14th Asia Pacific Vibration Conference, 5-8 December 2011, The Hong Kong Polytechnic University

Estimation of Interfacial Forces in a Multi-Degree of Freedom Nonlinear Vibration System

Rajendra SINGH Acoustics and Dynamics Laboratory, Smart Vehicle Concepts Center

Department of Mechanical and Aerospace Engineering

The Ohio State University, Columbus, Ohio 43210 USA

E-mail: singh.3@osu.edu

Abstract Direct measurement of dynamic forces is not practical in many real-life applications since the interfacial conditions may change. Thus indirect force estimation methods must be developed, though they pose special difficulty for nonlinear mounts or isolators since the transfer path type methods are applicable for linear systems only. The hydraulic engine mount (as a dynamic load sensor) is examined as an illustrative example in this article. This device is inherently nonlinear (with spectrally-varying and amplitude-sensitive parameters). The proposed nonlinear models for force transmitted to the mount base are found to correlate well with measured forces under harmonic excitation. Then interfacial forces are estimated in a multi-degree of freedom system, and a laboratory experiment consisting of a powertrain, three powertrain mounts including a dynamic load sensing hydraulic mount, a sub-frame, and four bushings, is constructed to verify the proof of the concept. The proposed indirect force estimation methods utilize a combination of mathematical models and operating motion and/or pressure measurements. Overall, interfacial force predictions match well with the direct force measurements.

Key words: Isolation systems, Force measurements, Powertrain mounts, Nonlinear analysis

1. Introduction

Precise knowledge of the dynamic forces at sub-system junctions or interfaces is of vital interest in the dynamic and vibro-acoustic design of mechanical systems, vehicles, buildings, and power plants(1), (2). In general, it is difficult to install force transducers at the sub-system junctions without altering interfacial or boundary conditions unless there is a mobility mismatch(3). Thus, indirect measurement or force reconstruction methods must be adopted to estimate dynamic forces. For instance, one could employ transfer path approaches, though they are applicable primarily in the frequency domain for a linear time-invariant system(4), (5), (6). Also, dynamic forces could be estimated by using other measured signals such as operating motions, but then dynamic stiffness must be known a priori. Such indirect force estimation methods pose special difficulty for nonlinear mounts or isolators. For instance, hydraulic engine mounts exhibit spectrally-varying and amplitude-sensitive parameters(7), (8), (9), (10), (11), (12). There is a significant body of literature on the frequency domain based transfer path analysis methods (4), (5), (6) where the interfacial forces are implicitly or explicitly estimated. For instance, several indirect frequency domain formulations have been established to estimate interfacial forces or moments in discrete systems (4), (5), (6). These employ frequency response functions of the subsequent sub-system only at the driving points, assuming that

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14th Asia Pacific Vibration Conference, 5-8 December 2011, The Hong Kong Polytechnic University

the six-dimensional motions are somehow known (or measurable) at any location. Experimental measurement burden could be significantly reduced provided that the interfacial forces are estimated only from the forced responses and transfer functions at the driving point(4). However, most of the available indirect or inverse force estimation methods are valid only for linear time-invariant systems since they employ transfer functions or similar frequency domain concepts.

2. Problem Formulation

Underlying issues can be conceptually illustrated by the generic source-path-receiver system of Fig. 1. Multiple linear and nonlinear isolators (a parallel path configuration) are shown. For the sake of illustration, first consider a linear time-invariant (LTI) system, and define interfacial forces in the time domain for path II as follows.

tpAtktctf ur2ES2ESIT (1)

tpAtktctf ur2ES2ESIB (2)

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

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 ξ(t) = xE(t) – xS(t) is the relative displacement. When the above equations are compared, the top and bottom forces (in path II) are identical except for the sign as long as there are no inertial elements in the fluid path. This is obviously not the case in many practical devices. In order to estimate interfacial forces on both source and receiver sides, one must recognize and resolve some difficulties. First, the above equations require a precise knowledge of in-situ parameters such as nonlinear stiffness kES2, damping cES2 (in the rubber force path), and effective piston area Ar (in the hydraulic force path). Second, note that the relative motions of the adjoining structures and in-situ interfacial properties must be precisely known if Eqs. (1-2) were to be experimentally or computationally implemented (assuming a linear system of course). Now consider the case when the contact (interfacial) condition is very hard (i.e. k is very high). The estimated force could contain significant errors due to difficulties in measuring relative

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14th Asia Pacific Vibration Conference, 5-8 December 2011, The Hong Kong Polytechnic University

responses (ξ(t) = xE(t) – xS(t)) across the spring and damping elements. Thus very small differences in the displacements could induce a large error in the force. Further, in-situ stiffness and damping values are often unknown and thus implementation of Eqs. (1-2) is rather elusive in practical systems. Third, forces could be estimated by using measured operating motions and/or pressure signals, but then effective (dynamic) stiffness and damping properties (or calibration parameters) must be known a priori. The latter poses a special difficulty for both elastomeric and hydraulic isolators. The scope of this article is limited to steady state experiments under sinusoidal motion or force excitation. First consider the hydraulic mount (as a component itself) as shown in Fig. 2, and xM(t), puM(t), and fTM(t) are measured on the elastomer test machine. Both linear time-invariant (LTI) and nonlinear (NL) models are developed, and their fT(t) predictions are compared with measurements. Then, a laboratory experiment is constructed and instrumented corresponding to Fig. 1; one load sensing hydraulic mount is embedded. Finally the interfacial forces at both ends of the load sensing hydraulic mount are estimated by employing quasi-linear (QL) and non-linear (NL) mechanical and fluid models of the load sensing mount and compared with direct force measurements in frequency and time domains.

(a)

(b)

Fig. 2. Force transmitted fT(t) by a hydraulic mount in the context of non-resonant elastomeric test: (a) Fluid model of the hydraulic mount and its parameters for rubber and hydraulic paths; (b) sinusoidal displacement excitation x(t) and dynamic forces transmitted by two paths (fTr and fTh).

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14th Asia Pacific Vibration Conference, 5-8 December 2011, The Hong Kong Polytechnic University

3. Methods

To illustrate the concepts, consider Fig. 2(a) that displays the internal configuration of the hydraulic engine mount and its fluid system model. Figure 2(b) shows the non-resonant dynamic stiffness test concept. Here, fm is the preload, ti

moeX

~Rextx is the

excitation displacement, xm is the mean displacement, xiXeX~ is the complex valued

excitation amplitude, X is the amplitude of displacement, φ is the phase of x(t), ωo is the excitation (fundamental) frequency (rad s-1), and Re[] is the real value operator; tilde (~) over a symbol implies that is complex valued. In addition, a pressure transducer, pu(t), is also installed in the upper chamber in our laboratory experiments. As shown in Fig. 2(b), the steady state force transmitted fT(t), in time (t) domain, is related to the excitation displacement x(t) and upper chamber response pu(t): tftftf ThTrT ,

txktxctf rrTr , and tpAtf urTh . Here, fTr(t) is the rubber path force (subscript r), fTh(t) is the hydraulic path force (subscript h), kr and cr are the rubber stiffness and damping coefficient respectively, and Ar is the effective piston area. The experiments show that pu(t) deviates from the sinusoidal shape depending on X and ωo. Likewise, kr and cr vary as well with X and ωo. Accordingly, linear time-invariant (LTI), nonlinear (NL), and quasi-linear (QL) models are utilized to predict the force time history under sinusoidal excitation conditions given measured (or calculated) motion and/or internal pressure time histories. In particular, the super-harmonic contents in pu(t) and fT(t) time histories are investigated using both measurements and mathematical models. The nonlinear differntial equations for the fluid system mount of Fig. 2(a) are developed. The following nonlinearities(7), (8), (9), (10), (11) are described: 1. The discontinuous motion of the decoupler element is given in terms of a switching mechanism; 2. The nonlinear functions for compliances Cu(pu(t)), Cl(pl(t)), and inertia track resistance Ri(qi(t)) are described based on prior work(7), (8), (9), (10), (11); 3. The nonlinear model of the fixed decoupler mount is also developed by ignoring decoupler elements. The nonlinear model is solved numerically using the Runge Kutta integration technique. The initial values of pu(t) and fT(t) are selected from measurements and the mean values xm and fm. Table 1 provides an overview of models and prediction methods used. For further details, see recent articles by Yoon and Singh(13), (14), (15).

Table 1. Model designations and prediction methods

Model Type Description (see Yoon and Singh(13), (14), (15) for details)

Linear Time-Invariant (LTI) Model Quasi-Linear (QL) Model Nonlinear (NL) Model

Transfer function formulation Parameters vary with frequency and excitation amplitude; calculations conducted in frequency domain, reverse spectral path method and Fourier composition also utilized Nonlinear differential equations numerically solved in time domain

Fluid Model of Mount or System Mechanical Model of Mount or System

Based on continuity and momentum equations; lumped parameter formulation Based on Newton’s Second Law; analogous mechanical system representation of the fluid system

Force Estimation Methods Predicted by models without using any

measured data Predictions based on operating

measurements

Given motion and pressure data Given only motion data Given only pressure data

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14th Asia Pacific Vibration Conference, 5-8 December 2011, The Hong Kong Polytechnic University

Figure 3 illustrates the laboratory experimental setup with powertrain (assembly of engine and transmission) and sub-frame; it is based on the generic multi-degree of freedom isolation system of Fig. 1. The powertrain is connected to the sub-frame by two hydraulic mounts though it is supported by the third rubber mount that is grounded. The sub-frame is supported by 4 identical elastomeric bushings. The experiment is excited by an electrodynamic shaker that is located on the powertrain. A signal generator is used to generate multiple excitations. One piezoelectric force transducer is between the shaker and powertrain to measure the excitation force fE(t); it is also used as a reference signal. Here, the excitation force is expressed as ti

EEoeF

~Retf where Fi

EE eFF~ is the complex

valued excitation amplitude, FE is the amplitude of force, φF is the phase of excitation force fE(t), and ωo is the excitation (fundamental) frequency (rad s-1). The following sensors are employed as illustrated in Fig. 3: (1) two piezoelectric accelerometers are located on top of two hydraulic mounts and two accelerometers are placed on the sub-frame near the hydraulic mounts; (2) two piezoelectric force transducers are located for directly measuring the interfacial forces to on both the source and receiver side; and (3) one piezoelectric pressure transducer is installed within the upper chamber of the load sensing hydraulic mount.

(a)

(b)

Fig. 3. Laboratory experimental setup with load sensing hydraulic mount; (a) powertrain and sub-frame assembly with instrumentation; (b) front view laboratory experiment with focus on vertical motions.

4. Results

Figure 4 compares the nonlinear (NL) model with two quasi-linear models (schemes II-C and III-C) in terms of super-harmonic contents. Specifically, the nonlinear (NL) models

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14th Asia Pacific Vibration Conference, 5-8 December 2011, The Hong Kong Polytechnic University

predict lower magnitudes at 4ωo and 5ωo for both fixed and free decouplers; conversely, two quasi-linear (QL) schemes match experimental data very well. Figures 5 and 6 compare measured and predicted interfacial forces by using both mechanical and fluid system QL models. The fluid model predicts the interfacial forces better than the 2DOF or 3DOF analogous mechanical model. Specifically, the fluid model reflects the asymmetrical characteristics well. Observe that MoIT ,F

~ of the 2DOF mechanical model follows the measurement quite well. However, the 2DOF model is unable to predict MoIB ,F

~ since it assumes symmetric forces. The force estimation with the 3DOF mechanical model shows the asymmetrical behavior for both magnitude and phase spectra as observed in Figs. 5 and 6. However, its force magnitudes do not match well with direct force measurements since the effective mechanical parameters are not well assessed. Figure 7 compares measured fIT(t) and fIB(t) with estimated forces in the time domain by using the 2DOF mechanical and fluid system QL models. Note that only the fundamental ωo term is included here. For the top forces fIT(t), errors from the 2DOF mechanical and fluid system models are found as 71.0% and 9.0% respectively. Errors for the bottom forces fIB(t) from the 2DOF and fluid system models are 111.0% and 3.7% respectively. Thus, the fluid system model is more accurate than the 2DOF mechanical model.

Fig. 4. Comparison of super-harmonics between nonlinear (NL) and quasi-linear (QL) models with experiment given

ti oeX~

Retx at ωo/2π = 8.5 Hz and X = 1.5 mm: (a)

fixed decoupler; (b) free decoupler. Key: , experiment;

, QL scheme II-C; , QL scheme III-C; , nonlinear (NL) model.

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14th Asia Pacific Vibration Conference, 5-8 December 2011, The Hong Kong Polytechnic University

Fig. 5. Comparison between measured and predicted

interfacial force spectra MoIT ,F~ (for the system of

Fig. 3) using mechanical or fluid system model, given

tiEE

oeF~

Retf with EF~

= 100 N, and by employing

measured oM~ and MouM ,P

~ . Key: ,

measured; , 2DOF mechanical model; , 3DOF mechanical model; , fluid system model.

Fig. 6. Comparison between measured and predicted

interfacial force spectra MoIB ,F~ (for the system of

Fig. 3) using mechanical or fluid system model, given

tiEE

oeF~

Retf with EF~

= 100 N, and by employing

measured oM~ and MouM ,P

~ . Key: ,

measured; , 2DOF mechanical model; , 3DOF mechanical model; , fluid system model.

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

Phase (o)

ITF~

(N)

Frequency

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

Phase (o)

ITF~

(N)

Frequency

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14th Asia Pacific Vibration Conference, 5-8 December 2011, The Hong Kong Polytechnic University

(a)

(b)

Fig. 7. Comparison between predicted and measured interfacial forces

in time domain given tiEE

oeF~

Retf with EF~

= 100 N and

ωo/2π = 25 Hz. Key: , measurement; , fluid system model; , 2DOF mechanical system model.

5. Conclusion

This article has proposed new methods for estimating dynamic forces (in both time and frequency domains) that are transmitted by a hydraulic mount under sinusoidal excitation. First, the super-harmonic contents of measured force fTM(t) are examined and correlated. The hydraulic mount also exhibits a sub-harmonic term response. This period-doubling effect and a more refined nonlinear model will be the subject of a future article. Second, a new concept is successfully developed to indirectly measure interfacial forces by employing the hydraulic mount as a dynamic force sensor. The proposed method utilizes a combination of models and operating motion and/or pressure measurements. Overall, the proposed fluid system model yields a better prediction of the interfacial forces though the mechanical models provide some useful insights. There is significant room for further research in this area; in fact, work is underway to estimate interfacial forces or contact stiffnesses in bearings(16), brakes, etc.

References

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their correlation to sound radiation over a broad frequency range, Journal of Sound and

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(2) S. Kim and R. Singh, Multi-dimensional characterization of vibration isolators over a

wide range of frequencies, Journal of Sound and Vibration, Volume 245, 2001, pp. 877-

913.

Time (sec)

fIB (N)

Time (sec)

fIT(N)

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14th Asia Pacific Vibration Conference, 5-8 December 2011, The Hong Kong Polytechnic University

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Acknowledgement

The author is 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. Also, the work of former and current doctoral students (especially J.Y. Yoon) is acknowledged.

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