ISO 18431-4:2007

INTERNATIONAL ISO
STANDARD 18431-4
First edition
2007-02-01
Mechanical vibration and shock — Signal
processing —
Part 4:
Shock-response spectrum analysis
Vibrations et chocs mécaniques — Traitement du signal —
Partie 4: Analyse du spectre de réponse aux chocs

Reference number
©
ISO 2007
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©  ISO 2007
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ii © ISO 2007 – All rights reserved

Contents Page
Foreword. iv
Introduction . v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions. 1
4 Symbols and abbreviated terms . 2
5 Shock-response spectrum fundamentals . 2
6 Shock-response spectrum calculation. 7
7 Sampling frequency considerations. 12
Bibliography . 16

Foreword
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(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
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rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 18431-4 was prepared by Technical Committee ISO/TC 108, Mechanical vibration, shock and condition
monitoring.
ISO 18431 consists of the following parts, under the general title Mechanical vibration and shock — Signal
processing:
⎯ Part 1: General introduction
⎯ Part 2: Time domain windows for Fourier Transform analysis
⎯ Part 4: Shock-response spectrum analysis
The following parts are under preparation:
⎯ a part 3, dealing with bilinear methods for joint time-frequency analysis
⎯ a part 5, dealing with methods for time-scale analysis
iv © ISO 2007 – All rights reserved

Introduction
In the recent past, nearly all data analysis has been accomplished through mathematical operations on
digitized data. This state of affairs has been accomplished through the widespread use of digital signal-
acquisition systems and computerized data processing equipment. The analysis of data is, therefore, primarily
a digital signal-processing task.
The analysis of experimental vibration and shock data should be thought of as a part of the process of
experimental mechanics that includes all steps from experimental design through data evaluation and
understanding.
ISO 18431 (all parts) assumes that the data have been sufficiently reduced so that the effects of instrument
sensitivity have been included. The data covered in ISO 18431 (all parts) are considered to be a sequence of
time samples of acceleration describing vibration or shock. Experimental methods for obtaining the data are
outside the scope of ISO 18431 (all parts).
This part of ISO 18431 is concerned with methods for the digital calculation of a shock-response spectrum.
The analysis is by no means restricted to signals that can be characterized as shocks. On the contrary, it is, in
a strict sense, meaningless to analyze a shock according to the definition in ISO 2041, where a shock is
defined as a sudden event, taking place in a time that is short compared with the fundamental periods of
concern. Such a shock has no frequency characteristics in the frequency range of concern. It is only
characterized by its time integral, the impulse, corresponding to constant frequency content. The notation
“shock-response spectrum” has been kept, however, although a better term would be maximum-response
spectrum.
Historically, the shock-response spectrum was initially used to describe transient phenomena, at the time
called shocks.
Response analysis in general is a method to characterize a vibration or shock when other frequency analysis
methods are inadequate. This can be the case, for instance, when different kinds of vibration are compared.
Spectrum analysis based on the Fourier Transform produces spectra that are incompatible when the signals
analyzed are of different kinds, such as periodic, random or transient.
The typical use of a shock-response spectrum is to characterize a dynamic mechanical environment. The
vibration (or shock) characterized is recorded in digital form, commonly as acceleration. The data are
analyzed into a shock-response spectrum. This spectrum can then be used to define a test for equipment that
is required to endure the environment in question. There exist International Standards that describe how to
design tests from given shock-response spectrum specifications, for example IEC 60068-2-81. (See the
bibliography for additional information.)
When measurements to characterize a vibration and/or shock environment are performed, it is necessary to
take certain measures, for instance to ascertain a proper dynamic load in the measurement points. These
measures are beyond the scope of this part of ISO 18431. There are many good handbooks and
[1],[2]
recommended practices that are helpful in this area .

INTERNATIONAL STANDARD ISO 18431-4:2007(E)

Mechanical vibration and shock — Signal processing —
Part 4:
Shock-response spectrum analysis
1 Scope
This part of ISO 18431 specifies methods for the digital calculation of a shock-response spectrum (SRS) given
an acceleration input, by means of digital filters. The filter coefficients for different types of shock-response
spectra are given together with recommendations for adequate sampling frequency.
NOTE The definition of a shock-response spectrum given in ISO 2041, implies that a shock-response spectrum can
be defined in terms of an acceleration, velocity or displacement transfer function. This part of ISO 18431 deals only with
acceleration input.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 2041, Vibration and shock — Vocabulary
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 2041 and the following apply.
3.1
maximax shock-response spectrum
SRS where the maximum absolute value of the response is taken
3.2
negative shock-response spectrum
SRS where the maximum value is taken in the negative direction of the response
3.3
positive shock-response spectrum
SRS where the maximum value is taken in the positive direction of the response
3.4
primary shock-response spectrum
SRS where the maximum value is taken during the duration of the input
3.5
residual shock-response spectrum
SRS where the maximum value is taken after the duration of the input
4 Symbols and abbreviated terms
a(s) Laplace transform of acceleration (m/s )·s
c damping constant in SDOF system N/(m/s)
d(s) Laplace transform of displacement m·s
f natural frequency for SDOF system Hz
n
f sampling frequency, sampling rate Hz
s
G(s) transfer function in s domain
H(z) transfer function in z domain
k spring constant in SDOF system N/m
m mass in SDOF system kg, N/(m/s )
Q Q-value, resonance gain
s Laplace variable, complex frequency rad/s
SDOF single-degree-of-freedom system
SRS shock-response spectrum
T sampling time interval s
v(s) Laplace transform of (vibration) velocity (m/s)·s
z z-transform variable
α digital filter denominator coefficient
β digital filter numerator coefficient
ω angular natural frequency rad/s
n
ζ damping ratio, fraction of critical damping
5 Shock-response spectrum fundamentals
5.1 Introduction
In this part of ISO 18431, a shock-response spectrum is the response to a given acceleration of a set of
single-degree-of-freedom, SDOF, mass-damper-spring oscillators. The given acceleration is applied to the
base of all oscillators, and the maximum responses of each oscillator versus the natural frequency make up
the spectrum; see Figure 1.
Each single-degree-of-freedom system in Figure 1 has a unique set of defining parameters; mass, m, damping
constant, c, and spring constant, k. The parameters of the system are the conventional ones, given in
Clause 4.
A given acceleration, a , is applied to the base. If the response is measured as acceleration, a , then the
1 2
transfer function, G(s), for a SDOF system is given by Equation (1):
as()
cs +k
Gs()== (1)
as()
ms ++cs k
2 © ISO 2007 – All rights reserved

where s is the Laplace variable (complex frequency) in radians per second. The single-degree-of-freedom
system is normally characterized by its (undamped) natural frequency, f , in hertz, as given in Equation (2),
n
and the resonance gain, Q (Q-factor), as given in Equation (3):
1 k
f = (2)
n
2π m
km
Q = (3)
c
a
input motion
b
response motion
NOTE The responses of a set of single-degree-of-freedom (SDOF) mechanical systems define the shock-response
spectrum. The combination of m, c and k differs among the systems.
Figure 1 — Responses of a set of single-degree-of-freedom (SDOF) mechanical systems
The transfer function may then be rewritten, as given in Equation (4):
ω s
n
+ ω
n
as() Q
Gs()== (4)
ω s
as()
n
s++ ω
n
Q
with ω =π2 f being the angular natural frequency in radians per second.
nn
The transfer function is given versus frequency in Figure 2, where the natural frequency is set to 1 Hz and
Q = 10 as an example. Note the gain of Q at resonance.
NOTE Equation (4) defines the transfer function used. The maximum is approximately Q and the maximum occurs
approximately at f Hz. The larger the Q-value, the more accurate the approximation.
n
Key
X frequency, expressed in hertz
Y transfer function
Figure 2 — Transfer function of SDOF system as function of frequency
Instead of the resonant gain, Q, the damping ratio, fraction of critical damping, ζ, may be used. ζ is often
expressed in “percent of critical damping,” as given in Equation (5):
1 c
ζ== (5)
2Q
2 km
NOTE Critical damping, c , is defined as ck=2.m
crit
crit
To calculate the shock-response spectrum, the acceleration signal to be analyzed is applied to the base of a
set of SDOF systems characterized by their natural frequencies and Q-values. The responses are calculated;
the maximum responses as a function of the natural frequencies compose the shock-respon
...