Study of crankshaft torsional deformation under steady...

Study of crankshaft torsional deformation understeady-state and transient operation of turbochargeddiesel enginesE G Giakoumis, C D Rakopoulos*, and A M Dimaratos

Department of Thermal Engineering, School of Mechanical Engineering, National Technical University of Athens,

Athens, Greece

The manuscript was received on 24 May 2007 and was accepted after revision for publication on 6 December 2007.

DOI: 10.1243/14644193JMBD113

Abstract: The modelling of transient operation of turbocharged diesel engines appeared in theearly 1970s, and continues to be in the focal point of research due to the importance of transientresponse in the everyday operating conditions of engines. The majority of studies have focusedso far on thermodynamics, as this directly affects heat release predictions and consequentlyperformance and pollutants emissions. On the other hand, issues concerning the dynamicsof engine operation are often disregarded or over-simplified. In the present work, an exper-imentally validated diesel engine simulation code is used to study and evaluate the importanceof a notable engine dynamic issue, i.e. the crankshaft torsional (angular) deformations duringturbocharged diesel engine operation owing to the difference between instantaneous engineand load (resistance) torques. The analysis aims ultimately in studying the phenomena underthe very demanding, and often critical, transient operating conditions. Detailed crankshaftangular momentum equilibrium is formulated that takes into account instantaneous gas, iner-tia, friction, load as well as stiffness, and damping torque contributions. Details are providedconcerning the underlying mechanism of the crankshaft torsional deformations duringsteady-state and transient operation. This deformation can assume significant values depend-ing on the engine-load configuration (load change, crankshaft stiffness, kind of aspiration of theengine), and as such it is of great importance for safe engine operation.

Keywords: diesel engine, crankshaft, torsional deformation, transient operation

1 INTRODUCTION

The turbocharged diesel engine is nowadays the mostpreferred primemover in medium andmedium-largeunits applications (truck driving, land traction, shippropulsion, electrical generation). Moreover, it con-tinuously increases its share in the highly competitiveautomotive market, owing to its reliability that iscombined with excellent fuel efficiency. Particularly,its transient operation is of paramount importancein the everyday operating conditions of engines,

being often linked with offdesign (e.g. turbochargerlag) and, consequently, non-optimum performanceand increased exhaust emissions.During the last decades, mathematical simulation

has paved the way for an in-depth study of dieselengine processes under both steady-state and transi-ent conditions [1–7]. However, the majority ofresearch has focused so far on thermodynamics, asthis directly affects heat release predictions and con-sequently performance and pollutants emissions. Onthe other hand, issues concerning engine dynamics,e.g. connecting rod complex movement, kinematicsof the slider-crank mechanism, crankshaft defor-mations, torsional vibrations etc., are often disre-garded or over-simplified, possibly for the sake ofspeeding up thermodynamic simulation programexecution time [7].

*Corresponding author: Internal Combustion Engines Laboratory,

Department of Thermal Engineering, School of Mechanical Engin-

eering, National Technical University of Athens, 9 Heroon

Polytechniou Street, Zografou Campus 15780, Athens, Greece.

email: [email protected]

17

JMBD113 # IMechE 2008 Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics

An important aspect of internal combustion engineoperation is that instantaneous engine torque fluctu-ates significantly during an engine cycle, even understeady-state conditions. The main mechanismbehind this is the cyclic nature of gas pressures andinertia reciprocating forces [8]. This fluctuation maybe of considerable magnitude, particularly duringturbocharged diesel engine operation, where thecylinder pressures assume very high values. On theother hand, resistance (load) torque remains practi-cally constant during a cycle, owing to the adequatelylow non-uniformity of rotation; the latter being,mainly, determined by the flywheel mass momentof inertia (as will be shown in section 2.4 loadtorque depends mainly on engine speed). As aresult, a significant fluctuation occurs in the instan-taneous net (engine minus load) torque that even-tually leads to cyclic speed irregularities, twistsbetween individual cranks of a multi-cylinderengine and, finally, torsional (angular) deformationof the whole elastic crankshaft. Particularly inmodern automotive engines, the above phenomenonis intensified by the flexibility of light-weight crank-shafts that may even lead to ‘engine roughness’ [9,10]. The crankshaft deformation is further enhancedduring transient operation, owing to the dynamicinstability induced by the considerable deficit oftorque, during the early cycles of the transient eventafter the new, increased, load or fuelling has beenapplied.The instantaneous values of crankshaft torsional

deformation should be maintained overall low, inorder to avoid excessive stress on the crankshaftand its bearings so that safe engine operation isensured. In order to be able to study and quantifythis deformation, a first satisfactory approach lies inthe formulation of detailed crankshaft angularmomentum equilibrium. Apart from the well knowngas, inertia and load terms, this equilibrium musttake into account the instantaneous values of alltorque contributors, including stiffness, damping,and (engine) friction. A further significant contri-bution comes from the whirl of engine bearings,which is usually accounted for in multi-bodydynamic studies; the latter are based on Lagrangiandynamics and, typically, comprise all rigid bodyinertia members, support bearings, joints, couplers,and other connections between the various enginecomponents, as well as means of vibration damping[11, 12].Past research on steady-state engine dynamics can

prove useful in describing the detailed crankshaftangular momentum balance needed for torsionaldeformation studies [13–16]. However, in all theseworks the object was the estimation of indicatedtorque using, for example, sliding mode observers[13], or the reconstruction of cylinder pressures

from crankshaft angular velocity measurements [14,15], e.g. for diagnosis purposes [16], rather than theinvestigation of the instantaneous crankshaft tor-sional deformation itself. On the other hand, classictorsional vibration handbooks deal with the subjecton a cycle rather than degree crank angle (8CA)basis, for the special, but very important, casewhere the engine operates at resonance with someharmonic order of the exciting engine forces [17, 18].As regards the very demanding transient diesel

engine operation no data are available, since in allprevious simulations the crankshaft has beenassumed sufficiently short and rigid. Consequently,no address was made for crankshaft torsional defor-mation effects [7].In this work, the crankshaft angular momentum

equilibrium analysis is expanded and theinstantaneous torsional deformation duringsteady-state and, ultimately, transient conditionsare quantified. It is, therefore, believed that usefulinformation will be drawn regarding the underlyingthermodynamic-dynamic mechanisms. In order tofulfil this goal, an experimentally validated dieselengine simulation code that follows the filling andemptying approach will be used; this incorporatessome important features to account for thepeculiarities of transient operation. Improvedrelations concerning fuel injection, combustion,dynamic analysis, heat transfer to the cylinder walls,friction and multi-cylinder engine operation duringtransients have been developed, which contribute toan in-depth modelling [7, 19, 20]. Special care ispaid to the formulation of the crankshaft torque bal-ance by taking into account, among other things, theinstantaneous engine friction torque, using a recentlydeveloped detailed friction model, as well as theinstantaneous gas torque contribution separatelyfrom each cylinder when studying the transient oper-ation of a multi-cylinder engine.Owing to the narrow speed range of the engine in

hand, mainly load increases under constant governorsetting will be investigated, which, nonetheless, playa significant role in the European or American transi-ent cycles of heavy duty vehicles. Fundamentalaspects of crankshaft torsional deformation will bepursued with relation to the initiating in-cylinderpressure and gas torque build-up. A sensitivity analy-sis will also be carried out in order to establishwhether by incorporating a detailed crankshaft sub-model, this leads in more accurate predictions oftransient engine speed response.

2 SIMULATION ANALYSIS

The block diagram of the simulation model devel-oped is illustrated in Fig. 1, referring to both

18 E G Giakoumis, C D Rakopoulos, and A M Dimaratos

Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics JMBD113 # IMechE 2008

steady-state and transient turbocharged dieselengine operation. A brief description of the equationsinvolved will be presented in the next subsections.

2.1 General process description

The present analysis does not, at the moment,include predictions of exhaust gas emissions and,on the other hand, deals with transient operation cal-culations on a 8CA basis. Therefore, a single-zonemodel following the filling and emptying approachis used for the thermodynamic processes evaluation.This approach is believed to be the best compromisebetween accuracy and limited PC program executiontime [7]. The fuel is dodecane (C12H26) with a lowerheating value, LHV ¼ 42 500 kJ/kg. Perfect gas beha-viour is assumed. Polynomial expressions are usedfor the species considered, concerning the evaluationof internal energy and specific heat capacities forfirst-law applications to the engine cylinder contents[3]. The species considered are O2, N2, CO2, H2O, andCO; the latter is taken into account, using the corre-sponding chemical equilibrium scheme, only whenthe mixture is rich and for gas temperatures exceed-ing 1400 K, as for example during the early cycles of

the transient event where the turbocharger lag isprominent [7].For heat release rate predictions, the fundamental

model proposed by Whitehouse and Way [21] isused. Especially during transients, the constant K inthe (dominant) preparation rate equation of themodel is correlated with the Sauter mean diameter(SMD) of the fuel droplets, through a formula of thetype K/(1/SMD)2.5 [1].The improved model of Annand and Ma [22] is

used to simulate heat loss QL to the cylinder walls

dQL

dt¼A

�kg

DReb

�aðTg � TwÞ þ

a0

v

dTg

dt

�

þ cðT4g � T 4

wÞ

�ð1Þ

where a, a0, b, and c are constants evaluated afterexperimental matching at steady-state conditions.Further, A ¼ 2Apist þ A0, with Apist ¼ (pD2/4) thepiston cross-section area, and A0 ¼ pDx with xthe instantaneous cylinder height in contact withthe gas (see equation (2) below). kg is the gas thermalconductivity, while the Reynolds number Re is calcu-lated with a characteristic speed derived from a k–1

Fig. 1 Block diagram of developed simulation code

Study of crankshaft torsional deformation 19


turbulence model and a characteristic length equal tothe piston diameter. During transient operation, thethermal inertia of the cylinder wall is taken intoaccount, using a detailed heat transfer scheme thatmodels the temperature distribution from the gas tothe cylinder wall up to the coolant.

2.2 Dynamics of the slider-crank mechanism –detailed connecting rod modelling

At each instant of time, the displacement ofthe piston from the top dead centre (TDC) is givenby [3]

xðwÞ ¼ rð1� cos wÞ þ Lrod 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� l2 sin2 w

q� �ð2Þ

where r is the crank radius, l ¼ r/Lrod with Lrod theconnecting rod length (see also Fig. 2), and thecrank angle w is measured from the TDC position.Differentiation of the above equation with

respect to time, gives the instantaneous pistonvelocity

upistðwÞ ¼ v � r sinw 1þlcos wffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� l2 sin2 wp

!ð3Þ

and by differentiating once again with respect totime, the instantaneous piston acceleration is yielded

bðwÞ ¼ v2r coswþ lcos2wþ l2sin4w

ð1� l2sin2wÞ3=2þ

1

v21upist

rv

� �ð4Þ

The last term on the right hand side of equation (4)takes into account the influence of the crank’s angu-lar acceleration, 1 ¼ dv/dt on the piston linearacceleration.The connecting rod is usually modelled in the

literature as equivalent to two-lumped masses con-centrated at its ends, i.e. one reciprocating with thepiston assembly and the other rotating with thecrank pin. This approach is widely adopted for sim-plicity reasons. However, it induces errors in theslider-crank mechanism dynamics by miscalculatingthe actual rod’s moment of inertia and the variousforces of the kinematic mechanism. For a more accu-rate computation of engine (inertia) torque, adetailed model of the connecting rod has been devel-oped based on rigid body dynamics [23]. Here, thecomplex, elliptical movement of the rod’s centre ofgravity that is produced by its reciprocating androtating motion is analysed.For the connecting rod angle b, it holds (see also

Fig. 2)

L sinb ¼ r sinw)Ldb

dtcos b ¼ r

dw

dtcosw ð5Þ

Solution of the above equation with respect to db/dt,yields

db

dt¼ vrod ¼ lv

cos w

cos b¼ lv

cos w

ð1� l2sin2 wÞ1=2ð6Þ

Differentiation of the previous equation with respectto time, gives

d2b

dt2¼

dvrod

dt¼ 1rod ¼ lv2sin w

l2 � 1

ð1� l2sin2 wÞ3=2

þ l1cos w

ð1� l2sin2 wÞ1=2ð7Þ

In the above relations, vrod and 1rod are the angularvelocity and acceleration of the connecting rod’scentre of gravity, respectively.With reference to Fig. 2(a) (moving) frame of refer-

ence of orthogonal axes (n, t) is considered, which isfixed on the rod, with directions parallel and perpen-dicular to the rod’s axis, respectively; they have

Fig. 2 Slider crank mechanism illustrating forces and

torques for the computation of the total inertia

force



components F4 and F3 of the crankpin force on them,respectively.A balance of all forces acting on the connecting rod,

analysed in n and t directions, givesn-axis forces

F4 þ Fpist cos bþ Fthr sin b ¼ Frod � F1cos b ð8Þ

t-axis forces

F3 � Fpist sin bþ Fthr cos b ¼ F2 þ F1 sin b ð9Þ

A balance of torques with respect to the crankpinaxis, gives

� Fthr � ðLrod cos bÞ þ Fpist � ðr sin wÞ

¼ Grod � 1rod � F1 � ðxrod sin bÞ � F2xrod ð10Þ

In the above equations, torque trod ¼ Grod1rod withGrod the connecting rod’s real mass moment of inertiawith respect to its centre of gravity, Fpist ¼ 2mpistb isthe longitudinal force along the cylinder axis actingon the piston assembly to produce its acceleration[8] (piston acceleration b(w) has been defined inequation (4)), and F1 ¼ 2mrodb is the force acting onthe connecting rod centre of gravity due to the linearacceleration of the piston. Also, F2 ¼mrod(L2 xrod)1rodand Frod ¼mrod(L2 xrod)v

2rod, with (L2 xrod)1rod the

tangential component and (L2 xrod)v2rod the normal

component of the rod’s centre of gravity accelerationwith respect to the piston pin.The system of equations (8) to (10) can be solved

for the unknown forces Fthr, F3, and F4. Force Fthr isthe inertia component of the thrust force acting bythe piston on the side wall of the cylinder. The sumof the projections of forces F3 and F4 on an axis per-pendicular to the crank radius produces the tangen-tial (inertia) force FTin, due to the inertia of themoving parts (piston assembly and connecting rod)acting on the crank, that is

FTin ¼ F3 cosðwþ bÞ þ F4 sinðwþ bÞ ð11Þ

The latter will be used in section 2.4 for the compu-tation of the engine indicated torque.

2.3 Friction

For the calculation of friction inside the cylinder, themodel proposed by Taraza et al. [24] is adopted. Itdescribes the non-steady profile of friction torqueduring each cycle, based on fundamental frictionanalysis. Here, the total amount of friction is dividedinto four parts, i.e. piston rings assembly, loadedbearings, valve train, and auxiliaries. Total friction

torque, Tfr, at each 8CA is the sum of the aboveterms. The important outcome here is that thisfriction torque varies during the engine cycle,especially around ‘hot’ TDC, unlike the usuallyapplied ‘mean’ f.m.e.p. equations where frictiontorque remains constant throughout each cycle [7].It should be mentioned at this point, that the par-

ticular model is a generic friction model that wasdeveloped using simplified equations of the variousprincipal friction components. More detailed,analytical methods are available in the literature,especially for the cases of piston rings and pistonskirt, both of which suffer mixed regime of lubricationaround top and bottom dead centres [25].

2.4 Crankshaft torque equilibrium

The crankshaft is considered as a flexible, elastic bodythat may deform during engine operation; it is simu-lated using the lumped mass model of Fig. 3. A con-densed crankshaft model was chosen, i.e. rigidenough between the cylinders and elastic betweenflywheel and load. The elastic crankshaft rotarymotion is excited by the gas and inertia torque fluctu-ation. The following two, non-linear differentialequations apply [7, 13, 26]

TeðwÞ�T frðwÞ�TS�TD¼ðGeþGflþGcouplÞdv

dtð12aÞ

tSþtD�tLðwLÞ¼GLdvL

dtð12bÞ

Here,Ge,Gfl,Gcoupl, andGL are the engine, flywheel,elastic coupling, and load mass moments of inertia,respectively (considered constant, although, in thegeneral case, the engine moment of inertia mayvary with crank angle [27]*), v ¼ dw/dt is the engineangular velocity, vL ¼ dwL/dt is the load angular vel-ocity, and the torsional deformation of the crankshaftdue to the torque difference between engine andload is defined as w2 wL. Also, Te denotes theengine indicated torque that includes gas, inertiaand (the negligible) gravitational forces contri-butions. Engine torque is mostly dependent on

*For the present engine-load configuration, the total mass

moment of inertia is very high due to the dominant contribution

of the flywheel (see data in Table 1), so that the differentiation of

(engine) mass moment of inertia with crank angle is neglected in

the two lumped-mass model considered in Fig. 3. Nonetheless, for

the analysis of Fig. 8 in section 4, where the angular deformation

between cylinders 3 and 4 will be studied, the variation of cylinder

moment of inertia with crank angle has been taken into account.



accurate combustion modelling and is given expli-citly by

TeðwÞ ¼ TgðwÞ þ T inðwÞ þ TgrðwÞ

¼ pgðwÞ � Apist �upistðwÞ

rv

� �þ FTinðwÞ

�

þ mlgupistðwÞ

rvþmrg sin w

� �� r ð13Þ

In the above relation, pg(w) is the instantaneouscylinder pressure, ml ¼ mpist þ mrod.l is the recipro-cating mass, mr ¼ mcrank þ mrod.r is the rotatingmass, and FTin is the torsional inertia force calculatedfrom equation (11).For the torsional stiffness and damping torques it

holds, respectively,

TS ¼ kðw� wLÞ ð14aÞ

TD ¼ Cðv� vLÞ ð14bÞ

with k the stiffness coefficient, which is given by

k ¼Q � JP‘

¼Q

‘�pd4

32ð15Þ

and C the damping coefficient, which is defined bythe characteristics of the (viscous or rubber)damper mounted on the engine’s crankshaft. Inequation (15), Q is the shear modulus (in N/m2), JPis the polar moment of inertia (in m4) of the shaftcross section and d, ‘ are the shaft diameter andlength, respectively, between engine flywheel andload. Both coefficients are assumed constant throughthe analysis.It is reminded here that if the crankshaft is

assumed short and rigid enough (as for the usualcase in transient simulations), w ; wL. Thus, byadding both sides of equations (12a) and (12b),these are replaced by the simple equation: Te(w) 2Tfr(w)2 TL(w) ¼ Gtot(dv/dt), where Gtot representsthe total mass moment of inertia of the engine-loadconfiguration reduced to the crankshaft axis.

Finally, TL is the load torque, which isapproximated by the following relation

TLðwLÞ ¼ C1 þ C2vC3

L ð16Þ

For a linear load-type (i.e. electric brake, generator)C3 ¼ 1, while for a quadratic load-type (i.e. hydraulicbrake, fixed pitch propeller, vehicle aerodynamicresistance) C3 ¼ 2, with C1 the speed-independentload term (e.g. road slope).The differential equations (12a) and (12b) are solved

at each 8CA, using the instantaneous torque termsfrom all six cylinders of the present engine. Particu-larly, as regards the gas torque during transients, theindividual cylinder gas pressures derive from the ther-modynamicmodel, with application of the ‘truemulti-cylinder’ approach described later in subsection 2.6.Solution of equations (12) provides, at each 8CA, theindividual engine-side and load-side rotationalspeeds v and vL as well as the respective angles w

and wL and, hence, the torsional deformation w2 wL.Previous researchers [14, 16] have described an

even more detailed angular momentum balance, bytaking into account all possible crankshaft defor-mations between pulley, each cylinder of a multi-cylinder diesel engine, and flywheel. This eventuallyled to a system of 6–10 differential equations,depending on the number of cylinders of the enginein hand. They also preferred solution of the respectivedifferential equations using Fourier analysis of thegas and inertia torque harmonics. This approach isnot, however, suitable for transient operation cases,which forms the main goal of investigation in thepresent work.

2.5 Fuel pump operation

Instead of applying the steady-state fuel pump curvesduring transients, a fuel injection model, experimen-tally validated at steady-state conditions, is used.Thus, simulation of the fuel pump-injector lift mech-anism is accomplished, taking into account the deliv-ery valve and injector needle motion [28]. Theunsteady gas flow equations are solved using themethod of characteristics, providing the dynamicinjection timing as well as the duration and the rateof injection for each cylinder at each transient cycle.The obvious advantage here is that the transient oper-ation of the fuel pump is also taken into account. Thisis mainly accomplished through the fuel pumpresidual pressure value, which is built up togetherwith the other variables during the transient event.

2.6 Multi-cylinder engine modelling

At steady-state operation, the performance of eachcylinder is essentially the same, due to the

Fig. 3 Schematic arrangement of the engine-load

dynamic system for the crankshaft angular

momentum equilibrium analysis



quasi-steady position of the governor clutch resultingin the same amount of fuel being injected per cycle,and the quasi-steady turbocharger compressor oper-ating point. Under transient operation, however, eachcylinder experiences different fuelings and air massflowrates during the same engine cycle. This happensdue to the combined effect of: (a) the continuousmovement of the fuel pump rack that is initiated bya load or speed change, and (b) the continuous move-ment of the turbocharger compressor operatingpoint. As regards speed changes, only the firstcycles are practically affected. However, when loadchanges are investigated, significant variations canbe experienced throughout the whole transientcycle. The usual approach, here, is the solution ofthe governing equations for one cylinder and the sub-sequent use of suitable phasing images of this cylin-der’s behaviour. This approach is widely popular forlimiting the computational time [7]. Unlike this, thepresent research group has developed a true ‘multi-cylinder’ engine model. Here, all the governing differ-ential and algebraic equations are solved individuallyfor every one cylinder of the six-cylinder engineunder study, according to the current values of thefuel pump rack position and turbocharger compres-sor flow. This results in (significant) differentiationsin both fuelling and air mass flowrates for each cylin-der during the same cycle of a transient event, affect-ing, among other things, the crankshaft torsionaldeformation results.

3 EXPERIMENTAL VALIDATION

The objective of the experimental test bed developedwas to investigate the steady-state and transient per-formance of the engine in hand. To accomplish thistask the engine was coupled to a hydraulic brake(dynamometer). Strategic measuring points wereconnected to a computer data logging system forrecording and processing engine and turbocharger

variables. The experimental facilities can be dividedinto two parts: (a) the engine test bed, and (b) thedata logging and processing system. The basic datafor the engine and turbocharger are given inTable 1. A schematic arrangement of the test bedinstrumentation is illustrated in Fig. 4.

3.1 Engine test bed instrumentation

The experimental investigation was conducted on anMWM TbRHS 518S, six-cylinder, turbocharged andaftercooled, medium-high speed diesel engine ofmarine duty.Apart from the engine and dynamometer, the main

parts of the test installation include:

(a) tank and flow-meter for measuring diesel fuelconsumption rate;

(b) turbocharger compressor boost pressuremanometer;

(c) compressor inlet and exit thermometers andaftercooler exit air thermometer;

(d) magnetic pick-up TDC marker and r/minindicator;

(e) Kistler piezotron pressure transducer with a vol-tage amplifier for continuously measuring theturbocharger compressor boost pressure;

(f) Kistler piezoelectric transducer with voltageamplifier for continuously measuring the cylin-der pressure diagram;

(g) Bosch RTT-100 smokemeter fitted into theexhaust pipe downstream of the turbine;

(h) Schaevitz–Lucas linear variable displacementtransducer (LVDT) for continuously measuringthe fuel pump rack position;

(i) AEG stroboscope for the measurement of(steady-state) turbocharger speed;

(j) J and K type thermocouples for measuring thetemperatures of the exhaust gas after each cylin-der and downstream of the turbine;

(k) Tektronix storage oscilloscope.

The data acquisition and processing system for theengine test bed comprised:

(a) a Metrabyte DASH-16F, high-speed data acqui-sition card for measuring pressures, LVDT andTDC signals;

(b) an Advantech PCL-818HD, high-speed dataacquisition card with PCLD-789D expansioncard for various temperature measurements;

(c) two IBM compatible PCs, properly interfaced forfast data acquisition and recording, on the busesof which the two cards mentioned above weremounted.

Both sampling systems use a fast 12-bit converterwith an 8 ms analogue-to-digital conversion time,

Table 1 Basic data for engine and turbocharger

Engine Model andType

MWM TbRHS 518S

In-line, six-cylinder, four-stroke,compression ignition, turbocharged,aftercooled, heavy-duty

Speed range 1000/1500 r/minBore/stroke 140/180 mmCompression ratio 17.7:1Maximum power 320 HP (236 kW) @ 1500 r/minMaximum torque 1520 Nm @ 1250 r/minTurbocharger Single-stage, centrifugal compressor,

single-stage, twin entry, axial turbineMoment of inertia Engine & brake: 15.60 kg m2 Turbocharger:

7.5 � 1024 kg m2

Shaft rigidity 240 000 Nm/rad



giving a maximum throughput rate of 100 kHz indirect memory access mode.

3.2 Experimental results

The first requirement from the engine test bed instru-mentation was to investigate the steady-state per-formance of the engine in hand. For this purpose,an extended series of steady-state trials was con-ducted in order on the one hand to examine themodel’s predictive capabilities and, on the other, tocalibrate successfully the individual submodelsdescribed in the previous sections. By so doing, theconstants for the combustion, heat transfer, friction,etc., submodels were made possible to be estimated.The investigation of transient operation was the nexttask. Since the particular engine is one with a rela-tively small speed range, mainly load changes(increases) with constant governor setting wereexamined [19]. A typical example of a conductedtransient experiment is given in Fig. 5. Here, theinitial load was 10 per cent of the full engine load at1180 r/min. The final applied load was almost50 per cent of the full engine load. The non-linearcharacter of the load application, which could notbe accounted for in the simulation, is responsible

for the differences observed in boost pressure andengine speed response. The particular hydraulicbrake has a very high mass moment of inertia, ofthe order of 5.375 kg m2, resulting in long, abruptand non-linear actual load-change profiles. Nonethe-less, the matching between experimental and pre-dicted transient responses seems satisfactory for allengine and turbocharger variables measured(engine speed, fuel pump rack position, and boostpressure); it is believed to form a sound basis forthe theoretical investigation that follows.

4 RESULTS AND DISCUSSION

Figure 6 illustrates the development of crankshaft tor-sional deformation, i.e. term (w2 wL) from equations(12), during typical, steady-state, diesel engine oper-ation. A ‘single-cylinder version’ of the currentengine is initially chosen for the analysis, in orderfor the results to be directly comparable to thein-cycle pressure and torque build-up. The data pre-sented in Fig. 6 are in accordance with the findings ofprevious works at steady-state conditions [16, 29], asregards both in-cycle profile and absolute values.

Fig. 4 Schematic arrangement of engine test bed instrumentation



Figure 6 should be studied in conjunction with Fig. 7that depicts the development of all torquecontributors during the same engine cycle.During compression (0–1808CA), a deficit of gas

torque exists that leads to engine speed decrease(upper subdiagram of Fig. 7) and to the ‘negative’crankshaft deformation shown in Fig. 6; torque,speed, and deformation all peak around ‘hot’ TDC.After the start of combustion, there is a surplus oftorque, as now the engine enters the power produ-cing phase of operation. Consequently, ‘positive’deformation is established, while the instantaneous

engine speed increases. This lasts for the wholeexpansion stroke. The considerably higher amountof engine torque produced during expansion leadsto the greater local peak in crankshaft deformation,i.e. 0.168 occurring at about 55 8CA after ‘hot’TDC, compared to the local minimum of 20.038 at180 8CA (70 per cent load operation).The main mechanism behind the crankshaft tor-

sional deformation profile, over an engine cycle, isclearly the gas torque because of its direct impacton the total engine torque. This confirms the resultsof Chen and Chen [16], who concluded that torsionalangle amplitude is, practically, a linear function of thecorresponding gas torque. Closer examination ofFig. 6 reveals that inertia torque influence is also pre-sent, mainly during the open part of the cycle wherethe cylinder pressure is low, as well as duringthe second half of compression. For the presentengine the inertia contribution is rather small, dueto the low engine speed (recall that inertia forcesvary as the square of the engine speed). Crankshafttorsional deformation may assume again negativevalues during the open part of the cycle, due to theinertia torque effects.Of significant importance is the variation of friction

torque over the engine cycle. The usual approach,when studying engine friction, is the use of ‘mean’f.m.e.p. relations. This is not the case, however, inthis work, as application of the detailed Taraza et al.[24] friction model assisted in establishing the real

Fig. 5 Experimental and predicted engine transient

response to an increase in load

Fig. 7 Engine speed and various torque contributors

build-up during steady-state, single-cylinder

engine operation

Fig. 6 Development of the crankshaft torsional

deformation during steady-state, single-

cylinder engine operation for two different loads



friction profile during the engine cycle. Notice, inFig. 7, the increased values for a period of almost180 degrees CA around ‘hot’ TDC. These differentiatethe deformation results obtained, compared to thecase where friction would have been computed bythe ‘mean’ f.m.e.p. relations.Another interesting finding is that stiffness and

damping instantaneous torques may reach highvalues during an engine cycle. The profile of stiffnesstorque follows the torsional deformation as dictatedby equation (14a). The profile of damping torque isindicative of the instantaneous difference betweenengine and load speeds (v2 vL). Owing to the‘internal inertia’ of these ‘processes’, the respectivetorque peaks observed in the centre subdiagram ofFig. 7 may be delayed compared to the initiating gastorque or cylinder pressure. However, despite theirrelatively high instantaneous values, both stiffnessand damping torques’ mean (over the engine cycle)values are negligible compared to the gas or loadtorque. As a result, the usually applied assumptionis justified, i.e. of neglecting these torques for thecalculation of mean engine speed.The torsional deformation for the 25 per cent load

is also depicted in Fig. 6 for comparison purposes.Here, lower maxima of crankshaft deformation are,overall, observed. This was intuitively expected,because of the lower values of fuelling and conse-quently engine gas torque produced during thiscycle (inertia forces retain the same values as in the70 per cent load case, since the engine speed is thesame). Similarly, smaller deformations are to beexpected for naturally aspirated diesel or for sparkignition (Otto) engines, where the cylinder pressuresaremuch lower. In the latter cases, a greater influenceof inertia torque is also expected, particularly for thesmall Otto (car) engines operating at high rotationalspeeds.The main findings of the ‘single-cylinder’ configur-

ation can be extended to the (real) multi-cylinderengine operation, which is depicted in Fig. 8 for thesame engine load*. The effect of the number of cylin-ders is the dominant factor in the multi-cylinderanalysis of Fig. 8. Owing to the relatively high load(70 per cent), there is always an adequate surplus ofgas torque that prevents the crankshaft torsionaldeformation from reaching negative values.An interesting case is illustrated at the lower part of

Fig. 8. Here, the operation of cylinders 3 and 4 hasbeen isolated and the torsional deformation of thecrankshaft section that is surrounded by these

cylinders is studied. Again, the deformation profileclosely follows the gas torque development, butnow both cylinders’ torques are in phase (there isno such thing as ‘resistance’ torque in this case).The distance between the two peaks observed corre-sponds to the firing interval between these two cylin-ders, i.e. 360 8CA. Owing to the much smaller lengthof the crankshaft axis between cylinders 3 and 4, thestiffness coefficient, k34, is now one order of magni-tude higher, thus leading to much smaller defor-mations during the 70 per cent load operationcompared to the engine-load results (upper graphin Fig. 8).The investigation of the results during transient

operation was the next task. When studying thedevelopment of crankshaft deformation, it is impera-tive that instantaneous values for each torque contri-butor are taken into account. In the present analysis,the dominant gas torque term is computed duringtransients separately for each cylinder, using themulti-cylinder engine model described earlier. Thus,during the same engine cycle, differentiations areobserved between the gas torques of individual cylin-ders due to the continuous movement of the fuelpump rack.Figure 9 depicts the development of the maximum

and the mean, over each engine cycle, deformationand stress for a typical load increase transient of10–75 per cent commencing from an engine speedof 1180 r/min. The response of the crank anglewhere the maximum deformation occurs (corre-sponding to the second of the six ‘similar’ peaks ofthe six-cylinder engine under study) is also depictedin the same figure.Initially, the deformation is negligible due to the

low engine load (see Fig. 6). As the governor respondsto the drop in engine speed caused by the abrupt loadincrease, fuelling increases too (right-hand side


deformation during steady-state multi-cylinder

engine operation

* Since the mass moments of inertia of the ‘single’ cylinder and the

six-cylinder engines’ are not proportional, no conclusion is meant

to be drawn concerning the instantaneous maxima or minima of

the angular deformations observed in Figs 6 and 8.



subdiagram of Fig. 9) leading in higher gas pressuresand torques throughout the cycle. This resultsin greater maximum and mean, over the enginecycle, deformations, as well as in a gradual shift ofthe crank angle where the peak occurs earlier in thecycle. It is important to note that the instantaneousmaximum deformation is considerably higher (upto 50 per cent for the cases examined in this work)than the respective mean value in the same cycle;this justifies the analysis on a 8CA basis, in order tobe able to estimate the ‘true’ maximum stress (leftsubdiagram of Fig. 9) that is instantaneously experi-enced by the (crank)shaft, that is

smax ¼Q

2

d

‘Dwmax ¼

Q

2

d

‘ðwe � wLÞmax ð17Þ

It should be pointed out here that the evolution oftransient maximum or mean, over the engine cycle,deformation and, hence, stress develop in a differentway compared to the corresponding steady-stateengine operation points (i.e. at the same enginespeed and fuel pump rack position). The differenceis, mainly, attributed to:

(a) the different air-fuel equivalence ratios experi-enced during transients owing to the turbochar-ger lag, which significantly affects the air-massflowrate, particularly during the early cycles ofthe transient event;

(b) the transient operation of the fuel pump thatdifferentiates from the steady-state fuel pumpcurves.

The above results are expanded in Fig. 10, by showingthe ‘wave’ of crankshaft deformation build-up forseveral cycles of the same transient event. It can be

further observed that the increase in loading/fuellingleads in greater in-cycle fluctuations of the defor-mation too (see Fig. 6).From the analysis it has been revealed that, in prin-

ciple, the main parameters affecting the profile andvalues of crankshaft torsional deformations duringtransients, are the applied engine load and the crank-shaft stiffness.The effect of the magnitude of the applied load is

demonstrated in Fig. 11. A higher applied load leadsto higher fuelling rates and, thus, cylinder press-ures/gas torques. Consequently, after the start ofcombustion, the surplus of net torque is muchhigher now resulting in greater peaks of crankshaftdeformation (cf. the two load curves in Fig. 6 forsteady-state engine operation).

Fig. 9 Development of the maximum and mean, over an engine cycle, crankshaft torsional

deformation, and stress during a 10–75 per cent load increase transient event


deformation during various cycles of the

10–75 per cent load increase transient



The effect of crankshaft stiffness on the torsionaldeformation is also presented in Fig. 11. As expected,the more rigid the crankshaft (greater values of ‘k’in equation (14a)), the smaller the torsional defor-mation observed throughout the whole event.An increase of the order of 160 per cent in the defor-mation peak is observed when comparing the resultsbetween the double and the half stiffness coefficientvalues. Low stiffness (e.g. due to great physicallength of the crankshaft) acts, practically, in thesame way as the low mass moment of inertia, i.e. itallows higher acceleration rates throughout thecycle, thus, lowering the engine non-uniformity ofrotation. In modern automotive engines, the trendof lighter crankshaft configurations may intensifythe above-mentioned deformations, leading even torough engine operation. The above phenomenonhas been studied in multi-body dynamics studies[9] or by applying a dynamic stiffness matrixmethod, or transfer matrix method as, for example,in Okamura and Morita [10].Finally, a sensitivity analysis was carried out, which

showed that the detailed formulation of the crank-shaft angular momentum (equations (12)) does notdiversify the overall (transient) engine speedresponse predictions. Hence, its incorporation intoa transient simulation code is necessary only if thestudy of torsional deformation is the object ofresearch, as it is actually the case for this work.

5 CONCLUSIONS – FUTURE WORK

An experimentally validated simulation codedeveloped has been used to study the development

of crankshaft torsional deformations duringsteady-state and transient operation of a turbo-charged diesel engine.When formulating the crankshaft angular

momentum equilibrium, care was taken to applyinstantaneous values for all torque terms, includ-ing engine friction (using a detailed model),stiffness and damping, whereas the gas torquecontribution was computed separately for eachcylinder using a ‘real multi-cylinder’ enginemodel.Application of the detailed crankshaft angular

momentum equilibrium helped in quantifying crank-shaft torsional deformation values duringsteady-state and transient operation. Improvedunderstanding of contributing and controlling factorsis believed to have been achieved. From the analysisof the present engine-load configuration, the follow-ing results were reached.

1. Engine torque was identified as themain contribu-tor in crankshaft deformation profile and peakvalues. Inertia torque influences the crankshaftdeformation mainly during the open part of thecycle (of a single-cylinder engine), but it was over-all limited for the present engine due to its, rela-tively, low speed.

2. Stiffness and damping torques can assume signifi-cant instantaneous values during a cycle. How-ever, their mean, over the engine cycle, valuesare negligible compared to the engine and loadterms.

3. Local deformation between individual cylinders isalways of much lesser importance, because ofthe considerably higher stiffness coefficientsinvolved.

4. Mean and maximum, over an engine cycle, defor-mation response can assume significant valuesduring transient operation, depending on theload increase schedule. The instantaneous maxi-mum deformation can be up to 50 per centhigher compared to the respective mean value inthe same transient cycle.

5. The transient deformations develop in a differentway compared to the corresponding steady-stateoperating points, owing to the differentiated fuel-ling and air-mass flowrates experienced duringtransients.

6. The assumption of the crankshaft assumedsufficiently rigid is well justified, as regards accu-racy of the mean, over the engine cycle, enginespeed response predictions during transientoperation.

7. Smaller load changes as well as more rigidconstruction of the crankshaft are identified askey parameters for reducing the crankshafttorsional deformation during transients.

Fig. 11 Effect of the load change magnitude and the

crankshaft stiffness on the crankshaft

torsional deformation during transient

operation after a load increase



REFERENCES

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11 Rahnejat, H.Multi-body dynamics: historical evolutionand application. Proc. Instn Mech. Engrs, Part C:J. Mechanical Engineering Science, 2000, 214(C1),149–173.

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13 Wang, Y. Y., Krishnaswami, V., and Rizzoni, G. Event-based estimation of indicated torque for I.C. enginesusing sliding mode observers. Control Eng. Pract.,1997, 5(8), 1123–1129.

14 Taraza, D., Henein, N. A., and Bryzik, W. Determi-nation of the gas-pressure torque of a multicylinderengine from measurements of the crankshaft’s speedvariation. SAE paper 980164, 1998.

15 Zweiri, Y. H., Whidborne, J. F., and Seneviratne, L. D.Numerical inversion of the dynamic model of a single-cylinder diesel engine. Commun. Numer. Methods.Eng., 2000, 16, 505–517.

16 Chen, S. K. and Chen, S. Engine diagnostics by dynamicshaft measurement: a progress report. SAE paper932412, 1993.

17 Ker Wilson, W. Practical solution of torsional vibrationproblems, 3rd edition, 1956–1969 (Chapman and Hall,London).

18 Haug, K. Die Drehschwingungen in Kolbenmaschinen,1952 (Springer-Verlag, Berlin/Goettingen/Heidelberg).

19 Rakopoulos, C. D., Giakoumis, E. G., Hountalas, D. T.,and Rakopoulos, D. C. The effect of various dynamic,thermodynamic and design parameters on theperformance of a turbocharged diesel engine operating

under transient load conditions. SAE paper2004-01-0926, 2004.

20 Rakopoulos, C. D. and Giakoumis, E. G. Sensitivityanalysis of transient diesel engine simulation. Proc.IMechE, Part D: J. Automobile Engineering, 2006,220(D1), 89–101.

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APPENDIX

Notation

A surface area (m2)b piston acceleration (m/s2)BDC bottom dead centreC damping coefficient (Nm s/rad)8CA degrees crank angled shaft diameter (m)D cylinder bore (m)f.m.e.p. friction mean effective pressure (bar)F force (N)g gravitational acceleration ¼ 9.81 m/s2

G mass moment of inertia (kg m2)k torsional stiffness coefficient (Nm/rad),

or thermal conductivity (W/m/K)K combustion model preparation rate

constant



‘ shaft length (m)L connecting rod length (m)N engine speed (rpm)p pressure (bar)Q heat loss (J)r crank radius (m)r/min revolutions per minuteS piston stroke (m)SMD Sauter mean diameter (mm)t time (s)T temperature (K)TDC top dead centreu velocity (m/s)x piston displacement (m)

b connecting rod angle (8)Dw crankshaft torsional deformation (8 or

rad)1 angular acceleration (rad/s2)Q shear modulus (modulus of rigidity)

(N/m2)l ratio of crank radius to connecting rod

length

s torsional (shear) stress (N/m2)T torque (Nm)w crank angle (deg)v angular velocity (rad/s)

Subscripts

D dampinge engine (indicated)fl flywheelfr frictiong gasgr gravitationalin inertial reciprocatingL load, or heat losspist pistonr rotatingrod connecting rodS stiffnessT tangentialw wall



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