Influence of Flexible Foundation on Isolator Wave Effects

Shock and Vibration, Jul 2018

This article deals with the interaction between wave effects in mounts and resonances of foundations inflexible vibration isolation systems. A new model is proposed that is represented as a rigid mass supported by two linear unidirectional isolators on a flexible foundation beam, whose closed-form solutions for transmissibility and response ratio are then obtained, with which the influence of wave effects coupled with the flexibility of the foundation on the effectiveness of isolation is discussed. The wave effects on flexible isolation systems are analyzed under various parametric conditions and compared with those in rigid systems. In addition, several special cases are presented to show the transition between various limiting cases. Some approaches to control wave effects are also proposed.

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Influence of Flexible Foundation on Isolator Wave Effects

Shock and Vibration, Vol. Influence of Flexible Foundation on Isolator Wave Effects Kong-Jie Song Xing Ai 0 0 Department of Mechanical Engineering Shandong University of Technology finan , 250014 , China This article deals with the interaction between wave effects in mounts and resonances offoundations inflexible vibration isolation systems. A new model is proposed that is represented as a rigid mass supported by two linear unidirectional isolators on a flexible foundation beam, whose closed-form solutions for transmissibility and response ratio are then obtained, with which the influence of wave effects coupled with the flexibility ofthe foundation on the effectiveness ofisolation is discussed. The wave effects on flexible isolation systems are analyzed under various parametric conditions and compared with those in rigid systems. In addition, several special cases are presented to show the transition between various limiting cases. Some approaches to control wave effects are also proposed. © /996 John Wiley & SOilS, Inc. - INTRODUCTION Because today machines run at high speed with flexible components of lighter and thinner struc­ ture, some mounted on upper floors in multistory workshops, the problems for vibration and noise control are challenging. In these cases, vibration isolation may not be adequately predicted at higher frequencies, because their isolators may not behave as ideal resilient members in the high frequency range where the so-called wave effects will be apparent in their supports. The effective­ ness, therefore, will be considerably reduced by approximately 20 dB as compared to those pre­ dicted in a massless spring dashpot mount. Therefore, it is necessary to consider the distrib­ uted mass and stiffness in the isolators when dealing with the problem of isolation at higher frequencies. The wave effects in isolators has been studied theoretically and experimentally by Harrison (1952) , Snowdon (1978) , and Sykes (1960) . How­ ever, attention was focused predominantly on the mounting itself, and theoretical models were limited to either a single-mount system or with a rigid foundation. These approaches might lead to overoptimistic, or even erroneous predictions for the efficiency of isolation for three reasons: the coupling between mounting points being ne­ glected; the interaction between mount and foun­ dation being overlooked; the wave effects in the mount being studied separately from the reso­ nances of the foundation. In this study, these three factors are considered simultaneously; and closed forms for transmissibility and response ra­ tios are derived from a new model in which the system is regarded as a rigid mass on a resilient foundation via four isolators. The influence of both wave effects and flexibility of the founda­ tion on the efficiency of isolation under various parametric conditions were analyzed. Some new phenomena about wave effects were found and corresponding approaches to control them were presented, MODEL AND EQUATIONS Models The model in Fig. lea) is the proposed parallel type isolation system, which could be used in common engineering practice, in which the ideal rigid machine M is supported by two identical mounts laid on a nonrigid foundation simulated by an end clamped beam with internal solid damping E* = E(1 + jo), where 0 is the damping factor of the beam. The machine vibrates under the excitation of a sinusoidally varying force P. The performance of each mount is described by its complex stiffness, K* = K(1 + j 'Y), where K and 'Y) are its respective stiffness and damping factor. Other isolation system models in Fig. l(b-d) are shown for comparison. Substructure Mobility Analysis Considering the system consisting of three sub­ structures, i.e., machine A, isolator system B, and flexible foundation C, we can obtain the transfer matrices of mobility for each substruc­ ture in the following forms: A12] A 22 = [11M + 1I(2~) 11M - 11(2 J) -11M 11M - 11(2]) IIM + 11(2]) -11M -11M] -11M 11M B = [ BII B2J Bll] B22 = [ cos(n*h)J fL* Sin(n*h)I] -(fL*)-1 sin(n*h)J cos(n*h)I 'Pm(hJ'Pm(hj.J w;,,(1 + jo) (i,k=1,2) where M = jwM, J = jw JI(2b) , J is the moment of inertia of the machine, J is a 2 x 2 unit matrix, fL* = jwml(n*h), n* = wVpIE*, (f3*)4 = (1 + jo)(wmlw)2, 'Pm(h;) is the normal function of the beam, Mb is the mass of the beam, and p is the density of the isolator. Considering the conditions for force equilibrium and motion compatibility at the junction, we can obtain the force transmitted to the foun­ dation from the isolators as follows: Q = {QJ, QzV = -[Bll + All· BII + Bl2 . C + AIIB lz . C]-I . AI2 . P. F p M h Mb F p M p M The response ratio of the system may be derived in the same manner. Noticing that when machine A is directly laid on the foundation, the four-pole parameters for the isolator system B will be changed into Accordingly, the force directly transmitted to the foundation can be expressed as Q" = {QuI. Qd T = -[All + C]-I . AI2 . P. (8) To calculate the response ratio, one should first obtain the responses of the foundation at the mounting points with and without isolators. By using the transfer matrix method, we obtain the required responses V= CQ, Vu = CQu' Therefore, the response ratio of the system can be derived as Bil = B22 = /zx2, BJ2 = B2l = 02X2. (7) Case Study (9a) (9b) (10) Ri = I:'~il (i = 1, 2), I 1 - A* Co - A*[Co + So/(nn*h)] - n(n*h)Soi I RI = R2 = 211 - A*ITI (symmetricity), (11) where Co = cos(n*h), So = sin(n*h) , "II = M/(2m) , and (12) (13) (14) In addition, the output force F transmitted to the termination of the flexible foundation can be used to define the overall transmissibility To, which describes a companion force transmissibility across the entire system. Based on the superpo­ sition principle, To is then derived as and the overall transmissibility Te is defined as where !Pi(X) = ch(nix) - cos(nix) - OMh(nix) sin(nix)] and Oi = [ch(nJ) - cos(nil)]/[sh(nJ) sin(nil)]. This quantity differs significantly from either TJ or Ri in Eqs. (5) and (10). Four special cases of the above-mentioned gen­ eral equations are discussed below. Classical Model. If the exciting frequencies are much lower than the fundamental frequencies of both isolator and the flexible foundation, the wave effects then may not be apparent in the mounts. Moreover, as the foundation being com­ paratively rigid in general, the distributed mass of isolators can be neglected, and the mobilities of the foundation can be taken as zero. The model for this case is shown in Fig. I(b). Let m ~ 0, Cij ~ 0 (i, j = 1,2), then Eq. (6) reduces to where n = w/wo, Wo = V2KIM. This is identical to the formula derived based on the classical theory. Harrison's Model. When a machine is mounted on a very stiff and heavy foundation (such as vibration-free concrete blocks on the ground floor), the resonances of the foundation may not be taken into account. In this case, Mb ~ x, thus Cij ~ 0, and A * ~ O. Therefore, Eq. (5) reduces to T = Icos(n*h) - n(n*h)sin(n*h)I-I. This equation coincides with Harrison's (1952) formula. Snowdon's Model. For machines mounted on upper floors of workshops, the interaction be­ tween machine and its foundation should be con­ sidered. If wave effects of the isolators are ne­ glected, i.e., m ~ 0, thus 'YI ~ x, n*h ~ 0, then we find Eqs. (5) and (10) are simplified to, respec­ tively, T' I I I = "2 :I - A * - (O*)~I I - A* I R; = I1 - A* - (0*)2 . These are the solutions studied by Snowdon (1973) , which is also a special case (N = 2, p., = 0.5) in another study (Xiong et aI., 1990) . Single Mounting System. Snowdon (1978) also studied wave effects in isolators for a simple mounting system in which the coupling between different mounting points was obviously ignored. Here b ~ 0, and noticing that ell = e l2 = e 21 = e 22 = I/ZF, All = A22 = I/(jwM) = I/ZM, AI2 = A2J = - I/ZM Bij = aij (four-pole parameters) (i, j = 1,2). We can express Eqs. (5) and (10) in the following ways (16) (17) (18) (19) where ZF and ZM are the impedance of founda­ tion and machine, respectively. It can be seen that Snowdon's formulas (1973) are also a special case of the present model shown in Fig. I(d). RESUlIS AND DISCUSSION To reveal the influences of wave effects coupled with resonances of a flexible foundation on the effectiveness of vibration isolation, representa­ tive results were numerically evaluated under >. -40 -.,.... .+,..>.. ..n ·til -80 's til ~~ -120 - - current model ----m-O Mb--OO ~ '- I" E-t -160 ~......--...................-;;-,-.J....J...J"""""<L...,-"'--'--"--'---'..u...d..--'-...........~ 10-1 100 101 10 2 10' Frequency ratio OCoo/ooo) various parametric conditions. The transmissibil­ ity and the response ratio are plotted in terms of frequency ratio O(wlwo) on a decibel scale, in which Wo is the natural frequency of the system with a rigid foundation. The results for several special cases are also given to show their relation and comparison with the present results, which are shown in Figs. 2-8 where the dashed line represents the values obtained through classical theory [i.e., case (1)]. Based on the theoretical results, both the wave effects in the flexible isolation system and the coupling effects of the related system on transmissibility and response ratio will be dis­ cussed. Wave Effects in Flexible Isolation System It can be seen from Figs. 2 and 3 that when the exciting frequency becomes relatively high, ::c 40 '"0 '" 0 ~ .+Q> -40 Cil I-< 3l -80 ~ o ~-120 CIJ ~ -!6O.1:-::-t-'-........~=-'"~....................,.7"--'-'-'-'-'-"'"'-;;,-'--"~~ 10- 10' 10 1 10 2 10 Frequency ratio OCoo/ooo) FlGURE 3 Comparison of RI and TI : 'YI = 100, 'Y/ = 0.05, 8 = 0.01, A = 10, M1Mb = 10. Harrison's Model Classical Model -160L-:;-:;i'"'"'"""~~S~~------:"::-T-~-~..l:-~~~ to-1 10' 10' 10' 10' Frequency ratio n ( OJ! OJ 0) standing waves will occur in the isolator. The peaks in Rand T are not only influenced by self­ resonances of the flexible foundation, but also are more significantly influenced by wave effects in the isolators, which may be troublesome and lead to poor performance. It has been observed that the higher the frequency, the greater the de­ viation of the transmissibility or the response ra­ tio curves from that predicted by the classical theory (dashed line). This demonstrates that the classical theory, which was based on the assump­ tion of massless resilient element for isolators, is not valid in the higher frequency range. On the other hand, the results obtained from different theories are almost the same in the low frequency range due to isolators continuously be­ having as lumped elements. That is to say, the Ol---~ 40 ::0 '" ~ .~ -40 ~ '" '" -80 § ",. '" : >'": -120 -16 Current Model -.-.- Snowdon's Model ---- Classical Model v~I~O~-I~~~I~O'~-----I·O~I-------IO~'------~lif Frequency ratio O( m/mo) Frequency ratio O( OJ/OJol FIGURE 5 Comparison of R in three models: "II = 25, 'YJ = 0.05, a = 0.01, M1Mb = 10, A = 10. -40 t> ;.:: ;§ I _r-l- 80!r -.----- 25 Ei ~ t: -12D r ------- 50 - - 2 5 0 -1601'O=--!=-:';-'-~~'___:_"'-'--~~.....,...~~~_,c_'_~~.... 10' 10' 10' 10' Frequency ratio O( OJ/OJol simple classic theory is still applicable with re­ markable accuracy for the prediction of isolation in the low frequency ranges. Figure 3 shows a comparison of the R, curve with the T, curve. We found that both levels will increase due to wave resonances. However, the resonances in the R, curve are much more obvi­ ous than those in the T, curve, meaning that the response ratio is more sensitive to the changes of system parameters than that in transmissibility. From this point of view, response ratios are often suggested to evaluate the effectiveness of isola­ tion for systems with flexible foundations. Effects on Standing-Wave Frequency The frequency at which the standing-wave reso­ nance occurs depends strongly on the dynamic 40 :D :s E"o< 0 ~ -'l ~t' i ·s'" '<"1 -80. 'k" Eo< ;;;; -120 '" " > 0 -160 - - Current Model - - - Snowdon's Model 10-' 10° 10' characteristics of the foundation. Comparing dif­ ferent curves in Fig. 4, it is obvious that wave effects will occur when w/wo > 10. Moreover, we find that the standing wave frequencies shift to the lower frequency range as compared with those predicted by Harrison (1952) . This shows the flexibility of the foundation causing an earlier ?ccurrence of wave effects, which may directly mfluence the reliability of the isolation design. It is further revealed that the lower the stiff­ ness of the foundation, the earlier the wave ef­ fects will occur when other parameters remain unchanged. In Table 1, the frequency ratio wIVlw (the first standing-wave frequency to the exciting frequency) varies with A(A = WI/WO, WI is the fun­ damental frequency of the foundation) and is shown for mass ratios "II = 25 and 200. Effectiveness of Isolation T~e influence of a flexible foundation coupled WIth wave effects over the efficiency of isolation can be observed from Fig. 5. The figure shows that the interaction between these two factors makes the peaks of the curve pronounced and closer to each other, seriously reducing the effectiveness of the high frequency isolation. Only in t~e lower frequency range (D < 10) do the pre­ dIcted values produced by the three models coin­ cide well. On the other hand, the results obtained from the three models differ from each other by about 20-50 db at higher frequencies. This dem­ onstrates the interaction between wave effects in the mounts and the flexibility of the foundation. It is suggested that these two factors should be taken into account for accurate prediction of the efficiency of high frequency isolation. Effect of Foundation Internal Damping Although many pronounced wave resonances can be suppressed reasonably well by internal damping of the rubber mounts, the performance of mounts becomes less effective with the in­ crease of frequency. Figures 4-8 show the ef­ fects of internal damping of the foundation on the suppression of wave effects. With sufficient damping in the foundation, the wave effects may be smoothed and may cease to exhibit any obvi­ ous resonance peaks. Also, increase of the damping of the foundation will decrease the vi­ brationallevel of the foundation and suppress the resonance peaks at standing-wave frequencies. When high-speed running machines mounted on upper floors or decks supported by relatively flexible and lightly damped structures, the com­ pound system will be liable to vibration. In this c~se, it is important for the foundation to be spe­ cmlly treated with damping material to increase its damping capacity to facilitate the control of wave effects. Effect of Mass Ratio The ratio of machine mass to the mass of the isolator is also an important parameter in deter­ mining the standing-wave frequencies. Figures 6-8 show how the mass ratio "II influences trans­ missibilities when other system parameters re­ main unchanged. With the decrease of mass ratio "II, the resonance peaks shift toward the lower frequency range. Consequently, the smaller the value of "II, the lower the frequency at which the first wave resonance occurs, and the more appar­ ent the T curve deviation from the prediction of classic theory. Moreover, the occurrence of wave resonance is of less concern as YI becomes larger. Wave effects calculations from Eq. (15) are also plotted for different mass ratios of YI = 25 and 100, which are shown in Figs. 7 and 8. The overall transmissibility curves obtained from Snowdon's (1973) theory are redrawn for com­ parison. Again, the curves show how the levels to which Te are increased by the wave reso­ nances depends upon the value of mass ratio 'YI. As shown in the figures, the occurrence of wave effects become less and scattered as 'YI becomes larger. Therefore, it is proposed that isolators be used as near to their maximum rated load as pos­ sible to make 'YI a relatively large quantity. CONCLUSIONS 1. Wave resonance peaks and the levels of the transmissibility and the response ratio are much higher in a flexible isolation system as compared to those of traditional system models. For accurate prediction of the ef­ fectiveness of vibration isolation at higher frequencies, the interaction between wave effects in isolators and the flexibility of foundations should be considered simulta­ neously. 2. Wave effects are influenced by the charac­ teristics of the isolator, the frequency of the vibration source, mass ratio, and, more importantly, the impedance of the founda­ tion. For high-speed running machines on flexible foundations, the standing-wave fre­ quencies are lower than those predicted by rigid foundation theory. The earlier occur­ rence of wave resonances will directly af­ fect the reliability of isolation design. 3. The increase of the foundation damping can not only decrease vibration responses of elastic foundation, but also can suppress the resonance peaks at standing-wave fre­ quencies. 4. With a decrease of mass ratio, the wave resonance peaks become more pronounced and shift toward the lower frequency range. It is proposed that isolators be used as near to their maximum rated loads as possible to prevent earlier occurrence of wave effects and decrease the magnitude of the transmissibility and response ratio. The Scientiifc World Journal Journal of Sensors Machinery International Journal of Hindawi Publishing Corporation ht p:/ www.hindawi.com Antennas and Advances in Civil Engineering Hindawi Publishing Corporation ht p:/ www.hindawi.com Journal of Robotics Hindawi Publishing Corporation ht p:/ www.hindawi.com Advances your manuscr ipts VLSI Design Hindawi Publishing Corporation ht p:/ www.hindawi.com Hindawi Publishing Corporation ht p:/ www.hindawi.com Hindawi Publishing Corporation Hindawi Publishing Corporation Navigation and Observation Hindawi Publishing Corporation ht p:/ www.hindawi.com Modelling ulation & Engineering International Journal of Distributed Control Science Engineering Electrical and Computer Aerospace Engineering Harrison , M. , 1952 , "Wave Effects in Isolation Mounts," Journal of the Acoustical Society of America , Vol. 24 , pp. 62 - 70 . Snowdon , J. c. , 1973 , "Isolation and Absorption of Machinery Vibration," Journal of Acoustics , Vol. 28 , pp. 307 - 317 . Snowdon , J. c. , "Vibration Isolation: Use and Characterisation," in CL John , NBS Handbook, 1978 , McGraw-Hill , New York, pp. 75 - 189 . Sykes , A. 0 ., 1960 , "Isolation of Vibration When Machine and Foundation Are Resilient and When Wave Effects Occur in the Mount," Noise Control , Vol. 23 , pp. 115 - 130 . Xiong Yeping , et al., "Asymmetric Vibration Isolation System with Multiple Mountings on Nonrigid Foundation," in Proceedings of the International Conference on Vibration Problems in Engineering , 1990 , International Academic Publishers, Beijing, Vol. 2 , pp. 973 - 978 .


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Ye-Ping Xiong, Kong-Jie Song, Xing Ai. Influence of Flexible Foundation on Isolator Wave Effects, Shock and Vibration, DOI: 10.3233/SAV-1996-3107