Understanding strain sensitivity effects in vibration isolators.
The natural frequency of an isolator is related to the dynamic stiffness of the isolator, as follows
[f.sub.n]=3.13 ([K.sub.d]/W)[1/2] (1) where Kd = dynamic stiffness and W = isolated weight.
When an isolator is being designed for a specific resonant frequency, Equation 1 is rearranged to determine the design dynamic stiffness ([K.sub.d]) of the isolator, [K.sub.d] = W ([f.sub.n] / 3.13) (2)
Dynamic stiffness of an isolator is the force required to produce a unit displacement, and is a function of both the isolator configuration and the elastomeric material.
The dynamic shear stiffness of an isolator is directly proportional to the dynamic shear modulus (G') of the elastomer, as shown by
[K.sub.ds] = G' A / t (3) where [K.sub.ds] = dynamic shear stiffness; G' = dynamic shear modulus; A = load area; and t = elastomer thickness.
One of the important characteristics of elastomers is the fact that they are essentially incompressible. This means that when the elastomer is subjected to a compressive load, the volume remains constant. If the loaded surfaces are bonded to a rigid plate, the compressive load will cause the rubber to bulge at the ends, resulting in a potentially significant increase in compressive stiffness. The degree of bulging is related to the shape factor (S) of the isolator the ratio of the compressive load area to the area free to bulge.
The dynamic compression stiffness of an isolator is of the form (refs. 1-3)
[K.sub.dc]= [E' (1 + b [S.sup.2])] A / t, (4)
where [K.sub.dc] = dynamic compression stiffness; E' = Young's modulus (compression); b = constant, 1 - 2, depending upon material and fillers; S = shape factor, dimensionless;
A = load area; and t = elastomer thickness.
Equations 3 and 4 assume a linear load deflection behavior of the isolator. While this assumption is never precisely correct, it is reasonably accurate for most practical vibration applications.
From Equations 3 and 4, it is seen that a change in dynamic modulus (G' or E') will result in a proportional change in dynamic stiffness. Since natural frequency is proportional to the square root of stiffness (Equation 1), it can further be seen that natural frequency is proportional to the square root of dynamic modulus.
From the classical theory of elasticity (ref. 1), compression modulus is related to shear modulus and Poisson's ratio [micro] by
E'= 2G'(1 + [micro]) (5) For elastomer elements which are not highly restrained, [micro] can be taken as equal to 0.5. Thus,
E' = 3 G' (6)
This relationship is valid for untilled polymers, but as fillers are added to the base polymer to increase stiffness, the constant can vary from three to four (refs. 2 and 3).
In addition to modulus, the designer is interested in the degree of damping in the elastomer. Transmissibility is a dimensionless expression, and is defined as the ratio of output force to input force. Transmissibility is a measurable characteristic of vibration mounts, and is related to several frequently used measures of damping, as shown below
Transmissibility = 1/[2(C/[C.sub.c])] (7a)
= 1/loss factor [Pi] ) (7b)
= 1/tan delta (7c)
These relationships are approximate, but are generally used as is for transmissibility greater than three.
Both modulus and damping characteristics of elastomers, whether in shear or compression, are dependent upon frequency, strain and temperature. Since every polymer and filler can affect these values, the isolator manufacturer needs detailed knowledge of these characteristics for the materials he uses. To be useful to the designer, this information must be genetic, so that it can be used for isolators of any shape and size.
From Equation 6, compression modulus is related to shear modulus, and both parameters exhibit similar physical dependencies (frequency, temperature and strain). As a result, measurements may be made for either mode of loading. Shear loading is most frequently used, since elastomer in shear provides a linear load deflection response over a greater range of strain than does compression loading. However, compression tests may be used, provided allowance is made for inherent geometry related nonlinearities in the specimen.
As generally understood, strain sensitivity of elastomers means that the dynamic modulus of an elastomer (or the stiffness of the resultant isolator) varies with dynamic input. In general, it is an inverse relationship, in which dynamic modulus increases with decreasing dynamic input (i.e., vibration input). The degree of sensitivity is a function of both the polymer and the additives which comprise the elastomeric compound. The amount of damping can also vary, although this variation is frequently a secondary consideration.
The fact that elastomers exhibit strain sensitivity has been recognized for many years. Reference 4 presents a thorough discussion of the dynamic mechanical properties of several common elastomers, including definition of variations in these properties with frequency and dynamic strain amplitude. For all the elastomers tested in reference 4, it was noted that all elastomers went through the same transition of dynamic mechanical properties in an identical range of dynamic shear strain. This transition occurred in the narrow range of dynamic shear strain amplitude from 0.001 to 0.005 inches per inch (.1% strain to .5% strain), regardless of frequency. Below this range of strain, modulus and damping did not vary, whereas, at higher strains, dynamic modulus and damping exhibited strain sensitivity to varying degrees, depending upon the material. For many vibration applications, the imposed strain is above this transition region, and strain sensitivity is the rule, rather than the exception.
Reference 5 addresses the effects of both geometry related nonlinearities (load deflection characteristics) and material related nonlinearities (strain and frequency sensitivity) on the vibration isolation characteristics of rubber components. Reference 5 states that nonlinearity present in a material is indicated by the degree of distortion that occurs in the hysteresis loop. Figure 1 (a) (from reference 5) shows hysteresis curves in shear for a range of untilled rubbers; the distortion is negligible, and the shapes are unaffected by strain amplitude. Figure 1 (b) shows hysteresis loops for a material exhibiting strain sensitivity. Realizing that the slope of the end points is proportional to dynamic modulus, it is seen that the slopes increase with decreasing strain amplitudes. Consistent with reference 4, reference 5 also identifies a transition zone of .2% strain below which the material behaved almost linearly.
Reference 5 also presents vibration transmissibility data for an experimental vulcanizate under three constant shear strain dynamic conditions. For the three test conditions, measured natural frequencies occur at approximately 28 Hz, 40 Hz and 70 Hz for shear strains of .1 in./in. (10% strain), .02 in./in. (2% strain) and .002 in./in. (.2% strain), respectively. In addition to these changes in natural frequency, there is an obvious increase in maximum transmissibility for the lower shear strain condition. In terms of vibration isolation, these curves indicate that if the vibration isolator were selected as a 30 Hz isolator, and if it were then subjected to the extremely light .002 in./in. shear strain condition, a significant loss in vibration isolation would result. For those applications in which fairly precise vibration control is required, the isolator designer requires accurate information on the dynamic environment, as well as for the materials he is using.
References 6 and 7 present additional data regarding the strain sensitivity of elastomers. Reference 7 also identifies the effect of strain sensitivity on vibration transmissibility when the dynamic condition is not a constant shear strain (constant input displacement), but a constant force input (constant input acceleration). As shown in figure 2, the transmissibility "curves lean backward, an effect which is obtained with mechanical systems in which stiffness decreases with amplitude." The increased stress decreases the dynamic modulus of the elastomer (i.e., stiffness of the mount), and thereby decreases the resonant frequency of the isolation system.
Reference 8 also discusses strain sensitivity effects in elastomers, and specifically addresses the difference between a constant acceleration input and a constant displacement input. Figure 3 shows a significant increase in the complex shear modulus over the frequency range 0 to 100 Hz, when the input is a constant acceleration. It should be noted that displacement is related to acceleration as the inverse of the square root of frequency. Thus, on figure 3, the input displacement at 100 Hz would be one fourth the input displacement at 50 Hz.
As previously stated, the dynamic modulus and damping characteristics of elastomers are dependent upon frequency, strain and temperature. In the characterization of dynamic properties of elastomers, it is standard practice to report one of these variables as a function of another, holding the third variable constant. This is a necessary simplification, as the number of permutations would approach astronomical limits.
In practice, however, nature does not recognize these simplifications, and vibration isolators may be subjected to any combination of these parameters. For most applications, either the equipment is not so delicate, or the changes in dynamic performance are not so great, that these sensitivities are not extremely critical. However, there are times when each variable needs to be considered in the design stage.
Acknowledging that both frequency and temperature play a role in vibration isolation, this article concentrates on the effect of strain on the performance of vibration isolators. It should be recognized that every elastomeric compound has its own unique set of frequency/strain/temperature sensitivities, and that these may combine in a different manner than for the material presented herein.
This article presents dynamic modulus data for a highly damped silicone compound as a means of defining material characteristics separate from final molded product performance. Vibration data are then introduced to demonstrate what strain sensitivity means in terms of vibration isolation. General comments are also included which identify potential strain sensitivity effects on temperature sensitivity and fatigue.
Material characterization - modulus and damping
The silicone compound has been evaluated dynamically using compression specimens. A static preload is applied to the specimen, and a dynamic strain superimposed on the specimen over a range of frequencies. The prestrain and the dynamic oscillations are selected to minimize the effect of geometrical nonlinearities in the specimen.
The test equipment used to perform the dynamic modulus tests is an MTS 831, with accessories and software designed to evaluate elastomeric materials.
Figure 4 shows dynamic compression modulus data (E') for the subject compound over the frequency range of 5 Hz to 100 Hz at imposed dynamic strains of 1%, 2% and 5%. As indicated, dynamic modulus increases with frequency. For instance, at 2% strain, the dynamic modulus at 100 Hz is approximately 30% greater than that at 5 Hz. Figure 4 also indicates the strain sensitivity of the material. For instance, at a frequency of 20 Hz, E' varies from 9.9 MPa to 4.6 MPa over the 1% to 5% strain range.
From the preceding, it is clear that strain sensitivity can have a larger effect on changes in dynamic modulus than frequency sensitivity.
Figure 4 also shows tan delta for the same material and test conditions, and indicates a minor frequency and strain sensitivity. Minimum damping occurs at 1% strain, with tan delta approximately .39 over the frequency range 5 Hz to 100 Hz.
Vibration results - constant displacement
Figure 5 shows sinusoidal vibration transmissibility curves for several constant input displacements. The elastomer is the highly damped silicone previously discussed. Results are summarized in table 1.
From figure 5, the strain sensitivity is apparent, as the entire curve translates left to fight with decreasing input. There is very little change in damping. Several specific observations can be made from figure 5 and table 1:
* Natural frequency increases with decreasing input displacement; there is no apparent distortion in the transmissibility curves. This is as would be anticipated from previous discussions;
* The crossover frequency is the frequency at which vibration isolation begins. Theoretically, the crossover frequency is 1.414 times the natural frequency of the isolation system. For the five test conditions, the crossover frequency varied from 1.40 to 1.51 times the natural frequency, in reasonable agreement with theory. Thus, for the constant input displacement, the crossover frequency moves consistently with the isolator natural frequency.
* The theoretical roll off rate for a vibration isolator is 12 dB/octave. For the five test conditions, the roll off rate varied from 12.3 to 14.4 dB/octave, again, reasonably close to theoretical values. Thus, for the constant displacement input, the slope of the transmissibility curve does not change significantly.
From figure 5 and table 1, it is clear that strain sensitivity affects transmissibility over the entire frequency range not just at the isolator natural frequency. For the constant displacement condition, the entire curve translates left or right. This raises the natural question - what happens if the strain changes over a frequency range removed from the isolator natural frequency?
This condition is shown in figure 6, which presents a single vibration transmissibility curve for a series of four discontinuous constant displacement inputs shown in table 2.
This artificial test condition is helpful in depicting the effect of strain sensitivity, since it graphically demonstrates that dynamic response at higher frequencies can change independent of the isolator frequency, if dynamic inputs differ. As shown in figure 6, there are obvious breaks in the transmissibility curve which coincide with the breaks in the constant displacement input. It is thus seen that although the natural frequency is measured at 37 Hz, the response at higher frequencies reacts as if the natural frequency were higher, the degree of variation depending upon the decrease in the dynamic input.
Vibration results - constant acceleration
There are many practical situations in which the vibration input is not a constant displacement, but a constant acceleration. For a sinusoidal input, displacement (X) is related to acceleration (g) by:
X(D.A.) = g(pk) / [(k) [f.sup.2]] (8)
where g = acceleration (pk); X = displacement, (D.A.); f =
frequency, Hz; and k = constant - .051 [for X(D.A.) in inches], .002 [(for X(D.A.) in mm].
Figure 7 shows the relationship of displacement and acceleration as a function of frequency for several constant acceleration inputs. Referring to figure 7, if an isolator which is subjected to a 1 g (pk) vibration input has a natural frequency of 30 Hz (where the input displacement is .56 mm (D.A.)), at 60 Hz it will respond as if to a lighter displacement input of 0.14 mm (D.A.). For the lighter input, the modulus will be higher, and the transmissibility curve at this particular frequency will shift to the fight. Similarly, at 90 Hz, the input displacement would be .06 mm (D.A.), the modulus would be higher still, and the transmissibility at this particular frequency would again shift to the fight.
From the preceding, when the vibration isolator is subjected to a constant acceleration input, one might expect it to have a crossover frequency greater than the theoretical value of 1.414.
Figure 8 shows vibration transmissibility curves for four constant acceleration inputs - 1/4 g(pk), 1/2 g(pk), 1 g(pk) and 2 g(pk). Results are summarized in table 3.
Consistent with previous results, the natural frequency decreases with increasing input. For a constant acceleration input, the transmissibility curve above resonance shifts to the right and the crossover frequency is greater than the theoretical value of 1.414. For the four test conditions, the crossover frequency varied from 1.72 to 2.12 times the natural frequency, compared to the theoretical crossover frequency ratio of 1.414. Comparison of figure 8 to figure 5 indicates that the constant acceleration transmissibility curves are broadened out - extended to the fight.
A further review of figure 8 shows that although the natural frequencies for the different inputs vary significantly - from 37.7 Hz to 79.0 Hz - at higher frequencies, the transmissibility curves appear to be converging near a frequency of 200 Hz. In other words, the roll off rates are not constant with acceleration input. For these conditions, the roll off rates vary from 15.0 dB/octave to 7.8 dB/octave, up to a frequency of 200 Hz.
All elastomers exhibit temperature sensitivity to varying degrees. At low temperatures, dynamic modulus increases, and at high temperatures, dynamic modulus decreases. Damping also varies with temperature - increasing (i.e., transmissibility decreases) at low temperatures until a transition temperature is reached, and decreasing at high temperatures.
When a vibration isolator is subjected to a constant acceleration input at temperature extremes, an additional set of sensitivities can come into play. To illustrate, if an isolator has a natural frequency of 30 Hz when subjected to a 1g (Pk) sinusoidal vibration input at room temperature, the input displacement at this frequency is .56 mm (D.A.) (refer to figure 7). When the temperature is decreased to, say, -65[degrees]F, the dynamic modulus and isolator stiffness increases, increasing the isolator natural frequency. This increase in frequency causes the isolator to function in a region of the acceleration/displacement curve in which the input displacement is reduced. From the preceding example, if the isolator natural frequency increases from 30 Hz to 55 Hz due to low temperature stiffening of the elastomer (at a constant strain), the "new" displacement input is .15 mm (D.A.). This reduced displacement input decreases the strain on the isolator, resulting in a secondary increase in dynamic modulus and natural frequency. Finally, at low temperatures, any increase in damping further decreases strain on the elastomer, contributing to the secondary increase in dynamic modulus.
The net result of these effects is an increase in natural frequency which is greater than would have been anticipated based solely on constant strain dynamic modulus values at various temperatures. For the constant acceleration input, this increase is due to a combination of temperature and strain effects.
Fatigue in elastomeric vibration isolators is a complex subject, and is well beyond the scope of this article. Nevertheless, since dynamically induced strain almost invariably contributes significantly to those isolator failures which are not climatic, there are some aspects of strain sensitivity which should be mentioned.
In normal operation, most vibration isolators are exposed to a benign-to-moderate vibration environment. However, high amplitude sinusoidal vibration accelerated life tests are frequently specified to provide some assurance that the isolator will function satisfactorily over the intended life of the isolated product.
If the high amplitude input is a constant displacement input, a normal reduction in natural frequency may occur due to strain sensitivity effects, but no secondary strain sensitivity effects would be expected (frequency changes, heat build up and isolator geometry considerations may still require review.) However, if the high amplitude input is a constant acceleration, secondary strain sensitivity effects may occur.
Many accelerated life tests consist of several cycles in which each cycle would consist of a sweep up in frequency immediately followed by a sweep down in frequency. Referring again to figure 7, one cycle could include a 2 g (Pk) sweep from 10 Hz to 100 Hz, followed immediately by a downward sweep in frequency from 100 Hz to 10 Hz. For the constant acceleration input, the downward frequency sweep typically has a different transmissibility curve than the upward frequency sweep. Figure 9 shows transmissibility curves for a vibration mount with the highly damped silicone subjected to an upward and downward sweep with a constant acceleration input of 2 g (Pk). The characteristic difference lies in the fact that the amplification region becomes broader, and the natural frequency is lower for the downward sweep in frequency. This characteristic appears related to the previously discussed aspect regarding constant acceleration inputs. Specifically, as frequency decreases during the downward frequency sweep, the input displacement increases, resulting in a reduction in dynamic modulus and natural frequency. As seen in figure 9, the difference for this particular mount, material and strain condition is not large, but a mount with a different material or a higher strain condition may exhibit substantial differences. In some circumstances, internal heat build up within the elastomer can contribute to additional softening with potentially catastrophic failures. When tests such as these are intended to simulate an accelerated life test, it is not uncommon to either provide for external cooling of the isolator or to permit the test to be stopped to allow the mount to cool down.
The fatigue characteristics of isolators are frequently evaluated by random vibration tests. One of the advantages of random vibration testing is that the vibration input (power spectral density, or PSD) can be tailored to the specific platform being evaluated. The platform can be an aircraft, marine vessel or a land based vehicle. Each platform can have a unique vibration profile, which can be measured, recorded and specified for the components on the platform, including vibration mounts. The vibration profile may consist of a single power spectral density over a specified frequency range, or it may include any number of amplitude steps with increasing or decreasing amplitudes at higher or lower frequencies. This flexibility in defining dynamic environments provides unique challenges to the designers of vibration isolators, because there are now an infinite number of strain conditions to which a mount may be subjected.
For purposes of illustration, figure 10 shows three random vibration input curves, plotting power spectral density ([g.sup.2]/Hz) as a function of frequency.
Curve A shows a constant PSD over a wide range of frequencies. This should not be confused with the constant displacement input previously discussed for sinusoidal vibration, since the input is in terms of power. For a constant PSD such as this, relative displacement increases with decreasing frequency, if damping remains constant.
For a flat PSD such as curve A, the RMS displacement is given by
[delta]rms = 12.25 [(PSD) T / [f.sup.3] ]1/2 (9)
where [delta]rms = RMS displacement of isolator; PSD = random vibration input, power spectral density; f = frequency, Hz; and T = transmissibility at resonance.
From equation 9, it is seen that for a flat PSD input, displacement is inversely proportional to [f.sup.1.5], compared to [fsup.2] for sinusoidal vibration with an acceleration input (see equation 8). As a result, strain sensitivity effects are not generally severe for such an input.
Curve B shows a random vibration input with a PSD which decreases rapidly with increasing frequency. Selection of an isolator to function on the sloping portion of the curve requires careful consideration of materials and isolator size. The strain sensitivity effects are comparable to those previously discussed under "Vibration results constant acceleration," except that the effects can be magnified when the specified random vibration input has a steep slope. Under these conditions, a mount may soften, resulting in a decrease in natural frequency. This would cause the mount to operate at a natural frequency where the input is more severe, resulting in a further decrease in stiffness and natural frequency. In severe tests, the elastomer may heat up, due to internal hysteresis, accelerating this softening process. For these conditions, not only should strain sensitivity effects be evaluated, but vibration induced internal heat generation must be considered. There are too many variables involved in this dynamic environment to provide general guidelines, but the designer should be alert to the considerations identified.
Curve C shows a random vibration input with a PSD which increases with increasing frequency. A curve such as this does not generally result in any strain sensitivity effects which are damaging to the vibration mount. For instance, if there is a reduction in stiffness of the mount, the natural frequency would decrease to a region where the dynamic input is reduced, and the dynamic modulus would increase, tending to move the natural frequency to its original value.
Fatigue of vibration isolators is generally controlled by the vibration input in the frequency range near the isolator natural frequency. Failures are often related to high amplitude strain at resonance, and seldom related to high amplitude inputs at frequencies well removed from the isolator natural frequency. Consequently, potential strain sensitivity effects should be identified in the early stages of design.
The use of elastomers in vibration isolators requires an understanding of the dynamic characteristics (modulus and damping) of the materials being used. For many applications, material information alone is not sufficient, as the designer must be able to relate material data to the specific dynamic environment under evaluation.
Without an understanding of the material data and the dynamic environment, interactions between frequency, strain and temperature can compromise the performance of an isolation system.
1. P.K. Freakley and A.R. Payne, "Theory and practice of engineering with rubber, "Applied Science Publishers Ltd., London, 1978.
2. P.B. Lindley, "Engineering design with natural rubber," Natural Rubber Producers Research Association, London, 1970, 3rd edition.
3. J.C. Snowdon, "Vibration and shock in damped mechanical systems, "John Wiley & Sons, New York, 1968
4. G.E. Warnaka, "Dynamic strain effects in elastomers," Rubber Chemistry and Technology, Vol. XXXVI, No. 2, April - June, 1963.
5. J. Harris and A. Stevenson, "On the role of nonlinearity in the dynamic behavior of rubber components," Rubber Chemistry and Technology, Volume 59.
6. J.C., Snowdon, "Vibration isolation: use and characterization," The Journal of the Acoustical Society of America, Vol. 66. No. 5, November 1979.
7. A.R. Payne and R.E. Whittaker, "Low strain dynamic properties of filled rubbers," Rubber Chemistry and Technology, Volume 44, 1971.
8. M.J. Gregory, "Dynamic properties of rubber in automotive engineering,, Elastomerics, November 1985.
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|Author:||Andrews, Francis J.|
|Date:||Nov 1, 1992|
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