and SLS Compared with case 1 SLS 1 of the 1F modedecreased from the original 56550 rpm to 55078 rpm andmade the 1F mode enter the unstable region ahead of timee 2F mode change was more obvious and the same SLS 2decreased from the original 97946 rpm to 95969 rpm ereason for this can be explained by the following formulaξr (α2ωhr) + (βωhr2) Here ξr is the damping ratiocoefficient of the rth mode α is the mass damping coeffi-cient β is the stiffness damping coefficient and ωhr is thenatural frequency of the rth mode In most cases α (α 0)can be disregarded that is ξr βωhr2 β did not changewhereas ωhr decreased because of the effect of temperatureresulting in a decrease in ξr In other words SLS 1 and SLS 2in the 1F and 2F modes were reduced erefore whenpredicting the stability limit reset and dynamic perfor-mance of a turbocharger rotor the effect of temperatureshould have a great influence on the dynamic design of therotor
323 Unbalance Response Analysis We applied a dummyunbalance mass of 001 kg middot mm at node 2 and the fre-quency range of the steady-state synchronous responseanalysis was 0 to 2500Hz (corresponding speed range0 to 150 000 rpm) We then measured the response am-plitude of the turbine end floating ring bearing centerposition node 5 In Figure 21 case 1 is represented by ablack curve (normal temperature) and case 2 by a redcurve (high temperature) In case 1 (case 2) the curvesrepresent 6F 6B and 20F modes e frequency at thewave crest corresponding to the frequency correspondingto the fast reverse rotation (the frequency of the rotationspeed within one minute) in the case 1 (case 2) Campbelldiagram as shown in Figures 22 and 23) is similar to the
12 Shock and Vibration
black (red) number in Figure 20 Because the steeringmode is dominant in the response diagram the criticalspeeds Ωcr1 and Ωcr2 in case 1 (case 2) 1F and 2F modesare not shown in the diagram and followed a phenom-enon similar to that reported by Chouksey and Roy[29 30] Because of the effect of temperature the steering
frequency of case 2 shifted to the left making the changeof response amplitude more obvious e responsevalue at the peak of the 6F and 6B modes decreased from0147mm to 0014 mm and that at the peak of the20F and 20B modes decreased from 0036 mm to00037 mm
324 Verification of Rotor FEM Natural frequency anal-ysis is very important as the basis of a dynamic analysisWe obtained the first two natural frequencies of the rotorby hammering experiments and verified them by nu-merical simulation Figure 24 shows the measured parts(rotor system) force hammer and piezoelectric acceler-ation sensor To reduce the measurement error rate we
adopted a sensor with a nonlinearity of less than 12 anda sensitivity of 10 mvg and placed the sensor in themiddle of the rotor shaft Table 6 compares the naturalfrequency experimental value with the simulated valuee experimental and simulated values of the first naturalfrequency were 1386Hz and 1453 Hz respectively esecond-order natural frequencies were 4100 Hz and
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
To adapt to the more complicated outer contour of theblade tetrahedral pyramid and prismmethods were used togenerate the unit in the blade and the remaining partFigure 9 shows the (i + 1) (i + 2) and (i + 3) elementsgenerated by three methods
e effects of rotational inertia translational inertia gyromoment support stiffness support damping materialdamping rotational damping and thermal stress stiffnesswere considered in the model to establish the equation ofmotion e rotor model was discretised into Ne elementsand the total number of nodes became a e node dis-placement vector of the ith element is
qe xiI yiI ziI xiJ yiJ ziJ xiμ yiμ ziμ xiA1113966
yiA ziA xiB yiB ziB1113865T
(5)
where μ I J K L M N O P Q R S T U V W
X Y Z A B is the node number and the superscript T isthe matrixvector transpose ese writing formats are alsomentioned later in the text
After combining the governing equations of all elementsand combining the boundary conditions the equation ofmotion of the rotor-bearing system is
Meuroq (t) + C _q(t) + Kq (t) f (t) (6)
where M M trs + M rot C [minusΩG + Ccon + Cbrg] andK [Kbrg + Ksh + Ktem + Ktemσ]
HereM is a mass matrix that includes mass componentscaused by translation along the axis (Mtrs) and rotation alongthe axis (Mrot) C is a damping matrix (asymmetric matrixcoefficient of the velocity vector) that includes a skew-symmetric gyro matrix (G) a constant structural dampingmatrix (Ccon βKcon) with hysteresis characteristics causedby internal friction of the material and damping caused bythe floating ring bearing support matrix (Cbrg) K is thestiffness matrix (matrix coefficient of the displacementvector) that includes the stiffness matrix (Kbrg) caused by thesupport of the floating ring bearing the spin-softeningbending matrix (Ksh) generated by the rotor rotation thestiffness matrix (Ktem) of the elastic modulus the stiffnessmatrix (Ktemσ) caused by the nonuniform thermal stress andthermal deformation caused by temperature changes esize of all matrices is 3a times 3a Ω is the rotor speed in rpm βis the material damping (Rayleigh damping) stiffnessdamping coefficient
e total displacement and global force vectors q(t) andf(t) are expressed as
where matrices A and B state vector w(t) and force vectorF(t) are
A 0 M
M C1113890 1113891
6atimes6a
B M 0
0 minusK1113890 1113891
6atimes6a
w(t) _q (t)
q (t)1113896 1113897
6atimes1F(t)
0
f(t)1113896 1113897
6atimes1
(9)
where 0 is an empty matrix of size 3a times 3a and an emptyvector of size 3a times 1
Figure 10 shows a simplified finite element model of therotor-shaft system e grey-blue points in the figure areused to observe the modal shapes line trajectories andunbalanced harmonic response analysis
Figure 11 shows the rotor finite element global grid etotal number of grids was 3176485 with 546067 gridnodes In the geometry software each portion of the rotorwas treated as a part to avoid extra contact stiffness in thefinite element simulation and prevent affecting the simu-lation accuracy
To aid the comparison we considered two cases in thisstudy case 1 the rotor-shaft system without the influence oftemperature and case 2 the rotor-shaft system affected bytemperature In the following sections r-th forward and r-thbackward modes are represented as ldquor-Frdquo and ldquor-Brdquo modesrespectively
3 Results and Discussion
31 Heat Transfer Results and Experimental VerificationAfter the calculation we verified the numerical results of themass flow and the supercharger ratio (expansion ratio) enumerical results agreed with the experimental data and theerror control was approximately 7 as Figure 12 shows
e temperature distribution of the rotor was obtainedthrough postprocessing as Figure 13 shows the temperaturedistribution law was the same as that in the Bohn [19] studyresult and is shown in Figure 14
Figures 13 and 14 show that the highest temperature wasat the outer edge of the turbine wheel blades and that thelowest temperature was at the compressor wheel Overallthe rotor had a large temperature gradient
32 Modal Results
321 Campbell Diagram and Stability Diagram of NormalTemperature Environment Figure 15 shows the Campbelldiagram of the rotor-shaft system in case 1 that iswithout considering the temperature e figure was
Figure 5 Material properties of rotor-shaft system as a function of temperature (Youngrsquos modulus Poisson ratio specific heat capacity andthermal conductivity) (a) K418 (b) ZL105 (c) 42CrMo (d) 316 stainless steel [25 26]
Table 4 Damping coefficient of each partPart name Damping coefficientTurbo impeller 0002Compressor impeller and nut 0001Shaft 0005rust bearing and seal sleeve 0005
Table 5 Parameters of floating ring bearing that supported therotor shaftParameter name Parameter valueFloating ring mass 517 gInner length 514mmOuter length 796mmInner diameter 750mmOuter diameter 129mm
Shock and Vibration 7
Y
Divide
Z
X
14 cylinder
Figure 7 Rotor shaft divided into four parts and 14 cylinder
Figure 6 Total impedance of floating ring bearing (equivalent stiffness and damping coefficients) (a) Turbine side (b) Compressor side
8 Shock and Vibration
obtained from the whirl frequency (obtained from theimaginary part of the eigenvalue) and the rotational speedTo present a clear figure and express its basic
characteristics only the first four whirling directions aredescribed In the figure there are two forward whirls (thewhirl direction of the rotor agrees with the rotation
cyx cyx
cxy
cyy cyykyxkyx
kyy
kxxkxx kxycxx
cxy cxxkxy
kyy
1 2 3 4 5 6 7 8 9 10 11
Y
X Z
Figure 10 Finite element model of rotor-shaft system
N
Z
U
J
QM
Y
T L
B
P
IY
Z
X
V ON V
U A
RK X
S
M
YQ
I
O
W
P
B
LT
Z
JTransit (sweep)A
K
SX
R W
Figure 8 Solid186 element generated by the sweep method
N
Z
U
J
QM
Y
T L
B
P
I
V O
A
K
SX
R W
YZ
X
Transit (tetrahedral) (i + 1)
(i + 2)
(i + 3)
J K O N R AV Z
J K O N R A V Z
O N V
W U
P M XT
I
QY SK
BL
AR
P M X
W U
I
Q
Q
I
UM X
W
PB
L
SY
T
J Z
YS
T L B
Transit (pyramid)
Transit (prism)
Figure 9 Solid186 elements generated by tetrahedral pyramid and prism methods
Shock and Vibration 9
direction of the speed) which are marked as 1F and 2F inorder of increased frequency ere are also two backwardwhirls (the whirl direction of the rotor is opposite to therotation direction of the speed) that are marked as 3B and
4B respectively e synchronous whirl line is called thesynchronous excitation line In the figure the velocitiescorresponding to the intersection of the synchronouswhirl line and the frequency line of the forward whirl are
Temperature8177
Turbineside
Compressorside
(K)
7688
7199
6710
6222
5733
5244
4755
4266
3777
Figure 13 Postprocessing rotor temperature distribution
Figure 11 Global grid of rotor-shaft system
008
007
006
005
004
003
002
001
000
60000rpm
80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
12 13 14 15 16 17 18
ExperimentalCFD
(a)
014
012
010
008
006
004
002
000
60000rpm80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
10 11 12 13 14 1615 17
ExperimentalCFD
(b)
Figure 12 Numerical simulation verified by experiments (a) Turbine side (b) Compressor side
10 Shock and Vibration
Compressorside
Turbineside035 10
TToT times103 (K)
Figure 14 Rotor temperature distribution of Bohn [19]
Figure 16 Stability diagram of rotor-shaft system without considering temperature
Shock and Vibration 11
marked as Ωcr1 and Ωcr2 respectively and were calculatedas follows Ωcr1 5 0115 rpm and Ωcr2 43 622 rpmBecause the backward whirl could not be excited by theunbalanced force it is not marked or considered
When the speed was in the critical speed range be-tween Ωcr1 and Ωcr2 in the 1F mode the whirling fre-quency had a clear increasing trend at trend may havebeen caused by floating ring bearings because in thecritical speed range of Ωcr1 and Ωcr2 the stiffness anddamping of the oil film changed sharply When the criticalspeed of Ωcr2 was exceeded the stiffness and damping ofthe oil film changed smoothly and the eddy frequency inthe corresponding 1F mode also increased at a constantrate
Figure 16 shows that the curve of the damping ratio ofthe four modes in case 1 changed with the rotation speede modal damping ratio is the ratio of actual damping tothe critical damping of a certain mode and the dampingratio is the decisive parameter to describe the stability Apositive damping ratio means that the rotor-shaft systemwas in a stable region when the vibration energy wasdissipated A negative damping ratio means that therotation energy supported the rotor rotation by addingvibration energy resulting in the rotor-shaft systembecoming unstable e point where the damping ratiocurve intersects with the zero line is the stable limit speed(SLS) e figure shows that the rotor-shaft system be-came unstable when SLS 1 56 550 rpm in 1F mode andthen became unstable when SLS 2 97 946 rpm in 2Fmode
Figure 17 shows the four modes close to the SLS 1stability limit speed at 55000 rpm e black dotted linerepresents the axis the combination of the orange line andthe orange sphere represents the starting position of thetrajectory and the incomplete track curve represents thewhirl direction e figure shows that the 1F mode wascylindrical thus combining rigid body motion with rotorbending Furthermore the rotor was close to the unstablestate at this time In contrast the 2F mode was a conicalbending mode e figure shows that the 3B mode wasconical with both rigid body motion and rotor bendingLastly the 4B mode was cylindrical with both rigid bodymotion and rotor bending
Figure 18 shows the four modes close to the SLS 2stability limit speed at 97946 rpme rotor was close to theunstable state at this time e 1F and 3B modes were cy-lindrical and conical combining rigid body motion androtor bending In contrast the 2F mode was a conicalbending mode
322 Effect of Temperature on the Campbell Diagram andStability Diagram We exported the temperature data ofeach node in Figure 12 and inserted the temperature valueinto the corresponding rotor-shaft system node through the
Ansys software commands functionWe analysed the criticalspeed and stability of the rotor-shaft system in the thermalenvironment
Figure 19 shows that temperature affected the intrinsicfrequency (whirl frequency) Compared with case 1 thewhirl-frequency curves in the 1F 2F and 3B modes wereslightly lower whereas for the 4B mode the whirl-fre-quency curve was much lower In the figure the startingpoint at the left end (when the rotating speed was 0 rpm)was taken as the observation object In case 1 the frequencywas 23009 Hz whereas in case 2 the frequency decreasedto 21825 Hz Because the temperature changed the ma-terialrsquos elastic modulus and Poissonrsquos ratio it affected itsstructural stiffness For higher-order frequencies the effecton frequency became greater as the order increasedHowever the effect on the critical speed Ωcr1 did notchange much whereas Ωcr2 decreased from 43622 rpm to41815 rpm is decrease was also caused by the intrinsicfrequency drop
Figure 20 shows the stability diagram in case 2 and thatthe temperature had an important effect on the dampingratio and SLS Compared with case 1 SLS 1 of the 1F modedecreased from the original 56550 rpm to 55078 rpm andmade the 1F mode enter the unstable region ahead of timee 2F mode change was more obvious and the same SLS 2decreased from the original 97946 rpm to 95969 rpm ereason for this can be explained by the following formulaξr (α2ωhr) + (βωhr2) Here ξr is the damping ratiocoefficient of the rth mode α is the mass damping coeffi-cient β is the stiffness damping coefficient and ωhr is thenatural frequency of the rth mode In most cases α (α 0)can be disregarded that is ξr βωhr2 β did not changewhereas ωhr decreased because of the effect of temperatureresulting in a decrease in ξr In other words SLS 1 and SLS 2in the 1F and 2F modes were reduced erefore whenpredicting the stability limit reset and dynamic perfor-mance of a turbocharger rotor the effect of temperatureshould have a great influence on the dynamic design of therotor
323 Unbalance Response Analysis We applied a dummyunbalance mass of 001 kg middot mm at node 2 and the fre-quency range of the steady-state synchronous responseanalysis was 0 to 2500Hz (corresponding speed range0 to 150 000 rpm) We then measured the response am-plitude of the turbine end floating ring bearing centerposition node 5 In Figure 21 case 1 is represented by ablack curve (normal temperature) and case 2 by a redcurve (high temperature) In case 1 (case 2) the curvesrepresent 6F 6B and 20F modes e frequency at thewave crest corresponding to the frequency correspondingto the fast reverse rotation (the frequency of the rotationspeed within one minute) in the case 1 (case 2) Campbelldiagram as shown in Figures 22 and 23) is similar to the
12 Shock and Vibration
black (red) number in Figure 20 Because the steeringmode is dominant in the response diagram the criticalspeeds Ωcr1 and Ωcr2 in case 1 (case 2) 1F and 2F modesare not shown in the diagram and followed a phenom-enon similar to that reported by Chouksey and Roy[29 30] Because of the effect of temperature the steering
frequency of case 2 shifted to the left making the changeof response amplitude more obvious e responsevalue at the peak of the 6F and 6B modes decreased from0147mm to 0014 mm and that at the peak of the20F and 20B modes decreased from 0036 mm to00037 mm
324 Verification of Rotor FEM Natural frequency anal-ysis is very important as the basis of a dynamic analysisWe obtained the first two natural frequencies of the rotorby hammering experiments and verified them by nu-merical simulation Figure 24 shows the measured parts(rotor system) force hammer and piezoelectric acceler-ation sensor To reduce the measurement error rate we
adopted a sensor with a nonlinearity of less than 12 anda sensitivity of 10 mvg and placed the sensor in themiddle of the rotor shaft Table 6 compares the naturalfrequency experimental value with the simulated valuee experimental and simulated values of the first naturalfrequency were 1386Hz and 1453 Hz respectively esecond-order natural frequencies were 4100 Hz and
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
where matrices A and B state vector w(t) and force vectorF(t) are
A 0 M
M C1113890 1113891
6atimes6a
B M 0
0 minusK1113890 1113891
6atimes6a
w(t) _q (t)
q (t)1113896 1113897
6atimes1F(t)
0
f(t)1113896 1113897
6atimes1
(9)
where 0 is an empty matrix of size 3a times 3a and an emptyvector of size 3a times 1
Figure 10 shows a simplified finite element model of therotor-shaft system e grey-blue points in the figure areused to observe the modal shapes line trajectories andunbalanced harmonic response analysis
Figure 11 shows the rotor finite element global grid etotal number of grids was 3176485 with 546067 gridnodes In the geometry software each portion of the rotorwas treated as a part to avoid extra contact stiffness in thefinite element simulation and prevent affecting the simu-lation accuracy
To aid the comparison we considered two cases in thisstudy case 1 the rotor-shaft system without the influence oftemperature and case 2 the rotor-shaft system affected bytemperature In the following sections r-th forward and r-thbackward modes are represented as ldquor-Frdquo and ldquor-Brdquo modesrespectively
3 Results and Discussion
31 Heat Transfer Results and Experimental VerificationAfter the calculation we verified the numerical results of themass flow and the supercharger ratio (expansion ratio) enumerical results agreed with the experimental data and theerror control was approximately 7 as Figure 12 shows
e temperature distribution of the rotor was obtainedthrough postprocessing as Figure 13 shows the temperaturedistribution law was the same as that in the Bohn [19] studyresult and is shown in Figure 14
Figures 13 and 14 show that the highest temperature wasat the outer edge of the turbine wheel blades and that thelowest temperature was at the compressor wheel Overallthe rotor had a large temperature gradient
32 Modal Results
321 Campbell Diagram and Stability Diagram of NormalTemperature Environment Figure 15 shows the Campbelldiagram of the rotor-shaft system in case 1 that iswithout considering the temperature e figure was
Figure 5 Material properties of rotor-shaft system as a function of temperature (Youngrsquos modulus Poisson ratio specific heat capacity andthermal conductivity) (a) K418 (b) ZL105 (c) 42CrMo (d) 316 stainless steel [25 26]
Table 4 Damping coefficient of each partPart name Damping coefficientTurbo impeller 0002Compressor impeller and nut 0001Shaft 0005rust bearing and seal sleeve 0005
Table 5 Parameters of floating ring bearing that supported therotor shaftParameter name Parameter valueFloating ring mass 517 gInner length 514mmOuter length 796mmInner diameter 750mmOuter diameter 129mm
Shock and Vibration 7
Y
Divide
Z
X
14 cylinder
Figure 7 Rotor shaft divided into four parts and 14 cylinder
Figure 6 Total impedance of floating ring bearing (equivalent stiffness and damping coefficients) (a) Turbine side (b) Compressor side
8 Shock and Vibration
obtained from the whirl frequency (obtained from theimaginary part of the eigenvalue) and the rotational speedTo present a clear figure and express its basic
characteristics only the first four whirling directions aredescribed In the figure there are two forward whirls (thewhirl direction of the rotor agrees with the rotation
cyx cyx
cxy
cyy cyykyxkyx
kyy
kxxkxx kxycxx
cxy cxxkxy
kyy
1 2 3 4 5 6 7 8 9 10 11
Y
X Z
Figure 10 Finite element model of rotor-shaft system
N
Z
U
J
QM
Y
T L
B
P
IY
Z
X
V ON V
U A
RK X
S
M
YQ
I
O
W
P
B
LT
Z
JTransit (sweep)A
K
SX
R W
Figure 8 Solid186 element generated by the sweep method
N
Z
U
J
QM
Y
T L
B
P
I
V O
A
K
SX
R W
YZ
X
Transit (tetrahedral) (i + 1)
(i + 2)
(i + 3)
J K O N R AV Z
J K O N R A V Z
O N V
W U
P M XT
I
QY SK
BL
AR
P M X
W U
I
Q
Q
I
UM X
W
PB
L
SY
T
J Z
YS
T L B
Transit (pyramid)
Transit (prism)
Figure 9 Solid186 elements generated by tetrahedral pyramid and prism methods
Shock and Vibration 9
direction of the speed) which are marked as 1F and 2F inorder of increased frequency ere are also two backwardwhirls (the whirl direction of the rotor is opposite to therotation direction of the speed) that are marked as 3B and
4B respectively e synchronous whirl line is called thesynchronous excitation line In the figure the velocitiescorresponding to the intersection of the synchronouswhirl line and the frequency line of the forward whirl are
Temperature8177
Turbineside
Compressorside
(K)
7688
7199
6710
6222
5733
5244
4755
4266
3777
Figure 13 Postprocessing rotor temperature distribution
Figure 11 Global grid of rotor-shaft system
008
007
006
005
004
003
002
001
000
60000rpm
80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
12 13 14 15 16 17 18
ExperimentalCFD
(a)
014
012
010
008
006
004
002
000
60000rpm80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
10 11 12 13 14 1615 17
ExperimentalCFD
(b)
Figure 12 Numerical simulation verified by experiments (a) Turbine side (b) Compressor side
10 Shock and Vibration
Compressorside
Turbineside035 10
TToT times103 (K)
Figure 14 Rotor temperature distribution of Bohn [19]
Figure 16 Stability diagram of rotor-shaft system without considering temperature
Shock and Vibration 11
marked as Ωcr1 and Ωcr2 respectively and were calculatedas follows Ωcr1 5 0115 rpm and Ωcr2 43 622 rpmBecause the backward whirl could not be excited by theunbalanced force it is not marked or considered
When the speed was in the critical speed range be-tween Ωcr1 and Ωcr2 in the 1F mode the whirling fre-quency had a clear increasing trend at trend may havebeen caused by floating ring bearings because in thecritical speed range of Ωcr1 and Ωcr2 the stiffness anddamping of the oil film changed sharply When the criticalspeed of Ωcr2 was exceeded the stiffness and damping ofthe oil film changed smoothly and the eddy frequency inthe corresponding 1F mode also increased at a constantrate
Figure 16 shows that the curve of the damping ratio ofthe four modes in case 1 changed with the rotation speede modal damping ratio is the ratio of actual damping tothe critical damping of a certain mode and the dampingratio is the decisive parameter to describe the stability Apositive damping ratio means that the rotor-shaft systemwas in a stable region when the vibration energy wasdissipated A negative damping ratio means that therotation energy supported the rotor rotation by addingvibration energy resulting in the rotor-shaft systembecoming unstable e point where the damping ratiocurve intersects with the zero line is the stable limit speed(SLS) e figure shows that the rotor-shaft system be-came unstable when SLS 1 56 550 rpm in 1F mode andthen became unstable when SLS 2 97 946 rpm in 2Fmode
Figure 17 shows the four modes close to the SLS 1stability limit speed at 55000 rpm e black dotted linerepresents the axis the combination of the orange line andthe orange sphere represents the starting position of thetrajectory and the incomplete track curve represents thewhirl direction e figure shows that the 1F mode wascylindrical thus combining rigid body motion with rotorbending Furthermore the rotor was close to the unstablestate at this time In contrast the 2F mode was a conicalbending mode e figure shows that the 3B mode wasconical with both rigid body motion and rotor bendingLastly the 4B mode was cylindrical with both rigid bodymotion and rotor bending
Figure 18 shows the four modes close to the SLS 2stability limit speed at 97946 rpme rotor was close to theunstable state at this time e 1F and 3B modes were cy-lindrical and conical combining rigid body motion androtor bending In contrast the 2F mode was a conicalbending mode
322 Effect of Temperature on the Campbell Diagram andStability Diagram We exported the temperature data ofeach node in Figure 12 and inserted the temperature valueinto the corresponding rotor-shaft system node through the
Ansys software commands functionWe analysed the criticalspeed and stability of the rotor-shaft system in the thermalenvironment
Figure 19 shows that temperature affected the intrinsicfrequency (whirl frequency) Compared with case 1 thewhirl-frequency curves in the 1F 2F and 3B modes wereslightly lower whereas for the 4B mode the whirl-fre-quency curve was much lower In the figure the startingpoint at the left end (when the rotating speed was 0 rpm)was taken as the observation object In case 1 the frequencywas 23009 Hz whereas in case 2 the frequency decreasedto 21825 Hz Because the temperature changed the ma-terialrsquos elastic modulus and Poissonrsquos ratio it affected itsstructural stiffness For higher-order frequencies the effecton frequency became greater as the order increasedHowever the effect on the critical speed Ωcr1 did notchange much whereas Ωcr2 decreased from 43622 rpm to41815 rpm is decrease was also caused by the intrinsicfrequency drop
Figure 20 shows the stability diagram in case 2 and thatthe temperature had an important effect on the dampingratio and SLS Compared with case 1 SLS 1 of the 1F modedecreased from the original 56550 rpm to 55078 rpm andmade the 1F mode enter the unstable region ahead of timee 2F mode change was more obvious and the same SLS 2decreased from the original 97946 rpm to 95969 rpm ereason for this can be explained by the following formulaξr (α2ωhr) + (βωhr2) Here ξr is the damping ratiocoefficient of the rth mode α is the mass damping coeffi-cient β is the stiffness damping coefficient and ωhr is thenatural frequency of the rth mode In most cases α (α 0)can be disregarded that is ξr βωhr2 β did not changewhereas ωhr decreased because of the effect of temperatureresulting in a decrease in ξr In other words SLS 1 and SLS 2in the 1F and 2F modes were reduced erefore whenpredicting the stability limit reset and dynamic perfor-mance of a turbocharger rotor the effect of temperatureshould have a great influence on the dynamic design of therotor
323 Unbalance Response Analysis We applied a dummyunbalance mass of 001 kg middot mm at node 2 and the fre-quency range of the steady-state synchronous responseanalysis was 0 to 2500Hz (corresponding speed range0 to 150 000 rpm) We then measured the response am-plitude of the turbine end floating ring bearing centerposition node 5 In Figure 21 case 1 is represented by ablack curve (normal temperature) and case 2 by a redcurve (high temperature) In case 1 (case 2) the curvesrepresent 6F 6B and 20F modes e frequency at thewave crest corresponding to the frequency correspondingto the fast reverse rotation (the frequency of the rotationspeed within one minute) in the case 1 (case 2) Campbelldiagram as shown in Figures 22 and 23) is similar to the
12 Shock and Vibration
black (red) number in Figure 20 Because the steeringmode is dominant in the response diagram the criticalspeeds Ωcr1 and Ωcr2 in case 1 (case 2) 1F and 2F modesare not shown in the diagram and followed a phenom-enon similar to that reported by Chouksey and Roy[29 30] Because of the effect of temperature the steering
frequency of case 2 shifted to the left making the changeof response amplitude more obvious e responsevalue at the peak of the 6F and 6B modes decreased from0147mm to 0014 mm and that at the peak of the20F and 20B modes decreased from 0036 mm to00037 mm
324 Verification of Rotor FEM Natural frequency anal-ysis is very important as the basis of a dynamic analysisWe obtained the first two natural frequencies of the rotorby hammering experiments and verified them by nu-merical simulation Figure 24 shows the measured parts(rotor system) force hammer and piezoelectric acceler-ation sensor To reduce the measurement error rate we
adopted a sensor with a nonlinearity of less than 12 anda sensitivity of 10 mvg and placed the sensor in themiddle of the rotor shaft Table 6 compares the naturalfrequency experimental value with the simulated valuee experimental and simulated values of the first naturalfrequency were 1386Hz and 1453 Hz respectively esecond-order natural frequencies were 4100 Hz and
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
Writing (6) in state-space form yields
A _w(t) Bw(t) + F(t) (8)
where matrices A and B state vector w(t) and force vectorF(t) are
A 0 M
M C1113890 1113891
6atimes6a
B M 0
0 minusK1113890 1113891
6atimes6a
w(t) _q (t)
q (t)1113896 1113897
6atimes1F(t)
0
f(t)1113896 1113897
6atimes1
(9)
where 0 is an empty matrix of size 3a times 3a and an emptyvector of size 3a times 1
Figure 10 shows a simplified finite element model of therotor-shaft system e grey-blue points in the figure areused to observe the modal shapes line trajectories andunbalanced harmonic response analysis
Figure 11 shows the rotor finite element global grid etotal number of grids was 3176485 with 546067 gridnodes In the geometry software each portion of the rotorwas treated as a part to avoid extra contact stiffness in thefinite element simulation and prevent affecting the simu-lation accuracy
To aid the comparison we considered two cases in thisstudy case 1 the rotor-shaft system without the influence oftemperature and case 2 the rotor-shaft system affected bytemperature In the following sections r-th forward and r-thbackward modes are represented as ldquor-Frdquo and ldquor-Brdquo modesrespectively
3 Results and Discussion
31 Heat Transfer Results and Experimental VerificationAfter the calculation we verified the numerical results of themass flow and the supercharger ratio (expansion ratio) enumerical results agreed with the experimental data and theerror control was approximately 7 as Figure 12 shows
e temperature distribution of the rotor was obtainedthrough postprocessing as Figure 13 shows the temperaturedistribution law was the same as that in the Bohn [19] studyresult and is shown in Figure 14
Figures 13 and 14 show that the highest temperature wasat the outer edge of the turbine wheel blades and that thelowest temperature was at the compressor wheel Overallthe rotor had a large temperature gradient
32 Modal Results
321 Campbell Diagram and Stability Diagram of NormalTemperature Environment Figure 15 shows the Campbelldiagram of the rotor-shaft system in case 1 that iswithout considering the temperature e figure was
Figure 5 Material properties of rotor-shaft system as a function of temperature (Youngrsquos modulus Poisson ratio specific heat capacity andthermal conductivity) (a) K418 (b) ZL105 (c) 42CrMo (d) 316 stainless steel [25 26]
Table 4 Damping coefficient of each partPart name Damping coefficientTurbo impeller 0002Compressor impeller and nut 0001Shaft 0005rust bearing and seal sleeve 0005
Table 5 Parameters of floating ring bearing that supported therotor shaftParameter name Parameter valueFloating ring mass 517 gInner length 514mmOuter length 796mmInner diameter 750mmOuter diameter 129mm
Shock and Vibration 7
Y
Divide
Z
X
14 cylinder
Figure 7 Rotor shaft divided into four parts and 14 cylinder
Figure 6 Total impedance of floating ring bearing (equivalent stiffness and damping coefficients) (a) Turbine side (b) Compressor side
8 Shock and Vibration
obtained from the whirl frequency (obtained from theimaginary part of the eigenvalue) and the rotational speedTo present a clear figure and express its basic
characteristics only the first four whirling directions aredescribed In the figure there are two forward whirls (thewhirl direction of the rotor agrees with the rotation
cyx cyx
cxy
cyy cyykyxkyx
kyy
kxxkxx kxycxx
cxy cxxkxy
kyy
1 2 3 4 5 6 7 8 9 10 11
Y
X Z
Figure 10 Finite element model of rotor-shaft system
N
Z
U
J
QM
Y
T L
B
P
IY
Z
X
V ON V
U A
RK X
S
M
YQ
I
O
W
P
B
LT
Z
JTransit (sweep)A
K
SX
R W
Figure 8 Solid186 element generated by the sweep method
N
Z
U
J
QM
Y
T L
B
P
I
V O
A
K
SX
R W
YZ
X
Transit (tetrahedral) (i + 1)
(i + 2)
(i + 3)
J K O N R AV Z
J K O N R A V Z
O N V
W U
P M XT
I
QY SK
BL
AR
P M X
W U
I
Q
Q
I
UM X
W
PB
L
SY
T
J Z
YS
T L B
Transit (pyramid)
Transit (prism)
Figure 9 Solid186 elements generated by tetrahedral pyramid and prism methods
Shock and Vibration 9
direction of the speed) which are marked as 1F and 2F inorder of increased frequency ere are also two backwardwhirls (the whirl direction of the rotor is opposite to therotation direction of the speed) that are marked as 3B and
4B respectively e synchronous whirl line is called thesynchronous excitation line In the figure the velocitiescorresponding to the intersection of the synchronouswhirl line and the frequency line of the forward whirl are
Temperature8177
Turbineside
Compressorside
(K)
7688
7199
6710
6222
5733
5244
4755
4266
3777
Figure 13 Postprocessing rotor temperature distribution
Figure 11 Global grid of rotor-shaft system
008
007
006
005
004
003
002
001
000
60000rpm
80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
12 13 14 15 16 17 18
ExperimentalCFD
(a)
014
012
010
008
006
004
002
000
60000rpm80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
10 11 12 13 14 1615 17
ExperimentalCFD
(b)
Figure 12 Numerical simulation verified by experiments (a) Turbine side (b) Compressor side
10 Shock and Vibration
Compressorside
Turbineside035 10
TToT times103 (K)
Figure 14 Rotor temperature distribution of Bohn [19]
Figure 16 Stability diagram of rotor-shaft system without considering temperature
Shock and Vibration 11
marked as Ωcr1 and Ωcr2 respectively and were calculatedas follows Ωcr1 5 0115 rpm and Ωcr2 43 622 rpmBecause the backward whirl could not be excited by theunbalanced force it is not marked or considered
When the speed was in the critical speed range be-tween Ωcr1 and Ωcr2 in the 1F mode the whirling fre-quency had a clear increasing trend at trend may havebeen caused by floating ring bearings because in thecritical speed range of Ωcr1 and Ωcr2 the stiffness anddamping of the oil film changed sharply When the criticalspeed of Ωcr2 was exceeded the stiffness and damping ofthe oil film changed smoothly and the eddy frequency inthe corresponding 1F mode also increased at a constantrate
Figure 16 shows that the curve of the damping ratio ofthe four modes in case 1 changed with the rotation speede modal damping ratio is the ratio of actual damping tothe critical damping of a certain mode and the dampingratio is the decisive parameter to describe the stability Apositive damping ratio means that the rotor-shaft systemwas in a stable region when the vibration energy wasdissipated A negative damping ratio means that therotation energy supported the rotor rotation by addingvibration energy resulting in the rotor-shaft systembecoming unstable e point where the damping ratiocurve intersects with the zero line is the stable limit speed(SLS) e figure shows that the rotor-shaft system be-came unstable when SLS 1 56 550 rpm in 1F mode andthen became unstable when SLS 2 97 946 rpm in 2Fmode
Figure 17 shows the four modes close to the SLS 1stability limit speed at 55000 rpm e black dotted linerepresents the axis the combination of the orange line andthe orange sphere represents the starting position of thetrajectory and the incomplete track curve represents thewhirl direction e figure shows that the 1F mode wascylindrical thus combining rigid body motion with rotorbending Furthermore the rotor was close to the unstablestate at this time In contrast the 2F mode was a conicalbending mode e figure shows that the 3B mode wasconical with both rigid body motion and rotor bendingLastly the 4B mode was cylindrical with both rigid bodymotion and rotor bending
Figure 18 shows the four modes close to the SLS 2stability limit speed at 97946 rpme rotor was close to theunstable state at this time e 1F and 3B modes were cy-lindrical and conical combining rigid body motion androtor bending In contrast the 2F mode was a conicalbending mode
322 Effect of Temperature on the Campbell Diagram andStability Diagram We exported the temperature data ofeach node in Figure 12 and inserted the temperature valueinto the corresponding rotor-shaft system node through the
Ansys software commands functionWe analysed the criticalspeed and stability of the rotor-shaft system in the thermalenvironment
Figure 19 shows that temperature affected the intrinsicfrequency (whirl frequency) Compared with case 1 thewhirl-frequency curves in the 1F 2F and 3B modes wereslightly lower whereas for the 4B mode the whirl-fre-quency curve was much lower In the figure the startingpoint at the left end (when the rotating speed was 0 rpm)was taken as the observation object In case 1 the frequencywas 23009 Hz whereas in case 2 the frequency decreasedto 21825 Hz Because the temperature changed the ma-terialrsquos elastic modulus and Poissonrsquos ratio it affected itsstructural stiffness For higher-order frequencies the effecton frequency became greater as the order increasedHowever the effect on the critical speed Ωcr1 did notchange much whereas Ωcr2 decreased from 43622 rpm to41815 rpm is decrease was also caused by the intrinsicfrequency drop
Figure 20 shows the stability diagram in case 2 and thatthe temperature had an important effect on the dampingratio and SLS Compared with case 1 SLS 1 of the 1F modedecreased from the original 56550 rpm to 55078 rpm andmade the 1F mode enter the unstable region ahead of timee 2F mode change was more obvious and the same SLS 2decreased from the original 97946 rpm to 95969 rpm ereason for this can be explained by the following formulaξr (α2ωhr) + (βωhr2) Here ξr is the damping ratiocoefficient of the rth mode α is the mass damping coeffi-cient β is the stiffness damping coefficient and ωhr is thenatural frequency of the rth mode In most cases α (α 0)can be disregarded that is ξr βωhr2 β did not changewhereas ωhr decreased because of the effect of temperatureresulting in a decrease in ξr In other words SLS 1 and SLS 2in the 1F and 2F modes were reduced erefore whenpredicting the stability limit reset and dynamic perfor-mance of a turbocharger rotor the effect of temperatureshould have a great influence on the dynamic design of therotor
323 Unbalance Response Analysis We applied a dummyunbalance mass of 001 kg middot mm at node 2 and the fre-quency range of the steady-state synchronous responseanalysis was 0 to 2500Hz (corresponding speed range0 to 150 000 rpm) We then measured the response am-plitude of the turbine end floating ring bearing centerposition node 5 In Figure 21 case 1 is represented by ablack curve (normal temperature) and case 2 by a redcurve (high temperature) In case 1 (case 2) the curvesrepresent 6F 6B and 20F modes e frequency at thewave crest corresponding to the frequency correspondingto the fast reverse rotation (the frequency of the rotationspeed within one minute) in the case 1 (case 2) Campbelldiagram as shown in Figures 22 and 23) is similar to the
12 Shock and Vibration
black (red) number in Figure 20 Because the steeringmode is dominant in the response diagram the criticalspeeds Ωcr1 and Ωcr2 in case 1 (case 2) 1F and 2F modesare not shown in the diagram and followed a phenom-enon similar to that reported by Chouksey and Roy[29 30] Because of the effect of temperature the steering
frequency of case 2 shifted to the left making the changeof response amplitude more obvious e responsevalue at the peak of the 6F and 6B modes decreased from0147mm to 0014 mm and that at the peak of the20F and 20B modes decreased from 0036 mm to00037 mm
324 Verification of Rotor FEM Natural frequency anal-ysis is very important as the basis of a dynamic analysisWe obtained the first two natural frequencies of the rotorby hammering experiments and verified them by nu-merical simulation Figure 24 shows the measured parts(rotor system) force hammer and piezoelectric acceler-ation sensor To reduce the measurement error rate we
adopted a sensor with a nonlinearity of less than 12 anda sensitivity of 10 mvg and placed the sensor in themiddle of the rotor shaft Table 6 compares the naturalfrequency experimental value with the simulated valuee experimental and simulated values of the first naturalfrequency were 1386Hz and 1453 Hz respectively esecond-order natural frequencies were 4100 Hz and
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
Y
Divide
Z
X
14 cylinder
Figure 7 Rotor shaft divided into four parts and 14 cylinder
Figure 6 Total impedance of floating ring bearing (equivalent stiffness and damping coefficients) (a) Turbine side (b) Compressor side
8 Shock and Vibration
obtained from the whirl frequency (obtained from theimaginary part of the eigenvalue) and the rotational speedTo present a clear figure and express its basic
characteristics only the first four whirling directions aredescribed In the figure there are two forward whirls (thewhirl direction of the rotor agrees with the rotation
cyx cyx
cxy
cyy cyykyxkyx
kyy
kxxkxx kxycxx
cxy cxxkxy
kyy
1 2 3 4 5 6 7 8 9 10 11
Y
X Z
Figure 10 Finite element model of rotor-shaft system
N
Z
U
J
QM
Y
T L
B
P
IY
Z
X
V ON V
U A
RK X
S
M
YQ
I
O
W
P
B
LT
Z
JTransit (sweep)A
K
SX
R W
Figure 8 Solid186 element generated by the sweep method
N
Z
U
J
QM
Y
T L
B
P
I
V O
A
K
SX
R W
YZ
X
Transit (tetrahedral) (i + 1)
(i + 2)
(i + 3)
J K O N R AV Z
J K O N R A V Z
O N V
W U
P M XT
I
QY SK
BL
AR
P M X
W U
I
Q
Q
I
UM X
W
PB
L
SY
T
J Z
YS
T L B
Transit (pyramid)
Transit (prism)
Figure 9 Solid186 elements generated by tetrahedral pyramid and prism methods
Shock and Vibration 9
direction of the speed) which are marked as 1F and 2F inorder of increased frequency ere are also two backwardwhirls (the whirl direction of the rotor is opposite to therotation direction of the speed) that are marked as 3B and
4B respectively e synchronous whirl line is called thesynchronous excitation line In the figure the velocitiescorresponding to the intersection of the synchronouswhirl line and the frequency line of the forward whirl are
Temperature8177
Turbineside
Compressorside
(K)
7688
7199
6710
6222
5733
5244
4755
4266
3777
Figure 13 Postprocessing rotor temperature distribution
Figure 11 Global grid of rotor-shaft system
008
007
006
005
004
003
002
001
000
60000rpm
80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
12 13 14 15 16 17 18
ExperimentalCFD
(a)
014
012
010
008
006
004
002
000
60000rpm80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
10 11 12 13 14 1615 17
ExperimentalCFD
(b)
Figure 12 Numerical simulation verified by experiments (a) Turbine side (b) Compressor side
10 Shock and Vibration
Compressorside
Turbineside035 10
TToT times103 (K)
Figure 14 Rotor temperature distribution of Bohn [19]
Figure 16 Stability diagram of rotor-shaft system without considering temperature
Shock and Vibration 11
marked as Ωcr1 and Ωcr2 respectively and were calculatedas follows Ωcr1 5 0115 rpm and Ωcr2 43 622 rpmBecause the backward whirl could not be excited by theunbalanced force it is not marked or considered
When the speed was in the critical speed range be-tween Ωcr1 and Ωcr2 in the 1F mode the whirling fre-quency had a clear increasing trend at trend may havebeen caused by floating ring bearings because in thecritical speed range of Ωcr1 and Ωcr2 the stiffness anddamping of the oil film changed sharply When the criticalspeed of Ωcr2 was exceeded the stiffness and damping ofthe oil film changed smoothly and the eddy frequency inthe corresponding 1F mode also increased at a constantrate
Figure 16 shows that the curve of the damping ratio ofthe four modes in case 1 changed with the rotation speede modal damping ratio is the ratio of actual damping tothe critical damping of a certain mode and the dampingratio is the decisive parameter to describe the stability Apositive damping ratio means that the rotor-shaft systemwas in a stable region when the vibration energy wasdissipated A negative damping ratio means that therotation energy supported the rotor rotation by addingvibration energy resulting in the rotor-shaft systembecoming unstable e point where the damping ratiocurve intersects with the zero line is the stable limit speed(SLS) e figure shows that the rotor-shaft system be-came unstable when SLS 1 56 550 rpm in 1F mode andthen became unstable when SLS 2 97 946 rpm in 2Fmode
Figure 17 shows the four modes close to the SLS 1stability limit speed at 55000 rpm e black dotted linerepresents the axis the combination of the orange line andthe orange sphere represents the starting position of thetrajectory and the incomplete track curve represents thewhirl direction e figure shows that the 1F mode wascylindrical thus combining rigid body motion with rotorbending Furthermore the rotor was close to the unstablestate at this time In contrast the 2F mode was a conicalbending mode e figure shows that the 3B mode wasconical with both rigid body motion and rotor bendingLastly the 4B mode was cylindrical with both rigid bodymotion and rotor bending
Figure 18 shows the four modes close to the SLS 2stability limit speed at 97946 rpme rotor was close to theunstable state at this time e 1F and 3B modes were cy-lindrical and conical combining rigid body motion androtor bending In contrast the 2F mode was a conicalbending mode
322 Effect of Temperature on the Campbell Diagram andStability Diagram We exported the temperature data ofeach node in Figure 12 and inserted the temperature valueinto the corresponding rotor-shaft system node through the
Ansys software commands functionWe analysed the criticalspeed and stability of the rotor-shaft system in the thermalenvironment
Figure 19 shows that temperature affected the intrinsicfrequency (whirl frequency) Compared with case 1 thewhirl-frequency curves in the 1F 2F and 3B modes wereslightly lower whereas for the 4B mode the whirl-fre-quency curve was much lower In the figure the startingpoint at the left end (when the rotating speed was 0 rpm)was taken as the observation object In case 1 the frequencywas 23009 Hz whereas in case 2 the frequency decreasedto 21825 Hz Because the temperature changed the ma-terialrsquos elastic modulus and Poissonrsquos ratio it affected itsstructural stiffness For higher-order frequencies the effecton frequency became greater as the order increasedHowever the effect on the critical speed Ωcr1 did notchange much whereas Ωcr2 decreased from 43622 rpm to41815 rpm is decrease was also caused by the intrinsicfrequency drop
Figure 20 shows the stability diagram in case 2 and thatthe temperature had an important effect on the dampingratio and SLS Compared with case 1 SLS 1 of the 1F modedecreased from the original 56550 rpm to 55078 rpm andmade the 1F mode enter the unstable region ahead of timee 2F mode change was more obvious and the same SLS 2decreased from the original 97946 rpm to 95969 rpm ereason for this can be explained by the following formulaξr (α2ωhr) + (βωhr2) Here ξr is the damping ratiocoefficient of the rth mode α is the mass damping coeffi-cient β is the stiffness damping coefficient and ωhr is thenatural frequency of the rth mode In most cases α (α 0)can be disregarded that is ξr βωhr2 β did not changewhereas ωhr decreased because of the effect of temperatureresulting in a decrease in ξr In other words SLS 1 and SLS 2in the 1F and 2F modes were reduced erefore whenpredicting the stability limit reset and dynamic perfor-mance of a turbocharger rotor the effect of temperatureshould have a great influence on the dynamic design of therotor
323 Unbalance Response Analysis We applied a dummyunbalance mass of 001 kg middot mm at node 2 and the fre-quency range of the steady-state synchronous responseanalysis was 0 to 2500Hz (corresponding speed range0 to 150 000 rpm) We then measured the response am-plitude of the turbine end floating ring bearing centerposition node 5 In Figure 21 case 1 is represented by ablack curve (normal temperature) and case 2 by a redcurve (high temperature) In case 1 (case 2) the curvesrepresent 6F 6B and 20F modes e frequency at thewave crest corresponding to the frequency correspondingto the fast reverse rotation (the frequency of the rotationspeed within one minute) in the case 1 (case 2) Campbelldiagram as shown in Figures 22 and 23) is similar to the
12 Shock and Vibration
black (red) number in Figure 20 Because the steeringmode is dominant in the response diagram the criticalspeeds Ωcr1 and Ωcr2 in case 1 (case 2) 1F and 2F modesare not shown in the diagram and followed a phenom-enon similar to that reported by Chouksey and Roy[29 30] Because of the effect of temperature the steering
frequency of case 2 shifted to the left making the changeof response amplitude more obvious e responsevalue at the peak of the 6F and 6B modes decreased from0147mm to 0014 mm and that at the peak of the20F and 20B modes decreased from 0036 mm to00037 mm
324 Verification of Rotor FEM Natural frequency anal-ysis is very important as the basis of a dynamic analysisWe obtained the first two natural frequencies of the rotorby hammering experiments and verified them by nu-merical simulation Figure 24 shows the measured parts(rotor system) force hammer and piezoelectric acceler-ation sensor To reduce the measurement error rate we
adopted a sensor with a nonlinearity of less than 12 anda sensitivity of 10 mvg and placed the sensor in themiddle of the rotor shaft Table 6 compares the naturalfrequency experimental value with the simulated valuee experimental and simulated values of the first naturalfrequency were 1386Hz and 1453 Hz respectively esecond-order natural frequencies were 4100 Hz and
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
obtained from the whirl frequency (obtained from theimaginary part of the eigenvalue) and the rotational speedTo present a clear figure and express its basic
characteristics only the first four whirling directions aredescribed In the figure there are two forward whirls (thewhirl direction of the rotor agrees with the rotation
cyx cyx
cxy
cyy cyykyxkyx
kyy
kxxkxx kxycxx
cxy cxxkxy
kyy
1 2 3 4 5 6 7 8 9 10 11
Y
X Z
Figure 10 Finite element model of rotor-shaft system
N
Z
U
J
QM
Y
T L
B
P
IY
Z
X
V ON V
U A
RK X
S
M
YQ
I
O
W
P
B
LT
Z
JTransit (sweep)A
K
SX
R W
Figure 8 Solid186 element generated by the sweep method
N
Z
U
J
QM
Y
T L
B
P
I
V O
A
K
SX
R W
YZ
X
Transit (tetrahedral) (i + 1)
(i + 2)
(i + 3)
J K O N R AV Z
J K O N R A V Z
O N V
W U
P M XT
I
QY SK
BL
AR
P M X
W U
I
Q
Q
I
UM X
W
PB
L
SY
T
J Z
YS
T L B
Transit (pyramid)
Transit (prism)
Figure 9 Solid186 elements generated by tetrahedral pyramid and prism methods
Shock and Vibration 9
direction of the speed) which are marked as 1F and 2F inorder of increased frequency ere are also two backwardwhirls (the whirl direction of the rotor is opposite to therotation direction of the speed) that are marked as 3B and
4B respectively e synchronous whirl line is called thesynchronous excitation line In the figure the velocitiescorresponding to the intersection of the synchronouswhirl line and the frequency line of the forward whirl are
Temperature8177
Turbineside
Compressorside
(K)
7688
7199
6710
6222
5733
5244
4755
4266
3777
Figure 13 Postprocessing rotor temperature distribution
Figure 11 Global grid of rotor-shaft system
008
007
006
005
004
003
002
001
000
60000rpm
80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
12 13 14 15 16 17 18
ExperimentalCFD
(a)
014
012
010
008
006
004
002
000
60000rpm80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
10 11 12 13 14 1615 17
ExperimentalCFD
(b)
Figure 12 Numerical simulation verified by experiments (a) Turbine side (b) Compressor side
10 Shock and Vibration
Compressorside
Turbineside035 10
TToT times103 (K)
Figure 14 Rotor temperature distribution of Bohn [19]
Figure 16 Stability diagram of rotor-shaft system without considering temperature
Shock and Vibration 11
marked as Ωcr1 and Ωcr2 respectively and were calculatedas follows Ωcr1 5 0115 rpm and Ωcr2 43 622 rpmBecause the backward whirl could not be excited by theunbalanced force it is not marked or considered
When the speed was in the critical speed range be-tween Ωcr1 and Ωcr2 in the 1F mode the whirling fre-quency had a clear increasing trend at trend may havebeen caused by floating ring bearings because in thecritical speed range of Ωcr1 and Ωcr2 the stiffness anddamping of the oil film changed sharply When the criticalspeed of Ωcr2 was exceeded the stiffness and damping ofthe oil film changed smoothly and the eddy frequency inthe corresponding 1F mode also increased at a constantrate
Figure 16 shows that the curve of the damping ratio ofthe four modes in case 1 changed with the rotation speede modal damping ratio is the ratio of actual damping tothe critical damping of a certain mode and the dampingratio is the decisive parameter to describe the stability Apositive damping ratio means that the rotor-shaft systemwas in a stable region when the vibration energy wasdissipated A negative damping ratio means that therotation energy supported the rotor rotation by addingvibration energy resulting in the rotor-shaft systembecoming unstable e point where the damping ratiocurve intersects with the zero line is the stable limit speed(SLS) e figure shows that the rotor-shaft system be-came unstable when SLS 1 56 550 rpm in 1F mode andthen became unstable when SLS 2 97 946 rpm in 2Fmode
Figure 17 shows the four modes close to the SLS 1stability limit speed at 55000 rpm e black dotted linerepresents the axis the combination of the orange line andthe orange sphere represents the starting position of thetrajectory and the incomplete track curve represents thewhirl direction e figure shows that the 1F mode wascylindrical thus combining rigid body motion with rotorbending Furthermore the rotor was close to the unstablestate at this time In contrast the 2F mode was a conicalbending mode e figure shows that the 3B mode wasconical with both rigid body motion and rotor bendingLastly the 4B mode was cylindrical with both rigid bodymotion and rotor bending
Figure 18 shows the four modes close to the SLS 2stability limit speed at 97946 rpme rotor was close to theunstable state at this time e 1F and 3B modes were cy-lindrical and conical combining rigid body motion androtor bending In contrast the 2F mode was a conicalbending mode
322 Effect of Temperature on the Campbell Diagram andStability Diagram We exported the temperature data ofeach node in Figure 12 and inserted the temperature valueinto the corresponding rotor-shaft system node through the
Ansys software commands functionWe analysed the criticalspeed and stability of the rotor-shaft system in the thermalenvironment
Figure 19 shows that temperature affected the intrinsicfrequency (whirl frequency) Compared with case 1 thewhirl-frequency curves in the 1F 2F and 3B modes wereslightly lower whereas for the 4B mode the whirl-fre-quency curve was much lower In the figure the startingpoint at the left end (when the rotating speed was 0 rpm)was taken as the observation object In case 1 the frequencywas 23009 Hz whereas in case 2 the frequency decreasedto 21825 Hz Because the temperature changed the ma-terialrsquos elastic modulus and Poissonrsquos ratio it affected itsstructural stiffness For higher-order frequencies the effecton frequency became greater as the order increasedHowever the effect on the critical speed Ωcr1 did notchange much whereas Ωcr2 decreased from 43622 rpm to41815 rpm is decrease was also caused by the intrinsicfrequency drop
Figure 20 shows the stability diagram in case 2 and thatthe temperature had an important effect on the dampingratio and SLS Compared with case 1 SLS 1 of the 1F modedecreased from the original 56550 rpm to 55078 rpm andmade the 1F mode enter the unstable region ahead of timee 2F mode change was more obvious and the same SLS 2decreased from the original 97946 rpm to 95969 rpm ereason for this can be explained by the following formulaξr (α2ωhr) + (βωhr2) Here ξr is the damping ratiocoefficient of the rth mode α is the mass damping coeffi-cient β is the stiffness damping coefficient and ωhr is thenatural frequency of the rth mode In most cases α (α 0)can be disregarded that is ξr βωhr2 β did not changewhereas ωhr decreased because of the effect of temperatureresulting in a decrease in ξr In other words SLS 1 and SLS 2in the 1F and 2F modes were reduced erefore whenpredicting the stability limit reset and dynamic perfor-mance of a turbocharger rotor the effect of temperatureshould have a great influence on the dynamic design of therotor
323 Unbalance Response Analysis We applied a dummyunbalance mass of 001 kg middot mm at node 2 and the fre-quency range of the steady-state synchronous responseanalysis was 0 to 2500Hz (corresponding speed range0 to 150 000 rpm) We then measured the response am-plitude of the turbine end floating ring bearing centerposition node 5 In Figure 21 case 1 is represented by ablack curve (normal temperature) and case 2 by a redcurve (high temperature) In case 1 (case 2) the curvesrepresent 6F 6B and 20F modes e frequency at thewave crest corresponding to the frequency correspondingto the fast reverse rotation (the frequency of the rotationspeed within one minute) in the case 1 (case 2) Campbelldiagram as shown in Figures 22 and 23) is similar to the
12 Shock and Vibration
black (red) number in Figure 20 Because the steeringmode is dominant in the response diagram the criticalspeeds Ωcr1 and Ωcr2 in case 1 (case 2) 1F and 2F modesare not shown in the diagram and followed a phenom-enon similar to that reported by Chouksey and Roy[29 30] Because of the effect of temperature the steering
frequency of case 2 shifted to the left making the changeof response amplitude more obvious e responsevalue at the peak of the 6F and 6B modes decreased from0147mm to 0014 mm and that at the peak of the20F and 20B modes decreased from 0036 mm to00037 mm
324 Verification of Rotor FEM Natural frequency anal-ysis is very important as the basis of a dynamic analysisWe obtained the first two natural frequencies of the rotorby hammering experiments and verified them by nu-merical simulation Figure 24 shows the measured parts(rotor system) force hammer and piezoelectric acceler-ation sensor To reduce the measurement error rate we
adopted a sensor with a nonlinearity of less than 12 anda sensitivity of 10 mvg and placed the sensor in themiddle of the rotor shaft Table 6 compares the naturalfrequency experimental value with the simulated valuee experimental and simulated values of the first naturalfrequency were 1386Hz and 1453 Hz respectively esecond-order natural frequencies were 4100 Hz and
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
direction of the speed) which are marked as 1F and 2F inorder of increased frequency ere are also two backwardwhirls (the whirl direction of the rotor is opposite to therotation direction of the speed) that are marked as 3B and
4B respectively e synchronous whirl line is called thesynchronous excitation line In the figure the velocitiescorresponding to the intersection of the synchronouswhirl line and the frequency line of the forward whirl are
Temperature8177
Turbineside
Compressorside
(K)
7688
7199
6710
6222
5733
5244
4755
4266
3777
Figure 13 Postprocessing rotor temperature distribution
Figure 11 Global grid of rotor-shaft system
008
007
006
005
004
003
002
001
000
60000rpm
80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
12 13 14 15 16 17 18
ExperimentalCFD
(a)
014
012
010
008
006
004
002
000
60000rpm80000rpm
100000rpm
Pressure ratio
Mas
s flo
w ra
te (k
g sndash1
)
10 11 12 13 14 1615 17
ExperimentalCFD
(b)
Figure 12 Numerical simulation verified by experiments (a) Turbine side (b) Compressor side
10 Shock and Vibration
Compressorside
Turbineside035 10
TToT times103 (K)
Figure 14 Rotor temperature distribution of Bohn [19]
Figure 16 Stability diagram of rotor-shaft system without considering temperature
Shock and Vibration 11
marked as Ωcr1 and Ωcr2 respectively and were calculatedas follows Ωcr1 5 0115 rpm and Ωcr2 43 622 rpmBecause the backward whirl could not be excited by theunbalanced force it is not marked or considered
When the speed was in the critical speed range be-tween Ωcr1 and Ωcr2 in the 1F mode the whirling fre-quency had a clear increasing trend at trend may havebeen caused by floating ring bearings because in thecritical speed range of Ωcr1 and Ωcr2 the stiffness anddamping of the oil film changed sharply When the criticalspeed of Ωcr2 was exceeded the stiffness and damping ofthe oil film changed smoothly and the eddy frequency inthe corresponding 1F mode also increased at a constantrate
Figure 16 shows that the curve of the damping ratio ofthe four modes in case 1 changed with the rotation speede modal damping ratio is the ratio of actual damping tothe critical damping of a certain mode and the dampingratio is the decisive parameter to describe the stability Apositive damping ratio means that the rotor-shaft systemwas in a stable region when the vibration energy wasdissipated A negative damping ratio means that therotation energy supported the rotor rotation by addingvibration energy resulting in the rotor-shaft systembecoming unstable e point where the damping ratiocurve intersects with the zero line is the stable limit speed(SLS) e figure shows that the rotor-shaft system be-came unstable when SLS 1 56 550 rpm in 1F mode andthen became unstable when SLS 2 97 946 rpm in 2Fmode
Figure 17 shows the four modes close to the SLS 1stability limit speed at 55000 rpm e black dotted linerepresents the axis the combination of the orange line andthe orange sphere represents the starting position of thetrajectory and the incomplete track curve represents thewhirl direction e figure shows that the 1F mode wascylindrical thus combining rigid body motion with rotorbending Furthermore the rotor was close to the unstablestate at this time In contrast the 2F mode was a conicalbending mode e figure shows that the 3B mode wasconical with both rigid body motion and rotor bendingLastly the 4B mode was cylindrical with both rigid bodymotion and rotor bending
Figure 18 shows the four modes close to the SLS 2stability limit speed at 97946 rpme rotor was close to theunstable state at this time e 1F and 3B modes were cy-lindrical and conical combining rigid body motion androtor bending In contrast the 2F mode was a conicalbending mode
322 Effect of Temperature on the Campbell Diagram andStability Diagram We exported the temperature data ofeach node in Figure 12 and inserted the temperature valueinto the corresponding rotor-shaft system node through the
Ansys software commands functionWe analysed the criticalspeed and stability of the rotor-shaft system in the thermalenvironment
Figure 19 shows that temperature affected the intrinsicfrequency (whirl frequency) Compared with case 1 thewhirl-frequency curves in the 1F 2F and 3B modes wereslightly lower whereas for the 4B mode the whirl-fre-quency curve was much lower In the figure the startingpoint at the left end (when the rotating speed was 0 rpm)was taken as the observation object In case 1 the frequencywas 23009 Hz whereas in case 2 the frequency decreasedto 21825 Hz Because the temperature changed the ma-terialrsquos elastic modulus and Poissonrsquos ratio it affected itsstructural stiffness For higher-order frequencies the effecton frequency became greater as the order increasedHowever the effect on the critical speed Ωcr1 did notchange much whereas Ωcr2 decreased from 43622 rpm to41815 rpm is decrease was also caused by the intrinsicfrequency drop
Figure 20 shows the stability diagram in case 2 and thatthe temperature had an important effect on the dampingratio and SLS Compared with case 1 SLS 1 of the 1F modedecreased from the original 56550 rpm to 55078 rpm andmade the 1F mode enter the unstable region ahead of timee 2F mode change was more obvious and the same SLS 2decreased from the original 97946 rpm to 95969 rpm ereason for this can be explained by the following formulaξr (α2ωhr) + (βωhr2) Here ξr is the damping ratiocoefficient of the rth mode α is the mass damping coeffi-cient β is the stiffness damping coefficient and ωhr is thenatural frequency of the rth mode In most cases α (α 0)can be disregarded that is ξr βωhr2 β did not changewhereas ωhr decreased because of the effect of temperatureresulting in a decrease in ξr In other words SLS 1 and SLS 2in the 1F and 2F modes were reduced erefore whenpredicting the stability limit reset and dynamic perfor-mance of a turbocharger rotor the effect of temperatureshould have a great influence on the dynamic design of therotor
323 Unbalance Response Analysis We applied a dummyunbalance mass of 001 kg middot mm at node 2 and the fre-quency range of the steady-state synchronous responseanalysis was 0 to 2500Hz (corresponding speed range0 to 150 000 rpm) We then measured the response am-plitude of the turbine end floating ring bearing centerposition node 5 In Figure 21 case 1 is represented by ablack curve (normal temperature) and case 2 by a redcurve (high temperature) In case 1 (case 2) the curvesrepresent 6F 6B and 20F modes e frequency at thewave crest corresponding to the frequency correspondingto the fast reverse rotation (the frequency of the rotationspeed within one minute) in the case 1 (case 2) Campbelldiagram as shown in Figures 22 and 23) is similar to the
12 Shock and Vibration
black (red) number in Figure 20 Because the steeringmode is dominant in the response diagram the criticalspeeds Ωcr1 and Ωcr2 in case 1 (case 2) 1F and 2F modesare not shown in the diagram and followed a phenom-enon similar to that reported by Chouksey and Roy[29 30] Because of the effect of temperature the steering
frequency of case 2 shifted to the left making the changeof response amplitude more obvious e responsevalue at the peak of the 6F and 6B modes decreased from0147mm to 0014 mm and that at the peak of the20F and 20B modes decreased from 0036 mm to00037 mm
324 Verification of Rotor FEM Natural frequency anal-ysis is very important as the basis of a dynamic analysisWe obtained the first two natural frequencies of the rotorby hammering experiments and verified them by nu-merical simulation Figure 24 shows the measured parts(rotor system) force hammer and piezoelectric acceler-ation sensor To reduce the measurement error rate we
adopted a sensor with a nonlinearity of less than 12 anda sensitivity of 10 mvg and placed the sensor in themiddle of the rotor shaft Table 6 compares the naturalfrequency experimental value with the simulated valuee experimental and simulated values of the first naturalfrequency were 1386Hz and 1453 Hz respectively esecond-order natural frequencies were 4100 Hz and
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
Compressorside
Turbineside035 10
TToT times103 (K)
Figure 14 Rotor temperature distribution of Bohn [19]
Figure 16 Stability diagram of rotor-shaft system without considering temperature
Shock and Vibration 11
marked as Ωcr1 and Ωcr2 respectively and were calculatedas follows Ωcr1 5 0115 rpm and Ωcr2 43 622 rpmBecause the backward whirl could not be excited by theunbalanced force it is not marked or considered
When the speed was in the critical speed range be-tween Ωcr1 and Ωcr2 in the 1F mode the whirling fre-quency had a clear increasing trend at trend may havebeen caused by floating ring bearings because in thecritical speed range of Ωcr1 and Ωcr2 the stiffness anddamping of the oil film changed sharply When the criticalspeed of Ωcr2 was exceeded the stiffness and damping ofthe oil film changed smoothly and the eddy frequency inthe corresponding 1F mode also increased at a constantrate
Figure 16 shows that the curve of the damping ratio ofthe four modes in case 1 changed with the rotation speede modal damping ratio is the ratio of actual damping tothe critical damping of a certain mode and the dampingratio is the decisive parameter to describe the stability Apositive damping ratio means that the rotor-shaft systemwas in a stable region when the vibration energy wasdissipated A negative damping ratio means that therotation energy supported the rotor rotation by addingvibration energy resulting in the rotor-shaft systembecoming unstable e point where the damping ratiocurve intersects with the zero line is the stable limit speed(SLS) e figure shows that the rotor-shaft system be-came unstable when SLS 1 56 550 rpm in 1F mode andthen became unstable when SLS 2 97 946 rpm in 2Fmode
Figure 17 shows the four modes close to the SLS 1stability limit speed at 55000 rpm e black dotted linerepresents the axis the combination of the orange line andthe orange sphere represents the starting position of thetrajectory and the incomplete track curve represents thewhirl direction e figure shows that the 1F mode wascylindrical thus combining rigid body motion with rotorbending Furthermore the rotor was close to the unstablestate at this time In contrast the 2F mode was a conicalbending mode e figure shows that the 3B mode wasconical with both rigid body motion and rotor bendingLastly the 4B mode was cylindrical with both rigid bodymotion and rotor bending
Figure 18 shows the four modes close to the SLS 2stability limit speed at 97946 rpme rotor was close to theunstable state at this time e 1F and 3B modes were cy-lindrical and conical combining rigid body motion androtor bending In contrast the 2F mode was a conicalbending mode
322 Effect of Temperature on the Campbell Diagram andStability Diagram We exported the temperature data ofeach node in Figure 12 and inserted the temperature valueinto the corresponding rotor-shaft system node through the
Ansys software commands functionWe analysed the criticalspeed and stability of the rotor-shaft system in the thermalenvironment
Figure 19 shows that temperature affected the intrinsicfrequency (whirl frequency) Compared with case 1 thewhirl-frequency curves in the 1F 2F and 3B modes wereslightly lower whereas for the 4B mode the whirl-fre-quency curve was much lower In the figure the startingpoint at the left end (when the rotating speed was 0 rpm)was taken as the observation object In case 1 the frequencywas 23009 Hz whereas in case 2 the frequency decreasedto 21825 Hz Because the temperature changed the ma-terialrsquos elastic modulus and Poissonrsquos ratio it affected itsstructural stiffness For higher-order frequencies the effecton frequency became greater as the order increasedHowever the effect on the critical speed Ωcr1 did notchange much whereas Ωcr2 decreased from 43622 rpm to41815 rpm is decrease was also caused by the intrinsicfrequency drop
Figure 20 shows the stability diagram in case 2 and thatthe temperature had an important effect on the dampingratio and SLS Compared with case 1 SLS 1 of the 1F modedecreased from the original 56550 rpm to 55078 rpm andmade the 1F mode enter the unstable region ahead of timee 2F mode change was more obvious and the same SLS 2decreased from the original 97946 rpm to 95969 rpm ereason for this can be explained by the following formulaξr (α2ωhr) + (βωhr2) Here ξr is the damping ratiocoefficient of the rth mode α is the mass damping coeffi-cient β is the stiffness damping coefficient and ωhr is thenatural frequency of the rth mode In most cases α (α 0)can be disregarded that is ξr βωhr2 β did not changewhereas ωhr decreased because of the effect of temperatureresulting in a decrease in ξr In other words SLS 1 and SLS 2in the 1F and 2F modes were reduced erefore whenpredicting the stability limit reset and dynamic perfor-mance of a turbocharger rotor the effect of temperatureshould have a great influence on the dynamic design of therotor
323 Unbalance Response Analysis We applied a dummyunbalance mass of 001 kg middot mm at node 2 and the fre-quency range of the steady-state synchronous responseanalysis was 0 to 2500Hz (corresponding speed range0 to 150 000 rpm) We then measured the response am-plitude of the turbine end floating ring bearing centerposition node 5 In Figure 21 case 1 is represented by ablack curve (normal temperature) and case 2 by a redcurve (high temperature) In case 1 (case 2) the curvesrepresent 6F 6B and 20F modes e frequency at thewave crest corresponding to the frequency correspondingto the fast reverse rotation (the frequency of the rotationspeed within one minute) in the case 1 (case 2) Campbelldiagram as shown in Figures 22 and 23) is similar to the
12 Shock and Vibration
black (red) number in Figure 20 Because the steeringmode is dominant in the response diagram the criticalspeeds Ωcr1 and Ωcr2 in case 1 (case 2) 1F and 2F modesare not shown in the diagram and followed a phenom-enon similar to that reported by Chouksey and Roy[29 30] Because of the effect of temperature the steering
frequency of case 2 shifted to the left making the changeof response amplitude more obvious e responsevalue at the peak of the 6F and 6B modes decreased from0147mm to 0014 mm and that at the peak of the20F and 20B modes decreased from 0036 mm to00037 mm
324 Verification of Rotor FEM Natural frequency anal-ysis is very important as the basis of a dynamic analysisWe obtained the first two natural frequencies of the rotorby hammering experiments and verified them by nu-merical simulation Figure 24 shows the measured parts(rotor system) force hammer and piezoelectric acceler-ation sensor To reduce the measurement error rate we
adopted a sensor with a nonlinearity of less than 12 anda sensitivity of 10 mvg and placed the sensor in themiddle of the rotor shaft Table 6 compares the naturalfrequency experimental value with the simulated valuee experimental and simulated values of the first naturalfrequency were 1386Hz and 1453 Hz respectively esecond-order natural frequencies were 4100 Hz and
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
marked as Ωcr1 and Ωcr2 respectively and were calculatedas follows Ωcr1 5 0115 rpm and Ωcr2 43 622 rpmBecause the backward whirl could not be excited by theunbalanced force it is not marked or considered
When the speed was in the critical speed range be-tween Ωcr1 and Ωcr2 in the 1F mode the whirling fre-quency had a clear increasing trend at trend may havebeen caused by floating ring bearings because in thecritical speed range of Ωcr1 and Ωcr2 the stiffness anddamping of the oil film changed sharply When the criticalspeed of Ωcr2 was exceeded the stiffness and damping ofthe oil film changed smoothly and the eddy frequency inthe corresponding 1F mode also increased at a constantrate
Figure 16 shows that the curve of the damping ratio ofthe four modes in case 1 changed with the rotation speede modal damping ratio is the ratio of actual damping tothe critical damping of a certain mode and the dampingratio is the decisive parameter to describe the stability Apositive damping ratio means that the rotor-shaft systemwas in a stable region when the vibration energy wasdissipated A negative damping ratio means that therotation energy supported the rotor rotation by addingvibration energy resulting in the rotor-shaft systembecoming unstable e point where the damping ratiocurve intersects with the zero line is the stable limit speed(SLS) e figure shows that the rotor-shaft system be-came unstable when SLS 1 56 550 rpm in 1F mode andthen became unstable when SLS 2 97 946 rpm in 2Fmode
Figure 17 shows the four modes close to the SLS 1stability limit speed at 55000 rpm e black dotted linerepresents the axis the combination of the orange line andthe orange sphere represents the starting position of thetrajectory and the incomplete track curve represents thewhirl direction e figure shows that the 1F mode wascylindrical thus combining rigid body motion with rotorbending Furthermore the rotor was close to the unstablestate at this time In contrast the 2F mode was a conicalbending mode e figure shows that the 3B mode wasconical with both rigid body motion and rotor bendingLastly the 4B mode was cylindrical with both rigid bodymotion and rotor bending
Figure 18 shows the four modes close to the SLS 2stability limit speed at 97946 rpme rotor was close to theunstable state at this time e 1F and 3B modes were cy-lindrical and conical combining rigid body motion androtor bending In contrast the 2F mode was a conicalbending mode
322 Effect of Temperature on the Campbell Diagram andStability Diagram We exported the temperature data ofeach node in Figure 12 and inserted the temperature valueinto the corresponding rotor-shaft system node through the
Ansys software commands functionWe analysed the criticalspeed and stability of the rotor-shaft system in the thermalenvironment
Figure 19 shows that temperature affected the intrinsicfrequency (whirl frequency) Compared with case 1 thewhirl-frequency curves in the 1F 2F and 3B modes wereslightly lower whereas for the 4B mode the whirl-fre-quency curve was much lower In the figure the startingpoint at the left end (when the rotating speed was 0 rpm)was taken as the observation object In case 1 the frequencywas 23009 Hz whereas in case 2 the frequency decreasedto 21825 Hz Because the temperature changed the ma-terialrsquos elastic modulus and Poissonrsquos ratio it affected itsstructural stiffness For higher-order frequencies the effecton frequency became greater as the order increasedHowever the effect on the critical speed Ωcr1 did notchange much whereas Ωcr2 decreased from 43622 rpm to41815 rpm is decrease was also caused by the intrinsicfrequency drop
Figure 20 shows the stability diagram in case 2 and thatthe temperature had an important effect on the dampingratio and SLS Compared with case 1 SLS 1 of the 1F modedecreased from the original 56550 rpm to 55078 rpm andmade the 1F mode enter the unstable region ahead of timee 2F mode change was more obvious and the same SLS 2decreased from the original 97946 rpm to 95969 rpm ereason for this can be explained by the following formulaξr (α2ωhr) + (βωhr2) Here ξr is the damping ratiocoefficient of the rth mode α is the mass damping coeffi-cient β is the stiffness damping coefficient and ωhr is thenatural frequency of the rth mode In most cases α (α 0)can be disregarded that is ξr βωhr2 β did not changewhereas ωhr decreased because of the effect of temperatureresulting in a decrease in ξr In other words SLS 1 and SLS 2in the 1F and 2F modes were reduced erefore whenpredicting the stability limit reset and dynamic perfor-mance of a turbocharger rotor the effect of temperatureshould have a great influence on the dynamic design of therotor
323 Unbalance Response Analysis We applied a dummyunbalance mass of 001 kg middot mm at node 2 and the fre-quency range of the steady-state synchronous responseanalysis was 0 to 2500Hz (corresponding speed range0 to 150 000 rpm) We then measured the response am-plitude of the turbine end floating ring bearing centerposition node 5 In Figure 21 case 1 is represented by ablack curve (normal temperature) and case 2 by a redcurve (high temperature) In case 1 (case 2) the curvesrepresent 6F 6B and 20F modes e frequency at thewave crest corresponding to the frequency correspondingto the fast reverse rotation (the frequency of the rotationspeed within one minute) in the case 1 (case 2) Campbelldiagram as shown in Figures 22 and 23) is similar to the
12 Shock and Vibration
black (red) number in Figure 20 Because the steeringmode is dominant in the response diagram the criticalspeeds Ωcr1 and Ωcr2 in case 1 (case 2) 1F and 2F modesare not shown in the diagram and followed a phenom-enon similar to that reported by Chouksey and Roy[29 30] Because of the effect of temperature the steering
frequency of case 2 shifted to the left making the changeof response amplitude more obvious e responsevalue at the peak of the 6F and 6B modes decreased from0147mm to 0014 mm and that at the peak of the20F and 20B modes decreased from 0036 mm to00037 mm
324 Verification of Rotor FEM Natural frequency anal-ysis is very important as the basis of a dynamic analysisWe obtained the first two natural frequencies of the rotorby hammering experiments and verified them by nu-merical simulation Figure 24 shows the measured parts(rotor system) force hammer and piezoelectric acceler-ation sensor To reduce the measurement error rate we
adopted a sensor with a nonlinearity of less than 12 anda sensitivity of 10 mvg and placed the sensor in themiddle of the rotor shaft Table 6 compares the naturalfrequency experimental value with the simulated valuee experimental and simulated values of the first naturalfrequency were 1386Hz and 1453 Hz respectively esecond-order natural frequencies were 4100 Hz and
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
black (red) number in Figure 20 Because the steeringmode is dominant in the response diagram the criticalspeeds Ωcr1 and Ωcr2 in case 1 (case 2) 1F and 2F modesare not shown in the diagram and followed a phenom-enon similar to that reported by Chouksey and Roy[29 30] Because of the effect of temperature the steering
frequency of case 2 shifted to the left making the changeof response amplitude more obvious e responsevalue at the peak of the 6F and 6B modes decreased from0147mm to 0014 mm and that at the peak of the20F and 20B modes decreased from 0036 mm to00037 mm
324 Verification of Rotor FEM Natural frequency anal-ysis is very important as the basis of a dynamic analysisWe obtained the first two natural frequencies of the rotorby hammering experiments and verified them by nu-merical simulation Figure 24 shows the measured parts(rotor system) force hammer and piezoelectric acceler-ation sensor To reduce the measurement error rate we
adopted a sensor with a nonlinearity of less than 12 anda sensitivity of 10 mvg and placed the sensor in themiddle of the rotor shaft Table 6 compares the naturalfrequency experimental value with the simulated valuee experimental and simulated values of the first naturalfrequency were 1386Hz and 1453 Hz respectively esecond-order natural frequencies were 4100 Hz and
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
324 Verification of Rotor FEM Natural frequency anal-ysis is very important as the basis of a dynamic analysisWe obtained the first two natural frequencies of the rotorby hammering experiments and verified them by nu-merical simulation Figure 24 shows the measured parts(rotor system) force hammer and piezoelectric acceler-ation sensor To reduce the measurement error rate we
adopted a sensor with a nonlinearity of less than 12 anda sensitivity of 10 mvg and placed the sensor in themiddle of the rotor shaft Table 6 compares the naturalfrequency experimental value with the simulated valuee experimental and simulated values of the first naturalfrequency were 1386Hz and 1453 Hz respectively esecond-order natural frequencies were 4100 Hz and
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
4383 Hz respectively and the error rate was within 7But the higher mass of the accelerometer is the maincause of error value between the experimental value andthe simulated value
A more favorable verification was the oil film data shownin Figure 6e oil film data were experimental data providedby the turbocharger company In the simulation the oil filmwas the key factor in supporting or restraining the rotor isalso illustrates the reliability of the simulation method
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
Figure 23 Partial enlargement of Campbell diagram (case 2)
Figure 24 Hammering experiment for a rotor
16 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
4 Conclusions
is study used the CHT numerical simulation method toobtain the temperature of a rotor-shaft system and foundthat there was a large temperature gradient when the systemwas in use
Splitting the rotation shaft solved the problem that asolid model cannot obtain an axis orbit on the AnsysWorkbench software platform
Analysing the Campbell diagram shows that the tem-perature reduced the stiffness of the rotor structure causingthe intrinsic frequency to decrease especially for higher-order frequencies e decrease in stiffness was greater andthe critical speed (Ωcr1) in the 1F mode did not changemuch whereas the critical speed (Ωcr2) in the 2F modedecreased a large amount erefore dynamic design andstability prediction of a turbocharger rotor must considerthe temperature during its operation
For the analysis of an unbalanced response the responsecorresponding to the critical speed in the 1F and 2F modesdoes not appear in Figure 21 because these two modes havehigher damping and because the two steering modes weredominant However the response diagram can well reflectthe rotor steering mode Also the temperature effect on theresponse was mainly reflected in the response amplitudeand the response frequency was slightly shifted to the left(reduced) which was similar to the decreased intrinsicfrequency relation
sh Shafttrs Translationrot Rotationtem Temperaturetot Total
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
is work was supported by the Key Research amp Devel-opmental Program of Shandong Province (Grant no2019GGX104104) e authors are grateful for technicalsupport and the experimental data from Kangyue Tech-nology Co Ltd
References
[1] Z Yang and A Bandivadekar Light-Duty Vehicle GreenhouseGas and Fuel Economy Standards ICCT Washington DCUSA 2017
[2] S B Arab J D Rodrigues S Bouaziz and M HaddarldquoStability analysis of internally damped rotating compositeshafts using a finite element formulationrdquo Comptes RendusMecanique vol 346 no 4 pp 291ndash307 2018
[3] H Zinberg and M F Symonds ldquoe development of anadvanced composite tail rotor driveshaftrdquo in Proceedings ofthe 26th Annual National Forum of the American HelicopterSociety pp 55ndash63 1970
[4] D G Lee H Sung Kim J Kook Kim and J K Kim ldquoDesignand manufacture of an automotive hybrid aluminumcom-posite drive shaftrdquo Composite Structures vol 63 no 1pp 87ndash99 Washington DC USA 2004
[5] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquoComposites Science andTechnology vol 68 no 2 pp 337ndash345 2008
[6] H L Wettergren and K-O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous dampingsupported in anisotropic bearingsrdquo Journal of Sound andVibration vol 195 no 1 pp 75ndash84 1996
[7] O Montagnier and C Hochard ldquoDynamic instability ofsupercritical driveshafts mounted on dissipative supports-effects of viscous and hysteretic internal dampingrdquo Journal ofSound and Vibration vol 305 no 3 pp 378ndash400 2007
[8] F Vatta and A Vigliani ldquoInternal damping in rotating shaftsrdquoMechanism and Machine Deory vol 43 no 11 pp 1376ndash1384 2008
[9] B L Newkirk ldquoShaft whippingrdquo Genernal Electric Reviewvol 27 pp 169ndash178 1924
[10] G Genta ldquoOn a persistent misunderstanding of the role ofhysteretic damping in rotordynamicsrdquo Journal of Vibrationand Acoustics vol 126 no 3 pp 459ndash461 2004
[11] L S Andres J C Rivadeneira K Gjika C Groves andG LaRue ldquoA virtual tool for prediction of turbochargernonlinear dynamic response validation against test datardquo
Table 6 Experimental and calculated values of natural frequencies
Order number Experimental value Simulated value Error rate1 1386Hz 1453Hz 00462 4100Hz 4383Hz 0064
Shock and Vibration 17
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
18 Shock and Vibration
Journal of Engineering for Gas Turbines and Power vol 129no 4 pp 1035ndash1046 2007
[12] P Zych and G Zywica ldquoOptimisation of stress distribution ina highly loaded radial-axial gas microturbine using FEMrdquo DeGruyter vol 10 pp 318ndash335 2020
[13] P Jeyaraj N Ganesan and C Padmanabhan ldquoVibration andacoustic response of a composite plate with inherent materialdamping in a thermal environmentrdquo Journal of Sound andVibration vol 320 no 1-2 pp 322ndash338 2009
[14] Y Guo L Li and D Zhang ldquoDynamic modeling and vi-bration analysis of rotating beams with active constrainedlayer damping treatment in temperature fieldrdquo CompositeStructures vol 226 Article ID 111217 2019
[15] R D Burke ldquoAnalysis and modeling of the transient thermalbehavior of automotive turbochargersrdquo Journal of Engineeringfor Gas Turbines and Power vol 136 pp 1ndash10 Article ID10511 2014
[16] A Romagnoli A Manivannan S Rajoo et al ldquoA review ofheat transfer in turbochargersrdquo Renewable and SustainableEnergy Reviews vol 79 pp 1442ndash1460 2017
[17] G Tanda S Marelli G Marmorato and M Capobianco ldquoAnexperimental investigation of internal heat transfer in anautomotive turbocharger compressorrdquo Applied Energyvol 193 pp 531ndash539 2017
[18] M P Bulat and P V Bulat ldquoComparison of turbulencemodels in the calculation of supersonic separated flowsrdquoWorld Applied Sciences Journal vol 27 no 10 pp 1263ndash12662013
[19] D Bohn T Heuer and K Kusterer ldquoConjugate flow and heattransfer investigation of a turbo chargerrdquo Journal of Engi-neering for Gas Turbines and Power vol 127 no 3pp 663ndash669 2005
[20] L Gang K Karsten H A Anis B Dieter and S TakaoldquoConjugate heat transfer analysis of convection-cooled tur-bine vanes using c-Reθ transition modelrdquo InternationalJournal of Gas Turbine Propulsion and Power Systems vol 6no 3 pp 9ndash15 2014
[21] R D Burke C R M Vagg D Chalet and P Chesse ldquoHeattransfer in turbocharger turbines under steady pulsating andtransient conditionsrdquo International Journal of Heat and FluidFlow vol 52 pp 185ndash197 2015
[22] H Chung H-S Sohn J S Park K M Kim and H H Choldquoermo-structural analysis of cracks on gas turbine vanesegment having multiple airfoilsrdquo Energy vol 118pp 1275ndash1285 2017
[23] D Bohn B Bonhoff H Schonenborn and H WilhelmiPrediction of the Film-Cooling Effectiveness in Gas TurbineBlades using a Numerical Model for the Coupled Simulation ofFluid Flow and Diabatic Walls Publikationsserver der RWTHAachen University Aachen Germany 1995
[24] S M Moosania and X Zheng ldquoEffect of internal heat leakageon the performance of a high pressure ratio centrifugalcompressorrdquo Applied Dermal Engineering vol 111pp 317ndash324 2017
[25] China Aeronautical Materials Handbook Editorial Commit-tee China Aeronautical Materials Handbook China Stan-dards Press Beijing China 2002
[26] Z Wang Structural Reliability of Vehicle Turbocharger Sci-ence Press Beijing China 2013
[27] L Tian W J Wang and Z J Peng ldquoDynamic behaviours of afull floating ring bearing supported turbocharger rotor withengine excitationrdquo Journal of Sound and Vibration vol 330no 20 pp 4857ndash4874 2011
[28] A A Alsaeed and D J Inman Dynamic Stability Evaluationof an Automotive Turbocharger Rotor-Bearing System Vir-ginia Polytechnic Institute and State University BlacksburgVA USA 2005
[29] M Chouksey J K Dutt and S V Modak ldquoModal analysis ofrotor-shaft system under the influence of rotor-shaft materialdamping and fluid film forcesrdquo Mechanism and MachineDeory vol 48 pp 81ndash93 2012
[30] H Roy and S Chandraker ldquoDynamic study of viscoelasticrotor modal analysis of higher order model consideringvarious asymmetriesrdquo Mechanism and Machine Deoryvol 109 pp 65ndash77 2017
