Vibration and Shock Handbook 12

Vibration and Shock Handbook 12 Every so often, a reference book appears that stands apart from all others, destined to become the definitive work in its field. The Vibration and Shock Handbook is just such a reference. From its ambitious scope to its impressive list of contributors, this handbook delivers all of the techniques, tools, instrumentation, and data needed to model, analyze, monitor, modify, and control vibration, shock, noise, and acoustics. Providing convenient, thorough, up-to-date, and authoritative coverage, the editor summarizes important and complex concepts and results into “snapshot” windows to make quick access to this critical information even easier. The Handbook’s nine sections encompass: fundamentals and analytical techniques; computer techniques, tools, and signal analysis; shock and vibration methodologies; instrumentation and testing; vibration suppression, damping, and control; monitoring and diagnosis; seismic vibration and related regulatory issues; system design, application, and control implementation; and acoustics and noise suppression. The book also features an extensive glossary and convenient cross-referencing, plus references at the end of each chapter. Brimming with illustrations, equations, examples, and case studies, the Vibration and Shock Handbook is the most extensive, practical, and comprehensive reference in the field. It is a must-have for anyone, beginner or expert, who is serious about investigating and controlling vibration and acoustics.

Shock and Vibration III III-1 © 2005 by Taylor & Francis Group, LLC 12 Mechanical Shock 12.1 Definitions 12-2 Shock † Simple (or Perfect) Shock † Half-Sine Shock Versed-Sine (or Haversine) Shock † Terminal Peak Sawtooth Shock or Final Peak Sawtooth Shock † Rectangular Shock † 12.2 Description in the Time Domain 12-3 12.3 Shock Response Spectrum 12-4 Need † Shock Response Spectrum Definition † Response of a Linear One-Degree-of-Freedom System † Definitions † Standardized Shock Response Spectrum † Choice of Damping † Shock Response Spectra Domains † Algorithms for Calculation of the Shock Response Spectra † Choice of the Digitization Frequency of the Signal † Use of Shock Response Spectra for the Study of Systems with Several Degrees of Freedom 12.4 Pyroshocks 12-17 12.5 Use of Shock Response Spectra 12-18 Severity Comparison of Several Shocks † Test Specification Development from Real Environment Data 12.6 Standards 12-24 Types of Standards † Installation Conditions of Test Item † Uncertainty Factor † Bump Test 12.7 Damage Boundary Curve 12-26 Definition † Analysis of Test Result 12.8 Shock Machines 12-28 Main Types † Impact Shock Machines † Shock Simulators (Programmers) † Limitations † Pneumatic Machines † High Impact Shock Machines † Specific Test Facilities 12.9 Generation of Shock Using Shakers 12-44 Principle Behind the Generation of a Simple Shape Signal versus Time † Main Advantages † Pre- and Postshocks † Limitations of Electrodynamic Shakers † The Use of Electrohydraulic Shakers 12.10 Control by a Shock Response Spectrum 12-52 Principle † Principal Shapes of Elementary Signals † Comparison of WAVSIN, SHOC Waveforms, and Decaying Sinusoid † Criticism of Control by a Shock Response Spectrum 12.11 Pyrotechnic Shock Simulation 12-58 Christian Lalanne Engineering Consultant Simulation Using Pyrotechnic Facilities † Simulation Using Metal-to-Metal Impact † Simulation Using Electrodynamic Shakers † Simulation Using Conventional Shock Machines 12-1 © 2005 by Taylor & Francis Group, LLC 12-2 Vibration and Shock Handbook Summary Transported or on-board equipment is very frequently subjected to mechanical shocks in the course of its useful lifetime (in material handling, transportation, etc.) This kind of environment, although of extremely short duration (from a fraction of a millisecond to a few dozen milliseconds), is often severe and cannot be neglected What is presented in this chapter is summarized here After a brief recapitulation of the shock shapes most widely used in tests, the shock response spectrum (SRS) is presented with its numerous definitions and calculation methods The main properties of the spectrum are described, showing that important characteristics of the original signal can be drawn from it, such as its amplitude or the velocity change associated with the movement during a shock The SRS is the ideal tool for comparing the severity of several shocks and for drafting specifications Recent standards require writing test specifications from real environment measurements associated with the life profile of the material (test tailoring) The process that makes it possible to transform a set of recorded shocks into a specification of the same severity is detailed Packages must protect the equipment contained within them from various forms of disturbance related to handling and possible free fall drop and impact onto a floor A method to characterize the shock fragility of the packaged product, using the “damage boundary curve” (DBC), and to choose the characteristics of the cushioning material constituting the package is described The principle of shock machines that are currently most widely used in laboratories is described To reduce costs by restricting the number of changes in test facilities, specifications expressed in the form of a simple shock (halfsine, rectangle, sawtooth with a final peak) can occasionally be tested using an electrodynamic exciter The problems encountered, which stress the limitations of such means, are set out together with the consequences of modifications, that have to be made to the shock profile on the quality of the simulation Determining a simple shape shock of the same severity as a set of shocks on the basis of their response spectrum is often a delicate operation Thanks to progress in computerization and control facilities, this difficulty can sometimes be overcome by expressing the specification in the form of a response spectrum and by controlling the exciter directly from that spectrum In practical terms, as the exciter can only be driven with a signal that is a function of time, the software of the control rack determines a time signal with the same spectrum as the specification displayed The principles of composition of the equivalent shock are described with the shapes of the basic signals commonly used, while their properties and the problems that can be encountered, both in the generation of the signal and with respect to the quality of the simulation obtained, are emphasized Pyrotechnic devices or equipment (cords, valves, etc.) are frequently used in satellite launchers due to the very high degree of accuracy that they provide in operating sequences Shocks induced in structures by explosive charges are extremely severe, with very specific characteristics It is shown that they cannot be correctly simulated in the laboratory by conventional means and that their simulation requires specific tools 12.1 12.1.1 Definitions Shock Shock occurs when a force, a position, a velocity, or an acceleration is abruptly modified and creates a transient state in the system considered The modification is normally regarded as abrupt if it occurs in a time period that is short compared with the natural period concerned (AFNOR, 1993) Shock is defined as a vibratory excitation having a duration between the natural period of the excited mechanical system and two times that period (Figure 12.1) 12.1.2 Simple (or Perfect) Shock A shock whose signal can be represented exactly in simple mathematical terms is called a simple (or perfect) shock Standards generally specify one of the three following: half-sine (approached by a versed sine waveform), terminal peak sawtooth, and rectangular shock (approached by a trapezoidal waveform) © 2005 by Taylor & Francis Group, LLC Mechanical Shock 12.1.3 12-3 Half-Sine Shock This is a simple shock for which the acceleration – time curve has the form of a half-period (part positive or negative) of a sinusoid Acceleration (m/s2) 150 12.1.4 Versed-Sine (or Haversine) Shock This is a simple shock for which the acceleration – time curve has the form of one period of the curve representative of the function ẵ1 cos ị ; with this period starting from the zero value of this function It is thus a signal ranging between two minima of a sine wave 100 50 −50 −100 −150 10 Time (ms) 12 14 FIGURE 12.1 Example of a shock (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.) 12.1.5 Terminal Peak Sawtooth Shock or Final Peak Sawtooth Shock This is a simple shock for which the acceleration –time curve has the shape of a triangle, where acceleration increases linearly up to a maximum value and then instantly decreases to zero 12.1.6 Rectangular Shock This is a simple shock for which the acceleration–time curve increases instantaneously up to a given value, remains constant throughout the signal, and decreases instantaneously to zero In practice, what is carried out are trapezoidal shocks 12.1.6.1 Trapezoidal Shock This is a simple shock for which the acceleration–time curve grows linearly up to a given value, remains constant during a certain time period, after which it decreases linearly to zero 12.2 Description in the Time Domain Three parameters are necessary to describe a shock in the time domain: its amplitude, its duration, t; and its form The physical parameter expressed in terms of time is generally an acceleration, x€ ðtÞ, but can be also a velocity, vðtÞ; a displacement, xðtÞ, or a force, FðtÞ: In the first case, which we will consider in particular in this chapter, the velocity change corresponding to the shock movement is equal to (Table 12.1) DV ẳ â 2005 by Taylor & Francis Group, LLC ðt x€ ðtÞdt ð12:1Þ 12-4 Vibration and Shock Handbook TABLE 12.1 Main Simple Shock Waveforms (Amplitude, xm ; Duration, t; Velocity Change, DV) Waveform 12.3 Function Half-sine xtị ẳ xm sin Versed-sine xtị ¼ Rectangle xðtÞ ¼ xm Terminal peak sawtooth xðtÞ ¼ xm p t t 2p xm cos t t DV x t p m x t m xm t t t x t m Shock Response Spectrum 12.3.1 Need Very often, the problem is to evaluate the relative severity of several shocks (shocks measured in the real environment, measured shocks with respect to standards, establishment of a specification, etc.) A shock is an excitation of short duration, which induces transitory dynamic stress in structures These stresses are a function of the following: * * The characteristics of the shock (amplitude, duration, and shape) The dynamic properties of the structure (resonant frequencies, Q factors; see Chapter 19) The severity of a shock can thus be estimated only according to the characteristics of the system that undergoes it The evaluation of this severity requires in addition the knowledge of the mechanism leading to a degradation of the structure The two most common mechanisms are as follows: * * The exceeding of a value threshold of the stress in a mechanical part can lead to either a permanent deformation (acceptable or not) or a fracture, or at any rate, a functional failure If the shock is repeated many times (e.g., the shock recorded on the landing gear of an aircraft, the operation of an electromechanical contactor), the fatigue damage accumulated in the structural elements can lead in the long term to fracture (Lalanne, 2002c) The comparison would be difficult to carry out if one used a fine model of the structure, and in any case this is not always available, particularly at the stage of the development of the specification of dimensioning One searches for a method of general nature, which leads to results that can be extrapolated to any structure 12.3.2 Shock Response Spectrum Definition In a thesis on the study of earthquakes’ effects on buildings, Biot (1932) proposed a method consisting of applying the shock under consideration to a “standard” mechanical system, which thus does not claim to be a model of the real structure It is composed of a support and of N linear one-degree-offreedom (one-DoF) resonators, comprising each one are a mass, mi a spring of stiffness, p kiffiffiffiffiffi and ffi a damping device, ci ; chosen such that the fraction of critical damping (damping ratio) j ¼ ci ki mi is the same for all N resonators A model for the shock response spectrum (SRS) is shown in Figure 12.2 (also see Chapter 17) When the support is subjected to the shock,pffiffiffiffiffiffi each mass, mi ; has a specific movement response according to its natural frequency, f0i ẳ 1=2pị ki =mi and to the chosen damping ratio, j; while a © 2005 by Taylor & Francis Group, LLC Mechanical Shock 12-5 z(t) t f1 ξ f2 ξ fN−1 ξ fN ξ x(t) t FIGURE 12.2 permission.) Model of the SRS (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With stress, si ; is induced in the elastic element The analysis consists of seeking the largest stress, smi ; observed at each frequency in each spring For applications deviating from the assumptions of definition of the SRS (linearity, only one DoF), it is desirable to observe a certain prudence if one wishes to estimate quantitatively the response of a system starting from the spectrum (Bort, 1989) The response spectra are more often used to compare the severity of several shocks It is known that the tension static diagram of many materials comprises a more-or-less linear arc on which the stress is proportional to the deformation In dynamics, this proportionality can be allowed within certain limits for the peaks of the deformation If a mass –spring–damper system is supposed to be linear, it is then appropriate to compare two shocks by the maximum response stress, sm , that they induce or by the maximum relative displacement, zm ; that they generate This occurs since it is supposed sm ẳ Kzm 12:2ị zm is a function only of the dynamic properties of the system, whereas sm is also a function, via K; of the properties of the materials which constitute it The curve giving the largest relative displacement, zsup multiplied by v20 (v0 ¼ 2p f0 ) according to the natural frequency, f0 ; for a given damping ratio j; is the SRS 12.3.3 Response of a Linear One-Degree-of-Freedom System 12.3.3.1 Shock Defined by a Force Consider a mass – spring – damping system subjected to a force, FðtÞ; applied to the mass (Figure 12.3) The differential equation of the movement is written as m dz dz ỵ kz ẳ Ftị ỵc dt dt Mass m ð12:3Þ where zðtÞ is the relative displacement of the mass, m; relative to its support in response to the shock, FðtÞ: This equation can be expressed in the form (Lalanne, 2002b): d2 z dz FðtÞ ỵ v20 z ẳ 12:4ị ỵ 2jv0 dt dt m pffiffiffiffi where pffiffiffiffiffi j ¼ c=2 km (damping ratio) and v0 ẳ k=m (natural frequency) â 2005 by Taylor & Francis Group, LLC Force Damping constant c Stiffness k Fixed support FIGURE 12.3 Linear one-Dof system subjected to a force (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.) 12-6 Vibration and Shock Handbook 12.3.3.2 Shock Defined by an Acceleration Let us set x€ ðtÞ as an acceleration applied to the base of a linear one-DoF mechanical system, with y€ðtÞ the absolute acceleration response of the mass, m; and zðtÞ the relative displacement of the mass, m; with respect to the base (Figure 12.4) The equation of the movement is written as above: m d2 y dy dx ¼ 2kðy xÞ c dt dt dt Mass m Damping constant c Stiffness k Moving base ð12:5Þ that is d2 y dy dx ỵ v20 y ẳ v20 xtị ỵ 2jv0 ỵ 2jv0 dt dt dt 12:6ị or while setting ztị ẳ ytị xtị d2 z dz d2 x ỵ 2jv0 ỵ v20 z ẳ 2 dt dt dt Absolute reference FIGURE 12.4 Linear one-DoF system subjected to acceleration (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.) ð12:7Þ The differential equation (Equation 12.7) can be integrated by parts or by using the Laplace transformation If the excitation is an acceleration of the support, the response relative displacement is given, for zero initial conditions, by an integral called Duhamel’s integral: qffiffiffiffiffiffiffiffi ðt 21 p ztị ẳ x aịe2jv0 t2aị sin v0 j2 ðt aÞda ð12:8Þ v j2 where a is an integration variable homogeneous with time The absolute acceleration of the mass is given by qffiffiffiffiffiffiffiffi t v0 y tị ẳ p x aịe2jv0 t2aị ẵ1 2j2 Þsin v0 j2 ðt aÞ j2 q q ỵ 2j j2 cos v0 j2 ðt aÞ da 12.3.4 Definitions 12.3.4.1 Response Spectrum ð12:9Þ This is a curve representative of the variations of the largest response of a linear one-DoF system subjected to a mechanical excitation, plotted against its natural frequency, f0 ¼ v0 =2p; for a given value of its damping ratio (see Chapter 17) 12.3.4.2 Absolute Acceleration Shock Response Spectrum In the most usual cases where the excitation is defined by an absolute acceleration of the support or by a force applied directly to the mass, the response of the system can be characterized by the absolute acceleration of the mass (which can be measured using an accelerometer fixed to this mass) The response spectrum is then called the absolute acceleration SRS 12.3.4.3 Relative Displacement Shock Spectrum In similar cases, we often calculate the relative displacement of the mass with respect to the displacement of the base of the system This displacement is proportional to the stress created in the © 2005 by Taylor & Francis Group, LLC Mechanical Shock 12-7 spring (since the system is regarded as linear) In practice, one generally expresses in ordinates the quantity v20 zsup ; which is called the equivalent static acceleration (Biot, 1941) This product has the dimensions of acceleration, but does not represent the absolute acceleration of the mass, except when damping is zero However, when damping is close to the current values observed in mechanics, and in particular when j ¼ 0:05; as a first approximation one can assimilate v20 zsup to the absolute acceleration y€ sup of the mass, m (Lalanne, 1975, 2002b) The quantity v20 zsup is termed pseudo-acceleration In the same way, one terms the product v0 zsup pseudo-velocity The spectrum giving v20 zsup vs the natural frequency is named the relative displacement shock spectrum In each of these two important categories, the response spectrum can be defined in various ways according to how the largest response at a given frequency is characterized 12.3.4.4 Primary Positive Shock Response Spectrum or Initial Positive Shock Response Spectrum This is the highest positive response observed during the shock 12.3.4.5 Primary (or Initial) Negative Shock Response Spectrum This is the highest negative response observed during the shock 12.3.4.6 Secondary (or Residual) Shock Response Spectrum This is the largest response observed after the end of the shock Here also, the spectrum can be positive or negative Example An example giving standardized primary and residual relative displacement SRS curves for a half-sine pulse is shown in Figure 12.5 12.3.4.7 Positive (or Maximum Positive) Shock Response Spectrum This is the largest positive response due to the shock, without reference to the duration of the shock It thus corresponds to the envelope of the positive primary and residual spectra Half-sine (1 m/s2 - s) 2.0 Primary positive spectrum w 20 zsup (m/s2) 1.5 1.0 Residual positive spectrum 0.5 0.0 −0.5 Primary negative spectrum −1.0 −1.5 0.0 Residual negative spectrum 0.5 1.0 1.5 2.0 2.5 3.0 Frequency (Hz) 3.5 x = 0.05 4.0 4.5 5.0 FIGURE 12.5 Standardized primary and residual relative displacement SRS of a half-sine pulse (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.) © 2005 by Taylor & Francis Group, LLC 12-8 12.3.4.8 Vibration and Shock Handbook Negative (or Maximum Negative) Shock Response Spectrum This is the largest negative response due to the shock, without reference to the duration of the shock As before, it corresponds to the envelope of the negative primary and residual spectra 12.3.4.9 Maximax Shock Response Spectrum This is the envelope of the absolute values of the positive and negative spectra 12.3.4.10 Choice of Shock Response Spectrum Which spectrum must be used? Absolute acceleration SRS can be useful when absolute acceleration is the parameter easiest to compare with a characteristic value (as in a study of the effects of a shock on a man, a comparison with the specification of an electronics component, etc.) In practice, it is very often the stress (and thus the relative displacement) which seems the most interesting parameter The spectrum is primarily used to study the behavior of a structure, to compare the severity of several shocks (the stress created is a good indicator), to write test specifications (as it is also a comparison between the real environment and the test environment), or to dimension a suspension (relative displacement and stress are then useful) The damage is assumed to be proportional to the largest value of the response, i.e., to the amplitude of the spectrum at the frequency considered, and it is of little importance for the system whether this maximum, zm ; takes place during or after the shock The most interesting spectra are thus the positive and negative spectra that are most frequently used in practice, with the maximax spectrum The distinction between positive and negative spectra must be made each time the system, if dissymmetrical, behaves differently, for example under different tension and compression It is, however, useful to know these various definitions so as to be able to correctly interpret the curves published The Shock Response Spectrum is a curve representative of the variations of the largest response of a linear one-DoF system subjected to a mechanical excitation, plotted against its natural frequency, for a given value of its damping ratio The response can be defined by the pseudo-acceleration, v20 zsup (relative displacement shock spectrum) or by the absolute acceleration of the mass (absolute acceleration SRS) For the usual values of Q; the spectra are very close The most interesting spectra are the positive and negative spectra, which are most frequently used in practice, with the maximax spectrum The relation between the various types of SRS that have been discussed here is shown in Figure 12.6 Primary (initial) positive SRS Primary (initial) negative SRS Positive SRS Secondary (residual) positive SRS Negative SRS Maximax SRS Secondary (residual) negative SRS FIGURE 12.6 © 2005 by Taylor & Francis Group, LLC Relation between the different types of SRS Mechanical Shock 12-9 12.3.5 Standardized Shock Response Spectrum 12.3.5.1 Definition * * 1.4 1.2 SRS / xm For a given shock, the spectra plotted for various values of the duration and the amplitude are similar in shape It is thus useful, for simple shocks, to have a standardized or reduced spectrum plotted in dimensionless coordinates, while plotting on the abscissa the product f0 t (instead of f0 ) or v0 t and on the ordinate the spectrum/shock pulse amplitude ratio, v20 zm =€xm ; which, in practice, amounts to tracing the spectrum of a shock of duration equal to sec and amplitude m/sec2 This is shown in Figure 12.7 These standardized spectra can be used for two purposes: 1.0 0.8 0.6 0.4 Q = 10 0.2 0.0 0.0 1.0 2.0 f0t 3.0 4.0 5.0 FIGURE 12.7 Standardized positive SRS of a terminal peak sawtooth pulse (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.) Plotting of the spectrum of a shock of the same form, but of arbitrary amplitude and duration Investigating the characteristics of a simple shock of which the spectrum envelope is a given spectrum (resulting from measurements from the real environment) 12.3.5.2 Standardized Shock Response Spectra of Simple Shocks Figure 12.8 to Figure 12.15 give the reduced SRSs for various pulse forms, with unit amplitude and unit duration, for several values of damping To obtain the spectrum of a particular shock of arbitrary amplitude, x€ m ; and duration, t (different from 1) from these spectra, it is enough to regraduate the scales as follows: * * For the amplitude, multiply the reduced values by x€ m : For the abscissae (x-axis values), replace each value fẳ f0 tị by f0 ẳ f=t: We will see later on how these spectra can be used for the calculation of test specifications 2.0 1.5 Half-sine (1 m/s2 - s) 0.05 1.0 w 20 zsup (m/s2) 0.025 0.1 0.25 0.5 Positive spectra 0.5 0.5 0.0 −0.5 0.25 −1.0 0.1 −1.5 −2.0 0.0 0.5 0.05 0.025 1.0 1.5 Negative spectra 2.0 2.5 3.0 Frequency (Hz) 3.5 4.0 4.5 5.0 FIGURE 12.8 Standardized positive and negative relative displacement SRS of a half-sine pulse (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.) © 2005 by Taylor & Francis Group, LLC 12-50 Vibration and Shock Handbook Rest Rest Maximum displacement during the shock movement FIGURE 12.55 Displacement of the coil of the shaker (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.) * * The maximum velocity is also limited (Young, 1964): 1.5 to m/sec in sine mode (in shock, one can admit a larger velocity with nontransistorized amplifiers (electronic tubes), because these amplifiers can generally accept a very short overvoltage) During the movement of the moving element in the air-gap of the magnetic coils, there is an electromotive force (emf) produced which is opposed to the voltage supply The velocity must thus have a value such that this emf is lower than the acceptable maximum output voltage of the amplifier The velocity must in addition be zero at the end of the shock movement (Smallwood and Witte, 1972; Galef, 1973) There is a limit to maximum acceleration, related to the maximum force McClanahan and Fagan (1965) consider that the realizable maxima shock levels are approximately 20% below the vibratory limit levels in velocity and in displacement The majority of authors agree that the limits in force are, for the shocks, larger than those indicated by the manufacturer (in sine mode) The determination of the maximum force and the maximum velocity is based, in vibration, on considerations of the fatigue of the shaker mechanical assembly Since the number of shocks that the shaker will carry out is very much lower than the number of cycles of vibrations that it will undergo during its life, the parameter maximum force can be, for the shock applications, increased considerably Another reasoning consists of considering the acceptable maximum force, given by the manufacturer in random vibration mode, expressed by its rms value Knowing that one can observe random peaks being able to reach 4.5 times this value (limitation of control system), one can admit the same limitation in shock mode One finds other values in the literature, such as: * * # times the maximum force in sine mode, with the proviso of not exceeding 300g on the armature assembly (Hug, 1972) times the maximum force in sine mode in certain cases (very short shocks; 0.4 msec, for example; Gallagher and Adkins, 1966) Dinicola (1964) and Keegan (1973) give a factor of about ten for the shocks of duration lower than msec The limits of velocity, displacement, and force are not affected by the mass of the specimen 12.9.4.1.1 Abacuses For a given shock and for given pre- and postshocks shapes, the velocity and the displacement can be calculated as a function of time by integration of the expressions of the acceleration, as well as the maximum values of these parameters, in order to compare them with the characteristics of the facilities From these data, abacuses can be established allowing quick evaluation of the possibility of realization of a specified shock on a given test facility (characterized by its limits of velocity and © 2005 by Taylor & Francis Group, LLC Mechanical Shock 12-51 10 Shock amplitude (m/s2) FORCE LIMITATION A 10 C VELOCITY D LIMITATION p = 0.05 E F 0.10 G 0.25 0.50 10 10 DISPLACEMENT LIMITATION 10 −4 10 C′ 10−3 10−2 Shock duration (s) A′ 1.00 F′ D′ E′ G′ 10−1 FIGURE 12.56 Abacus of the realization domain of a shock (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.) of displacement) These abacuses are made up of straight-line segments on logarithmic scales (Figure 12.56) * * AA0 corresponding to the limitation of velocity: the condition vm # vL (vL ¼ acceptable maximum velocity on the facility considered) results in a relationship of the form x€ m t # constant (independent of p; the ratio of the absolute values of the pre- and postshocks amplitude and of the principal shock amplitude) CC, DD, and so on, the greater slope corresponding to the limitation in displacement for various values of p ( p ¼ 0.05, 0.10, 0.25, 0.50, and 1.00) A particular shock will be thus realizable on the shaker only if the point of coordinates t; x€ m (duration and amplitude of the shock considered) is located under these lines, this useful domain increasing when p increases Example 103 The TPS shock pulse (340 m/sec2, msec) is realizable on this shaker with p ẳ 0:05: â 2005 by Taylor & Francis Group, LLC 0.25 340 0.5 1.0 m/s2 TPS shock pulse, 340 m/sec , msec (example of Section 12.5.2.5) Unit table mass: 192 kg Shaker: 135 kN (maximum velocity: 1.78 m/sec, maximum stroke: ^12.7 mm; see Figure 12.57) Test item ỵ xture mass: 150 kg Maximum acceleration without load: (135,000/192) < 703 m=sec2 Maximum acceleration with test item and xture: (135,000/(192 ỵ 150)) < 395 m=sec2 No load Test item + fixture mass: 150 kg 102 p = 0.05 0.1 0.009 101 −4 10 10−3 10−2 Shock Duration (s) 10−1 FIGURE 12.57 Shaker 135 kN, ^12.7 mm — TPS pulse with half-sine symmetrical pre- and postshocks 12-52 Vibration and Shock Handbook The limitation can also be due to: * * The resonance of the moving element (a few thousands Hertz; although it is kept to the maximum by design, the resonance of this element can be excited in the presence of signals with very short rise time) The strength of the material (very great accelerations can involve a separation of the coil of the moving component) Mechanical limitations of electrodynamic shakers for shocks: * * * 12.9.4.2 Maximum stroke of the coil-table unit: 25.4 to 75 mm peak-to-peak Maximum acceleration, related to the maximum force: according to the author, # times the maximum force in sine mode, with the proviso of not exceeding 300g on the armature assembly, more than eight times the maximum force in sine mode in certain cases (very short shocks; 0.4 msec, for example) Maximum velocity: 1.5 to m/sec in sine mode Electronic Limitations Limitation of the output voltage of the amplifier (Smallwood, 1974), which limits coil velocity Limitation of the acceptable maximum current in the amplifier, related to the acceptable maximum force (i.e., with acceleration) Limitation of the bandwidth of the amplifier Limitation in power, which relates to the shock duration (and the maximum displacement) for a given mass Current transistor amplifiers make it possible to increase the low frequency bandwidth but not handle even short overtensions well, and thus are limited in mode shock (Miller, 1964) 12.9.5 The Use of Electrohydraulic Shakers Shocks are realizable on the electrohydraulic exciters, but with additional stresses * * Contrary to the case of the electrodynamic shakers, one cannot obtain via these means shocks of amplitude larger than realizable accelerations in the steady mode The hydraulic vibration machines are in addition strongly nonlinear (Favour, 1974) However, their long stroke, required for long duration shocks, is an advantage 12.10 Control by a Shock Response Spectrum 12.10.1 Principle The exciters are actually always controlled by a signal that is a function of time An acceleration –time signal gives only one SRS However, there is an infinity of acceleration –time signals with a given spectrum The general principle thus consists in searching out one of the signals, x€ ðtÞ; having the specified spectrum Historically, the simulation of shocks with spectrum control was first carried out using analog and then digital methods (Smallwood and Witte, 1973; Smallwood, 1974) © 2005 by Taylor & Francis Group, LLC Time (a) Shock response spectrum 12-53 Acceleration Mechanical Shock Frequency f0 (b) FIGURE 12.58 Elementary shock (a) and its SRS (b) (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.) From the data of selected points on the shock spectrum to be simulated, the calculator of the control system uses an acceleration signal with a very tight spectrum For that, the calculation software proceeds as follows (Lalanne, 2002b): * * * At each frequency, f0 , of the reference shock spectrum, the software generates an elementary acceleration signal, for example, a decaying sinusoid Such a signal has the property of having a SRS presenting a peak of the frequency of the sinusoid whose amplitude is a function of the damping of the sinusoid (Figure 12.58) With an identical shock spectrum, this property makes it possible to realize shocks on the shaker that would be unrealizable with a control carried out by a temporal signal of simple shape For high frequencies, the spectrum of the sinusoid tends roughly towards the amplitude of the signal All the elementary signals are added by possibly introducing a given delay (and variable) between each one of them in order to control to a certain extent the total duration of the shock, which is primarily due to the lower frequency components (Figure 12.59) The total signal being thus made up, the software proceeds to processes correcting the amplitudes of each elementary signal so that the spectrum of the total signal converges towards the reference spectrum after some iterations Shock response spectrum (m/s2) 60 50 40 30 20 10 0 100 200 300 Frequency (Hz) 400 500 FIGURE 12.59 SRS of the components of the required shock (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.) © 2005 by Taylor & Francis Group, LLC 12-54 Vibration and Shock Handbook The operator must provide to the software with, at each frequency of the reference spectrum: * * * * The frequency of the spectrum Its amplitude A delay The damping of sinusoids or other parameters characterizing the number of oscillations of the signal When a satisfactory spectrum time signal has been obtained, it remains to be checked that the maximum velocity and displacement during the shock are within the authorized limits of the test facility (by integration of the acceleration signal) Lastly, after measurement of the transfer function of the facility, one calculates the electric excitation which will make it possible to reproduce on the table the acceleration pulse with the desired spectrum, as in the case of control from a signal according to the time (Powers, 1974) 12.10.2 Principal Shapes of Elementary Signals 12.10.2.1 Decaying Sinusoid The shocks measured in the field environment are very often responses of structures to an excitation applied upstream, and are thus composed of the superposition of several modal responses of a damped sine type (Smallwood and Witte, 1973; Crimi, 1978; Boissin et al., 1981; Smallwood, 1985) Electrodynamic shakers are completely adapted to the reproduction of this type of signal According to this, one should be able to reconstitute a given SRS from such signals of the form: t $ 0= t , 0; atị ẳ A e2hVt sin Vt atị ẳ 12:43ị where V ẳ 2pf , f ẳ frequency of the sinusoid, and h ¼ damping factor Velocity and displacement are not zero at the end of the shock with this type of signal These nonzero values are very awkward for a test on a shaker Compensation can be carried out in 150 several ways: © 2005 by Taylor & Francis Group, LLC 100 50 m/s2 By truncating the total signal until it is realizable on the shaker This correction can, however, lead to an important degradation of the corresponding spectrum (Smallwood and Witte, 1972) By adding to the total signal (sum of all the elementary signals) a highly damped decaying sinusoid at low frequency, shifted in time, defined to compensate for the velocity and the displacement (Smallwood and Nord, 1974; Smallwood 1975, 1985) By adding to each component two exponential compensation functions, with a phase in the sinusoid (Nelson and Prasthofer, 1974; Smallwood, 1975) −50 −100 −150 −200 −250 −300 −350 20 40 60 Time (ms) 80 100 FIGURE 12.60 Shock pulse generated from decaying sinusoids, compensated by a decaying sinusoid (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.) Mechanical Shock 12-55 −50 −100 −150 −200 ZERD Function The use of a decaying sinusoid with its compensation waveform modifies the response spectrum at the low frequencies and, in certain cases, can harm the quality of simulation Fisher and Posehn (1977) proposed using a waveform, which they named “ZERD” (Zero Residual Displacement), defined by the expression: sin Vt t cosðVt ỵ fị atị ẳ A e2hVt V 12:44ị where f ¼ arc tanð2h=1 h2 Þ: This function resembles a damped sinusoid and has the advantage of leading to zero velocity and displacement at the end of the shock (Figure 12.62) Example The reference SRS is that of Figure 12.24 (Section 12.5.2.5) Figure 12.63 shows an example of acceleration signal generated from ZERD functions having approximately the same SRS 12.10.2.3 50 −250 −300 10 20 60 70 80 1.5 f = Hz h = 0.05 A=1 1.0 0.5 0.0 −0.5 −1.0 −1.5 12 Time (s) WAVSIN Waveform Yang (1970, 1972) and Smallwood (1974, 1975, 1985) proposed (initially for the simulation of the earthquakes) a signal of the form: 30 40 50 Time (ms) FIGURE 12.61 Acceleration signal generated from decaying sinusoids, compensated by two exponentia functions (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.) Acceleration (m/s2) 12.10.2.2 100 m/s2 Example The reference SRS is that of Figure 12.24 (Section 12.5.2.5) Examples of acceleration signals generated from decaying sinusoids and having approximately the same SRS are shown in Figure 12.60 and Figure 12.61 in the cases of a compensation by a decaying sinusoid and by two exponential functions 16 20 FIGURE 12.62 ZERD waveform of D.K Fisher and M.R Posehn (example) atị ẳ am sin 2pbt sin 2pft 0#0#t atị ẳ elsewhere ) 12:45ị where f ẳ Nb 12:46ị 2b 12:47ị tẳ â 2005 by Taylor & Francis Group, LLC 12-56 200 150 100 50 m/s2 where N is an integer (which must be odd and higher than on 1) The first term of aðtÞ is a window of half-sine form of half-period t: The second describes N half-cycles of a sinusoid of greater frequency ð f Þ; modulated by the preceding window (Figure 12.64) This function leads also to zero velocity and displacement at the end of the shock Vibration and Shock Handbook −50 Example −100 Figure 12.65 shows an example of acceleration signal generated from WAVSIN functions having approximately the same SRS as the reference SRS of Figure 12.24 (Section 12.5.2.5) −150 The cases treated by Smallwood (1974) seem to show that these three methods give similar results It is noted, however, in practice, that, according to the shape of the reference spectrum, one or other of these waveforms allows a better convergence The ZERD waveform very often gives good results © 2005 by Taylor & Francis Group, LLC 20 30 40 50 60 70 80 90 FIGURE 12.63 Acceleration signal generated from ZERD functions 1.5 WAVSIN (N = f = t = 2.5) 1.0 0.5 0.0 −0.5 −1.0 −1.5 0.0 12.10.4 Criticism of Control by a Shock Response Spectrum 0.5 1.5 1.0 Time t (s) 2.0 2.5 FIGURE 12.64 Example of WAVSIN waveform (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.) 150 100 50 m/s2 Whatever the method adopted, simulation on a test facility of shocks measured in the real world requires the calculation of their SRSs and the search for an equivalent shock If the specification must be presented in the form of a time-dependent shock pulse, the test requester must define the characteristics of shape, duration, and amplitude of the signal, with the already quoted difficulties If the specification is given in the form of an SRS, the operator inputs in the control system the given spectrum, but the shaker is always controlled by a signal according to the time calculated and according to procedures described in the preceding sections It is known that the transformation shock spectrum signal has an infinite number of solutions, and that very different signals can have identical SRSs This phenomenon is related to the loss of most of the information initially contained in the signal, x€ ðtÞ; during the calculation of the spectrum (Metzgar, 1967) 10 Time (ms) Acceleration (m/s2) 12.10.3 Comparison of WAVSIN, SHOC Waveforms, and Decaying Sinusoid 0 −50 −100 −150 20 40 60 80 100 120 140 160 180 Time (ms) FIGURE 12.65 Acceleration signal generated from WAVSIN functions Mechanical Shock 12-57 The oscillatory shock pulses have a spectrum that presents an important peak to the frequency of the signal This peak can, according to choice of parameters, exceed by a factor of five the amplitude of the same spectrum at the high frequencies; that is, five times the amplitude of the signal itself Being given a point of the specified spectrum of amplitude, S; it is thus enough to have a signal vs time of amplitude S=5 to reproduce the point For a simple shaped shock, this factor does not exceed two in the most extreme case All these remarks show that the determination of a signal of the same spectrum can lead to very diverse solutions, the validity of which one can question If any particular precaution is not taken, the signals created by these methods have, in a general way, one duration much larger and an amplitude much smaller than the shocks that were used to calculate the reference SRS (a factor of about ten in both cases) Figure 12.66 and Figure 12.67 give an example In the face of such differences between the excitations, one can legitimately wonder whether the SRS is a sufficient condition to guarantee a representative test It is necessary to remember that this equivalence is based on the behavior of a linear system that one chooses the Q factor a priori One must be aware of the following 60 40 Shock A m/s2 20 Shock B −20 −40 −60 0.5 FIGURE 12.66 1.5 2.5 Seconds 3.5 4.5 × 10E-2 Example of shocks having spectra near the SRS 90 80 Shock A 70 m/s2 60 Shock B 50 40 30 20 Q = 10 10 0 0.2 0.4 FIGURE 12.67 © 2005 by Taylor & Francis Group, LLC 0.6 0.8 Hz 1.2 1.4 1.6 1.8 × 10E3 SRS of the shocks shown in Figure 12.66 12-58 * * * Vibration and Shock Handbook The behavior of the structure is in practice far from linear and that the equivalence of the spectrum does not lead to stresses of the same amplitude Another effect of these nonlinearities appears sometimes by the inaptitude of the system to correct the drive waveform to take account of the transfer function of the installation Even if the amplitudes of the peaks of acceleration and the maximum stresses of the resonant parts of the tested structure are identical, the damage by the fatigue generated by accumulation of the stress cycles is rather different when the number of shocks to be applied is significant The tests carried out by various laboratories not have the same severity These questions did not receive a really satisfactory response By prudence rather than by rigorous reasoning, many agree, however, on the need for placing parapets, while trying to supplement the specification defined by a spectrum with complementary data (DV; duration of the shock, require SRS at two different values of damping, etc.; Favour, 1974; Smallwood, 1974, 1975, 1985) There is an infinity of acceleration –time signals with a given spectrum Several elementary waveforms can be used to build a signal of acceleration having a given SRS They give similar results Without particular precaution, the signals thus obtained generally have one duration much larger and an amplitude much smaller than the shocks which were used to calculate the reference SRS A complementary parameter (shock duration, velocity change, etc.) is often specified with the SRS to limit this effect 12.11 Pyrotechnic Shock Simulation Many works have been published on the characterization, measurement, and simulation of shocks of pyrotechnic origin, generated by bolt cutters, explosive valves, separation nuts, and so on (Zimmerman, 1993) The test facilities suggested are many, ranging from traditional machines to very exotic means The tendency today is to consider that the best simulation of shocks measured in near-field can be obtained only by subjecting the material to the shock produced by the real device For shocks in the far-field, simulation can be carried out either using the real pyrotechnic source and a particular mechanical assembly, using specific equipment using explosives, or by impacting metal to metal if the structural response is more important When the real shock is practically made up only of the response of the structures, a simulation on a shaker is possible (when performances by this means are allowed) 12.11.1 Simulation Using Pyrotechnic Facilities The simplest solution consists of making functional, real pyrotechnic devices on real structures The simulation is perfect but (Conway et al., 1976; Luhrs, 1976): * * It can be expensive and destructive One cannot apply an uncertainty factor (Lalanne, 2002d) to cover the variability of this shock without being likely to create unrealistic local damage (a larger load, which requires an often expensive modification of the devices and can be much more destructive) To avoid this problem, an expensive solution consists of carrying out several tests in a statistical matter © 2005 by Taylor & Francis Group, LLC Mechanical Shock 12-59 One often prefers to carry out a simulation on a reusable assembly, the excitation still being pyrotechnic in nature Several devices have been designed, some examples of which are as follows: * * * A test facility made up of a cylindrical structure (Ikola, 1964), which comprises a “consumable” sleeve cut out for the test by an explosive cord Preliminary tests are carried out to calibrate the facility while acting on the linear charge of the explosive cord and/or the distance between the FIGURE 12.68 Plate with resonant system subjected equipment to be tested (fixed on the to detonation (Source: Lalanne, Chocs Mecaniques, structure as in the real case if possible) Hermes Science Publications With permission.) and the explosive cord A greater number of small explosive charges near the equipment to be tested on the structure in “flowers pots.” The number of pots to be used on each axis depends on the amplitude of the shock, the size of the equipment, and the local geometry of the structure The shape of the shock can be modified within certain limits by use of damping devices, placing the pot closer to or further from the equipment, or by putting suitable padding in the pot (Aerospace Systems pyrotechnic shock data, 1970) A test facility made up of a basic rectangular steel plate (Figure 12.68) suspended horizontally This plate receives on its lower part, directly or by the intermediary of an “expendable” item, an explosive load (chalk line, explosive in plate or bread) A second plate supporting the test item rests on the base plate via four elastic supports (Thomas, 1973) The reproducibility of the shocks is better if the explosive charge is not in direct contact with the base plate 12.11.2 Simulation Using Metal-to-Metal Impact The shock obtained by a metal-to-metal impact has similar characteristics to those of a pyrotechnical shock in an intermediate field: great amplitude; short duration; high frequency content; SRS comparable with a low frequency slope of 12 dB per octave, etc The simulation is in general satisfactory up to approximately 10 kHz The shock can be created by the impact of a hammer on the structure itself, a Hopkinson bar or a resonant plate (Bai and Thatcher, 1979; Luhrs, 1981; Davie, 1985; Dokainish and Subbaraj, 1989; Davie and Batemen, 1992) With all these devices, the amplitude of the shock is controlled while acting on the velocity of impact The frequency components are adjusted by modifying the resonant geometry of system (changing the length of the bar between two points of fixing, adding or removing runners, etc.) or by the addition of a deformable material between the hammer and the anvil 12.11.3 Simulation Using Electrodynamic Shakers The limitation relating to the stroke of the electrodynamic shaker is not very constraining for the pyrotechnical shocks since they are at high frequencies The possibilities are limited especially by the acceptable maximum force and then concern the maximum acceleration of the shock © 2005 by Taylor & Francis Group, LLC 12-60 Vibration and Shock Handbook (Conway et al., 1976; Luhrs, 1976; Powers, 1976; Caruso, 1977) If one agrees to cover only part of the SRS where material has resonant frequencies, then when one makes a possible simulation on the shaker, which gives a better approach to matching the real spectrum Exciters have the advantage of allowing the realization of any signal shape such as shocks of simple shapes (Dinicola, 1964; Gallagher and Adkins, 1966), but also of reproducing a specified SRS (direct control from an SRS; see Section 12.10.1) It is possible, in certain cases, to reproduce the real SRS up to 1000 Hz If one is sufficiently far away from the source of the shock, the transient has a lower level of acceleration and the only limitation is the bandwidth of the shaker, which is about 2000 Hz Certain facilities of this type were modified to make it possible to simulate the effects of pyrotechnical shocks up to 4000 Hz One can thus manage to simulate shocks whose spectrum can reach 7000g (Moening, 1986) 12.11.4 Simulation Using Conventional Shock Machines We saw that, generally, the method of development of a specification of a shock consists of replacing the transient of the real environment, whose shape is in general complex, with a simple shape shock, such as half-sine, triangle, trapezoid, and so on, starting from the SRS equivalence criterion, with the application of a given or calculated uncertainty factor (Lalanne, 2002d) to the shock amplitude (Luhrs, 1976) With the examination of the shapes of the response spectra of standard simple shocks, it seems that the signal best adapted is the TPS pulse, whose spectra are also appreciably symmetrical SRSs of the pyrotechnical shocks with, in general, averages close to zero have a very weak slope at low frequencies The research of the characteristics of such a triangular pulse (amplitude, duration) having an SRS envelope of that of a pyrotechnical shock led often to a duration of about msec and to an amplitude being able to reach several tens of thousands of msec22 Except in the case of very TABLE 12.12 Advantages and Drawbacks of Various Test Facilities for the Pyroshock Simulation Shock Facility Field Real pyrotechnic devices on real structures Near Very good simulation Expensive, generally destructive, no uncertainty factor/test factor Reusable assembly with pyrotechnic excitation Near Good simulation Necessity of preliminary tests, no uncertainty factor/test factor, use of explosive (specific conditions to ensure safety), expensive Metal to metal impact Near Good simulation, no explosive charge Necessity of preliminary tests, limitations in acceleration and frequency (approximately 10 kHz) Electrodynamic shaker Far Easy implementation, control using any time history signal or SRS, possibility of using an uncertainty factor or a test factor Necessity of one test by axis, maximum frequency up to about to kHz Conventional shock-test machine Far Easy implementation, possibility of using an uncertainty factor or a test factor Use of a shock pulse with velocity change instead of an oscillatory shock pulse (over test at low frequency), necessity of one test by axis, shock duration higher than msec (0.1 msec using a specific device for very light test item), limitation in amplitude, useable for very small test items only © 2005 by Taylor & Francis Group, LLC Advantages Drawbacks Mechanical Shock 12-61 small test items, it is in general not possible to carry out such shocks on the usual drop tables due to certain limitations: * * * Limitation in amplitude (acceptable maximum force on the table) Duration limit: the pneumatic shock simulators not allow it to go below to msec; even with the lead shock simulators, it is difficult to obtain a duration of less than msec and a larger shock duration leads to a significant overtest at low frequency The SRSs of the pyrotechnical shocks are much more sensitive to the choice of damping than simple shocks carried out on shock machines A comparison of some pyroshock-test facilities is given in Table 12.12 References Aerospace Systems pyrotechnic shock data (Ground test and flight), Final Report, Contract NAS 5, 15208, June 1966, March 1970 ASTM D3332, Standard Test Method for Mechanical-Shock Fragility of Products, Using Shock Machines American Society for Testing and Materials, Philadelphia, PA Bai, M and Thatcher, W., High G pyrotechnic shock simulation using metal-to-metal impact, Shock Vib Bull., 49, Part 1, 97 –100, 1979 Benioff, H., The physical evaluation of seismic destructiveness, Bull Seismol Soc Am., 398 –403, 1934 Biot, M.A 1932 Transient oscillations in elastic systems, Thesis No 259, Aeronautics Department, California Institute of Technology, Pasadena Biot, M.A., A mechanical analyzer for the prediction of earthquake stresses, Bull Seismol Soc Am., 31, 2, 151–171, 1941 Boissin, B., Girard, A., and Imbert, J.F 1981 Methodology of uniaxial transient vibration test for satellites, In Recent Advances in Space 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November Young, F.W., Shock testing with vibration systems, Shock Vib Bull., 34, Part 3, 355–364, 1964 Zimmerman, R.M., Pyroshock — bibliography, IES Proc., 471 –479, 1993 © 2005 by Taylor & Francis Group, LLC ... corresponding to the shock movement is equal to (Table 12. 1) DV ẳ â 2005 by Taylor & Francis Group, LLC ðt x€ ðtÞdt 12: 1Þ 12- 4 Vibration and Shock Handbook TABLE 12. 1 Main Simple Shock Waveforms.. .12 Mechanical Shock 12. 1 Definitions 12- 2 Shock † Simple (or Perfect) Shock † Half-Sine Shock Versed-Sine (or Haversine) Shock † Terminal Peak Sawtooth Shock or Final Peak Sawtooth Shock. .. Conventional Shock Machines 12- 1 © 2005 by Taylor & Francis Group, LLC 12- 2 Vibration and Shock Handbook Summary Transported or on-board equipment is very frequently subjected to mechanical shocks