INTRODUCTIONSuspension bushings are often expected to provide a high

viscous damping coefficient (c) and stiffness (k) for largeamplitude excitations at lower frequencies [1,2,3,4,5].Further, lower k and c values are needed for controllingstructure-borne noise at moderate to higher frequencies[1,2,7]. Since elastomeric bushings cannot satisfy suchconflicting requirements, many fluid-filled bushing designshave been developed and utilized in vehicles [1-2], as evidentfrom 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 performance features,they provide no analytical justifications or even measureddynamic properties. Further, very few scholarly articles haveaddressed hydraulic bushing characterization and modelingissues [7,8,9,10]. Moreover, prior articles [1,7,8,9,10] haveonly examined the frequency domain properties, and mostdesigns [1,7,8,9,10,11,12,13,14,15,16,17,18,19,20] are basedon spectral characteristics. Therefore, this article aims topropose new or improved models based on the linear timeinvariant system theory and study the time domaincharacteristics of hydraulic bushings.

The schematic of a typical fluid-filled bushing is shown inFigure 1. Such a device consists of two almost identicalelastomeric chambers between inner and outer metal sleeves.These chambers are filled with the anti-freeze mixture, and

they communicate fluid via a long passage (inertia track),and/or a short passage (or a controlled leakage orifice [1]).The relative deflection of the inner and outer metal causeschamber pressures to vary, and thus fluid flows back andforth through the inertia track and orifice-like passages,which provides effective damping at the desired frequencyrange.

Figure 1. Fluid model I of the hydraulic path with a longflow passage (inertia track, #i) and a short flow passage

with restriction (#s).

2013-01-1927Published 05/13/2013

Copyright © 2013 SAE Internationaldoi:10.4271/2013-01-1927saepcmech.saejournals.org

Transient Response of Hydraulic Bushing with Inertia Trackand Orifice-Like Elements

Tan Chai, Jason T. Dreyer and Rajendra SinghThe Ohio State University

ABSTRACTHydraulic bushings are widely used in vehicle applications, such as suspension and sub-frame systems, for motion

control and noise and vibration isolation. To study the dynamic properties of such devices, a controlled laboratory bushingprototype is designed and fabricated. This device has the capability of varying different combinations of long and shortflow passages and flow restriction elements. Transient experiments with step-up and step-down excitations are conductedon the prototype, and the transmitted force responses are measured. The transient properties of several commonly seenhydraulic bushing designs are experimentally studied. Analytical models for bushings with different design features aredeveloped based on the linear system theory. System parameters are then estimated for step responses based on theory andmeasurements. Finally, the linear models are utilized to analyze the step force measurements, from which somenonlinearities of the bushing system are identified.

CITATION: Chai, T., Dreyer, J. and Singh, R., "Transient Response of Hydraulic Bushing with Inertia Track and Orifice-Like Elements," SAE Int. J. Passeng. Cars - Mech. Syst. 6(2):2013, doi:10.4271/2013-01-1927.

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Although fluid-filled bushings exhibit frequency-dependent and amplitude-sensitive properties [7], a lineartime-invariant model must be developed first to gain physicalinsight into the bushing system and systematically examineits dynamic properties. Only radial damping is consideredbecause the relative displacement between the inner and outermetal parts is assumed to be only in the radial direction.Accordingly, specific objectives for this article are asfollows: (a) Design a laboratory device with controlledinternal configurations and conduct experiments on thisprototype under transient excitations with step-up and step-down displacement excitations; (b) Develop lumpedparameter linear models of fluid-filled bushings andanalogous mechanical models; (c) Analyze the measuredtransient responses by utilizing the linear system theory.

EXPERIMENTAL STUDIESFluid passages in production hydraulic bushings usually

have irregular geometry and are constructed with alternatematerial from elastomers to metals. Moreover,experimentation with many bushing samples from differentmanufacturers would pose a difficult task. Thus, a laboratorydevice which can provide insights into various aspects isdesigned and fabricated for dynamic characterization andscientific examination. This device is able to providedifferent combinations of long fluid passage (inertia track)and short passage with adjustable flow restriction.Experiments are conducted on five configurations (designatedB1 to B5, as listed in Figure 2 and Table 1) of this prototypeusing the MTS elastomer test machine [21]. Additionally, twolimiting cases (B0 with all fluid passages closed and BR withthe drained bushing) are also evaluated in the experiment.

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.

Table 1. Configurations of the hydraulic bushingprototype

In the transient experiment, a step-up and step-downexcitation is applied to the prototype device, and thetransmitted force is measured. As shown in Figure 3 (a), firsta compressive mean load fm corresponding to displacementxM (t) = Ao is applied, and then the step-up process increasesthe load to reach xM (t) = Ao + Ax and holds it for Δt = t2 − t1;then xM (t) is released to Ao − Ax via the step-down processand is held for another Δt =t3 − t2. Finally, the load isincreased back to around fm such that now xM (t) ≈ Ao.However, an ideal step function as shown in Figure 3(a) isextremely difficult to generate given the practical limitationsof the actuator. Thus, only step-like functions such assmoothened step events are realized as shown in Figure 3(b).Note that the step-down excitation has twice the step heightAe of the step-up, and their initial loading levels are different.Thus, this measurement will investigate the effect of Ae andthe mean load on transient responses. Experimental resultswill be discussed in detail along with analytical analysis inthe later sections.

Fig. 3. Excitation profiles for the transient experiment.(a) Ideal step-up and step-down pulse; (b) Smoothenedstep-up excitation. Key: , measured step excitation;

, analytical approximation; , ideal step function.

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Fig. 3 (cont). Excitation profiles for the transientexperiment. (a) Ideal step-up and step-down pulse; (b)

Smoothened step-up excitation. Key: , measured stepexcitation; , analytical approximation; , ideal step

function.

MATHEMATICAL MODELSModel I for Two Parallel Flow Passages

A fluid system model of the hydraulic path of a bushingwith two passages in parallel is proposed in Figure 1 based onthe lumped parameter method. These include an internal longpassage such as the inertia track (#i), though shown asexternal tubing for the sake of clarity in Figure 1, and/or ashort passage (#s) with a flow restriction or controlledleakage element (such as the orifice). The long and short flowpassages are represented by the fluid inertances Ii and Is andfluid resistance Ri and Rs, respectively. The rubber element(#r) is modeled by rubber stiffness kr and viscous dampingcoefficient cr. The compliances of the two fluid chambers (#1and #2) are C1 and C2, respectively, with effective pumpingareas A1 and A2.

For better illustration of the mechanism of a hydraulicbushing, an analogous mechanical system model is developedin Figure 4, where the hydraulic path includes long (inertiatrack) and short passages in parallel. The reason for selectingthe mechanical model is that it should facilitate an easierimplementation into structural models of the vehicle system.The rubber path is still approximated by the Kelvin-Voigtmodel with rubber stiffness kr and damping coefficient cr.The effective fluid path parameters based on the fluid systemmodel are defined as follows: Equivalent mechanical stiffness

of two fluid chamber compliances and

, effective cross-sectional area of the inertia track

, effective fluid velocity in theinertia track , effective mass of the inertia

track fluid , effective viscous damping of inertia

track fluid , effective cross-sectional area of the

short fluid passage , effectivefluid velocity in the short passage , effective

mass of the short passage fluid , and effective

viscous damping of the short passage fluid .

Figure 4. Analogous mechanical model of the hydraulicpath with long and short flow passages in parallel

Since the mass (mr) of the hydraulic bushing is usuallynegligible, the transmitted (dynamic) force is given as:

(1)

Application of Newton's second law to the effectivemasses of the inertia track and short passage fluid elementsyields the following equations:

(2)

(3)

By transforming Eqs. (1), (2), (3) into the Laplacedomain, the dynamic stiffness Kd is obtained. Since mse isusually negligible at the lower frequencies, Eq. (3) issimplified and expressed by a reduced order form as follows:

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(4)

Further, convert Eq. (4) into a standard form as follows,where γ is the static stiffness of two fluid chambers, τ is thetime constant in the numerator, and ωn2 and ζ2 are the naturalfrequency and damping coefficient of the denominatorexpression, respectively.

(5a,b)

(5c,d)

(5e)

Model II for a Single Long PassageA hydraulic bushing design with only an inertia track can

be analyzed as a sub-set of the mechanical system in Figure4. The following governing equations are found by assumingxse = 0, mse = 0 and cse → ∞:

(6)

(7)

Then the dynamic stiffness of a bushing with one inertiatrack is obtained by transforming Eqs. (6) and (7) into theLaplace domain:

(8)

Also, Eq. (8) is converted into a standard form as followswhere γ = k1+k2 and τ = cie/mie :

(9a)

(9b,c)

IDENTIFICATION OF SYSTEMPARAMETERS

System parameters are estimated by linear fluid systemtheory and bench experiments. The effective pumping areasA1 and A2 could be assumed to be similar, which isapproximated as the area A at the location of excitation x(t).For the calculation of C1 and C2, three components must beconsidered: the flexibility of the chamber (the rubbercontainer), the compliance of the fluid itself, and that of theentrapped air. The fluid inertances of the long and shorttracks are calculated as Ii = ρli/Ai and Is = ρls/As, where ρ isthe fluid density, Ai and As are the cross-sectional areas of thelong track and short passage, respectively, and li and ls aretheir lengths [22]. Rubber stiffness and effective damping ofeach flow passage are estimated based on the transientresponses due to their sensitivity to the excitationcharacteristics and flow conditions. Consider the stepresponses of B1 as shown in Figure 5. Since the transmittedforce fT(t) converges to Axkr as t → ∞, rubber stiffness kr isidentified from the asymptotic value of the step response. Theeffective damping of the long flow passage is estimated fromthe decaying transient amplitudes in the step response of B1.Both the oscillation period and the envelope curve depend onthe effective damping value. Likewise, cse of the short fluidpassage is estimated from the step response of B2.

Figure 5. Typical step-up force response of B1. Key:, measured step-up response; , envelope curve

Ξ(t).

Dynamic forces transmitted by a hydraulic bushing areestimated using numerical simulations and analyticalapproaches of lumped system models in the time domain.Based on the dynamic stiffness transfer function and giventhe step excitation, predictions for fT(t) are first numericallysolved using the Runge-Kutta 4th/5th order algorithm.Further, analytical solutions are also derived in terms of theestimated system parameters. For instance, for an ideal stepdisplacement input x(t) = Ax u(t), the dynamic forcetransmitted by the rubber path fTr(t) is:

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(10)The force transmitted by the hydraulic path is derived by

the convolution (*) of x(t) and hdh(t), where hdh(t) is theimpulse response corresponding to Kdh. For a bushing withonly an inertia track, the fTh(t) given an ideal step excitation

is obtained as follows, where ,

(11)Then the total force transmitted to the ideal step excitation

is obtained by fT(t) = fTr(t) + fTh(t).

ANALYSIS OF TRANSIENTMEASUREMENTS

As mentioned earlier, the step height Ae of the step-upexcitation is half that of the step-down. Thus, to compare thethree step-like responses, both the excitation xM(t) andresponse fT(t) are normalized by the step height Ae and thensuperposed in the same plot. The normalized step responsesof configuration B1 are divided into three regimes as shownin Figure 6. Regime 1 focuses on the overshoot which ismainly controlled by the rubber stiffness and damping andthe stiffness of two hydraulic chambers. Regime 2 isdominated by the hydraulic path; both oscillations andexponential decays of the step responses are seen. Finally,regime 3 exhibits the asymptotic value, which is affectedmostly by the rubber stiffness. Comparison of the threeregimes shows that the normalized step-down response withAe = 2Ax has a lower oscillation peak value and longer periodthan the step-up event with Ae = Ax. This implies that fluidresistance (effective damping) of the long flow passage maydepend on the step-height Ae. The lower overshoot value ofthe normalized step-down response may also be introducedby the lower abruptness in its step-like excitation. In addition,a variation in the asymptotic value may be caused by thepreload since the rubber path properties are highly dependenton the mean load.

To examine the experimental results, the predictions ofmodel II with three effective damping values cie = 1.0cie-n,1.8cie-n, 3.0cie-n are compared in Figure 7 with measuredstep-up and step-down responses of B1, where cie-n is thenominal effective damping estimated from a bench flow test.Observe that cie-n underestimates the damping in stepresponses, while 1.8cie-n gives a good approximation of thestep-up response trends with Ae = Ax; similarly, 3.0cie-nworks for the step-down with Ae = 2Ax. This suggests that theeffective damping increases when a higher step heightinduces more flow, especially during the first two oscillation

periods. Furthermore, the envelope curve Ξ(t) of the stepresponse of B1 does not follow the exponential decay of alinear system, and its oscillation period decreases as thetransmitted force reaches asymptotic value. These againimply that the fluid resistance is a function of flow rate, andthe bushing system is indeed a nonlinear device.

Figure 6. Normalized step responses for B1configuration. Key: , first step-up response with Ae

= Ax; , step-down response with Ae = 2Ax; ,second step-up response with Ae = Ax; , regimes.

Figure 7. Comparison of step responses for B1 withalternate system parameters. (a) Step-up response withAe = Ax; (b) Step-down response with Ae = 2Ax. Key:

, measured response; , prediction of model IIwith cie = 1.0ci-n; , prediction of model II with cie =

1.8ci-n; , prediction of model II with cie = 3.0ci-n.

The step responses of configuration B4 with two flowpassages in parallel is well predicted by model I with

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parameters estimated from the methods discussed above. Forinstance, the step-up prediction with Ae = Ax is comparedwith measured transmitted force in Figure 8. Since cse ishigher than cie, the response is first dominated by an inertiatrack effect when the step excitation takes place. As thepressure drop between two chambers decreases, the shortpassage starts to communicate the most flow; thus, increaseddamping diminishes the oscillations.

Figure 8. Comparison of step responses for B4. Key:, measured response; , prediction of model I.

For both configurations, discrepancies between thepredicted and measured responses suggest that an exponentialdecay term that is not represented by the Kelvin-Voigt modelmay exist in the rubber path. Moreover, discrepancies on theoscillation amplitudes and periods may be caused by thechanges of effective damping during the step event asdiscussed above.

CONCLUSIONThis paper focuses on the analysis of the transient

responses of hydraulic bushings. A laboratory prototype withthe capability of varying key design features in a controlledmanner is built and instrumented. Transient experiments withstep-up and step-down excitations are conducted on thisdevice, and the transmitted force is measured. Analogousmechanical models are developed for a fluid-filled bushingwith both a long inertia track and a short passage withrestriction; also the corresponding dynamic stiffness transferfunctions are found. Model parameters are estimated by aprocedure of parameter identification based on bushinggeometry and bench experiments. Then linear modelpredictions are utilized to analyze the transient experimentalresults and to identify the main nonlinearities exhibited intransient responses. The predictions agree with measuredtrends. Further, the step-up and step-down responses suggestthat the effective damping of a hydraulic bushing is highlydependent on the step height and has significant changeduring the decaying transient. Further, such models can beutilized to examine frequency domain responses as describedin the companion paper [23]. Future work should examine

alternate models of the rubber path and develop nonlinearmodels of the hydraulic paths.

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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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.

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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Chai, T., Dreyer, J. and Singh, R., “Dynamic Stiffness of HydraulicBushing with Multiple Internal Configurations,” SAE Int. J. Passeng.Cars – Mech. Syst. 6(2):1209-1216, 2013, doi:10.4271/2013-01-1924.

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CONTACT INFORMATIONProfessor Rajendra SinghAcoustics and Dynamics LaboratoryNSF I/UCRC Smart Vehicle Concepts CenterDept. of Mechanical and Aerospace EngineeringThe Ohio State University, Columbus, OH 43210singh.3@osu.eduPhone: (614)292-9044www.AutoNVH.org

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

Ae - step height

Aie - effective cross-sectional area of the inertia track

Ase - effective cross-sectional area of the short passage

Ax - nominal step height

cie - effective viscous damping of the inertia track

cse - effective viscous damping of the short passage

cr - damping coefficient of rubber

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

f - frequency, Hzfm - mean load

fT - transmitted fore

fTh - fore transmitted by the hydraulic path

fTr - fore transmitted by the rubber path

hdh - impulse response of Kdh

Ii - inertance of the long fluid passage (inertia track)

Is - inertance of the short restricted fluid passage

Kd - dynamic stiffness

Kdh - dynamic stiffness of the hydraulic path

Kdr - dynamic stiffness of the rubber path

kr - stiffness of rubber

k1, k2 - equivalent stiffness of two fluid chambers

li - length of the long inertia track

ls - length of the short fluid passage

mie - effective mass of the inertia track fluid

mse - effective mass of the short passage fluid

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 variableTd - oscillation period

t - timeu(t) - unit step functionx - excitation displacementxie - effective fluid velocity in the inertia track

xse - effective fluid velocity in the short passage

X - peak to peak value of sinusoidal displacementγ - stiffness of two fluid chambersΔp - pressure differential between two fluid chambers

δ(t) - Dirac delta functionω - circular frequency, rad/sωn - natural frequency, rad/s

Ξ(t) - envelope functionρ - fluid densityτ - time constantζ - damping ratio

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