2892-6-T REDUCTION of the RESPONSE to VIBRATION of STRUCTURES POSSESSING FINITE MECHANICAL IMPEDANCE Part II J. C. SNOWDON January 1960 FLUID AND SOLID MECHANICS LABORATORY T H E U N I V E'R S I T Y O F M I C H I G A N Ann Arbor, Michigan

This project is conducted for the Bureau of Ships, Noise and Vibration Branch (Code 345) under Navy Contract Number NObs 77072, Index No. NS/713/017o University contract administration is provided to the Willow Run Laboratories through The University of Michigan Research Institute.

ACKNOWLEDGMENTS The author wishes to acknowledge with gratitude the assistance in computing provided by Mr. J. K. Chrow, and the assistance of Mr. W. A. Richardson in. preparing the figures for this report. The curves referred to in Section 3 of the report were computed for the most part by the Computation Department of the Willow Run Laboratories. iii

TABLE OF CONTENTS Page LIST OF FIGURES vii ABSTRACT 1 1. INTRODUCTION 2 2. MASS-LOADING OF A FOUNDATION POSSESSING FINITE MECHANICAL IMPEDANCE 3 2.1. Introduction 3 2.2. The Simple Mounting System 4 2.3. The Parallel Mounting System 6 3. THE COMPOUND MOUNTING SYSTEM 7 3.1. Introduction 7 3.2. The Transmissibility of the Compound Mounting System 7 5.3. The Response Ratio of the Compound Mounting System 8 4. SUMMARY AND CONCLUSIONS 9 5. APPENDIX 11 5.1. Introduction 11 5.2. Mass-Loading of a Foundation Possessing Finite Mechanical Impedance 13 5.3. Transmissibility of the Compound Mounting System 14 5.4. Response Ratio of the Compound Mounting System 16 REFERENCES 18 v

LIST OF FIGURES No. Page 1. The frequency dependence of (a) the dynamic shear modulus and (b) the damping factor possessed by a natural rubber vulcanizate. 19 2. The frequency dependence of (a) the dynamic shear modulus and (b) the damping factor possessed by Thiokol R. D. 20 3. The transmissibility of simple and parallel mountings of vulcanized hevea and Thiokol R. D. at a temperature of 20~C. 21 4. The response ratio of a simple mounting of vulcanized hevea. Mounted item ten times more massive than the foundation. Foundation damping defined by 6f = 0.01, 0.1, and 1.0. 22 5. The response ratio of a simple mounting of vulcanized hevea. Mounted item twice, ten, and fifty times more massive than the undamped foundation. 23 6. The frequency dependence of the velocity with which a nonrigid foundation responds to mechanical vibration. 24 7. The response ratio of a simple mounting of vulcanized hevea supported by a mass-loaded foundation. Mounted item fifty times more massive than the foundation. Loading mass one-tenth, onefifth, and equal to the mass of the mounted item. Foundation damping defined by 6f = 0.01. 25 8. The response ratio of a simple mounting of Thiokol R. D. supported by a mass-loaded foundation. Mounted item fifty times more massive than the foundation. Loading mass one-tenth, onefifth, and equal to the mass of the mounted item. Foundation damping defined by 6f = 0.01. 26 9. The response ratio of a simple mounting of vulcanized hevea supported by a mass-loaded foundation. Mounted item ten and fifty times more massive than the foundation. Loading mass equal to 0.08 of the mass of the mounted item. Foundation damping defined by 5f = 0.01. 27 vii

LIST OF FIGURES (Continued) No. Page 10. The response ratio of a simple mounting of vulcanized hevea supported by a mass-loaded foundation. Mounted item ten times more massive than the foundation. Loading mass one-tenth, one-fifth, and equal to the mass of the mounted item. Foundation damping defined by 6f = 0.1. 28 11. The response ratio of a simple mounting of Thiokol R. D. supported by a mass-loaded foundation. Mounted item ten times more massive than the foundation. Loading mass one-tenth, one-fifth, and equal to the mass of the mounted item. Foundation damping defined by &f = 0.1. 29 12. The response ratio of simple and parallel mountings of vulcanized hevea and Thiokol R. D. supported by a mass-loaded foundation. Mounted item fifty times more massive than the foundation. Loading mass one-fifth of the mass of the mounted item. Foundation damping defined by &f = 0.01. 30 13. The response ratio of simple and parallel mountings of vulcanized hevea and Thiokol R. D. supported by a mass-loaded foundation. Mounted item ten times more massive than the foundation. Loading mass one-tenth of the mass of the mounted item. Foundation damping defined by &f = 0.1. 31 14. The response ratio of simple and parallel mountings of vulcanized hevea and Thiokol R. D. supported by a mass-loaded foundation. Mounted item ten times more massive than the foundation. Loading mass equal to the mass of the mounted item. Foundation damping defined by 6f = 0.1. 32 15. The transmissibility of a compound system employing vulcanized hevea mounts. Secondary mass one-tenth, one-fifth, and equal to the mass of the mounted item. 33 16. The transmissibility of a compound system employing Thiokol R. D. mounts. Secondary mass one-tenth, one-fifth, and equal to the mass of the mounted item. 34 viii

LIST OF FIGURES (Continued) No. Page 17. The transmissibility of a compound system employing parallel mounts comprised of vulcanized hevea and Thiokol R. D. Secondary mass one-tenth, one-fifth, and equal to the mass of the mounted item. 35 18. The transmissibility of a compound system employing vulcanized hevea, Thiokol R. D., and parallel mounts. Secondary mass onetenth of the mass of the mounted item. 36 19. The transmissibility of a compound system employing vulcanized hevea, Thiokol R. D., and parallel mounts. Secondary mass equal to the mass of the mounted item. 37 20. The response ratio of a compound system employing vulcanized hevea mounts. Mounted item fifty times more massive than the foundation. Secondary mass one-tenth, one-fifth, and equal to the mass of the mounted item. Foundation damping defined by 5f = 0.01. 38 21. The response ratio of a compound system employing Thiokol R. D. mounts. Mounted item fifty times more massive than the foundation. Secondary mass one-tenth, one-fifth, and equal to the mass of the mounted item. Foundation damping defined by bf = 0.01. 39 22. The response ratio of a compound system employing parallel mounts comprised of vulcanized hevea and Thiokol R. D. Mounted item fifty times more massive than the foundation. Secondary mass one-tenth, one-fifth, and equal to the mass of the mounted item. Foundation damping defined by 5f = 0.01. 40 23. The response ratio of a compound system employing vulcanized hevea, Thiokol R. D., and parallel mounts. Mounted item fifty times more massive than the foundation. Secondary mass onetenth of the mass of the mounted item. Foundation damping defined by 5f = 0.01. 41 24. The response ratio of a compound system employing vulcanized hevea, Thiokol R. D., and parallel mounts. Mounted item fifty times more massive than the foundation. Secondary mass equal to the mass of the mounted item. Foundation damping defined by b6 = 0.01. 42 ix

LIST OF FIGURES (Concluded) No. Page 25. The response ratio of a compound system employing vulcanized hevea mounts, and the response ratio of a simple mounting of vulcanized hevea supported by a mass-loaded foundation. Mounted item fifty times more massive than the foundation. Secondary mass one-fifth of the mass of the mounted item ( 0 = 0.2). Loading mass one-fifth of the mass of the mounted item (m/M = 0.2). Foundation damping defined by 6f = 0.01. 43 26. The response ratio of a compound system employing vulcanized hevea mounts, and the response ratio of a simple mounting of vulcanized hevea supported by a mass-loaded foundation. Mounted item fifty times more massive than the foundation. Secondary mass equal to the mass of the mounted item (5 = 1.0). Loading mass equal to the mass of the mounted item (m/M = 1.0). Foundation damping defined by 5f = 0.01 44 x

ABSTRACT The performance of simple and compound mounting systems supported by a foundation of finite mechanical impedance, and the performance of the simple mounting supported by a mass-loaded foundation, have been theoretically determined and compared. A simply supported damped beam has been employed to simulate the behavior of the foundation. The dynamic mechanical properties of natural rubber and a high-damping rubber have been employed to describe the behavior of antivibration mount materials. When the ratio of the mass of the mounted item to the mass of the foundation is large, the isolation afforded by the simple mounting is much less than predicted by its transmissibility curve, which relates to an ideally rigid foundation. The isolation provided by the simple mounting is increased significantly at high frequencies when the foundation of the mounting system is mass-loaded, being largest for a natural rubber mounting. In the example considered, large, but not greater, isolation is provided at high frequencies by the compound mounting utilizing a secondary mass equal to this loading mass, and mountings composed of natural and high-damping rubber in parallel. 1

1. INTRODUCTION This report describes the second phase of a theoretical investigation concerned with the isolation of machinery vibration from structures possessing finite mechanical impedance. Particular attention has been devoted to representing realistically the dynamic mechanical properties of rubber-like materials employed as vibration isolators. The mechanical properties of natural rubber and a synthetic rubber, Thiokol R. D., have again been considered to typify the mechanical properties of low- and high-damping rubbers, respectively. The dynamic shear modulus and damping factor possessed by natural rubber (Fig. 1) and Thiokol R. D. (Fig. 2) have been deduced from the experimental results of other workers.2 The parallel mounting1 discussed here is comprised of elements of these materials in parallel, arranged such that both materials experience the same strain. The transmissibility1 of simple mountings of natural rubber and Thiokol R. Do and the transmissibility1 of three parallel mountings composed of the same materials are shown in Fig. 3 at a temperature of 20~C. It is assumed that the mounting systems possess natural frequencies of 5 cps. The parameter "a" represents the ratio of the cross-sectional area of the Thiokol R. D. element to that of the natural rubber element. General equations have been derived from which the response ratios1 of different mounting systems may be computed when the variation with frequency of the mechanical impedance Z of their foundations is known. The response ratios of the mounting systems considered here have been evaluated for a nonrigid foundation, the impedance of which is simulated by the driving point impedance of a supported-supported beam excited by a sinusoidal force at its mid-point. It has been assumed that the beam possesses damping of the solid type, which is described1 by a constant damping factor 6f. A series of response ratios for a natural rubber mounting have been determined (Fig. 4) for values of foundation damping defined by 6f = 0.01, 0.1, and 1. The mass of the mounted item M was assumed to be ten times greater than the mass of the foundation Mf. The magnitude of 6f is seen to have little influence upon the value of the response ratio at frequencies between the resonant modes of foundation vibration. The isolation afforded by the mount at these frequencies appears to be determined primarily by the value of the mass ratio M/Mf, even when the foundation is heavily damped. It has previously been shownl that the isolation afforded by a resilient mounting will become smaller as the mass ratio M/Mf becomes larger. The response ratio of a natural rubber mounting is plotted in Fig. 5 for values of M/Mf = 2, 10, and 50, with the assumption that foundation damping is negligibleO As the ratio M/Mf increases, the isolation afforded by the mounting is seen to depart progressively further from the isolation obtained when the foundation is completely rigid (Fig. 3). 2

The mounted item is supposed to behave purely as a lumped mass. The antivibration mountings are assumed to employ only linear rubber-like materials. "Wave effects" which may occur in the mountings have been disregarded. 2. MASS-LOADING OF A FOUNDATION POSSESSING FINITE MECHANICAL IMPEDANCE 2.1. Introduction An expression for the response ratio of a simple or parallel mounting system supported by a mass-loaded foundation of finite mechanical impedance is presented in the Appendix (Section 5.2). Substitution has been made in this expression for the driving point impedance of a damped supported-supported beam, and the resulting response ratios of several mounting systems have been computed. Before these results are discussed, however, an attempt will be made to explain in general terms the mechanism by which a mass m, placed between the bottom of a resilient mounting and a supporting nonrigid foundation, increases the isolation afforded by the mounting system. Reference is made to any multi-resonant sub-structure of mechanical impedance Z. It is assumed that the structure responds to a sinusoidally varying force Fo at the driving point with a velocity Vo, which is shown, diagrammatically as a function of frequency in Fig. 6a. When an item of mass M is supported by the structure, such that a sinusoidally varying force F1 acting upon M, or generated within, M, excites the structure in the same position and fashion as before, then the structure will respond with the velocity V1 shown as a function of frequency in Fig. 6b. The response of the second mode of foundation vibration has been drawn with a broken line. The resonant frequencies of vibration possessed by the structure are removed to lower frequencies by an amount which will depend upon the magnitude of M, but which will not exceed the original (Fig. 6a) frequency separation of the particular mode of vibration and the next lower mode. When the vibrating item is resiliently mounted, as in Fig. 6c, the resonant frequencies of the foundation return again to higher frequencies. In fact, the higher modes of vibration occur at frequencies which are essentially identical to those of the unloaded structure (Fig. 6a), because in this frequency region the inertia of the mounted item is large, and to a first approximation the motion of the structure is restrained only by a resilient element, the other end of which is attached to the "stationary" mass M. An additional region of high foundation velocity is introduced at low frequencies due to the resonant motion of the resiliently mounted item M. This resonance has been assumed to occur at a lower frequency than the fundamental mode of foundation vibration. The response ratio of the mounting system is shown in Fig. 6d, this quantity having been defined3 as the magnitude of the velocity ratio V2/V1. Essen5

tially, the maximi-un and mi:nimum values of the response ratio occur at the frequencies for which the velocitLes V2 and V1, respectively, possess maximum values. The response ratios which have been presented in Figs. 4 an.d 5 for simple systems employing natural rubber mounts are seen to be of this form. It follows that the greater the mass M~i the greater the shift of the resonant modes of the foundation to h"igher frequencies occurring when M is mounted resiliently, and therefore the smaller the over-all reduction in. foundation velocity afforded by the mounting system compared with the level observed when M, is rigidly supported. When a mass is employed to load the foundation (Fig. 6e), the resonant frequencies of the structure do not remove as far towards higher frequencies as before. In aac, w hvern r is an appreciable fraction of the resiliently supported mass M-the case depicted n Fig. 6e-the resonant frequencies of the foundation only slightly exceed the values observed when M is rigidly supported (Fig. 6b). It follows that the response ratio of the mounting system, namely, the magnitude of the velocity ratio Vs/VT (Fig. 6f) is reduced by the introduction of tm. In fact, when nm becomes comparable in magnitude to M_, the isolation afforded by the mounting system approaches the value predicted by the simple transmissibility curve (which refers to an ideally rigid foundation) except at frequencies in close proximity to the resonant modes of founldation vibration. 2.i2 The Simple Mounting System Following the qualitative description of the manner in which a mass situated below a resilient mounting favorably decreases the response ratio of the mlounting system,, a specific foundation, the mechanical impedance of which is represented by the driving point impedance of a damped beam (Section 1), is disciussed. This foundation has previously been consideredl to possess a mass Mf. and solid-type damping described by a constant damping factor The response ratios of natural rubber and the high-damping rubber Thiokol Ro D. have bee:n computed from Eq. (5.14), assuming that the mass ratio M/Mf = 50 and that the damping inherently possessed by the foundation may be described by a damping factor bf = Oo0l. The response ratios are presented in Figs. 7 and 8, respectively, fior values of the mass ratio m/M = 0.1O 0.2, and 1.O. The broken curves refer *to the response ratios of the mounting systems when the foundatcrion is not loaded by an additional mass m. Figures 7 and 8 show that an extremely large gain in isolation is afforded by the mounting systems when the mass ratio m/M approaches unityo When m/M = 1, the foundation velocity is reduced (negative logarithmic values of the response ratio) by introducti on of the natural rubber mount at all frequencies above 9 cps, and by the inltroduction of the Thiokol R. Do mount at all frequencies above 5 cps. Moreover9 the isolation afforded by these mounts at frequencies above the fundamental resonant frequency of the foundatior is essentially equal to the isolation they afford in. a simple mounting system supported by an ideally rigid foundli.ation (Fig 5).3

While the value of the response ratio R at the fundamental mode of foundation vibration is hardly influenced by the introduction of m, the magnitude of R at the second, and the higher modes of vibration is reduced significantly. It is interesting to note that the magnitude of the response ratio at the second mode of foundation vibration relative to the value taken by R at neighboring frequencies is only slightly influenced by the magnitude of the mount damping. As demonstrated previously,l the maximum compression experienced by the mounts and, consequently, the influence of mount damping at the higher modes of foundation vibration will be small. It may be questioned if the response ratio of a mounting system would be improved equally well if the additional mass m were employed to increase the integral mass of the foundation. This is not the case, as is evident from Fig. 9, which presents the response ratios for a simple system utilizing natural rubber mounts. The two broken curves of this figure relate to foundations which are one-tenth and one-fiftieth as massive as the mounted item, namely, foundations for which M/Mf = 10 and 50. The full line relates also to the least massive foundation for which M/Mf = 50, but, in addition, this foundation is loaded by a mass m chosen such that the ratio M/(m+Mf) is equal to 10. A value of the mass ratio m/M = o0.8 has therefore been considered. In consequence, it is possible to compare the response ratios for the mounting system when it is supported by foundations of "equal" mass, namely, foundations for which M/Mf = 10 (broken line) and M/(m+Mf) = 10 (full line). It is evident from Fig. 9 that while the response ratio at the fundamental mode of foundation vibration is of the same order of magnitude in each case, the response ratio at the second mode, and presumably at the higher modes of foundation vibration, is significantly less for the mass-loaded foundation. Again, the over-all level of the response ratio at frequencies greater than 35 cps is generally lower for the mass-loaded foundation, especially in the neighborhood of the resonant modes of foundation vibration. The response ratios of natural rubber and Thiokol R. D. mountings have been computed also for a second foundation, and are presented in Figs. 10 and 11, respectively. The curves of these figures relate to a mass-loaded foundation to which a damping treatment has been applied. It has been assumed that the effect of the damping treatment may be described1 by a damping factor bf = 0.1. The response ratio curves have been computed for values of the mass ratios M/Mf = 10 and m/M = 0.1, 0.2, and 1.0. The response ratios of these mounting systems are seen to be very similar in form to those illustrated by Figs. 7 and 8, which have previously been discussed. The significant increase in isolation resulting from the introduction of the mass m is again apparent. In fact, when the ratio m/M = 1, the isolation afforded by the mounts becomes equal, as before, to the isolation which they afford in a simple mounting system supported by an ideally rigid foundation (Fig. 3). The greater foundation damping favorably suppresses the response ratio of the natural rubber mounting at the fundamental mode of foundation vibration, particularly for the smaller values of the ratio m/M (compare with the curves of Figs. 4 and 7). 5

It is suggested by the results discussed in this section that the conclusion drawn from an elementary consideration of the transmissibility curves of simple mounting systems, namely, that high mount damping is undesirable, may in fact be completely valid when the mounting system is supported by a massloaded foundation of finite mechanical impedance. (Compare the response ratios of natural rubber mountings shown in Figs. 7 and 10 with the response ratios of the highly damped mounts shown in Figs. 8 and 11, respectively.) 2.3. The Parallel Mounting System The response ratios for several mounting systems employing a parallel mount comprised of natural and Thiokol R. D. rubbers have been computed. The performance of the parallel mount, for which a = 0.2 (Section 1), has been evaluated solely at a temperature of 20~C. The results obtained relate to a mounting system which is supported by one of two mass-loaded foundations. The response ratio of the parallel mounting supported by a mass-loaded foundation for which m/M = 0.2, M/Mf = 50, and 6f = 0.01 is shown in Fig. 12. Although the resonant motion of the mounted item and the fundamental mode of foundation vibration are favorably suppressed by the parallel mounting, the natural rubber mount is seen to afford the greatest isolation at higher frequencies. In fact, the relative values of the response ratio at any one frequency are then very similar to the relative values of the transmissibility which the mounts possess (Fig. 3) at the same frequency. It should be noticed, however, that the response ratios of the parallel and natural rubber mountings do not diverge appreciably until the absolute value of the isolation afforded by both mountings is quite large. The response ratios of a parallel mounting supported by a mass-loaded foundation for which M/Mf = 10, bf = 0.1, and m/M = 0.1 and 1.0 are shown in Figs. 13 and 14, respectively, Because of the high foundation damping, the amplitudes of the fundamental and second modes of foundation vibration are seen to be virtually unaffected by the magnitude of the mount damping, substantiating an earlier conclusion1 that there is little merit in employing any mount material other than natural rubber when the foundation damping is large. The relative values of the response ratio at any frequency above the fundamental mode of foundation vibration are again practically identical to the relative values of mount transmissibility at the same frequency. 6

3. THE COMPOUND MOUNTING SYSTEM 3.1. Introduction The transmissibility of a two-stage or compound mounting system314 has been determined in general terms [Eq. (5.15)], assuming that different rubber-like materials are employed in each stage of the mounting system. The transmissibility of the compound mounting may therefore be computed when the dependence upon frequency of the dynamic elastic moduli and the damping factors of the materials is known, although only a reduced form of the equation referring to mounts of the same material has been employed here. The transmissibility of mountings of natural rubber and Thiokol R. D., and of a parallel mounting composed of the same materials [for which a = 0.2 (Section 1)] has been evaluated for a temperature of 20~C. The response ratios of the compound systems utilizing mounts of these materials have also been determined, since mount performance cannot be judged satisfactorilyl by reference to transmissibility alone. Expressions for the response ratio have been derived for any foundation of finite mechanical impedance Z [see Eq. (5.21)], and for the particular nonrigid foundation previously discussed [see Eq. (5.24)]. 3.2. The Transmissibility of the Compound Mounting System The transmissibility of compound mountings of natural rubber, Thiokol R. D., and the parallel mounting is shown in Figs. 15, 16, and 17, respectively. In each figure, the performance of a compound mounting employing three different secondary masses M defined by P = 0.1, 0.2, and 1.0 has been compared with the transmissibility of a simple mounting utilizing the same mount material. The parameter P is equal to M/M, where M is the mass of the mounted item. The natural mounting frequencies of the simple systems are again assumed to occur at a frequency of 5 cps. The transmissibility of the compound mountings utilizing hevea mounts (Fig. 15) compares closely with the form predicted by simple theory.3Y5 The highfrequency isolation afforded by the compound and simple systems does increase at essentially 24 db per octave and 12 db per octave, respectively, and the primary and secondary resonant frequencies l1 and (D2 do comply with the relation:5 i2 = (1+P) + (1+P)1/2 (. c2 (1+) (1) so that

P 0.1 0.2 1.0 )2/W1 6.48 4.69 2.41 The superior high-frequency isolation afforded by the compound mounting systems employing large intermediate masses may easily be recognized from Fig. 15. The reduction or loss in isolation at the secondary resonance is, however, an undesirable feature of the compound mounting system.3)5 The greater damping possessed by the Thiokol R. D. mounts effectively suppresses both the primary and secondary resonances of the compound systems shown in Fig. 16. The frequency ratio 02/01 is seen to be greater than Eq. (3.1) predicts, since 02 is displaced towards higher frequencies by the increase in stiffness of the mount with increasingl5 frequency. The rapid increase in the dynamic modulus of high-damping mount materials with frequency is also responsiblel5 for the relatively poor isolation afforded by these materials at high frequencies. The transmissibility of the compound system utilizing parallel mountings for which the parameter a = 0.2 is shown in Fig. 17. The secondary resonance is favorably suppressed by the mount damping yet, because the mount stiffness increases more slowly1 with frequency than a mount comprised solely of Thiokol R. D., the high-frequency isolation afforded by the mounting increases relatively rapidly with frequency, and the displacement of u2 to higher frequencies is small. The performance of the three mountings discussed here may be compared more satisfactorily by reference to Figs. 18 and 19, where the transmissibility of the compound systems employing mass ratios P = 0.1 and 1.0, respectively, are redrawn. These figures confirm that the parallel mounting suppresses the secondary resonance of the compound systems while affording appreciably greater isolation at high frequencies than the Thiokol R. D. mount. In fact, the highfrequency isolation afforded by the parallel mounting resembles that afforded by the hevea mounts, the correspondence being least close when the isolation afforded by the mountings is very large. 3.3. The Response Ratio of the Compound Mounting System The response ratios of compound mountings of natural rubber, Thiokol R. D., and a parallel mounting composed of these rubbers (for which a = 0.2) have been computed assuming that the same mount materials are employed in each stage of the compound systems. The response ratios have been determined from Eq. (5.24), and relate to a single foundation which is one-fiftieth as massive as the mounted item (M/Mf = 50) and to values of the mass ratio P = 0.1, 0.2, and 1.0. The damping inherently possessed by the foundation has again been described by a damping factor 5f = 0.01. 8

The response ratios computed for natural rubber mountings are shown in Fig. 20, the broken curve referring to a simple mounting system. It is evident that the isolation afforded at high frequencies by the compound system increases very rapidly with frequency, and at any one frequency becomes larger when the mass ratio P is increased. In particular, the compound mounting greatly reduces the value of the response ratio at the fundamental and second modes of vibration of the foundation. As mentioned previously, however, an undesirable loss in isolation occurs at, and in the neighborhood of, the secondary resonant frequency of the compound mounting system. In fact, the peak values of the response ratio introduced at the frequencies 9.2 cps, 18.6 cps, and 25.5 cps, when P = 1.0, 0.2, and 0.1, respectively, are comparable to the value of the response ratio observed at the fundamental mode of foundation vibration. Response ratios computed for compound systems utilizing high-damping rubber mounts and parallel mountings are shown in Figs. 21 and 22, respectively. The Thiokol R. D. mounts are seen to suppress greatly the primary and secondary resonances of the compound system, and the fundamental mode of vibration of the foundation. The isolation afforded by the compound mounting again increases rapidly with frequency in relation to the isolation afforded by the simple system but, for a given value of P, the absolute magnitude of the isolation is generally much less than provided by the compound system utilizing hevea mounts (Fig. 20). In fact, even though the second mode of foundation vibration is greatly suppressed by the Thiokol R. D. mountings, greater isolation is provided at this resonant frequency by the compound system employing hevea mounts. In contrast, the parallel mounting is seen (Fig. 22) to suppress both the mounting and the foundation resonances, and to afford an isolation at intervening frequencies which resembles the isolation provided by the hevea mounts more closely than the isolation afforded by the Thiokol R. D. mounts. The relative performance of the hevea, Thiokol R. D., and parallel mountings can more easily be examined in Figs. 23 and 24, where the response ratios of the compound mountings defined by P = 0.1 and 1.0, respectively, are redrawn. These figures show. clearly that while the compound system employing parallel mounts does not afford as large an isolation as the compound system employing hevea mounts-other than in the neighborhood of the resonant frequencies of the mounting system and the foundation-the difference is relatively small except at high frequencies, where the isolation afforded by both mountings is very large. 4. SUMMARY AND CONCLUSIONS This report describes the second phase of a theoretical investigation that has suggested and examined methods by which machinery vibrationn n. be isolated from structures possessing finite mechanical impedance. Equations have been derived from which the performance of various mounting systems may be determined when the mechanical impedance of the nonrigid structure 9

which supports them is known. In order to compare the performance of the mounting systems, substitution has been made in these equations for the driving point impedance of a damped supported-supported beam excited by a sinusoidally varying force at its mid-point. Substitution has also been made for the dynamic mechanical properties of anti-vibration mount materials, namely, for the dynamic elastic moduli and damping factors of natural rubber and the high-damping synthetic rubber Thiokol R. D. The mechanical properties of these rubbers have been deduced as functions of frequency from the experimental results of other workers. It is shown that as the mass of a mounted item M increases with respect to the mass of its nonrigid foundation Mf, the isolation afforded by the mounting system will become smaller and depart further from the isolation that would be observed if the foundation were completely rigid. The analysis suggests that a damping treatment applied to a nonrigid foundation will only increase the isolation afforded by the mounting system at the resonant frequencies of the foundation. The damping treatment will have little influence upon the isolation afforded at other frequencies, the isolation then being determined primarily by the magnitude of the mass ratio M/Mf. Since a damping treatment applied to the foundation would probably have only secondary influence upon the over-all level of the isolation provided by the mounting system, and since it would probably be impractical to damp the foundation heavily, most of the results refer to an "untreated" foundation, the inherent solid-type damping of which is defined by a damping factor bf = 0.01. When the ratio of the mass of the mounted item to the mass of the foundation is large, the over-all level of the isolation afforded by a simply mounted item is found to be appreciably less than predicted by the transmissibility curve of the mounting system (the curve relating to an ideally rigid foundation), not regarding the loss in isolation occurring as expected at the resonant frequencies of the foundation. This report discusses two ways in which the over-all level of the isolation afforded by a mounting system, for which the ratio M/Mf is large, can be made to approach the isolation predicted by its transmissibility curve. Both methods introduce a mass which is employed either to load the foundation supporting the mounting system, or to form a compound mounting3'4 as an intermediate or secondary mass. It is desirable in both cases that the mass introduced be as large a fraction of the mass of the mounted item as possible. It is shown that the isolation afforded by the simple mounting system supported by a mass-loaded foundation becomes larger as the loading mass m is increased. In fact, when the loading mass is equal to the mass of the mounted item, the mounting system affords the isolation which is predicted by its transmissibility curve at frequencies above the fundamental resonant frequency of the mass-loaded foundation. The conventional conclusion drawn from simple consideration of the transmissibility curves of rubber-like materials is therefore valid in these circumstances, that is to say, low-damping rubbers such as natural rubber will be the most suitable anti-vibration mount materials. 10

It is also shown that the isolation afforded by the compound mounting system becomes larger as the secondary mass is increased. In fact, when the secondary mass is equal to the mass of the mounted item, the isolation afforded at frequencies above the secondary resonant frequency of the mounting system is very large. In the example considered, it approaches that provided by the simple mounting supported by the mass-loaded foundation (for which m = M) discussed above. The isolation afforded by the compound system is detrimentally reduced at the resonant frequencies of the nonrigid foundation, but the loss in isolation at the fundamental mode of vibration is not large when parallel mountings are employed, although at most other frequencies natural rubber mountings would provide somewhat greater isolation. The relative performance of the simple mounting, the foundation of which is mass-loaded, and the compound mounting may be compared more readily with the help of Figs. 25 and 26, which refer to a loading mass or a secondary mass which is one-fifth and equal to the mass of the mounted item, respectively. Figure 26 illustrates the extremely large isolation afforded by the simple mounting system when supported by a heavily mass-loaded foundation (for which m = M). This level of isolation is greater than that afforded by either the compound or simple mounting systems at nearly all frequencies above the fundamental frequency of the massloaded foundation. The performance of the compound system is superior at some high frequencies, but the isolation is theoretically so large at these frequencies that comparison is trivial, since it is most probable that the isolation will be impaired by mechanical "shorts" linking the vibrating machine to the foundation other than through its resilient mountings. It should be realized that when the mass ratio M/Mf decreases below the value considered in this example, the over-all isolation afforded by the compound mounting system may increase more rapidly than that of the simple mounting with a mass-loaded foundation. For sufficiently smaller values of M/Mf, therefore, the compound mounting may provide the greatest over-all isolation, particularly at high frequencies. 5. APPENDIX 5.1. Introduction It has been shown that the response ratio of a parallel mounting supported by a foundation of finite mechanical impedance Z is given by the following equation: R2 (.1) {l \1 +-. aGo) /1 + ijM/Z + 11

where the dynamic moduli and damping factors of the low- and the high-damping rubbers comprising the mounting are Gj1 and 61I, and G2) and 62CD, respectively. The parameter "a" represents the ratio of the cross-sectional area of the highdamping material to that of the low-damping rubber. The cross-sectional areas are assumed to be uniform. The parameters Glo and G20 refer, respectively, to the values of GIw and G2W at the natural mounting frequency wo. This angular frequency is given by the relation: o2 = kl (Go + aG20) (5.2) 0 M where M is the mass of the mounted item and kl is a constant equal to the ratio of the cross-sectional area of the low-damping rubber to the mount thickness. The damping factor of the parallel mounting is given by the following relation: Gzj 61z + aG2o 62) (/2); G d + aG2 5|3) The equation for the response ratio of a simple mounting supported by a foundation of mechanical impedance Z may simply be obtained by equating the parameter a to zero. The impedance of a damped supported-supported beam excited by a sinusoidal force at its mid-point has been employed1 to simulate the mechanical properties of a nonrigid foundation. The beam was assumed to possess a complex elastic modulus with constant real and imaginary parts and, consequently, a constant damping factor bf.1 It has been shownl that: [l + J4 = 1 - g Jl ndbn [sin(ndb)cosh(ndb)-cos(ndb)sinh(ndb) L Zi 1 Mf(2 / cos(ndb)cosh(ndb) where Mf is the mass of the beam, b the half-length of the beam, and nd a parameter defined by the relation: 2 4 _ (C p, d k2E (1 + j 6f) * The quantities p, k, and E are constants representing the density, the radius of gyration of the cross section, and the real part of the complex Young's modulus of the beam, respectively. It is convenient to express the product (ndb) as the complex number (p +jq), where p and q are given by the relations: 12

p = (nb) + (1+I /]- (5.6) 2Afi/2 2 f4 ) /f q = (nb) (1 + Af)1/1]/ (, 7 _2Ab~f1/2f 2 1/2 Af where n = (cop/k2E)1/4 and A- = (1 + 6b)1/2 Equation (5.4) may then be written in the form: 1 + J = [1 - (A + jB)] (5.8) where (1-A) = 1- -P ( M (P sinp cosp - Q sinhp coshp) 2 (Q sinq cosq - P sinhq coshq) (5.9) 2PQ, \Mf_ B = K.L (j) (P sinp cosp - Q sinhp coshp) 2PQ f 2PQ- i- (Q sinq cosq - P sinhq coshq),10) 2PQ jf. j and P = (cosh2p - sin2q) (5.11) Q = (cosh2q - sin2p). (5.12) 5.2. Mass-Loading of a Foundation Possessing Finite Mechanical Impedance A qualitative interpretation of the mechanism by which a mass m placed between the bottom of a resilient mounting and a nonrigid foundation (Fig. 6e) increases the isolation afforded by the mounting system has been given in Section 2. It can be shown that the response ratio of a parallel mounting supported by a mass-loaded foundation of finite mechanical impedance Z is given by the 15

following equation: 1R^~~~~~~2 ____(1 + A2) { F-W () Gio+aG2o ( 1 >] + (m ( j a/Z Vl U (Gjo+aG20oV ~~~~R2 ~l2 L c Gl(+aG2o G1+jwM/Z/j \VlJ +j/ZjL Cj VGiU\+aGaUE/J + (D [j +(M) < (j (5.13) From this equation the response ratio of the parallel mounting system (or that of any simple mounting if the parameter a is equated to zero) may be evaluated when the dependence of the mechanical impedance Z upon frequency is known. Substitution in this expression for the mechanical impedance possessed by the nonrigid foundation discussed previously [Eq. (5.8)] leads to the equation: =R2 = ______(1 + A2) [(1-A)2 + B2] J4 (x2+,2) (lo+aG20 2 OG1aG2 ( 2 +) olo+aG2 - 2 2. (X-A) +. (+B) + Bb G+aG2o Loo \GlG+aG2cD/ c L j \G1Lco+aG2uD + (1 + A) [(X-A)2 + ( 5+B)2], (5.14) where the parameters A and B are defined by Eqs. (5.9) and (5.10), respectively, and \ = [1 - (m/M)A] and 5 = (m/M)B. 5.3. Transmissibility of the Compound Mounting System The transmissibility of the compound mounting has been determined most generally with the assumption that a different rubber-like material is employed in each stage of the mounting system. The materials for the primary and secondary mountings are described by dynamic moduli and damping factors equal to G3W and 63X, and G44 and 64W, respectively. (The secondary mounting supports the intermediate mass M.) The optimum value for the ratio of the secondary to the primary mount stiffnesses assumed in the derivation of the expression for transmissibility ensures that the separation of the two resonant frequencies of the mounting system is a minimum, providing that the mount damping factors are not large. The optimum value of this stiffness ratio equals (1+8), where: = M/M, M again being the mass of the mounted item. The general transmissibility equation is: 14

(G3w4 2\( [ (1+23C) (1+&25C)]...2 3 - --,- (5.15) [E2 + F2] where F(4E = W( ),40 G3 )0};T - f + CD _2 __ _ L'(D0 \\2+/ 2+P) GG3o) G3C o 2+P 3 VG)30) G3)C + G4) (1-6364D)] (5.16) \G \ F = ) ( ) ( ) 3 + 64D) _ ( (63 + 64CD) * (5 17) O 2+/ %G3) \G30 G3C / -4 The quantities G30 and G40 refer, respectively, to the values of G35 and G4W at the natural mounting frequency w0, which is given by the relation: 2 kG30+ (5.18) (1 + 6b ) +[ ) -G1j (461X), (5.19) where 2 - Glo X+) (5.20) The transmissibility of identical parallel mountings in the compound system may be obtained from this equation simply by substituting [(Glo+aG20)/(Glcn+aG2w)] for (Glo/Glw) and A for 61S, where these parameters have the same significance as in Section 5.1. 15 Wo = 2i _ +) * 5.0

5.4. Response Ratio of the Compound Mounting System The response ratio of the compound mounting system supported by a foundation of finite mechanical impedance Z has been determined with the assumption that the same rubber-like material is utilized in both the primary and secondary mountings. The optimum value of the mount stiffness ratio (Section 5.3) has been employed in the derivation of the following equation: 22 2 2 = (1 + 61X) 1(1 + j /Z) (5.21) ER --- (H- jl)|2 -- (5.21) (H -JI) where H 2 } rH {L(4) (Q )2 (23 (2 2 (j 2 ( ) (+ +) (1 - 51X)] zLoW \oG/ 92+PJ 2+p r \G1a 2+J z- [ F(- i K (G1O ( ) 2- (l- )] (l+}){ (5.22) Z Lkwo/ \Gj1W \2+^/ i J,oI =X 2 fX Glo P -1 (2601) +jM1 [- ( -X (P ]6 I 2W+^ -P/ J^l) Z )1 2 p (5.23) Equation (5.21) relates to a single rubber-like material, but again the response ratio for the system employing the parallel mounting may be obtained when appropriate substitution is made for the ratio (Gio/GiL) and 61-. When the mechanical impedance of the nonrigid foundation considered previously [Eq. (5.8)] is substituted for Z in this equation, the response ratio may be written: R2 = (1 + 62)2 [(1-A)2 + B2] (.4) [j2 + K2] where J rl U \2 o1 2 ~ F/ \4 ) \2 + / ) -(l +2 (\2 + + + (1-( 11-i(^ (L ( 2 1 1 (.25) L coo 1/ 2+ p y2+py \C~o/ G/ 2+p16 16

K -= A(1+)o6F [W (G) (" l -21 +B(1+P) F(G)2 (G ( )- (1- 2) I ~LUo/ \Glc/ ^W Lwo/ \GIcJw \2+P/ 26 1w Gfo )2l+P f \ - 1 1 (5.26) and the parameters A and B are defined by Eqs. (5.9) and (5.10). 17

REFERENCES 1. J. C. Snowdon, Reduction of the Response to Vibration of Structures Possessing Finite Mechanical Impedance, Part I, The University of Michigan Willow Run Laboratories Technical Report 2892-4-T, November, 1959. 2. W. P. Fletcher and A. N. Gent, Brit. J. Appl. Phys., 8, 194 (1957). 3. J. C. Snowdon, Akust. Beih., No. 1, 118 (1956). 4. J. C. Snowdon and G. G. Parfitt, J. Acoust. Soc. Am., 51, 967 (1959). 5. J. C. Snowdon, Brit. J. Appl. Phys., 9, 461 (1958). 18

5 VULCANIZED HEVEA CN 2 5 10 20 50 102 52 103 210 5x103 104 ((9.5_ 2 Ix 106E 5x l05 --,' I I I I 0.05 VULCANIZED HEVEA 0.02 >" 5 /0 oc / 0.02 0.01 I I I I I I I I 2 5 10 20 50 102 2xl02 5x102 103 2xlO3 5x103 104 FREQUENCY (CPS) (b) Fig 1. The frequency dependence of (a) the dynamic shear modulus and (b) the damping factor possessed by a natural rubber vulcanizate. 19

2 / THIOKOL R.D., Ix10/ |0 C-. / 520~ C:5 z.,107 3xlO6 I I I I I I 1 2 5 10 20 50 10 2x102 5x102 10 2x103 5x103 10 FREQUENCY (CPS) (a) o5 C? 0.2 - 0.2 0.1 _ 0.05 I —---— I I i! 0.05 1 2 5 10 20 50 102 2x102 5xl02 103 2x103 5x3 10 FREQUENCY (CPS) (b)'ig. 2. The frequency dependence of (a) the dynamic shear modulus and (b) the damping factor possessed by Thiokol R. D. 20

40 8,= 0.023 30 Ao. 0.091 I Ao.2- 0.142 20 Ao.3= 0.182 82= 0.515 10: A - r.^,. 0 -10 \, /- VULCANIZED HEVEA 3-.~~~~ f^^s / /- HIOKOL RD 1 -20 \./. —-- -- -30 --— \ 2 z~ -40 A/- a= 0.1 5 10 50 100 500 1000.2'___ Ao,:o.505 —----- y/r-' -:0.3 I Ao.2= 0.407 —- - Ao.,:= 0.262,g. T0.024 -50 -60 -— SIMPLE MOUNTING ~2=1.775 A'""".3i2-,, vulca e —d PARALLEL MOUNTINGokol R. at a tee of -80 AO., = 0.867 —-/ /\ 8, 0.038 -— / \ -90 1 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 3. The transmissibility of simple and parallel mountings of vulcanized hevea and Thiokol R. D. at a temperature of 20~C. 21

40 30 20 I0 0 -10 -20 -o ~0 c, -30 w 0 * " -40 -50 -60 I M/M 10 0. -70 -80 -90 -100 I I I I i 1 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 4. The response ratio of a simple mounting of vulcanized hevea. Mounted item ten times more massive than the foundation. Foundation damping defined by 6f = 0.01, 0.1, and 1.0 22

40 _'30 20 10 t O -40 -5o 0- - --- -ti -- \A\/l\ -70 -80 I 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 5. The response ratio of a simple mounting of vulcanized hevea. Mounted item twice, ten, and fifty times more massive than the undamped foundation. 25

FoeijWt Vo Iz (b)'A iwt'OJ? (d); Iogw) c\ 24 (e) lo'W) Fig. 6. The frequency dependence of the velocity with which a nonrigid foundation responds to mechanical vibration. 24

40 30 20 - T I II I! -~~~~~~~~~~~~~10~~I 2 0I r~~~~~~~~I 10 -S 1- 0 1 I I I % \/\\' -30o cn -40 I! II fudton Loading mas ----- n, -20 and Ieq - th as -50 o/ -0M= -70 -80 ---- M/Mf=50 - 8f=0.01 mountdaitio. Foadi atone-teng oned by eu = 0.01. -90 -100 L L2 I 5 10 50 I00 500 I000 FREQUENCY (CPS) Fig. 7. The response ratio of a simple mounting of vulcanized hevea supported by a mass-loaded foundation. Mounted item fifty times more massive than the foundation. Loading mass one-tenth, one-fifth, and equal to the mass of the mounted item. Foundation damping defined by 5f - 0.01. 25

40 30 20 10 o?, —---------------- s'"30 =I \0 \ m~ I \ I \ 0 -100I Y l l lll 1 \, / \1 w,_0. m/M= 0 0.1 -50 0 — 1.0 -60 -70 -80 M/Mf= 50 f 0.01...... -90 15 5 0 0 00 500 1000 FREQUENCY (CPS) Fig. 8. The response ratio of a simple mounting of Thiokol R. D. supported by a mass-loaded foundation. Mounted item fifty times more massive than the foundation. Loading mass one-tenth, one-fifth, and equal to the mass of the mounted item. Foundation damping defined by 6f = 0.01. 26

30 I. I II I I I I II II 20 m=.- "I >' " >"II gI II I II I I I 1.50 50-1005000I0 II Fo 9. The___ rs n rai o.lo n fi II I \ \' I \' ) ) -I0 II cn -30 -I0 1 \1 1 > \ \';, CI 0) y /\ \III If(I,~~~~~~~~~~~~~~I.50 10 - -- \ I I, II \ -60 II -80 ------ m/M= 0.08 ---- S^O.OI 0.0 f: —I 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 9. The response ratio of a simple mounting of vulcanized hevea supported by a mass-loaded foundation. Mounted item ten and fifty times more massive 0FRQEC (CPS Fig 9.Th repnertooasipemutnofvlaieheaspotd by~~~ ~ ~ ~ ~~ aI aslae onaion one tmtnadffytmsmr asv'90~~~~~~~~

*T'0 = SQ peauTJap BuTdu-ep uor;epunog'uau;T peaunoui ea; jSo ss-euu eaq oq einbe puse'qq;S-jj-uo'q;ue;-euo sseu SuTppeol'uoe;-punoj 9aq; ueq%; aAss-ewu a9Jo saumL. ua9 muaqT p9auno14'uo0.4epunoj papPeo-sseui e Xq pa;.ioddns 9A0aq pazlus-DTnA jo uT;unom 9adumTs 9 jio oTe a9suodse a 9qd *01 01 J (Sd3) A3N3nf3lJ 0001 OO 001 OS 01 S I 00106-'o —- ------- o0 =n/ - \ \ \ ci )V —/V 08OL-'01 o\ \ \ 09I x \ i I \ \ \ I \ 04- ci' -o' 0 o 01-'I I \I _______ at'~O

40 30 20 10. 0 01^^ ^ ^ ----- -\ ---------- ^/-\ \ -10 I' " \ i- /-20 \ \\ t,-40 -, ^ \ -30 m/M =O - 1 w0 -40 1.0 -70 -8 M/M 0 M= 10.1 -90 -100 I I; 1 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 11. The response ratio of a simple mounting of Thiokol R. D. supported by a mass-loaded foundation. Mounted item ten times more massive than the foundation. Loading mass one-tenth, one-fifth, and equal to the mass of the mounted item. Foundation damping defined by 6bf = 0.1. 29

40 - 30 ell, \ I, V VULCANIZED HEVEA L -10 I,"W't", I II I 20, II I I______ __I___________ I 20 ------ --- ---------------— II II II otI I: /n \'I \ \= 0 - -- 0MO PARALLEL MOUNTING -80 0 50 100 500 1000 FREQUENCY (CPS) Fig. 12. The response ratio of simple and parallel mountings of vulcanized hevea and Thiokol R. D. supported by a mass-loaded foundation. Mounted item fifty times more massive than the foundation. Loading mass one-fifth c, -40- -- - %, -60of the mass of the m/Munted0.2 M/Mem.50 Foundatio0.0n damping defined by -- -701..... -------------—,3--— SIMPLE MOUNTING, — PARALLEL MOUNTING " -90...... 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 12. The response ratio of simple and parallel mountings of vulcanized hevea and Thiokol R. D. supported by a mass-loaded foundation. Mounted item fifty times more massive than the foundation. Loading mass one-fifth of the mass of the mounted item. Foundation damping defined by bf = 0.01. 30

40 — 30 A II 0 20 -2 lo0 L 11 ~ —a= 0.2 o L\ /~ I ^~/ — VULCANIZED HEVEA _^^^ -'\ ILA / // — THIOKOL RD -0 -3 \\ / //?i \ /0 -50 -602m/M= 0.1 M/M~~ 10 — — 0 —---------- - - ~. /' \- ^ --— SIMPLE MOUNTING— PARALLEL MOUNTING -80 -100 I I I 5 10 50 100 500 1000 FREQUENCY (CPR) Fig. 13. The response ratio of simple and parallel mountings of vulcanized hevea and Thiokol R. D. supported by a mass-loaded foundation. Mounted item \ \ \ i~i'\ \ X / \\ mas-0s of the mounted item. Foundation damping defined by = - 0O. 1. 1\ \1 \ \ \ —..SIMPLE MOUNTING \ -— PARALLEL MOUNTING I 5 10 50 100 500 1000 FREQUENCY (cPSg Fig. 13. The response ratio of simple and parallel mountings of vulcanized hevea and Thiokol R. D. supported by a mass-loaded foundation. Mounted item ten times more massive than the foundation. Loading mass one-tenth of the mass of the mounted item. Foundation damping defined by 6f = 0.1. 51

40 _ 30 20 10,_ I a:t~~~ ^\@~ ~ \ / rVULCANIZED HEVEA 0 F I I ^^ \^ / // — THIOKOL R D, -30 m /M = —----------- z 0 cJn -40 -60 Ill \ %\ -90 5f 1 50 10 o-8 5 10 50 100 500 1000 FRF3UENCY (CPS\ Fig. 14. The response ratio of simple and parallel mountings of vulcanizec hevea and Thiokol R. D. supported by a mass-loaded foundation. Mounted item ten times more massive than the foundation. Loading mass equal to the mass 52

40 30 20 i,1 -10 -20,- E X X {t r.1 90 \ \ \ 0.2 oo _ _ _ __ _ _ _ __ _____\ \ \ \__ _ _ 0. J -30 —------ — \ \ —7/r 0-2 33 cU \ -40 -50 -60 -70 -—.SIMPLE MOUNTING COMPOUND MOUNTING -80 -90 -100 I 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 15. The transmissibility of a compound system employing vulcanized hevea mounts. Secondary mass one-tenth, one-fifth, and equal to the mass of the mounted item. 33

40 30 20 10 -10 -20 -30 z -40 [-50,50 0 --— SIMPLE MOUNTING — COMPOUND MOUNTING -70 -80 -90 -100 I I I I 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 16. The transmissibility of a compound system employing Thiokol R. D. mounts. Secondary mass one-tenth, one-fifth, and equal to the mass of the mounted item. 34

40 30 20 0 -10O -20 I-60 - l -\ \\- ^ - -- I-7 \ \\\ \ 1-0 H -~ ---— \ \ / —H 2~~~~~~~~~~~~~~~~~0.1 _ -0\ \ \ 2 Z -40 -60 -70 --— SIMPLE MOUNTING -COMPOUND MOUNTING -80 -90 -100 1 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 17. The transmissibility of a compound system employing parallel mounts comprised of vulcanized hevea and Thiokol R. D. Secondary mass one-tenth, one-fifth, and equal to the mass of the mounted item. 35

40 30 20 -I0 - HEVEA MOUNTING n -20. —-- -- / —-- PARALLEL MOUNTING - THIOKOL RD MOUNTING I -30 co o") -40 -50 -60 -70 /= -0.1 -80 -90 1 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 18. The transmissibility of a compound system employing vulcanized hevea, Thiokol R. D., and parallel mounts. Secondary mass one-tenth of the mass of the mounted item. 36

40 30 20 10 0 ~8^ i~ ~ HEVEA MOUNTING -10 - PARALLEL MOUNTING THIOKOL RD MOUNTING -20 \ -40 -50 -70 -— 1.0%- \ -80 _ -30 so \ —-- \ -50 -100 15 50 10 50 100000 FREQUENCY (CPS) Fig. 19. The transmissibility of a compound system employing vulcanized hevea, Thiokol R. D., and parallel mounts. Secondary mass equal to the mass of the mounted item. 37

40 81 -4~~~~~~~~~o ------ --------- / \\ - 1 —II i /A/\\ H 3I 20C 10 9''\I I in II /I I5, -9~I I I I Sr I IMPLE (ICS \ \ c oM I M \., /. \ -90n\,' +1 \ ~~~LU ~ ~ ~ ~ ~ ~ ~ ~.2: -40 I o-~//\ I I /I t! -60 tion. Secondary mass one-tenth, one-fifth, and equal to the mass of 70the mounted M/M50 Foundation damping defined by = 00.01 ~-80~8 -90 -100 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 20. The response ratio of a compound system employing vulcanized hevea mounts. Mounted item fifty times more massive than the foundation. Secondary mass one-tenth, one-fifth, and equal to the mass of the mounted item. Foundation damping defined f f 0.01. 58

40'_______ 30 20 -iO //'I \\ -20 _o i Ir \\ 11 / \ i -9 _____://_ \\I I if) -40 -50 1.0 -60 -70 -- M/Mf= 50 --- 0.01 -80 --— SIMPLE MOUNTING -— COMPOUND MOUNTING -90 1 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 21. The response ratio of a compound system employing Thiokol R. D. mounts. Mounted item fifty times more massive than the foundation. Secondary mass one-tenth, one-fifth, and equal to the mass of the mounted item. Foundation damping defined by bf = 0.01. 39

40 / oI \ \\ A' 70 0 M/M\ / /, 0.01'i' 1/ \ -100 o \ \' 1 1 ~- CIP v / \; 40nt ______ ______of\\ vue a I~~~~~0 ~~ 4 -:0 _o.~ \ crw~ ~ ~ ~ ~~~~( 50 -40 //! UO 0.1 0.2 M/M^ 5 \ I I -70 M/M- = 50 -- 0.01 -80 --— SIMPLE MOUNTING COMPOUND MOUNTING -90 -100 I 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 22. The response ratio of a compound system employing parallel mounts comprised of vulcanized hevea and Thiokol R. D. Mounted item fifty times more massive than the foundation. Secondary mass one-tenth, one-fifth, and equal to the mass of the mounted item. Foundation damping defined by 5f = 0.01. 40

40 30 -- 20 10 0 -10 0 HEVEA MOUNTING -- X \ \ \ / \ -50 rr6 -70 - 3=O.I -- M/Mf= 50 -- 0.01 \ / -80 1 5 10 50 n10 500 1000 FREQUENCY (CPS) Fig. 23. The response ratio of a compound system employing vulcanized hevea, Thiokol R. D., and parallel mounts. Mounted item fifty times more massive than the foundation. Secondary mass one-tenth of the mass of the mounted item. Foundation damping defined by 16 = 0.01. 41 41

40 30 20 10 HEVEA MOUNTING 0 \- 1- — \- PARALLEL MOUNTING THIOKOL RD MOUNTING -10 v. -20 0 -30 0 -40 -50 -60 - /31.0 - M/M= 50 f =0.0 \ \ I. /| -80 -90 \ -100 -0oo I I I I I 11111 i1 1 111 il I il i i 1 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 24. The response ratio of a compound system employing vulcanized hevea, Thiokol R. D., and parallel mounts. Mounted item fifty times more massive than the foundation. Secondary mass equal to the mass of the mounted item. Foundation damping defined by bf = 0.01. 42

40 - _ — BEm\ 30 II 20 -I\ ii \ 10 ~~~~~~~~~~I -100 XI\\ II II -20 SUPPORTED BY MASS LOADED FOUNDATION m/M= 0.2 ~ " \ ", HEVEA MOUNTING IN COMPOUND SYSTEM/m= 0.2 -— F/ \ (; \ -60 -70 M/Mf= 50 f 0.01 L~~~~~~i~~ -80 -- HEVEA MOUNTING IN SIMPLE SYSTEM m/M=6= -.-HEVEPARALLEL MOUNTING IN COMPOUND SYSTEM, /= 0.2 2 -90 I 5 10 50 100 500 1000 FREQUENCY (CPS) Fig. 25. The response ratio of a compound system employing vulcanized hevea mounts, and the response ratio of a simple mounting of vulcanized hevea supported by a mass-loaded foundation. Mounted item fifty times more massive than the foundation. Secondary mass one-fifth of the mass of the mounted item (5 = 0.2). Loading mass one-fifth of the mass of the mounted item (m/M = 0.2). Foundation damping defined by bf = 0.01. 45

40 30 20 II II,I -\0II I -20, II II L 100 l l1 l l l l l l I I I I l l Il Ii, I %'zF ________________ ( \ ) more massHEVEA MOUNTING IN SIMPLE SYSTEM, mass e t t I 0I -50 -he mountPR BYd MAS LeDpons radio ma equal e m ass of FONDTIN med HEVEA MOUNTING IN COMPOUND SYSTEM,.0 item (m/M = 1.0). Foundation damping defined by = 0.01. HEVEA MOUNTING IN SIMPLE SYSTEM, M -90 t -100 / I I5 10 50 100 500 1000 the moute iem.0. Ladngmas eua i t i-te SI = 1.0). Fou t dapn define by 1, 0.1 44

i + AD Div. 25/5 UNCLASSIFIED AD Div. 25/5 UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Mountings- Willow Run Laboratories, U. of Michigan, Ann Arbor 1. MountingsREDUCTION OF THE RESPONSE TO VIBRATION OF STRUCTURES Performance REDUCTION OF THE RESPONSE TO VIBRATION OF STRUCTURES Performance POSSESSING FINITE MECHANICAL IMPEDANCE, PART II, by 2. Rubber-MeChanical POSSESSING FINITE MECHANICAL IMPEDANCE, PART II, by 2. Rubber-Mechanical J. C. Snowdon. Technical rept. January 1960. properties J. C. Snowdon. Technical rept. January 1960. properties 44 p. incl. illus., 5 refs. I. Snowdon, J. C. 44 p. incl. illus., 5 refs. I. Snowdon, J. C. (Rept. no. 2892-6-T) II. Bureau of Ships (Rept. no. 2892-6-T) II. Bureau of Ships (Contract NObs 77072) Unclassified report Noise and Vibra- (Contract NObs 77072 tion Branch tion Branch The performance of simple and compound mounting sys- III. Contract NObs 77072 The performance of simple and compound mounting sys- III. Contract NObs 77072 tens supported by a foundation of finite mechanical tems supported by a foundation of finite mechanical impedance, and the performance of the simple mounting impedance, and the performance of the simple mounting supported by a mass-loaded foundation, have been the- supported by a mass-loaded foundation, have been theoretically determined and compared. A simply supported oretically determined and compared. A simply supported damped beam has been employed to simulate the behavior damped beam has been employed to simulate the behavior of the foundation. The dynamic mechanical properties of the foundation. The dynamic mechanical properties of natural rubber and a high-damping rubber have been of natural rubber and a high-damping rubber have been employed to describe the behavior of anti-vibration employed to describe the behavior of anti-vibration mount materials. mount materials. When the ratio of the mass of the mounted item to Armed Services When the ratio of the mass of the mounted item to Armed Services the mass of the foundation is large, the isolation Technical Information Agei the mass of the foundation is large, the isolation Technical Information Agency (over) UNCLASSIFIED (over) UNCLASSIFIED AD Div. 25/5 UNCLASSIFIED.D Div. 25/5 UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Mountings- Willow Run Laboratories, U. of Michigan, Ann Arbor 1. MountingsREDUCTION OF THE RESPONSE TO VIBRATION OF STRUCTURES Performance REDUCTION OF THE RESPONSE TO VIBRATION OF STRUCTURES Performance POSSESSING FINITE MECHANICAL IMPEDANCE, PART II, by 2. Rubber-Mechanical POSSESSING FINITE MECHANICAL IMPEDANCE, PART II, by 2. Rubber-Mechanical J. C. Snowdon. Technical rept. January 1960. properties J. C. Snowdon. Technical rept. January 1960. properties 44 p. incl. illus., 5 refs. I. Snowdon, J. C. 44 p. incl. illus., 5 refs. I. Snowdon, J. C. (Rept. no. 2892-6-T) II. Bureau of Ships (Rept. no. 2892-6-T) II. Bureau of Ships (Contract NObs 77072) Unclassified report Noise and Vibra- (Contract NObs 77072) Unclassified report Noise and Vibration Branch tion Branch The performance of simple and compound mounting sys- III. Contract NObs 77072 The performance of simple and compound mounting sys- III. Contract NObs 77072 tems supported by a foundation of finite mechanical tems supported by a foundation of finite mechanical impedance, and the performance of the simple mounting impedance, and the performance of the simple mounting supported by a mass-loaded foundation, have been the- supported by a mass-loaded foundation, have been theoretically determined and compared. A simply supported oretically determined and compared. A simply supported damped beam has been employed to simulate the behavior damped beam has been employed to simulate the behavior of the foundation. The dynamic mechanical properties of the foundation. The dynamic mechanical properties of natural rubber and a high-damping rubber have been of natural rubber and a high-damping rubber have been employed to describe the behavior of anti-vibration employed to describe the behavior of anti-vibration mount materials. mount materials. When the ratio of the mass of the mounted item to Armed Services When the ratio of the mass of the mounted item to Armed Services the mass of the foundation is large, the isolation Technical Information Agency the mass of the foundation is large, the isolation Technical Information Agency (over) UNCLASSIFIED (over) UNCLASSIFIED -f 4- ->_

AD UNCLASSIFIED ADUNCLASSIFIED afforded by the simple mounting is much less than pre- UNITERMS afforded by the sipl mounting is much less than pre- UNTE dicted by its transmissibility curve, which relates tn dicted by its transmissibility curve, which relates to an ideally rigid foundation. The isolation provided Performance an ideally rigid foundation. The isolation provided Performance by the simple mounting is increased significantly at Simple mounting system by the simple mountin is increased significantly at Simple mounting system high frequencies when the foundation of the mounting Compound mounting system high frequencies when he foundation of the mounting Compound mounting system system is mass-loaded, being largest for a natural Finite mechanical impedance system is mass-loaded being largest for a natural Finite mechanical impedance rubber mounting. In the example considered, large, Damped beam rubber mounting. In the example considered, large, Damped beam but not greater, isolation is provided at high fre- Foundation but notgreater, isolation is provided at high fre- Foundati quencies by the compound mounting utilizing a second- Natural rubber quencies by the compound mounting utilizing a second- Natural ary mass equal to this loading mass, and mountings Anti-vibration ary mass equal to thi loading mass, and mountings Anti-vibration composed of natural and high-damping rubber in paral- Mass composed of natural and high-damping rubber in paral- Mass lel. Transmissibility curve lel. Transmissibility curve UNCLASSIFIEDUNCLASSIFIED AD UNCLASSIFIED ADUNCLASSIFIED afforded by the simple mounting is much less than pre- UNITERMS afforded by the simple mounting is much less than pre- UNITEEMS diected by its transmissibility curve, which relates to an ideally rigid foundation. The isolation provided Performance by the simple mounting is increased significantly at Simple mounting system by the simple mounting is increased significantly at Simple mounting system high frequencies when the foundation of the mounting Compound mounting system high frequencies when the foundation of the mounting Compound mounting system system is mass-loaded, being largest for a natural Finite mechanical impedance system is mass-loaded, being largest for a natural Finite mechanical impedance rubber mounting. In the example considered, large, Damped beam rubber mounting. In the example considered, large, Damped beam but not greater, isolation is provided at high fre- Foundation but not greater, isolation is provided at high fre- Foundation quencies by the compound mounting utilizing a second- Natural rubber quencies by the compound mounting utilizing a second- Natural rubber ary mass equal to this loading mass, and mountings Anti-vibration ary mass equal to this loading mass, and mountings Anti-vibration composed of natural and high-damping rubber in paral- Mass composed of natural and high-damping rubber in paral- Mass lel. Transmissibility curve lel. Transmissibility curve UNCLASSIFIED UNCLASSIFIED

i-~~~~~~~~~~~~~~-1- - AD Div. 25/5 UNCLASSIFIED AD Div. 2/5 UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Mountings- Willow Run Laboratories, U. of Michigan, Ann Arbor 1. MountingsREDUCTION OF THE RESPONSE TO VIBRATION OF STRUCTURES Performange REDUCTION OF THE RESPONSE TO VIBRATION OF STRUCTURES Performance POSSESSING FINITE MECHANICAL IMPEDANCE, PART II, by 2. Rubber-Mechanical POSSESSING FINITE MECHANICAL IMPEDANCE, PART II, by 2. Rubber-Mechanical J. C. Snowdon. Technical rept. January 1960. properties J. C. Snowdon. Technical rept. January 1960. properties 44 p. incl. illus., 5 refs. I. Snowdon, J. C. 44 p. incl. illus., 5 refs. I. Snowdon, J. C. (Rept. no. 2892-6-T) II. Bureau of Ships (Rept. no. 2892-6-T) II. Bureau of Ships (Contract NObs 77072) Unclassified report Noise and Vibration Branch tion Branch The performance of simple and compound mounting sys- III. Contract NObs 77072 The performance of simple and compound mounting sys- III. Contract NObs 77072 tens supported by a foundation of finite mechanicaltems supported by a foundation of finite mechanical impedance, and the performance of the simple mounting impedance, and the performance of the simple mounting supported by a mass-loaded foundation, have been the- supported by a mass-loaded foundation, have been theoretically determined and compared. A simply supported oretically determined and compared. A simply supported damped beam has been employed to simulate the behavior damped beam has been employed to simulate the behavior of the foundation. The dynamic mechanical properties of the foundation. The dynamic mechanical properties of natural rubber and a high-damping rubber have been of natural rubber and a high-damping rubber have been employed to describe the behavior of anti-vibration employed to describe the behavior of anti-vibration mount materials. mount materials. When the ratio of the mass of the mounted item to Armed Services When the ratio of the mass of the mounted item to Armed Services the mass of the foundation is large, the isolation Technical Information Agel the mass of the foundation is large, the isolation Technical Information Agency (over) UNCLASSIFIED (over) UNCLASSIFIED AD Div. 25/5 UNCLASSIFIED RD Div. 25/5 UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Mountings- Willow Run Laboratories, U. of Michigan, Ann Arbor 1. MountingsREDUCTION OF THE RESPONSE TO VIBRATION OF STRUCTURES Performance DEDUCTION OF THE RESPONSE TO VIBRATION OF STRUCTURES Performance POSSESSING FINITE MECHANICAL IMPEDANCE, PART II, by 2. Rubber-Mechanical POSSESSING FINITE MECHANICAL IMPEDANCE, PART II, by 2. Rubber-Mechanical J. C. Snowdon. Technical rept. January 1960. properties J. C. Snowdon. Technical rept. January 1960. properties 44 p. incl. illus., 5 refs. I. Snowdon, J. C. 44 p. incl. illus., 5 refs. I. Snowdon, J. C. (Rept. no. 2892-6-T) II. Bureau of Ships (Rept. no. 2892-6-T) II. Bureau of Ships (Contract NObs 77072) Unclassified report Noise and Vibra- (Contract NObs 77072) Unclassified report Noise and Vibration Branch tion Branch The performance of simple and compound mounting sys- III. Contract NObs 77072 The performance of simple and compound mounting sys- III. Contract NObs 77072 tems supported by a foundation of finite mechanical tems supported by a foundation of finite mechanical impedance, and the performance of the simple mounting impedance, and the performance of the simple mounting supported by a mass-loaded foundation, have been the- supported by a mass-loaded foundation, have been theoretically determined and compared. A simply supported oretically determined and compared. A simply supported damped beam has been employed to simulate the behavior damped beam has been employed to simulate the behavior of the foundation. The dynamic mechanical properties of the foundation. The dynamic mechanical properties of natural rubber and a high-damping rubber have been of natural rubber and a high-damping rubber have been employed to describe the behavior of anti-vibration employed to describe the behavior of anti-vibration mount materials. mount materials. When the ratio of the mass of the mounted item to Armed Services When the ratio of the mass of the mounted item to Armed Services the mass of the foundation is large, the isolation Technical Information Agency the mass of the foundation is large, the isolation Technical Information Agency (over) UNCLASSIFIED (over) UNCLASSIFIED + -^~~~~~~~~~~~~~+ A

AD UNCLASSIFIED ADUNCLASSIFIED afforded by the simple mounting is much less than pre- UNITERMS afforded by the simpl mounting is much less than predicted by its transmissibility curve, which relates to dicted by its transmissibility curve, which relates to an ideally rigid foundation. The isolation provided Performance an ideally rigid foundation. The isolation provided Performace by the simple mounting is increased significantly at Simple mounting system by the simple mounting is increased significantly at Simple mounting system high frequencies when the foundation of the mounting Compound mounting system high frequencies when he foundation of the mounting Compound system is mass-loaded, being largest for a natural Finite mechanical impedance rubber mounting. In the example considered, large, Damped beam rubber mounting. In the example considered, large, Damped beam but not greater, isolation is provided at high fre- Foundation quencies by the compound mounting utilizing a second- Natural rubber quencies by the compound mounting utilizing a second- Natural rubber ary mass equal to this loading mass, and mountings Anti-vibration ary mass equal to this loading mass, and mountings Anti-vibration composed of natural and high-damping rubber in paral- Mass composed of natural and high-damping rubber in paral- Mass lel. Transmissibility curve lel. Transmissibility curve UNCLASSIFIED UNCLASSIFIED AD UNCLASSIFIED ADUNCLASSIFIED afforded by the simple mounting is much less than pre- UNITERMS afforded by the simple mounting is much less than pre- UNITERMS dicted by its transmissibility curve, which relates to dicted by its transmissibility curve, which relates to an ideally rigid foundation. The isolation provided Performance an ideally rigid foundation. The isolation provided Performance by the simple mounting is increased significantly at Simple mounting system by the simple mounting is increased significantly at Simple mounting system high frequencies when the foundation of the mounting Compound mounting system high frequencies when the foundation of the mounting Compound mounting system system is mass-loaded, being largest for a natural Finite mechanical impedance system is mass-loaded, being largest for a natural Finite mechanical impedance rubber mounting. In the example considered, large, Damped beam rubber mounting. In the example considered, large, Damped beam but not greater, isolation is provided at high fre- Foundation but not greater, isolation is provided at high fre- Foundation quencies by the compound mounting utilizing a second- Natural rubber quencies by the compound mounting utilizing a second- Natural rubber ary mass equal to this loading mass, and mountings Anti-vibration ary mass equal to this loading mass, and mountings Anti-vibration composed of natural and high-damping rubber in paral- Mass composed of natural and high-damping rubber in paral- Mass lel. Transmissibility curve lel. Transmissibility curve UNCLASSIFIED UNCLASSIFIED

UNIVERSITY OF MICHIGAN 3 9015 03524 4857