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Application study of nonlinear viscoelastic constitutive model for dynamic behavior of suspension arm bushing.

ABSTRACT

Ride quality is an important purchasing consideration for consumers. It is typically defined in terms of noise, vibration and harshness. These phenomena are a result of vibrations caused at the engine/powertrain and from the road surface, which are transmitted to the passenger cabin. To minimize such vibrations, rubber parts are used extensively at mounting points for the cabin, such as engine mountings and suspension bushings. The vehicle development process increasingly requires performance testing, including rubber parts using CAE, prior to prototype evaluation. This in turn requires a rubber material model that can accurately describe dynamic characteristics of rubber components, particularly frequency and amplitude dependency. Conventional rubber models using commercially available structural analysis solvers cannot solve for both frequency and amplitude dependency at the same time, and are unable to predict transient phenomena such as harshness that involve inputs of varying amplitude. The authors have proposed a new rubber material model that is able to reproduce both frequency and amplitude dependency simultaneously, based on the rubber material model developed by Simo, J.C. [1]. Previous studies have demonstrated the accuracy of the new model under quasi-static and harmonic input conditions. Actual vehicle evaluation involves several input directions, with simultaneous translational and rotational inputs that are transient. In this paper, the new rubber material model is applied to a suspension arm bushing to confirm bushing force when subjected to complex inputs. The model was shown to predict bushing stiffness with greater accuracy and therefore was validated.

CITATION: Ueda, M., Ito, S., and Suzuki, D., "Application Study of Nonlinear Viscoelastic Constitutive Model for Dynamic Behavior of Suspension Arm Bushing," SAE Int. J. Passeng. Cars - Mech. Syst. 9(1):2016.

INTRODUCTION

Ride quality in a vehicle is expressed in terms of factors that directly affect the occupants, such as vibration, noise and harshness (unpleasant sensations such as jolting). Ride quality is therefore considered a key selling point in terms of vehicle performance. Vibration is transmitted from the engine and powertrain as well as the road surface to the vehicle cabin, and thence to the occupants. In order to minimize the level of vibration reaching the cabin, rubber parts such as engine mounts and suspension bushings are deployed extensively in components used to join the engine and suspension systems to the cabin.

Given that rubber parts used in the joint portions between components have a significant bearing on ride quality, the vehicle development process will ideally include a CAE analysis of ride quality including rubber parts to provide a preliminary evaluation of ride quality prior to the prototype testing stage. This analysis requires a rubber material model that can accurately reproduce the dynamic characteristics of the rubber parts, particularly with respect to frequency and amplitude dependency. However, conventional rubber material models that use commercially available structural analysis solvers cannot reproduce both frequency and amplitude dependency at the same time. This makes it difficult to generate accurate predictions of transient phenomena such as harshness, where the amplitude is constantly changing.

The authors have developed a rubber material model [2] (hereafter denoted the "new rubber material model") based on the nonlinear viscoelastic rubber model proposed by Simo, J.C. [1] that is capable of reproducing both amplitude and frequency dependency at the same time. The accuracy of the new rubber material model has been demonstrated in both quasi-static condition and harmonic oscillation condition. [2]

Bushing components in an actual vehicle are generally subject to simultaneous and transient inputs in the translational as well as rotational directions. It is rare that an input is confined to a specific frequency in a uniaxial direction. At SAE, Hartley, C. et al. (2012) [3] and Nakahara, J. et al. (2015) [4] have previously described methods that are capable of reproducing both frequency and amplitude dependency at the same time. However these studies tend to concentrate on bushing force generated by inputs in the uniaxial direction. There is no literature on bushing force for multi-axial inputs or transient inputs.

This study reports on the reproducibility of bushing forces associated with multi-axial inputs and transient inputs in the new rubber material model applied to bushing components in a suspension arm as shown in Figure 1. The ultimate aim is to improve the accuracy of harshness analysis using the vehicle model.

RUBBER MATERIAL MODEL

Basic Concept of the New Rubber Material Model

Figure 4 illustrates the concept of the rubber material model proposed by Simo, J.C. [1], while Equation 1 shows the associated equation of evolution for stress and strain.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Equation 1

where

[G.sub.i] is the modulus of elasticity for the network

[[gamma].sub.i] is the modulus ratio for the network

[[tau].sub.i] is relaxation time

[Q.sub.i] is stress-like internal variable

[W.sub.0] denotes the deviatoric parts of the initial elastic stored energy

C denotes the deviatoric part of the right Cauchy-Green deformation tensor

The new rubber material model describes amplitude dependency by defining relaxation time [[tau].sub.i] in Equation 2 as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Equation 2

where

E' is a norm of the strain rate tensor

[A.sub.i] [m.sub.i] are material coefficients

Note that these must be identified along with [G.sub.i] and [[gamma].sub.i] in Figure 4. Identification of material coefficients is described below.

Identification of Material Coefficients

The network material coefficients were determined from the experimental observations using the test pieces of the material used in the bushing components. Figure 5 shows a sample procedure for calculating the material coefficients.

The hyperelastic portion corresponding to [G.sub.0] was modeled with the Yeoh model [5] [6]. In the Yeoh model, the strain energy per unit of reference volume (the elastic potential that indicates strain energy per unit volume) is expressed as shown in Equation 3.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Equation 3

where

[I.sub.1] is the first deviatoric strain invariant

[J.sub.el] is the elastic volume ratio

In the Yeoh model, the coefficients of elastic behavior [C.sub.10], [C.sub.20] and [C.sub.30] are determined from the uniaxial tensile test results, while the coefficient for volumetric change [D.sub.i] assumes linear volumetric change and is defined for [D.sub.1] only. The [G.sub.0] value is used for [G.sub.l] through [G.sub.i]. Next, the results from the harmonic oscillation test were used to determine the values of [m.sub.i] and [A.sub.i]. The proportion of [[gamma].sub.i] in each network was determined via optimization with the storage modulus and loss modulus from the harmonic oscillation test as the objective function. Table 1 lists the resulting material coefficients.

Validation of Material Coefficients

This section concerns validation of material coefficients derived as per the procedure detailed above. We used the Abaqus [6] 6-14.1 solver for calculation, and figure 6 shows the single-element validation model used to define the material coefficients thus obtained.

The first validation was of the quasi-static uniaxial tensile test characteristics. This involved comparing the stress-strain calculations with the corresponding experimental observations, as illustrated in Figure 7 (bushing 1) and Figure 8 (bushing 2). From Figures 7 and 8, it can be seen that the calculation results accurately represent the quasi-static uniaxial tensile strength characteristics.

The next validation was of harmonic oscillation characteristics. The validation model was the same as in Figure 6, with the dynamic modulus as the performance index. Figures 9 and 10 compare calculated values with experimental data for materials of bushings 1 and 2 respectively. It can be seen that the two sets of values diverge by up to around 10% for the material of bushing 1, but are largely reproducible for the material of bushing 1 and bushing 2. Amplitude dependency, meanwhile, is reproducible for both materials, and this demonstrates the validity of the new rubber material model.

Thus it was concluded that the material coefficients determined above faithfully reproduce the quasi-static characteristics and harmonic oscillation characteristics.

STAND-ALONE VALIDATION OF BUSHING COMPONENT

Bushing Model

This study evaluates the response of the bushing component when the point where the ball joint is attached moves in the vehicle X-direction as in Figure 1. A validation of the stand-alone bushing model was performed for bushing 1 in the axial direction and the radial direction, and for bushing 2 in the void direction. Figure 11 shows the standalone model (including the definition of axis) for bushing 1, while Figure 12 shows the same for bushing 2. For the purpose of calculation, the outer cylinder was restrained and the inner cylinder was subject to displacement.

Validation of Static Characteristics

This section seeks to validate the static characteristics of the standalone bushing model by examining the impact of input displacement on force in the bushing.

The first validation concerns static characteristics in the axial direction and the radial direction to the axis in bushing 1. Figure 13 compares calculations for bushing 1 in the radial direction with the experimental observations, while Figure 14 shows the corresponding figures for the axial direction. The calculations are highly consistent with the experimental observations in both directions.

The next validation examines the characteristics of bushing 2 in the void direction. Figure 15 shows experimental data and calculation results for bushing 2 in the void direction. The calculated rigidity tends to be lower than the measured rigidity in the experiments. Given that bushing characteristics are faithfully reproduced in the region where there is no contact with the void ([+ or -] 2 mm), we can expect to see some divergence between the theoretical model and the measured data with respect to the reproducibility of the void shape in the bushing model and the characteristics of friction where the void is in contact. This study demonstrates that the calculations can reproduce changes in rigidity associated with the collapse of the void.

Validation of Harmonic Oscillation Characteristics

This section validates harmonic oscillation characteristics for the stand-alone bushing model. Figures 11 and 12 show the validation models, with the dynamic modulus as the performance index.

Figure 16 compares the calculation results and experimental data for the bushing 1 stand-alone model in the radial direction, and Figure 17 shows the corresponding comparison in the axial direction. It can be seen that the dynamic modulus in the calculations tends to be slightly higher than the observed value, while frequency and amplitude dependency (Figure 17) are reproduced consistently.

Figure 18 compares calculation results with experimental data for the bushing 2 stand-alone model in the void direction. There is a discrepancy at an amplitude of 1.0 mm, but the two sets of results are fairly consistent at an amplitude of 0.1 mm. Figure 18 also illustrates good reproducibility of amplitude dependency.

Validation of Transient Characteristics

This section validates transient characteristics for the stand-alone bushing model. Figure 11 and Figure 12 show the validation model, with the bushing force as the performance index when the transient displacement was input.

Figure 19 shows the input displacement for the bushing 1 stand-alone model in the radial direction. Figure 20 and Figure 21 compare the calculation results and experimental data for the bushing 1 in the radial direction, observed by the time axis and frequency axis respectively. For Figure 21, there is a slight difference between calculation results and experimental data in low frequency, but it can be seen that the reproducibility of bushing 1 radial force is good.

Figure 22 shows the input displacement for the bushing 1 stand-alone model in the axial direction. Figure 23 and Figure 24 compare the calculation results and experimental data for the bushing 1 in the axial direction, observed by the time axis and frequency axis respectively. For Figure 24, there is a slight difference between calculation results and experimental data in low frequency and around 80Hz, but it can be seen that the reproducibility of bushing 1 axial force is good.

Figure 25 shows the input displacement for the bushing 2 stand-alone model in the void direction. Figure 26 and Figure 27 compare the calculation results and experimental data for the bushing 2 in void direction, observed by the time axis and frequency axis respectively. For Figure 27, we can see the difference between calculation results and experimental data over 70Hz, but it can be seen that the reproducibility of bushing 2 in void direction force is good. Therefore, both the material model and the bushing model have been validated.

SUSPENSION ARM VALIDATION

Suspension Arm Model

This section describes the suspension arm assembly model used for validation. The bushing model validated above was incorporated into the suspension arm model shown in Figure 28 and used for the purpose of accuracy validation involving harmonic oscillation and transient inputs. A conventional bushing model (see Figure 29) was also created, and this was compared with the results using the new rubber material model.

Validation Testing

For validation testing, jigs were attached to bushings 1 and 2 as shown in Figures 30 and 31, and displacement input was applied in the vicinity of the ball joint. Figure 32 shows the original position of the suspension arm (unloaded). In the suspension arm test, the ball joint was initially positioned higher in the Z direction than the original position of suspension arm, as shown in Figure 33, in order to emulate the state of a loaded vehicle. The inputs were sine wave harmonic oscillation and a time-series waveform designed to replicate harshness.

Accuracy Validation with Harmonic Oscillation

This section describes validation of the harmonic oscillation characteristics of the suspension arm model. The performance index is bushing force relative to bushing displacement.

Figure 34 compares calculated values for force and displacement of bushing 1 in the Y direction with the corresponding measured values. Clearly the new rubber material model is more accurate than the conventional model at reproducing experimental bushing rigidity.

This can be attributed to the fact that bushing 1 was subjected to multi-axial inputs in the experiment-translational displacement in the Y direction and rotational displacement around the Z axis-which resulted in higher bushing rigidity than for uniaxial load. This phenomenon was successfully reproduced by the new rubber material model.

Figure 35 compares calculated and measured values for force in bushing 2 in the X direction. As area A shows, the new rubber material model produces a higher bushing rigidity value than the conventional model. This can be attributed to increasing rigidity in bushing 2 associated with multi-axial inputs on the bushing at the start of the experiment, namely, rotational displacement around the X axis as shown in Figure 36 together with translational displacement through excitation. These combined to increase rigidity, in the same manner as bushing 1. The results demonstrated that the new rubber material model accurately reproduces changes in bushing rigidity caused by multi-axial inputs for both bushing 2 and bushing 1.

The new rubber material model also models the bushing shape, as shown in area B of Figure 35, and faithfully replicates the timing of the void collapse and the point at which rigidity begins to change. Conventional models can also do this, but only when measurement and modeling is predicated on the bushing characteristics depicted in Figure 36. To perform the analysis for any given initial state using a conventional model would require such a large amount of experimentation to measure the bushing characteristics that it becomes unfeasible.

The new rubber material model, on the other hand, can accommodate any initial state, provided that the material coefficients and bushing shape are known, and can also be used to analyze ongoing changes in vehicle status over time.

Accuracy Validation for Transient Inputs

The compliance of the suspension arm in the vehicle X-direction is controlled by bushing 2. Thus the force in bushing 2 has a significant effect on ride quality. This section focuses on validation of the model for the relationship between bushing displacement and force, particularly in bushing 2 in the X and Y directions.

Figures 30 and 31 illustrate the validation experiments. A shaker was used to impart transient inputs in the vicinity of the ball joint. The displacement waveform at the excitation point determined from the experiments (see Figure 37) was incorporated into the calculation model.

Figure 38 shows bushing displacement in the X direction and Figure 39 compares bushing force in the X direction. It can be seen that the new rubber material model is able to describe the bushing displacement and force waveforms more accurately than the conventional model. In particular, the new rubber material model describes the bushing force peak in the experiment with considerable accuracy. Figure 40 shows bushing displacement in the Y direction and Figure 41 shows bushing force in the Y direction. Once again, the new rubber material model provides excellent reproduction of experimental results in the Y direction.

There are two main reasons why the new rubber material model is better at reproducing the results of transient input experiments. First, the model is able to reproduce frequency and amplitude dependency simultaneously. This means that it can accurately reproduce bushing force under the experimental conditions shown in Figure 37, where the amplitude is constantly varying. Second, it can accurately reproduce bushing spring characteristics when the initial conditions (as per Figure 36) are subject to translational displacement.

Finally, this study has demonstrated that the new rubber material model provides improved accuracy in describing bushing forces generated by transient displacement inputs acting on the suspension arm.

SUMMARY

This paper has given an overview of the new rubber material model, together with examples of the procedure for finding the material coefficients of bushings 1 and 2. The material coefficients were used to validate the quasi-static uniaxial tensile characteristics and harmonic oscillation characteristics for the material test piece model, as well as static characteristics, harmonic oscillation characteristics and transient characteristics for the stand-alone bushing model. The results demonstrated that a bushing model featuring the new rubber material model can be used to reproduce quasi-static characteristics as well as frequency and amplitude dependency.

The stand-alone bushing model was incorporated into a model for the suspension arm, which was used to validate bushing displacement and force through experiments involving harmonic oscillation and transient displacement inputs acting on the suspension arm. The results demonstrated that the new rubber material model is capable of reproducing changes in bushing rigidity caused by multi-axial inputs as well as nonlinear bushing characteristics caused by phenomena such as void contact.

Future research will concentrate on the application of these findings to vehicle models.

REFERENCES

[1.] Simo, J.C.: On a fully three dimensional finite-strain viscoelastic damage model :Formulation and computational aspects [J], Computer Methods in Applied Mechanics and Engineering, 1987, 60: 153-173

[2.] Long, Phan Vinh., Satoshi, Ito., Kouhei, Shintani.: A nonlinear viscoelastic constitutive model for dynamic behaviors of rubber FISITA 2014 World Automotive Congress, F2014-IVC-006

[3.] Hartley, C. and Choi, J., "Finite Element Overlay Technique for Predicting the Payne Effect in a Filled-Rubber Cab Mount," SAE Int. J. Passeng. Cars - Mech. Syst. 5(1):413-424, 2012, doi:10.4271/2012-01-0525.

[4.] Nakahara, J., Yamazaki, K., and Otaki, Y. , "Rubber Bushing Model for Vehicle Dynamics Performance Development that Considers Amplitude and Frequency Dependency," SAE Int. J. Commer. Veh. 8(1):117-125, 2015, doi:10.4271/2015-01-1579.

[5.] Marckmann , G. and Verron, E.: Comparison of hyperelastic models for rubber-like materials, Rubber Chemistry and Technology Vol.79(5), 2006, 835-858

[6.] Dassault Systemes Simulia Corp.: Abaqus User Manual

Masahiro Ueda, Satoshi Ito, and Daichi Suzuki

Toyota Motor Corporation

Table 1. The material coefficients

                                 Bushing 1
Hyperelasticity  C1        C2      C3      D1
parameters       5.4413   -0.6388  0.247S  0.01
                 Network  Ai       yi      mi
                 No.

                 0            -    0.1000
Viscoetasticity  1           10    0.0989
                 2          100    0.0743  0.897
                 3        10000    0.3525
                 4        50000    0.3744

                                 Bushing 1
Hyperelasticity  C1       C2      C3     D1
parameters       4.403   -0.014  -0.269  0.01
                 Network  Ai     yi      mi
                     No.

                  0           -  0.1000
Viscoetasticity   1          10  0.1795
                  2         100  0.1857  0.790
                  3         500  0.2365
                  4       50000  0.2483
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Article Details
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Author:Ueda, Masahiro; Ito, Satoshi; Suzuki, Daichi
Publication:SAE International Journal of Passenger Cars - Mechanical Systems
Article Type:Report
Date:Apr 1, 2016
Words:3333
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