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12.3 Shock Response Spectrum

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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, ki and a

damping device, ci; chosen such that the fraction of critical damping (damping ratio) j ¼ ci

􀀋􀀍

ffiffiffiffiffiffi

kimi

p 􀀎

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, each mass, mi; has a specific movement response

according to its natural frequency, f0i ¼ ð1=2pÞ

􀀍 ffiffiffiffiffiffi

ki=mi

p 􀀎

and to the chosen damping ratio, j; while a

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

Waveform Function DV

Half-sine x􀀊ðtÞ ¼ x􀀊m sin

􀀄p

􀀅 2

x􀀊mt

Versed-sine x􀀊ðtÞ ¼

x􀀊m

􀀄

1 2 cos

􀀅 1

x􀀊mt

Rectangle x􀀊ðtÞ ¼ x􀀊m x􀀊mt

Terminal peak sawtooth x􀀊ðtÞ ¼ x􀀊m

x􀀊mt

12-4 Vibration and Shock Handbook

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 ¼ Kzm ð12:2Þ

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

d2z

dt2 þ c

dt þ kz ¼ FðtÞ 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):

d2z

dt2 þ 2jv0

dt þ v20

z ¼

FðtÞ

m ð12:4Þ

where j ¼ c=2

ffiffiffiffi

ffiffiffiffiffi pkm (damping ratio) and v0 ¼

k=m p (natural frequency).

f1 ξ f2 ξ fN−1 ξ fN ξ

x(t)

z(t)

FIGURE 12.2 Model of the SRS. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With

permission.)

Force

Mass m

Stiffness k

Damping

constant c

Fixed support

FIGURE 12.3 Linear one-Dof system subjected to a

force. (Source: Lalanne, Chocs Mecaniques, Hermes

Science Publications. With permission.)

Mechanical Shock 12-5

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:

d2y

dt2 ¼ 2kðy 2 xÞ 2 c

􀀏 􀀐

ð12:5Þ

that is

d2y

dt2 þ 2jv0

dt þ v20

y ¼ v20

xðtÞ þ 2jv0

ð12:6Þ

or while setting zðtÞ ¼ yðtÞ 2 xðtÞ

d2z

dt2 þ 2jv0

dt þ v20

z ¼ 2

d2x

dt2 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:

zðtÞ ¼

ffiffiffiffiffiffiffiffi

1 2 j2

ðt

x€ðaÞe2jv0 ðt2aÞ sin v0

ffiffiffiffiffiffiffiffi

1 2 j2

ðt 2 aÞda ð12:8Þ

where a is an integration variable homogeneous with time.

The absolute acceleration of the mass is given by

y€ðtÞ ¼

ffiffivffiffi0ffiffiffiffi

1 2 j2

ðt

x€ðaÞe2jv0 ðt2aÞ½ð1 2 2j2Þsin v0

ffiffiffiffiffiffiffiffi

1 2 j2

ðt 2 aÞ

þ 2j

ffiffiffiffiffiffiffiffi

1 2 j2

cos v0

ffiffiffiffiffiffiffiffi

1 2 j2

ðt 2 aÞ􀀉da ð12:9Þ

12.3.4 Definitions

12.3.4.1 Response Spectrum

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

Absolute reference

Damping

constant c Stiffness k

Mass m

Moving base

FIGURE 12.4 Linear one-DoF system subjected to

acceleration. (Source: Lalanne, Chocs Mecaniques,

Hermes Science Publications. With permission.)

12-6 Vibration and Shock Handbook

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 v0zsup

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.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

2.0

1.5

1.0

0.5

0.0

−0.5

−1.0

−1.5

Frequency (Hz)

w 2

0 zsup (m/s2)

Half-sine (1 m/s2 - 1 s)

Primary positive spectrum

Residual positive spectrum

Primary negative spectrum

Residual 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.)

Mechanical Shock 12-7

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 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 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 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.

Primary (initial)

positive SRS

Primary (initial)

negative SRS

Secondary (residual)

negative SRS

Secondary (residual)

positive SRS

Positive SRS

Negative SRS

Maximax SRS

FIGURE 12.6 Relation between the different types of SRS.

12-8 Vibration and Shock Handbook

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 ) or v0t and

on the ordinate the spectrum/shock pulse amplitude

ratio, v20

zm=x€m; 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 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 fð¼ f0tÞ by f0 ¼ f=t:

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

0.0 1.0 2.0 3.0 4.0 5.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

SRS / xm

f0t

Q = 10

FIGURE 12.7 Standardized positive SRS of a terminal

peak sawtooth 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)

2.0

1.5

1.0

0.5

0.0

−0.5

−1.0

−1.5

−2.0

w 2

0 zsup (m/s2)

Half-sine (1 m/s2 - 1 s)

Positive spectra

Negative spectra

0.5

0.25

0.1

0.05 0

0.025

0.5

0.25

0.1

0.05

0.025

FIGURE 12.8 Standardized positive and negative relative displacement SRS of a half-sine pulse. (Source: Lalanne,

Chocs Mecaniques, Hermes Science Publications. With permission.)

Mechanical Shock 12-9

12.3.5.2.1 Half-Sine Pulse

Figure 12.8 and 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 12.11 and Figure 12.12 show the standardized SRS curves for terminal peak sawtooth (TPS) pulse.

12.3.5.2.4 Rectangular Pulse

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

12.3.5.2.5 Trapezoidal Pulse

Figure 12.14 presents the standardized SRS curve for a trapezoidal pulse. A comparison of various SRS

curves is given in Figure 12.15.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Frequency (Hz)

2.0

1.5

1.0

0.5

0.0

−0.5

−1.0

−1.5

−2.0

Absolute acceleration ysup (m/s2)

Half-sine (1 m/s2 - 1 s)

Positive spectra

Negative spectra

0.5 0.25

0.1

0.05

0 0.025

0.5

0.25

0.1

0.05

0 0.025

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)

2.0

1.5

1.0

0.5

0.0

−0.5

−1.0

−1.5

−2.0

0 zsup (m/s2)

Versed-sine (1 m/s2 - 1 s)

Positive spectra

Negative spectra

0.5

0.25

0.1

0 0.05

0.025

0.5

0.25

0.1

0.05

0.025

FIGURE 12.10 Standardized positive and negative relative displacement SRS of a versed sine pulse. (Source:

Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

12-10 Vibration and Shock Handbook

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 calculated

spectra, 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. Unless

otherwise specified, as noted on the curve, it is the value chosen conventionally. With the spectra varying

relatively little with damping, this choice is often not very important. To limit possible errors, the selected

value 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 the

response) 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)

1.5

1.0

0.5

0.0

−0.5

−1.0

−1.5

0 zsup (m/s2)

T.P.S. (1 m/s2−1s−td = 0.1s)

Positive spectra

Negative spectra

0.5

0.25

0.1 0.05

0 0.025

0.5

0.25

0.1

0.05

0 0.025

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)

1.5

1.0

0.5

0.0

−0.5

−1.0

−1.5

w 2

0 zsup (m/s2)

T.P.S. (1 m/s2 - 1 s)

Positive spectra

Negative spectra

0.5

0.25

0.1

0.05

0 0.025

0.5

0.25

0.1

0.05

0.025

FIGURE 12.11 Standardized positive and negative relative displacement SRS of a TPS pulse. (Source: Lalanne, Chocs

Mecaniques, Hermes Science Publications. With permission.)

Mechanical Shock 12-11

in this impulse region that it would be advisable to choose the natural frequency of an isolation system to

the shock, from which we can deduce the stiffness envisaged of the insulating material:

k ¼ mv20

¼ 4p2f 2

0 m ð12:10Þ with 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 this

impulse region ð0 # f0t # 0:2Þ:

* The form of the shock has little influence on the amplitude of the spectrum. Only (for a given

damping value) the velocity change DV associated with the shock, equal to the algebraic surface

under the curve x€ðtÞ 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

dðv20

zsupÞ

df0 ¼ 2pDV ð12:11Þ

This relation is approximate if damping is small.

2.5

2.0

1.5

1.0

0.5

0.0

−0.5

−1.0

−1.5

−2.0

−2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Negative spectra

Positive spectra

0.5

0.1

0.25

0.05

0 0.025

0.25

0.5

0 0.025 0.05 0.1

Frequency (Hz)

Rectangle (1 m / s2 - 1s)

w 2

0 zsup (m/s2)

FIGURE 12.13 Standardized positive and negative relative displacement SRS of a rectangular pulse.

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

Negative spectra

Positive spectra

Frequency (Hz)

Trapezoid (1 m/s2−1 s−t r = 0.1 s−t d = 0.1 s)

2.5

2.0

1.5

1.0

0.5

0.0

−0.5

−1.0

−1.5

−2.0

−2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0 zsup (m/s2)

0 0.025

0.05

0.1

0.25

0.5

0.25

0.1

0 0.025 0.05

FIGURE 12.14 Standardized positive and negative relative displacement SRS of a trapezoidal pulse.

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

12-12 Vibration and Shock Handbook

* The positive and negative spectra are in general the residual spectra (it is sometimes necessary that

the frequency of spectrum is very small, and there can be exceptions for certain long shocks in

particular). 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 towards

the amplitude of the shock whatever the damping: All occurs here as if the excitation were a static

acceleration (or a very slowly varying acceleration), as the natural period of the system is small

compared with the duration of the shock. This does not apply to rectangular shocks or to the shocks with

zero rise time. Real shocks having necessarily a rise time different from zero, this restriction remains

theoretical.

3. An intermediate domain in which there is dynamic amplification of the effects of the shock, the

natural 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 exceed

1.77 for shocks of traditional simple shape (half-sine, versed sine, TPS). Much larger values are reached in

the case of oscillatory shocks, made up, for example, by a few periods of a sinusoid.

Various domains of an SRS are illustrated in Figure 12.16.

Example

Consider a half-sine shock pulse, amplitude x€m ¼ 50 m=sec2; duration t ¼ 11 msec; positive SRS

(relative displacement) for a damping ratio j ¼ 0:05:

The slope p of the SRS (Figure 12.16) at the origin is equal to p ¼ 30:6=15 ¼ 2:04 m=sec; yielding

DV < p

2p ¼

2:04

< 0:325 m=sec

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

DV ¼

x€mt ¼

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;

2.0

1.5

1.0

0.5

0.0

−0.5

−1.0

−1.5

−2.0

0.0 1.0 2.0 3.0 4.0 5.0

Frequency (Hz)

w20

zsup (m/s2)

ξ = 0.05

FIGURE 12.15 Comparison between the SRS of the three main simple shock waveforms.

Mechanical Shock 12-13

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) frequency

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 frequency,

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 associated

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).

50 100 150 200 250 300 350 400 450 500

Intermediate

Domain

Impulse

Domain

30.6 Slope

Static Domain

Shock amplitude

p ≈ 2 π ΔV

m/s2

Q factor : 10

FIGURE 12.16 Shock response spectra domains.

Response

Shock pulse

FIGURE 12.18 Sampling frequency sufficient for the

shock pulse and too low for the response (error on the

SRS at high frequency).

12-14 Vibration and Shock Handbook

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 spectrum).

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

SF ¼

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)

eS ¼ 100 1 2 cos

􀀒 􀀏 􀀐􀀓

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 computing

time, it could be interesting to determine a

sampling frequency varying with each natural

frequency (Smallwood, 2002).

Sampled data

Error

Response

(sinusoid)

FIGURE 12.19 Error made in measuring the amplitude

of the peak. (Source: Lalanne, Chocs Mecaniques,

Hermes Science Publications. With permission.)

10 15

Sampling factor

Error (%)

20 25 30

FIGURE 12.20 Error made in measuring the amplitude

of the peak plotted against sampling factor. (Source:

Lalanne, Chocs Mecaniques, Hermes Science Publications.

With permission.)

Note: 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%.

TABLE 12.2 Some Sampling Factors with Corresponding

Error on the SRS

SRS Maximum Frequency Multiplied by Error (%)

23 1

16 2

10 5

Mechanical Shock 12-15

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 system

subjected to a shock. If the real structure is comparable to such a system, the SRS can be used to evaluate

this response directly. This approximation is often possible, with the displacement response being mainly

due to the first mode. In general, however, the structure comprises several modes, which are

simultaneously excited by the shock. The response of the structure consists of the algebraic sum of the

responses 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. The

phase relationships between the various modes are not preserved and the exact way in which the

modes are combined cannot be known simply.

* The SRS is plotted for a given constant damping over all the frequency range, whereas this

damping 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 one

mode. However, it happens that this is the only possible means. The problem is to know how to combine

these “elementary” responses so as to obtain the total response and to determine, if need be, any suitable

participation factors dependent on the distribution of the masses of the structure, of the shapes of the

modes, 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 that

the values of the maximum responses take place all at the same moment with the same sign, the

real maximum response is lower than the sum of the absolute values. This method gives an upper

limit 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 showed

that, in the majority of the practical problems, the distribution of the modal frequencies and

the 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, or

increase 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 realistic

evaluation 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 a

system of weighted averages. For example, the relative displacement response of the mass j is

estimated by

max

t$0

lzjðtÞl ¼

ffiffiffiffiffiffiffiffiffiffiffi

i¼1

􀀍

􀀎

þ p

i¼1

lzmli

p þ 1 ð12:13Þ

where the terms lzmli are the absolute values of the maximum responses of each mode and p is a

weighting factor (Merchant and Hudson, 1962).

12-16 Vibration and Shock Handbook

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12.3 Shock Response Spectrum

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