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.

III Shock and Vibration

III-1

12 Mechanical Shock

Christian Lalanne

Engineering Consultant

12.1 Definitions 12-2Shock † Simple (or Perfect) Shock † Half-Sine Shock †

Versed-Sine (or Haversine) Shock † Terminal Peak Sawtooth Shock or Final Peak Sawtooth Shock †

Rectangular Shock12.2 Description in the Time Domain 12-312.3 Shock Response Spectrum 12-4Need † 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 Freedom12.4 Pyroshocks 12-1712.5 Use of Shock Response Spectra 12-18Severity Comparison of Several Shocks † Test

Specification Development from Real Environment Data12.6 Standards 12-24Types of Standards † Installation Conditions of Test

Item † Uncertainty Factor † Bump Test12.7 Damage Boundary Curve 12-26Definition † Analysis of Test Result

12.8 Shock Machines 12-28Main Types † Impact Shock Machines † Shock

Simulators (Programmers) † Limitations † Pneumatic Machines † High Impact Shock Machines † Specific Test Facilities

12.9 Generation of Shock Using Shakers 12-44Principle Behind the Generation of a Simple Shape Signal

versus Time † Main Advantages † Pre- and Postshocks † Limitations of Electrodynamic Shakers † The Use of Electrohydraulic Shakers12.10 Control by a Shock Response Spectrum 12-52Principle † 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 12-58Simulation Using Pyrotechnic Facilities † Simulation

Using Metal-to-Metal Impact † Simulation Using Electrodynamic Shakers † Simulation Using Conventional Shock Machines

12-1

Transported or on-board equipment is very frequently subjected to mechanical shocks in the course of its usefullifetime (in material handling, transportation, etc.) This kind of environment, although of extremely shortduration (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 mostwidely used in tests, the shock response spectrum (SRS) is presented with its numerous definitions and calculationmethods The main properties of the spectrum are described, showing that important characteristics of theoriginal signal can be drawn from it, such as its amplitude or the velocity change associated with the movementduring a shock

The SRS is the ideal tool for comparing the severity of several shocks and for drafting specifications Recentstandards require writing test specifications from real environment measurements associated with the life profile ofthe material (test tailoring) The process that makes it possible to transform a set of recorded shocks into aspecification of the same severity is detailed

Packages must protect the equipment contained within them from various forms of disturbance related tohandling and possible free fall drop and impact onto a floor A method to characterize the shock fragility of thepackaged product, using the “damage boundary curve” (DBC), and to choose the characteristics of the cushioningmaterial 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 sine, rectangle, sawtooth with a final peak) can occasionally be tested using an electrodynamic exciter The problemsencountered, 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

(half-Determining a simple shape shock of the same severity as a set of shocks on the basis of their responsespectrum is often a delicate operation Thanks to progress in computerization and control facilities, thisdifficulty can sometimes be overcome by expressing the specification in the form of a response spectrum and bycontrolling the exciter directly from that spectrum In practical terms, as the exciter can only be driven with asignal that is a function of time, the software of the control rack determines a time signal with the samespectrum as the specification displayed The principles of composition of the equivalent shock are described withthe shapes of the basic signals commonly used, while their properties and the problems that can beencountered, both in the generation of the signal and with respect to the quality of the simulation obtained, areemphasized

Pyrotechnic devices or equipment (cords, valves, etc.) are frequently used in satellite launchers due to thevery high degree of accuracy that they provide in operating sequences Shocks induced in structures by explosivecharges are extremely severe, with very specific characteristics It is shown that they cannot be correctlysimulated in the laboratory by conventional means and that their simulation requires specific tools

12.1 Definitions

12.1.1 Shock

Shock occurs when a force, a position, a velocity, or an acceleration is abruptly modified and creates atransient 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 systemand 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 aversed sine waveform), terminal peak sawtooth, and rectangular shock (approached by a trapezoidalwaveform)

12.1.3 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

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 ½1 2 cosð Þ ; with this

period starting from the zero value of this function

It is thus a signal ranging between two minima of a

sine wave

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, whereacceleration 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 givenvalue, remains constant throughout the signal, and decreases instantaneously to zero In practice, what iscarried 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, remainsconstant 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; andits form

The physical parameter expressed in terms of time is generally an acceleration, €xðtÞ, but can be also avelocity, 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 ¼ðt

15010050

12.3 Shock Response Spectrum

* 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 thatundergoes 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 permanentdeformation (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, theoperation of an electromechanical contactor), the fatigue damage accumulated in the structuralelements 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 anycase this is not always available, particularly at the stage of the development of the specification ofdimensioning One searches for a method of general nature, which leads to results that can beextrapolated 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 notclaim to be a model of the real structure It is composed of a support and of N linear one-degree-of-freedom (one-DoF) resonators, comprising each one are a mass, mi a spring of stiffness, ki and adamping device, ci; chosen such that the fraction of critical damping (damping ratio)j ¼ ci 2pffiffiffiffiffiffikimi isthe same for all N resonators A model for the shock response spectrum (SRS) is shown inFigure 12.2(also see Chapter 17)

When the support is subjected to the shock, each mass, mi; has a specific movement responseaccording to its natural frequency, f0i¼ ð1=2pÞ pffiffiffiffiffiffiki=mi and to the chosen damping ratio, j; while a

TABLE 12.1 Main Simple Shock Waveforms (Amplitude, x m ; Duration, t; Velocity Change, DV)

stress,si; is induced in the elastic element The analysis consists of seeking the largest stress,sm i; observed

at each frequency in each spring

For applications deviating from the assumptions of definition of the SRS (linearity, only one DoF), it isdesirable to observe a certain prudence if one wishes to estimate quantitatively the response of a systemstarting from the spectrum (Bort, 1989) The response spectra are more often used to compare theseverity of several shocks

It is known that the tension static diagram of many materials comprises a more-or-less linear arc onwhich the stress is proportional to the deformation In dynamics, this proportionality can be allowedwithin 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 twoshocks 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

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

md2z

dt2 þ cdz

dt þ kz ¼ FðtÞ ð12:3Þwhere 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

Fixed supportFIGURE 12.3 Linear one-Dof system subjected to a force (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.)

12.3.3.2 Shock Defined by an Acceleration

Let us set €xðtÞ as an acceleration applied to

the base of a linear one-DoF mechanical system,

with €yðtÞ the absolute acceleration response of

the mass, m; and zðtÞ the relative displacement

of the mass, m; with respect to the base

The absolute acceleration of the mass is given by

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 aforce applied directly to the mass, the response of the system can be characterized by the absoluteacceleration of the mass (which can be measured using an accelerometer fixed to this mass) The responsespectrum 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 thedisplacement of the base of the system This displacement is proportional to the stress created in the

spring (since the system is regarded as linear) In practice, one generally expresses in ordinates thequantity v2zsup; which is called the equivalent static acceleration (Biot, 1941) This product has thedimensions of acceleration, but does not represent the absolute acceleration of the mass, except whendamping is zero However, when damping is close to the current values observed in mechanics, and inparticular whenj ¼ 0:05; as a first approximation one can assimilate v2zsupto the absolute acceleration

€ysupof the mass, m (Lalanne, 1975, 2002b)

The quantity v2zsup is termed pseudo-acceleration In the same way, one terms the productv0zsuppseudo-velocity The spectrum givingv2zsupvs the natural frequency is named the relative displacementshock spectrum

In each of these two important categories, the response spectrum can be defined in various waysaccording to how the largest response at a given frequency is characterized

12.3.4.4 Primary Positive Shock Response Spectrum or Initial Positive Shock ResponseSpectrum

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 ornegative

Example

An example giving standardized primary and residual relative displacement SRS curves for a half-sinepulse 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

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

2.01.51.00.50.0

Residual positive spectrum

Primary negative spectrumResidual negative spectrum x = 0.05

FIGURE 12.5 Standardized primary and residual relative displacement SRS of a half-sine pulse (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.)

12.3.4.8 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 theparameter easiest to compare with a characteristic value (as in a study of the effects of a shock on a man, acomparison with the specification of an electronics component, etc.)

In practice, it is very often the stress (and thus the relative displacement) which seems the mostinteresting parameter The spectrum is primarily used to study the behavior of a structure, to comparethe severity of several shocks (the stress created is a good indicator), to write test specifications (as it isalso a comparison between the real environment and the test environment), or to dimension asuspension (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 ofthe spectrum at the frequency considered, and it is of little importance for the system whether thismaximum, zm; takes place during or after the shock The most interesting spectra are thus the positiveand 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, ifdissymmetrical, 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 relation between the various types of SRS that have been discussed here is shown in Figure 12.6

The Shock Response Spectrum is a curve representative of the variations of the largest response of alinear 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,v2zsup (relative displacement shockspectrum) or by the absolute acceleration of the mass (absolute acceleration SRS) For the usualvalues of Q; the spectra are very close

The most interesting spectra are the positive and negative spectra, which are most frequentlyused in practice, with the maximax spectrum

Primary (initial)positive SRSPrimary (initial)negative SRS

Secondary (residual)negative SRS

Secondary (residual)positive SRS

12.3.5 Standardized Shock Response Spectrum

12.3.5.1 Definition

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 f0t (instead of f0) orv0t and

on the ordinate the spectrum/shock pulse

ampli-tude ratio,v2zm=€xm; which, in practice, amounts to

tracing the spectrum of a shock of duration equal to

1 sec and amplitude 1 m/sec2 This is shown in

Figure 12.7

These standardized spectra can be used for two

purposes:

* 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 givenspectrum (resulting from measurements from the real environment)

12.3.5.2 Standardized Shock Response Spectra of Simple Shocks

Figure 12.8 toFigure 12.15give the reduced SRSs for various pulse forms, with unit amplitude and unitduration, for several values of damping To obtain the spectrum of a particular shock of arbitraryamplitude, €xm; 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 €xm:

* For the abscissae (x-axis values), replace each valuefð¼ f0tÞ by f0¼f=t:

We will see later on how these spectra can be used for the calculation of test specifications

1.41.21.00.80.60.40.20.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Frequency (Hz)

2.01.51.00.50.0

0.5 0.25 0.1 0.05 0.025 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.)

12.3.5.2.1 Half-Sine Pulse

Figure 12.8and Figure 12.9 show the standardized SRS curves in this case

12.3.5.2.2 Versed Sine Pulse

Figure 12.10 shows the standardized SRS curves in this case

12.3.5.2.3 Terminal Peak Sawtooth Pulse

Figure 12.11andFigure 12.12show the standardized SRS curves for terminal peak sawtooth (TPS) pulse.12.3.5.2.4 Rectangular Pulse

Figure 12.13gives the standardized SRS curves for a rectangular pulse shock

FIGURE 12.9 Standardized positive and negative absolute acceleration SRS of a half-sine pulse (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Frequency (Hz)

2.01.51.00.50.0

0.025

0.5 0.25 0.1 0.05 0.025 0

FIGURE 12.10 Standardized positive and negative relative displacement SRS of a versed sine pulse (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.)

12.3.6 Choice of Damping

The choice of damping should be carried out according to the structure subjected to the shock When this

is not known, or studies are being carried out with a view to comparison with other already calculatedspectra, the outcome is that one plots the shock response spectra with a damping ratio equal to 0.05 (i.e.,

Q ¼ 10; see Chapter 19).It is an approximately average value for the majority of structures Unlessotherwise specified, as noted on the curve, it is the value chosen conventionally With the spectra varyingrelatively little with damping, this choice is often not very important To limit possible errors, the selectedvalue should, however, be systematically noted on the diagram

12.3.7 Shock Response Spectra Domains

Three domains can be schematically distinguished for shock spectra

1 An impulse domain at low frequencies, in which the amplitude of the spectrum (and thus of theresponse) is lower than the amplitude of the shock: The system reduces the effects of the shock It is thus

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Frequency (Hz)

1.51.00.50.0

0.5 0.25 0.1 0.05 0.025 0

FIGURE 12.12 Standardized positive and negative relative displacement SRS of a TPS pulse with nonzero decay time (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Frequency (Hz)

1.51.00.50.0

Negative spectra

0.50.250.1 0.05

0 0.025

0.5 0.25 0.1 0.05 0.025 0

FIGURE 12.11 Standardized positive and negative relative displacement SRS of a TPS pulse (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.)

in this impulse region that it would be advisable to choose the natural frequency of an isolation system tothe shock, from which we can deduce the stiffness envisaged of the insulating material:

k ¼ mv2¼ 4p2f2m ð12:10Þwith m being the mass of the material to be protected

The shock here is of very short duration with respect to the natural period of the system In thisimpulse region ð0 # f0t # 0:2Þ:

* The form of the shock has little influence on the amplitude of the spectrum Only (for a givendamping value) the velocity change DV associated with the shock, equal to the algebraic surfaceunder the curve €xðtÞ is important

* The slope p at the origin of the spectrum plotted for zero damping in linear scales is proportional

to the velocity change DV corresponding to the shock pulse (Lalanne, 2002b):

p ¼ dðv2zsupÞ

This relation is approximate if damping is small

2.52.01.51.00.50.0

Negative spectra

Positive spectra

0.5 0.1 0.25

0.05 0.025 0

0.25 0.5

0.1 0.05 0.025 0

Frequency (Hz)

Trapezoid (1 m/s2−1 s−tr= 0.1 s−td= 0.1 s)2.5

2.01.51.00.50.0

0.5 0.25 0.1

0.05 0.025 0

FIGURE 12.14 Standardized positive and negative relative displacement SRS of a trapezoidal pulse (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.)

* The positive and negative spectra are in general the residual spectra (it is sometimes necessary thatthe frequency of spectrum is very small, and there can be exceptions for certain long shocks inparticular) They are nearly symmetrical so long as damping is small.

2 A static domain in the range of the high frequencies, where the positive spectrum tends towardsthe amplitude of the shock whatever the damping: All occurs here as if the excitation were a staticacceleration (or a very slowly varying acceleration), as the natural period of the system is smallcompared with the duration of the shock This does not apply to rectangular shocks or to the shocks withzero rise time Real shocks having necessarily a rise time different from zero, this restriction remainstheoretical

3 An intermediate domain in which there is dynamic amplification of the effects of the shock, thenatural period of the system being close to the duration of the shock: This amplification, which is more

or less significant depending on the shape of the shock and the damping of the system, does not exceed1.77 for shocks of traditional simple shape (half-sine, versed sine, TPS) Much larger values are reached inthe case of oscillatory shocks, made up, for example, by a few periods of a sinusoid

Various domains of an SRS are illustrated inFigure 12.16

2:042p < 0:325 m=sec

a value to be compared with the surface under the half-sine shock pulse(Table 12.1):

DV ¼ p2€xmt ¼

2

p £ 50 £ 11 £ 1023< 0:35 m=sec

12.3.8 Algorithms for Calculation of the Shock Response Spectra

Various algorithms have been developed to solve the second-order differential equation (Equation 12.7;O’Hara, 1962; Gaberson, 1980; Smallwood, 1981; Cox, 1983; Hughes and Belytschko, 1983; Irvine, 1986;

2.01.51.00.50.0

Dokainish and Subbaraj, 1989; Colvin and Morris, 1990; Hale and Adhami, 1991; Mercer and Lincoln,1991; Seipel, 1991; Merritt, 1993; Grivelet, 1996) Very reliable results are obtained in particular with those

of Cox (1983) and Smallwood (1986, 2002)

12.3.9 Choice of the Digitization

Frequency of the Signal

The SRS is obtained by considering the largest

peak of the response of a one-DoF system This

response is in general calculated by the algorithms

with the same temporal step as that of the shock

signal

First of all, the digitization (sampling)

fre-quency must be sufficient to correctly represent the

signal itself and in particular not to truncate its

peaks Two cases are shown in Figure 12.17 and

Figure 12.18

When the natural frequency of the one-DoF

system is lower than the smallest shock

fre-quency, the detection of peaks of the response

can be carried out accurately even if the signal

digitization (sampling) frequency is insufficient

for correctly describing the shock (Figure 12.17)

The error on the SRS is then only related to the

poor digitization (sampling) of shock and results

in an inaccuracy on the velocity change

associ-ated with the shock, i.e., on the SRS slope at low

frequency

Even if the sampling frequency allows a good

representation of the shock, it can be insufficient

for the response when the natural frequency of the

system is higher than the maximum frequency of

Shock pulse

Response

FIGURE 12.17 Sampling frequency sufficient for the response and too low for the shock pulse (error on the slope of SRS at low frequency).

90 80 70 60 50 40 30 20 10 0

15 50 100 150 200 250Hz 300 350 400 450 500

IntermediateDomain

Impulse Domain

FIGURE 12.18 Sampling frequency sufficient for the shock pulse and too low for the response (error on the SRS at high frequency).

the signal(Figure 12.18).The error is here related

to the detection of the largest peak of the response,

which occurs throughout shock (primary

spec-trum)

Figure 12.19 shows the error made in the more

stringent case when the points surrounding the

peak are symmetrical with respect to the peak

If we set

SF¼ Sample frequency

SRS maximum frequency

it can be shown that, in this case, the error made

according to the sampling factor, SF; is equal to

(Sinn and Bosin, 1981; Wise, 1983)

eS¼ 100 1 2 cos p

SF ð12:12Þ

The sampling frequency must be higher than 16

times the maximum frequency of the spectrum so

that the error made at high frequency is lower than

2% (23 times the maximum frequency for an error

lower than 1%) The rule of thumb often used to

specify a sampling factor equal to ten can lead to an

error of about 5% Percentage error as a function of

the sampling factor is plotted in Figure 12.20 Also

see Table 12.2

Algorithms use generally the same sampling

frequency for the shock input and the response

of the one-DoF system This choice led to define

the sampling frequency according to the highest

SRS frequency In order to decrease the

comput-ing time, it could be interestcomput-ing to determine a

sampling frequency varying with each natural

frequency (Smallwood, 2002)

Sampled dataError

Response(sinusoid)

FIGURE 12.19 Error made in measuring the tude of the peak (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.)

ampli-5 0 1 2 3 4 5

ampli-Note: The sampling frequency must be higher than 16 times the maximum frequency of thespectrum so that the error made at high frequency is lower than 2%

TABLE 12.2 Some Sampling Factors with sponding Error on the SRS

Corre-SRS Maximum Frequency Multiplied by Error (%)

12.3.10 Use of Shock Response Spectra for the Study of Systems with Several Degrees of Freedom

By definition, the response spectrum gives the largest value of the response of a linear single-DoF systemsubjected to a shock If the real structure is comparable to such a system, the SRS can be used to evaluatethis response directly This approximation is often possible, with the displacement response being mainlydue to the first mode In general, however, the structure comprises several modes, which aresimultaneously excited by the shock The response of the structure consists of the algebraic sum of theresponses of each excited mode

The maximum response of each one of these modes can be read on the SRS, but the following apply

* One does not have any information concerning the moment of occurrence of these maxima Thephase relationships between the various modes are not preserved and the exact way in which themodes are combined cannot be known simply

* The SRS is plotted for a given constant damping over all the frequency range, whereas thisdamping varies from one mode to another in the structure

It thus appears difficult to use an SRS to evaluate the response of a system presenting more than onemode However, it happens that this is the only possible means The problem is to know how to combinethese “elementary” responses so as to obtain the total response and to determine, if need be, any suitableparticipation factors dependent on the distribution of the masses of the structure, of the shapes of themodes, etc

When there are several modes, several proposals have been made to limit the value of the total response

of the mass j of the one of the DoF starting from the values read on the SRS, as follows

* Add the values with the maxima of the responses of each mode, without regard to the phase(Benioff, 1934)

* Sum the absolute values of the maximum modal responses (Biot, 1932) As it is not probable thatthe values of the maximum responses take place all at the same moment with the same sign, thereal maximum response is lower than the sum of the absolute values This method gives an upperlimit of the response and thus has a practical advantage: the errors are always on the side of safety.However, it sometimes leads to excessive safety factors (Shell, 1966)

* Perform an algebraic sum of the maximum responses of the individual modes A study showedthat, in the majority of the practical problems, the distribution of the modal frequencies andthe shape of the excitation are such that the possible error remains probably lower than 10%(Rubin, 1958; Fung and Barton, 1958)

* Add to the response of the first mode a fixed percentage of the responses of the other modes, orincrease in the response of the first mode by a constant percentage (Clough, 1955)

* Combine the responses of the modes by taking the square root of the sum of the squares to obtain

an estimate of the most probable value (Merchant and Hudson, 1962) This criterion gives values

of the total response lower than the sum of the absolute values and provides a more realisticevaluation of the average conditions (Ostrem and Rumerman, 1965; Ridler and Blader, 1969)

* Average the sum of the absolute values and the square root of the sum of the squares (Jennings,1958) One can also choose to define positive and negative limiting values starting from asystem of weighted averages For example, the relative displacement response of the mass j isestimated by

max

t$0lzjðtÞl ¼

ffiffiffiffiffiffiffiffiffiffiffi

Xn i¼1

12.4 Pyroshocks

The aerospace industry uses many pyrotechnic

devices such as explosive bolts, squib valves, jet

cord, and pin pushers During their operation,

these devices generate shocks which are

character-ized by very strong acceleration levels at very high

frequencies that can be sometimes dangerous for

the structures, but especially for the electric and

electronic components involved An example of a

pyroshock is given in Figure 12.21

Pyroshock intensity is often classified

accord-ing to the distance from the point of detonation of

the device Agreement on classifying

intensity according to this criterion is not

unanimous Two fields are generally considered

(Table 12.3):

* The near-field, close to the source (material

within about 15 cm of point of detonation

of the device, or about 7.5 cm for less

intense pyrotechnic devices), in which the

effects of the shocks are primarily related to

the propagation of a stress wave in the material

* The far-field (material beyond about 15 cm for intense pyrotechnic devices, or beyond 7.5 cm forless intense devices) in which the shock is then propagated whilst attenuating in the structure andfrom which the effects of this wave combine with a damped oscillatory response of the structure atits frequencies of resonance (or the structural response only)

Three fields are sometimes suggested: the near-field, the mid-field (same definition as the far-fieldabove, between 15 and 60 cm, or 7.5 and 15 cm for the less intense shocks), and the far-field, where onlythe structural response effect persists

An investigation by Moening (1986) showed that the causes of observed failures on the Americanlaunchers between 1960 and 1986 (63 due to pyroshocks) are mainly the difficulty in evaluating theseshocks a priori, especially the lack of consideration of these excitations during design and the absence ofrigorous test specifications

Such shocks have the following general characteristics

* The levels of acceleration are very important; the shock amplitude is not simply related to thequantity of explosive used (Hughes, 1983) The quantity of metal cut by a jet cord is, for example,

a more significant factor than the mass of the explosive

FIGURE 12.21 Example of a pyroshock (Source: Lalanne, Chocs Mecaniques, Hermes Science Publi- cations With permission.)

TABLE 12.3 Characteristics of Each Pyroshock Intensity Domain

the Source (cm) Intense PyrotechnicDevices (cm) Frequency ContentShock AmplitudeNear field (stress wave

Far-field (stress wave propagation effect

Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.

* The signals assume an oscillatory shape.

* The shocks have very close components according to three axes; their positive and negativeresponse spectra are curves that are roughly symmetrical with respect to the axis ofthe frequencies They begin at zero frequency with a very small slope at the origin, growwith the frequency until a maximum located at some kHz, even a few tens of kHz, isreached, and then tend according to the rule towards the amplitude of the temporal signal.Due to their contents at high frequencies, such shocks can damage electric or electroniccomponents

* The a priori estimate of the shock levels is neither easy nor precise

These characteristics make pyroshocks difficult to measure, requiring sensors that are able to acceptamplitudes of 100,000g, frequencies being able to exceed 100 kHz, with important transversecomponents They are also difficult to simulate

The dispersions observed in the response spectra of shocks measured under comparable conditions areoften important, 3 dB with more than 8 dB compared with the average value, according to the authors(Smith, 1984, 1986) The reasons for this dispersion are in general related to inadequate instrumentationand the conditions of measurement (Smith, 1986):

* The fixing of the sensors on the structure using insulated studs or wedge which act like mechanicalfilters

* Zero shift, due to the fact that high accelerations make the crystal of the accelerometer work in atemporarily nonlinear field (this shift can affect the calculation of the SRS)

* Saturation of the amplifiers

* Resonance of the sensors

With correct instrumentation, the results of measurements carried out under the same conditions areactually very close The spectrum does not vary with the tolerances of manufacture and the assemblytolerances

12.5 Use of Shock Response Spectra

12.5.1 Severity Comparison of Several Shocks

A shock, A, is regarded as more severe than a shock, B, if it induces in each resonator a larger stress.One then carries out an extrapolation, which is certainly open to criticism, by supposing that, ifshock A is more severe than shock B when it is applied to all the standard resonators, it isalso more severe with respect to an arbitrary real structure (which cannot be linear nor have asingle DoF)

12.5.2 Test Specification Development from Real Environment Data

frequency band of analysis will have to envelop the principal resonant frequencies of the structure(known or foreseeable frequencies).

* If the number of measurements is sufficient, calculate the mean spectrum, m (mean of thepoints at each frequency) as well as the standard deviation spectrum (s), then the meanspectrum þa standard deviations, according to the frequency; if it is insufficient, make theenvelope of the spectra

The value of a can be either arbitrary (for example 2.5 or 3) or the result of a statisticalcalculation It is often considered that the SRS amplitudes obey to a log-normal distribution If yiisthe logarithm of the SRS amplitude, yj¼ log10SRSj; the real environment envelope (for a givenprobability P0 of not exceeding at the confidence level p0) can be defined by

SRSEnv ¼ 10m y þ a s y ð12:14Þwhere my and sy are, respectively, the mean and the standard deviation of the yj values:

The number of standard deviations, a; is given in Table 12.4 for different values of ri; P0; andp0:SRSEnv can also be defined as the upper one-sided normal tolerance interval for which 100 P0% ofthe values will lie below the limit with 100p0% confidence

* Apply the mean spectrum or the mean spectrum þa standard deviations a statistical uncertaintycoefficient (Lalanne, 2002d), calculated for a probability of tolerated maximum failure (takinginto account the uncertainties related to the dispersion of the real environment and of themechanical strength), or contractual (if one uses the envelope)

Event # 1Handling shock

r1measured dataEvent # 2Loading shock

r2measured dataEvent # pIgnition shock

or envelopeMean and standarddeviation spectra

Each event thus being synthesized in only one spectrum, one proceeds to an envelope of all the spectraobtained to deduce from it an SRS covering the totality of the shocks of the life profile Aftermultiplication by a test factor, which takes account of the number of tests performed to demonstrate theresistance of the equipment (Lalanne, 2002d), this spectrum will be used as reference “real environment”for the determination of the specification.

The reference spectrum can consist of the positive and negative spectra or the envelope of theirabsolute value (maximax spectrum) In this last case, the specification will have to be applied according

to the two corresponding half-axes of the test item

12.5.2.2 Nature of the Specification

There is an infinity of shocks having a given response spectrum This property is related to the very greatloss of information in computing the SRS, since one retains only the largest value of the responseaccording to the time to constitute the SRS at each natural frequency

According to the characteristics of the spectrum and available means, the specification can beexpressed in the forms given below

* It can be a simple shape signal according to the time realizable on the usual shock machines(half-sine, TPS, rectangular pulse)

One can thus try to find a shock of simple form, in which the spectrum is closed to the referencespectrum, characterized by its form, its amplitude, and its duration It is in general desirable thatthe positive and negative spectra of the specification, respectively, cover the positive and negativespectra of the field environment If this condition cannot be obtained by application of only one shock(owing to the particular shape of the spectra, and the limitations of the facilities), the specification will bemade up of two shocks, one on each half-axis The envelope must be approaching the reference SRS aswell as possible, if possible on all the spectrum in the frequency band retained for the analysis, if not in afrequency band surrounding the resonant frequencies of the test item (if they are known)

* It can be a SRS In this case, the specification is directly the reference SRS

12.5.2.3 Choice of Shape

The choice of the shape of a shock is carried out by a comparison of the shapes of the positive andnegative spectra of the real environment with those of the spectra of the usual shocks of simple shape(half-sine, TPS, rectangle;Figure 12.23)

TABLE 12.4 Number of Standard Deviations Corresponding to a Given Probability of not Exceeding, P 0 ; at the Confidence Level, p 0

12.5.2.4 Amplitude

The amplitude of a shock is obtained by

plotting the horizontal straight line that closely

envelops the positive reference SRS at high

frequency (Figure 12.24) This line cuts the

y-axis at a point which gives the amplitude

sought (here, one uses the property of the

spectrum at high frequencies, which tends in

this zone towards the amplitude of the signal in

the time domain)

12.5.2.5 Duration

The shock duration is given by the coincidence

of a particular point of the reference spectrum

(Figure 12.24) and the reduced spectrum of the

simple shock selected above

One in general considers the abscissa, f01, of

the first point which reaches the value of

the asymptote at the high frequencies (amplitude

of shock) as shown in Figure 12.25 Table 12.5

joins together some values of this abscissa for the

most usual simple shocks according to the Q

factor (Lalanne, 2002b) Another possibility is to

use the coordinates of the first (higher) peak of the SRS, as given inTable 12.6

Notes:

1 If the calculated duration must be rounded (in milliseconds), the higher value should always beconsidered, so that the spectrum of the specified shock remains always higher or equal to thereference spectrum

Envelope at the high frequencies

Real environment

x = 0.05

400300

200Frequency (Hz)

10049.50

340400500

FIGURE 12.24 Determination of the amplitude and duration of the specification (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.)

2 It is in general difficult to carry out shocks of

duration lower than 2 msec on standard

shock machines (except for very light

equipment)

One will validate the specification by checking

that the positive and negative spectra of the

shock thus determined will envelop the respective

reference spectra and one will verify, if the

resonant frequencies of the test item are known,

that one does not overtest exaggeratedly at these

frequencies

Example

As an example, let us consider the positive and

negative spectra characterizing the real

environ-ment plotted inFigure 12.24, which is a result of a

true synthesis It is noted that the negative

spectrum preserves a significant level throughout

the entire frequency domain (the beginning of the

spectrum being excluded)

The most suitable simple shock shape is the TPS The shock amplitude, whatever its form, is equal to 340 m/sec2 The abscissa, f01, of the first point that reaches the value of theasymptote is equal to 0.415 From this value, f0¼ 49:5 Hz; the duration is given by t ¼0:415=49:5 ¼ 0:0084 sec:

wave-The duration of the shock will thus be (rounding up) t ¼ 9 msec; which slightly moves thespectrum towards the left and makes it possible to better cover the low frequencies Figure 12.26

shows the spectra of the environment and those of the TPS pulse thus determined

The main steps of deriving a shock test specification from the SRS of a real environment are outlined

inTable 12.7

T.P.S dimensionless S.R.S

1.51.00.50.0

−0.5

−1.0

−1.50.00.415

TABLE 12.5 Values of the Dimensionless Frequency Corresponding to the First

Passage of the SRS by the Amplitude Unit

12.5.2.6 Difficulties

The response spectra of shocks measured in the real environment often have a complicated shapewhich is impossible to envelop by the spectrum of a shock of simple shape realizable with the usualtest facilities of the drop table type This problem arises in particular when the spectrum presents animportant peak (Smallwood and Witte, 1972) The spectrum of a shock of simple shape will be (seeFigure 12.27): either an envelope of the peak, which will lead to significant overtesting comparedwith the other frequencies, or envelope of the spectrum except the peak, consequently leading toundertesting at the frequencies close to the peak, if one knows that the material does not have anyresonance in the frequency band around the peak The simulation of shocks of pyrotechnic originleads to this kind of situation

TABLE 12.6 Values of the Dimensionless Frequency and Amplitude Corresponding to the First Peak of the SRS

Shock pulses of simple shape (half-sine, TPS) have, in logarithmic scales, a slope of 6 dB/octave(i.e., 458) at low frequencies incompatible with those larger ones, of spectra of pyrotechnic shocks(.9 dB/octave) When the levels of acceleration do not exceed the possibilities of the shakers, simulationwith control using spectra is of interest (Section 12.10).

Note: In general, it is not advisable to choose a simple shock shape as a specification when the realshock is oscillatory in nature In addition to overtesting at low frequencies (the oscillatory shock is withvery small velocity change), the amplitude of the simple shock thus calculated is more sensitive to thevalue of the Q factor in the intermediate frequency range

12.6 Standards

12.6.1 Types of Standards

There are two types of standards: (1) those which specify arbitrary shock pulses (IEC, ISO, MIL STD

810 C, …), and (2) those which require test tailoring (GAM EG 13, 1986; DEF STAN, 1999; MIL STD

810 F, 2000; NATO, 2000)

For the first case, the most frequently specified shock shapes are the half-sine, the TPS, and therectangular (or trapezoidal) waveforms In these standards, a table proposes several values of levels anddurations, with preferred combinations (for example, 30g, 18 msec or 50g, 11 msec)

To take account of the limitations of test facilities and unavoidable signal distortions, the shock carriedout is regarded as acceptable if the time acceleration signal lies between two tolerance limits Two shocksincluded within these limits can, however, have very different effects (which can be evaluated with theSRS; Lalanne, 2002b)

2 211

Shock waveform Comparing the shape of the SRS of the real environment

(reference) to that of the SRS of the simple shape shocks (half-sine, TPS or rectangular waveform)

Shock amplitude The SRS amplitude at high frequency Shock duration Writing that the abscissa of the first point which reaches

the value of the asymptote at high frequencies (amplitude of shock) is the same for the reduced SRS

of the chosen simple shock and the reference SRS

Some examples of standards are given in

Figure 12.28 and Figure 12.29

In the second case, the test specification is

preferably written from real measurements

corre-sponding to the life profile of the material The

data to be used can be any of the following, in the

preferential order:

* Functional real environment time history

measurements of the material

* Data measured under similar conditions

and estimated to be representative

* Data issued from prediction or calculation

* Default values (fallback levels), obviously

more arbitrary in character, to be used if

measured data not available (classical pulse

shock or SRS)

Derivation of the test specification and

subsequent test should be carried out, in the

preferential order as follows:

* For measured data of the same event, if the

measured pulse shapes are very similar, use

direct reproduction of the measured data

under shaker waveform control (if

poss-ible) If the measured shock shapes are very

different, use the following method

* For measured data of different shock events

use a synthesis of measurements using

SRS (see Section 12.5.2) Test on shaker

with SRS control if possible If not possible,

test on shock machine with a classical

pulse having the same SRS

* If there is no measured data of the real shock, but measured data under similar conditions, use themethod as above

* If there are no measured data, fallback levels and provisional values are to be replaced by results ofmeasurement as soon as possible

The transformation shock spectrum-signal has an infinite number of solutions, and very differentsignals can have identical response spectra Standards often require specifying in addition to thespectrum other complementary data such as the duration of the signal time, the velocity change duringthe shock or the number of cycles (less easy), in order to deal with the spectrum and the coupleamplitude/duration of the signal at the same time (see Section 12.10.4)

It is not correct to decompose a SRS into two separate domains in order to be able to meet a shockrequirement (a low frequency component and a high frequency component) If the specimen has nosignificant low natural frequency, it is permissible to allow the low frequency domain of the SRS to fallout of tolerance in order to satisfy the high frequency part of the requirement

The tolerance on the SRS amplitude should be, for example (MIL STD 810 F), 21.5 dB, þ3 dB over thespecified frequency range; a tolerance of þ3 dB, þ6 dB being permissible over a limited frequency range

It is generally required to determine the positive and negative spectra (absolute acceleration or relativedisplacement) at Q ¼ 10; at at least 1/12-octave frequency intervals

Integration time

Nominal pulse

Limits of tolerances

0.8 A A

0.4 D

2.4 D = T1

0.1 D D D

6 D = T2

FIGURE 12.28 Half-sine pulse (NATO Stanag 4370, AECTP 403) T 1 : minimum time during which the pulse shall be monitored for shocks produced using a conventional shock-testing machine; T 2 : minimum time during which the pulse shall be monitored for shocks produced using a vibration generator.

Ideal Sawtooth Pulse Tolerance Limits 0.15 A

0.07 D 0.02 A

1.15 A

A 0.05 A

0.05 A

FIGURE 12.29 TPS pulse (MIL STD 810 F) D: duration of nominal pulse; A: peak acceleration of nominal pulse.

In the absence of accurate information on the number of shocks which the material will undergo in itsservice life, a minimum is often required of three shocks in both directions along each of the threeorthogonal axes, a total of 18 shocks.

12.6.2 Installation Conditions of Test Item

* The test item should be mechanically fastened to the shock machine, directly by its normal means

of attachment or by means of a fixture

* The mounting configuration should enable the test item to be subjected to shocks along thevarious axes and directions as specified

* External connections necessary for measuring purposes should add minimum restraint and mass

* The fixture should not modify the dynamic behavior of the test item

* Material intended for use with isolators should be tested with its isolators

* The direction of gravity or any loading factors (mechanisms, shock isolators, etc.) must be takeninto account by compensation or by suitable simulation

12.6.3 Uncertainty Factor

An uncertainty factor may be added to the resulting envelope if confidence in the data is low or in order

to take account of the dispersion of levels in the real environment when the data set is small This factorcan be arbitrary, of the order of 3 to 6 dB, for example, or determined from a reliability computation,taking account of the statistical distributions of the real environment and of the material strength(Lalanne, 2002d)

It is important that all uncertainties be clearly defined and that uncertainties are not superimposedupon estimates that already account for uncertainties

Note: The purpose of the test is to demonstrate that the equipment has at least the specified strength atthe time of its design However, for obvious reasons of cost, this demonstration is generally conductedonly on one specimen To take into account the variability of the strength of the material, it is possible toincrease the test severity by applying a “test factor.” This second factor depends on the number of tests to

be conducted and on the coefficient of variation of the material strength (Lalanne, 2002d)

12.6.4 Bump Test

Abump test is a test inwhich a simple shock is repeated many times (DEF STAN, 1999; IEC, 1987b; AFNOR,1993) Standardized severities are proposed For example, half-sine, 10 g, 16 ms, 3000 bumps (shock) peraxis, 3 bumps a second

The purpose of this test is not to simulate any specific service condition It is simply considered that itcould be useful as a general ruggedness test to provide some confidence in the suitability of equipment fortransportation in wheeled vehicles It is intended to produce in the specimen effects similar of thoseresulting from repetitive shocks likely those encountered during transportation

In this test, the equipment is always fastened (with its isolators if it is normally used with isolators) tothe bump machine during conditioning

12.7 Damage Boundary Curve

12.7.1 Definition

Products are placed in a package to be protected from possible free-fall drops and impacts onto a floor or

a shipping platform during transport or handling This packaging is often made up of a cushioningmaterial (for example, honeycomb or foam) which absorbs the impact energy (related to the impactvelocity) either by inelastic deformation, and which generates a shock at the entry of the material, whose

shape is often comparable to a rectangular or a trapezoid pulse (Figure 12.30) Alternatively, it can bemade of an elastic material, which produces at the material entry a shock with a near half-sine waveform.After determination of the shock environment, a statistical analysis allows one to specify the designdrop height, with a given percentage of loss tolerated.

To choose the characteristics of the cushioning material constituting the package, it is first of allnecessary to determine the shock fragility of the product that would be subjected to a shock with one ofthese two forms

It can be considered that the severity of a shock is related to its amplitude and to its associated velocitychange (we saw that these two parameters intervene in the SRS) We thus determine the largestacceleration and the largest velocity change that the unpackaged product subjected to these shocks cansupport

At the time of two series of tests carried out on a shock machine, we note, for a given acceleration, thecritical velocity change or, for a given velocity change, the critical maximum shock acceleration that leads to

a damage on the material (deformation, fracture, faulty operation after the shock, etc.)

Results are expressed on a diagram of the acceleration–velocity change by a curve defined as thedamage boundary curve (DBC; ASTM D3332), as shown in Figure 12.31

Variable velocity change tests begin with a short-duration shock, then the duration is increased (bypreserving constant acceleration) until the appearance of damage (functional or physical) The criticalvelocity change is equal to the velocity change just lower than that producing damage (ASTM, 1994).The variable acceleration tests are performed on

a new material, starting with a small acceleration

level and with a rather large velocity change

(at least 1.5 times the critical velocity change

previously determined)

The tests should be carried out in the more

penalizing impact configuration (unit orientation)

12.7.2 Analysis of Test Results

Damage can occur if the acceleration and the

velocity change are together higher than the

critical acceleration and the critical velocity

change

From the critical velocity change, the critical

drop height can be calculated If Viis the impact

velocity, VR is the rebound velocity, and a is the

Input Shock

CushionMaterial

PackageProduct

rate of rebound ðVR¼ 2aViÞ; the velocity change

If this critical height is lower than the design

height defined from the real use conditions of the

product, it is necessary to use a package with a

medium cushioning and then to define its

characteristics (crush stress, thickness) so that maximum acceleration at the time of impact is lowerthan the critical acceleration If not, no protection is necessary

Tests are in general carried out with a rectangular shock waveform, for two reasons

* As the rectangular shock is most severe (see SRSs), the result is conservative, as seen in Figure 12.32

* The DBC is made up only of two lines, which makes it possible to determine the curve from only twoset of tests (saving time) by destroying only two specimens A much more significant number of sets

of tests would be necessary to determine the curve from a half-sine shock waveform

Note: If, for cost reasons, the same product is used to determine the critical velocity change or thecritical acceleration, it undergoes several shocks before failure The test result is usable only if the productfails in a brittle mode If the material is ductile, each shock damages the product by an effect of fatigue,which should be taken into account (Burgess, 1996, 2000)

12.8 Shock Machines

12.8.1 Main Types

A shock machine, whatever its standard, is primarily a device allowing modification over a short timeperiod of the velocity of the material to be tested(also, see Chapter 15) Two principal categories areusually distinguished (Lalanne, 2002b):

* The first category is that of impulse machines, which increase the velocity of the test item duringthe shock The initial velocity is in general zero The air gun, which creates the shock during thesetting of the velocity in the tube, is an example

* The second category is that of impact machines, which decrease the velocity of the test itemthroughout the shock and/or which change its direction

The test facilities now used are classified as follows:

* In free fall machines, the impact is made on a shock simulator (in American literature thesedevices are termed “shock programmers”) adapted to the shape of the specified shock (elastomerdiscs, conical or cylindrical lead pellets, pneumatic shock simulators, etc.) To increase the impactvelocity, which is limited by the drop height, that is, by the height of the guide columns, the fallcan be accelerated by the use of bungee cords

* In pneumatic machines, the velocity is derived from a pneumatic actuator

* In electrodynamic exciters, the shock is specified either by the shape of a temporal signal, itsamplitude and its duration, or by a SRS

Rectangle shock pulseHalf-sine shock pulse

Velocity change

FIGURE 12.32 DBC comparison of half-sine and rectangular shock pulses.

* Exotic machines are designed to carry out shocks that are nonrealizable by the preceding methods,generally because their amplitude and duration characteristics are not compatible with theperformances from these means The desired shapes, not being normal, are not possible with theshock simulators delivered by the manufacturers.

We will try to show in the following sections how mechanical shocks could be simulated on materials

in the laboratory The facilities described are the most current, but the list is far from being exhaustive.Many other processes were or are still used to satisfy particular needs (Nelson and Prasthofer, 1974;Powers, 1974, 1976; Conway et al., 1976)

12.8.2 Impact Shock Machines

Most machines with free or accelerated drop

testing belong to the category of impact shock

machines The machine itself allows the setting of

velocity of the test item

The shock is carried out by impact, with the

help of the shock simulator (programmer), which

formats the acceleration of braking according to

the desired shape The impact can be without

rebound when the velocity is zero at the end of the

shock, or with rebound when the velocity changes

sign during the movement Laboratory machines

of this type consist of two vertical guide rods on

which the table carrying the test item slides

(Figure 12.33)

The impact velocity is obtained by gravity,

after the dropping of the table from a certain

height or using bungee cords allowing one to

obtain a larger impact velocity

Let us consider a free fall shock machine for

which the friction of the shock table on the

guidance system can be neglected The necessary

drop height, H; to obtain the desired impact

velocity, vi, is given by

H ¼ v2i

2g ð12:19Þwhere g is the acceleration of gravity

(9.81 m/sec2)

These machines are limited by the possible drop height, that is, by the height of the columns and theheight of the test item when the machine is provided with a gantry It is difficult to increase the height ofthe machine due to overcrowding and problems with guiding the table

However, the impact velocity can be increased using a force complementary to gravity by means ofbungee cords tended before the test and exerting a force generally directed downwards The accelerationproduced by the cords is in general much higher than gravity, which then becomes negligible This ideawas used to design horizontal (Lonborg, 1963) and vertical machines (Marshall et al., 1965; La VerneRoot and Bohs, 1969), this last configuration being less cumbersome

During impact, the velocity of the table changes quickly and forces of great amplitude appear betweenthe table and machine bases To generate a shock of a given shape, it is necessary to control the amplitude

of the force throughout the stroke during its velocity change This is carried out using a shock simulator(programmer)

FIGURE 12.33 Elements of a shock-test machine (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.)

12.8.2.1 Universal Shock-Test Machines

12.8.2.1.1 Impact Mode

As an example, the MRLwCompany (Monterey

Research Laboratory) has marketed a machine

allowing the carrying out of shocks according to

two modes: impulse and impact (Bresk and Beal,

1966) In the two test configurations, the test item

is installed on the upper face of the table The table

is guided by two rods that are fixed at a vertical

frame

To carry out a test according to the impact mode

(the general case), one raises the table by the height

required by means of a hoist attached to the top of

the frame, using the intermediary assembly for

raising and dropping (see Figure 12.34) By

opening the blocking system in a high position,

the table falls under the effect of gravity or owing

to the relaxation of elastic cords if the fall is

accelerated After rebound, as seen on the shock

simulator (programmer), the table is again

blocked to avoid a second impact

12.8.2.1.2 Impulse Mode

The impulse mode shocks(see Figure 12.35) are

obtained while placing the table on the piston of

the shock simulator (used for the realization

of initial peak sawtooth shock pulses) The

piston of this hydropneumatic shock simulator

(programmer) propels the table upward according

to an appropriate force profile to produce the

specified acceleration signal The table is stopped in

its stroke to prevent its falling down for a second

time on the shock simulator (programmer)

md2x

FIGURE 12.34 MRL universal shock-test machine (impact mode) (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.)

the shock durationt can be deduced:

t ¼ p

ffiffiffiffimk

r

ð12:21Þwhere m is the mass of the moving assembly

(table þ fixture þ test item) and k is the stiffness

constant of the shock simulator (programmer)

This expression shows that, theoretically, the

duration can be regarded as a function alone of the

mass, m; and of the stiffness of the target It is, in

particular, independent of the impact velocity The

mass, m; and the duration, t; being known, we

deduce from it the stiffness constant, k; of the target:

k ¼ mp2

t2 ð12:22Þ12.8.3.1.2 Impact Velocity

Let us set vias the impact velocity of the table and

vR as the velocity of rebound The elastomeric

shock simulators often have a coefficient of

restitution, a ðvR¼ 2aviÞ; of about 50% In a

first approximation, we will consider that the

rebound is perfect ða ¼ 1Þ: The impact velocity is

then equal to DV=2; where DV is the velocity

change given byTable 12.1:

DV ¼ 2

p€xmt12.8.3.1.3 Maximum Deformation of the

Shock Simulator (Programmer)

If xm is the maximum deformation of the shock

simulator (programmer) during the shock, it

becomes, by equalizing the kinetic loss of energy

and the deformation energy during the

com-pression of the shock simulator (programmer)

s

ð12:25Þ

FIGURE 12.35 MRL universal shock-test machine (impulse mode) (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications With permission.)