# Evaluating dynamic properties of polymeric isolators.

Polyurethane cellular materials have been used in track insulator and other vibration insulting systems in the railtrack (ref. 1). They have offered good vibration and noise insulating performance. In order to achieve effective anti-vibration performance, certain specifications have to be laid down to define the products. however, due to the complex nature of polymeric materials, the dynamic properties of polymeric isolators are not well presented in some specifications. Some criteria have been defined as critical parameters with tight control limits. This may mislead the design engineers who are assessing different materials or design. This article intends to explain the relation between the dynamic properties and anti-vibration performance of polymeric isolators. The static stiffness are also the important topics in this discussion.

General dynamic properties of polymeric isolators

The polymers have quite complicated dynamic properties which includes viscoelasticity, nonlinearity and hysteresis (refs. 2 and 3). Despite the complexity of the material nature, it is possible to use a generalized dynamic model to present or describe the dynamic behaviors of polymeric isolators. The model consists of two elements - elastic spring element and general damping element. One is in-phase component and the other is out-phase component (figure 1).

The dynamic load-deflection equation is given as:

c([omega], T, ...) dX/dt + K([omega],T,...) X = F (1)

where X and F are complex. C is the damping and K is the spring stiffness. Both are the functions of temperature T, frequency [omega] and other factors.

When the dynamic signals are sinusoidal, the dynamic stiffness can be presented as |F|/|X| - the ratio of load amplitude |F| to displacement amplitude |X|.

|F|/|X| = [[[C[omega]).sup.2]=[(K).sup.2].sup.1/2] = DS([omega],T,...) (2)

Inequation s, the dynamic stiffness is a function of multifactors. Those main factors are listed as:

* Frequency;

* Temperature;

* Amplitude of displacement;

* Preload level which determines the mean position of deformation.

In order to demonstrate the frequency-dependent stiffness, an isolator of PU foam has been tested up to 30 Hz. The results are given in figure 2. Over this relatively low frequency range, the stiffness varies from 0.51 kN/mm to 0.68 kN/mm (0.51/1 Hz, 0.54/5 Hz, 0.54/5 Hz, 0.60/10 Hz, 0.65/20 Hx, 0.68/30 Hz). The question is, what effect on isolation performance can be caused by the frequency dependent stiffness? This will be discussed in the next section.

The relation between C and K is given as

Tan [alpha] = (C[omega])/(K) (3)

[alpha] is called the loss angle which is the difference between the load and displacement signals. Tan [alpha] is an important value which is commonly used by engineers in vibration control systems. It is the indicator for the ratio of dissipated energy to the elastic energy. the dissipated energy is irreversible and elastic energy is reversible. Unfortunately, this parameter is usually neglected in the specification for isolator.

Discussion on static stiffness and dynamic stiffness

The static stiffness is usually defined as the slope on static load-deflection curve which is df/dx/. Most polymeric isolators have a non-linear load deflection curve. Therefore, the static stiffness is a variable. Atypical load-deflection curve usually has "S" shape (figure 3). Through design, the shape of the curve can be changed. The load-deflection curve is obtained through a large deformation cycle, which may present 30% compression of original height of isolator. For example, the load-deflection of a 30 mm height pad is given in figure 3. At 8 mm compression, the static stiffness is given as 0.262 kN/mm.

the dynamic stiffness is a quite different concept for polymers. There is a little relation between static stiffness and dynamic stiffness on polymeric materials, that is quite opposite to steel coil springs. In some specifications for polymeric isolators, an artificial value is given for the ratio of the dynamic stiffness to the static stiffness. The value is usually in between 1.5 to 2.5. This value has been strictly laid down and sometimes becomes the only criterion to assess the dynamic properties. To make this assessment even more difficult, in the specification, very little explanation is given in the detail procedure of dynamic testing. The data of dynamic stiffness can vary form one testing condition to another. This condition can be easily ignored. However, the testing method is not the topic in this article.

An experiment was designed to investigate the difference between the static stiffness and dynamic stiffness. The static load-deflection curve is given in figure 3. A dynamic test was also conducted at 8 mm pre-compression with 0.5 mm amplitude sinusoidal deformation superimposed on the pre-compression. As we know, the static load-deflection curve is measured under very low speed of loading, which is regarded as static load. If we use low frequency such as 1 Hz, we may expect that the dynamic stiffness under this test condition should be the same to the static stiffness which is the tangent of the load-deflection curve at 8 mm. At 1Hz, the dynamic stiffness at 8 mm is 0.51 kN/mm, which nearly doubles the static stiffness .262 kN/mm. at 16 mm pre-compression the static stiffness is 2.5 kN/mm and the dynamic stiffness at 1 Hz and 0.5 mm amplitude is 2.66 kN/mm which is rather close to the static stiffness. From this result, the low speed of dynamic loading does not produce the static stiffness obtained in the static test. This implies that the static stiffness does not gave any indication of the dynamic stiffness. There is no apparent relation between the two stiffnesses. Therefore, why should we make a link between the static stiffness and dynamic stiffness?

Effect of dynamic stiffness on vibration insulating performance

In this section, a single degree of freedom system with a polymeric isolator is under study. It is quite common to use the transmissiblility to evaluate the isolation performance of polymeric isolators. The transmissibility indicates the ratio of response force to the excitation force.

Tr = F2/F1 = [[[(1+ [Tan].sup.2] [alpha] - [[omega].sup.2] M/K).sup.2] + [[omega].sup.4] +[M.sup.2] [Tan.sup.2] [alpha]].sup.0.5]/[[(1-[[omega].sup.2] M/K).sup.2] + [Tan .sup.2]

where Tan [alpha] is given in equation 3.

The typical curve of transmissibility is given in figure 5. The resonant frequency is mainly determined by the dynamic stiffness. Most polymeric material have frequency dependent dynamic properties to a certain extent In general, the frequency dependent characteristics can be tolerated for engineering application. Now, we try to find out the variation of transmissibility caused by the frequency dependent stiffness over a frequency range up to 30 Hz. The reference system is assumed to have constant dynamic stiffness. the transmissibility curve is shown in the curve a in figure 5. If another system has a frequency dependent stiffness, the dynamic stiffness gradually increases to double the initial stiffness through a linear function. the transmissibility curve of this system is given by the curve B. There is a little difference between the tow curves. But the resonant frequency is about the same. the difference in isolation performance between the two system is negligible.

From the above analysis, it is not quite right to say the frequency dependent stiffness is not suitable for anti-vibration systems. Through a proper design method, the target of isolation performance can be achieve with alternative material with different dynamic properties.

Conclusions

It is not necessary to link the static stiffness obtained from the tangent line of load-deflection curve. There is no clear relation between these two parameters. It is even worse to imposed a fixed ratio of dynamic stiffness to static stiffness. This ratio does not guarantee the optimum isolation performance. It may impose a constraint in the freedom of material selection to achieve optimum design.

The isolation performance should be evaluated through transmissibility and other values with similar nature. The frequency dependent properties can be therefore evaluated according to its influence in isolation performance. different materials with different characteristics may achieve similar performance through proper design methods.

[Figure 1 - damping and elastic elements]

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[Figure 2 - frequency dependent dynamic stiffness

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[Figure 3 - typical shape of load-deflection curve of polymeric isolators

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[Figure 4 - single degree-of-freedom system

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[Figure 5 - transmissibility curves with different dynamic stiffness

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References

(1.} F. Ohishi, et al. Rubber World. vol. 204, no. pp. 16-18, Sept. 1991. (2.) J.D. Ferry, Viscoelastic properties of polymers 3rd ed. Wiley, N.Y. 1980. (3.) I.M. Ward, Mechanical properties of solid polymers, Wiley, N.Y. 1971.
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