ISO 18431-1:2005

Mechanical vibration and shock — Signal processing — Part 1: General introduction

ISO 18431-1:2005 defines the mathematical transformations, including the physical units, that convert each category of vibration and shock data into a form that is suitable for quantitative comparison between experiments and for quantitative specifications. It is applicable to the analysis of vibration that is deterministic or random, and transient or continuous signals. The categories of signals are defined in Clause 6. Extreme care is to be exercised to identify correctly the type of signal being analysed in order to use the correct transformation and units, especially with the frequency domain analysis. The data may be obtained experimentally from measurements of a mechanical structure or obtained from numerical simulation of a mechanical structure. This category of data is very broad because there is a wide variety of mechanical structures, for example, microscopic instruments, musical instruments, automobiles, manufacturing machines, buildings and civil structures. The data can determine the response of machines or of humans to mechanical vibration and shock.

Vibrations et chocs mécaniques — Traitement du signal — Partie 1: Introduction générale

General Information

• Status: Published
• Publication Date: 10-Nov-2005
• ICS: 17.160 - Vibrations, shock and vibration measurements
• Technical Committee: ISO/TC 108 - Mechanical vibration, shock and condition monitoring
• Drafting Committee: ISO/TC 108 - Mechanical vibration, shock and condition monitoring
• Current Stage: 9060 - Close of review
• Completion Date: 02-Sep-2029

Overview

ISO 18431-1:2005 - Mechanical vibration and shock - Signal processing - Part 1: General introduction provides a structured, practical framework for converting vibration and shock measurements into comparable, quantitative results. The standard defines the mathematical transformations and physical units required to process deterministic, random, transient and continuous vibration signals, whether obtained experimentally or from numerical simulation. Key themes include signal conditioning, sampling, frequency-domain care (Nyquist/aliasing), and correct identification of signal type before analysis.

Key topics and requirements

• Signal taxonomy and identification: Classifies signals (deterministic vs. random, periodic vs. non‑periodic, stationary vs. non‑stationary) and stresses correct identification because analysis methods and units depend on signal category.
• Signal conditioning: Guidance on pre‑digitization amplification, offset removal and analog filtering to prevent distortion and loss of dynamic range.
• Anti-aliasing and Nyquist: Emphasizes low‑pass filtering and sampling choices to avoid aliasing; recalls the Nyquist limit (maximum usable frequency = sampling frequency/2).
• Sampling and record length: Specifies considerations for sampling frequency, sampling interval and record length to capture transient events and spectral content.
• Time-domain, frequency-domain and time–frequency analysis: Introduces preprocessing, Fourier analysis, time‑frequency distributions and averaging methods for random stationary and ergodic signals.
• Units and spectral representations: Defines the mathematical transforms and physical units needed so results from different experiments or simulations are quantitatively comparable.
• Practical cautions: Advises checks (e.g., examining spectral energy near Nyquist) and attention to filter phase characteristics when phase information matters.

Applications

ISO 18431-1 is applicable across a broad range of vibration and shock contexts where accurate signal processing is required:

• Laboratory modal testing, modal parameter extraction and modal analysis
• Noise, vibration and harshness (NVH) testing in automotive and aerospace
• Shock response and durability testing of components and consumer electronics
• Structural health monitoring and condition monitoring of machinery and civil structures
• Validation of numerical simulations and comparison with measured data
• Human response and biomechanics studies involving vibration and shock

Who should use this standard

• Vibration and shock test engineers
• Instrumentation and data‑acquisition specialists
• Structural, mechanical and aerospace engineers
• R&D and product qualification teams
• Applied researchers and consultants in experimental mechanics

Related standards

• ISO 18431-2: Time domain windows for Fourier Transform analysis
• ISO 18431-4: Shock response spectrum analysis
• (Parts 3 and 5 - bilinear time-frequency and time-scale methods - noted as under preparation in the standard)

Keywords: ISO 18431-1:2005, mechanical vibration, shock, signal processing, sampling, anti-aliasing, Nyquist frequency, Fourier Transform, time-frequency analysis, vibration testing.

Standards Content (Sample)

INTERNATIONAL ISO
STANDARD 18431-1
First edition
2005-11-15
Mechanical vibration and shock — Signal
processing
Part 1:
General introduction
Vibrations et chocs mécaniques — Traitement du signal
Partie 1: Introduction générale

Reference number
©
ISO 2005
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ii © ISO 2005 – 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 . 3
5 Signal conditioning. 4
5.1 Cautionary overview. 4
5.2 Filtering. 4
5.3 Sampling. 5
6 Determination of signal type . 5
6.1 Signal taxonomy . 5
6.2 Deterministic signals. 6
6.3 Random signals . 7
7 Analysis of signals . 8
7.1 Preprocessing of signals . 8
7.2 Time domain analysis. 9
7.3 Frequency domain analysis of signals. 13
7.4 Time-frequency distributions . 17
7.5 Averages of random stationary, ergodic signals . 18
Bibliography . 20

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 18431-1 was prepared by Technical Committee ISO/TC 108, Mechanical vibration and shock.
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:
 Part 3: Bilinear methods for joint time-frequency analysis
 Part 5: Methods for time-scale analysis
iv © ISO 2005 – 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.
This part of ISO 18431 assumes that the data have been sufficiently reduced so that the effects of instrument
sensitivity have been included. The data considered in this part of ISO 18431 are considered to be a
sequence of time samples of a physical quantity, such as a component of velocity, acceleration, displacement
or force. Experimental methods for obtaining these data are outside the scope of this part of ISO 18431.

INTERNATIONAL STANDARD ISO 18431-1:2005(E)

Mechanical vibration and shock — Signal processing
Part 1:
General introduction
1 Scope
This part of ISO 18431 defines the mathematical transformations, including the physical units, that convert
each category of vibration and shock data into a form that is suitable for quantitative comparison between
experiments and for quantitative specifications. It is applicable to the analysis of vibration that is deterministic
or random, and transient or continuous signals. The categories of signals are defined in Clause 6.
Extreme care is to be exercised to identify correctly the type of signal being analysed in order to use the
correct transformation and units, especially with the frequency domain analysis.
The data may be obtained experimentally from measurements of a mechanical structure or obtained from
numerical simulation of a mechanical structure. This category of data is very broad because there is a wide
variety of mechanical structures, for example, microscopic instruments, musical instruments, automobiles,
manufacturing machines, buildings and civil structures. The data can determine the response of machines or
of humans to mechanical vibration and shock.
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 2041:1990, 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
aliasing
false representation of spectral energy caused by mixing of spectral components above the Nyquist frequency
with those spectral components below the Nyquist frequency
3.2
confidence interval
range within which the true value of a statistical quantity will lie, given a value of the probability
3.3
data
sampled measurements of a physical quantity
3.4
statistical degrees of freedom
number of independent variables in a statistical estimate of a probability
3.5
frequency resolution
difference between two adjacent spectral lines
3.6
number of lines
number of spectral lines that are displayed
3.7
Nyquist frequency
maximum usable frequency available in data taken at a given sampling rate
ff= 2
Ns
where
f is the Nyquist frequency;
N
f is the sampling frequency
s
3.8
record length
number of data points comprising a contiguous set of sampled data points
3.9
sampling
measurement of a varying physical quantity at a sequence of values of time, angle, revolutions or other
mechanical, independent variable
3.10
sampling frequency
number of samples per unit of time for uniformly sampled data
3.11
sampling interval
number of units (e.g. time, angle, revolutions) between two successive samples
3.12
sampling period
duration of time between two successive samples
3.13
sampling rate
number of samples per unit of time, angle, revolutions or other mechanical, independent variable for uniformly
sampled data
3.14
side-lobes
sequence of peaks in the frequency domain caused by the use of a finite time window with the Fourier
Transform
3.15
signal bandwidth
interval over frequency between the upper and lower frequencies of interest
2 © ISO 2005 – All rights reserved

3.16
spectral leakage
width of the peak in the power spectrum due to a single spectral component caused by using a finite window
with the Fourier Transform
4 Symbols and abbreviated terms
ADC analog-to-digital converter
B signal bandwidth
B equivalent noise bandwidth
e
C amplitude scaling factor
a
DFT Discrete Fourier Transform
E{ } expectation operator that computes the statistical mean value or average value
F(n) time-dependent force
H (m) frequency response function of the first type
H (m) frequency response function of the second type
K summation limit of time delay k or length of window w(k)
I number of data blocks
L record length
L level in units U for amplitude histogram of signal x(n)
i x
N data block length: the number of sampled points that are transformed
O (k,m) wavelet transform of x(n)
x
P (m) power spectral density of signal x(n)
xx
P (m) low frequency part of the power spectral density of signal x(n)
xx,low
P (m) low frequency part of the power spectral density of signal x (n)
x2,low
P (m) cross power spectral density of signal x(n) with y(n)
xy
Q quality factor of a single degree-of-freedom system
R (m) r.m.s. spectrum of signal x(n)
xx
S (m,n) short-time Fourier Transform of x(n)
x
T total time of a block of digital data = N∆t
V(k,m) Cohen class filter for smoothing the Wigner distribution
X(m) Discrete Fourier transform of x(n)
Y(m) Discrete Fourier transform of y(n)
b number of increments, also known as bits, in an ADC
c (k,n) auto-covariance of x(n)
xx
c (k,n) cross-covariance of x(n) with y(n)
xy
e (m) energy spectral density of signal x(n)
xx
e (m) cross energy spectral density of signals x(n) and y(n)
xy
f frequency = m∆f
f Nyquist frequency, the highest frequency present in a sampled signal
N
f natural frequency of a single degree-of-freedom system
n
f sampling frequency = 1/∆t
s
i index of data block
k index of time shift
l index of summation
m index of frequency or scale
xn() mean of non-stationary signal x(n)
x mean of stationary signal x(n)
n index of time
p lower limit of summation
q upper limit of summation
r upper limit of summation
r (k,n) auto-correlation of non-stationary data x(n)
xx
r (k) auto-correlation of stationary data x(n)
xx
r (k,n) cross-correlation of non-stationary data x(n) with y(n)
xy
r (k) cross-correlation of stationary data x(n) with y(n)
xy
t time = n∆t
v (n) variance of the non-stationary data x(n)
x
v variance of the data x(n)
x
w(n) window function
x(n) physical data in the time domain
y(n) physical data in the time domain
∆t sample period
∆f frequency resolution
ε relative random error
r
γ (m) coherence function
xy
ν(n) noise component of measured signal
ψ(n) mother wavelet
σ statistical variance of x
x
Ξ (m,n) Cohen class Wigner distribution using Cohen class filter V(n,m)
x
Ω (m,n) Wigner distribution of signal x(n)
x
5 Signal conditioning
5.1 Cautionary overview
The electrical signal from a transducer shall be properly conditioned for digitization by an analog-to-digital
converter (ADC). This signal conditioning requires the determination of several parameters associated with
amplification, filtering and digitization. The selection of these parameters is very important for the acquisition
of data that is appropriate for signal processing.
5.2 Filtering
Before the signal can be successfully digitized by the ADC, the signal shall be low-pass filtered to prevent
aliasing. Aliasing occurs when there are components of the signal at a frequency that is too high. The highest
frequency in the signal is limited by the sampling frequency, f , of the ADC. The range of settings of f are
s s
found in the specifications of the ADC. The highest frequency component of the signal may be no greater than
f = f /2, which is known as the Nyquist frequency. The upper frequency of the low-pass filter depends on the
N s
roll-off characteristics of the filter and the spectral properties of the signal. If the phase of the data is important,
attention shall also be paid to the phase characteristics of the filter.
The following test should be performed to check the adequacy of the low-pass filter. A signal should be
digitized and recorded. Then a Fourier transform should be performed on the data. The amplitude of the
Fourier-transformed signal at the Nyquist frequency should be less than or equal to the expected noise level
of the Fourier-transformed signal at the frequency of interest. If this is not the case, then the sampling rate
should be increased or the upper frequency of the low-pass filter should be lowered.
4 © ISO 2005 – All rights reserved

In addition to the low-pass filter, a high-pass filter may also be required because a non-negligible d.c.
component of the signal may reduce the useful range of the ADC. Reducing or eliminating this offset prior to
digitizing is preferable unless the d.c. component or low-frequency components are important.
The external analog anti-aliasing filtering considerations for a sigma delta ADC are different. The analog
signal shall meet the Nyquist criterion for the high frequency 1-bit digitizer, not the frequency for the end result.
With sigma delta digitizers, the manufacturer usually includes the low-pass filter needed for the analog input
and an internal digital low-pass filter to match the output sample rate.
5.3 Sampling
The ADC converts an analog signal into a sequence of integers. The output integers are proportional to the
input over a range of voltage. This range of voltage is given in the specifications of the ADC and determines
the proper gain setting discussed in 5.1.
NOTE The number of increments b in the largest output number determines the dynamic range of the ADC, which is
specified in terms of decibels, 6b + 1,8 dB.
The sequence of numbers is sampled at a rate called the sampling frequency, f , discussed in 5.1. A signal
s
may be resampled to order track the signal into samples that are equal increments of units other than time, for
example angular displacement or degrees.
Another parameter to be selected is the number of samples, the record length. The record length shall be
large enough to capture the whole signal if the signal is transient or limited in time.
The sampling frequency, f , fixes the following parameters:
s
 the maximum (Nyquist) frequency f = f /2
N s
 the sampling interval ∆t = 1/f
s
6 Determination of signal type
6.1 Signal taxonomy
The signals that make up the data are considered to be approximate members of idealized categories. In this
part of ISO 18431, the signals are categorized by the taxonomy shown in Table 1.The category of signal often
determines the methods of analysis. If an inappropriate analysis is used, then the results may be misleading
or inconclusive. Usually data contain a mixture of two types of signals. For example, a signal may be the sum
of a deterministic, non-periodic transient signal and a random, stationary, continuous signal. The triggering,
filtering and processing to determine the characteristics of the two signals are very different. The type of signal
to be described determines the signal conditioning digitization and data analysis. For example, specifying a
mechanical environment for equipment requires random, non-stationary transients to be sufficiently described
so that the mechanical conditions can be experimentally modelled with deterministic, non-periodic transients.
The signal taxonomy shown in Table 1 implies the decision tree required to determine the nature of the signal
of interest and also the subsequent data analysis.
Table 1 — Signal taxonomy
Decision Deterministic Random
tree
Periodic Non-periodic Stationary Non-stationary
Sinusoidal Harmonic Non- Transient Ergodic Non-ergodic Transient Continuous
harmonic
sinusoidal
Example Vibration Vibration Vibration Vibration Fluid Vibration Vehicle Vibration
signals from from gears from from modal dynamic from jet vibration during
imbalanced rolling impact noise engine with from pothole rocket take-
rotor bearings several off
operating
states
Appropriate r.m.s. r.m.s. r.m.s. Energy Power Power Energy Power
spectral
transform
6.2 Deterministic signals
6.2.1 Description of deterministic signals
Deterministic signals do not contain components that average to zero when subsequently captured signals are
averaged together.
NOTE A deterministic signal can reoccur over time or it can occur only once. Deterministic signals are caused by an
event. If the event recurs over time, a deterministic signal has the same behaviour each time. The event can be accessible
for use or not. The event can be periodic, continuous or impulsive in time.
6.2.2 Periodic signals
6.2.2.1 Description of periodic signals
These signals are generated from a periodic excitation and appear steady over subsequent signal captures.
Also they do not decay significantly with time over the duration of the signal capture.
6.2.2.2 Sinusoidal signals
Sinusoidal signals consist of single frequency components. This single frequency component is described by
its amplitude and phase relative to a reference.
NOTE Sinusoidal signals are usually caused by a sinusoidal event in a linear system.
6.2.2.3 Harmonic signals
Complicated periodic signals contain harmonically related components, the phase of which is related to a
reference excitation signal. The Fourier series may be applied to this case with the lowest frequency being the
inverse of the period of the signal.
A time history that is periodic may be represented by a superposition of sinusoids whose frequencies are
integral multiples of the fundamental frequency. The Fourier series includes a constant, which represents the
mean value and the sinusoidal terms and frequencies f, 2 f, …, nf, where f is the fundamental frequency and n
is an integer limited in magnitude by the frequency range of interest or the nature of the physical phenomenon
being analysed. The average values and the amplitudes of the sinusoidal components are obtained from the
integrals defined in ISO 2041:1990, A.18, under the definition of Fourier coefficients.
6 © ISO 2005 – All rights reserved

6.2.3 Non-periodic signals
6.2.3.1 Description of non-periodic signals
These signals are of two types. One type contains frequency components that are not harmonically related.
The other type decays over time.
6.2.3.2 Non-harmonic sinusoidal signals
These signals contain a non-harmonic sequence of spectral components that are related by the internal
components of the machine, e.g. gear ratios that have rational but not integer values.
These signals are composed of sinusoids whose frequencies are not all integral multiples of a common
fundamental frequency. Such a vibration can be viewed as a superposition of a number of periodic vibrations
where each periodic vibration includes those of the sinusoids that are harmonically related. Fourier series
cannot be used since periodicity is not present. The discrete frequency components of either periodic or multi-
sinusoidal vibration data may be isolated individually by narrowband pass filtering provided the bandwidth of
the filter is smaller than the difference of adjacent discrete frequencies (see 5.2 and 5.3).
6.2.3.3 Transient signals
Transient signals decay rapidly and therefore contain a continuous distribution of spectral components. The
power spectral density is an inappropriate description of these signals.
6.3 Random signals
6.3.1 Description of random signals
Random signals can originate with or without any reference excitation signal. They differ from deterministic
signals by varying either over time or over subsequent measurements.
For the purpose of analysis, a set of statistically independent blocks of data is required.
6.3.2 Stationary signals
6.3.2.1 Description of stationary signals
A stationary signal possesses statistical characteristics that do not change over time. Thus they do not
increase or decrease over time.
6.3.2.2 Ergodic signals
An ergodic signal has statistical properties that permit averages over time to replace averages over ensemble.
6.3.2.3 Non-ergodic signals
A non-ergodic signal has statistical properties that must be generated by the performance of several
experiments.
EXAMPLE There may be statistical variations between machines that cause the signal of interest to be
characterized through an average of measurements taken with a sequence of machines. Averages over time of a signal
taken from a single machine do not produce a meaningful statistical average over a set of machines.
6.3.3 Non-stationary signals
6.3.3.1 Description of non-stationary signals
A non-stationary signal has time-dependent statistical properties.
6.3.3.2 Continuous signals
A continuous, non-stationary random signal must be described by statistical characteristics that are functions
of time. Because the signal is continuous, the signal must be decomposed into the power spectrum.
6.3.3.3 Transient signals
A transient, non-stationary random signal must be described by statistical characteristics that are functions of
time. This time-limited signal should be decomposed into an energy spectrum.
This may be characteriz
...