proc imeche part d: nonlinear dynamic properties of hydraulic … · 2015-08-24 · nonlinear...

Original Article

Proc IMechE Part D:J Automobile Engineering2015, Vol. 229(10) 1327–1344� IMechE 2014Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0954407014561048pid.sagepub.com

Nonlinear dynamic properties ofhydraulic suspension bushing withemphasis on the flow passagecharacteristics

Tan Chai, Jason T Dreyer and Rajendra Singh

AbstractHydraulic bushings, which are often employed in vehicle suspension systems, exhibit significant excitation-dependentproperties. However, previous analyses were mainly based on the linear system theory. To overcome this void, nonlinearcharacteristics of common hydraulic bushing configurations are examined in this article, with focus on the componentproperties as excited by sinusoidal or step displacements of various amplitudes. First, a nonlinear model for a laboratoryprototype with a long passage and a short passage (in parallel) is developed using a lumped-parameter approach. Thenthe system parameters and nonlinearities are identified using experimental and computational methods, with an empha-sis on characterization of the flow passage resistances. Steady-state harmonic and transient step experiments are con-ducted on the prototype, and the dynamic pressures inside two fluid chambers and the force transmitted to the baseare measured. Numerical solution of the nonlinear model shows that the proposed model predicts both steady-statesinusoidal responses and transient responses well for single-passage and dual-passage configurations; significant improve-ment over a corresponding linear model is observed. Finally, approximate analytical and semi-analytical solutions of thenonlinear model are obtained by using the harmonic balance method.

KeywordsSuspension bushings, vibration and ride control, hydraulic device, parameter identification, analytical methods

Date received: 26 December 2013; accepted: 31 October 2014

Introduction

Hydraulic bushings are widely used in vehicle suspen-sion systems for improved ride and handling perfor-mance and to isolate vibration and structure-bornenoise.1–4 Their design features have been described inmany patents5–14 but without any analytical justifica-tions. Further, very few scholarly articles haveaddressed their characterization and modelingissues.1,15–20 For instance, Sauer and Guy1 havenumerically simulated a hydraulic bushing with aparallel inertia track and bypass track (a controlledleakage path), but no results were provided. Lu andAri-Gur15,16 have derived a simple expression for thenatural frequency of hydraulic bushing based on a lin-ear model, without presenting any experimental valida-tion. Sevensson and Hakansson17 have proposedempirical nonlinear models, but such models are notaccurate, especially at the low excitation amplitudes.Arzanpour and Golnaraghi18 have presented a linearmodel of the hydraulic engine subframe bushing, but

their approach is virtually identical with that for a fixeddecoupler hydraulic engine mount.21–23

Although hydraulic bushings exhibit significantfrequency-dependent and amplitude-sensitive proper-ties,1,2,15–20 previous articles1,15–20 and designs5–14

essentially employed a linear system theory to examinetheir spectral characteristics. Conversely, hydraulicshock absorbers and hydraulic engine and transmissionmounts24–26 have received much attention regardingtheir nonlinear properties. Therefore, this articleintends to overcome this particular void in the litera-ture and to examine rigorously the nonlinear dynamic

Acoustics and Dynamics Laboratory, Smart Vehicle Concepts Center,

Department of Mechanical Engineering, The Ohio State University,

Columbus, Ohio, USA

Corresponding author:

Rajendra Singh, Acoustics and Dynamics Laboratory, Smart Vehicle

Concepts Center, Department of Mechanical Engineering, The Ohio

State University, Columbus, OH 43210, USA.

Email: [email protected]

at OHIO STATE UNIVERSITY LIBRARY on August 24, 2015pid.sagepub.comDownloaded from

http://pid.sagepub.com/

properties of hydraulic bushings. Accordingly, the spe-cific objectives (with focus on the component only) areas follows.

1. Formulate the nonlinear model of common fluid-filled bushing configurations, and identify fluidpassage nonlinearities via experiments on a labora-tory prototype.

2. Validate the nonlinear model by comparing har-monic and transient response predictions withmeasurements.

3. Construct analytical steady-state sinusoidal solu-tions of the nonlinear model by using a single-termor a multi-term harmonic balance method.

This article will also extend the linear system formu-lation for time domain responses as proposed in arecent publication.20

Nonlinear fluid system model

A typical hydraulic bushing usually consists of innerand outer metal sleeves, a rubber element, and twohydraulic chambers filled with a mixture of anti-freezeand water.1,2,20 The two chambers are usually con-nected by one or more flow passages which communi-cate fluid between two chambers when a relativedeflection of the inner and outer metal sleeves causesthe pressures to vary. Note that, although hydraulicbushings are somewhat similar to the hydraulic enginemount, their application, design, construction, anddynamic performance are different.1,2,20,21 Hydraulicmounts, which are usually equipped with fixed or freedecouplers, are widely employed to support vehicleengines and to control motion mostly at lower frequen-cies. The excitation is applied to the mount top. Itslower chamber is very compliant, and thus the pressurein the lower chamber can usually be neglected in

comparison with the pressure in the upper chamber. Incontrast, fluid-filled bushings are mainly used in auto-motive suspension and chassis systems. They usuallydo not have any decoupler mechanisms. The outermetal sleeve can be assumed to be fixed in most appli-cations, and the displacement excitation is applied tothe inner metal part. The two fluid chambers arealmost identical, and some bushing designs have a leak-age (bypass) path in parallel with an inertia track tocommunicate fluid under high-amplitude deflection.1,20

Overall, hydraulic engine mount models cannot bedirectly applied to fluid-filled bushings because of thedifferences discussed above.

In the fluid-filled bushing model, the hydraulic ele-ments are represented by several control volumes, andincompressible flow is assumed. Moreover, only radialdamping is considered because the relative displacementbetween the inner metal part and the outer metal part isassumed to be only in the radial direction. The fluidmodel for a bushing with a long flow passage in parallelwith a short flow passage is developed in Figure 1(a).Note that the internal long fluid passage (the inertiatrack i) is shown as external tubing for clarity. Staticdisplacement (xm) and dynamic displacement (x(t) fromthe static equilibrium) excitations are applied to theinner metal part while the outer metal sleeve is fixedrelative to the inner sleeve. The rubber element r isassumed to be represented by a Kelvin–Voigt modelwith an excitation displacement-dependent rubber stiff-ness kr(x) and a viscous damping coefficient cr(x).Unlike the fluid chambers in the work reported by Chaiet al.,20 the two fluid chambers 1 and 2 are described bynonlinear compliance elements C1(p1) and C2(p2)respectively, with effective pumping areas A1 and A2.Since the fluid passages are usually made of hard plasticor thin rubber-coated metal and can accommodate onlysmall fluid volumes, they are represented by nonlinearfluid inertances Ii(qi) and Is(qs) and nonlinear fluid

Inner Metal

Outer Metal Sleeve

A1 C1(p1) p1(t)# 1

A2 C2(p2) p2(t)# 2

x(t)

fT(t)

Outer Metal Sleeve

# r

Inertia Track # i

# s

Inner Metal x(t)

Outer Metal Sleeve

2rk

2rc

2rk

2rc

Outer Metal Sleeve

(a) (b)

(( )

i i)ii

R qI q

( )sq t

( )iq t

2sk

2sc

2sk

2sc

( )2rc x

( )2rc x( )

2rk x

( )2rk x

( ), ( )s s s sR q I q

Figure 1. Lumped-parameter nonlinear model of a hydraulic bushing: (a) a fluid system model consisting of two complianceelements (denoted by # 1 and # 2), a long passage (say inertia track, # i) and a short passage with an orifice-like restriction (# s); (b)alternative rubber path formulation that could replace the nonlinear Kelvin–Voigt model of the rubber path in (a).

1328 Proc IMechE Part D: J Automobile Engineering 229(10)



resistances Ri(qi) and Rs(qs) respectively, where q is thevolumetric flow rate. Note that Ri and Rs also includethe minor momentum losses at the tube fittings andbends, etc.27

Under the static condition, the mean force fm trans-mitted to the outer sleeve under a static displacementxm is given by

fm = kr(xm)xm + A2 � A1ð Þpm ð1Þ

where pm is the fluid chamber pressure under staticequilibrium. The dynamic displacement excitation x(t),from the static equilibrium, is applied to the bushingunder a mean load fm and it induces flow through bothpassages. By applying the momentum equation to eachflow passage, the equations obtained are

p1(t)� p2(t)= Ii(qi) _qi(t)+Ri(qi)qi(t) ð2aÞp1(t)� p2(t)= Is(qs) _qs(t)+Rs(qs)qs(t) ð2bÞ

where qi tð Þ and qs tð Þ are the dynamic volumetric flowrates through the long track and the short passagerespectively and p1 tð Þ and p2 tð Þ are the dynamic pres-sures inside the two fluid chambers 1 and 2 respectively.

The application of the continuity equation to twocompliance elements yields

� A1 _x(t)� qi(t)� qs(t)=C1(p1) _p1(t) ð3aÞA2 _x(t)+ qi(t)+ qs(t)=C2(p2) _p2(t) ð3bÞ

When only one passage is used (say, the long inertiatrack), the governing equations are obtained by settingqs(t)=0 and Rs ! ‘. With respect to the total dynamicforce transmitted to the outer sleeve, fT(t) is dividedinto the rubber path (subscript r) and the hydraulicpath (subscript h) according to

fT(t)= fm + fTr(t)+ fTh(t) ð4aÞfTr(t)= kr(x)x(t)+ cr(x) _x(t) ð4bÞfTh(t)=A2p2(t)� A1p1(t) ð4cÞ

Experimental studies with sinusoidaldisplacement excitation

A laboratory device, as described by Chai et al.,20 isutilized to characterize bushing nonlinearities and todevelop a nonlinear mathematical model. As shown inFigure 2(a),20 this device consists of two similar cham-bers filled with water and a customized mid-plate whichaccommodates an external long flow passage and ashort flow passage. The long flow passage, which simu-lates the inertia track, is facilitated via an externalmetal tube of inside diameter di and length li’95di, andits opening is controlled by a ball valve placed at theend of a metal tube. Also, the short flow passage withdiameter ds =3di=2 and length ls’9ds is formed by uti-lizing a needle valve that is embedded in the mid-plate.By adjusting the orifice diameter do (inside the needlevalve), a short passage with different flow restrictionscan be constructed (see the paper by Chai et al.20 for

more details). To simulate typical real-life fluid-filledbushing designs, five configurations of the prototype,as illustrated in Figure 2(b) to (d), are evaluated. First,configuration B1 is investigated when the two hydraulicchambers are connected by only one long passage.Second, configuration B2 is examined when the longpassage is closed and the effective diameter do of arestriction in the short passage is set to do1=2di /3with a fully opened needle valve. Like configurationB2, configuration B3 is formed when the restrictiondiameter is reduced to do2=2di /9. Note that config-uration B3 is designed to simulate an abrupt reductionin the flow area via a needle valve. Finally, the combi-nation of a long passage and a short passage in parallelis examined by using configuration B4 with do= do1and configuration B5 with do= do2.

The steady-state sinusoidal experiments are firstconducted on the above-mentioned configurationsusing a non-resonant elastomer MTS 831.50 testmachine.28 A sinusoidal displacement excitationx(t)=Ax sin (2pOt) is applied to the prototype deviceunder a mean load, where Ax is the zero-to-peak ampli-tude and O is the excitation frequency (Hz). Thedynamic transmitted force fTd(t) and the dynamic pres-sures p1(t) and p2(t) inside the two chambers are mea-sured from 1Hz to 60Hz in 1Hz increments for twoamplitudes X=0.1mm and 1.0mm, where X=2Ax isthe peak-to-peak value. The magnitude Kdj j and theloss angle fK of the dynamic stiffness Kd are defined asthe amplitude ratio and the phase difference betweenfTd(t) and x(t) respectively. Accordingly, define

Kd vð Þ= Kd(v)j j cos fK(v)½ �+i Kd(v)j j sin fK(v)½ �

where i=ffiffiffiffiffiffiffi�1p

and v=2pf is the circular frequency(rad/s).

The measured dynamic stiffness spectra for config-urations B1, B2, B3, and B4 with X=0.1mm and1.0mm are shown in Figure 3. Here Kdj j is normalizedby its static stiffness kstatic. The frequency ratio �O isdefined as the excitation frequency (Hz) divided by areference frequency, at which the loss angle of config-uration B1 achieves its maximum value (about 10Hz).Significant amplitude and frequency dependences areobserved. It can be observed that the loss angle of con-figuration B1 reaches its peak at O�f as the frequency isincreased and then gradually decreases with increasingO, while Kdj j continues to increase until the frequencyO�Kj j, which corresponds to maximum Kdj j, and thensettles down. For configurations B2 and B4, their peakloss angles are at higher frequencies but in a broaderrange than for configuration B1, while the magnitudeand the loss angle of configuration B3 are relativelylow compared with those of other configurations.Furthermore, both Kd(v)j j and fK(v) decrease for ahigher value of X, and their peaks become less abrupt;thus both O�Kj j and O�f both decrease. Accordingly, thenonlinear system parameters must be judiciously identi-fied to model the frequency- and amplitude-dependent

Chai et al. 1329



properties. The methods and procedures for estimatingthe model parameters, including kr and cr, the hydrau-lic chamber compliances C1 and C2, the effective pump-ing areas A1 and A2, the inertances Ii and Is of eachflow passage, and the resistances Ri, Rs, and Ro of eachfluid passage, will be discussed in detail in the followingsection although the focus will be on determination ofthe flow resistance terms.

Identification of the nonlinear systemparameters

The rubber path is approximated by the Kelvin–Voigtmodel, as shown in Figure 1(a), with kr and cr as therubber stiffness and the viscous damping elementsrespectively. To examine the rubber path properties,the dynamic stiffness Kdr of a bushing with fluiddrained is measured for X=0.1mm, 0.5mm, and1.0mm (peak-to-peak value) from 1Hz to 60Hz, as

shown in Figure 4. Observe that Kdrj j decreases as X isincreased, but fr slightly increases. Nevertheless, bothKdrj j and fr gradually increase at higher frequencies.Based on the assumption that Kdr = kr+ivcr from theLaplace transform of fTr(t)= krx(t)+ cr _x(t) (equation(4b)), both kr and cr are first estimated from very-low-frequency responses. Since fr varies by only about 1–2�from 1Hz to 60Hz, cr(X) is assumed to be only ampli-tude dependent. Accordingly, empirical values ofkr v,Xð Þ are estimated by interpolating the measuredKdr data.

In order to estimate C1 or C2, a laboratory experi-ment, as shown in Figure 5(a), is carried out to measurea change DV in the chamber’s internal volume due tothe increase Dp in the pressure. By applying an externalforce to the cylinder piston with cross-sectional areaApiston, DV is calculated as Apistonxpiston, where xpiston isthe piston displacement. Dp is measured by a pressuretransducer. Then the fluid compliance around an oper-ating point o is estimated as C’DV=Dpjo. From the

1

1

4

5

3

2

(a)

(b) (c) (d)

Figure 2. Configurations of the hydraulic bushing constructed for laboratory studies: (a) schematic diagram of the prototype withexternal fluid passages, where 1 is the fluid chamber, 2 is the mid-plate, 3 is a long fluid passage, 4 is a short fluid passage withrestriction, and 5 is a flow control valve; (b) configuration B1 with a long fluid passage of diameter di; (c) configuration B2 with ashort flow passage with restriction of the diameter do1 = 2di /3 and configuration B3 with do2 = 2di/9; (d) configuration B4 with a longflow passage in parallel with a short flow passage with do1 = 2di /3 and configuration B5 with do2 = 2di/9.




Figure 3. Measured dynamic stiffness spectra of the four bushing configurations (a) B1, (b) B2, (c) B3, and (d) B4 for two excitationamplitudes. Key: ———, X = 0.1 mm; – – –, X = 1.0 mm.

PressureTransducer

Cylinder

Piston

Mid-plate

Fluid chamber

External force

piston pistonV AxΔ =

pistonApistonx

(a) (b)

Figure 5. Estimation of the effective chamber compliance. (a) The bench experiment for chamber compliance measurement and (b)the measured Dp versus DV plot for chamber compliance under a mean load. Key: measured; – – –, linear curve-fit.

Figure 4. Dynamic stiffnesses of the rubber path (bushing with the fluid drained) with three excitation amplitudes. Key: .........,X = 0.1 mm; ———, X = 0.5 mm; – – –, X = 1.0 mm.

Chai et al. 1331



measured DV versus Dp data in Figure 5(b), it is seenthat, as the pressure is increased from 0kPa to around180kPa, the variation in DV/Dp is less than 15%.Therefore, C1 and C2 are assumed to be linear, andtheir values are estimated on the basis of a linear poly-nomial fit of measurements. The effective pumpingareas A1 and A2 are examined next by employing astructural finite element code, as shown in Figure 6(a).A static load fm is applied to the top metal part of theelastomeric chamber, and the corresponding deflection�x and the chamber’s internal volume change DV are cal-culated. Then A1 and A2 are estimated using DV=�x. Theresults in Figure 6(b) show that the variations in A1 andA2 are less than 3% when the load is varied from 100Nto 1000N. Thus, A1 and A2 are also assumed to be con-stant values.

The inertances of a long passage and a short passageare calculated as Ii = rli=Ai and Is = rls=As respec-tively by assuming unsteady turbulent flow.29 For avery long passage, fully developed laminar flow mayexist, and then the effective inertance29 is given byIi =4rli=3Ai. However, the transition length of 115dinecessary to achieve a fully developed laminar flow27 islonger than li. In addition, turbulence may be generatedaround the pipe fittings and the sharp entrances orexits. Thus, a constant value Ii = rli=Ai based on geo-metric li and Ai values is employed and no coefficient isapplied. Similarly, the flow restriction in a short pas-sage requires a much lower Reynolds number todevelop laminar flow fully,27 and thus a constantIs = rls=As is assumed for the short passage as well.

Assume that the flow resistance obtained fromsteady flow calculations or measurements is applicablefor an oscillatory flow condition at lower frequen-cies.30,31 A bench experiment is conducted to measurethe steady flow rate q through each fluid passage undera pressure differential Mp. As shown in Figure 7(a), theinlet (embedded in the mid-plate) is connected to a flowsource (city water) by a smooth plastic tube, while theoutlet is open to the ambient atmosphere; the proper-ties of the anti-freeze mixture are close to those of water

at room conditions. The flow source is controlled by aregulator, and the pressure at the inlet is measured witha pressure gauge. By adjusting the flow source and reg-ulator, steady flow in a pressure range 6.9–206.8 kPacan be generated. The steady flow rate q is estimated bymeasuring the accumulated fluid volume (using a

Graduated cylinder

Flow source (city water)

Flow passage

Plastic tube

Flow control device

Pressure gauge

Mid-plate

hFlow source (city water)

Flow passage

Graduated cylinder

Flow control valve

Mid-plate

(b)

(a)

Figure 7. Steady flow experiments (a) over a higher pressuredifferential range and (b) over a lower pressure differentialrange.

Figure 6. Estimation of the effective pumping area: (a) finite element analysis of the fluid chamber; (b) computed DV versus �x plotfrom the finite element model.




graduated cylinder) over a known time duration. Inorder to reduce the test error, multiple measurementsof q with a specific Dp are taken and an averaged valueis obtained. Experimental results for a long passageand a short passage with effective restriction diametersdo1 and do2 are shown in Figure 8. It is evident that therelationship of Dp to q is nonlinear for each passage.Accordingly, the nonlinear resistance Ri(qi) of a longpassage is defined on the basis of a fully developed tur-bulent flow in a smooth circular pipe27 as

Ri(qi)=Dp

qi

= li0:242lim

0:25r0:75

d4:75i

q0:75i

ð5Þ

where an empirical scaling factor li ’ 1.4 is employedto account for momentum losses at the tube bends andfittings, and r and m are the water density and viscosityrespectively. The nonlinear resistance Rs(qs) for a shortpassage of effective diameter do1 is defined by usingthe well-known orifice formula27 with an empiricalfactor ls

Rs(qs)= lsr

2C2dA

2o1

qsj j ð6Þ

where Cd is the discharge coefficient and Ao1 =pd2o1�4

is the geometric cross-sectional area. When the effectivediameter is reduced to do2, the short passage should actmore like a sharp-edge orifice element, and the non-linear resistance is

Ro(qo)=r

2C2dA

2o2

qoj j ð7Þ

where lo! 1.0 and Cd should be about 0.61 for an idealorifice.

The Dp versus q relationship predicted by the aboveequations is compared with the measurements inFigure 8. For a long passage, a curve estimated byequation (7) with Cd=0.61 is also compared with thatfrom equation (5) in Figure 8(a). It is observed thatequation (5) gives a much better representation of themeasured data. Figure 8(b) compares the measured Dpversus q relation for a short passage with equation (6)predictions for three ls values. It is evident that theideal orifice formula (ls=1.0) underestimates the

Figure 8. Steady flow measurements versus theory under high pressure differentials for (a) a long passage (***, measurements;– –, turbulent flow formula; – � –, orifice formula; ———, turbulent flow formula with li), (b) a short passage with a diameter do1

(***, measurements; – –, orifice formula with ls’1:3; ———, ideal orifice formula; – –, orifice formula with ls’1:7), and (c) anorifice-like element (–*–, measurements; ———, ideal orifice formula).

Chai et al. 1333



short-passage resistance. The curve with ls’1:3, whichis a quadratic relation provided by the needle valvesupplier,32 yields a better representation of the mea-surements. To account for minor momentum lossesaround the valve fittings, ls’1:7 is utilized for dynamicanalysis. Finally, Figure 8(c) shows that equation (7)with Cd=0.61 yields a good prediction for the steady-state flow characteristics of the orifice-like element.

Estimation of linearized systemparameters

The nonlinear resistance formulation of the previoussection can be linearized by assuming that the variationaround an operating point is small. For instance, bylinearizing the Dp versus q relations in equations (5) to

(7) at an operating point qop, the linear resistance para-meters for configurations B1, B2, and B3 are29

Ri =d Dpð Þdqi

��qop

= li0:4235lim

0:25r0:75

d4:75i

q0:75op

ð8aÞ

Rs = lsr

C2dA

2o1

qop ð8bÞ

Ro =r

C2dA

2o2

qop ð8cÞ

Accordingly, a simplified bench experiment is con-ducted over the low pressure range, as shown inFigure 7(b). By adjusting the flow rate of the source,

Figure 9. Comparison of the nonlinear model and the linear model for the steady-state harmonic responses of (a) configuration B1 at10 Hz with X = 0.1 mm, (b) configuration B1 at 10 Hz with X = 1.0 mm, (c) configuration B2 at 10 Hz with X = 0.1 mm, and (d)configuration B2 at 10 Hz with X = 1.0 mm. Key: ———, nonlinear model prediction; – � – �, linear model prediction; – – –,measurement.




a steady water level h can be maintained, whereDp =rgh and g is the acceleration due to gravity.Then the linear resistance is estimated as d Dpð Þ=dqaround qop (where qop’10ml=s, 8ml/s, and 3ml/s forRi, Rs, and Ro respectively). Note that this experimentis limited to a lower Dp range owing to the difficultyof maintaining a high water level.

By using the linear resistance terms and assumingthat kr and cr are constant values estimated from thedynamic stiffness Kdr measurements at lower frequen-cies, equations (2) to (4) are linearized as

� A1 _x(t)� qi(t)� qs(t)=C1 _p1(t) ð9aÞA2 _x(t)+ qi(t)+ qs(t)=C2 _p2(t) ð9bÞp1(t)� p2(t)= Ii _qi(t)+Riqi(t) ð9cÞ

p1(t)� p2(t)= Is _qs(t)+Rsqs(t) ð9dÞfTd(t) = fTr(t)+ fTh(t)

= krx(t)+ cr _x(t)½ �+ A2p2(t)� A1p1(t)½ �ð10Þ

where fTd(t) is the dynamic transmitted force. By trans-forming equations (9) and (10) to the Laplace (s)domain and assuming zero initial conditions, the crosspoint dynamic stiffness Kd (s) is defined as

Kd(s)=FTd

X(s)

=a4s

4 +a3s3 +a2s

2 +a1s+a0

b3s3 +b2s

2 +b1s+b0

ð11aÞ

a4 = crC1C2IiIs ð11bÞa3 = krC1C2IiIs + crC1C2 IiRs + IsRið Þ ð11cÞ

Figure 10. Comparison of the nonlinear model and the measurements for the harmonic responses of configuration B4 (a) withX = 0.1 mm at 15 Hz and (b) with X = 1.0 mm at 15 Hz. Key: ———, nonlinear model prediction; – – –, measurements.

Table 1. Fourier amplitudes of fT(t) and Dp(t) for configuration B2 given sinusoidal excitation with two excitation amplitudes. Here,the predicted results are from the nonlinear model.

Fourier amplitudes of B2, with X = 0.1 mm (peak-to-peak value) and O= 10 Hz

O 2O 3O 4O 5O 6O 7O

DP (kPa) Measured 0.99 3.54 3 10–2 0.24 2.15 3 10–2 1.45 3 10–2 4.69 3 10–3 1.44 3 10–2

Predicted 1.11 1.57 3 10–3 0.34 7.81 3 10–4 4.52 3 10–2 3.35 3 10–4 7.16 3 10–3

FT (N) Measured 25.98 0.10 0.52 7.33 3 10–2 4.93 3 10–2 1.74 3 10–2 1.30 3 10–2

Predicted 24.39 2.81 3 10–2 0.53 1.34 3 10–2 7.49 3 10–2 8.21 3 10–3 1.68 3 10–2

Fourier amplitudes of B2, with X = 1.0 mm (peak-to-peak value) and O= 10 Hz

O 2O 3O 4O 5O 6O 7O

DP (kPa) Measured 36.95 0.94 2.32 0.28 0.66 0.13 0.27Predicted 43.33 0.11 2.42 1.59 3 10–2 0.58 1.51 3 10–2 0.20

FT (N) Measured 281.12 0.49 3.31 0.36 1.78 0.47 0.63Predicted 279.32 0.58 4.24 0.21 1.05 0.14 0.39

Chai et al. 1335



a2 = krC1C2ðIiRs + IsRiÞ+ cr½C1C2RiRs

+ ðC1 +C2ÞðIi + IsÞ�ð11dÞ

a1 = kr½C1C2RiRs + ðC1 +C2ÞðIi + IsÞ�+ crðC1 +C2ÞðRi +RsÞ

ð11eÞ

a0 = kr C1 +C2ð Þ Ri +Rsð Þ ð11fÞb3 =C1C2IiIs ð11gÞb2 =C1C2 IiRs + IsRið Þ ð11hÞb1 =C1C2RiRs + C1 +C2ð Þ Ii + Isð Þ ð11iÞb0 = C1 +C2ð Þ Ri +Rsð Þ ð11jÞ

For a bushing with only one long passage (inertiatrack), Kd (s) is simplified by setting Rs!N in equation(11) to give

Kd(s)=a3s

3 + a2s2 + a1s+ a0

b2s2 + b1s+ b0ð12aÞ

a3 = crC1C2Ii ð12bÞa2 =C1C2 krIi + crRið Þ+ A2

2C1 +A21C2

� �Ii ð12cÞ

a1 = cr C1 +C2ð Þ+ A22C1 +A2

1C2 + krC1C2

� �Ri

ð12dÞa0 = kr C1 +C2ð Þ+ A1 � A2ð Þ2 ð12eÞb2 =C1C2Ii ð12fÞb1 =C1C2Ri ð12gÞb0 =C1 +C2 ð12hÞ

The sinusoidal stiffness Kd(v) is obtained by substitut-ing s=iv. Also, harmonic responses in the timedomain can be obtained by using the convolutionmethod or numerical simulations as described by Chaiet al.20 The linear model predictions will be comparedwith the nonlinear model responses and the experimen-tal measurements in the following section.

Validation of the nonlinear model usingsinusoidal measurements

The governing nonlinear equations (equations (2) to(4)) are represented by

_p1(t)= � 1

C1qi(t)+ qs(t)+A1 _x(t)½ � ð13aÞ

_p2(t)=1

C2qi(t)+ qs(t)+A2 _x(t)½ � ð13bÞ

_qi(t)=1

Iip1(t)� p2(t)� Ri(Dp, qi)qi(t)½ � ð13cÞ

_qs(t)=1

Isp1(t)� p2(t)� Rs(Dp, qs)qs(t)½ � ð13dÞ

_fTd(t)=A1

C1+

A2

C2

� �½qi(t)+ qs(t)�+ ½kr(x)

+ g� _x(t)+ cr(x)€x(t)

ð13eÞ

Figure 11. Comparison of the dynamic flow rate spectra predicted by various formulations for configuration B1 (a) with X=0.1mm and(b) with X=1.0mm. Key: ———, numerical integration method; – � – �, multi-term harmonic method; .........., analytical approximation.




where g =A21

�C1 +A2

2

�C2. Based on the identified

system parameters, this nonlinear model is numericallysolved using an explicit fourth- and fifth-order Runge–Kutta method. The steady-state harmonic responses ofconfigurations B1 and B2, as predicted by the nonlinearmodel, are compared with the measured pressure andtransmitted force at 10Hz in Figure 9. Excellent agree-ment between the nonlinear model and the measure-ments of configuration B1 is observed in Figure 9(a)with X=0.1mm. Since the maximum Dp between thetwo fluid chambers is below 22 kPa with X=0.1mm,the model with a linear Ri value also agrees well withmeasurements. When X is increased to 1.0mm, thenonlinear model still yields a very good match withmeasurements, as shown in Figure 9(b). Conversely,the linear model overestimates the measured Dp(t) andfT(t) responses. Since the maximum Dp between the twofluid chambers is now around 90kPa with X=1.0mm,the linear resistance value needs to be re-estimated overthe higher Dp range; the effective Ri with X=1.0mm isfound to be about three times the Ri value withX=0.1mm. For the steady harmonic responses of con-figuration B2, as seen in Figure 9(c) and (d) for 10Hzexcitation, the nonlinear model provides excellentagreement with the measurements at both low excita-tion amplitudes and high excitation amplitudes, espe-cially when compared with Dp(t) predicted by the linearmodel with a constant Rs value.

The fast Fourier transform results of the predictedtime histories and the measured time histories are com-pared in Table 1. Measured Dp(t) signals of configura-tion B2 contain significant odd superharmonics. Forinstance, the third harmonic magnitude is around 24%of the fundamental frequency (10Hz) with X=0.1mm.Overall, both Figure 9 and Table 1 show that the pro-posed nonlinear model is capable of estimating the oddsuperharmonic terms and harmonic distortions inDp(t). The sinusoidal responses for configurations B4and B5 with a long passage and a short passage (in par-allel) are finally examined using the nonlinear model.For instance, the harmonic time histories of configura-tion B4 at 15Hz with X=0.1 and 1.0mm are shown inFigure 10. Good agreement between the nonlinearmodel and the measurements validates the nonlinearformulation for both single-passage configurations anddual-passage configurations.

Construction of the analytical sinusoidalsolution to the nonlinear model

Although numerical methods may yield harmonicresponses for a specific design, they are computation-ally time intensive, especially when responses at manyfrequencies are desired. Moreover, the starting transi-ents in the simulations may lead to inaccurate spectralpredictions as it is difficult to identify when the steady

Figure 12. Comparison of the dynamic flow rate spectra predicted by various formulations for configuration B2 (a) with X = 0.1 mmand (b) with X = 1.0 mm. Key: ———, numerical integration method; – � – �, multi-term harmonic method; ......., analyticalapproximation.

Chai et al. 1337



state begins at a particular frequency. Thus, approximateanalytical solutions are needed to calculate the amplitude-dependent frequency responses efficiently. First, a generalform of R(q) for all passages is assumed as

Ri(qi)= lier

2C2dA

2i

qij j

=R�i qij jð14Þ

where R�i = lier�2C2

dA2i is a resistance coefficient and

lie’ 4.75 is a scaling coefficient, although the r.m.s.error of equation (14) for the Dp calculation is higherthan that of equation (5). Then, on the basis of equa-tions (13a) to (13d), the equations for a single-flow-passage configuration are obtained as

� A1 _x(t)� q(t)=C1 _p1(t) ð15aÞA2 _x(t)+ q(t)=C2 _p2(t) ð15bÞp1(t)� p2(t)= I _q(t)+R� q(t)j jq(t) ð15cÞ

where I and R� are the inertance and the resistancerespectively of each passage; for the short-flow-passageconfigurations, R�s = lsr

�2C2

dA2o1 and R�o = r

�2C2

dA2o2.

By eliminating p1(t) and p2(t), a single-degree-of-free-dom nonlinear model in terms of x(t) and q(t) is derivedas

kq(t)+2R�q(t) _q(t) sgn q(t)½ �+ I€q(t)=ax _x(t) ð16Þ

where ax = � A1=C1 +A2=C2ð Þ and k=1=C1

+1=C2.The harmonic displacement excitation

x(t)=Ax sin (vt� f) is applied. The flow rate responseis approximated by only one harmonic term, namelyq(t)=Q1 sin (vt), where Q1 is the amplitude of the fun-damental term and f is the phase difference betweenx(t) and q(t). By substituting these expressions intoequation (16) and applying trigonometry, the equationobtained is

Figure 13. Displacement excitation profiles and typical force measurements for the transient experiments. (a) An ideal step-up andstep-down pulse, (b) smoothened step-up excitation implemented in the experiments and (c) normalized step responses for B1configuration. Key: ——— , first step-up response with Ae = Axt; – – –, step-down response with Ae = 2Axt; ........., second step-upresponse with Ae = Axt.




kQ1 sin (vt)+R�Q21v sin (2vt) sgn sin (vt)½ �

� Iv2Q1 sin (vt)

=axAxv cos (vt) cosf+ sin (vt) sinu½ �ð17Þ

where sgn is the sign function.The nonlinear term in equation (17) is expanded

using the Fourier series as

sin (2vt)sgn sin (vt)½ �=X‘

n=1

an cos (nvt) ð18aÞ

an =1

p

1

2+ n+

1

2� n

� �1� (� 1)n½ � ð18bÞ

The coefficient a1 =8=3p when n=1. Substitutingequation (18) into equation (17) and then equating the

coefficients of sin vtð Þ and cos vtð Þ yield the tworelations

kQ1 � Iv2Q1 =axAxv sinf ð19aÞ

8R�vQ21

3p=axAxv cosf ð19bÞ

By solving the above, the amplitude Q1 and the phase f

at the frequency v are obtained as

Q21 =� k� Iv2� �2

+

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik� Iv2ð Þ4 +4 8axAxR�v2=3pð Þ2

q2 8R�vð Þ2

.3pð Þ2

ð20aÞ

Figure 15. Comparison of the predicted forces and the measured forces for configuration B1 in the transient experiments for (a) astep-up response with Ae = Axt and (b) a step-down response with Ae = 2Axt. Key: ———, nonlinear model; – � – �, linear model; – – –,measurements.

Figure 14. The predicted and the measured forces transmitted through the rubber path in a step-like experiment. Key: .........,measured; ———, predicted by the original Kelvin–Voigt model of Figure 1(a); – – –, predicted by the modified model of Figure 1(b).

Chai et al. 1339



f= arctan

ffiffiffi2p

k� Iv2� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� k� Iv2ð Þ2 +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik� Iv2ð Þ4 +4 8axAxR�v2=3pð Þ2

qr2664

3775

ð20bÞ

Note that the k� Iv2 term mainly controls the reso-nance frequency. An increase in k� Iv2 causes the

peaks to be more abrupt and to shift to the left. Also,the 8R�v=3p term dictates the damping; it is observedthat the peak amplitude decreases when R� is increased.Next, the sinusoidal volume flow rate is defined as

Q(v)=Q1 cosf+iQ1 sinf ð21Þ

Then relations between q(t), p1(t), p2(t) and fT(t) fromequations (13e), (15a), and (15b) are employed, theLaplace transform is applied and s=iv is substituted

Figure 16. Comparison of the predicted forces and the measured forces for configuration B2 in the transient experiments for (a) astep-up response with Ae = Axt and (b) a step-down response with Ae = 2Axt. Key: ———, nonlinear model; –� –�, linear model; – – –,measurements.

T ( ) (N)f t

(s)t

T ( ) (N)f t

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

900

950

1000

1050

1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7

-200

-100

0

100

(a)

(b)

Figure 17. Comparison of the predicted forces and the measured forces for configuration B4 for (a) a step-up response withAe = Axt and (b) a step-down response with Ae = 2Axt. Key: ———, nonlinear model; – – –, measurements.




to yield, at v (for a given excitation amplitude X), thedynamic stiffness and pressure amplitude expressions

Kd(v)= kr(v,X)+ ivcr(v,X)+ g +A1

C1+

A2

C2

� �Q(v)

iAxv

ð22aÞ

P1(v)= � AxA1

C1�Q(v)

iC1vð22bÞ

P2(v)=AxA2

C2+

Q(v)

iC1vð22cÞ

Although Kd(v) can be approximated by Q(v), theeffect of kr will still induce a difference between thenumerical method and the analytical method. Thus, theQ(v) values (peak to peak) are calculated by usingequation (20) from 1Hz to 60Hz, with X=0.1mm and1.0mm, and then compared with the numerical integra-tion results in Figure 11. Both methods yield almost thesame results for the long passage (configuration B1),although the peak magnitude predicted by equation(20) with X=1.0mm is slightly higher. However, somediscrepancies are observed for configuration B2, asshown in Figure 12, especially with the X=1.0mmcase. This can be explained by referring to Figure 9 andTable 1. Since configuration B2 exhibits more signifi-cant superharmonics than configuration B1 does, theusage of only one harmonic term is not sufficient.Nevertheless, the analytical solution still provides a verygood approximation.

Semi-analytical solution using a multi-term harmonic balance method

A multi-term harmonic balance method (MHBM)33,34

is utilized next to calculate the steady-state response ofthe nonlinear equation (16), given a sinusoidal excita-tion of x(t)=Ax sin (vt). First, using a truncatedFourier series with N odd harmonics, the solution forq(t) is written

q(t)=XNn=1

fa2n+1 sin½(2n+1)vt�+ b2n+1

cos½(2n+1)vt�gð23Þ

where n is a positive integer. The even harmonics areneglected because the Fourier transforms of the mea-sured Dp(t) and fT(t) signals show that the odd harmo-nics are dominant, as also suggested by equation (18).By defining u=vt, q0(u)=dq=du, q00(u)=d2q

�du2,

and x0(u)=dx=du, the derivatives of q(t) and x(t) are

obtained as _q(t)=vq0, €q(t)=v2q00, and _x(t)=vx0.Then equation (16) is converted into

kq(u)+2R�q(u)vq0(u)sgn qð Þ+ Iv2q00(u)=axvx0(u)

ð24Þ

Suppose that the discrete u signal consists of L datapoints, say from u1to uL. Equation (23) is discretized asq=Ca where a is the vector of Fourier coefficients,and C is the discrete Fourier transform matrix definedas

C=

sin u1ð Þ cos u1ð Þ sin 3u1ð Þ cos 3u1ð Þ � � � sin (2N+1)u1½ � cos (2N+1)u1½ �sin u2ð Þ cos u2ð Þ sin 3u2ð Þ cos 3u2ð Þ � � � sin (2N+1)u2½ � cos (2N+1)u2½ �

..

. ... ..

. ... . .

. ... ..

.

sin uLð Þ cos uLð Þ sin 3uLð Þ cos 3uLð Þ � � � sin (2N+1)uL½ � cos (2N+1)uL½ �

26664

37775 ð25Þ

By using a hyperbolic tangent function to approximatethe discontinuous sign function, the nonlinear term isexpressed using the truncated Fourier series asqq0 tanh sqð Þ=Cb where b is the Fourier coefficientvector of the nonlinear function and s is a regularizing(smoothening) factor. The derivatives of q with respectto u are obtained as q0=CDa and q00=CD2a, whereD is the differential operator matrix defined as

D=

0 �11 0

0 �33 0

. ..

0 �(2N+1)2N+1 0

2666666664

3777777775ð26Þ

By substituting the above Fourier series expansionsand derivatives of q into equation (24), the discretizedequations in the time domain are given in the matrixform

kCa+2R�vCb+ Iv2CD2a=axvCDX ð27Þ

where X= Ax 0 � � � 0½ �T is the input vector corre-sponding to the displacement excitationx(t)=Ax sin (vt). The vector a can be found by balan-cing the right-hand side and the left-hand side of theequation. First, by pre-multiplying equation (27) withthe pseudo-inverse of C as C+ = CTC

� ��1CT, the

residue function R is defined as

R= ka+2R�vb+ Iv2D2a� axvDX ð28Þ

Then the correct solution of a is calculated when R isminimized. The Broyden method,35 which requires onlythe calculation of the initial Jacobian matrix, is appliedto minimize R. The Jacobian matrix J is calculated as

J=∂R

∂a

= k+2R�v∂b

∂a+ Iv2D2

ð29aÞ

Chai et al. 1341



∂b

∂a=C+ ∂½qq0 tanh(sq)�

∂a

=C+

(q0 tanh (sq)

∂q

∂a+ q tanh (sq)

∂q0

∂a

+ qq0s½1� tanh2 (sq)� ∂q∂a

) ð29bÞ

Note that ∂q=∂a=C and ∂q0=∂a=CD. The initialJacobian matrix J0 is obtained by substituting a0 intoequation (29), and a first approximation of a is calcu-lated using the Newton method35 as a1 = a0 � J�10 R0.The Jacobian matrix for the mth iteration is updated as

Jm = Jm�1 +UWT ð30aÞW= am � am�1 ð30bÞ

W=Rm � Rm�1 � Jm�1W

WTWð30cÞ

The inverse of the Jacobian matrix is calculated byusing the Sherman–Morrison formula35 J�1m = ½I�J�1m�1UWT=ð1+WTJ�1m�1UÞ�J�1m�1. Then, a is updated ateach step with am+1 = am � J�1m Rm.

The iterations continue until a converges to a satis-factory solution when Rk k\ e, i.e. the L2 norm of vec-tor R is less than a very small number e.

Q(v) and fQ(v), which are calculated by the multi-term harmonic method for N=3 with X=0.1mm and1.0mm, are shown in Figures 11 and 12 from 1Hz to60Hz. The difference between the MHBM solutionsand the numerical integrations is negligible for bothconfiguration B1 and configuration B2, but there areminor discrepancies for the response of configurationB2 with X=1.0mm. Unlike the analytical solution ofthe previous section, the jQ(v)j spectra given by theMHBM are much closer to the numerical results. Thephase angle spectra of the three methods are very closesince they are calculated on the basis of the phase differ-ence between the fundamental harmonic terms of x(t)and q(t). Overall, both single-term solutions and multi-term solutions approximate the harmonic responseswell. Moreover, Dp(t) and fT(t) can be easily calculated,given q(t) and equation (13).

Study of the transient responses usingexperiments and the nonlinear model

A step-up and step-down displacement excitation, asshown in Figure 13(a), shows the ideal input signalwhich can be expressed in terms of the unit step functionu(t) as xM(t)=Axto + Axtuðt � t1Þ � 2Axtuðt� t2Þ+Axtuðt� t3Þ

Since it is extremely difficult to generate an ideal stepsignal given the practical limitations of an actuator,only smoothened step events are realized in the experi-ment, such as the step-up-like excitation shown inFigure 13(b). Since the step-down excitation has twicethe step amplitude Ae of the step-up excitation, themeasured fT(t) responses are normalized by Ae and then

superposed in the same plot for easier comparison.Figure 13(c) shows the normalized force response ofconfiguration B1. Comparison shows that the two step-up responses (with Ae=Axt) exhibit similar transients,although their asymptotic values differ according to theinitial loading levels; this also suggests that kr and cr arehighly dependent on fm. However, the normalized step-down response (with Ae=2Axt) has a longer oscillationperiod and lower oscillation peak value than the step-up event does (with Ae=Axt). This again implies thatthe dynamic responses depend highly on the step ampli-tude, and thus the proposed nonlinear model is appliednext to analyze the transient force measurements.

Since the Kelvin–Voigt model cannot represent theexponential decay term in fTr(t), a Maxwell element(with stiffness ks and damping cs) is added to the rub-ber path model, as shown in Figure 1(b). Load-depen-dent kr(x) is estimated from the asymptotic value offTr(t), while ks and cs are identified by a curve-fit proce-dure of the measurements. The force response predictedby the modified rubber path model, given a step-likeexcitation, is compared with the measured fTr(t) inFigure 14. It can be observed that the modified model(with Maxwell element) yields much better agreementwith the measurement than does the original Kelvin–Voigt model.

The nonlinear model predictions of the step-upresponse and the step-down response of configurationB1 are compared in Figure 15 with the transmittedforce measurements; a linear model with an effectiveresistance R_ei is also compared. Here R_ei is estimatedon the basis of the settling time, the oscillation period,and the peak values of the step response at each ampli-tude.20 Similarly, the effective resistances R_es and R_eo

are estimated from the step responses of configurationsB2 and B3 respectively.20 As shown in Figure 15, thepredicted fT(t) matches the measured forces well, and ityields much better agreement of the oscillatory peaksand periods than the linear model does. Similarly, thepredicted forces and the measured forces of configura-tion B2 are compared in Figure 16, and excellentagreement is observed for both step-up responses andstep-down responses. In particular, the nonlinear modelrepresents the higher-frequency small-amplitude oscil-lations well in the fT(t) responses of configuration B2,which are neglected in the linear model predictions.Furthermore, configurations with a long passage and ashort passage (in parallel) are investigated. Figure 17compares the nonlinear model and the measurementsof configuration B4, and excellent predictions for bothstep-up responses and step-down responses areobserved. These comparisons confirm that the non-linear model represents the nonlinearities well in thetransient responses for both single-passage configura-tions and dual-passage bushing configurations. Somediscrepancies between the predictions and the measure-ments exist, especially for higher Dp conditions,although the error is less than 20%. These may becaused by a variation in the assembly and disassembly




process of the prototype device and by the Dp measure-ment errors when q is high. Nevertheless, the effect ofnonlinearities is more evident in the transient responseswhen compared with the sinusoidal excitations becausea step response induces a broader operating range. Forinstance, Dp is around 220kPa in the step-downresponses of configuration B1 but only up to 22kPaand 90 kPa for X=0.1mm and 1.0mm respectively insteady-state harmonic responses. Thus the linear modelwith constant effective parameters is incapable of accu-rately predicting the step response although it is appro-priate for steady-state harmonic responses over anarrow operating range.

Conclusion

Unlike previous literature that has focused on only thelinear system analysis of fluid-filled bushings, this articlehas developed a refined nonlinear model for a bushingwith a long passage and a short passage and identifiedkey nonlinear parameters using theory and experiments.The system nonlinearities are found to be dominated bythe fluid resistance terms, and thus these are defined onthe basis of the turbulent flow formulation. Five config-urations of the laboratory prototype device are examinedby using the nonlinear models. The nonlinear model isfirst solved by using the numerical integration method,and its predictions match both the steady-state harmonicmeasurements and the transient measurements well foreither a single-passage configuration or a multi-passage configuration. Finally, this article obtainedan analytical approximation and a semi-analyticalsolution of the nonlinear flow rate model by using theharmonic balance method, given a sinusoidal displa-cement excitation. These solutions may be utilized toestimate the amplitude-dependent responses effi-ciently in the frequency domain. Such nonlinear mod-els should lead to better industrial design and tuningprocesses. Future work is needed to characterize theamplitude dependence of the rubber path and toexamine the bushings in the context of specific vehiclesuspension systems.

Acknowledgement

The authors thank C Gagliano and PC Detty for theirhelp with experimental studies.

Declaration of conflict of interest

The authors declare that there is no conflict of interest.

Funding

This work was supported by the Transportation ResearchCenter Inc., Honda R&D Americas, Inc., and YUSACorporation of the Smart Vehicle Concepts Center36 andthe National Science Foundation, Industry & UniversityCooperative Research Program37.

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Appendix 1

Notation

a, b equation coefficientsa, b vectors of Fourier coefficientsA area or displacement excitation amplitudec damping coefficient or nonlinear

resistance coefficientC chamber complianced diameterD differential operator matrixf, F transmitted forceg acceleration due to gravityh liquid level in the bench experimenti =

ffiffiffiffiffiffiffi�1p

I fluid inertance

J Jacobian matrixk stiffnessK dynamic stiffnessl length of fluid passageL number of data pointsn indexN number of harmonicsp pressure in the time domainP dynamic pressure in the s or v domainq flow rate in the time domainQ dynamic flow rate in the s or v domainR fluid resistanceR residue vectors Laplace domain variablet timeV volumex displacement excitation in the time

domainX peak-to-peak value of displacement

excitation amplitudeX excitation vector

a, b equation coefficientsg static stiffness of the hydraulic chambersu angular displacementl scaling factor for the nonlinear resistance

modelm fluid viscosityr densityf phase angleC discrete Fourier transform matrixv circular frequency (rad/s)O frequency (Hz)

Superscripts

* peak magnitude or peak phase angle+ pseudo-inverse� normalized value

Subscripts

d dynamice effective or equivalenth hydraulic pathi long flow passage (inertia track)K dynamic stiffnessjKj magnitude of the dynamic stiffnessm mean or static valueo orifice-like elementr rubber or rubber paths short flow passage with restrictionT transmittedx displacement excitationf phase angle1 chamber 12 chamber 2



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