Vibration Isolation: A Review, I. Sinusoidal and Random Excitations

Shock and Vibration, Jul 2018

This article reviews the engineering theory and technology associated with the use of dicrete Isolators to protect Vibration-sensitive equipment or machinery against sinusoidal or random excitation. Special attention is given to protection against low frequency (<5 Hz) and high-frequency (>100 Hz) excitation. Both passive and active isolators systems are discussed.

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Vibration Isolation: A Review, I. Sinusoidal and Random Excitations

Received September Vibration Isolation: A Review, I. Sinusoidal and Random Excitations 0 F. C. Nelson Col/ege of Engineering Tufts University Medford , MA 02155 , USA VIBRATION CONTROL Vibration control is the reduction of unwanted vibration in a machine or in a structural system. Like noise control, vibration control is usually categorized into methods that affect the source, the path, or the receiver, see Fig. 1. An example of source control is machinery balancing (Harris, 1987, Chap. 39) . An example of receiver control is detuning, that is, assuring that the receiver nat­ ural frequencies do not coincide with any forcing frequencies. An example of path control is the use of vibration isolators, which are represented in Fig. 1 by discrete connections between the receiver and the path and between the source and the path. As such, vibration isolation is a subset of vibration control, being only one of an array of methods available to the vibration control engi­ neer. Although most vibration engineers spend most of their time on vibration control, it is a topic that receives little coverage in textbooks or mono­ graphs. The only textbook that gave extensive coverage to vibration control has long been out of print (MacDuff and Curreri, 1958) . Most cur­ rent coverage is found in handbooks, for example Beranek and Ver (1992 , Chap. 28), although a new monograph is under preparation by Mead (1993) . ISOLATION AGAINST SINUSOIDAL VIBRATION Vibration isolation is the technique of controlling vibration by interposing compact, resilient con­ nections between the vibration source and its surrounding structure, source isolation, or be­ tween the surrounding structure and the vibra­ tion receiver, receiver isolation. In this context, compact means small compared to the vibration wavelength. These compact resilient connec­ tions will be called isolators, although other terms, such as antivibration mounts, are com­ mon. Isolators gain their resilience from their shape or their material, or both. Typical isolators employ metallic springs, polymer blocks, or trapped volumes of air (Harris, 1987, Chap. 29) . The simplest analytical model for an isolator is the single degree-of-freedom (SDOF) transla­ tional spring-mass system (Fig. 2). The force transmissibility for the source isola­ tion problem is defined as (1) Internal Components Casing r-Jgi§£~~£~:f:=:::;.Connection Cam-driven mass Casing (frame) Connection j J J (2) Source Path Receiver and the motion transmissibility for the receiver isolation problem is defined as All textbooks point out that TF = TM but few show that this result is not fortuitous but is a necessary consequence of the reciprocity inher­ ent in all linear elastic systems (Ungar, 1991) . Because of this identity, the symbol TM will be used for force transmissibility as well as motion transmissibility. Figure 3 shows how TM varies with frequency, more precisely, with frequency ratio, o'lw, and with damping (both viscous damping, {, and hys­ teretic damping, YJ). Versions of Fig. 3 have appeared in undergraduate vibration textbooks from Den Hartog (1934 ) to Thomson (1993) . It is unfortunate that even contemporary vibration textbooks tend to stop with Fig. 3, more or less where Den Hartog stopped 60 years ago. In fact, it has been argued elsewhere (Nelson, 1991) that the SDOF model is not a good design model and that a two degree-of-freedom (2DOF) model is preferable (Fig. 4). Nevertheless, the SDOF model has an extensive literature (e.g. Snowdon, 1979, contains over 200 references) and many useful trends can be learned from its study. For example • in the vicinity of resonance (o'lw = 1), the larger the damping, the smaller the TM ; Machine (Source) Isolator Surrounding Structure (Path) Equipment (Receiver) Excitation Force Transmitted Force x Response motion Excitation motion (b) 0.001 f-------1-----+--~~___l • isolation (TM < 1) begins at f!/w = v2 for all values of damping; • in the region of isolation (f!/w > v2) the larger the damping, the larger the TM (however, for hysteretic damping this effect is so small as to be a negligible consideration in design). In addition, for the frequency region f!/w ~ 1: • if the isolator is undamped, TM falls off at 40 dB/decade (12 dB/octave); • if the isolator has hysteretic damping, TM falls off at 40 dB/decade; • if the isolator has appreciable viscous damp­ ing, TM falls off at 20 dB/decade (6 dB/oc­ tave). The 2DOF system shown in Fig, 4 can also be used as a model of a two-stage isolation system (Fig, 5), Two-stage isolation systems are widely used to isolate equipment against high frequency vi­ bration because TM falls off at 80 dB/decade at high frequencies (Fig, 6). Snowdon (1968 , Chap. 3) contains an analysis of a two-stage isolation system that employs elastomeric isolators. Isolator staging can be ex­ tended beyond two; in at least one case, a nine­ stage isolation system has been designed and used, Machine mouonting { Fioo' { (a) { (b) on mounting F - - - - /1\V ,------'-------'---, - - - - /1\V x y y=~y cos 0 t The inertia block is a special case of Fig. 5 (Fig. 7). Protecting equipment against low fre­ quency vibrations «5 Hz) with metal springs or polymer blocks can lead to very high aspect ratio isolators and concomitant problems with high de­ flection and low lateral stability. By using a mas­ sive inertia block (m2 ~ ml), the isolator aspect ratio can be reduced and, in addition, the amount of static and dynamic displacement can be de­ creased. The thing to remember is that inertia blocks permit the design of more practical and equipment stage 2 isolator raft structure stage 1 isolator vibrating base structure more stable isolator systems; they seldom lead to appreciably lower values of TM. Low frequency isolation without inertia blocks can be obtained by using air springs (Harris, 1987, Chap. 33) . Special mechanical sys­ tems, usually variations of the Stewart Platform or the Mallock Suspension, can also provide low frequency isolation without using an inertia block. Lately, a mechanical isolator that contains a negative stiffness element has shown the ability to isolate vibrations down to 1 Hz (Platus, 1993) . Only a few real systems are dominated by mo­ tion in one dimension and thus allow the use of SDOF models. Most real systems are better modeled as a piece of equipment vibrating about its center of mass in two or more of the six possi­ ble rigid mode DOF (Fig. 8). In general, all six modes are coupled. Harris (1987, Chap. 3) gives the six coupled equations of motion. Symmetry and judicious choice of isola­ tor stiffness can uncouple some, and in special cases, all of the DOF (Macinante, 1984, Chap. 7) . Such uncoupling is desirable because then one can make the uncoupled natural frequencies equal and thus simplify the problem of achieving good isolation of these DOF. These particular results, both for high and low frequency, contain a generic assumption; namely, that the excitation force or the excita­ tion motion is ideal, that is, the dynamics of the K FIGURE 7 Diagram of a piece of equipment (ml) hard mounted to an inertia block (m2). ,-.... Q) "D '--' I>00-' :D en en E en c: 0 'I20 10 0 -10 -20 -30 -40 -50 -60 -70 -80 -90 Compound system 1\ / '\ Simple system '\'\'\ '{ '\ '\ '\ '\ \ --F;- x FIGURE 8 Model of a rigid piece of equipment (m) mounted on four isolators of equal stiffness (k). There are six degrees-of-freedom about the center of mass of the equipment: three in translation (x, y, z) and three in rotation (8, $, cp). excitation are not affected by the dynamics of the response. This assumption becomes more and more untenable as the excitation frequency in­ creases further and further into the audio range. Simulation of high frequency (> 100 Hz) isolator systems is better served by replacing the ideal source by a linear source (Ungar and Dietrich, 1966) . Figure 9 shows a piece of equipment, a receiver (R), being driven by a linear source (S) through a massless isolator (I). It is also conven­ tional (White and Walker, 1982, Chap. 26) to re­ place the motion transmissibility, TM, by the in­ sertion loss, D. D = 20 log ~R,HM VR,SM where OR,HM = the velocity amplitude of the re­ ceiver (R) when it is hard mounted (HM) to the source; and OR,SM = the velocity amplitude of the receiver (R) when it is soft mounted (SM) to the source. For the case of Fig. 9, it can be shown that (3) (4) where Mr = mobility of the isolator (for a mass­ less spring of stiffness k, Mr = ifllk); Ms = mo­ bility of the linear source (ratio of the free veloc­ ity of the source to the blocked force of the source); and MR = mobility of the receiver. Even though the simulation of Fig. 9 is still simplified, Eq. (4) gives much better guidance for achieving isolation at high frequencies. In partic­ ular 1. if one lets the receiver (the vibration sensi­ tive equipment) have a natural frequency of w, then when fl = W, MR --,)-00 and D --,)- 0, that is, there will be no isolation at un­ damped equipment resonances. 2. A high level of isolation (D ;? 1) will be achieved if k is chosen small enough so that (5) In other words, the role of an isolator is to introduce a mobility mismatch between the source and the receiver. The larger the mis­ match, the higher the isolation. 3. If Ms is not too large, one can also increase the mismatch by making MR smaller. This can be done in three ways: (a) if the equipment is stiffness controlled (i.e., fl/w <;g 1), make the equipment stiffer; (b) if the equipment is mass controlled (i.e., fl/w ;? 1), make the equipment more massive; (c) if the equipment is damping controlled (i.e., fl/w = 1), augment the equipment damping. Linear Source !solator F B:ocked Force SCLJrce ' .., , / Characteristic Receiver (t:!sqouliaptmedent) , , , Free Velocity FIGURE 9 Schematic of a linear source (S) driving a piece of equipment (R) through a massless isolator (I). Blocking Moss Linear Source Isolator This high-frequency model can be expanded to include an isolator with mass and damping (Snowdon, 1979, Section 6.2) . A comprehensive method for designing isolators based on this ap­ proach is given in SAE (1962) . At even higher frequencies, isolators begin to behave as wave guides and it is helpful to think of the vibration as structure-borne noise and to treat the problem with the methods described in Cremer and Heckl (1988) . In particular, at these frequencies a concentrated mass that is inserted into an isolator can effectively block the trans­ mission of wave energy, especially if the waves are flexural. Figure 10 is a schematic of such a blocking mass. A practical example of an isolator with a self-contained blocking mass is described by Young and Hanners (1973) . A second generic assumption of the above technology is that the isolators or isolator sys­ tems are passive, that is, they receive no external energy or information. When they do receive such inputs, the system is active. Active vibra­ tion has received much impetus recently from progress in control algorithms (e.g., neural, fuzzy, or adaptive) and in sensors and actuators (e.g., piezoelectric polymer sheets). A schematic of an active isolation system ap­ plied to a SDOF system is shown in Fig. 11 and a typical enhancement of isolation performance is shown in Fig. 12. Active isolators were first used to protect in­ struments against low frequency vibration, for example, Tsutsumi (1964) describes an early ex­ ample of the active isolation of an instrument test platform against slow tilting caused by earth mo­ tion and building distortion. With the develop­ ment of actuators with high frequency response, it has become possible to extend the performance of active isolators into the low audio range. Also, in using active isolators, it must be remembered that the isolation system may have to transmit static loads. Hence active isolators with low stiff­ ness at low frequency are usually used in parallel with passive isolators, see Watters et aI., (1988 ) where active isolators are used in parallel with passive elastomeric isolators to reduce the TF of a diesel engine by an additional 20 dB in the fre­ quency band 20-100 Hz. It is more difficult to achieve broad-band ac­ tive isolation than narrow-band and, as a result, active isolation is most likely to be used in cases where the excitation is periodic. This is espe­ cially true for those cases where high perfor­ mance demands justify the added initial cost and maintenance. A review of active control of ma­ chinery isolators can be found in von Flotow (1988 ). As was mentioned previously, the textbook and monograph literature in vibration control is thin; this is also true in vibration isolation. Crede M isolation (spring) Mass (vibration sensitive equipment) Sensor (accelerometer) Force actuator 1------(5(l~- Command Signal (value 0) (1965) is an acknowledged classic but has been out of print for sometime. Fortunately, Mac­ inante (1984) is in print and is a worthy succes­ sor. More recently Frolov and Furman (1990) has become available, giving insight into Russian techniques and technology in this field. For vi­ bration isolation of mechanical equipment in buildings, Jones (1984) is extremely helpful be­ cause it provides pictures and schematics of a wide selection of practical isolation systems. In addition, it is unique in providing a long litany (backed up with photographs) of misinstalled, misaligned, and misselected isolators. Much can be learned from studying the mistakes and over­ sights of others. Among these few books and monographs on vibration isolation, there are even fewer that dis­ cuss the design and use of nonlinear isolators. Frolov and Furman (1990) present a highly clas­ sical discussion of this topic while Mustin (1968) addresses one of its more practical aspects, the use of nonlinear cushioning material in the pack­ aging of vibration-sensitive equipment. ISOLATION AGAINST RANDOM VIBRATION The previous discussion has been limited to vi­ bration isolation against sinusoidal excitation, or, at most, against highly tonal excitation. A design acceptance test would usually be a sine wave whose frequency was swept slowly (2 octaves/ min) over the frequency range of interest. Prior to the 1950s when power plants, particularly air­ craft power plants, were primarily of the reciprocating type, the above knowledge and associated technology were adequate to solve the problems at hand. With the advent of jet engines and rocket mo­ tors, vibration problems arose, especially in the aerospace industry where the sources of tonal excitation were overwhelmed by sources of ran­ dom excitation. Examples of random sources are boundary layer turbulence and the exhaust from jet and rocket engines. These sources are charac­ terized by features not found in the sinusoidal or highly tonal world: smooth frequency spectra over wide frequency bands and large variations in amplitude over short periods of time. The un­ derlying theory of random vibration of mechani­ cal systems has been well codified (Argyris and Mlejnek, 1991, Chap. 8) . Vibration testing using random inputs was ap­ plied to missile electronics in the early 1960s and extended to aircraft electronics in the 1980s. At present, random vibration testing is recognized as a good way of evaluating electronics reliabil­ ity, even if the operational environment is non­ random or even nonvibrating. For a discussion of such vibration screening see Tustin and Mercado (1984 , Section 30). Note that vibration screening is usually done with the equipment or system hard mounted, that is, with isolators removed. A fundamental result of random vibration the­ ory is where SR is the power spectral density (PSD) of the response, SE is the PSD of the excitation, and H is the (complex) frequency response charac­ teristic between response and excitation. If SR is the velocity PSD of the response and if SE is the force PSD of the excitation, then H is the random equivalent to the mobility of the system. Simi­ larly, if SR is the acceleration PSD of the re­ sponse and SE is the acceleration PSD of the ex­ citation, IHI is the random equivalent of the sinusoidal motion transmissibility denoted by the symbol, TM. Furthermore, as shown in Fig. 13, the re­ sponse of a spring-mass system to a narrow-band random excitation is very closely approximated by an amplitude-modulated sine wave. This sug­ gests that TM = TM for narrow-band response and thus for this case, Eq. (6) can be rewritten as (6) (7) If the excitation is wide band and the system is linear, the wide-band response can be obtained by using Eq. (7) in a series of contiguous narrow bands (e.g., 1 Hz) that span the complete excita­ tion bandwidth. The result of such a calculation is shown in Fig. 14. CONCLUSIONS It is hoped that this survey gives the practicing engineer an understanding of some of what is needed to protect vibration-sensitive equipment or machinery against sinusoidal or random exci­ tation. However, it is worth asking several questions before using the theory and practice described above. 1. Are vibration isolators really necessary? Not all equipment needs to be isolated to survive its vibration environment. To de­ termine if isolators are necessary, one must know, or determine, the fragility (sensitiv­ ity) of the equipment (see Kornhauser, 1964, Chap. 5) and then compare that fra­ gility to the excitation of the equipment. 2. What are the dominant power paths? There is little point in inserting isolators into load paths that convey only a minor portion of the vibrational power flow. Some thought, analysis, or testing can usually identify the major load paths and save expensive rede­ sign. 3. Is the design governed by impulsive excita­ tion rather than sinusoidal or random exci­ tation? Vibration isolation against impul­ sive excitation, or shock, is an extensive subject and is not as well codified as are the subjects of this review. Part II will discuss shock isolation. The author would like to acknowledge helpful discus­ sion with D.S. Nokes. The treatment of isolation from random vibrations especially benefitted from his in­ sight and experience. Journal of Engineering The Scientiifc Hindawi Publishing Corporation ht p:/ Journal of Sensors Machinery Volume 2014 International Journal of Hindawi Publishing Corporation ht p:/ and Passive Advances in Civil Engineering Hindawi Publishing Corporation ht p:/ Journal of Robotics Hindawi Publishing Corporation ht p:/ Advances ctronics Submit your manuscr ipts VLSI Design Hindawi Publishing Corporation ht p:/ Hindawi Publishing Corporation Hindawi Publishing Corporation Navigation and Observation Hindawi Publishing Corporation ht p:/ Modelling ulation & Engineering International Journal of Distributed Control Science Engineering Electrical and Computer Argyris , J. , and Mlejnek , H.-P., 1991 , Dynamics of Structures, North-Holland, Amsterdam. Beranek , L. L. , and Ver , I. L. , Editors, 1992 , Noise & Vibration Control Handbook, John Wiley, New York. Crandall , S. H. , and Mark , W. D., 1963 , Random Vibration in Mechanical Systems, Academic Press, New York. Crede , C. 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J. , 1993 , Personal Communication . Mustin , G. S. , Theory and Practice of Cushion Design , 1968 , Shock & Vibration Information Center Monograph No. 2 . Nelson , F. C. , 1991 , "Shock and Vibration Isolation: Breaking the Academic Paradigm," ASME PVP222 (Seismic, Shock, and Vibration Isolation), pp. 1 - 6 . Platus , D. L. , February 26 , 1993 , "Smoothing Out Bad Vibes," Machine Design , pp. 123 - 130 . SAE Committee G- 5 , 1962 , Design of Vibration Isolation Systems , Society of Automotive Engineers. Snowdon , J. C. , 1979 , Vibration Isolation , NBS Handbook 128, U.S. Department of Commerce. Snowdon , J. C. , 1968 , Vibration and Shock in Damped Mechanical Systems , John Wiley, New York. Thomson , W. T., 1993 , Theory of Vibration with Applications , 4th ed., Prentice-Hall , New York. Tsutsumi , K. , 1964 , "A Ground Tilt Isolation Platform," MIT Instrumentation Lab Report E- 1508 . 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J. , 1973 , "CompoundTwo-Stage Resilient Isolation Mounting for Use in Attenuating Mechanical Vibration," U.S. Patent 3 , 764 , 100 .

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F.C. Nelson. Vibration Isolation: A Review, I. Sinusoidal and Random Excitations, Shock and Vibration, DOI: 10.3233/SAV-1994-1508