FREE-FREE DYNAMICS OF SOME VIBRATION ISOLATORS James A. Forrest Maritime Platforms Division (MPD), Defence Science & Technology Organisation (DSTO) PO Box 433, Melbourne VIC 3, Australia. Email: james.forrest@dsto.defence.gov.au. INTRODUCTION This paper discusses the experimentally measured free-free dynamics of some simple vibration-isolator assemblies. If these dynamics are known, the behaviour of the vibration isolators can be incorporated into a mathematical model of a machine mounted onto a foundation via those isolators. Frequency-response functions (FRFs) for two single-stage isolators are presented, the isolators comprising two steel endplates and one or four rubber elements respectively. The FRFs were measured with the isolators suspended on a light string and excited by a modal hammer. From these, mode shapes are deduced and natural frequencies and modal damping are estimated. The actual isolator dynamics prove to be more complicated than initially assumed. The measurements described were undertaken to provide data for comparison to mathematical models being developed by Defence Research & Development Canada (DRDC). The shortterm aim is to extend this work to a two-stage isolator. The longer-term aim is to obtain accurate models for realistic vibration-isolator assemblies which can be used in larger models of machinery-excited structural vibration transmission in ships and submarines. An overview of foundation design issues, including isolator impedance, is given in Tso et al. The dynamic characteristics of rubber depend primarily on static pre-load, temperature and the frequency and amplitude of vibration. These issues with others are discussed by Lindley, temperature effects by Kari et al. 3 and static load effects by Dickens 4. Warley 5 gives a method for selecting rubber materials for a particular isolator performance. The dynamics of an isolator may be expressed as a blocked impedance as in Verheij 6, or as its four-pole parameters as described by Snowdon 7 and Norwood and Dickens 8. A method and testing machine to determine the four-pole parameters is described by Dickens and Norwood 9. While all these approaches are able to include the effects of various parameters, they deal either with the basic material properties of rubber or with one-dimensional dynamics of vibration isolators in the axial direction. A free-free modal analysis does not allow for any pre-load for example, but does make it possible to easily examine dynamics in all directions and check for coupling. The following sections describe the isolators, the measurement approach and the results obtained thus far for the two single-stage isolators.. VIBRATION-ISOLATOR MODELS AND OTHER EQUIPMENT The two single-stage isolators for which FRFs were measured are shown in Figures (a) and (b). These are referred to as the small and medium isolator respectively. Figure (c) shows the two-stage isolator of three plates and eight rubber elements, to which the current work will be extended. The three principal directions X, Y and Z are shown Figure. These
Figure : Dimensions (mm) and construction of the (a) small, (b) medium and (c) two-stage vibration isolator models. The two-stage isolator is essentially the medium isolator with another layer added. The end plates are steel and the rubber elements are shown shaded. are also referred to as the lateral, transverse and axial directions respectively, based on the two shear and one compressive directions of translation of the rubber elements. Each plate has three threaded holes in each edge and one threaded hole in each outward face. For the results presented here, the single-stage isolators were suspended from a string threaded through a hollow screw screwed into one of the outward faces. The mass of each steel plate for the small isolator is 64 gram and for each plate in the larger isolators 687 gram. Each rubber element is 3 gram and the suspension screw is 9 gram. The suspended isolator model was excited by a PCB modally tuned impulse force hammer, model 86C5, of sensitivity.4 mv/n, with a teflon tip. Responses were measured with two PCB model 353B8 accelerometers of sensitivity mv/g and mass.3 gram each. They were secured with wax. Data was acquired using an HP 3566A FFT analyser connected to a controlling PC. The measured data was processed using the Matlab software package. 3. METHOD To simulate the free-free state, the isolator models were suspended axially from a thin string, as shown in Figure (a). This suspension modifies the zero-frequency rigid-body modes that would otherwise be seen for a truly free-free set of boundary conditions. For example, the rigid-body translations in the lateral (X) and transverse (Y) directions are replaced by the pendulum motion of Figure (b); others are shown in Figure. Nevertheless, the frequencies associated with these modified rigid-body modes were found to be very low, typically a few Hz. These modes therefore have negligible effect on the higher modes that are of interest. It was initially assumed for the small isolator with its one rubber element that there would be six basic non-rigid-body modes. These modes are shown in Figure 3: a translational and a rotational mode for each of the three principal directions. This assumes that the steel plates are rigid and that the rubber element can be construed as a set of uncoupled springs aligned in the different directions. It was assumed that the medium isolator would exhibit a similar set of modes. Given these six modes, it should be possible to arrange two accelerometers such that they detect motion due to only one translation and one rotation. This is illustrated in Figure 4(a) for picking up translation in the lateral (X) direction and rotation about the axial (Z) axis, and
Figure : The suspension (a) of the isolator on a string to give conditions similar to free-free. Various modified rigid-body modes (b)-(d). Figure 3: The six assumed basic non-rigid-body modes based on motion in the directions of the three principal axes, shown for the small isolator. (a) (b) (c) Figure 4: The two-accelerometer approach for (a) shear modes of the rubber element and (b) compressive modes of the rubber element, and (c) the comprehensive approach for axial modes, shown for the small isolator. in Figure 4(b) for picking up axial (Z) translation and transverse (Y) rotation. To separate the modes, a sum of the two accelerations would give the translation, while a difference would give the rotation. The approach of Figure 4(b) for the modes producing displacement in the axial direction was not successful. The accelerometers for compressive modes are always aligned axially, so will pick up more than two modes at once if others are excited, unlike for the shear modes. Also, the simple compressive modes postulated in Figure 3 are not the only ones which appeared in the compressive measurements. Thus a more comprehensive approach was required, given in Figure 4(c). The top plate was designated A, the bottom B, and all the corners were numbered, with A directly above B and so on. One accelerometer was attached to A and the other to B upside down. Then impulse forces were applied to each corner of A in turn. By reciprocity, the responses at A to forces at each corner of A are equivalent to the responses at each corner of A to a force at A. Similarly, by symmetry, reciprocity and a negation (for the upside-down positioning), the responses at B to forces at each corner of A are equivalent to the responses at the four corners of B to a force applied at A. This fully characterises the motion of the isolator excited by a force applied at one corner. A separated mode can be treated as the response of a simple harmonic oscillator to determine the modal quantities. The position of its maximum magnitude gives the natural frequency of
the mode. Strictly speaking this is the damped natural frequency ω d, but if the modal damping ratio ζ is small, it is very close to the undamped natural frequency ω n because ω = ζ () d ω n The damping ratio ζ can be estimated from the mode s halfpower bandwidth ω n, which is the width of the modal peak on the magnitude plot at of the peak s maximum height, or 3 db down from the maximum. This is shown graphically on the results in the next section. The relationship for finding the damping ratio is, from Newland ω n ω n ζ () where angular frequency ω could be replaced by πf with the frequency f in Hz. These relationships can also be applied to individual modal peaks in a multi-mode response if the modal overlap is low, so that adjacent modes do not influence one another significantly. 4. RESULTS FOR SHEAR MODES Figure 5 shows examples of the frequency-response functions (FRFs), H and H, obtained for the small isolator in the transverse direction following the approach of Figure 4(a). That is, the measurements are of transfer FRFs from one end of the isolator to the other. H is the FRF for the accelerometer in line with the off-centre applied force. Two main peaks can be seen, with some variation below Hz due to the modified rigid-body modes. Simple addition and subtraction of the two FRFs does not separate the modes, because the modes are present in various proportions. One approach which does work is to look at the imaginary parts of the FRFs and apply the quadrature peak-pick method as described in Hewlett-Packard s application note. This relies on the fact that there is a 9 phase change at resonance, so that the imaginary part goes from zero to some peak value at the natural frequency. At the same time, the real part goes to zero. This means the total magnitude of the mode at resonance is represented by the imaginary part of its FRF. Given FRFs for different parts of a structure, the imaginary parts at a given resonance are thus a measure of the modal coefficients for that mode, i.e. they give the mode shape. All this is valid for well-spaced modes; with high modal overlap, the imaginary maxima and real zeros are obscured. The imaginary parts for the measurement of the shear modes are shown in Figures 6 and 7. Separation of the two modes appearing in each case can be achieved by noting that if the isolators are linear, the FRFs H and H must be a linear combination of the modes such that stt: H H mag. (db re g/n) 4 mag. (db re g/n) 4 Figure 5: The two transfer FRFs obtained for the small isolator with the two accelerometers aligned in the transverse (Y) direction. H is the FRF derived from accelerometer and H is the FRF from accelerometer.
H H = ay + by = cy + dy where Y is the lower and Y the higher of the two underlying modes and a, b, c and d are the modal coefficients denoting each mode s contribution to the FRFs. If the modes are assumed to be normalised to unity, the coefficients can be taken to be the signed magnitudes of the peaks from the plots of FRF imaginary parts. From equations (3), it can be seen that the two modes are separated by the operations ( d b) H H = ( ad b c) Y (4) ( c a) H H = ( bc a d) Y The values for the coefficients are given in Table, obtained by graphically selecting the imaginary parts peaks when plotted on the computer screen. The value for b for the medium transverse case was estimated from the other three coefficients using the result that the new imaginary part of the first mode is equal to ( ad b c) when the first of equations (4) is applied to obtain Y. Since it is an estimate, it appears in brackets. The fact that the peak at about 39 Hz for the small isolator (69 Hz for the medium) appears in both the lateral and transverse cases shows that this is the axial rotation mode. This means that the other peak in each case belongs to the relevant translation mode. This is also borne out by the signs of the modal coefficients. For the lower, rotational, natural frequency f, a and c have opposite signs, meaning the two adjacent corners where accelerometers and were attached are moving in opposite directions. Similarly, b and d have the same sign, so the two corners are moving in the same direction for the higher mode of frequency f. However, the magnitudes of these coefficients are not the same, which means the modes are neither purely rotational nor purely translational. This indicates that there is coupling between the axial torsion and shear translation of the single rubber element in the case of the (3) (a).5 sll: H H.5.5 (b) stt: H.5 H.5.5 Figure 6: Imaginary parts of the two FRFs obtained for the measurement on the small isolator in the (a) lateral (X) direction and (b) transverse (Y) direction.
(a) (b). mll: H.4. mtt: H.4. H.. H. Figure 7: Imaginary parts of the two FRFs obtained for the measurement on the medium isolator in the (a) lateral (X) direction and (b) transverse (Y) direction.. Results from H Results from H Test f (Hz) a f (Hz) b f (Hz) c f (Hz) d small lateral small transverse 39. 39. -.743 -.6. 9.7 -.45 -.334 38.8 39..36.58 99.9 9.6 -.7 -.67 med. lateral med. transverse 69. 69.3 -.397 -.387 76.6 (7.8) -.45 (-.3) 68.9 68.9.93.78 77. 7.8 -. -.3 Table : Natural frequencies and modal coefficients obtained from the imaginary parts of the relevant FRFs. small isolator. Therefore the rubber element cannot be considered as a set of uncoupled springs as originally assumed. In the medium isolator, the disposition of the four rubber elements implies that shearing of each individual element would have to be involved in an overall rotational mode in any case, but the coefficients show that this also is not pure rotation. The separated modes obtained by applying equations (4) with the modal coefficients from Table to the FRFs in Figures 6 and 7 are shown in Figures 8 and 9. What are labelled as Y and Y on the graphs are actually the expressions on the right-hand side of equations (4), i.e. Y and Y multiplied by combinations of modal coefficients. Shown on the magnitude plots are the 3 db lines on each peak which define the halfpower bandwidth. The natural frequencies, bandwidths and corresponding modal damping ratios calculated from () are listed in Table. There is about 5% difference between the two values of damping ratio given for the axial rotation in both cases, an indication of the damping values uncertainty. While the separation is quite good, the curves show traces of the mode which was being eliminated. This demonstrates the approximate nature of the quadrature peak-picking method to obtain the modal coefficients. Apart from errors in determining the imaginary peak magnitudes (especially for significantly overlapping modes), it is possible that the modal coefficients are in fact complex rather than real, since the modes are damped. Better results would require the use of more sophisticated curve-fitting approaches.
(a) (b) magnitude (db re g/n) magnitude (db re g/n) 4 6 4 sll stt Y Y Y Y.5.5.5.5 Figure 8: Magnitudes and imaginary parts of the separated modes for the small isolator in the (a) lateral (X) direction and (b) transverse (Y) direction. Halfpower bandwidth indicated by lines on magnitude peaks. (a) magnitude (db re g/n) 4 mll Y Y 6...4 (b) magnitude (db re g/n) 4 6 mtt Y Y...4 Figure 9: Magnitudes and imaginary parts of the separated modes for the medium isolator in the (a) lateral (X) direction and (b) transverse (Y) direction. Halfpower bandwidth indicated by lines on magnitude peaks. small isolator medium isolator mode fn (Hz) f (Hz) ζ 38.9 3.58, 3.5 9.7 9.7. 9.4 axial rotation transverse transl. lateral transl. axial rotation transverse transl. lateral transl. 69. 7.9 76.9 5.45, 4.73 4.7 6.9.46,.4.5.45.39,.34.3.4 Table : Modes, natural frequencies, half-power bandwidths and corresponding modal damping factors for the small and medium isolator s shear modes. The natural frequency is the average of the values for that mode from Table. There are two bandwidths / factors for the rotation modes because they appear twice for each isolator.
5. RESULTS FOR COMPRESSIVE MODES The imaginary parts of the corner FRFs of the small isolator for excitation at A are given in Figure. These were measured in the manner of Figure 4(c) and applying the reciprocity and symmetry reasoning given in Section 3. The peak values with their corresponding natural frequencies are listed in Table 3. The 48.6 Hz mode is the simple lateral rotation (about X) and the 34.9 Hz mode is the simple axial (Z) translation which were postulated in Figure 3. However, the modes in between do not correspond to the transverse rotation (about Y) that was expected. They are instead modes where the rotation of the end plates is about a face diagonal instead of one of the principal axes. The 93.4 Hz mode comprises mainly of a rocking motion of A and A3 and B and B4, opposite diagonals on the opposite faces. Likewise, the.4 Hz mode comprises mainly of a rocking of A and A4 and B and B3. These diagonal modes are sketched in Figure. It can be seen that the lower mode is more like rotation about X with a narrow width of rubber, while the higher one is more like rotation about Y, with a wide width of rubber and therefore greater stiffness. Small: A A A3 A4 B B B3 B4 Figure : Imaginary parts of the small isolator s corner FRFs in the axial (Z) direction, for an excitation applied at corner A.
Position: f (Hz) Im(H) f (Hz) Im(H) f 3 (Hz) Im(H) f 4 (Hz) Im(H) A A A3 A4 48.6 48.8 48.4 48.6 -.8 -.44.39.4 93. 9.9 -.54. 99.7? 98.5.4 +? -. 34.5 35.5 34.7 34.9 -.74 -.7 -.63 -.73 B B B3 B4 48.6 48.7 48.5 48.6.3.4 -.39 -.7 93.6 94.7 9.6 94.4 -.53 -.68..73?.9.3.5 -?.5.47 -.7 34.8 35.4 34.5 35. av. f n (Hz) 48.6 93.4.4 34.9 mode lateral (X) rotat n diagonal rotation diagonal rotation axial (Z) transl n Table 3: Natural frequencies and modal coefficients for the small isolator, obtained from the FRF imaginary parts of Figure. The average natural frequency for each mode is given. A dash indicates the mode is not visible and a question mark that its peak is not clear..68.78.75.76 Figure : Schematic of the diagonal modes of the small isolator at the natural frequency of (a) 93.4 Hz and (b).4 Hz. The rubber element is not shown. The arrows indicate the main motion only. The imaginary parts of the corner FRFs for the medium isolator are plotted in Figure and are listed numerically with their natural frequencies in Table 4. Note, however, that only the five most prominent peaks have been tabulated this time. In addition, there may be other modes, but these are unclear. For example, A4 and B4 show a side bulge on the 5 Hz peak at about 8 Hz, and A3 and B3 show a similar bulge at about 75 Hz. Since there is such a high degree of apparent modal overlap, the imaginary parts listed in Table 5 are not necessarily good estimates of the actual modal coefficients. Nevertheless, the 4.9 Hz mode would seem to be a form of transverse rotation (about Y), and the 57. Hz mode a lateral rotation (about X). The imaginary parts for the 74.5 Hz mode are all about the same magnitude with the signs for plate A matching those of plate B, which suggests that this might be a rigid-body type of rotation about Y, with the suspension string acting as a pivot. This would have to be checked by changing the suspension point to an edge instead of the centre of the plate face. The simple axial (Z) translation mode does not seem to be in the Hz frequency range analysed. No modal damping ratios are given for either the small or the medium isolator. While it would be possible to estimate some of these by the halfpower bandwidth method where the modes are already well separated, as for the 48.6 Hz mode of the small isolator, high modal overlap elsewhere prevents this. Since there are now many modes and much overlap, the separation procedure is more complicated than the straightforward two-mode situations of the previous section. A more comprehensive curve-fitting procedure would be needed to estimate the damping for most of the compressive modes identified here. Such a procedure would also allow more accurate determination of the mode shapes.
Medium: A A A3 A4 B B B3 B4 Figure : Imaginary parts of the medium isolator s corner FRFs in the axial (Z) direction, for an excitation applied at corner A. Posit n f (Hz) Im(H) f (Hz) Im(H) f 3 (Hz) Im(H) f 4 (Hz) Im(H) f 5 (Hz) Im(H) A A A3 A4 64.6 64.9 64.8 65.5 -.7 -.4.5.7 74.4 74.3 74.4 74.8 -.7.4.6 -. 3.7 -.9 5.9 3.3 5. 5.6 -.9.3.43 -.45 57.5 57.4 56.7 57. -. -..8.6 B B B3 B4 64.7 64.7 65. 65.4 -..3.5 -. 74.5 74.5 74.3 74.9 -.9.3.5 -. 3.5.6 5.9 3. 5. 5.7.84 -.4 -.4.4 57.5 57.3 56.7 57.3.. -.7 -.6 f n (Hz) 65. 74.5 3.6 4.9 57. mode??? transv. (Y) rot n lateral (X) rot n Table 4: Natural frequencies and modal coefficients of the prominent modes for the medium isolator, obtained from the FRF imaginary parts of Figure. The same format as Table 3 applies.
6. CONCLUSIONS The use of fairly simple modal analysis techniques has allowed the identification of several modes with their natural frequencies for two single-stage vibration isolator models. In some cases, modal damping ratios have also been able to be estimated with confidence. However, the isolators exhibited more modes than originally assumed. A major reason for this is the coupling of translational and rotational motion through the rubber elements, which would therefore need to be included in any mathematical modelling of the isolators. The greater number and more complicated nature of the modes means that there was considerable modal overlap in many of the measured FRFs. A more sophisticated curve-fitting approach could be used to determine the parameters of the modes in these cases. 7. REFERENCES. Y. Tso, J. Forrest & C. Norwood (), Foundation design methods for ship noise reduction, Proceedings Pacific International Maritime Conference, 9-3 January, Sydney, Australia, 56-65.. P.B. Lindley (964), Engineering design with natural rubber, NR Technical Bulletin, Malaysian Rubber Producers Research Association, Welwyn Garden City, Herts., UK. 3. L. Kari, P. Eriksson & B. Stenberg (), On the temperature dependence of vibration isolator stiffness in the audible frequency range, Proceedings Seventh International Congress on Sound and Vibration, 4-7 July, Garmisch-Partenkirchen, Germany, V, 947-954. 4. J.D. Dickens (), Phase velocity of rubber element in vibration isolator under static load, Journal of Sound and Vibration, 34(), -4. 5. R.L. Warley (996), Dynamic properties of elastomers as related to vibration isolator performance, Rubber World, 3(6), 33-39. 6. J.W. Verheij (98), Multi-path sound transfer from resiliently mounted shipboard machinery, PhD thesis, Institute of Applied Physics, TNO-TH, Delft, Netherlands. 7. J.C. Snowdon (979), Vibration isolation: use and characterisation, Journal of the Acoustical Society of America, 66(5), 45-74. 8. C.J. Norwood & J.D. Dickens (998), The effect of vibration isolator properties and structural stiffness on isolator performance, Journal of Vibration and Control, 4, 53-75. 9. J.D. Dickens and C.J. Norwood (), Universal method to measure dynamic performance of vibration isolators under static load, Journal of Sound and Vibration, 44(4), 685-696.. D.E. Newland (989), Mechanical Vibration Analysis and Computation, Longman, Harlow, Essex, UK.. Anon. (986), The Fundamentals of Modal Testing, Application Note 43-3, Hewlett- Packard, Palo Alto, California, USA.