Dynamic Stiffness of Hydraulic Bushing with Multiple ... et al_2013.pdfdefined as the excitation...

INTRODUCTIONElastomeric and hydraulic bushings are widely used in

vehicle suspension, body and engine sub-systems forimproved handling and ride performance characteristics andto reduce vibration and noise [1,2,3,4,5,6]. Such devices areoften expected to provide a high viscous damping coefficient(c) and stiffness (k) for large amplitude excitations at lowerfrequencies. Further, lower k and c values are needed forcontrolling structure-borne noise at moderate to higherfrequencies [1,2,7]. Since elastomeric bushings cannot satisfysuch conflicting requirements, many fluid-filled bushingdesigns have been developed and utilized in vehicles [1-2], asevident from some articles [7,8,9,10] and patents[11,12,13,14,15,16,17,18,19,20]. Even though many patents[11,12,13,14,15,16,17,18,19,20] claim certain performancefeatures, they provide no analytical justifications or specificstiffness properties. Further, very few scientific articles haveaddressed the hydraulic bushing characterization andmodeling issues [7,8,9,10]. Therefore, this article aims to fillthis void and experimentally and analytically study thedynamic characteristics of a family of hydraulic bushingdesign.

A typical fluid-filled bushing consists of two almostidentical elastomeric chambers between inner and outer metalsleeves. These chambers are usually filled with the anti-

freeze mixture, and they communicate fluid via a longpassage (inertia track), and/or a short passage (or a controlledleakage orifice [1]). The relative deflection between the innerand outer metal sleeves causes chamber pressures to vary,and thus fluid flows back and forth through inertia track typeand orifice-like passages, which provides effective dampingover the desired frequency range, depending on the fluidsystem configuration and flow conditions. An accurateestimation of the dynamic stiffness of bushings is importantfor the prediction of dynamic loads in a vehicle sub-frameand suspension system. Although fluid-filled bushings exhibitfrequency-dependent and amplitude-sensitive properties, alinear time-invariant (LTI) model must be developed first toestimate dynamic properties and to diagnose and tune suchdevices over the lower frequency regime (say from 1 to 50Hz). Several different configurations of the hydraulic bushingshould be experimentally and analytically examined to gain abetter understanding of the underlying physics. Only radialdamping is considered because the relative displacementbetween inner and outer metal parts is assumed to be only inthe radial direction. Accordingly, specific objectives for thisarticle are as follows: (a) Design a laboratory device withcontrolled internal configurations and conduct experimentson a production bushing and the laboratory prototype undersinusoidal excitation; (b) Analyze measured dynamic

2013-01-1924Published 05/13/2013

Copyright © 2013 SAE Internationaldoi:10.4271/2013-01-1924saepcmech.saejournals.org

Dynamic Stiffness of Hydraulic Bushing with Multiple InternalConfigurations

Tan Chai, Rajendra Singh and Jason DreyerThe Ohio State University

ABSTRACTFluid filled bushings are commonly used in vehicle suspension and sub-frame systems due to their spectrally-varying

and amplitude-dependent properties. Since the literature on this topic is sparse, a controlled laboratory prototype bushing isfirst designed, constructed, and instrumented. This device provides different internal combination of long and short flowpassages and flow restriction elements. Experiments with sinusoidal displacement excitations are conducted on theprototype, and dynamic stiffness spectra along with fluid chamber pressure responses are measured. The frequency-dependent properties of several commonly seen hydraulic bushing designs are experimentally studied and compared undertwo excitation amplitudes. Further, new linear time-invariant models with one long and one short flow passages (in parallelor series) are proposed along with the limiting cases. The analytical models are validated in frequency and time domainsusing transmitted force and chamber pressure measurements. The proposed formulation provides insights into variousperformance features.

CITATION: Chai, T., Singh, R. and Dreyer, J., "Dynamic Stiffness of Hydraulic Bushing with Multiple InternalConfigurations," SAE Int. J. Passeng. Cars - Mech. Syst. 6(2):2013, doi:10.4271/2013-01-1924.

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THIS DOCUMENT IS PROTECTED BY U.S. AND INTERNATIONAL COPYRIGHT.It may not be reproduced, stored in a retrieval system, distributed or transmitted, in whole or in part, in any form or by any means.

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stiffness results and compare the dynamic properties ofdifferent configurations and examine their spectrally-varyingproperties amplitude-dependence; (c) Develop linear time-invariant (LTI) models of a fluid-filled bushing with alternateconfiguration using the lumped fluid parameter approach; (d)Validate the proposed models by comparing predictions withmeasurements under sinusoidal excitation only.

EXPERIMENTAL STUDIESThe static and dynamic properties of a practical fluid-

filled bushing (designated Y1) is evaluated first using thenon-resonant elastomer test machine (MTS 831.50, [21])under a mean load fm. A sinusoidal displacement x(t) = A sinωt is applied to the inner metal part in the radial direction,from 1 to 50 Hz in 1 Hz increments under amplitude X, whereω is the excitation frequency (rad/s), and X = 2A is the peakto peak (pp) amplitude. The force transmitted to the rigidbase fT(t) is typically measured. The dynamic stiffness isdefined as:

(1)

where , and |Kd(ω)| and ϕK(ω) are the magnitude andloss angle of Kd (ω), respectively. In this paper, results arepresented in the dimensionless form, and thus |Kd| isnormalized by its static stiffness. The frequency ratio isdefined as the excitation frequency (f = ω/2π) divided by the

frequency , at which the loss angle of a laboratory bushingprototype is maximum. One typical set of the measurementsunder three excitation amplitudes (X = 0.1, 0.5 and 1.0 mm(pp)) is shown in Figure 1. Significant amplitude andfrequency dependence is observed from the Kd (ω) spectra.Both |Kd| and ϕk decrease as X is increased. The loss angle

achieves its peak at and then gradually decreases with ω,

while |Kd| continues to increase until , which is about two

times , and then settles down.

Figure 1.

Figure 1 (cont). Measured dynamic stiffness spectra of atypical production bushing (Y1) for three excitation

amplitudes (X). Key: , X = 0.1 mm; , X = 0.5 mm;, X = 1.0 mm.

Fluid passages in practical bushings usually have irregulargeometry and may be constructed with elastomer or metal.Moreover, experimentation with many bushing samples fromdifferent manufacturers would pose a difficult task. Thus, alaboratory device which can provide insights into variousaspects is designed and fabricated for dynamiccharacterization and scientific examination. This device isable to provide different combinations of long fluid passage(inertia track) and short passage with adjustable flowrestriction. Initial experiments are conducted on fiveconfigurations (designated B1 to B5, as listed in Table 1 andFigure 2) of this prototype using the MTS elastomer testmachine. Additionally, two limiting cases (B0 with all thefluid passages closed and BR as the drained bushing) are alsoevaluated in the experiment. Similar to the dynamic stiffnessmeasurements of the production bushing, a sinusoidaldisplacement excitation x(t) = A sin ωt is applied to theprototype device under a mean load. The dynamic stiffnessKd(ω) spectra, along with the steady state time histories ofthe transmitted force fT(t) are measured from 1 to 60 Hz with1 Hz increment at two amplitudes, X = 0.1 and 1 mm. Inaddition, the internal pressures of two hydraulic chambers,p1(t) and p2(t), are measured by the dynamic pressuretransducers installed within the fluid chambers of theprototype device.

Table 1. Configurations of the prototype hydraulicbushing

Chai et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 6, Issue 2(July 2013)



Figure 2. Multiple configurations of the hydraulicbushing constructed for controlled laboratory studies. (a)Configuration B1 with long fluid passage of diameter di;

(b) Configuration B2 with short flow passage withrestriction of diameter do and configuration B3 of

diameter do/3; (c) Configuration B4 with parallel longand short flow passages with do and B5 with do/3.

ANALYTICAL MODELSFluid Model I with Two Parallel FlowPassages

A fluid system model of the hydraulic path of a bushingwith two passages in parallel is proposed in Figure 3, withmany simplifying assumptions, based on lumped parametermethod. These include an internal long passage such as theinertia track (#i) though shown as external tubing for the sakeof clarity in Figure 3, and/or a short passage (#s) with a flowrestriction or controlled leakage element (such as the orifice).Assuming the linear system, the long and short flow passagesare represented by the fluid inertances Ii and Is and fluidresistance Ri and Rs, respectively. The compliances of twofluid chambers (#1 and #2) are C1 and C2, respectively, witheffective pumping areas A1 and A2. The rubber element (#r,parallel with the hydraulic path, not shown in Figure 3) ismodeled by rubber stiffness kr and viscous dampingcoefficient cr.

Figure 3. Fluid model I of the hydraulic path with a longflow passage (inertia track, #i) and a short flow passage

with restriction (#s).

Static and dynamic displacement excitations are appliedto the inner metal part while the outer metal sleeve is

relatively fixed. The mean force transmitted to the outersleeve under a static displacement is as follows, where isthe fluid chamber pressure under static equilibrium:

(2)

The dynamic displacement excitation x(t), from the static

equilibrium, is applied to the bushing under a mean load .By applying the continuity equation to two fluid controlvolume elements (1 and 2), the following equations areobtained, where qi(t) and qs(t) are the volumetric flow ratesthrough the long track and short passage, and p1(t) and p2(t)are the dynamic pressures inside the two fluid chambers,

(3a)

(3b)

The application of the momentum equation to two flowpassages (i and s) yields the following:

(4a)

(4b)




The dynamic force is transmitted to the outer sleeve by

rubber and hydraulic paths. Thus, the total transmitted forcefT(t) is defined below where fTr(t) is the rubber path and fTh(t)the hydraulic path force:

.(5)By transforming Eqs. (3), (4), (5) to the Laplace (s)

domain and assuming zero initial conditions, the followingdynamic stiffness Kd (s) is obtained, where Kdh(s) and Kdr(s)are the dynamic stiffness of the hydraulic and rubber path,respectively, and Zi = Iis + Ri and Zs = Iss + Rs are the fluidimpedance (Z) of the long (inertia track) and short (orifice-like) passages respectively,

(6)When the short fluid passage behaves like a sharp-edged

orifice, Is ≈ 0 can be assumed at the lower frequencies. Thus,Eq. (6) can then be converted into a reduced order form withstandard parameters as follows, where Ro is the fluidresistance of the orifice-like element, γ is the static stiffnessof the hydraulic chambers, ωn1 and ωn2 (rad/s) are the naturalfrequencies, and ζ1 and ζ2 are the damping ratios:

(7a)

(7b,c)

(7d)

(7e,f)The frequency response Kd(ω) is obtained by substituting

s =iω into the Laplace transfer functions, where .

Fluid Model II with Only Inertia TrackSome practical fluid-filled bushings have only one long

passage (inertia track) [11,12,13,14,17,18,19,20]. Thus, fluidmodel II is developed without the short passage by settingqs(t) = 0 in Eqs. (3) and (4) and transforming them into theLaplace domain with zero initial conditions. The resulting Kd(s) expression is now given as follows:

(8)

Eq. (8) is also converted into a standard form as follows,where γ is the same as that in Eq. (7):

(9a)

(9b)

(9c)

(9d)

(9e)

ANALYSIS OF MEASUREMENTSAND FREQUENCY DOMAIN

PROPERTIESOne typical set of dynamic stiffness measurements under

excitation amplitude X = 0.1 mm (p-p) is shown in Figure 4.As mentioned above, |Kd(ω)| and ϕK (ω) are defined as thestiffness magnitude (modulus) and phase (loss angle) ofKd(ω). Significant frequency dependence is observed fromthe Kd(ω) spectra of each configuration. Responses of thefive configurations (of Table 1) can be categorized into threegroups as seen in Figure 4. First, the dynamic stiffnessspectra of configurations B1 and B5 have high peak stiffnessand loss angles at lower frequency, which implies that thesetwo configurations are mainly controlled by the long inertiatrack and can provide higher damping levels at lowerfrequencies in a narrow bandwidth. Second, the Kd(ω) of B2and B4 have lower peaks |Kd| and ϕK at higher frequencies,which are dominated by the short, restricted fluid passage.These introduce a broader damping peak at higherfrequencies. Third, the measurements of B3 yield relativelylower |Kd| and ϕK values since the small diameter of therestricted short fluid passage only permits limited fluidpumping effect. In addition, B4 and B5 exhibit two differentfunctions of a bushing with parallel long (inertia track) andshort (leakage) paths; here the short path allows only limitedflow (and significant resistance) when excitation amplitude issmall (B5) and is forced to be wide open when the amplitudeis large (B4).




Figure 4. Measured dynamic stiffness spectra of fiveprototype configurations at low excitation amplitude.Key: , B1; , B2; , B3; , B4; , B5.

To examine the real (Re) and imaginary (Im) parts ofKd(ω), Nyquist diagrams of B1 to B5 are displayed in Figure5. For all configurations, the imaginary part of Kd(ω) is closeto zero near ω ≈ 0 and the real part is approximated as

from Equation 7. As the frequencyincreases, the Nyquist plot assumes the form of a circle, anellipse or semi-circle depending on the fluid passagecharacteristics.

Figure 5. Measured Nyquist diagrams of five prototypeconfigurations at low excitation amplitude. Key: ,

B1; , B2; , B3; , B4; , B5.

Asymptotically, Re(Kd)→ kr + γ and Im(Kd) → crω as ω→ ∞. Comparison of the five configurations leads to thefollowing observations. The Nyquist plot of B1 has a much

larger radius, and its shape is closer to a circle. ConfigurationB5 has a similar shape to B1 with a slightly smaller radius,which implies that the inertia track plays a dominant role inthis configuration. The Nyquist diagrams of B2 and B4 aresimilar and closer to an ellipse, which suggest that the shortfluid passage dominates and thus yields broader dampingproperties. Only an arc is formed for the B3 configurationsince the value of imaginary part is relatively low.

The measured dynamic stiffness of the two limiting cases,BR and B0 are shown in Figure 6 and compared with the B3configuration. The loss angles of BR and B0 are trivialcompared with other configurations which exhibit the fluidpumping effect. The |Kd| of B0 can be viewed as the sum ofrubber stiffness (kr) and the static stiffness of two fluidchambers (γ). Observe that stiffness magnitudes of B0 andBR both slowly increase with frequency, but with differentslopes. This shows that the stiffness of rubber and two fluidchambers are also frequency-dependent, though it is not assignificant in B1 to B5 cases. Moreover, |Kd| of B3 is close tothat of BR at lower frequencies, but gradually converges tothat of B0 as the frequency is increased. This implies thatsome air might be constrained in the flow passage and only asmall amount of fluid commutes between two chambers athigher frequencies.

Figure 6. Measured dynamic stiffness spectra of twolimiting cases compared with B3. Key: , B3; ,

BR; , B0.

Amplitude DependenceThree configurations (B1, B3 and B4) are selected from

the three categories mentioned above to examine theamplitude dependence of hydraulic bushings. Their dynamicstiffness spectra under two excitation amplitudes X = 0.1 and1.0 mm are compared in Figure 7. Both |Kd| and ϕK decreaseas X is increased, especially for the long inertia trackconfiguration. The frequencies of peak magnitude and loss

angle and both decrease as well. The stiffness




magnitude difference of B3 at high frequencies under X = 0.1and 1 mm is higher than that of BR (the drained bushing),which implies that the two chamber compliances mayincrease when the amplitude is increased. The Nyquist plotsof three configurations at low and high amplitudes arecompared in Figure 8. When X is increased, the values ofboth the real and imaginary parts reduce and the shape ofNyquist plots becomes less circular.

Figure 7. Measured dynamic stiffness spectra of threeconfigurations of the prototype bushing. Key: , B1with X = 0.1 mm; , B1 with X = 1.0 mm; , B3

with X = 0.1 mm; , B3 with X = 1.0 mm; , B4 withX = 0.1 mm; , B4 with X = 1.0 mm.

Figure 8. Nyquist diagrams of measured stiffness data.Key: , B1 with X = 0.1 mm; , B1 with X = 1.0mm; , B3 with X = 0.1 mm;, , B3 with X = 1.0

mm; , B4 with X = 0.1 mm; , B4 with X = 1.0 mm.

EXPERIMENTAL VALIDATION OFANALYTICAL MODELS

Frequency domain measurements of B4 and B1configurations are compared with the dynamic stiffnesspredictions of linear models I and II. Parameters areestimated by linear fluid system formulae and/or benchexperiments. However, theoretical formulation of the fluidresistance [22] tends to underestimate the Ri and Rs values.This implies that the momentum losses at fitting, valves, andsharp entrances and exits are significant. Consequently,simple bench experiments are conducted (with the watermedium at room temperature) to measure the pressure dropΔp and mean flow rate q under steady flow conditions foreach fluid passage [22]. Empirical Ri and Rs are then obtainedby relating Δp and q. Figure 9 compares the predictions ofmodel II (for configuration B1) based on analytical andempirical Ri. Stiffness and loss angle spectra predicted withanalytical Ri have much higher amplitudes compared with themeasurements, but the dynamic responses with empirical Ri(from the bench test) match well with experimental results.Small discrepancies in the stiffness magnitude rise beyond

due to frequency dependent properties of the rubber path.Similarly, the dynamic stiffness predictions of model I arecompared with measurements of configuration B4. Goodagreement between theory and experiment is also observed.

Figure 9. Validation of model II (with long fluid passageonly). Key: , predicted by linear model II with

measured Ri; , predicted with theoretical Ri; ,

measured dynamic stiffness of B1; , measured and

; , predicted peak magnitude and loss anglefrequencies.

The measured time histories of fT(t) and Δp(t) = p2(t) −p1(t) at several excitation frequencies and amplitudes areexamined using the predictions of the linear models for eachconfiguration. For instance, the transmitted force and




differential pressure time histories estimated by model II (B1configuration) are compared in Figure 10 with measurementsunder 0.1 mm (p-p) excitation at 10 Hz. Figure 11 comparesthe measurements and predictions of fT(t) and Δp(t) for B2configuration with X = 0.1 mm (p-p) at 30 Hz. Excellentagreements are observed for the transmitted force andpressure differential. The measured signals of fT(t) and Δp(t)are transformed to the frequency domain by the Fast FourierTransform (FFT). Only the fundamental frequencycomponent is found to be significant, and the Fouriermagnitudes at the second and third harmonics are at least 30dB lower than the first one. This implies that the time historyof transmitted force can be approximated by a singleharmonic term and the sinusoidal responses of fT(t) are wellrepresented by the proposed linear models.

Figure 10. Time domain sinusoidal responses of B1 withX = 0.1 mm at 10 Hz. Key: , measured response;

, prediction of model II.

Figure 11. Time domain sinusoidal responses of B2 withX = 0.1 mm at 30 Hz. Key: , measured response;

, prediction of model II.

CONCLUSIONThis paper focuses on the dynamic analysis of hydraulic

bushings. The main contribution is the combinedexperimental and analytical study of a laboratory prototypewith the capability of varying key design features in acontrolled manner. Frequency- and amplitude- dependentcharacteristics of each configuration are analyzed based onthe experimental measurements of the dynamic stiffnessspectra and Nyquist diagrams. It is observed that the dynamicstiffness spectra dominated by the short fluid passages withrestriction have lower peak stiffness and loss angle butexhibit a broader bandwidth compared with those mainlycontrolled by the long fluid passage (inertia track). A linearmodel for a fluid-filled bushing with both a long inertia trackand a short passage with restriction is developed, and themodel for a device with only an inertia track has beenexamined as well. Further, the proposed models are validatedby comparing predictions of dynamic stiffness and timedomain sinusoidal responses with the measurements of theprototype device. The linear models presented in this article(with many assumptions of course) are capable of analyzingmost practical designs[1-2,11,12,13,14,15,16,17,18,19,20].Additionally, suchmodels can be utilized to examine transient responses asdescribed in the companion paper [23]. Future work willfocus on an improvement of the rubber path model andanalysis of fluid path nonlinearities.

ACKNOWLEDGMENTSThe authors acknowledge the member organizations such

as Transportation Research Center Inc., Honda R&DAmericas, Inc., and YUSA Corporation of the Smart VehicleConcepts Center (www.SmartVehicleCenter.org) and theNational Science Foundation Industry/University CooperativeResearch Centers program (www.nsf.gov/eng/iip/iucrc) forsupporting this work. The authors also thank C. Gagliano andP. C. Detty for their help with experimental studies.

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3. Duan, C. and Singh, R., “Analysis of the vehicle brake judder problemby employing a simplified source-path-receiver model,” Proc. IMechEPart D: Automobile Engineering 225: 141-149, 2011.

4. Ibrahim, R. A., “Recent advances in nonlinear passive vibrationisolators,” Journal of Sound and Vibration 314: 371-452, 2008.

5. Gil-Negrete, N., Vinolas, J. and Kari, L., “A nonlinear rubber materialmodel combining fractional order viscoelasticity and amplitudedependent effects,” Journal of Applied Mechanics 76, 2009.

6. Gil-Negrete, N., Rivas, A. and Vinolas, J., “Predicting the dynamicsbehaviour of hydro bushings,” Shock and Vibration 12: 91-107, 2005.

7. Lu, M. and Ari-Gur, J., “Study of dynamic properties of automotivehydrobushing,” ASME Design Engineering Division Proceedings 106:135-140, 2000.




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http://www.nsf.gov/eng/iip/iucrc

http://www.sae.org/technical/papers/2003-01-1475

http://dx.doi.org/10.4271/2003-01-1475


http://dx.doi.org/10.4271/2007-01-0850

8. Lu, M. and Ari-Gur, J., “Study of hydromount and hydrobushing withmultiple inertia tracks,” JSAE Annual Congress Proceedings 68-02: 5-8,2002.

9. Sevensson, M. and Hakansson, M., “Hydrobushing model for multi-body simulation,” MS Thesis, Lund University, 2004.

10. Arzanpour, S. and Golnaraghi, M., “A novel semi-activemagnetorheological bushing design for variable displacement engines,”Jounal of Intelligent Material Systems and Structures 19 (2008):989-1003.

11. Hipsher, G. L., “High damping resilient bushing,” U.S. Patent 3642268,1972.

12. Konishi, K., “Fluid-filled resilient bushing,” U.S. Patent 4588174, 1986.13. Kanda, R., “Fluid-filled resilient bushing with circumferential orifice,”

U.S. Patent 4693456, 1987.14. Kanda, R., “Fluid-filled resilient bushing having damping means within

fluid chambers,” U.S. Patent 4749173, 1988.15. Kanda, R., “Resilient bushing structure filled with viscous fluid,” U.S.

Patent 4786036, 1988.16. Kanda, R., “Fluid-filled cylindrical elastic connector having two orifice

passages with different cross sectional areas,” U.S. Patent 5060918,1991.

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19. Tabata, T., “Fluid-filled insulating bushing,” U.S. Patent 5156379, 1992.20. Roth, S. F. and Henry, H. H.III, “Fluid-filled elastomeric suspension

bushing,” U.S. Patent 5397112, 1995.21. MTS Elastomer Test System Model 831.50 (1000 Hz), http://

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Bushing with Inertia Track and Orifice-Like Elements,” SAE TechnicalPaper 2013-01-1927, 2013, doi:10.4271/2013-01-1927.

CONTACT INFORMATIONProfessor Rajendra SinghAcoustics and Dynamics LaboratoryNSF I/UCRC Smart Vehicle Concepts CenterDept. of Mechanical and Aerospace EngineeringThe Ohio State University, Columbus, OH [email protected]: (614)292-9044www.AutoNVH.org

LIST OF SYMBOLSA1, A2 - effective pumping areas of fluid chambers #1 and #2

cr - damping coefficient of rubber

C1, C2 - lumped compliances of fluid chambers #1 and #2

f - frequency, HzfT - transmitted fore

- frequency of maximum stiffness magnitude

- frequency of maximum loss angleIi - inertance of the long fluid passage (inertia track)

Is - inertance of the short restricted fluid passage

Kd - dynamic stiffness

|Kd| - magnitude of dynamic stiffness

kr - stiffness of rubber

p1, p2 - pressure in fluid chambers #1 and #2

qi - volumetric flow rate of the inertia track

qs - volumetric flow rate of the short fluid passage

Ri - fluid resistance of the inertia track

Rs - fluid resistance of the short restricted fluid passage

s - Laplace transformation variablet - timex - excitation displacementX - peak to peak value of sinusoidal displacementZ - impedance of the fluid passageγ - stiffness of two fluid chambersΔp - pressure differential between two fluid chambersϕK - phase (loss angle) of dynamic stiffness

ω - circular frequency, rad/sωn - natural frequency, rad/s

ζ - damping ratio




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