A semiactive control scheme for a rotor-bearing systemusing a squeeze film damper with MR fluid

L.A. Dominguez-Nunez, G. Silva-Navarro, O.A. Garcia-PerezCentro de Investigacion y de Estudios Avanzados del I.P.N.Departamento de Ingenierıa Electrica, Seccion de MecatronicaAv. Instituto Politecnico Nacional 2508, Col. Zacatenco, CP 07360 Mexico D.F., Mexicoe-mail: ladominguez@cinvestav.mx

AbstractSqueeze-film dampers (SFD) are commonly used to passively control the vibration response in rotor-bearingsystems because they can provide flexibility, damping and extend the so-called stability thresholds. Morerecently, SFD are combined with Magneto-Rheological (MR) or Electro-Rheological (ER) fluids to intro-duce a control mechanism to modify the rotordynamic coefficients and deal with the robust performance ofthe overall system response for higher operating speeds. There are, however, some theoretical and technolo-gical problems that complicate a more extensive use in industry. In this work the mathematical model of aNewtonian fluid SFD hydrodynamic behavior through the Reynolds equation is shown and it is considered aSFD using MR fluid to model, analyze and semiactively control a rotor-bearing system using finite elementmethods, including also the controllable viscoelastic properties associated to the MR fluid. This semiactivecontrol scheme is combined with an acceleration scheduling of the rotor speed to stabilize and reduce aspossible the overall system response in a Jeffcott-like rotor configuration passing through the first criticalspeeds. In addition, a Choi-Lee-Park polynomial model is analyzed and adapted in order to simplify theMR fluid dynamics and synthesized using sliding-mode control techniques. The SFD with Newtonian fluidis connected to a platform with a rotor-bearing system to validate and experimentally evaluate the systemperformance.

1 Introduction

The most common rotordynamics problems are the high synchronous vibration levels and subsynchronousrotor instabilities. The first problem may be reduced by improving the balancing, or by introducing modifi-cations into the rotor bearing system to move the critical speeds out of the operating range, or by introducingexternal damping to limit peak amplitudes at traversed critical speeds. Subsynchronous rotor instabilitiesmay be avoided by raising the natural frequency of the system as high as possible or by introducing dampingto raise the onset speed of instability above the operating speed range (Zeidan et al. [1]).

To deal with these two problems, the so-called Squeeze Film Dampers (SFD) have been successfully usedin some rotor-bearing systems. One of the most important design characteristics of these types of bearingsis the flexibility and damping bearing structure modification. As a result of these, the transmitted vibrationlevels are reduced and the bearing life is extended, mostly in machinery that works above the system naturalfrequency. Damping reduction in machinery that works above a natural frequency leads to a transmittedforce reduction, however, high damping is suitable when passing through the critical speeds.

The appropriate selection of a bearing type for each application is very important to minimize the risk ofearly failure by wear or fatigue, while reducing the vibration negative effects (Harnoy [2], Nicholas [3]).This is a challenge that has become more difficult with the design of lighter and smaller machinery and,therefore, the used of active or semiactive bearings instead of traditional ones has been spread.

223

Figure 1: Schematic diagram of a SFD and its geometric coordinates.

In the last two decades, the research efforts on intelligent fluids like Electro-Rheological(ER) and Magneto-Rheological (MR) have resulted in industrial semiactive devices. However, only some of them have beenfocused on the vibration control problem in rotor-bearing systems (Wang et al. [11]). The first studies onintelligent fluids and rotors were focused on ER fluids [4]-[10] and, more recently, some works about MRdevices have been reported [11]-[23]. Moreover, Kim et al. [20] designed and modeled a SFD with MR fluidwith a design method and a parametric identification based on magnetic bearings, working on a semiactivecontrol scheme [22], which reduces the unbalance response in a flexible rotor with multiple masses usinga MR-SFD. Yan-Hong et al. [21] designed and compared an adaptive MR-SFD with a traditional SFD.Jagadish et al. [23] studied the stiffness and damping phenomena on a large MR-SFD, based on the Reynoldsnumber and working with low eccentricity levels and different films thicknesses.

In this work the mathematical model of a Newtonian fluid SFD hydrodynamic behavior, through the Reynoldsequation, is developed. This model is then modified to consider a MR fluid and a semiactively control fora rotor-bearing system using finite element methods is proposed, including also the controllable viscoelas-tic properties associated to the MR fluid. This semiactive control scheme is combined with an accelerationscheduling of the rotor speed to stabilize and reduce as possible the overall system response in a Jeffcott-likerotor configuration passing through the first critical speeds. In addition, a Choi-Lee-Park polynomial model[24] is analyzed and adapted in order to simplify the MR fluid dynamics and synthesize using sliding-modecontrol techniques. The SFD with Newtonian fluid is connected to a platform with a rotor-bearing system tovalidate and experimentally evaluate the system performance.

2 Analysis of a SFD with Newtonian fluid

The SFD is a hydrodynamic bearing with a nonrotating journal provided by the outer race of a rolling-elementbearing, which is supported by a centering spring to prevent the rotation while allowing orbital translation.This device adds some extra damping to the system, which is a well-known desirable characteristic of hy-drodynamic bearings, thus avoiding rotordynamic instability [1].

In Fig. 1 is shown a schematic diagram of a conventional SFD, where it is appreciated the hydrodynamicspressure field. Here ω is the circular orbital velocity of the nonrotating journal.

To model a SFD with Newtonian fluid is first considered the Reynolds equation (see, e.g., Hori [25])

1R2

∂

∂θ(h3 ∂p

∂θ) +

∂

∂z(h3 ∂p

∂z) = 6µ[(−U

R

∂h

∂θ+

h

R

∂U

∂x+ 2V ] (1)

224 PROCEEDINGS OF ISMA2012-USD2012

where R is the radius of the nonrotating journal, θ is the rotation angle about the center of the bearing,p = p(Rθ, z) is the fluid pressure distribution, (Rθ, z) are the coordinates associated to a fluid film point, µis the fluid viscosity, U is the flow velocity in direction x, V is the flow velocity in direction y and h is thedistance between the nonrotating journal and the inner pad of the bearing, defined by

h = Cr(1 + ε cos θ) (2)

where ε = e/Cr, e is the SFD eccentricity and Cr is the radial clearance between the bearing inner and outerpad, where is located the fluid film.

Starting from equation (1) and after some assumptions (see Hori [25]), is obtained the so-called short SFDpressure distribution (i.e., L < R, with L the SFD length) as well as the radial and tangential forces with a πfluid film, that is, when the region of cavitation is π < θ < 2π (Vance [26])

Fr = − 2µRL3ωε2

C2r (1− ε2)2

(3)

Ft = − πµRL3ωε

2C2r (1− ε2)3/2

where w = θ. The forces in cartesian coordinates are described by

Fhx = Fr sin θ + Ft cos θ (4)

Fhy = −Fr cos θ + Ft sin θ

3 Analysis of a SFD with MR fluid

In order to modify the bearing dynamic characteristics the SFD is filled with a MR fluid, although these kindof fluids do not satisfy a Newtonian behavior and, therefore, equation (3) is no longer useful to model theresultant forces. MR fluids are smart materials that respond well to a magnetic field with significant changeson their rheological behaviors, increasing the yield shear stress (see Carlson [27]). Such changes can becontinuous and reversible, which makes possible the use of MR fluids for semiactive vibration control.

Under certain conditions, it is possible to express the SFD hydrodynamic forces by a linearized force coeffi-cient (Vance [26])

Fx = Fxx + Fxy (5)

Fy = Fyy + Fyx

where Fxx = Cxxx, Fyx = Cyxx, Fyy = Cyyy, Fxy = Cxyy with Cxx, Cyy like direct damping coefficientsand Cyx,Cxy like cross-coupled damping coefficients.

The MR-SFD dynamic behavior is modeled through virtual MR dampers, which will provide the direct andcross-coupled force coefficients to the rotor-bearing system (see Fig. 2). Hence, using four synchronizedvirtual dampers will make possible to obtain an equivalent SFD dynamic behavior. Moreover, each MRvirtual damper is characterized using the Choi-Lee-Park polynomial model (Choi et al. [24]), whose forcesare described by

FMRk(v, I) =

n=2∑i=0

(bik + cikI)vi (6)

v = x, y;k = xx, yx, yy, xy

ACTIVE NOISE AND VIBRATION CONTROL 225

Figure 2: SFD modeled by MR virtual dampers.

where x, y are the linear SFD velocities, and I the control electric current, which will be applied to theMR virtual dampers. The coefficients (bik , cik ) are given by positive (b+

ik, c+

ik) and negative (b−ik , c−ik ) parts,

described by

cik =(c+

ik+ c−ik) +

∣∣∣c+ik− c−ik

∣∣∣ sign(v)

2,i = 0, 1, 2 (7)

bik =(b+

ik+ b−ik) +

∣∣∣b+ik− b−ik

∣∣∣ sign(v)

2,v = x, y

The coefficients (b+ik

, c+ik

) and (b−ik , c−ik ) with i = 0, 1, 2 are going to be determinate experimentally, foreach virtual damper, to characterize the semi active SFD dynamic behavior with the forces FMRxx , FMRyx ,FMRyy and FMRxy . A similar work have been reported by Cabrera and Silva [30], but using a conventionaljournal bearing supported on two radial MR dampers and compression springs (i.e., a non hydrodynamicbearing).

4 Rotor-bearing system

The system consists of a plane disk with a mass mounted at the midspan of a elastic shaft, which is supportedin the left by a traditional journal bearing and the right support is a SFD.

The rotor is modeled using finite element methods [29], where two finite elements, three nodes and twoplanes are considered. The nodes are placed in the left bearing, the disk and the right SFD bearing. The finiteelements are considered as Euler beam type with lengths l = L/2, where L is the total length between thebearings, shaft density ρ, transversal section A, moment of inertia I and Young modulus E for an isotropicelastic shaft. Thus, the system model has 12 degrees of freedom with 6 degrees for each plane (x, y) andfour for each node, two for the radial displacements (x, y) and two for the angular deflections (βx, βy). Thecoordinates associated to left bearing are xi, yi, βxi, βyi, while the coordinates for the disk are xr, yr, βxr

and βyr and the ones placed in the SFD bearing are xd, yd, βxd and βyd (see Fig. 3).

The overall system has concentrated masses associated to the disk and bearing supports, which are punctualand added to the corresponding nodes. The system damping is added only to the disk node and this is obtainedby means of experimental run-up rotor operations, from rest to rotor speeds above the critical speeds, andapplication of modal identification techniques like Peak-Peaking methods (see, e.g., [28]).

226 PROCEEDINGS OF ISMA2012-USD2012

Figure 3: Finite element model.

The endogenous forces, generated by the unbalance disk, are expressed by

Fxr = mreθ2 cos θ + mreθ sin θ (8)

Fyr = mreθ2 sin θ −mreθ cos θ

The mass, stiffness and damping matrices are obtained as follows (see, e.g., Genta [32])

Ma =ρAl

140

156 22l 54 −13l 0 022l 4l2 13l −3l2 0 054 13l 420mr

ρAl + 312 0 54 −13l2

−13l −3l2 0 8l2 13l −3l2

0 0 54 13l 420mdρAl + 156 −22l

0 0 −13l −3l2 −22l 4l2

(9)

Ka =EI

l3

l3ki

EI+ 12 6l −12 6l 0 0

6ll3kaβi

EI+ 4l2 −6l 2l2 0 0

−12 −6l 24 0 −12 6l2

6l 2l2 0 8l2 −6l 2l2

0 0 −12 −6l l3kad

EI+ 12 −6l

0 0 6l 2l2 −6l 4l2

(10)

ACTIVE NOISE AND VIBRATION CONTROL 227

Ca =

0 0 0 0 0 00 0 0 0 0 00 0 ca 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

(11)

where a = x, y to denote the corresponding horizontal or vertical plane. The system global matrices aregiven by

M =

[Mx 00 My

],K =

[Kx 00 Ky

],C =

[Cx 00 Cy

](12)

Q =[

xi βxi xr βxr xd βxd yi βyi yr βyr yd βyd

]T

Finally, the rotor-bearing system model is described by

Mq + Cq + Kq = F (13)

where q, F ∈ R12 and M,C, K ∈ R12x12. Here F stands for a SFD with Newtonian fluid defined by

F =[

0 0 Fxr 0 −Fhx 0 0 0 Fyr 0 −Fhy 0]T

(14)

where Fxr and Fyt are the endogenous (synchronous) forces from the unbalance disk (8), while Fhx and Fhy

are the hydrodynamic forces produced by the SFD with Newtonian fluid (4).

On the other hand, when the SFD-MR is replaced, the above vector F is given by

F =[

0 0 Fxr 0 −FMRxx − FMRxy 0 0 0 Fyr 0 −FMRyy − FMRyx 0]T

(15)

where FMRxx , FMRyx , FMRyy and FMRxy are the virtual MR dampers forces described in equation (6).

5 Semiactive SFD-MR control

For control purposes is considered the so-called inverse model for the MR virtual damping forces, similarlyto the approach followed by Choi et al. [24], but with the consideration of the cross-coupled forces inequation (5). Therefore, the modified inverse model for the force is proposed as

Ij =Uj −∑n=2

i=0 bixj xid −

∑n=2i=0 biyj y

id∑n=2

i=0 cixj xid +

∑n=2i=0 ciyj yi

d

, j = x, y (16)

I = (Ix + Iy)/2

where (bijj and cijj) correspond to the values in (7) and Uj is the control input force obtained by the sliding-mode control scheme. It is important to remark that in order to avoid singularities in (16), the experimentalparameters (bijj , cijj) should be carefully obtained for each MR virtual damper.

Now, a robust control scheme using sliding-mode control techniques is synthesized for the overall rotor-bearing system (13), (15) and (6). An schematic diagram of the overall semiactive SFD-MR rotor-bearingsystem is depicted in Fig. 4.

The semiactive control scheme based on sliding-mode control employs feedback information from the sys-tem obtained from proximitors, thus computing the appropriate stabilizing forces Ux, Uy, to get the corres-ponding electrical current input I by using the modified inverse model (16). Then, the equivalent MR virtualforces FMRxx , FMRyx , FMRyy and FMRxy are estimated from the current input I .

228 PROCEEDINGS OF ISMA2012-USD2012

Figure 4: Schematic diagram for the semiactive control scheme.

Two sliding-modes controllers are applied to calculate the stabilizing forces Ux, Uy, each of them to attenuatethe synchronous vibrations (Utkin[31]). This is a well-known robust control strategy against exogenousperturbations and parameter uncertainty. The two switching surfaces are proposed as

σx = α1x(xr − x∗r) + α0x(xr − x∗r) (17)

σy = α1y(yr − y∗r ) + α0y(yr − y∗r )

where X∗r and Y ∗

r are the disk reference coordinates, typically zero constant references (X∗r = Y ∗

r ≡ 0). Thedesign parameters (α1x, α0x, α1y, α0y) are used to get an adequate closed-loop system dynamic behavior.

The sliding-mode controller scheme is obtained as follows

Ux =Mx55

Mx35(Mx31xi + Mx32βxi + Mx36βxd + Cx33xr + Kx31xi + Kx32βxi) (18)

+Mx55

Mx35(Kx33xr + Kx36βxd + Kx35xd)− (Mx53xr + Kx56βxd + Kx53xr)

−(Kx54βxr + Kx55xd + Kx56βxd)− α0xMx33Mx55

α1xMx35xr +

Mx33Mx55

α1xMx35ωxsign(σx)

Uy =My55

My35(My31yi + My32βyi + My36βyd + Cy33yr + Ky31yi + Ky32βyi)

+My55

My35(Ky33yr + Ky36βyd + Ky35yd)− (My53yr + My56βyd + Ky53yr)

−(Ky54βyr + Ky55yd + Ky56βyd)− α0yMy33My55

α1yMy35xr +

My33My55

α1yMy35ωysign(σy)

where Mxij , Myij , Kxij , Kyij , Cxij and Cyij are elements (i, j) from the mass, stiffness and dampingmatrices in (9-11), and ωx and ωy are design parameters for the discontinuous control action. The forcesgiven by equation (18) are the stabilizing forces used to compute the equivalent electrical current from theinverse model (16).

ACTIVE NOISE AND VIBRATION CONTROL 229

Figure 5: (a) Experimental results using a SFD with Newtonian fluid and rotor speed run-up to 10000 rpm(b) System responses using ramp and Bezier rotor speed trajectories.

6 Experimental validation of the rotor-bearing model via a SFD withNewtonian fluid

Some simulation and experimental results using a Bently Nevada rotor kit platform and a SFD bearing wereperformed to illustrate the damping properties of the device.

The parameters for the rotor-bearing system are given in Table 1.

Parameters ParametersShaft density ρ = 7850 kg

m3 disk eccentricity e = 1.4517x10−5m

Shaft length L = 0.44m Left bearing stiffness ki = 600MNm

Shaft Young modulus E = 216GPa System damping cx = 9.3166Nsm , cy = 14.2333Ns

m

Shaft moment of inertia I = 4.9x10−10m4 SFD stiffness kxd = 70kNm , kyd = 100kN

m

disk mass mr = 0.806kg SFD stiffness kxβi = 0.4 kNrad , kyβi = 0.14 kN

rad

SFD mass md = 0.060kg SFD radius R = 0.017325m

SFD radial clearance Cr = 0.00052m SFD length L = 0.015m

Table 1: Parameters of the rotor-bearing system

The tests carried out experimentally consist in a run-up operation of the rotor up to 10000 rpm (4000 rpm)in 60 seconds with two different speed profiles (ramp type and a smooth Bezier profile). These tests aredesigned to clearly observe the rotor response at traversed the first critical speed. The SFD was evaluatedwithout fluid, and with two different types of them type A (kinematic viscosity of µ = 1250 mm2/s) andtype B (kinematic viscosity of µ = 1500 mm2/s. The simulation results correspond to the coordinate xr inequations (13) and (14), and the experimental results correspond to the disk displacement on the x plane.

Experimental results are obtained with a speed ramp profile up to 10000 rpm and three different fluids, abovementioned, for the SFD (Fig. 5a). A FFT comparison is made with the two different speed profiles up to10000 rpm with the fluid type A (see Fig. 5b).

Some simulation results (Fig. 6a) and experimental tests (Fig. 6b) were performed using a ramp speed profileup to 4000rpm for the empty SFD and with the fluid type A.

A FFT comparison is made with the experimental and simulation data for the empty SFD (Fig. 7a) and withfluid type A (Fig. 7b), applying a speed ramp profile up to 4000 rpm.

230 PROCEEDINGS OF ISMA2012-USD2012

Figure 6: Newtonian fluid SFD tests speed up 4000 rpm (a) Simulation results (b) Experimental results.

Figure 7: Experimental FFT and rotor speed for a run-up to 4000 rpm (a) Empty SFD, (b) Fluid type A.

7 Simulation results using the SFD-MR

Considering the parameters in Table 1, the design parameters for the sliding-mode controller are proposedas α1x = 375, α0x = 32.69, α1y = 375, α0y = 32.69, ωx = 2000, ωy = 2000. The virtual MR damperparameters (Table 2) were taken and enclosed with the results listed above, using those in Cabrera-Amadoand Silva-Navarro [30].

FMRxx , FMRyy FMRyx , FMRxy

i b+i c+

i b−i c−i b+i c+

i b−i c−i0 0.0290 0.2108 0.0391 -0.2236 0.0087 0.0632 0.0117 -0.06711 -1.3176 83.2320 -1.3355 83.5920 -0.3953 24.9696 -0.4007 25.07762 1.3687 -40.4136 0.06207 -26.8200 0.4106 -12.1241 0.1862 -8.0460

Table 2: Parameters of the rotor-bearing system

The simulate tests consist in speed the rotor up to 4000 rpm in 60 s with two different speed profiles (ramptype and a smooth Bezier profile), with an electrical current of I = 0 A, I = 0.5 A, and a robust controlscheme saturated at 1 A.

The open-loop simulation results for constant currents in Figs. 8a and 8b correspond to the rotor disk co-ordinates xr and yr, respectively, when I = 0 A and I = 0.5A. In addition, the closed-loop simulationresults employing the sliding-mode controller and the modified inverse model are described in Figs. 8a and8b, where one can observe the reduction on the unbalance responses in the rotor-bearing system using theSFD-MR bearing. Moreover, the current control inputs to the MR fluid, generated by the controller and

ACTIVE NOISE AND VIBRATION CONTROL 231

Figure 8: Open-loop and closed-loop system performance (a) xr (b) yr.

Figure 9: Closed-loop SFD-MR (a) Control currents, (b) System response using ramp and Bezier rotor speedprofiles.

applied to the SFD-MR, are shown in Fig. 9a. In Fig. 9b is described a FFT comparison of the closed-loopresponse (xr) using two different (ramp and Bezier) speed profiles up to 4000 rpm.

8 Analysis and discussion

From the above results one can observe that, in general, the viscosity modification in the Newtonian fluid SFDaffects mainly reducing the (peak) amplitude response during the first rotor critical speed. Note, however,that this is not always the case, because different viscosity values can yield to higher amplitudes. See, e.g.,Fig. 5a for a fluid type B, with a viscosity higher than the ideal, resulting in higher vibration amplitudeswith respect to the empty SFD or in case of a fluid type A. This attenuation properties can be used whenone substitutes an MR fluid to synthesize a semiactive vibration control scheme for the overall rotor-bearingsystem.

The rotor-bearing system response can be attenuated using a proper speed profile. A comparison responseusing two different speed profiles is shown in Fig. 5b, where the amplitude is reduced by means of a run-upwith a Bezier polynomial rotor speed trajectory leading to a smoother acceleration scheduling. Moreover,when a Newtonian fluid with kinematic viscosity µ = 1250 mm2/s is used, the system response is reducedup to 26% (see Fig. 6).

The experimental and numerical Frequency Response Functions (FRF) obtained with a conventional SFDwith Newtonian fluid are shown in Fig. 7. These comparison is useful to validate the proposed finite element

232 PROCEEDINGS OF ISMA2012-USD2012

model, over which is synthesized a robust semiactive vibration control scheme using the SFD-MR. Theoverall closed-loop system performance, applying the robust sliding-mode control, with respect to open-loop control for constant current inputs I = 0 A and I = 0.5 A, is presented in Fig. 8. The actual currentcontrol input and the rotor speed dynamics are shown in Figs. 9a. Note that, the closed-loop system responseresults in an unbalance response with reductions up to 48%, and the open-loop system response (I0.5 A)leads to reductions up to 39%. Furthermore, an acceleration scheduling using a Bezier speed profile yieldsan extra reduction of 2.5% (see Fig. 9b).

Finally, it is important to note that, the application of a SFD-MR bearing could employ powers less than17 W to modify the rheological properties of the MR fluid into the SFD and, as a consequence, one cansemiactively control the unbalance response.

9 Conclusions

In the present work the finite element model and the experimental validation of a rotor with a SFD withNewtonian fluid are described obtaining amplitude reductions up to 26%. A novel bearing using a SFDwith MR fluid, modeled via MR virtual dampers, is proposed to attenuate the unbalance response in a rotor-bearing system. The semiactive control scheme is synthesized using a robust sliding-mode controller andacceleration scheduling, obtaining amplitude reductions up to 48% in unbalance response. Further work isbeing completed to get the characterization, integration and experimental evaluation of the SFD-MR bearing.

References

[1] F. Y. Zeidan, L. San Andres, J. M. Vance, Design and application of squeeze film dampers in rotatingmachinery, Proceedings of the Twenty-Fifth Turbomachinery Symposium.

[2] A. Harnoy, Bearing Design in Machinery Engineering Tribology and Lubrication, Marcel Dekker Inc(2003).

[3] J.C. Nicholas, journal bearings - types, characteristics and applications, Rotating Machinery Technol-ogy, 0th Annual Meeting, Vibrations Institute, Willowbrook, IL (1996).

[4] J. L. Nikolajsen, An electroviscous damper for rotor applications, Trans. ASME, J. Vib. Acoust (1990),112 440-3.

[5] S. Morishita, Mitsui, Controllable squeeze film damper (an application of electro-rheological fluid),Trans. ASME, J. Vib. Acoust, 114 354-7.

[6] J. A. Tichy, Behavior of a squeeze film damper with an electrorheological fluid, STLE Tribol. Trans(1993), 36 127-33.

[7] J. A. Tichy, Hydrodinamic lubrication theory for the Bingham plastic flow model, J. Rheol 35(4).

[8] S. Y. Jung, S. B. Choi, Analysis of a short squeeze-film damper operating with electrorheological fluids,STLE Tribol. Trans (1995), 38 857-62.

[9] G. Z. Yao, G. Meng, Vibration control of a rotor system by disk type electrorheological damper, J. ofSound and Vibration (1999), 219175-88.

[10] G. Yao, Y. F. Fah, G. Chen, Electro-rheological multi-layer squeeze film damper and its application tovibration control of rotor system, Trans. ASME, J. Vib.Acoust (2000), 122 7-11.

[11] J. Wang, G. Meng, N. Feng, E. J. Hahn, Dynamic performance and control of squeeze mode MR fluiddamper-rotor system, Smart Materials and Structures (2005), 14 529-39.

ACTIVE NOISE AND VIBRATION CONTROL 233

[12] J. Wang, G. Meng, Study of the vibration control of a rotor system using a magnetorheological fluiddamper, J. Vib. Control (2005),11 263-76.

[13] J. Wang, G. Meng, Experimental study on stability of an MR fluid damper-rotor-journal bearing system,Journal of Sound and Vibration (2003), 262 999-1007.

[14] C. Zhu, Dynamics of a rotor supported on magnetorheological fluid squeeze film damper, Chinese J.Aeronaut (2001), 14 7-12.

[15] C. Zhu, D. A. Robb, D. J. Ewins, Magnetorheological fluid dampers for rotor vibration control,/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conf. (April 2001) (2001) vol3 pp 2121-7.

[16] P. Forte, M. Paterno and E. Rustighi, A magnetorheological fluid damper for rotor applications, . 6thInt. Conf. on Rotor Dynamics (Sept.–Oct., 2002), (2002) (Sydney: IFToMM), pp 63-70.

[17] C. Zhu, A disk-type magneto-rheological fluid damper for rotor system vibration control, Journal ofSound and Vibration 283 (2005) 1051-1069.

[18] C. Carmignani, P. Forte, E. Rustighi, Design of a novel magneto-rheological squeeze-film damper,Smart Materials and Structures 15 (2006), pp. 164-170.

[19] C. Zhu, Controllability of a magnetorheological fluid squeeze film damper under sinusoidal magneticfield, Key Engineering Materials vol. 334-335, pt.2 : 1089-92 (2007).

[20] K. J. Kim, C. W. Lee, J. H. Koo, Design and modeling of semi-active squeeze film dampers usingmagneto-rheological fluids, Smart Mater. Struct. 17 (2008) 035006.

[21] M. Yan-hong, W. Hong, H. Jie, Investigation of non-synchronous response of adaptive squeeze filmdamper with metal rubber oil film ring, Journal of Aerospace Power vol.24, no.2 : 390-5 (2009).

[22] K. J. Kim, C. W. Lee, J. H. Koo, Optimal positioning and control of a MR-squeeze film damper for re-ducing unbalanced vibrations in a rotor system with multiple masses, Journal of vibration and Acousticsvol. 131, no.4 (2009).

[23] H.P. Jagadish, L. Ravikumar, Calibration of the stiffness and damping characteristics of a magnetorhe-ological fluid long squeeze film damper in terms of Reynolds number, of the Institution of MechanicalEngineers Part C-Journal of Mechanical Engineering Science (2010), 224 (c10):2121-2128.

[24] S. B. Choi, S. K. Lee, Y. P. Park, A Hysteresis Model for the Field-dependent Damping Force of aMagnetorheological Damper, J. of Sound and Vibration (2001), 245-2, pp. 375-383.

[25] H. Yukio, Hydrodynamic Lubrication, Springer, Japan (2006).

[26] J. M. Vance, Rotordynamics of turbomachinery, John Wiley & Sons, USA (1988).

[27] J. D. Carlson, Magnetorheological fluid actuators. In Adaptronics and Smart Structures (Ed. H.Janocha), pp. 180–195, Springer-Verlag, Berlin (1999).

[28] W. Heylen, S. Lammens, P. Sas, Modal analysis theory and testing, Belgium: Katholicke UniversiteitLeuven.

[29] D. Childs, Turbomachinery Rotordynamics, John Wiley & Sons, USA (1993).

[30] A. Cabrera-Amado, G. Silva-Navarro, “Control of the Unbalance Response in a Rotor-Bearing Systemwith a Semiactive Suspension Based on MR Dampers”, of the 15th International Congress on Soundand Vibrations (ICSV15), pp. 3001-3008, Daejeon, Korea.

234 PROCEEDINGS OF ISMA2012-USD2012

[31] V. I. Utkin, Sliding Modes in Control and Optimization, Springer-Verlag, Berlin (1992).

[32] G. Genta, Dynamics of Rotating Systems, Springer-Verlag, NY (2005).

ACTIVE NOISE AND VIBRATION CONTROL 235

236 PROCEEDINGS OF ISMA2012-USD2012