Vibration Isolation Review: II. Shock Excitation

Shock and Vibration, Jul 2018

This is the second part of a two part review of shock and vibration isolation. It covers three distinct categories of shock excitation—pulselike shock, velocity shock, and complex shock—and discusses the means that are available in each case to measure the effectiveness of shock mitigation by the imposition of flexible connections between the isolated system and its base.

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Vibration Isolation Review: II. Shock Excitation

International Journal of II. Shock Excitation F. C. Nelson 0 0 College of Engineering Tufts University Medford , MA 02155 , USA This is the second part of a two part review ofshock and vibration isolation. It covers three distinct categories of shock excitation-pulselike shock, velocity shock, and complex shock-and discusses the means that are available in each case to measure the effectiveness ofshock mitigation by the imposition offlexible connections between the isolated system and its base © 1996 John Wiley & Sons, Inc. - Vibration Isolation Review: INTRODUCTION In a previous article vibration isolation against sinusoidal and random excitation was reviewed (Nelson, 1994) . This article continues that review by extending the consideration of vibration isola­ tion to systems excited by mechanical shock. A mechanical shock generally appears as an acceleration applied to the base of a machine, structure, or piece of equipment. In addition, the acceleration is applied suddenly, reaches a high level, and often persists for only a short time. As such, the system response is far from the steady­ state or stationary conditions considered in part I and, in particular, the concept of transmissibility introduced there must be redefined. This redefi­ nition is less codified than for the steady-state/ stationary case and a variety of measures of shock mitigation are in use. ISOLATION AGAINST PULSELIKE SHOCK The canonical pulselike shock is the rectangular pulse. As shown in Fig. 1, a rectangular pulse of acceleration applied to the base of a lumped parameter oscillator is dynamically equivalent to a rectangular pulse of force applied to the same oscillator with a fixed base. Subsequent discussion will be in terms of this fixed-base, force-shock model. The analytical so­ lution for the problem of Fig. l(b) can be found in many places. There is also a simple phase plane solution (Jacobsen and Ayre, 1958) that can be extended to pulses of arbitrary shape. For most purposes, it is best to represent these solutions in the frequency domain, in particular by means of the shock spectrum (SS). An SS is the locus of the global response maximums (i.e., the maxi­ mum of all the local response maximums) of a single degree of freedom (SOOF) oscillator plot­ ted in a nondimensional frequency space. As shown in Fig. 2, one can separately plot an SS for the period of time during which the shock force is acting, the initial shock spectrum (ISS), and an SS for the period of time after the force has been removed, the residual shock spectrum (RSS). The maximax SS is understood to be the upper bound of the combined ISS and RSS. The SS of a wave form should not be confused with its Fourier spectrum: the Fourier spectrum is an input quantity and the SS is an output quantity. However, if the quantity of interest is accelera­ tion, then it can be shown that the undamped equipment m x(t) • 1~ z(t) I I t --'------ : yet) yt 1--1'----.I rI - - - -t ao-L-..J (a) m (b) F(t) T= 2n 0) the frequency, w, and Zo is the maximum static deflection that would result if the force pulse were applied slowly. The variation of Ts with TIT for a rectangular pulse is shown in Fig. 3 where T is the pulse duration in time and T is the natural period of the SDOF oscillator. Ts for other pulse shapes are shown in Fig. 4. A more detailed study of Fig. 2 shows that Ts s 1 (shock isolation) if TIT s 1/6 or where D(w) is the maximax shock spectrum in terms of relative displacement as a function of w s -7T. 3T (3) 1'_1 "T-"6 RESIDUAL Shock Spectrum , l- I I-l 2.5 I I n ..J'tl0.5 1.0 1.5 2.0 3.0 3.5 Ratio Shock Pulse Time Duration ,-.1. Natural Period of System T The slightly conservative form w :s: lIT is easier to remember. Now consider an arbitrary (arb.) force pulse, F(t), that is circumscribed by the rectangular (rect.) force pulse Fo, T (see Fig. 5). Let Ts (arb.) be the shock transmissibility associated with F(t) and Ts (rect.) be the shock transmissibility for the circumscribing rectangular pulse. For an oscilla­ tor initially at rest, it can be shown (Frolov and Furman, 1990) that if Ts (rect.):s: 1, then Ts (arb.) :s: 1. In other words, if the circumscribing rectan­ gular pulse leads to isolation, the pulse F(t) will also be isolated. This result becomes obvious if T is much less than T. For this limiting case, the oscillator response is governed solely by the im­ pulse associated with each force pulse, i.e., by the area under their respective force vs. time functional forms; clearly the area under F(t) is less than Fo, T, On this basis, the rectangular pulse is widely used for the preliminary design of shock isolation systems even though it is unlikely to be encountered in practice. It can also be shown that w2D(w) = A(w), where A(w) is the SS in terms of absolute acceleration and where the equality is close for small damping and exact for zero damping. The term Yew) = wD(w) is equal to neither the absolute nor the relative velocity SS and hence is usually referred to as the pseudovelocity SS. These three SS, D(w), V(w) , andA(w), can be compactly displayed on a four-way logarithmic chart just like their sinusoidal counterparts (see Schiff, 1990) . ISOLATION AGAINST VElOCITY SHOCK The classical problem of velocity shock is a pack­ age falling freely through a distance h and impact­ ing a rigid surface (see Fig. 6). The classic article on this problem is that by Mindlin (1945) . Follow­ ing Mindlin, the expression for conservation of energy of a cushioned package in contact with a rigid surface is 21 mi2 + IX0 peg) dg = mg(h + x), where P(g) is the force in the cushion. The maxi­ mum cushion displacement, X m , will occur when i = o. If, in addition to i = 0, h ~ X m ' Eq. (4) simplifies to which gives Xm if P(g) is known. The maximum acceleration, Xm , will also occur at i = 0, hence where Pm is P(g) evaluated at g = Xm. If Gm is defined as the maximum acceleration in gs, then = Pm mg (4) (5) (6) (7) 2.0 o o II/ II I 1'4(,1 0.0} "'"to; 11'-. 1,-0.5 1.0 I 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Ratl·O Shock Pulse Time Duration ,-'t Natural Period of System T 2.0 E-<'" 1.5 ~ ;§ en en ·sen 1.0 !:: ~ ~ ..u><: 0.5 0 ..c:: (J') o If) 111 o f-( <0.01 mi r-... (1 P ! I I ~ - , li.l ~ ~ I I ~ ~ 1.0 I I DiS I -0.5 1.0 1.5 2.5 3.5 Ratl·O Shock Pulse Time Duration 't , Natural Period of System T The procedure for determining Gm is then 1. find Xm from Eq. (5), 2. find Pm from Pm = P(xm), and 3. find Gm from Eq. (7). For the case of a linear cushion, i.e., P = kx, this procedure yields Gm = J2hWg; , (8) where Wn = [klm]1I2 is the natural circular fre­ quency of the cushion-mass system. Rather than linear cushions, most packaged equipment is sup­ ported by a cushion that is made of a strongly nonlinear material such as plastic foam or latex(a) _I:::--. ! 2.0 (b) I I f \ ..J'tl I 1-1I I1 /1 ..J'tl-' J J 3.0 F Rectangular shock pulse / Arbitrary shock pulse t (9) bound fibers. While the above procedure is the same, its implementation is different. The nonlin­ ear equivalent of Eq. (8) was formulated by Jans­ sen (1952) as h Gm = 1-, t where h is the drop height, t is the thickness of the cushion, and 1 is the so-called cushion factor, the ratio of the peak stress developed in the cush­ ion to the energy stored per unit volume of the cushion. Mustin (1968) provided a proper derivation of Eq. (9) as well as a comprehensive review of velocity shock for items mounted on nonlinear cushions. As cushion strain increases, Gm decreases be­ cause of the increased ability of the cushion to absorb energy; however, at increased levels of strain, many nonlinear materials, such as plastic foam, begin to stiffen and this increases Gm . Hence, for such material J will have a minimum, denoted by 10 , This inference is supported by test Vihratioll holatioll R('['int': /I .J55; results (see Fig. 7, which is taken from Hcnny and Leslie, 1962) . Mustin shows how to determine .10 from a knowledge of the stress-strain behavior of the cushion material. If geometric and material constraints allow. the package engineer should design the cushion system to operate in the vicinity of 10 and to provide a Gm that does not exceed the fragility of the packaged item. In this sense, the figure of merit for velocity shock mitigation becomes Gm rather than transmissibility. The fragility, or damage sensitivity, of a piece of equipment or structure is the locus of motion parameters that first induce failure, malfunction, or, in the case of consumer products, loss of ac­ ceptable appearance. This locus is plotted in a space of Gm vs. LlV where LlV is the velocity change of the package falling from rest to impact. Figure 8 is a schematic fragility curve. A discussion of such curves is given by Kornhauser (1964) , and a testing protocol for measuring shock fragility is given in ASTM D3332 (1993). (Note that Korn­ hauser plots LlV vs. Gm .) An acceptable value of Gm is one that falls safely into the no-damage region of the equipment's fragility curve. Lacking the certainty of an experimentally de­ termined fragility curve, it is reasonable to substi­ tute the values contained in the qualification test specification for the packaged mechanical compo­ nent or electrical equipment. ISOLATION AGAINST COMPLEX SHOCK Pulselike shocks and velocity shocks are not os­ cillatory. However, the shock inputs associated with several types of shock events are highly osm // fT. rigid surface 7/////// (a) ~ t ~ h ~ rigid outer container cushioned item cushion (b) (c) Cushion Thickness =2 inches 1 2 3 4 5 6 Bearing Stress (psi) 7 8 9 10 cillatory. Such events are referred to as complex shocks. A specialized procedure was developed to assess the response of structures and equip­ ment subjected to complex shocks. This special­ ized procedure, called the response spectrum method (RSM), was first suggested by Biot in 1933 . In the 1950s the RSM was adapted by the Naval Research Laboratory to predict equipment "hardness" against the shocks (i.e., explosions) encountered by surface vessels and submarines during combat (see Belsheim and O'Hara, 1960) . A review of the subsequent evolution of the RSM in the Navy was provided by Remmers (1982) . In the 1970s it was widely used in the nuclear power industry to ensure protection of power stations from earthquakes (see Gupta, 1990) . In the 1980s the method was extended to cover the situation of large structures, e.g., buildings, that were iso­ lated from earthquake motions by rubber blocks (see Skinner et aI., 1993; Kelly, 1993) . The use of elastomeric shock isolators for entire buildings seems to have started in England (Waller, 1969). Now there are seismically isolated buildings in more than 17 countries and over 100 such struc­ tures have been built or are under construction; for a complete review see Buckle and Mayes (1990) . For a discussion of resiliently mounted Critical Velocity (Vc) rI.> -bl) Test No.7 do First Failure .~ ~ "0 < o _ _ _-Test No. 1- 6No Failure _ _ ~ I"i .~ ~ - - - - - - - - - - - - - - - - - - , I I I I I I Damage Region I test No.8 - 13=_ No Failure = Test No. 14 NO DAMAGE ~~~:r~ REGION Velocity Change in/sec (m/sec) ROOF 4 3 ,II I , , , I I ';""-k-.....,.......,.--Y mechanical equipment subjected to seismic shock, see Lama (1994) . The first base-isolated building in the United States is the Foothill Com­ munity Law and Justice Center in Rancho Cuca­ monga, California, built in 1985 (see Fig. 9). To illustrate the RSM for a base-isolated struc­ ture, consider the problem of Fig. 10, which is adapted from Kelly (1993) . The equations ofmo­ tion for the undamped, base-isolated structure in One Story Building where M = ms + mb' the total mass; y is the seismic excitation time history ofthe ground; and z the relative displacement with respect to the ground. Equation (10) can be rewritten in the form [MHz} + [KHz} = - [MHr}y. (11) Denote the eigenvectors (normal modes) associ­ ated with Eq. (11) by {CPI}' {CP2} and the eigenvalues (natural frequencies) by WI , W2. Using the coordi­ nate transformation are the modal coordinates, Eq. (11) is converted into two, uncoupled SDOF equations, i.e., where where are the modal participation factors. Further study of Eq. (14) reveals that i = 1 corresponds closely to a mode wherein the building acts as a rigid body on flexible isolators while the deformation of the building structure is confined principally to the i = 2 mode. In consequence, the usual design situation would be W2 ~ WI. Using Eq. (15) one can show that if W2 ~ WI' then L2 ~ L I • This result is the essence of the seismic isolation scheme shown in Fig. 10: low shear flexibility in the seismic isolators insures low participation by the modes that contain significant structural de­ formation. Guided by previous discussions, it can be shown that maximax{zJ = {cpJL;D(w;), (16) where D(w;) is the relative displacement shock spectrum associated with y evaluated at Wi. If Rij is the maximax response of the jth DOF when the structure is vibrating in its ith mode, Eq. (16) can be reduced to (13) (15) where CPij and Li come from the lumped parameter model of the structure and D(wi) is deduced from a study of measured complex excitations, be they earthquakes or explosions. There remains one uncertainty with the use of Eq. (17). The maximax operator used in Eq. (16) removes all phase information among the modes. So it is uncertain how best to sum Rij with respect to the modal index i. The obvious choice, a sum of the absolute values of Rij' is usually much too conservative for cost-effective design. The most common choice is the square root of the sum of squares, i.e., (17) (18) This choice effectively assumes that the various modes are uncorrelated, and it therefore works best when the modes are widely spaced. A discus­ sion of how to proceed when the modes are closely spaced can be found in Chopra (1995) . There seems no general agreement about a fig­ ure of merit for seismic excitation. One can bor­ row a term from acoustics and define an inser­ tion loss ILj = 20 log iR'.h ' j,S (19) where ILj is the insertion loss for the jth DOF, Rj,s is the combined modal response at the jth DOF for a soft mounted (i.e., isolated) structure, and Rj,h is the combined modal response at the jth DOF for a hard-mounted structure (i.e., a structure with the isolators replaced by rigid con­ nections). A 20-dB insertion loss would then cor­ respond to an order of magnitude reduction in response due to the presence of isolators. The design of large (1-m diameter) seismic shock isolators to carry large axial loads (e.g., 1000 tons) while permitting a low natural fre­ quency in shear (e.g., 1 Hz) is a specialized but well-developed technology. A design of growing acceptance is the use of a highly filled natural rubber prism with embedded horizontal steel plates for enhanced vertical stiffness. For exam­ ple, the design in Fig. 9 uses 98 such isolators. Design details and test results can be found in Skinner et al. (1993) and Kelly (1993) . CONCLUSION Three of the major fields of mechanical shock were surveyed: pulselike shocks, velocity shocks, and complex shocks. Shock mitigation figures of merit appropriate to each field were defined and discussed. There was no discussion of shock testing; however, a sampling ofthis field can be found in Kao (1975) , Harris (1988) , and Hudson (1991) . 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F.C. Nelson. Vibration Isolation Review: II. Shock Excitation, Shock and Vibration, DOI: 10.3233/SAV-1996-3605