ISO 21955:2021

INTERNATIONAL ISO
STANDARD 21955
First edition
2021-08
Acoustics — Experimental method
for transposition of dynamic forces
generated by an active component
from a test bench to a receiving
structure
Acoustique — Méthode expérimentale de transposition des forces
dynamiques générées par un composant actif d’un banc d’essai vers
une structure réceptrice
Reference number
©
ISO 2021
© ISO 2021
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ii © ISO 2021 – All rights reserved

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Principle of the method of transposition of the dynamic force . 5
4.1 General matters . 5
4.2 General formulae . 6
4.3 Geometrical considerations . 6
5 Operating mode . 7
5.1 General . 7
5.2 Synopsis of procedure . 7
5.3 Tasks and preliminary operations . 7
5.4 Transfer matrices determination . 9
5.4.1 General. 9
5.4.2 Final receiving structure transfer matrix determination Y . 9
RS
5.4.3 Test bench transfer matrix determination, Y . 9
TB
5.4.4 Connecting device spring-like matrix properties determination, S . 9
I
5.4.5 Active Component transfer matrix determination, Y . 9
AC
5.5 Measured dynamic forces transmitted to the test bench .10
5.6 Predicted dynamic forces transmitted to the final structure .10
5.6.1 General.10
5.6.2 Strong decoupling .10
5.6.3 Very similar bench and receiving structure .11
5.6.4 Case of a rigid receiving structure .11
5.6.5 Case of a non-rigid receiving structure .12
6 Requirements for data in test report .14
6.1 Specification of the integrator to the supplier .14
6.2 Data sent by the supplier to the integrator .14
Annex A (informative) Theoretical developments .16
Annex B (informative) Frequency response functions measurement .19
Annex C (informative) Dynamic forces measurement .22
Annex D (informative) Data processing .28
Annex E (informative) Study of a wiper system.31
Annex F (informative) Equivalent force torsor and block-sensor method .47
Bibliography .58
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.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
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expressions related to conformity assessment, as well as information about ISO's adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso .org/
iso/ foreword .html.
This document was prepared by Technical Committee ISO/TC 43, Acoustics, Subcommittee SC 1, Noise.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www .iso .org/ members .html.
iv © ISO 2021 – All rights reserved

Introduction
The vibroacoustic behaviour of products has become a major challenge not only in terms of user health
protection through regulations, but also in terms of sound quality for safety, quality perception, and
attractiveness.
At the same time, requirements on products development cycles are more and more stringent, reaching
the point where component suppliers and integrators should work independently, without physical
prototypes.
To master the transmission of dynamic forces (also called structure-borne noise), one needs to adapt
the components to the receiving structure, and hence exchange information prior to manufacturing
prototypes. This information will only be valuable for the integrator if it is clearly defined and intrinsic
to the component.
This document, issued from a French experimental standard, addresses this issue. It is a user guidance
to characterize an active source on a test bench and predict the effects of its integration on a passive
structure. The component is characterized on its own, which makes the document complementary to
the ISO 20270 that describes the measurement of “in-situ” characteristics (blocked forces), where the
component is connected to its receiving structure.
The intrinsic characterization of an active source requires measuring two quantities (expressed as
a function of the frequency): the first one characterizing the dynamic aspect, blocked forces, and the
second one describing “static” behaviour, such as the impedance or the mobility.
The objective of this document is to help the user predict the component behaviour in a particular
assembly. The theoretical background is laid in Annex A. The user is then guided (see 5.2) all along the
experimental procedure enabling to reach this objective:
— Static characterization of the component, the test bench and the receiving structure.
— Force measurement: the standard proposes here direct and indirect methods. Indirect methods are
generally easier to implement, but they need a particular focus on the measurement quality and
matrix inversion.
— Interface integration (connecting device).
— Prediction of the behaviour of the component/receiving structure assembly.
This whole procedure is based on a general formula expressing the dynamic forces in the assembly
as a function of blocked forces and static characteristics. Depending on these static characteristics,
simplifications are proposed (see 5.6).
Annex B and C guide the user to measure both transfer functions and dynamic forces. It should be noted
that, in general, these quantities are expressed in the 3 directions and 3 rotations, but the procedure
can be applied on a number of degrees of freedom chosen by the user.
The Annex D informs about data processing. The Annex E contains a test example and the Annex F
describes the method using a particular test bench (block sensor).
The data obtained and assessed in this document can be used:
— as part of a specification between suppliers and integrators;
— as input data of numerical vibroacoustic simulation models;
— to drive the modification of the physical structure or the interface in order to improve the
vibroacoustic behaviour.
INTERNATIONAL STANDARD ISO 21955:2021(E)
Acoustics — Experimental method for transposition of
dynamic forces generated by an active component from a
test bench to a receiving structure
1 Scope
This document specifies a method to predict the dynamic forces generated by an active component on a
receiving structure from measurement on a test bench.
It sets out the requirements applicable to test benches and setup measurement conditions of dynamic
forces: a criterion of validity of transfer functions measurements can be established for example.
The objective is to evaluate noise and vibrations generated by active components mounted on receiving
structures, including the possibility to optimise vibration isolators.
It can be applied to different systems connected to a building, such as a compressor or a power
generator, or to systems connected to a vehicle body, such as an engine powertrain or an electrical
actuator, for example.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/
3.1
active component
active substructure which generates dynamic forces
Note 1 to entry: See Figure 1.
3.2
connecting device
mechanical interface with a specific “spring like” matrix structure which allows connecting the active
component (3.1) to the receiving structure
Note 1 to entry: See Figure 1, Key 2.
Note 2 to entry: Insulators at fixation points are typical “spring like” connecting devices.
Note 3 to entry: A “spring like” connecting device is a structure with no internal degrees of freedom and internal
mass, see 3.10.
Note 4 to entry: In the case of a connecting point, active component and receiving structure share the same
location.
Note 5 to entry: In the case of seals at contact surfaces, direct connections or any other connection type, the
connecting device item 2 cannot be used, and a block diagram with items 1 and 3 and 4 shall be used (see
Figure 1).
Note 6 to entry: In case of direct connection, the hypothetical surface between the active component and
receiving structure is called interface.
3.3
receiving structure
passive substructure to which the dynamic forces are transmitted
Note 1 to entry: See Figure 1, Key 3 and 4.
Note 2 to entry: The receiving structure can be a test bench or the structure for which the dynamic forces will be
predicted.
Note 3 to entry: The “test bench” can be a specific structure designed to test the active component (3.1), or any
other receiving structure.
Note 4 to entry: active device, connecting device and receiving structures are deformable structures.
Key
1 active component
2 connecting device
3 receiving structure
4 test bench
NOTE An active component (left) connected via a connecting device (centre) transmits dynamic forces to a
receiving structure (right) which may vibrate and radiate sound.
Figure 1 — Schematic of the structure assembly
3.4
degree of freedom
n degrees of freedom through which structure-borne sound or vibration is transmitted from the active
component (3.1) to the receiving structure (3.3)
EXAMPLE A connection point can have up to 6 degrees of freedom (dof).
2 © ISO 2021 – All rights reserved

3.5
dynamic force
f
d
complex force associated to a structure or an interface with n degrees of freedom, arranged into a n × 1
vector at each frequency, according to
 f ()f 
d,1
 
f f
()
 d,2 
 

 
f ()f =
d
 

 
 
f f
()
d,n
 
 
 
th
where f ()f is the complex Fourier transform of the i component of dynamic force at frequency f
d,i
Note 1 to entry: Forces f can be considered as generalised forces, that is, including moments.
d
Note 2 to entry: Generalised forces units are Newtons for dynamic forces and N·m for dynamic moments
Note 3 to entry: In case pseudo-random signals, statistical tools can help into describing the dynamic forces into
set of amplitude and phase force vectors.
Note 4 to entry: In Table 1, the specific dynamic forces f applied at particular points used in this document are
d
defined.
Table 1 — Dynamic forces symbols
Symbols and abbrevia-
Definition
tions
Force generated by the active component at the interface of the connecting device
f
AC
in operational conditions.
Forces transmitted to the receiving structure: final receiving structure (RS) or
test bench (TB) in operational conditions. An additional “pred” or “meas” give
indication about how the force is obtained (predicted from formulae, or directly
measured):
f and f
RS TB
f and f .
RS_pred TB_meas
In the case of no presence of a connecting device, ff= .
AC RS
3.6
blocked force
dynamic force (3.5) applied by an active component (3.1) transmitted to a rigid receiving structure (3.3)
Note 1 to entry: Blocked forces indirect measurement methods are detailed in Annex F and ISO 20270.
3.7
velocity
v
d
complex vibration velocity associated to a structure or an interface with n degrees of freedom, arranged
into a n × 1 vector at each frequency, according to
v f
()
 
d,1
 
v ()f
d,2
 
 

 
v ()f =
d
 

 
 
v ()f
d,n
 
 
 
th
where v ()f is the complex Fourier transform of the i velocity component at frequency f
d,i
Note 1 to entry: Velocity units are meters per second (m/s).
Note 2 to entry: Associated complex acceleration a ()f can be defined via derivation of the velocity.
d
Note 3 to entry: Associated complex displacement x ()f can be defined via integration of the velocity.
d
Note 4 to entry: In Table 2, the displacement, velocity and acceleration xa,,v generated at particular points
dd d
used in this standard are defined.
Table 2 — Velocity, displacement and acceleration definitions
Symbols and abbreviations Definition
xa,v and dynamic displacement, velocity and acceleration generated by the active component.
AC AC AC
xa,v and dynamic displacement, velocity and acceleration on the receiving structure.
RS RS RS
3.8
frequency response function
FRF
frequency dependent ratio of the Fourier transform of the response to the Fourier transform of the
excitation of a linear system
Note 1 to entry: See ISO 2041.
Note 2 to entry: The FRF denomination and associated unit depends on the two vibration quantities of the ratio
(See Table 3).
Note 3 to entry: In this document, any reference to mobility Y or impedance Z is related to:
— the free mobility, Y , which is defined as a ratio of a dynamic velocity response in degree of freedom i to
free ij
an excitation force in degree of freedom j, with all degrees of freedom free, except the one of the excitation
forces; or
— the blocked impedance Z , which is defined as a ratio of the response force in degree of freedom j to
blocked ij
the dynamic velocity in degree of freedom i, with all degrees of freedom blocked, except the one of the
excitation velocity v
d, j
Table 3 — Denomination of frequency response functions FRF for various vibration quantities
(displacement, x, velocity, v and acceleration, a)
Dynamic Free Mobility Accelerance Dynamic Blocked imped- Effective
Compliance stiffness ance Mass
x v a
f f f
i i i
J J J
Y =
Denomination
freeij Z =
blocked ij
f f f
J J J x v a
i i i
4 © ISO 2021 – All rights reserved

Table 3 (continued)
Dynamic Free Mobility Accelerance Dynamic Blocked imped- Effective
Compliance stiffness ance Mass
m m m N Ns⋅
Ns⋅
Unit
N Ns⋅ m m
Ns⋅
m
Note 4 to entry: Thus, in terms of matrix writing, corresponding Y and Z matrices are related:
−1
ZY=
blockedfree
3.9
transfer matrix
set of FRF between multiple degrees of freedom systems
Note 1 to entry: In this document, the terms Y , Y and Y (see Table 4) can be related to free mobility,
AC RS TB
dynamic compliance or accelerances, depending of the quantities that are commonly used by the reader, with the
same boundary conditions as the free mobility.
Table 4 — Main transfer matrices used in the document
Symbols and abbreviations Definition
Y transfer matrix of the active component
AC
Y and Y transfer matrix of the receiving structure or the test bench
RS TB
3.10
connecting device transfer matrix
connecting device (case of insulators at fixation points) transfer matrix (3.9) (see Table 5) can be
obtained via different methods
Note 1 to entry: Such methods are described in ISO 10846.
Table 5 — Different expressions of connecting device transfer matrix versus dynamic stiffness
matrix
Formula Unit Homogeneous to
21*− −−12
accelerance
SK=ω mN⋅ s
II
*−1 −−11
mobility
SK= jω mN⋅ s
II
*−1 −1
compliance (or receptance)
SK= mN⋅
II
*
with K homogeneous to a dynamic stiffness complex matrix of the connecting device.
I
3.11
operational conditions
set of conditions under which the source operates for the operational test, including speed, load and any
other settings or conditions particular to the source which might affect source operation
4 Principle of the method of transposition of the dynamic force
4.1 General matters
This subclause explains how to predict the forces generated by an active component (which comes with
its own sources) on a receiving structure from a series of measurements on a test bench and on specific
data about the receiving structure.
Predicted or measured dynamic forces are required when
— there is no opportunity to measure directly any dynamic force of any sort (e.g. heavy and high cost
electrical machine to be duplicated in a new place),
— there is only the possibility to work on a test bench, because the final receiving structure is still not
available,
— the active source is provided by a component supplier to an integrator, and the integrator defines a
specification on a bench, with a target to comply, and
— internal forces matrix of a specific product is needed for noise comfort prediction, or for durability
purpose.
4.2 General formulae
The first Formula (1) which detail is given in Annex A can be written as follows:
−1
fY=+YS+ ⋅⋅YY++Sf (1)
[] []
RS RS AC ITBACI TB
where
is the transmitted force vector to the receiving structure;
f
RS
Y
is the transfer matrix of the receiving structure;
RS
Y
is the transfer matrix of the active component;
AC
Y
is the transfer matrix of the test bench;
TB
S
is the spring-like matrix representation of the connecting device;
I
is the transmitted force vector to the test bench.
f
TB
The purpose of Formula (2) is to enable the prediction of a force transmitted to a receiving structure
from the measurement or the estimation of 4 different FRFs matrices:
−1
fY=+[]YS+ ⋅⋅[]YY++Sf (2)
RS__predictRSACI TB AC ITBmeas
To build this formula, there are intermediate steps that are detailed in Annex A.
Instead of a link from bench to receiving structure, in certain cases, the need is to go from receiving
structure to bench; Formula (2) is then given as Formula (3):
−1
fY=+YS+ ⋅⋅YY++Sf (3)
[] []
TB__predictTBACI RS AC IRSmeas
4.3 Geometrical considerations
Sizes and quantities handled in this document are defined in a specific coordinate system, usually the
geometric coordinate system related to the receiving structure.
During FRF functions measurements (see Annex B), it can be more practical to use a local coordinate
system for certain attachment points. In this case, it will be necessary to re-project in a global reference
system.
6 © ISO 2021 – All rights reserved

5 Operating mode
5.1 General
In this subclause, an operating mode to apply this document is proposed, as an example.
This procedure is based on the general Formula (2) allowing to transpose the dynamic forces generated
by an active component from a test bench to a receiving structure. Depending on the assumptions on the
different transfer functions, this operating mode allows the use of simplified versions of Formula (2).
The frequency range(s) for which the formulated hypotheses and steps presented below are considered
as valid or invalid shall be mentioned.
5.2 Synopsis of procedure
The various special cases discussed below can be summarized in the form of a general diagram of the
procedure (see Figure 2).
5.3 Tasks and preliminary operations
A number of tasks and processes shall be performed previous to the application of this procedure:
a) During the development of the product, the component will determine the active component
transfer matrix at the connecting points. There are many cases for which the active component is
not only connected to its fixation points, but interacts with its environment through cables, rotation
axes, hoses, pipes, friction, which do not allow to measure the transfer matrix in free conditions for
all degrees of freedom. In this case, different alternatives are proposed in the document.
b) During the development of the product, the component chooses the properties of the connecting
device between the component and the receiving structure. It is remarked that this connecting
device matrix properties are of first order influence on the final transmitted forces: the product
shall ensure a perfect decoupling in order to minimize the vibration coupling between the
component and the receiving structure. Some advices are given hereunder in this operating mode.
c) To apply the methodology to predict the forces transmitted to the receiving structure in order to
check compliance with the specifications, a test bench is generally developed, transmitted forces to
the bench are measured. Usually, in the field of noise and vibration, an infinitely rigid bench, such
as a marble, is used, but this methodology is not mandatory. Therefore, the procedure covers the
case of a not infinitely rigid test bench.
The operating mode starts with the analysis of the general equation [Formula (2)] and attempts to
cover the different real cases that may be encountered in practice in the fields covered by the document.
The choices in the flow chart not only depend on the possibilities offered by the product, but also on
the relative order of scales of different transfer matrices in Formula (2). Three different examples are
described in Annexes E and F, to scan a wide range of applications.
Figure 2 — Synoptic of the steps to determine predicted force
8 © ISO 2021 – All rights reserved

5.4 Transfer matrices determination
5.4.1 General
Annex B is dedicated to frequency transfer functions measurement. These measurements are generally
performed with accelerometers and force sensors, leading to accelerance measurements.
5.4.2 Final receiving structure transfer matrix determination Y
RS
In Formula (2), the transfer matrix at the connecting points is required to predict the forces on the final
structure; whenever there is a component integrator specification about the predicted forces, which
means that the component integrator shall provide to the component supplier the transfer matrix values
at the connecting points. These values are generally available at early stages of a project development
via simulation tools.
In many cases, it is impossible to decouple the active component from the receiving structure to perform
a measurement. ISO 20270 can be applied, with an indirect measurement of blocked forces.
5.4.3 Test bench transfer matrix determination, Y
TB
At design stage for the test bench, it shall be considered to let some space to position the sensors for the
matrix determination.
5.4.4 Connecting device spring-like matrix properties determination, S
I
The connecting device matrix is generally determined on its own on a specific bench. In this case only
diagonal terms of the matrix are measured; ISO 10846 can be used.
Taking into account a connecting device is not adapted when:
— the active component is rigidly coupled to the receiving structure or test bench;
— the connecting devices are not a set spring-like point-like devices.
Taking into account a connecting device is not mandatory:
— the active component can be directly coupled to the receiving structure or test bench;
— the connecting device can be integrated in the Active component or in the Test Bench/ Receiving
structure part, whenever possible.
5.4.5 Active Component transfer matrix determination, Y
AC
Whenever possible to position the active component in free boundary conditions, the frequency
response functions of the transfer matrix shall be measured.
In many cases, the active component main function is to deliver a mechanical function to a system via
connections which are not fixation points or fixation surfaces. In this case, it is not possible to test the
active component in free boundary conditions.
Two options are possible to determine the active component transfer matrix:
a) Usage of a specific test bench called block sensor designed to measure indirect active component
transfer matrix and indirect blocked force (see Annex F);
b) Indirect determination of the active component transfer matrix associated with the connecting
device.
In this second case, only the case with no connecting device is relevant to be studied (with no connecting
device), and the Formula (2) can be written as:
−1
fY=+[]YY⋅⋅[]+Yf (4)
RS__predictRSACTBACTBmeas
At this stage, to measure indirectly the Y matrix, which should be tested in free conditions, but is not
AC
possible, it is needed to measure a transfer matrix of the coupled system: active component mounted on
the test bench: Y
AC:TB
It can be written that
−1
−−11
 
YY=+Y (5)
AC:TB AC TB
 
and then Y can be extracted:
AC
−1
−1
 
YY=−⋅⋅IY Y (6)
AC AC::TB TB ACTB
 
Same formulae can be reached from
ZZ=+ Z (7)
AC:TB AC TB
In the following subclauses, different cases of simplification are presented.
5.5 Measured dynamic forces transmitted to the test bench
Dynamic forces transmitted to the test bench can be measured by direct or indirect methods. Annex C
and Annex F propose different ways to measure these forces.
5.6 Predicted dynamic forces transmitted to the final structure
5.6.1 General
After determination of the dynamic force vector f and of the four transfer Matrices, the
TB_meas
dynamic forces transmitted to the final receiving structure can be predicted. Depending on the
characteristics of the test bench and/or the characteristics of the final receiving structure, Formula (2)
can be simplified or not.
5.6.2 Strong decoupling
This case arises when the connecting device is very flexible (impedance mismatch between active
component and connecting device transfer matrices, and between connecting device and receiving
structures RS and TB), that is:
YS and YS and YS (8)
AC I RS I BE I
Generally, an impedance mismatch is achieved when there is an order of scale higher than 10 (20 dB)
between each term of the transfer matrices for the different expressions of Formula (8).
This case is very useful because:
— the predicted force vector applied to the receiving structure is equal to the measured force vector
applied on the test bench, as given by Formula (9):
10 © ISO 2021 – All rights reserved

ff= (9)
RS TB
predictmeas
With such decoupling, it is very easy to estimate the transmitted forces.
— it minimizes the receiving structure force vector, which allows to transmit as less energy as possible
to the receiving structure.
5.6.3 Very similar bench and receiving structure
There is a particularly clear case for the prediction methodology that shall be addressed. As it is
generally necessary to work on a test bench on the active source design part (example of a component
supplier in charge of the development of the active component), choosing a test bench with a transfer
matrix similar to the transfer matrix of the receiving structure is to be pointed out:
YY~ (10)
RS TB
Then, from Formula (2), the predicted force vector applied to the receiving structure is equal to the
measured force vector applied to the test bench – see Formula (9) above.
ff=
RS TB
predictmeas
5.6.4 Case of a rigid receiving structure
5.6.4.1 General
It is common for the final receiving structure to be designed to have a transfer matrix much less influent
than the transfer matrix constituted by the active component and the connecting device. In this case, it
is given by Formula (11):
YY +S (11)
RS AC I
and the receiving structure can be considered as rigid.
Then, the test bench transfer matrix shall be studied.
5.6.4.2 Rigid test bench (marble)
At this stage, it is worth to try and previously design the test bench in order to be in the case written
in Formula (12). At the end, this formula is generally valid over certain frequency ranges, and not valid
over other frequency ranges, but easy to apply.
Ability to get access to a rigid bench enables to write:
YY +S (12)
TB AC I
Then, the prediction forces vector applied to the receiving structure is equal to the vector measurement
of the forces applied to the test bench – see Formula (9) above.
ff=
RS TB
predictmeas
5.6.4.3 Non- rigid test bench
If the test bench transfer matrix is not similar to the one of the receiving structure, then Formula (2)
can only be slightly simplified in Formula (13):
−1
fY=+[]SY⋅ []++YS ⋅ f (13)
RS__predictACI TB AC ITBmeas
5.6.5 Case of a non-rigid receiving structure
5.6.5.1 General
If the receiving structure cannot be considered as rigid (which means that Formula (11) is not satisfied),
the test bench transfer matrix that shall be studied.
5.6.5.2 Rigid test bench (marble)
If the test bench can be considered as rigid (which means that Formula (12) is satisfied), the predicted
force vector transmitted to the receiving structure is obtained by Formula (14):
−1
fY=+YS+ . YS+ ⋅ f (14)
[] []
RS __predictRSACI AC ITBmeas
5.6.5.3 Non-rigid test bench
If the test bench cannot be considered as rigid (which means that Formula (12) is not satisfied), the
predicted force vector transmitted to the receiving structure cannot be simplified and Formula (2) is to
be applied:
−1
fY=+[]YS+ . []YY++Sf
RS __predictRSACI TB AC ITBmeas
5.6.5.4 Evaluation of the quality of the predicted forces
The quality of the final predicted forces highly depends on the measurements and hypothesis that are
used:
— quality of the transfer functions that have been measured;
— quality of the operational measurements such as dynamic forces or accelerations;
— validity of the different hypothesis that enabled the formulae to be simplified;
— quality of matrix inversion.
These indicators of quality help to define frequency ranges for which the predicted force results are
reliable, see Table 6.
12 © ISO 2021 – All rights reserved

Table 6 — Indicators to be checked for quality evaluation of the predicted forces
Final predicted forc- Indicator To be checked
es depend on
Indicator of the frequency ranges for which non-lin-
earities will not ensure mastered FRF measurement
FRFs coherence uncertainties and of the frequency ranges for which
the signal to noise ratio is not high enough to ensure
repeatable results.
The principle of reciprocity of swapping the ex-
citation and response positions and comparing the
transfer functions Reciprocity
transfer functions is a very good indicator of the
(Annex B)
reproducibility of the measurement.
Based on the basic physical principle that the
damping factor of any system is a positive quan-
tity: if the real part of the complex mobility or the
Positive damping factor
imaginary part of the accelerances is not positive at
any frequency, the obtained measurement cannot
be considered as acceptable.
The associated modulus of the operational quantity
spectrum with the active component in operation
shall be considered towards the same spectrum
with active component not active. If the spectra
operational measure-
reproducibility
show similar levels, the confidence in the resulting
ments
measured quantities could be low.
Signal to noise ratio can be considered as reliable
when it exceeds 20 dB.
The conditioning number of the transfer matrix
can be used as an indicator of the quality of the
matrix inversion.
Force measurement by indirect To reach a good conditioning number, it is recom-
method mended to:
— have a matrix size with a number of sensors
higher than twice [ref] the number of matrix
minimum size
Use a regularization method.
The conditioning number of the transfer matrix
quality of matrix in-
should be used as an indicator of the quality of the
version
matrix inversion.
When very low values are obtained at certain fre-
quencies for the terms of the YY++S matrix,
[]
RS AC I
Predicted force on final receiving
matrix inversion introduces very big uncertainties
structure
in these frequency ranges: results are probably not
reliable in these frequency ranges.
When a quantity used in the formula used to predict
the receiving structure force is already obtained
through an indirect measurement process involv-
ing matrix inversion, the opportunity of a second
inversion shall be very carefully studied.
Table 6 (continued)
Final predicted forc- Indicator To be checked
es depend on
Inverting a matrix obtained from measured values tends to amplify measurement uncer-
tainties. It is generally recommended to discard the information detected as not reliable.
The conditioning number of the transfer matrix can be used as an indicator of the qual-
ity of the matrix inversion: to reach a good conditioning number, it is recommended to:
— have a matrix size with a number of sensors higher than twice the number of ma-
quality of matrix in-
trix minimum size
version
— use a regularization method.
As a good practice, it is recommended to validate that the final results are not too much
dependent on the inversion matrix tools. For example, comparing or sharing the results
of various inversions of a typical matrix is recommended.
6 Requirements for data in test report
Besides the usual indications about the contents of a test report (see ISO 17025), some advices are listed
hereunder.
Using this document, there are two types of test reports:
— Specifications of the integrator to the supplier;
— Data sent by the supplier to the integrator.
6.1 Specification of the integrator to the supplier
The test report shall include:
— a characterization of the final receiving structure (impedance, mobilities, or accelerances).
— if the integrator is responsible for the coupling elements at interface, the characteristics to be
implemented in the formula shall be sent.
— a list of specifications of the predicted dynamic force levels.
6.2 Data sent by the supplier to the integrator
In this case, a very accurate description and characteristics of the test bench shall be written, which
ensure a justification of the choice of bench design (impedance, mobilities, or accelerances).
The active component characteristics (impedance, mobilities, or accelerances….) shall be documented.
Forces measured on the test bench and the predicted forces on the final receiving structure shall be
included in the test report.
To ensure the quality of the measurement it is recommended to include (or in an annex) to plot measured
transfer functions and associated coherences; reciprocity and repeatability should be checked during
the different phases of measurements.
In the case of block sensor method (see Annex F), an auto validation can be added to the final report.
It is highly recommended to express the reasons for:
— reducing the number of DOF;
— not taking into account measurements with very low signal to noise ratio;
— reducing the number of sensors.
14 © ISO 2021 – All rights reserved

with the help of some quality indicators listed in Clause 5.
Annex A
(informative)
Theoretical developments
A.1 Introduction
The purpose of this annex is to develop theoretical developments that enable to obtain Figure 1 and
Formula (1). These theoretical developments will help the reader to better define how to design the
vibroacoustics properties of each component of the formula such as test bench related to final receiving
structure, or connecting device related to active component.
At first, direct mechanical coupling considerations are used for an active component coupled to a
receiving structure via a connecting device at connecting points, in order to help the reader to get a
better understanding of the different steps leading to this formula.
The electrical analogy is another pedagogical tool leading to unidimensional formulae.
Details about matrices organisation are given in Annex D.
An active component is coupled via a connecting device to a receiving structure, which does not
include any active structure (see Figure 1). This receiving structure can be a test bench or the final real
structure.
A.2 Components formulae
The following formulae are written with
— a dynamic movement u, can be indifferently an acceleration, a dynamic velocity or a displacement,
— frequency response functions Y which are respective to the dynamic movement accelerances, free
mobilities or dynamic compliances.
The quantities are
i c
— with exponent for internal points, and with exponent at the connecting points,
— with an index indicating the component.
Figure A.1 — Detailed schematic of the structure assembly
16 © ISO 2021 – All rights reserved

The active component vibrational behaviour is given in the frequency domain by the following linear
system given by Formula (A.1):
cc ci c c
    
YY f u
AC AC AC AC
    
= (A.1)
ic ii i i
    
YY f u
    
AC AC AC AC
As the receiving structure does not experience any internal excitations, its vibrational behaviour is
given in the frequency domain by the following linear system given by Formula (A.2):
cc c
Yf =u (A.2)
RS RS RS
The following hypothesis are made at the connection between the two components:
— the connecting device is not a structure with internal degrees of freedom and internal mass;
— the connecting points of the active component and of the receiving structure share the same
geometrical location.
Thus, with these hypothesis, the principle of action and reaction enables to obtain the following
Formula (A.3):
c c
...