A One‐Dimensional Lumped‐Mass Model for a Rubber Bushing and Its Application to the Calculation of D

www.ijm‐me.org International Journal of Material and Mechanical Engineering (IJMME), Volume 4, 2015 doi: 10.14355/ijmme.2015.04.003

A One‐Dimensional Lumped‐Mass Model for a Rubber Bushing and Its Application to the Calculation of Dissipation Energy Zhengui Zhang1, Haiyan H Zhang*2 Mechanical Engineering Technology, Purdue University, West Lafayette, IN, US zhang658@purdue.edu; *2hhzhang@purdue.edu

1

Abstract Modal analysis is employed in this paper using hammer impact testing to characterize the dynamic response of rubber bushing.

Based on multi‐body dynamic analysis and acceleration transfer function, the rubber core is decomposed into three pieces, which are joined by spring and dashpot considering the large deformation and damping properties. Parameters of mass, stiffness and damping coefficients are identified through fitting against the Bode plot of the transfer function. With those parameters, the stability of this structure can be evaluated by exploring the critical excitation frequency range, which attributes the failure mechanism of rubber to thermal effect. The dissipation energy in each piece of the rubber core is calculated to demonstrate the accumulation of energy in rubber bushing. This paper successfully employs multi‐body analysis to predict the dissipation energy of rubber components.

Keywords Modal Analysis; Rubber Bushing; Multi‐Body; Damping; Energy Dissipation; Heat Generation

Introduction Rubber bushing and mounting installed on vehicle systems play a critical role in many aspects, such as reducing vibration and noise, absorbing shocks and so on. They are also connection parts and allow a certain amount of movement to separate those metal objects in vehicles’ suspension systems. Rubber bushing working as vibration isolator provides an intermediate damping through the transmission of excitation force to the body of the vehicle[1]. By doing this, the transmission of noise and small vibration across the connecting components of the suspension system can be minimized. [2].

As the primary material working as automotive vibration isolators, rubber has several significant attributes[3]. In comparison with other materials, the viscoelastic rubber’s Young’s modulus is low and its yield strain is high.

Large strain without permanent deformation or fracture is allowed, furthermore, other special properties, like flexibility, extensibility, resiliency and durability turn rubber to be an unmatched material[4]. Rubber has better resilience and higher capacity of energy storage compared with many metal materials[5]. It can improve the riding comfort by reducing the harshness since rubber suspension can minimize the static friction[6]. The loading behaviours of rubber in suspension system are mainly compression, tension and shear[7]. For rubber bushing work as the suspension system, it is desirable to have higher hysteresis to absorb vibration and noise, and be stronger to support certain load. However, the high hysteresis is also related to a basic disadvantage of rubber bushing, which is the unsatisfactory life expectation attributed to fatigue of rubber[8]. While the primary reason accounted for the fatigue is the heat accumulation in the viscoelastic material, which is embodied in the stress‐strain curve. Because of high damping in the rubber material, the loading and response leave a hysteretic loop after cycle loading. The area of the hysteresis loop is dissipation energy, which is stored as heat source inside the rubber components [9].

The physical, chemical and mechanical properties of rubber can be significantly affected by the build‐up of dissipation energy inside the rubber components. The desired damping capacity of rubber components will lose after long term service as it becomes stiffer and ages. Furthermore, temperature rising within the rubber components leads to degradation of its physical and chemical properties, and losing of damping properties. In sum, heat generated within the rubber due to energy loss stimulates the aging process, weakens the material properties, and shortens the desired life expectation of rubber components[10].

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Possible relaxation of heat accumulation in rubber or rubber‐metal components can properly prolong their service life, but the lower heat transfer conductivity slows down the heat transfer to the environment. To predict the temperature distribution in rubber components, the amount of dissipative energy per cycle is needed for further transient‐state or steady state heat transfer analysis. Considering that the researched rubber bushing is installed on the suspension system of the vehicle, the driving speed of the automobile should be taken into consideration to reasonably to estimate the heat generation rate. For convenience, the excitation frequency of rubber bushing can be taken from the rotation frequency of tires, which can be converted from driving speed of vehicles[11]. Thus, the excitation frequency of the rubber bushing can be approximated according to the driving speed of the vehicles and the dimensions of the rolling tires.

One of the widely employed approaches to have the stress and thermal analysis of rubber components is the finite element method, especially with the fast development of the powerful computation ability of computers. However, the complex properties of rubber material result in the roadblock of describing dynamic behaviour of rubber components, which is amplitude and frequency dependent[12]. Appropriately characterized material properties and mathematic expression about the constitutive relationship are significant in predicting the energy losses or heat generation in the rubber components. The dependency is partly attributed to the compositions of rubber, the deformation rate and operating environment. Especially, stress‐strain performance is associated with the percentage of carbon black fillers and the interaction between the fillers and polymers. Thus, the FEA approach exploring the dissipation energy accumulation in the rubber components depends heavy on experimental measurement and computation demanding. As an alternative approach to solve this critical challenge, multi‐body dynamic analysis is employed in this research to investigate the dynamic behaviours and the dissipation energy distribution of rubber bushing.

Multi‐body dynamics is employed to investigate the dynamic behaviours of cylindrical bushing composed with steel sleeves and rubber core. The symbolical theory is created by Roberson and Wittenbury to describe the characteristics of multi‐body and they also developed the computer program MESA VERDE[13, 14]. In the current research, the tree structure developed by Wittenbury is built to construct bodies of bushing and their connections between bodies, while the coefficients of springs and dashpots are identified using the hammer impact modal testing method. The acceleration response of rubber bushing is utilized to detect the proper arrangement of multi‐ body pieces to represent the rubber core from the FFT of acceleration. With the Bode plot of bushing, the parameters of each rigid body, spring and dashpots are identified. Then, the working frequency range of the rubber bushing can be identified to explore the structure stability on the basis of the displacement transfer function.

The last step of this paper is the calculation of dissipation energy in each part of the rubber core using the estimated stiffness and damping coefficients from the impact testing. With this approach, the failure mechanism of rubber bushing under the normal driving condition can be specified between thermal damage and structure instability. That result is significant in providing suggestion about design optimization to prolong the service life of rubber bushing.

The Modal Testing of Rubber Bushing Due to the complexity of many material properties, simplifications become necessary for the pre‐processing part of finite element analysis. This is especially the case for viscoelastic material as it’s highly non‐linear and time‐ dependent. As a matter of fact, most current constitutive models describing viscoelastic material are empirical and experimental. Therefore, it is highly recommended to verify or compensate the numerical analysis with the conclusion derived from the experiment measurement and analysis. As there is no standard modulus of rubber material, the modal analysis is carried out to approximate the transfer function and for precisely calculating the dissipation energy. Hammer impact modal testing is a typical approach to process the structure dynamic problems[15]. One of the most applied and easiest testing measurements is the impulse technique by giving a known impulse force to collect the response of the tested structure. The typical structure of the rubber bushing installed on the suspension system is a sandwich ring with the rubber cord surrounded by the steel sleeves. The profile of the rubber bushing used in this research is sketched in fig 1 and the dimension and density of rubber are given in table 1.The viscoelastic cylindrical core is bonded by inner and outer steel sleeves to keep the required stiffness and minimize vibration. The method most commonly used to combine rubber and metallic or non‐metallic

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components is the use of adhesive cements. Prior to the use of these special adhesives, the surface of the insert must be clean and free of contamination [16, 17]. The displacement of the two rigid sleeves transferring to each other through the rubber tube is frequency dependent because of the frequency dependent modulus of rubber material.

To guarantee the reliable data to acquire the appropriate frequency response (FRF) function in fitting the modal, correct mechanical support is of a great concern. One end of the cylindrical rubber bushing’ inner steel sleeve is fixed to a thick steel column. The experimental platform is depicted in fig. 2 and the steel column is fixed on the ground. The purpose of current experiment modal analysis is trying to identify a theoretical model to represent the approximate dynamic behaviour of the tested structure. Dual channel analysis is applied in processing signals to get the auto spectrum and cross spectrum.

FIG. 1 THE SKETCH OF RUBBER BUSHING USED FOR MODAL TESTING TABLE 1: THE GEOMETRY AND MATERIAL PROPERTIES

Length 0.0508m

Rubber‐inner 0.015875m

Rubber‐outer 0.0319405

Steel‐outer 0.0349885m

Rubber density 952.54 (Kg/m3)

FIG. 2 THE ILLUSTRATION OF EXPERIMENT PLATFORM SETUP

FIG. 4 THE FFT OF ACCELERATION RESPONSE IN RADIAL IMPACT

FIG. 3 THE ILLUSTRATION OF IMPULSE FORCE AND RESPONSE

A softer impact hammer is chosen in this test considering the lower frequency of loading spectrum applied on the bushing during the field test. The hammer is applied in radial direction with similar amplitude to avoid the interference coming from the amplitude effect. The two accelerometers are glued on the rubber bushing to detect the dynamic response, which is used to estimate the FRF of the structure. The type of impact hammer is PCB

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208B01 and the ceramic accelerometer is PCB 333B40. The mass of the accelerometer is 7.5gm and the sensitivity (±10 %) is 51.0 mV/(m/s²). Roving hammer impact test (20 averages) is employed in this testing to minimize the error caused by the background noise. The impact force and output response collected with the acceleration monitor are shown in fig. 3. The sharp and short input can be represented as impulse function, and the general  ξωt

attenuate model of the radial signal can be expressed in exponential form Xe , where X is the initial amplitude of the signal, ω is angular frequency, ξ is the damping ratio and t is time. The two unknown parameters are estimated after fitting the acceleration response, which gives the overall ξ=0.08 in radial direction.

The Parameter Identification of Transfer Function and Structure Stability Fig.4 displays the result of single sided amplitude spectrum of acceleration in radial direction. With this amplitude spectrum, the damped natural frequency of rubber bushing can be identified. The FFT in radial direction indicates that the important frequencies are 51Hz, 132Hz, 310Hz, 337Hz, 355Hz, 397Hz, 413Hz, and 460Hz. With the definition of symbolical graph and tree structure in multi‐body dynamics, the body of the rubber bushing includes at least three parts, internal steel sleeve, rubber core and outer steel sleeve. Considering the balance between redundancy of calculation and accuracy in fitting the Bode plot, three most crucial frequencies are sufficient to develop the transfer function of rubber core. That means the rubber core can be divided into three pieces.

Fig.5 sketches the three layers rubber core of the bushing in view of the three important frequencies in the FFT plot.

According to the multi‐body dynamics, the three pieces of rubber core are defined as rigid body and the elasticity and viscosity of the rubber material are presented using the joints between bodies.

The DOF of rubber bushing shown in fig.6 displays the arrangement of those joints, which is usually composed of spring and dashpot. The mass of inner sleeves is taken as infinite as the inner surface of sleeves is clamped with the land to analogize the cantilever beam. To clarify the principle of partition, the state space equation of this structure is built to characterize its transfer function. In the hammer impact test, the known signals are the input and acceleration response of the outer sleeves, and the remaining variables are unknown and written in the following matrix, X   x1

v1

x2

v2

x3

v3

x4

v 4 18 (1) T

Where, the x i (i  1, 2,3, 4) and vi (i  1, 2,3, 4) are displacement and velocity of each lumped mass.

FIG. 6 THE DOF OF RUBBER BUSHING

FIG. 5 PARTION OF THE RUBBER CORE

With the state space equation, the force equilibrium of the system is presented to describe the relationship between input, output and state variables. The general state space equation of a linear system is given as, 

X  AX  Bu and y  CX  Du (2) 

Where the X  t  is the time derivative of state vector X   , y   is the output vector, u   is the input vector. The state matrix A   , input matrix B   , output matrix C   , and feed through matrix D   are given as,

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 0  k   ex  m ex   0  k ex   m A 1 0    0   0   0 

1

0

0

0

0

0

0

k ex m ex

0

0

0

0

0

0

1

0

0

0

 η1 m1

k1 m1

η1 m1

0

0

1

0

η1  η2 m2

k2 m2

0 

k ex  k1 m1

0

0

0

0

k1 m2

η1 m2

0

0

0

0

0

0

0

0

k2 m3

η2 m3

k k  2 3 m3

 B  0 

1 mex

0

 k C    ex  m ex



0

k1  k 2 m2



   0   0   0    (3) 0  η2  m2    1  η η  2 3 m3 88 0

T

 (4) 0 0 0 0 0 0 18

 0 0 0 0 0  (5) 18

k ex mex

 1  D  (6)  m ex 11

The input vector u   comes from hammer impact and can be described using impulse function δ(t) , t 0  t  0 u  Fex (t)  Fex δ(t) , δ(t)   and  δ(t) dt  1 (7) t 0 0 t  0 



Do the Laplace transform on the Equation (2) and yields the SX(S)  AX(S)  Bu(S) and y(S)  cX(S)  Du(S) (8)

Then, the output vector can be derived from Equation (8), y(S)  C  SI A  B D  u(S) (9)   1

As the transfer function of this system is the ratio of its output vector to its input vector, the transfer function of the clamped rubber bushing is written as,

G(S)  C  SI A  B (10) 1

With the parameters of stiffness, mass and damping coefficients k ex , k1 , k 2 , k 3 , m ex , m1 , m 2 , m3 , b1 , b 2 , and b3 , the transfer function in polynomial can be expressed as, Tf 

p8S8  p7S7  p6S6  p5S5  p 4S4  p3S3  p 2S2  p1S1  p0 q8S8  q 7S7  q 6S6  q 5S5  q 4S4  q 3S3  q 2S2  q1S1  q 0

(11)

Where the pi (i  0,1,,L , 7,8) and q i (i  0,1,,L , 7,8) are combinations of those stiffness, mass and damping coefficient of the three pieces of rubber. In order to estimate the stiffness, mass and damping coefficients in the proposed transfer function of rubber bushing, the transfer function is fitted with the experimental Bode plot as shown in fig.7. Fitting curve 1 is a simple fitting with three curves on the basis of the three crucial frequencies, which indicates that a three‐piece body is appropriate to simulate the Bode plot of rubber bushing. It is necessary to point out that Bode plot of acceleration transfer function is drawn here because the accelerometer is much more precise than the displacement measurement for such micro vibration testing. Fitting curve 2 is sketched on the

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basis of transfer function expressed in Equation (11). It is important to point out that the fitting is suitable for amplitude but not that satisfactory for the phase degree as the disturbance of noise. The fitting results are given in table 2.

FIG. 7 THE FITTING OF TRANSFER FUNCTION AGAINST THE EXPERIMENTAL BODE PLOT TABLE 2: THE IDENTIFIED PARAMETERS OF MODEL

Stiffness(N/m) Mass(g) Damping coefficients(N s/m)

k1=1.8×106 m1=69.9 b1=136.7

k2 = 9.0×106 m2=29.3 b2= 264.2

k3 =1.7×107 m3=48.5 b3=175.1

kex= 1.3×107 mex=255.6 /

With the identified coefficients from the fitting curve 2, the zero/pole/gain of the acceleration transfer function is,







3.914s 2 s 2  1038s  1.748e8 s 2  6320s  2.362 e8 s 2  1.734 e 4 s  6.728e8

s  232.4s 3.947 e  s  1901s 2.166 e s  5220s 2.343e s 2

6

2

8

2

8

2

4



 1.734 e s  6.73e8



(12)

Fig.8 shows the root locus of the estimated acceleration transfer function. In the low frequency range, the zeros and poles are close to each other and form open loop dipole. The effect of the zeros and poles pairs on the stability of the system is trivial due to the effect of cancellation. The more zeros in the transfer function, the root locus shifts more to the left, which greatly benefits the stability of the system. Furthermore, since the zeros are located at imaginary axis, the influence on the stability becomes more important. The zeros are more close to the imaginary axis than the poles in this plot and the dipole near the original or coordinates can enhance the static stability of the system. In sum, it is not convenient to detect the frequency range of structure instability from the acceleration transfer function.

FIG. 9 THE DISPLACEMENT BASED BODE PLOT OF THE CLAMPED RUBBER BUSHING

FIG. 8 THE ACCELERATION BASED BODE PLOT OF THE CLAMPED RUBBER BUSHING

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As a complement, the root locus of the system on the basis of displacement transfer function is plotted in fig.9. That plot shows the critical frequency of structure stability is located at 937.4Hz or 5890rad/s. However, the excitation force transferred to vehicles is around 10Hz, which is much lower than the critical frequency. Thus, the possible failure mechanism of the rubber bushing can be viewed in the aspect of thermal damage.

The Energy Dissipation Mechanism Under the harmonic external excitation, the energy dissipation of the system of rubber bushing in a cycle period T can be calculated using, T





T

2

T



T



 m x(t) x(t)dt   c x (t) dt   kx(t) x(t)dt   F0 cos( t) x(t)dt (13)

0

0

0

0

Where, F0 cos( t) is the sinusoidal force representing the real external excitation of the rubber bushing. The three terms at the left side of the equation are kinetic energy, dissipation energy or damping and strain energy of the system. Kinetic and strain energy are conservative energy and the accumulated energy in one cycle are zeros. Thus, the work done by the external force is equivalent to the dissipation energy inside of the rubber bushing.

FIG. 10 FORCE AGAINST DISPACEMENT OF THE THREE PARTS OF THE RUBBER CORE AND THE ACCUMULATED WORK AT 100Hz EXCITATION

Fig.10 shows the work (N.m) done on the three parts of the rubber bushing at excitation frequency of 100Hz. The left column in fig.10 plots the force against displacement of each piece of rubber. The right column in fig. 10 plots the accumulated work done by the external force calculated through Newton integration method. As to the three parts of the rubber core, the transfer function fitting according to Bode plot indicates lower stiffness of part1

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compared with stiffness of part 2 and part 3. As a result, the displacement of part 1 is larger than that of the other two. Again, the accumulated work or the dissipation energy in part 1 of rubber core is higher than the other two parts.

FIG. 11 FORCE AGAINST DISPACEMENT OF THE THREE PARTS OF THE RUBBER CORE AND THE ACCUMULATED WORK AT 200Hz EXCITATION

The natural frequency of the clamped structure can be taken as 310Hz or 2000rad/s according to the Bode plot of rubber bushing. In the real driving situation and in the following discussion, the external excitation frequency is less than 100Hz, and the dissipation energy loop formed by force/displacement or the loop area formed by stress/strain is also much smaller. Before reaching the natural frequency, the closer of the excitation frequency is to the natural frequency, the larger is the enclosed area of the loop because of the resonance effect. Again, the enclosed area is just the energy dissipation per cycle, at high frequency, more cycles’ dissipation energy are accumulated in unit time, which accounts for the accumulated energy at high frequency. When the excitation frequency is pretty high, which may approximate the first natural frequency, dissipation energy in per cycle and unit time can turn to pretty high. As to the complicated modal shape at higher order modal, strain energy needed to launch that deformation is also pretty large. In general, the dramatical accumulation of the dissipation energy at higher frequency is converted to the heat source inside the rubber components and causes the potential failure.

As comparison, the dissipation energy of the three parts of the rubber core accumulated at the excitation frequency 200Hz is given in fig.11. The higher generation rate of dissipation energy at 200Hz is expected from the higher displacements at that frequency. Since the 200Hz is closer to the first natural frequency of the clamped rubber bushing, the resonance effect can properly explain the higher displacements and corresponding higher dissipation energy.

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Using the multi‐body analysis, the dissipation energy of each part of the rubber core is estimated. With the preset loading and frequency, the displacement of each piece can be calculated using the fitted transfer function. Since stiffness of each part is higher than the whole piece of rubber core, the corresponding displacement of each part is lower than the whole piece. The dissipation energy of each piece of the rubber core is calculated individually from the loop formed by plotting force against displacement of each lumped mass. That approach is similar to the principle used in the FEA while the later usually estimates dissipation energy with loop area formed by plotting stress against strain. Especially, in order to predict the temperature distribution of the rubber bushing, the rubber core has better to split into many pieces as done in the finite element analysis. In that approach, each element is a unique piece with lumped mass and distinct dissipation energy calculated from the stress/strain of each interpolation points in each element.

Conclusions In the current analysis, one of the multi‐body dynamics analysis approach, symbolical graph is employed to format the structure of rubber bushing and characterize its dynamic properties. The tree structure of rubber bushing is constructed to facilitate the development of transfer function. Hammer impact modal testing is carried out to indicate the partition of rubber core for multi‐body analysis. With the FFT of the acceleration response of rubber bushing upon the hammer impact, three pieces of lumped mass to represent rubber core is proposed. The parameter of mass, spring and dashpot are identified with the curve fitting optimization method on the basis of Bode plot. The structure instability of rubber bushing is predicted using Bode plot of displacement transfer function assuming the zero initial value condition. Since the general driving frequency of vehicle is much lower than the critical frequency leading to the structure instability, the dominant failure mechanism can be attributed to the heat generation of rubber core. Furthermore, the dissipation energy accumulated in each part of the body is estimated, which shows dissipation energy over each loading cycle at different frequency. The lumped dissipation energy density gives the motivation to explore the more detailed density of dissipation energy and temperature distribution of rubber bushing during service.

REFERENCES

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stiffness of carbon‐black filled rubber bushings,ʺ Polymer testing, vol. 26, pp. 629‐638, 2007.

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Appendix: The scalar equations of motions for the model 















x1  v1 ; m ex v1  Fex  t   k ex  x1  x 2  ;

x 2  v 2 ; m1 v 2  k ex  x1  x 2   k1  x 2  x 3   η1  v 2  v3  ; x 3  v3 ; m 2 v3  k1  x 2  x 3   η1  v 2  v3   k 2  x 3  x 4   η2  v3  v 4  ; x 4  v 4 ; m3 v 4  k 2  x 3  x 4   η2  v3  v 4   k 3 x 4  η3 v 4 ; Zhengui Zhang: Ph.D. 2015, Mechanical Engineering Technology, Purdue University, West Lafayette, Indiana, USA Haiyan H Zhang: Ph.D. 1996, Associate professor at Mechanical Engineering Technology, Purdue University, West Lafayette, Indiana, USA

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