Dynamic properties of elastomers as related to vibration isolator performance.
An engine mount must perform two basic functions:
* It must support the static weight of the engine; and
* It must isolate the vibration caused by imbalances in the engine from the frame of the car. Similarly it must control motion of the engine due to vibration input from the frame of the car.
It must perform these functions under conditions which prevail under the hood of the car for long periods of time. These conditions include both high and low temperature and limited automotive fluids exposure.
Linear single degree of freedom system The simplest depiction of a vibration isolator is a linear single degree of freedom system represented by a mass supported by a spring and dashpot in a parallel combination. The name conveys a great deal of information about the system. Linear means that the spring has a unique spring constant that does not change with deflection, frequency or temperature. Single degree of freedom means that the mass is constrained to translate in one direction only, it is not permitted to rock, rotate or translate in other directions. Also implied is the condition that all of the mass in the system can be considered as being contained in the supported mass.
Isolators constructed using elastomeric mounts obviously do not meet these requirements. It is important however, to start with these idealizations in order to understand the behavior of more complex non-linear systems. Figure I is a plot of the response of a linear system as a function of excitation frequency. This curve was generated assuming a material with a tan [delta] = 0.4. The vertical axis represents the transmissibility of the system and is interpreted as in the ratio of the motion that the mass is experiencing to the motion input by the base attached to the spring and dashpot. The horizontal axis represents the ratio of the frequency of excitation to the natural frequency of the system. The natural frequency of the system is determined by the mass supported, the spring constant of the spring, and the viscosity of the dashpot fluid. In practical terms, it is the frequency at which the transmissibility curve peaks.
Damping in elastomers is hysteretic rather than viscous in nature. This simply states that the damping force in elastomers is proportional to the magnitude of the displacement rather than on the velocity of the displacement. The transmissibility equations for a linear viscoelastic system is given in equations 1 and 2 (ref 1):
T= ( 1+[tan.sup.2][delta] 1/2
(1-[r.sup.2).sup.2)+[tan.sup.2] [Phi]= [tan.sup.1]([-r.sup.2] tan [delta]/ (1+[tan.sup.2][delta]-[r.sup.2])
where T is the transmissibility, R is the ratio of the input frequency to the natural frequency, and [phi] is the phase angle of the displacement of the mass relative to the input displacement.
In order to assess the effect of non-linear elastomer behavior on the performance of an isolator it is important to look explicitly at how the strain on the isolator varies as the frequency, and hence transmissibility, vary. The strain is related to the magnitude of the relative displacement across the isolator
[delta](t) = [X.sub.mass](t) - Xbase(t)
which can be expressed in terms of the transmissibility of the system;
[delta](t) = [X.sub.mass]cos)omega]t-[phi] - [X.sub.base]cos)[omega]t)
[gamma](t) = ([X.sub.mass] cos([Phi]) - x.sub.base])2
[delta](t) = [X.sub.base] ([T.sup.2]-2Tcos[Phi]+1) 1/2 (5)
Equation 7 is a dimensionless form of equation 6 whereby the relative displacement is expressed as a ratio to the input displacement. The resulting numbers are then independent of the actual magnitude of the input displacement and can be interpreted more generally;
[gamma](t) = [delta](t) / [x.sub.base]=([T.sup.2]-2Tcos[Phi]+1)1/2
The phase angle of the strain with respect to the input displacement is given by Equation (8);
[beta] = [tan.sup.-](T sin([Phi]) / (T cos([Phi]) - 1)) (8)
Figure 1 is a plot of the response of a linear viscoelastic system to a constant displacement frequency sweep, with the relative displacement superimposed on the transmissibility curve.
At frequencies very low compared to the natural frequency (r <<1) there is very little relative displacement across the isolator and the resulting strain is very low such that the base and the mass are essentially moving together. The strain peaks at a frequency slightly higher than the natural frequency and then decreases to the value that would occur if the mass were artificially held stationary while the input displacement remained constant. This displacement is termed the isolation strain in this work and serves as a consistent reference point for a constant displacement frequency sweep test on an isolator. It is a condition of the system that depends only on the mechanical variables driving the vibration and the geometry of the isolator. The geometry of the isolator is needed only to express the relative displacement at isolation and the static displacement in terms of strain. This is required since the material properties G'([omega],[gamma]) and tan [delta]([omega],[gamma]) depend on strain and not on displacement.
Non-linear single degree of freedom system
The dynamic properties of most elastomers depend on both frequency and strain due to the well-known Payne effect (see figure 3). The transmissibility equations (ref. 2) for the situation in which the G' and tan [delta] of the isolator are not constant are given by equations 9 and 10;
(1-[r.sub.2](G]res/G'[omega],[gamma]))2 + [tan.sub.2][delta][omega],[gamma]
-tan[delta)([omega],[gamma])[r.sub.2]G'res/G'([omega],[gamma]) [Phi] = [tan.sup.-1]
(1-[r.sub.2](G'res/G'([omega],[gamma] + [tan.sub.2][delta]([omega][gamma]))
where [G'.sub.res] denotes the G'([omega],[gamma]) at the natural frequency conditions. It is useful to compare equations 1 and 9. The effect of a tan [delta]([omega],[gamma] is to change the magnitude of the transmissibility. The effect of the variation of G'([omega],[gamma]) is to shift the transmissibility curve by the factor ([G'.sub.res]/G'([omega],[gamma])1/2. This causes the system to behave like it would if the natural frequency were determined by G'([omega],[gamma]) rather than [G'.sub.res]. Since G'([omega],[gamma]) usually increases with increasing frequency and also with decreasing strain this has the effect of compressing the transmissibility curve in the direction of r=1 for the region r > 1.
Only two mechanical variables (such as natural frequency and isolation strain) can be independently specified for a given set of material properties (G'([omega],[gamma]), and tan [delta]([omega],[gamma]). For example, one can choose the value of the displacement under isolation conditions (which determines the isolation strain when coupled with the basic isolator geometry) and the natural frequency, from which the entire transmissibility curve can be generated for the simple system considered here. These are particularly useful since they lead to straightforward interpretation of the results and the data generation is simplified. The effect of material non-linearity on the detailed shape of the transmissibility curve has been shown previously (ref 3) and is not repeated in this work.
The goal of the remainder of this work is to provide a framework within which to understand how to compare the effect of varying the dynamic properties of an elastomer compound on isolator performance. This is most applicable to screening different materials for a particular isolator design or for evaluating different materials in a compounding study.
It is important to recognize that by specifying the natural frequency and isolation strain of a hypothetical isolator we have in effect specified the basic design of the isolator without looking at the details. Since the evaluations will focus mainly on dynamic property evaluation it may be that a material which looks spectacular in terms of the parameters developed in this work would not make a practical isolator.
Development of material evaluation parameters
A typical engine mount must both support the static weight of the engine and isolate the vibration of the engine from the supporting structure. For a conventional elastomeric mount both of these roles are performed with the same geometry. Since the material behavior is non-linear it is useful to derive expressions for parameters to help guide the material selection and design process. The parameters are necessarily based on simplifying assumptions are intended to provide insight into the isolator behavior and not to draw attention away from the very complex task of a practical isolator design. There are many factors that enter the isolator design process that are not considered in this work.
The need for meaningful parameters is apparent since it is difficult to look at a transmissibility curve together with plots of material properties vs. strain and frequency and determine a quantitative measure of the effect on isolator performance. It is certainly true that performance "gates" can easily be established for specific isolator or material property conditions but it is desired to develop more general measures of the effect of material properties on isolator performance.
An important factor is the balance between the static displacement of the mount and its natural frequency. Typically, only the geometry of the isolator and the "hardness" of the elastomer are varied in order to balance these requirements, since the supported mass is usually specified for a given mounting system. If we assume that the same geometric factor (C) is applied to both the static and vibration displacements then a parameter can be derived for the static displacement. This is not a restrictive assumption unless the static and dynamic loads are in different directions and the mount is kinematically non-linear. Equations 11 and 12 give the simple design relations for the vibration and static conditions respectively.
[omega]n = (CG'.sub.res] / M) 1/2 [delta]static = (Mg /[CG.sub.static] (t,[gamma])) 1 2
Eliminating the geometric factor C between equations and 12 yields an expression for die static displacement,(equation 13):
[delta]static = (g/([omega]n2)(G'res([omega],[gamma]))
It is important to recognize that the static modulus [G.sub.static] (t,[gamma]) is a function of strain in a manner quite similar to the way G'([omega],[gamma]) depends on strain. In fact, if the time scale of the static measurement is roughly equivalent to the time scale equivalent to the frequency at which G'([omega],[gamma]) is measured, the dependence is nearly identical (ref. 4).
Equation 13 shows that for an isolator with natural frequency ([omega]n), the static displacement is determined by the ratio of G'res/[G.static]. This ratio applies as long as the same geometric factor can be used to describe both the static and vibration stiffness of the isolator. Equation 13 is useful from a material selection standpoint since in this context [[delta].sub.static] is independent of die geometry of the isolator because of the way we have chosen to state the problem. The information about the geometry of the isolator is implicitly stated by having both [G'.sub.res]([omega],[gamma]) and con in the expression. The exact computation of the ratio is not trivial since [G'.sub.res]([omega],[gamma]) depends on the strain and frequency conditions at resonance which are determined by the isolation strain, the natural frequency of the isolator, and the behavior of the tan [delta]([omega],[gamma]) of the material see equation 17 and subsequent discussion for determination of die mechanical conditions at resonance). The calculation Of [[delta].sub.static] is thus iterative since [G.sub.static](t,[gamma]) is usually expressed in the form of a table of experimental data. If an isolator thickness is assumed then [[delta].sub.static] can be converted into a static strain and then equation 13 can be iterated to calculate a [[delta.sub.static] consistent with the [G.sub.static](t,[gamma]) data. Simple back-substitution is sufficient to solve the iteration problem.
The material data required for such a calculation is a strain sweep at the natural frequency (to determine the resonance conditions) and [G.sub.static](t,[gamma]) evaluated on an appropriate time scale as a function of the strain. The material properties at any given strain within the range of the data can be predicted using an interpolation scheme. The material properties are most easily interpreted on a logarithmic strain axis so it is advisable to space the strains in the strain sweep so they will be evenly spaced on a log scale. If it is necessary to interpolate to obtain both a particular strain and frequency then the two dimensional interpolation can be done sequentially by generating a value for the material property at the correct strain for each frequency in a set of frequencies, then interpolating the results from the interpolation in strain to the desired frequency.
Another useful measure is to determine the degree to which the non-linear material behavior affects the value of the transmissibility. There are many different possible scenarios under which to compare this parameter. The one chosen here is to compare the calculated transmissibility to a hypothetical transmissibility that the isolator would have if the material had constant material properties fixed at the resonance conditions. In this way the non-linear isolator and the linear isolator will have identical natural frequency and transmissibility at resonance values. The difference being the change in-isolation performance due to the non-linear material behavior. Since a linear isolator is the ultimate target of a conventional isolator, barring very unusual dynamic properties, this provides for a tangible measure of performance quality.
Taking the ratio of equations 9 and 1 and noting the specific conditions for the material properties yields
[T.sup.2] non-linear 1 + [tan.sub.2][delta]([omega],[gamma]) (1-[r.sub.2]2 + [tan.sub.2][delta]res [T.sup.2] linear 1 + [tan.sub.2][delta]res
(1-[r.sub.2]G]res 2+ [tan.sub.2][delta](omega],[gamma]
There are two material property related parameters of interest contained within this relation;
1 + [tan.sub.2][delta]9[omega],[gamma] [R.sub.tan][delta]
(1 + [tan.sub.2][delta]res
G]([omega],[gamma] Rg' G'res
The ratio of the strain at resonance to the strain at isolation can be derived by combining equations 1, 2 and 7 and noting that r = 1 at resonance which yields:
The current approach (ref 5) to screening material properties for suitability for isolator performance is done by plotting a parameter denoted Kd/ks versus tan [delta]. Where Kd is the G'([omega],[gamma]) at a condition meant to approximate isolation conditions, Ks is [G.sub.static](t,[gamma]) at a condition meant to represent static loading conditions, and tan [delta]([omega],[gamma]) is measured at conditions meant to simulate the resonance conditions of an isolator.
Since the material behavior clearly affects isolator performance and operating conditions, It is proposed that a more meaningful screening of material properties is done by plotting [T.sub.non-linear]/T linear vs. tan 8 at resonance. Further, it is proposed that it is important to calculate the strain at resonance from the isolation strain and the tan [delta]([omega][gamma] behavior of the material (equation 17).
This provides for a logical comparison of different elastomers in a way that simulates how they will perform in an isolator rather than comparing them under the same set of strain and frequency conditions. This requires only slightly more data than is required to generate numbers for Kd/ks vs. tan [delta] plots. The entire analysis can easily be done (including the iterative solutions) using a spreadsheet on a PC. The necessary data include a strain sweep at the natural frequency to determine [G'.sub.res] and tan [[delta].sub.res], and a measurement of G'([omega],[gamma]) and tan [delta](([omega],[gamma]) under isolation input conditions. Of course a measurement of [G.sub.static] is required in either case.
There are several benefits to evaluating material performance in this way: The measure is directly in terms of isolator performance.
* The measure involves both G'([omega],[gamma]) and tan [delta](([omega],[gamma]) effects.
* The measure includes the effect the material properties will have on isolator operating conditions.
* The well established trade-off of Kd/Ks as a function of tan [delta] is a statement that, within conventional compounding techniques, the hysteresis mechanism of reinforced elastomers is directly and inseparably tied to the mechanism underlying the Payne effect. Equation 14 is sufficient to describe this physical phenomena without the use of the static displacement relation (equation 13). The implicit independent variable underlying a plot of [T.sub.non-linear]/[T.sub.linear] linear vs. tan [delta] or Kd/Ks vs. tan [delta] is the composition and properties of the elastomer. Once the assumptions have been made sufficient to generate the [T.sub.non-linear]/[T.sub.linear] vs. tan [omega] curve enough information is available to plot [[delta].sub.static] vs. tan [delta].
Each plot carries different information about the non-linearity of the material and as such are not combined into a single parameter in this work. They could be combined for a specific design situation where the relative importance of dynamic performance relative to static deformation is known. Perhaps a better approach is to apply multi-variable optimization techniques (ref. 6) to simultaneously optimize [T.sub.non-linear]/[T.sub.linear] and [[delta].sub.static] based on a specific set of design value criteria, with the independent variables being formulation related variables. In this way the relative importance of static and dynamic performance is specified by the person performing the optimization calculation rather than an arbitrary combination of the two parameters. It would also be possible to simultaneously optimize other important physical properties in such a calculation.,
The compounds shown in table 1 were mixed in a laboratory scale internal mixer using the mixing procedure shown in table 2. The mixer used was a Brabender PrepCenter with internal mixer style rotors operating at 50[degrees]C, 50 rpm and a fill factor of 0.7.
Dynamic property testing was performed on a Rheometrics RDS 11 (10-10,000 g-cm transducer) at 23[degrees]C, using the parallel plate mode with a specimen geometry of 8mm diameter by 3.25 mm thick. The cured elastomer sample was bonded to brass endplates using a cyanoacrylate adhesive. The following conditions were used to establish values for Kd/Ks and tan [delta]: Kd = G' (0.2% SSA, 100 Hz), Ks= G(t) (30 sec, 20% SSA), tan [delta] = tan 8(2% SSA, 10 Hz). The test sequence used to generate the data consisted of:
* A three cycle conditioning step to [+ or -] 25% strain at 0.1 Hz;
* a stress relaxation test at 20% shear strain for 30 sec.;
* a strain sweep at 10 Hz from 0.1 to 20% strain;
* a frequency sweep at 0.2% strain from 0.08 to 80 Hz;
* a frequency sweep at 0.35% strain from 0.08 to 80 Hz.
Since the test machine exhibits some rolloff of displacement at the upper frequency ranges the two frequency sweeps were combined to yield at single frequency sweep curve at exactly 0.2% strain. The two particular strain levels were chosen to be as close together as possible yet make sure that the desired condition would be an interpolation between the two for all frequencies considered. This was done by interpolating between the actual strains from the two tests, to a strain value of 0.2%.
Results and discussion
Figures 2-5 show the dynamic property data for the NR compound used in this Study. Figures 6-9 show the same dynamic property data for the IIR compounds. One striking difference between the two sets of data is apparent in comparing Figures 3 and 7. The tan [delta]([omega]),[gamma]) for the IIR compound at very low strains decreases with increased carbon black loading rather than increases which would be expected. This has also been shown to occur with a BUR/N220 system (ref. 7). This difference between the two compounds would be completely ignored in the calculation of a plot of Kd/Ks vs. tan [delta]. It is explicitly accounted for in the [T.sub.non-linear]/[T.sub.linear] calculation, however. This is not to suggest that NR and IRZ would necessarily be considered for the same dynamic performance application since the tan [delta]([omega]),[gamma]) behavior of the two materials is quite different.
This can be clearly seen in figure 10 which demonstrates the difference between the Kd/Ks vs. tan [delta] plot and a plot of [T.sub.non-linear]/T[linear] vs. tan [delta] [delta]([omega]),([gamma]) at resonance. The first observation is that the plots are similar and the two materials are ranked in the same relative order. A subtle difference is shown in the range of the tan [delta] values contained in each plot. This is a direct consequence of using the fact that the strain at resonance will depend on the tan [delta]([omega]),[gamma]) behavior of the material in generating the plot. This is a very real observation reflecting the range of tan [delta]([omega]),[gamma]) values that would actually be observed from isolator testing, presuming that the assumptions made about comparing the hypothetical isolators are valid with respect to isolator construction and testing.
Figure 11 shows the values of the strain at resonance used to generate the tan [delta] and [G'.sub.res] values for the [T.sub.non-linear/[T.sub.linear] vs. tan [delta] plots assuming the isolation strain is 0.2%. There is a large difference between the NR and IIR strain at resonance values and there is also a significant difference within the IIR compound series.
Figure 12 is a direct comparison of the [T.sub.non-linear/[T.sub.linear] parameter in terms of Kd/Ks values. Each datum represents one compound of the compound series in either NR or UR. The HR formulations deviate significantly more from a linear relation primarily due to the larger variation of tan ([omega],[gamma]) with those materials and the inclusion of the effect of tan [delta]([omega],[gamma]) on the transmissibility.
Figure 13 shows the difference in behavior between the NR and IIR compounds. In both parameters ([R.sub.tan[delta]] and [R.sub.G']) a lower value provides a lower transmissibility at isolation. The IIR shows much larger decrease in [R.sub.tan] [delta] with increasing carbon black loading compared to the NR. The [R.sub.tan [delta]] term is simply a multiplicative factor applied in the [T.sub.non-linear]/[T.sub.linear] relation (equation 14). While the importance of neglecting the effect of tan [delta](([omega],[gamma]) is negligible in the case of NR, significant errors would be introduced in the IIR case (up to 40%). Thus it is clear that both tan [delta]([omega],[gamma]) and G'([omega],[gamma]) effects are important when considering the true effects of non-linear material behavior on isolator performance.
From the design standpoint the benefit to the [T.sub.non-linear]/[T.sub.linear] parameter is that it can be interpreted directly in terms of quantitative isolator performance. This is of paramount importance when decisions are being made about the relative tradeoffs due to material performance constraints and helps to determine the value of performance in real terms. There is much more intuitive leverage in a result that indicates a change in transmissibility as opposed to a less direct result indicating change in Kd/Ks.
A new method. to couple elastomer dynamic properties with isolator performance properties has been developed providing more quantitative information about isolator performance than previous parameters.
Table 1-elastomer formulations
IIR formulation Ingredient Specific phr gravity
Butyl 301 0.92 100 Sterling 6630 1.80 20, 45, 75, 90 Sunpar 2280 0.89 3.2, 7.2 12.0, 14.4 Zinc oxide 5.60 5 TMTD 1.27 2 MBT 1.54 1 Sulfur 2.07 1.5
NR formulation Ingredient Specific phr gravity SMR L 0.916 100 Sterling 6630 1.80 40,60, 80, 100 Zinc oxide 5.60 5 Stearic acid 0.84 2 Sunthene 4240 0.928 10 Wax 0.92 1.5 AgeRite DPPD 1.28 1.5 TBBS 1.32 1.5 Sulfur 2.07 2 Table 2-elastomer mixing procedures
NR mixing procedure 0 Add polymer 30 sec. Add wax, zinc oxide, stearic acid, antioxidant, 1/2 black 2.5 min. Add rest of black and oil 6 min. Sweep, add sulfur and accelerator 7.25 min. Dump 2.5 min. Add 1/2 carbon black, zinc oxide (pre-blend) and sulfur 2.5 min. Add rest of black and oil 6 min. Sweep, and add accelerators 7.25 min. Dump
[Figures 1 to 13 ILLUSTRATION OMITTED]
(1.) Mechanical vibrations, theory and applications, " 2nd Ed., Allyn and Bacon, 1978. (2.) "Vibration and shock in damped mechanical systems," Wiley 1968. (3.) Harris J.A., Rubber Chem. and Technol. 62, 515 (1988). (4.) Warley R.L., Ph.D. dissertation, Case Western Reserve University 1993. (5.) Nakauchi H., Intl. Polym. Sci. and Technol. 19, T/46 (1992). (6.) Kohli T.S., Paper given at ACS Rubber Division Meeting, Las Vegas, NV 1990. (7.) Dutta N.K., Tripathy D.K, Kautchuk and Gummi Kunstoffe 42, 665 (1989).
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|Author:||Warley, Russell L.|
|Date:||Mar 1, 1996|
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