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Journal of Sound and Vibration

Journal of Sound and Vibration 330 (2011) 4429–4446

0022-46

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jsvi

Estimation of interfacial forces in a multi-degree of freedomisolation system using a dynamic load sensing mount andquasi-linear models

Jong-Yun Yoon, Rajendra Singh n

Acoustics and Dynamics Laboratory, Smart Vehicle Concepts Center, Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210, USA

a r t i c l e i n f o

Article history:

Received 15 July 2010

Received in revised form

25 April 2011

Accepted 25 April 2011

Handling Editor: H. Ouyangpowertrain mounts (including a dynamic load sensing hydraulic mount), a sub-frame,

Available online 13 May 2011

0X/$ - see front matter & 2011 Elsevier Ltd.

016/j.jsv.2011.04.033

esponding author. Tel.: þ1 614 292 9044; fa

ail address: [email protected] (R. Singh).

a b s t r a c t

A new indirect measurement concept is developed to estimate interfacial dynamic

forces by employing the hydraulic mount as a dynamic force sensor. The proposed

method utilizes a combination of mathematical models and operating motion and/or

pressure measurements. A laboratory experiment consisting of a powertrain, three

and four bushings is then constructed to verify the proof-of-concept. Quasi-linear fluid

and mechanical system models of the experiment are proposed and evaluated in terms

of transfer functions and forced sinusoidal responses. The lower chamber pressure in

the hydraulic mount is estimated since it is not available from measurements. This

leads to an improved estimation of the effective rubber and hydraulic path parameters

with spectrally varying and amplitude-sensitive properties up to 50 Hz. Finally, the

reverse path spectral method is employed to predict interfacial forces at both ends of

the mount by using measured motions and upper chamber pressure signals. Overall, the

proposed quasi-linear fluid system model yields better indirect estimates of forces from

the measured responses when compared with direct force measurements, through a

simpler mechanical system model provides some insights. This work also advances

prior component and transfer path type studies by providing an improved multi-degree

of freedom system perspective.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Precise knowledge of the dynamic forces at sub-system junctions or interfaces is of vital interest in the dynamic andvibro-acoustic design of mechanical systems, vehicles, buildings, and power plants. In general, it is difficult to install forcetransducers at sub-system junctions without altering interfacial or boundary conditions unless there is a mobilitymismatch. Thus, indirect measurement or force reconstruction methods must be adopted to estimate dynamic forces[1–9]. This has been the subject of several recent articles. For instance, Inoue et al. [1] and Gunduz et al. [2] have suggestednew or improved transfer path methods to estimate forces in parallel structural paths of a discrete vibratory system. Yapand Gibbs [3] examined the forces at the machine–receiver interface by using the mobility method. Leclere et al. [4]assessed the internal loads on bearings with an inverse transfer function method. Carne et al. [5] utilized frequencyresponse function data to indirectly estimate the input force. Jacquelin et al. [6] employed a deconvolution technique to

All rights reserved.

x: þ1 614 292 3163.

www.elsevier.com/locate/jsvi

dx.doi.org/10.1016/j.jsv.2011.04.033

mailto:[email protected]

dx.doi.org/10.1016/j.jsv.2011.04.033

J.Y. Yoon, R. Singh / Journal of Sound and Vibration 330 (2011) 4429–44464430

reconstruct the dynamic force. Tao et al. [7] identified the excitation force in the center of an engine using the velocityamplitude and phase at the mounting points. Liu and Shepard [8] compared the truncated singular value decompositionand the Tikhonov filter approaches used to enhance the inverse process. Lin and Chen [9] identified the contact stiffnessand damping properties of mechanical interfaces. However, most of the available indirect or inverse force estimationmethods are valid only for linear time-invariant systems since they employ transfer functions or similar concepts.

In this article, an inherently nonlinear hydraulic mount will be embedded into a multi-degree of freedom system andthen utilized as a load sensing device to estimate interfacial forces (at both ends of the mount). In recent work from thecomponent perspective [10,11], forces that are transmitted by a hydraulic mount to a rigid base have been successfullyestimated by using linear, quasi-linear, and nonlinear models. In this paper, a system perspective is adopted and used toconstruct a laboratory experiment to examine the proof-of-concept. The scope of this article, however, is limited to theharmonic interfacial forces (in frequency or time domain) using measured or calculated motions and/or internal pressuresignals. Recent articles by Gunduz et al. [2] and Yoon and Singh [10,11] provide a comprehensive review of the relevantliterature on the identification of interfacial forces and hydraulic engine mount models, respectively. Additional reviewswill be reported as the material is further developed.

2. Problem formulation

Underlying issues can be conceptually illustrated by the generic source–path–receiver system of Fig. 1. This is amodified version of the systems analyzed by Inoue et al. [1] and Gunduz et al. [2]. Here, multiple linear and nonlinearisolators (paths) are shown. Path I has two separate force paths: one path is given by a linear spring (kES1) and viscousdamper (cES1); and the other path incorporates mass m1 which is connected to the source and receiver by two linearsprings (kES11 and kES12) and dampers (cES11 and cES12). Path II includes a nonlinear spring (kES2) and damper (cES2), alongwith a nonlinear fluid path designated by pu(t). A linear spring and damper (designated as kEG and cEG) also supportthe source. The receiver is connected to the ground by four linear springs (kSG1, kSG2, kSG3 and kSG4) and viscous dampers

Fig. 1. Schematic of nonlinear isolation system in the context of source, path and receiver network. Both paths are assumed to include hydraulic mounts,

and thus have parallel force paths. In particular, path II is assumed to possess spectrally varying and amplitude-sensitive properties.

J.Y. Yoon, R. Singh / Journal of Sound and Vibration 330 (2011) 4429–4446 4431

(cSG1, cSG2, cSG3 and cSG4). For the sake of illustration, assume a linear time-invariant (LTI) system, and define interfacialforces in the time domain for path II as follows:

fIT ðtÞ ¼ cES2_xðtÞþkES2xðtÞþArpuðtÞ: (1)

fIBðtÞ ¼�cES2_xðtÞ�kES2xðtÞ�ArpuðtÞ: (2)

Here, fIT(t) and fIB(t) are the interfacial forces on the source (subscript IT) and receiver (subscript IB) sides, respectively;Ar is the effective piston area in the fluid path; xE(t) and xS(t) are the displacements of the source and receiver, respectively;and x(t)¼xE(t)–xS(t) is the relative displacement. By transforming Eqs. (1) and (2) into the Laplace domain (s) and assumingthat the initial conditions are equal to zero, the path forces are:

FIT ðsÞ ¼ ðcES2sþkES2ÞXðsÞþArPuðsÞ: (3)

FIBðsÞ ¼�ðcES2sþkES2ÞXðsÞ�ArPuðsÞ: (4)

When the above equations are compared, the top and bottom forces (in path II) are identical except for the sign as longas there are no inertial elements in the fluid path. This is obviously not the case in many practical devices [12]. In order toestimate interfacial forces on both source and receiver sides, one must recognize and resolve some difficulties. First, theabove equations require a precise knowledge of in-situ parameters such as nonlinear stiffness kES2, damping cES2 (in therubber force path), and effective piston area Ar (in the hydraulic force path) [13,14]. Second, forces could be estimated byusing measured motions and/or pressure signals, but then effective (dynamic) stiffness and damping parameters must beknown a priori [15]. The latter poses a special difficulty for both elastomeric and hydraulic isolators [13]. For instance,hydraulic engine mounts exhibit spectrally varying and amplitude-sensitive parameters [16].

The specific objectives of this research are as follows: (1) construct and instrument a laboratory experimentcorresponding to Fig. 1; one load sensing hydraulic mount will be embedded. The system is excited in the verticaldirection by a steady-state sinusoidal force fE(t) of frequency oo, and motions in other directions are ignored; (2) conductexperiments under sinusoidal excitation and measure dynamic accelerations (at different points in the system), fluidpressure (in the top chamber of the load sending mount), and forces (at selected interfaces); (3) develop 2 and 3 degree offreedom (dof) linear and quasi-linear models (with spectrally varying and amplitude-sensitive parameters as suggested inprior component studies [10,11,16]); this would include the determination of the effective parameters, including upperand lower chamber compliances and (4) estimate the interfacial forces at both ends of the load sensing hydraulic mount byemploying mechanical and fluid models of the load sensing mount and compare them with direct force measurements inthe frequency and time domains.

3. Experiment with a dynamic load sensing hydraulic mount

Fig. 2 illustrates the laboratory experimental setup with powertrain (assembly of engine and transmission) and sub-frame; the setup is based on the generic multi-degree of freedom isolation system of Fig. 1. The powertrain is connected tothe sub-frame by two hydraulic mounts and is supported by a grounded third rubber mount. The sub-frame is supportedby four identical elastomeric bushings. The experiment is excited by an electrodynamic shaker located on the powertrain.A signal generator is used to generate multiple excitations. One piezoelectric force transducer located between the shakerand the powertrain measures the excitation force fE(t); it is also used as a reference signal. Here, the excitation force isexpressed as fEðtÞ ¼ Re½ ~F E eioot�, where ~F E ¼ FE eijF is the complex valued excitation amplitude, FE is the amplitude of force,

Fig. 2. Laboratory experimental setup with load sensing hydraulic mount: (a) powertrain and sub-frame assembly with instrumentation and (b) front

view laboratory experiment with focus on vertical motions.


jF is the phase of excitation force fE(t), oo is the excitation (fundamental) frequency (rad s�1), and Re[] is the real valueoperator; a tilde over a symbol implies that is complex valued. The scope of the experiment is limited to 50 Hz with fivedifferent excitation amplitudes (1, 10, 50, 100, and 150 N).

The following sensors are employed as illustrated in Fig. 2: two piezoelectric accelerometers located on top of twohydraulic mounts and two accelerometers placed on the sub-frame near the hydraulic mounts; two piezoelectric forcetransducers located between the hydraulic mounts and the sub-frame for directly measuring the interfacial forces; apiezoelectric force transducer located on top of the load sensing hydraulic mount to directly measure the interfacial forceson the source side; and one piezoelectric pressure transducer within the upper chamber of the load sensinghydraulic mount.

4. Development of a linear time-invariant (LTI) mechanical system model

Analogous 2- and 3dof mechanical system models of the experiment are shown in Fig. 3. Voigt models are used todescribe two hydraulic mounts, one rubber mount, and bushings. Here, the following symbols are designated: mE, mass ofthe powertrain; mS, mass of the sub-frame; cHF, viscous damping coefficient of the hydraulic mount at the front side; cHR,viscous damping coefficient of the dynamic load sensing hydraulic mount; mie, effective mass of the inertia track columnwhich is estimated from the fluid system model of the hydraulic mount; _xieðtÞ, effective velocity of the inertia track fluid;ku and kl, effective stiffness of the upper and lower chambers, respectively; kRr and cRr, rubber stiffness and viscousdamping coefficients of the dynamic load sensing hydraulic mount, respectively; cR, viscous damping coefficient of therubber mount; and kB and cB, stiffness and viscous damping coefficients of the bushings, respectively. The dynamic loadsensing hydraulic mount is described only by stiffness and viscous damping elements in the 2dof model. On the otherhand, the 3dof model includes effective mechanical properties (including fluid inertia) as derived from the fluidparameters [12,17,18].

Assume that both systems of Fig. 3 are linear time-invariant (LTI). First, the 2dof model (with subscript 2) is describedbelow. Here, x2(t) and f2(t) are the displacement and excitation force vectors, respectively, and M2, K2, and C2 are the mass,stiffness, and damping matrices.

M2 €x2ðtÞþC2 _x2ðtÞþK2x2ðtÞ ¼ f2ðtÞ, (5)

M2 ¼mE 0

0 mS

" #, K2 ¼

kRþkES2 �kES2

�kES2 kES2þkB

" #, C2 ¼

cRþcES2 �cES2

�cES2 cES2þcB

" #, (6a2c)

x2ðtÞ ¼xEðtÞ

xSðtÞ

" #, f2ðtÞ ¼

Re½ ~F E eioot�

0

" #: (7a,b)

Here, kES2¼kHFþkHR; cES2¼cHFþcHR. Similarly, the 3dof model (with subscript 3) is derived.

M3 €x3ðtÞþC3 _x3ðtÞþK3x3ðtÞ ¼ f3ðtÞ, (8)

Fig. 3. Alternate versions of the analogous mechanical system models: (a) schematic of the 2dof model and (b) schematic of the 3dof model with fluid

inertia in the mount.


M3 ¼

mE 0 0

0 mie 0

0 0 mS

264

375, K3 ¼

kRþkuþkES3 �ku �kES3

�ku kuþkl �kl

�kES3 �kl kES3þklþkB

264

375,

C3 ¼

cRþcES3 0 �cES3

0 cie 0

�cES3 0 cES3þcB

264

375, (9a2c)

x3ðtÞ ¼

xEðtÞ

xieðtÞ

xSðtÞ

264

375, f3ðtÞ ¼

Re½ ~F E eioot�

0

0

264

375, (10a,b)

where kES3¼kHFþkRr; cES3¼cHFþcRr.

5. Development of a quasi-linear (QL) mechanical system model

An interrelationship between the interfacial forces ~F IT ðoÞ and ~F IBðoÞ and internal pressure ~PuðoÞ can be observed fromthe spectral contents in Fig. 4 with respect to the fundamental and super-harmonic terms. Thus, a quasi-linear (QL) modelwith effective parameters must be developed to successfully predict the interfacial forces. Two analogous mechanicalsystem models, as illustrated in Fig. 3, will be employed to develop such models. Fig. 3(a) shows that the 2dof modelincludes the Voigt formulation in terms of kHR and cHR elements to represent the load sensing hydraulic mount. However,the 3dof model includes effective mechanical parameters ku, kl, and cie that effectively describe the internal fluid system.

First, the 2dof model (with subscript 2) is used to define the interfacial forces on both the top and bottom ends of theload sensing hydraulic mount as follows:

fIT2ðtÞ ¼ cHR_xðtÞþkHRxðtÞ; (11)

fIB2ðtÞ ¼�cHR_xðtÞ�kHRxðtÞ: (12)

Here, x(t)¼xE(t)–xS(t) is the relative displacement between xE and xS. By transforming Eqs. (11) and (12) into the Laplacedomain (s) with the assumption that the initial conditions are equal to zero, the interfacial forces are:

FIT2ðsÞ ¼ ðcHRsþkHRÞXðsÞ, (13)

FIB2ðsÞ ¼ �ðcHRsþkHRÞXðsÞ: (14)

Fig. 4. Comparison of spectral contents given fEðtÞ ¼ Re½ ~F E eioo t � at oo/2p¼23 Hz with 9 ~F E9¼ 100N.


Likewise, the interfacial forces with the 3dof model (with subscript 3) are calculated as follows:

fIT3ðtÞ ¼ cHR_xðtÞþkHRxðtÞþkuxuðtÞ, (15)

fIB3ðtÞ ¼�cHR_xðtÞ�kHRxðtÞ�klxlðtÞ, (16)

FIT3ðsÞ ¼ ðcHRsþkHRÞXðsÞþkuXuðsÞ, (17)

FIB3ðsÞ ¼ �ðcHRsþkHRÞXðsÞ�klXlðsÞ: (18)

Here, the relative displacements are defined as: xu(t)¼xE(t)–xie(t) and xl(t)¼xie(t)–xS(t). From Eqs. (15)–(18), the 3dofmodel shows that forces fIT(t) and fIB(t) differ by kuxu(t) and klxl(t), which signify different relative displacements andeffective stiffness values. Thus, the fluid element displacement xie(t) must be predicted to estimate fIT(t) and fIB(t). Therelationship between xie(t) and other known variables such as measured signals (with subscript M), including displace-ments xEM(t) and xSM(t) and force fEM(t), can be derived from Eqs. (5)–(10). Transforming Eqs. (5)–(10) into the Laplacedomain and ignoring initial conditions:

½mEs2þðcRþcHFþcRrÞsþðkRþkHFþkRrþkuÞ�XEðsÞ�½ðcHFþcRrÞsþðkHFþkRrÞ�XSðsÞ�kuXieðsÞ ¼ FEðsÞ, (19)

½mSs2þðcHFþcRrþcBÞsþðkHFþkRrþklþkBÞ�XSðsÞ�½ðcHFþcRrÞsþðkHFþkRrÞ�XEðsÞ�klXieðsÞ ¼ 0: (20)

From Eqs. (17)–(20), the interfacial forces FIT(s) and FIB(s) are derived as follows:

FIT3ðsÞ ¼ FEðsÞþðcHFsþkHFÞXSðsÞ�½mEs2þðcRþcHFÞsþðkRþkHF Þ�XEðsÞ, (21)

FIB3ðsÞ ¼ ðcHFsþkHF ÞXEðsÞ�½mEs2þðcHFþcBÞsþðkHFþkBÞ�XSðsÞ: (22)

Next, the interfacial forces in the frequency domain can be calculated by replacing s with io in the above equations.Thus, interfacial forces with the 2dof model in the frequency domain are as follows:

~F IT2ðoÞ ¼ ðiocHRþkHRÞXðoÞ, (23)

~F IB2ðoÞ ¼ �ðiocHRþkHRÞXðoÞ: (24)

Likewise, the interfacial forces with the 3dof model are described below:

~F IT3ðoÞ ¼ ~F EðoÞþðiocHRþkHRÞ~X SðoÞ�½�o2mEþ ioðcRþcHFÞþðkRþkHFÞ�

~X EðoÞ, (25)

~F IB3ðoÞ ¼ ðiocHFþkHFÞ~X EðoÞ�½�o2mEþ ioðcHFþcBÞþðkHFþkBÞ�

~X SðoÞ: (26)

Here, the displacements ~X EðoÞ and ~X SðoÞ are calculated directly from the measured accelerations ~CEMðoÞ and ~CSMðoÞ by therelationships ~X EMðoÞ ¼� ~CEMðoÞ=o2 and ~X SMðoÞ ¼� ~CSMðoÞ=o2 by avoiding double integration in the time domain. Thus, themeasured relative displacement ~XMðoÞ ¼ ~X EMðoÞ� ~X SMðoÞ is calculated by ~XMðoÞ ¼� ~CEMðoÞ=o2þ ~CSMðoÞ=o2. Fig. 5 showsthe measured ~XMðoÞ in the frequency domain under the sinusoidal excitation force fEðtÞ ¼ Re½ ~F E eioot� with 9 ~F E9¼ 100N.

Recall that Eqs. (19) and (20) are based upon the linear time-invariant (LTI) system concept, and thus their stiffness (kHF,kHR, kR and kB) and viscous damping coefficients (cHF, cHR, cR and cB) are assigned constant values. To develop quasi-linearmodels for either a 2- or 3dof mechanical system, spectrally varying parameters must be included. Fig. 6 shows measuredstiffness and viscous damping coefficient values of hydraulic and rubber mounts from the non-resonant dynamic stiffnesstesting procedure under the ISO standard 10846 [19]. Though such empirical data (typically supplied by the mountvendors) are limited in scope, some useful information can still be gained. Therefore, the quasi-linear model is developedby employing the following steps: (1) start with mechanical system models where the interfacial forces are derived fromthe LTI system formulation; (2) embed spectrally varying stiffness and damping elements in terms of kw(o) and cw(o)(where w is the mount index for HF, HR, R, and B) as shown in Fig. 6 into the frequency domain formulation; (3) employmeasured motions ~X EMðoÞ and ~X SMðoÞ in Eqs. (23) and (24) for the 2dof model and Eqs. (25) and (26) for 3dof model; and(4) calculate the time histories of the interfacial forces by using the complex exponential form as shown below. Note thatthis includes only the fundamental (excitation) frequency term as follows:

fITvðtÞ ¼ 9 ~F ITvðoÞ9Re½eiðootþjFITvÞ�, (27)

jFITv ¼+ ~F ITvðoÞ ðv¼ 2dof or 3dofÞ, (28)

fIBvðtÞ ¼ 9 ~F IBvðoÞ9Re½eiðootþjFIBvÞ�, (29)

jFIBv ¼+ ~F IBvðoÞ ðv¼ 2dof or 3dofÞ: (30)

10 15 20 25 30 35 40 45 5010-4

10-2

100

10 15 20 25 30 35 40 45 50-50

0

50

100

150

Pha

se (o )

Frequency (Hz)

(mm

)

Fig. 5. Relative displacement between xE and xS at the dynamic load sensing hydraulic mount from the measured accelerations ~CEMðoÞ and ~CSMðoÞgiven fEðtÞ ¼ Re½ ~F E eioo t � at 9 ~F E9¼ 100N in frequency domain.

10 20 30 40 501

1.2

1.4

1.6

1.8

2x 105

10 20 30 40 500

10

20

30

Frequency (Hz)

10 20 30 40 500

2

4

6

8x 105

10 20 30 40 500

2000

4000

6000

8000

Frequency (Hz)

|k HR

( ω)|

(N/m

)|c H

R(ω

)| (N

-s/m

)

|k HR

( ω)|

(N/m

)|c H

R(ω

)| (N

-s/m

)

Fig. 6. Measured dynamic properties of the hydraulic and rubber mounts given xðtÞ ¼ Re½ ~Xeioo t � with 9 ~X 9¼ 0:1mm: (a) stiffness and damping spectra for

the dynamic load sensing hydraulic mount and (b) stiffness and damping spectra for the rubber mount.


6. Development of a quasi-linear (QL) fluid system model

The linear system models for mounts were initially developed by Singh et al. [20]. Kim and Singh [21,22] and Tiwariet al. [17] measured nonlinear inertia track resistances and upper and lower chamber compliances using laboratory

Fig. 7. Experiment with a fluid system model of the dynamic load sensing hydraulic mount: (a) schematic used for modeling; (b) expanded view of the

fluid system model with parallel rubber and hydraulic force paths.


experiments. These properties were utilized by Adiguna et al. [18] and He and Singh [12] to predict steady state andtransient responses. Other hydraulic mount studies include work by Shangguan and Lu [23], Fan and Lu [24], Truong andAhn [25], Geisberger et al. [26], Mrad and Levitt [27], and Lee and Moon [28]. In particular, this article extends prior studiesby Yoon and Singh [10,11] and focuses on in-situ path forces in the context of Fig. 1. The interfacial forces will be dividedinto rubber and hydraulic force paths, and the quasi-linear model will be developed by embedding the spectrally varyingand amplitude-sensitive parameters and by using measured displacement and upper chamber pressure signals.

Fig. 7 illustrates the fluid system model of the load sensing hydraulic mount (fixed decoupler type). Here, fITr(t) andfIBr(t) are the top and bottom sides of the rubber path forces, respectively, and fITh(t) and fIBh(t) are the top and bottom sidesof the hydraulic path forces, respectively. The rubber forces are modeled using kRr and cRr elements, which are assumed tobe identical on the top and bottom sides of the hydraulic mount. However, the hydraulic path force is assumed to beasymmetric due to the nonlinearities from the fluid parameters. The designation of the fluid parameters in Fig. 7(b) are asfollows: pu(t) and pl(t) are the upper and lower chamber pressures, respectively; Cu and Cl are the upper (#u) and lower(#l) chamber compliances, respectively; Ii is the inertance of the inertia track (#i); Ri is the resistance of the inertia track;and qi is the fluid flow through inertia track. The interfacial forces for both the top and bottom sides of the mount arederived with the following sign conventions: pu(t) and pl(t) are positive (in compression) corresponding to the upward(positive) motion of xE(t), xS(t), qi(t), fIT(t), and fIB(t); and qd(t); and pu(t) and pl(t) are negative (in expansion) for thedownward motion of xE(t), xS(t), qi(t), fIT(t), and fIB(t)

fIT ðtÞ ¼ fIrðtÞþ fIThðtÞ, (31)

fIBðtÞ ¼ fIrðtÞþ fIBhðtÞ, (32)

fIrðtÞ ¼ cRr_xðtÞþkRrxðtÞ, (33)

fIThðtÞ ¼�2ArpuðtÞþArplðtÞ, (34)

fIBhðtÞ ¼�ArpuðtÞþ2ArplðtÞ: (35)

Here, the relative displacement x(t)¼xE(t)�xS(t). Also, the momentum and continuity equations for the hydraulic pathwith respect to x(t) are as follows [17,18,20]:

puðtÞ�plðtÞ ¼ Ii _qiðtÞþRiqiðtÞ, (36)

Cu _puðtÞ ¼ Ar_xðtÞ�qiðtÞ, (37)

Cl _plðtÞ ¼ qiðtÞ: (38)

Transform Eqs. (31)–(38) into the Laplace domain (s) and ignore the intial conditions:

FIT ðsÞ ¼ FIrðsÞþFIThðsÞ, (39)

FIBðsÞ ¼ FIrðsÞþFIBhðsÞ, (40)


FIrðsÞ ¼ ðcRrsþkRrÞXðsÞ, (41)

FIThðsÞ ¼�2ArPuðsÞþArPlðsÞ, (42)

FIBhðsÞ ¼�ArPuðsÞþ2ArPlðsÞ, (43)

PuðsÞ�PlðsÞ ¼ ðIisþRiÞQiðsÞ, (44)

CusPuðsÞ ¼ ArsXðsÞ�QiðsÞ, (45)

ClsPlðsÞ ¼QiðsÞ: (46)

Several upper and lower chamber pressure transfer functions (G1–G3) are determined from Eqs. (44)–(46), as expressedbelow along with the fluid system parameters

G1ðsÞ ¼Pu

XðsÞ ¼

Ar

CuþCl

ðs2=o2N1Þþð2z1=oN1Þsþ1


, (47)

G2ðsÞ ¼Pl

PuðsÞ ¼

1


, (48)

G3ðsÞ ¼Pl

XðsÞ ¼

Ar

CuþCl

1


, (49)

z1 ¼1

2

ffiffiffiffiffiffiffiffiffiClR

2i

Ii

s, oN1 ¼

ffiffiffiffiffiffiffi1

ClIi

s, (50,51)

z2 ¼1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCuClR

2i

IiðCuþClÞ

s, oN2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiCuþCl

CuClIi

s: (52,53)

Eqs. (34), (35), (42), and (43) reveal asymmetrical dynamic characteristics since the hydraulic path forces fITh(t) andfIBh(t) have different values. The following nominal (and constant) parameters are incorporated for calculations:kRr¼2�105 N m�1; cRr¼496.1 N s m�1; Ii¼4�106 kg m�4; Cu¼2.5�10�11 m5 N�1; Cl¼2.4�10�9 m5 N�1; andRi¼2�108 N s m�5. Measured motion xM(t) and upper chamber pressure puM(t) are employed to predict the interfacialforces fIT(t) and fIB(t). However, the lower chamber pressure spectrum Pl(o,X) is still unknown. Accordingly, it must beestimated by using Eqs. (39)–(43). First, by replacing s with io in Eqs. (39)–(43), the equations in the frequency domain areexpressed in terms of spectrally varying and amplitude-sensitive properties as follows:

~F IT ðo,XÞ ¼ ðiocRrþkRrÞ~XðoÞ�2Ar

~Puðo,XÞþAr~Plðo,XÞ, (54)

~F IBðo,XÞ ¼ ðiocRrþkRrÞ~XðoÞ�Ar

~Puðo,XÞþ2Ar~Plðo,XÞ: (55)

By subtracting Eq. (55) from (54), the lower chamber pressure ~Plðo,XÞ is derived as follows:

Ar~Plðo,XÞ ¼ �Ar

~Puðo,XÞ� ~F IT ðo,XÞþ ~F IBðo,XÞ: (56)

Second, the measured data set ~F ITMðo,XÞ and ~F IBMðo,XÞ from the laboratory experiment can be used to estimate~Plðo,XÞ, in the complex valued form as follows:

~F ITM ¼ FITREþ iFITIM , jITM ¼+ ~F ITMðo,XÞ, (57)

FITRE ¼ Re½ ~F ITM � ¼ 9 ~F ITM9cosðjITMÞ, (58)

FITIM ¼ Im½ ~F ITM � ¼ 9 ~F ITM9sinðjITMÞ, (59)

~F IBM ¼ FIBREþ iFIBIM , jIBM ¼+ ~F IBMðo,XÞ, (60a,b)

FIBRE ¼ Re½ ~F IBM � ¼ 9 ~F IBM9cosðjIBMÞ, (61)

FIBIM ¼ Im½ ~F IBM� ¼ 9 ~F IBM9sin ðjIBMÞ, (62)

~PuM ¼ PuREþ iPuIM , jPuM ¼+ ~PuMðo,XÞ, (63a,b)

PuRE ¼ Re½ ~PuM � ¼ 9 ~PuM9cosðjPuMÞ: (64)

10 15 20 25 30 35 40 45 500

20

40

60

80

10 15 20 25 30 35 40 45 50-200

-100

0

100

200

300

400

Pha

se (

o ) P u

P lor

(k

Pa)

Frequency (Hz)

Fig. 8. Predicted lower chamber pressure spectra 9 ~Plðoo ,XMÞ9 given fEðtÞ ¼ Re½ ~F E eioo t � with 9 ~F E9¼ 100N by using measured 9 ~XMðooÞ9 and 9 ~P uMðoo ,XMÞ9.Key: , ~PuMðoo ,XMÞ; , ~PlMðoo ,XMÞ.


PuIM ¼ Im½ ~PuM � ¼ 9 ~PuM9sin ðjPuMÞ: (65)

By substituing Eqs. (57)–(65) into Eq. (56), the real and imaginary parts of ~Plðo,XÞ are calculated as

PlRE ¼�PuREþ1

Arð�FITREþFIBREÞ, and (66)

PlIM ¼�PuIMþ1

Arð�FITIMþFIBIMÞ: (67)

Fig. 8 compares estimated ~PlMðoo,XMÞ measured ~PuMðoo,XMÞ spectra. Note that the ~PlMðoo,XMÞ spectrum seems to besimilar in terms of dynamic behavior and its range of magnitudes. This means that the load sensing hydraulic mount is alsoaffected by the lower chamber dynamic pressure ~Plðo,XÞ. Next, the effective stiffness and damping coefficient parametersin terms of kRre(o,X) and cRre(o,X), respectively, are estimated from the calculated ~PlMðoo,XMÞ as follows:

~F IRðo,XÞ ¼ ðiocRreþkRreÞ~XðoÞ

¼ FITREþ iFITIMþ2ArðPuREþ iPuIMÞ�ArðPlREþ iPlIMÞ, (68)

kRreðo,XÞ ¼ FITREþArð2PuRE�PlREÞ, (69)

cRreðo,XÞ ¼1

o FITIMþArð2PuIM�PlIMÞ½ �: (70)

Fig. 9 shows the results from Eqs. (69) and (70) by using the measured ~XMðooÞ and ~PuMðoo,XMÞ. Results show thatkRre(oo,XM) and cRre(oo,XM) not only include the dynamics of the component itself but also some coupling effects from thesub-system responses, since three resonances are observed in kRre(oo,XM). Note that cRre(oo,XM) has the highest valuearound 11 Hz. Finally, both effective lower and upper chamber compliances ~C leðoo,XMÞ and ~C ueðoo,XMÞ, respectively, are

10 15 20 25 30 35 40 45 500

0.5

1

1.5

2x 10

7

10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3x 10

5

Frequency (Hz)

k Rre

(N/m

)c Rre

(N-s

/m)

Fig. 9. Predicted effective stiffness kRre(oo,XM) and damping cRre(oo,XM) spectra given fEðtÞ ¼ Re½ ~F E eioo t �with 9 ~F E9¼ 100N by using measured ~XMðooÞ and~PuM ðoo ,XMÞ

�� .


estimated. The ~C leðoo,XMÞ term is expressed in the complex valued form as follows:

~C leðoo,XMÞ ¼ Cln~l lðoo,XMÞ ¼ Clnðaplþ ibplÞ, (71)

apl ¼ alðoo,XMÞ, bpl ¼ blðoo,XMÞ: (72,73)

Here, Cln is the nominal value of Cl, and ~l lðoo,XMÞ is the spectrally varying and amplitude-sensitive parameter for Cl

with the coefficients apl and bpl. From Eq. (48) by replacing s with io and employing Eqs. (71)–(73), the coefficients apl andbpl are estimated as follows:

PlREþ iPlIM

PuREþ iPuIM¼

1

1�o2IiClnðaplþ ibplÞþ ioRiClnðaplþ ibplÞ, (74)

apl ¼apl2o2

oþapl1oo

dclðI2i o4

oþR2i o2

o Þ, bpl ¼

bpl2o2oþbpl1oo

dclðI2i o4

oþR2i o2

oÞ: (75,76)

dcl ¼ ClnðP2lREþP2

lIMÞ, (77)

apl2 ¼ IiðP2lREþP2

lIM�PuREPlRE�PuIMPlIMÞ, (78)

apl1 ¼ Rið�PuREPlIMþPuIMPlREÞ, (79)

bpl2 ¼ IiðPuREPlIM�PuIMPlREÞ, (80)


bpl1 ¼ RiðP2lREþP2

lIM�PuREPlRE�PuIMPlIMÞ: (81)

The ~C ueðoo,XMÞ term is also estimated by substituting io for s with Eq. (49) and employing the results ~C leðoo,XMÞ fromEqs. (71)–(81). Next ~C ueðoo,XMÞ is defined in terms of an empirical spectrally varying and amplitude-sensitive parameter~luðoo,XMÞ along with coefficients apu and bpu:

~C ueðoo,XMÞ ¼ Cun~luðoo,XMÞ ¼ Cunðapuþ ibpuÞ, (82)

apu ¼ auðoo,XMÞ, bpu ¼ buðoo,XMÞ, (83,84)

r¼ 9 ~XMðooÞ9, jX ¼+ ~XMðooÞ, (85,86)

PuRE1 ¼ 9 ~PuM9cosðjPuM�jXÞ; (87)

PuIM1 ¼ 9 ~PuM9sinðjPuM�jXÞ: (88)

Here, ~Pueðoo,XMÞ is calibrated by using ~XMðoÞ as a reference value for the sake of deriving effective values

PuRE1þ iPuIM1

r ¼Ar½1�o2 ~C leIiþ io ~C leRi�

½Cunðapuþ ibpuÞþ~C le��o2Cunðapuþ ibpuÞ

~C leIiþ ioCunðapuþ ibpuÞ~C leRi

, (89)

apu ¼apu4o4

oþapu2o2oþapu1ooþapu0

dcuðdu4o4oþdu2o2

oþdu1ooþdu0Þ, bpu ¼

bpu4o4oþbpu2o2

oþbpu1ooþbpu0

dcuðdu4o4oþdu2o2

oþdu1ooþdu0Þ, (90,91)

dcu ¼ C2unðP

2uRE1þP2

uIM1Þ, du4 ¼ C2lnI2

i ða2plþb

2plÞ, (92,93)

du2 ¼ C2lnR2

i ða2plþb

2plÞ�2ClnIiapl, du1 ¼�2ClnRiðP

2uRE1þP2

uIM1Þbpl, (94,95)

du0 ¼ C2unðP

2uRE1þP2

uIM1Þ, apu4 ¼ ArrCunC2lnI2

i PuRE1ða2plþb

2plÞ, (96,97)

apu2 ¼ CunC2ln½ArrR2

i PuRE1þ IiðP2uRE1þP2

uIM1Þ�ða2plþb

2plÞ�2ArrCunClnIiPuRE1apl; (98)

apu1 ¼�2ArrCunClnRiPuRE1bpl, (99)

apu0 ¼�CunClnðP2uRE1þP2

uIM1ÞaplþArrCunPuRE1, (100)

bpu4 ¼�ArrCunC2lnI2

i PuIM1ða2plþb

2plÞ, (101)

bpu2 ¼�ArrCunC2lnR2

i PuIM1ða2plþb

2plÞþ2ArrCunClnIiPuIM1apl, (102)

bpu1 ¼ CunC2lnRiðP

2uRE1þP2

uIM1Þða2plþb

2plÞþ2ArrCunClnRiPuIM1bpl, (103)

bpu0 ¼�CunClnðP2uRE1þP2

uIM1Þbpl�ArrCunPuIM1: (104)

Fig. 10 compares the spectral contents of ~C ueðoo,XMÞ and ~C leðoo,XMÞ. When the ranges of 9 ~C ue9and 9 ~C le9 are compared,their magnitudes are similar, but the phase of ~C ueðoo,XMÞ shows a significant dynamic phenomenon above 20 Hz. Thus, thelower chamber dynamics must be modeled under in-situ conditions.

Overall, the QL model with effective fluid parameters kRre(o,X), cRre(o,X), ~C ueðo,XÞ and ~C leðo,XÞ is developed based onthe reverse path spectral method [29,30] as illustrated in Figs. 11 and 12. Fig. 11 describes the procedure used todetermine effective parameters from relevant transfer functions ~G1ðo,XÞ and ~G2ðo,XÞ and dynamic stiffness ~K Rðo,XÞ. Notethat ~PlMðoo,XMÞ is predicted first using ~F IT ðoo,XMÞ and ~F IBðoo,XMÞ. Then kRre(oo,XM) and cRre(oo,XM) are identified asillustrated in Fig. 11(a). Fig. 11(b) shows the procedure to determine ~C ueðoo,XMÞ and ~C leðoo,XMÞ by employing ~G1ðo,XÞ and~G2ðo,XÞ. Therefore, the interfacial forces are estimated as suggsted by Eqs. (54) and (55)

~F IT ðo,XÞ ¼ ~F IRðo,XÞþ ~F IThðo,XÞ, (105)

~F IBðo,XÞ ¼ ~F IRðo,XÞþ ~F IBhðo,XÞ, (106)

~F IRðo,XÞ ¼ ~K Rðo,XÞ ~XðoÞ, (107)

~K Rðo,XÞ ¼ iocRreðo,XÞþkRreðo,XÞ: (108)

Here, ~F IRðo,XÞ is the rubber path force with ~K Rðo,XÞ as its dynamic stiffness, and ~F IThðo,XÞ and ~F IBhðo,XÞ are thehydraulic path forces on the top and bottom sides of the hydraulic mount, respectively. The hydraulic path forces ~F IThðo,XÞ

10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

x 10-10

10 15 20 25 30 35 40 45 500

50

100

150

200

Pha

se (o )

Cue

Cle

or

(m5/N

)

Frequency (Hz)

Fig. 10. Predicted upper ð ~C ueðoo ,XMÞÞ and lower ð ~C leðoo ,XMÞÞ chamber compliance spectra given fEðtÞ ¼ Re½ ~F E eioo t � with 9 ~F E9¼ 100N by using measured~XMðooÞand ~PuM ðoo ,XMÞ. Key: , ~C ueðoo ,XMÞ; , ~C leðoo ,XMÞ; —, nominal value Cun.

Fig. 11. Identification of effective parameters for the reverse path spectral method: (a) prediction of lower chamber pressure ~P lMðoo ,XMÞ and

identification of effective stiffness (kRre) and viscous damping coefficient (cRre) for the fundamental harmonic term; (b) identification of effective lower

chamber compliance ~C leðoo ,XMÞand effective upper chamber compliance ~C ueðoo ,XMÞ.


and ~F IBhðo,XÞ may be estimated by using two different transfer functions as derived below. First, consider the forcetransmissibility concept:

~F IThðo,XÞ ¼ Ar½~G2eðo,XÞ�2� ~Pueðo,XÞ, (109)

~F IBhðo,XÞ ¼ Ar½2 ~G2eðo,XÞ�1� ~Pueðo,XÞ: (110)

Second, consider the dynamic stiffness concept:

~F IThðo,XÞ ¼ Ar ½�2 ~G1ðo,XÞþ ~G3ðo,XÞ� ~XðoÞ, (111)


~F IBhðo,XÞ ¼ Ar½�~G1ðo,XÞþ2 ~G3ðo,XÞ� ~XðoÞ: (112)

Fig. 12(a) and (b) compare both force estimation methods. The dynamic stiffness concept as illustrated in Fig. 12(b) issimpler than the force transmissibility concept since it needs only measured displacement ~XMðooÞ. Further, the method forboth force transmissibility and dynamic stiffness concepts, as illustrated in Fig. 12(a), needs two sets of measured data,~XMðooÞ and ~PuMðoo,XMÞ. However, the number of effective parameters is reduced compared with the case of Fig. 12(b).From the frequency domain results, the time histories of interfacial forces with the fluid system model are estimated as

Fig. 12. Force estimation in frequency domain using the reverse path spectral method: (a) force transmissibility ð ~G2eÞ and dynamic stiffness ð ~K ReÞ

concepts; and (b) dynamic stiffness ( ~G1e , ~G3e and ~K Re) concept.

10 15 20 25 30 35 40 45 5010

-1

100

101

102

103

10 15 20 25 30 35 40 45 50100

150

200

250

300

350

Phas

e (o )

FIT

(N)

Fig. 13. Comparison between measured and predicted interfacial force spectra ~F IT ðoo ,XMÞ using mechanical or fluid system model, given

fEðtÞ ¼ Re½ ~F E eioo t � with 9 ~F E9¼ 100N, and by employing measured ~XMðooÞ and ~PuM ðoo ,XMÞ. Key: , measured; – –, 2dof mechanical model; —, 3dof

mechanical model; , fluid system model.


follows:

fIT ðtÞ ¼ 9 ~F IT ðoo,XMÞ9Re½eiðootþjFIT Þ�, jFIT ¼+ ~F IT ðoo,XMÞ, (113,114)

fIBðtÞ ¼ 9 ~F IBðoo,XMÞ9Re½eiðootþjFIBÞ�, jFIB ¼+ ~F IBðoo,XMÞ: (115,116)

7. Experimental validation of quasi-linear (QL) models in frequency and time domains

Figs. 13 and 14 compare measured and predicted interfacial forces by using both the mechanical and fluid system QL

models described in Sections 5 and 6. As observed in Figs. 13 and 14, the fluid model predicts the interfacial forces betterthan the 2- or 3dof mechanical model. Specifically, the fluid model reflects the asymmetrical characteristics well. Observethat ~F IT ðoo,XMÞ of the 2dof mechanical model follows the measurement quite well. However, the 2dof model is unable topredict ~F IBðoo,XMÞ since it assumes symmetric forces as evident from Eqs. (23) and (24). The force estimation with the 3dofmechanical model shows the asymmetrical behavior for both the magnitude and phase spectra as observed in Figs. 13 and14. However, its force magnitudes do not match with direct force measurements since the effective mechanicalparameters are not well assessed.

Figs. 15 and 16 show the contributions of the rubber and hydraulic paths. The magnitudes of both the rubber andhydraulic path forces are similar to, but much higher than, the total force. This suggests that the interfacial forces areaffected by the relative phase between paths. For instance, Figs. 15(b) and 16(b) show that the ~F IT ðoo,XMÞ and ~F IBðoo,XMÞ

spectra are more affected by the rubber path force since the phases of ~F ITrðoo,XMÞ and ~F IBrðoo,XMÞ are the almost same as~F IT ðoo,XMÞ and ~F IBðoo,XMÞ.

10 15 20 25 30 35 40 45 5010

-2

100

102

104

10 15 20 25 30 35 40 45 500

100

200

300

400

Pha

se(o )

FIB

(N)

Frequency (Hz)

Fig. 14. Comparison between measured and predicted interfacial force spectra ~F IBðoo ,XMÞ using mechanical or fluid system model, given

fEðtÞ ¼ Re½ ~F E eioo t � with 9 ~F E9¼ 100N, and by employing measured ~XMðooÞ and ~P uMðoo ,XMÞ. Key: , measured; – –, 2DOF mechanical model; —, 3dof

mechanical model; , fluid system model.

10 15 20 25 30 35 40 45 5010

-1

100

101

102

103

10 15 20 25 30 35 40 45 50-200

-100

0

100

200

300

400

Frequency (Hz)

Pha

se(o )

FIT

(N)

Fig. 15. Comparison between rubber ~F ITrðoo ,XMÞ and hydraulic ~F IThðoo ,XMÞ paths with the fluid system model, given fEðtÞ ¼ Re½ ~F E eioo t � with 9 ~F E9¼ 100N,

and by employing measured ~XMðooÞ and ~P uMðoo ,XMÞ. Key: , ~F IT ðoo ,XMÞ; —, ~F ITrðoo ,XMÞ ; , ~F IThðoo ,XMÞ.


Fig. 17 compares measured fIT(t) and fIB(t) with estimated forces in the time domain by using the 2dof mechanical and fluidsystem QL models. Note that only the fundamental oo term is included here. To assess the validity of each model, define an error

e(t) between the measured force fTM(t) and predicted force fIT(t) and fIB(t) at any time t as eðtÞ ¼ ðfIwðtÞ�fTMðtÞÞ=fTMðtÞ (w¼T or B).

The overall root-mean-square error E(rms) is then given by E¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1=NmaxÞ

PNmax

v ¼ 1 ½evðtÞ�2

q, where Nmax is the maximum number of

points in the time domain. For the top forces fIT(t), errors from the 2dof and fluid system models are found to be 71.0 and 9.0percent, respectively. Errors for the bottom forces fIB(t) from the 2dof and fluid system models are 111.0 and 3.7percent,respectively. Thus, the fluid system model is more accurate than the 2dof mechanical model. Nevertheless, the fluid system modelstill shows discrepancies since it does not include any super- or sub-harmonics.

8. Conclusion

This article has made several contributions to the state of the art, as summarized below. First, a new concept wasdeveloped to indirectly measure interfacial forces by employing the hydraulic mount as a dynamic force sensor. Theproposed method utilized a combination of models and operating motion and/or pressure measurements. Second, alaboratory experiment consisting of a powertrain, three powertrain mounts (including a dynamic load sensing hydraulicmount), a sub-frame, and four bushings was then constructed to verify the proof-of-concept. Third, the lower chamberpressure pl(t) was estimated, as it was not measured in either this study or any prior articles [10,11,12,17,18]. This led toan improved estimation of effective lower chamber compliance ~C leðo,XÞ along with kRre(o,X), cRre(o,X) and ~C ueðo,XÞ.Overall, the proposed fluid system model yielded a better prediction of the interfacial forces, though the mechanicalmodels provided some useful insights. This work also advanced prior component studies [10,11] by providing an improvedmulti-degree of freedom isolation system perspective.

This article focused on the frequency domain analyses. Limited results in the time domain were also presented.Interfacial force measurements exhibited multi-harmonic terms (Fig. 4) under sinusoidal excitation. As part of future work,

10 15 20 25 30 35 40 45 5010

-2

100

102

104

10 15 20 25 30 35 40 45 50-200

-100

0

100

200

300

400

Frequency (Hz)

Pha

se(o )

FIB

(N)

Fig. 16. Comparison between rubber ~F IBrðoo ,XMÞ and hydraulic ~F IBhðoo ,XMÞ path with the fluid system model, given fEðtÞ ¼ Re½ ~F E eioo t � with 9 ~F E9¼ 100N,

and by employing measured ~XMðooÞ and ~PuM ðoo ,XMÞ. Key: , ~F IBðoo ,XMÞ; —, ~F IBrðoo ,XMÞ; , ~F IBhðoo ,XMÞ.

Fig. 17. Comparison between predicted and measured interfacial forces in time domain given fEðtÞ ¼ Re½ ~F E eioo t � with 9 ~F E9¼ 100N and oo/2p¼25 Hz.

Key: , measurement; , fluid system model; ——————, 2dof mechanical system model.


the super-harmonic responses in the time domain should be incorporated for a better estimation of the interfacial forces.Other excitations such as transients [2,18] should be considered as well.

Acknowledgement

The authors are grateful to the member organizations of the Smart Vehicle Concepts Center (www.SmartVehicleCenter.org) and the National Science Foundation Industry/University Cooperative Research Centers program (www.nsf.gov/eng/iip/iucrc) for supporting this work.

www.SmartVehicleCenter.org





www.nsf.gov/eng/iip/iucrc






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