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General framework for transfer path analysis: History, theory
and classification of techniques
$
Maarten V. van der Seijs
a,
n
, Dennis de Klerk
a,b
, Daniel J. Rixen
c
a
Delft University of Technology, Faculty of Mechanical, Maritime and Material Engineering, Department of Precision and Microsystems
Engineering, Section Engineering Dynamics, Mekelweg 2, 2628CD Delft, The Netherlands
b
Müller-BBM VibroAkustik Systeme GmbH, Robert-Koch-Strasse 13, 82152 Planegg, Germany
c
Technische Universität München, Faculty of Mechanical Engineering, Institute of Applied Mechanics, Boltzmannstr. 15,
85748 Garching, Germany
article info
Article history:
Received 20 March 2015
Received in revised form
29 July 2015
Accepted 3 August 2015
Available online 28 August 2015
Keywords:
Transfer path analysis
Source characterisation
Vibration transmission
Dynamic substructuring
abstract
Transfer Path Analysis (TPA) designates the family of test-based methodologies to study
the transmission of mechanical vibrations. Since the first adaptation of electric network
analogies in the field of mechanical engineering a century ago, a multitude of TPA
methods have emerged and found their way into industrial development processes.
Nowadays the TPA paradigm is largely commercialised into out-of-the-box testing
products, making it difficult to articulate the differences and underlying concepts that
are paramount to understan ding the vibration transmission problem. The aim of this
paper is to derive and review a wide repertoire of TPA techniques from their conceptual
basics, liberating them from their typical field of application. A selection of historical
references is provided to align methodological developments with particular milestones
in science. Eleven variants of TPA are derived from a unified framework and classified into
three categories, namely classical, component-based and transmissibility-based TPA .
Current challenges and practical aspects are discussed and reference is made to related
fields of research.
& 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC
BY-NC-ND lice nse (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Transfer Path Analysis (TPA) has been a valuable engineering tool for as long as noise and vibrations of products have
been of interest. A TPA concerns a product's actively vibrating components (such as engines, gearing systems or
turbochargers) and the transmission of these vibrations to the connected passive structures. TPA is particularly useful
when the actual vibrating mechanisms are too complex to model or measure directly, as it allows us to represent a source by
forces and vibrations displayed at the interfaces with the passive side.
In this way the source excitations can be separated from the structural/acoustic transfer characteristics, allowing us to
troubleshoot the dominant paths of vibration transmission. The engineer can then anticipate by making changes to either
the source itself or the receiving structures that are connected to it.
Contents lists available at ScienceDirect
journal homepage: www.elsevier.co m/locate/ymssp
Mechanical Systems and Signal Processing
http://dx.doi.org/10.1016/j.ymssp.2015.08.004
0888-3270/& 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
$
This research was funded and supported by the BMW Group.
n
Corresponding author.
E-mail addresses: m.v.vanderseijs@tudelft.nl (M.V. van der Seijs), ddeklerk@muellerbbm-vas.nl (D. de Klerk), rixen@tum.de (D.J. Rixen).
Mechanical Systems and Signal Processing 68-69 (2016) 217–244
A TPA often rises from the need to reduce some sort of undesired noise or vibration, for instance to improve product
comfort or lifetime, ensure safety or preserve stealthiness. Aside from automotive development, applications are also seen in
industries such as marine and aeroplane engineering, building acoustics and acoustic modelling of musical instruments. A
TPA is generally motivated by one of the following desires:
1. Secrecy: perhaps the earliest TP A studies were triggered by the need to reduce the transmission of engine vibrations in military
ships and submarines in order to make them stealthy. Man y publications in the 1 950s and 1 960s document on isolation of ship
engines by means of absorbers and decoupling mechanisms [1–5] to minimise the transmission through the interfaces.
2. Safety: along with the rapid development of aeroplanes and spacecraft in the 1960s, TPA concepts started to be of use to
study fatigue and stability (flutter) problems due to active or induced vibrations. As sources of vibrations are much more
persistent in aeronautics – think of vortex-induced vibrations – focus was on characterising the passive transfer paths by
means of modal analysis [6,7].
3. Comfort: over the last decades TPA tends to be particularly associated with noise, vibration and harshness (NVH)
engineering as commonly encountered in the automotive industry. The majority of recent developments and commercial
solutions have been tailored towards this engineering society or related industries, driven by the increasing customer
expectations on acoustic comfort [8–12].
In response to the evolving demands, TPA methods ha ve been under continuous development and their family members have
grown numerous. Some designations that found their w ay into the literature include Operational TP A (OTPA), Operational Path
Analysis with ex og enous inputs (OP AX), blocked-force TP A, Gear Noise Propagation, in situ Source Path Characterisation and Virtual
Acoustic Prototy ping. Very often those methods are presented from highly case-specific derivations. Not surprisingly , as the
underlying ph ysical concepts are similar , some of the above-mentioned show strong similarities or are even identical.
A TPA work flow can typically be subdivided in the following steps: (a) operational measurement on the active
component; (b) determination of the passive (sub)system characteristics (commonly by means of FRFs); (c) identification of
interface loads; (d) calculation of path contributions. The steps are shown schematically in Fig. 1. Depending on the TPA
method at hand, some or all of these steps may be performed in arbitrary order. The optimisation actions that follow from
such an analysis are generally not considered part of the work flow.
This paper presents a unified framework for derivation of a large range of TPA methods. It is chosen to present and
classify the methods separate from their typical fields of application, such that the underlying physical concepts are exposed
and can be compared. Section 2 presents an account of some early developments and their relation to currently established
TPA methods. This should by no means be regarded as a complete historical overview; rather it was chosen to highlight
some key publications that have inspired the methodological developments in different ways. In Section 3 a general
framework for TPA is introduced, starting by depicting the transfer problem using the Dynamic Substructuring paradigm
[13]. Hereafter the TPA methods are derived and classified along three families, namely the classical (Section 3.2),
component-based (Section 3.3) and transmissibility-based (Section 3.4) TPA methods, as depicted vertically in Fig. 1.
Nomenclature
DoF degree of freedom
FRF frequency response function
u dynamic displacements/rotations
f applied forces/moments
g interface forces/moments
Y admittance FRF matrix
Z impedance FRF matrix
T transmissibility matrix
⋆
AB
pertaining to the assembled system
⋆
A
; ⋆
B
pertaining to the active/passive component
⋆
R
pertaining to the test rig
⋆
1
source excitation DoF
⋆
2
interface DoF
⋆
3
receiver DoF
⋆
4
indicator DoF
⋆
ps
pseudo-force DoF
Fig. 1. The TPA work flow, depicted stepwise for the three TPA families.
M.V. van der Seijs et al. / Mechanical Systems and Signal Processing 68-69 (2016) 217–244218
The paper is concluded in Section 4 with a discussion of some practical aspects that are more common to TPA and partly
application-specific.
2. Historical overview
Transfer Path Analysis has been developed mostly during the second half of the 20th century, although some fundamental
concepts date back to the 1880s. As often occurs in science, inspiration was found in different fields of research. This section
starts with describing the adaptation of linear electric network theory to describe the transfer of structural vibrations. These
form the foundations on which many analytical, and later also experimental TPA techniques have been built.
2.1. Impedance and mobility concepts
The inherent task of analysing a transfer problem is to describe the relation between the inputs and outputs of systems,
preferably in a systematic or lumped way. Some of the major contributions in system theory originate from electric network
science, founded on the laws for electric circuits of Kirchhoff [14], the superposition principle and the definitions of electric
quantities such as admittance and impedance by Heaviside [15]. The equivalent source theorems of Thévenin [16] and
Norton [17] were stated
1
by the turn of the 20th century, allowing us to substitute a group of active and passive components
by an equivalent voltage/current and a single impedance. Altogether these discoveries provided a handful of tools to depict
complex electrical systems as a set of lumped subsystems, characterised by frequency dependent properties (e.g.
impedance, admittance) and interacting by the so-called “through” and “across” quantities, e.g. current and voltage.
The electric network principles appeared equally useful to describe structural vibrations of mechanical systems.
Gardonio and Brennan published an extensive review [21] of the system description based on impedance and mobility
(admittance) concepts in structural dynamics. They regard the article of Webster in 1914 [22] as the first effort to
demonstrate analogies between electrical and mechanical impedance properties. In an attempt to describe the acoustic
pressure in horns and musical instruments, Webster defined the acoustical impedance as the complex frequency-dependent
ratio between pressure and volume of flowing air. From thereon analogies have been derived for mechanical systems,
2
,
which is thoroughly reviewed in [21,25].
Impedance and mobility have since been well established as concepts to model and understand all sorts of vibratory
systems [26–29]. The adaptation of the four-pole matrix method [30] furthermore introduced means to model systems
consisting of a larger sequence of subsystems. However, most applications remained limited to fairly analytic cases [2–4].In
particular, the topic of experimental source characterisation has received little attention until the 1970s.
2.2. Advancing experimental techniques
Between 1971 and 1981 Bendat and Piersol provided a comprehensive set of spectral correlation and coherence functions
with special attention for digital data acquisition [31–34]. These publications have empowered multiple-input/multiple-
output (MIMO) measurement techniques that are instrumental to many advanced analyses, such as multi-reference modal
testing [35]. Indeed the engineers were now given the chance to analyse vibration problems in their full complexity (e.g.
multi-path, multi-DoF) rather than by simplified or analytical descriptions.
During the decades that followed, various simultaneous developments have been observed that led to a rapid expansion
of practical TPA methods:
The first exploration of techniques nowadays denoted as classical TPA is often attributed to the work of Verheij around
1980 who studied the transmission of ship machinery vibrations through resilient mounts [36,37]. Although theory on
mount stiffness had been around already for years [5], Verheij was one of the firs t to successfully determine interface
forces and moments by experiment. Although attractive from an academical point of view, practical engineering called
for less elaborate force determination methods. The matrix-inverse technique proved to be a good alternative [38–41]
and is up to to day one of the most popular cl assical TPA methods in practice. The theory of cl assical TPA is pre sented in
Section 3.2.
In 1981 Magrans proposed a general method of measuring transmissibility between terminals in a network [42]. The so-
called Global Transfer Direct Transfer (GTDT) method was further explored by Guasch [43–45] and later put into practice
as the Advanced TPA [46]. Independently, Liu and Ewins [47] and Varoto and McConnell [48] explored properties of
transmissibility matrices for structural vibrational problems, followed up by Ribeiro, Maia, Silva and Fontul [49–52].
Surprisingly though, the transmissibility-based method known as Operational TPA was first presented (again
1
Actually it was Helmholtz who posed the equivalent source theorem already in 1853 in one of his fundamental works [18]. Like many scientist in that
time, Thévenin was not aware of this early finding, as can be read in [19,20].
2
The initial analogy regarded force analogous to voltage, which seemed the most intuitive choice. Firestone however disqualified this mechanical-
impedance analogy in 1933 [23], arguing that it infringes Kirchhoff's definitions of the terms through and across. The mechanical-mobility analogy, linking
force with electrical current, is indeed the correct analogy for drawing mechanical circuits using parallel and series addition [24]. Nevertheless, the terms
impedance and mobility have never changed definition in common use.
M.V. van der Seijs et al. / Mechanical Systems and Signal Processing 68-69 (2016) 217–244 219
independently) by Noumura and Yoshida in Japan [53]. Section 3.4 discusses the theory and related techniques; reviews
and benchmark studies are found in [54–57].
Inspired on acoustics, Mondot and Petersson proposed a method in 1987 to depict the vibration transfer problem using the
charact eristic power of the source itself (the so-called source descrip tor) and a coupling function accounting for the added
dynamics of the recei ving structure [58]. This triggered the idea to chara cterise a source by means of blocked forces or free
velocities [59–61], as seen in e.g. the in situ method by Moorhouse and Elliott [62,63] and the pseudo-forces method by
Janssens and Verheij [64,65]. These and other strategies based on source component description are discussed in Section 3.3.
Most TPA methods require admittance of either the source, receiver or assembled structure. Dynamic Substructuring (DS)
techniques are particularly useful for this purpose, as they allow us to assemble systems from the dynamics of its
substructures [13]. Component Mode Synthesis (CMS) and model reduction [66–68] emerged in the 1960s as the first
application of DS. Rather than characterising an input/output problem, their main purpose was to compute natural
modes and frequencies of aerospace structures. This perhaps explains why DS techniques were hardly ever brought in
context with TPA.
3
Yet with the introduction of Frequency Based Substructuring (FBS) [72,73] in the late 1980s, methods
became available to assemble multiple substructures from FRFs, either obtained from numerical modelling or admittance
tests [74]. In fact, DS theory appeared very convenient to derive hybrid numerical/experimental TPA schemes and
perform component optimisation [75,76]. This is particularly effective in combination with component-based TPA
schemes [77,78], as discussed in Section 3.3.
2.3. Towards general TPA methodologies
The abundance of developments brought prosperity to the engineering community, yet at the same time raised
misunderstanding about the interrelations between the methods. In 1980 the ISO work group TC43/SC1/WG22 was
established, dedicated to investigating and standardising the present technologies for structure-borne TPA [79,80].An
intermediate report [81] already presented a comprehensive overview, addressing aspects such as the required number of
DoFs, source description by means of equivalent quantities (forces/velocities/power), reciprocal measurement techniques
and potential integration of Statistical Energy Analysis (SEA) principles. Although well accepted by acoustical engineers, the
standardisations failed to gain broad popularity in the field of structure-borne TPA.
Nevertheless, popular methods such as operational TPA and matrix-inverse TPA have nowadays been integrated into
many commercial noise and vibration solutions. Extensive literature is currently available, often discussing the application
of a particular technique in a case-specific fashion. From such perspective the relation with other TPA techniques can be
vague. In the remaining of this paper it is attempted to review the landscape of TPA techniques in a unified way, namely
from the framework as presented next.
3. Framework for transfer path analysis
The framework for Tr ansfer Path Analysis as presented here follows the notation and terminology of F req uency Based
Substructuring (FBS) [13]. Although different styles of deriv ation are encountered in the literature, it is the authors' belief that the
transfer path problem is best understood by describing the dynamic int er action between the activ e and passive subsyst ems.
In Section 3.1 the subsystem definition is introduced and the transfer problem is formulated based on the admittances
4
of
the active and passive subsystems. Thereafter a distinction is made between three families of TPA methods, respectively
denoted as classical TPA (Section 3.2), component-based TPA (Section 3.3)andtransmissibility-based TPA (Section 3.4). A
summary is presented in Section 3.5.
3.1. The transfer path problem
Let us consider the dynamic system AB as schematically depicted in Fig. 2a. Two subsystems can be distinguished: an active
subsystem A containing an excitation at node 1 and a passive subsystem B comprising the responses of interest at node 3. The
subsystems are rigidly interconnected at the interface node 2. For simplicity of deriv ation, the Degrees of F reedom (DoFs) in this
example are restricted to three distinct nodes. These may however represent a larger set of DoFs, repr esenting respectively
1. source: internal DoFs belonging to the active component that cause the operational excitation but are unmeasurable in
practice;
2. interface: coupling DoFs residing on the interface between the active and the passive component;
3. receiver: response DoFs at locations of interest at the passive component, possibly including acoustic pressures and other
physical quantities.
3
Early examples that suggest relations between TPA and dynamic substructuring are papers of Rubin [69,70] (who later published an important modal
reduction method [71]) and a NASA report [7].
4
Dynamic systems can be characterised and assembled using either impedance or admittance notations [27–29,82]. As admittances are obtained more
naturally in experimental practice, they are favoured throughout the derivations.
M.V. van der Seijs et al. / Mechanical Systems and Signal Processing 68-69 (2016) 217–244220
Hence, the example of Fig. 2a is illustrative for a wide range of practical problems, provided that the structure of interest
can be decomposed into an active and a passive part. In what follows, all methods assume that the operational excitation at
node 1 is unmeasurable in practice, but transmits vibrations through the interfaces at node 2 to receiving locations at node
3. The responses shall then be built up from a certain description of the vibrations measured at the interface (node 1-2)
and an appropriate set of transfer functions relating these vibrations to the receiving responses (node 2-3). The
fundamental choices herein dictate to which TPA family the method is classified.
3.1.1. Transfer path from assembled admittance
Let us first approach the transfer problem top-down for the assembled system AB, see Fig. 2a. W e are interested in the
response spectra at the receiving locations u
3
ðωÞ for source ex citations at node 1, given by the force spectra f
1
ðωÞ.Forthe
assembled problem, this is simply obtained from a superposition of the individual contributions, i.e. the ex citation force spectra
multiplied with their respective linear(ised) transfer functions, contained in the columns of admittance FRF matrix Y
AB
ωðÞ
u
i
ωðÞ¼∑
j
Y
AB
ij
ωðÞf
j
ωðÞ⟹ u
3
ωðÞ¼Y
AB
31
ωðÞf
1
ωðÞ ð1Þ
In the equations that follow the explicit frequency dependency ð
ωÞ will be omitted to improve readability. Also note that
the response set u can include displacements, velocities, accelerations or any other quantity, provided that the rows of the
FRF matrices are obtained accordingly. Furthermore, in order to keep the derivations brief and understandable, it is chosen
to only consider the structure-borne paths. Nevertheless, Eq. (1) can easily be extended to include contributions of airborne
paths if the application so requires. In that case Y and f need to be augmented with a set of (responses to) acoustic loads
such as volume velocities (m
3
/s), as further discussed in Section 4.3.
3.1.2. Transfer path from subsystem admittance
The same transfer function is now derived for an assembly of the individual subsystems, as depicted in Fig. 2b. Let us first
put the subsystem's FRF matrices Y
A
and Y
B
in a block-diagonal format. The force vector comprising the excitation force is
augmented with interface forces g
2
for both sides of node 2 that are yet to be determined. The obtained system of equations
resembles the admittance variant of dual assembly, which is a standard form of Dynamic Substructuring [13]
u
1
u
A
2
u
B
2
u
3
2
6
6
6
6
4
3
7
7
7
7
5
¼
Y
A
11
Y
A
12
00
Y
A
21
Y
A
22
00
00Y
B
22
Y
B
23
00Y
B
32
Y
B
33
2
6
6
6
6
6
4
3
7
7
7
7
7
5
f
1
0
0
0
2
6
6
6
4
3
7
7
7
5
þ
0
g
A
2
g
B
2
0
2
6
6
6
4
3
7
7
7
5
0
B
B
B
@
1
C
C
C
A
or u ¼ Yfþg
ðÞ
ð2Þ
The following physical explanation can now be reasoned to solve Eq. (2). The excitation force at node 1 induces a motion at
node 2 of subsystem A. As subsystem B is not directly affected by forces at A (due to the block-diagonal form of the global
FRF matrix), an incompatibility is caused between u
A
2
and u
B
2
. This is denoted by the interface “gap” δ, which can
conveniently be written using a signed Boolean matrix
5
B as shown in Eqs. (3a) and (3c). Next, assuming that no additional
mass is present between the subsystems, the equilibrium condition is satisfied, requiring the interface forces g
2
on both sides
to be equal in magnitude and opposing in sign. The interface forces are expressed by Eqs. (3b) and (3c), using a Lagrange
multiplier λ for the magnitude and the transposed Boolean matrix to account for the interface force direction
δ ¼ u
B
2
u
A
2
or δ9Bu ð3aÞ
g
A
2
¼g
B
2
¼ λ or g≜ B
T
λ ð3bÞ
with B ¼
0 II0
ð3cÞ
The interface forces
λ that pull the two subsystems together can be determined from Eq. (3a) by requiring δ ¼0, which
enforces the compatibility condition u
A
2
¼ u
B
2
. Considering Eq. (3b), the interface forces that ensure compatibility can be
Fig. 2. The transfer path problem: (a) based on the admittance of assembly AB and (b) based on the admittances of subsystems A and B.
5
The signed Boolean matrix B establishes the relations for all interface DoFs of A and B that are vectorially associated, e.g. u
A
2x
and u
B
2x
. Guidelines on
the construction and properties of the signed Boolean matrix are found in [13].
M.V. van der Seijs et al. / Mechanical Systems and Signal Processing 68-69 (2016) 217–244 221
determined by equating the second and the third line of Eq. (2) and solving for λ
Y
A
21
f
1
þY
A
22
g
A
2
¼ Y
B
22
g
B
2
Y
A
22
þY
B
22
λ ¼Y
A
21
f
1
λ ¼ðY
A
22
þY
B
22
Þ
1
Y
A
21
f
1
⇒ g
B
2
¼ðY
A
22
þY
B
22
Þ
1
Y
A
21
f
1
ð4Þ
Eq. (4) provides the interface forces at the coinciding interface DoFs caused by the operational excitation f
1
inside subsystem
A. The response at the receiving side u
B
3
is found by substituting Eq. (4) into the last line of Eq. (2)
u
3
¼ Y
B
32
g
B
2
¼ Y
B
32
ðY
A
22
þY
B
22
Þ
1
Y
A
21
hi
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Y
AB
31
f
1
ð5Þ
Comparing with Eq. (1), it follows that the terms between the brackets indeed represent the admittance of the assembly Y
AB
31
by coupling of the subsystems' admittances. It can be verified that this result is in accordance with Lagrange Multiplier
Frequency Based Substructuring (LM-FBS) assembly [13], which is further elaborated in Appendix A.
So far it has been assumed that the excitation at node 1 is measurable. In reality however it is impossible or impractical
to identify the exact force loading. This is solved in TPA by assuming that the dynamics at the interface node 2 due to this
excitation are measurable and may very well represent the source excitation. In what follows, different approaches are
examined to describe the transmission of vibrations, or rather, the response at the passive subsystem B for a non-
measurable excitation somewhere inside (or on) the active subsystem A. The notation of Dynamic Substructuring is used to
reveal the relations between different approaches.
3.2. Classical TPA
The family of Classical TPA methods is essentially intended to identify transfer path contributions in existing products.They
have nowadays become standard practice to troubleshoot NVH problems in automotive engineering [9,10]. A classical TPA
performs operational tests on the assembled product AB to obtain interface forces between the active and the passive side,
namely
λ in Eq. (4). It can be verified from Eqs. (2)and(5) that these interface forces fully determine the responses at the
passive side and are thus representative for the effects of the source vibrations at the receiver locations u
3
. To calculate the
receiver responses,
6
the passive-side interface forces g
B
2
¼λ are applied to the interfaces of subsystem B, as shown in Fig. 4a
u
3
¼ Y
B
32
g
B
2
ð6Þ
Both steps pose some challenges in practice. The FRFs of the passive side are typically determined from impact or shaker tests,
or in a reciprocal fashion using for instance an acoustic source at the receiving location and accelerometers at the interface
nodes [83–87].Eitherwayitrequiresdismounting of the active part(s) from the passive side. With respect to the determination
of operational interface forces g
B
2
, it could be impractical to mount force sensors between the active and the passive part.
Therefore a number of indirect methods have been developed to circumvent direct force measurement.
Hence, the variants of classical TPA are defined according to how g
B
2
is obtained from operational tests, which are
discussed next.
3.2.1. Classical TPA: direct force
The most straightforward technique to obtain the interface forces is from force transducers mounted directly between
the active and the passive side, as depicted in Fig. 3a. It was demonstrated by Eq. (4) that the interface force caused by the
operational excitation is given by
λ ¼ðY
A
22
þY
B
22
Þ
1
Y
A
21
f
1
⟹ g
B
2
¼λ ð7Þ
This is valid under the assumption that u
B
2
¼ u
A
2
, which requires the stiffness of the transducers to be high enough (relative
to the stiffness of the actual subsystems) in the frequency range of interest. In fact, the main drawback of the method is the
Fig. 3. Three approaches to determine the operational interface forces in classic TPA. (a) Direct force. (b) Mount stiffness. (c) Matrix inverse.
6
In this framework the resulting responses u
3
are formulated as a matrix–vector product, namely the sum of the partial responses. Techniques to
evaluate the individual transfer paths contributions are discussed in Section 4.5.
M.V. van der Seijs et al. / Mechanical Systems and Signal Processing 68-69 (2016) 217–244222
inconvenience of placing the transducers between the active component and the receiving structure. Lack of space,
distortion of the original mounting situation and the incapability to measure all desired degrees of freedom at a connection
point render the method impractical, especially for typical automotive applications. In case of large-scale systems such as
ship machinery, this method may still be preferred [37].
3.2.2. Classical TPA: mount stiffness
An effective w ay of reducing vibr ation transmission is by placing resilient mounts between the components instead of rigid
fixtures, as illustrated in Fig. 3b. By proper tuning of the mount flexibility (stiffness) and absorption (damping) properties, a high level
of vibration suppression can be achieved. The mount stiffness method uses these mount properties to determine the interface forces.
Assumed that the added mass of the mounts is negligible, the interface force equilibr ium condition Eq. (3b) is still satisfied. Howev er ,
the compatibility condition of Eq. (3a) is “weakened”,henceu
B
2
u
A
2
is no longer zero. Instead the m interface forces and coordinate
incompatibilities are related by the dynamic stiffnesses of the mounts, denot ed by z
mt
jj
, j representing a single interface DoF
g
A
j
g
B
j
2
4
3
5
¼z
mt
jj
1 1
11
u
A
j
u
B
j
2
4
3
5
⟹g
B
j
¼ z
mt
jj
ðu
A
j
u
B
j
Þ; jA 1; …; m
A spring-like stiffness matrix can be recognised, how ever with a minus sign because the interface forces g
j
act on the connected
subsystems A and B instead of the mount. Introducing the diagonal mount stiffness matrix Z
mt
,thefullsetofm interface forces g
B
2
can
be estimated from the differential interface displacements between the source and the receiver side, i.e. the measured displacements
at both sides of the mounts
g
B
2
¼ Z
mt
ðu
A
2
u
B
2
Þ with Z
mt
¼ diagðz
mt
11
…z
mt
mm
Þð8Þ
In most cases the flexible mounts are already integra ted in the design to attenua te the vibration transmission. If they are however
placed in the system for the mere purpose of TPA, it can be shown that the interface forces and thus the vibrations of the coupled
system are altered significantly [88,89],namely
g
B
2
¼ðY
A
22
þY
B
22
þY
mt
Þ
1
Y
A
21
f
1
with Y
mt
¼ðZ
mt
Þ
1
Although the mount stiffness method can be powerful and easy to conduct, the accuracy is highly subject to the terms
7
in Z
mt
.Typical
absorbers exhibit amplitude-dependent non-linearities and directional characteristics [1,2,91]. An adva nced TP A method that
estimates the mount properties from operational tests is discussed in Section 3.4.3.
3.2.3. Classical TPA: matrix inverse
The third and perhaps most popular classical TPA member is the matrix-inverse method [38– 41]. It was observed from
Eq. (6) that responses at the passive side are found from the application of the interface forces to the passive subsystem's
FRFs. Recalling these responses from Eq. (2)
u
B
2
u
3
"#
¼
Y
B
22
Y
B
32
"#
g
B
2
This problem can be inverted if the left-hand side contains sufficient independent responses to describe all m interface
forces and moments in g
B
2
. The set of receiver responses u
3
is typically too small in number and too distant from the
interfaces to be suitable for inversion. Inversion of the first row is theoretically sufficient, but requires complete
instrumentation of the assembled structure's interfaces to measure all DoFs u
B
2
associated with g
B
2
. In addition, a symmetric
FRF matrix Y
B
22
would be required for the passive subsystem's interfaces, which is challenging to obtain accurately.
8
Fig. 4. Application of forces representing the operational excitation: classical TPA and component-based TPA. (a) Classic TPA: application of interface forces
(measured under operation of assembled system AB) to the passive subsystem B. (b) Component-based TPA: application of equivalent forces (measured
under operation of subsystem A) to the assembled system AB.
7
Note that the terms in the dynamic stiffness matrix Z
mt
correspond to differential displacements of the associated interface DoFs u
A
2
u
B
2
and not the
coordinates of both A and B. Some implications for the terms in Z
mt
are discussed in [89,90].
8
See Section 4.4 for a discussion on obtaining a full and reciprocal set of translational and rotational DoFs.
M.V. van der Seijs et al. / Mechanical Systems and Signal Processing 68-69 (2016) 217–244 223
In practice, the passive side is equipped with the so-called indicator responses u
4
as shown in Fig. 3c. An amount of nZ m
responses shall be located in the proximity of the interfaces, such that the full set of m interface forces is properly observable
from these points (this is addressed below). As these indicator DoFs merely assist in the determination of the interface
forces, the sensor type can be chosen rather arbitrarily, although (tri-axial) accelerometers are the most common choice.
Two sets of measurements are now required to reconstruct the interface force spectra. First, responses u
4
are measured
on the assembled system AB during operational tests. These can be expressed in terms of subsystem admittances, similar to
Eq. (5) (see also Appendix A)
u
4
¼ Y
AB
41
f
1
¼ Y
B
42
Y
A
22
þY
B
22
1
Y
A
21
f
1
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
g
B
2
ð9Þ
Next, FRFs need to be measured for the passive subsystem B, relating the motion at these indicator points to forces at the
interface, namely Y
B
42
. Note that this requires dismounting of the active components from the assembly. Nevertheless, the FRFs
Y
B
42
(needed for the matrix-inverse force determination) and Y
B
32
(to calculate responses at the target locations, Eq. (6))canbe
obtained from the same FRF measurement campaign, as it only involves mounting of additional sensors. The operational
interface forces can be reconstructed from a pseudo-inverse of the indicator response spectra u
4
with the subsystem FRFs
g
B
2
¼ Y
B
42
þ
u
4
ð10Þ
If Y
B
42
is full rank, it holds that ðY
B
42
Þ
þ
Y
B
42
¼ I, such that in theory the correct interface forces from Eq. (4) are obtained by
subsequent application of Eqs. (9)and(10). Hence the conditioning of the FRF matrix is crucial, which should have (a)
sufficient rank to describe all interface forces g
B
2
independently from the set u
4
and (b) a reasonably low condition number in
order not to amplify measurement errors in the inversion. As a rule of thumb, it is common to use at least twice as many
response DoFs as strictly required to fully determine the interface forces. Much attention has been devoted to improving the
conditioning of the inverse problem by means of singular value decomposition, see for instance the work of Thite and Dobson
[39–41,92,93] or an early review of techniques [94]. Time domain implementations have also been developed, such as the
inverse structural filtering methods described in [38,95].
Recently there has been interest in the application of strain gages instead of the commonly used accelerometers at the
indicator points. It is argued that strain responses possess a more direct relation to interface forces and are better able to
capture the local phenomena of the structure. Consequently it is expected that strain measurements lead to better
conditioning of the matrix to be inverted [96,97]. More research is currently needed to further verify this assumption.
3.3. Component-based TPA
A fundamentally different class of methods is that of the component-based TPA .AsshowninEq.(7), the interface forces obtained
from a classical TPA are not a characteristic of the source alone but of the assembled dynamics. For that reason, a classical TP A
cannot predict the effects of subsystem modification, as one would need to conduct a new operational test for every change in
design. Hence, the interface forces measured in an assembly AB are not transferable to an assembly with anoth er receiving side B.
Component-based TPA tries to characterise the source excitation by a set of equivalent forces or velocities that are an
inherent property of the active component itself. The responses at the receiving side can be simulated by applying these
forces to the FRFs of an assembled system with the active part shut down, as illustrated in Fig. 4b. Hence, the dynamic
interaction with the passive side is accounted for in a later stage, at least not during operational measurements. This allows
defining a testing environment
9
that is ideal for operational measurement on the active component, explaining the
denotation component-based.
Interestingly, with respect to the origin of component-based TPA theory, literature has been very unambiguous. As
mentioned in Section 2, some researchers have found inspiration in acoustics or electronic network theory (particularly the
equivalent source identities of Thévenin and Norton), such as [58–63,98]. Others derived similar theories from a structural–
mechanical point of view [64,65,99] or dynamic substructuring techniques [77,78]. As a consequence, a wide variety of
component-based TPA methods have been derived, largely independent of each other. This section presents a more generic
derivation in order to unite all component methods and compare the various concepts.
3.3.1. The equivalent source concept
Approaching the problem top-down, one is looking for a set of equivalent forces f
eq
2
that, applied to the interface of the
assembled system AB at rest, yields the correct responses at u
3
. This can be directly formulated using the admittance of the
assembly Y
AB
32
, or expanded in terms of its subsystem admittances (see Appendix A)
u
3
¼ Y
AB
32
f
eq
2
¼ Y
B
32
Y
A
22
þY
B
22
1
Y
A
22
f
eq
2
ð11Þ
9
The dependency of the source excitation on the mounting conditions is discussed in Section 4.1.
M.V. van der Seijs et al. / Mechanical Systems and Signal Processing 68-69 (2016) 217–244224
The response u
3
, as a result of the equivalent forces, should equal the response obtained when the active component is
running in the same assembly, caused by f
1
in Eq. (5). Comparing Eqs. (11) with (5), it follows naturally that the equivalent
forces should take the form
Y
B
32
Y
A
22
þY
B
22
1
Y
A
22
f
eq
2
¼ Y
B
32
Y
A
22
þY
B
22
1
Y
A
21
f
1
⟹ f
eq
2
¼ Y
A
22
1
Y
A
21
f
1
ð12Þ
Eq. (12) shows that the equivalent forces are indeed a property on the active component A only. The existence of such an
equivalent source problem offers tremendous potential for practical component-based TPA methods. There is however one
important limitation: the equivalent forces only properly represent the operational excitations for responses on the receiving
structure or on the interface. Responses obtained on the source will be different and therefore of no use. This limitation was
already mentioned in [77,78] and can be understood by examining the system of Eq. (2): responses at the passive side B are
caused only by forces through or onto the interface, whereas the responses at the source side A are a result of both the direct
contribution of f
1
and its reflection through the coupled subsystem B. This is further substantiated in Appendix A.
In the next sections, several approaches are discussed that yield a set of equivalent forces from operational tests,
following the definition of Eq. (12).
3.3.2. Component TPA: blocked force
Consider the case where the boundary of subsystem A is rigidly fixed, as depicted in Fig. 5a. The operational excitation f
1
is entirely portrayed by the reaction forces at the “blocked” interface g
bl
2
, such that the interface displacements u
A
2
¼ 0.A
direct relation is found, leading to the following equivalent force:
u
1
u
2
¼ 0
"#
¼
Y
A
11
Y
A
12
Y
A
21
Y
A
22
"#
f
1
g
A
2
¼g
bl
2
"#
⟹
g
bl
2
¼ Y
A
22
1
Y
A
21
f
1
f
eq
2
¼ g
bl
2
8
>
<
>
:
ð13Þ
The so-called blocked-force TPA is perhaps the most commonly known variant of component-based TPA methods because of
its straight-forward applicability. It can be seen as an application of the Thévenin equivalent source problem [16], that found
its way into popular mechanics halfway the 20th century [2,27,70]. Mathematically one can regard the blocked-force
method as imposing a Dirichlet boundary condition on the active subsystem's interface.
The blocked-force method assumes the boundary to be infinitely stiff in all directions, which is in practice rarely the case.
10
Hence the accuracy of the blocked forces is highly subject to the stiffness of the boundary relative to the component at hand
[77,101,102]. An additional difficulty is the measurement of rotational moments, as most commonly used sensors are unable to
measure collocated 6-DoF interface loads. As a consequence, the blocked-force method is expected to perform best at the low
frequency end for which the rigid boundary assumption is most valid and rotational effects are in practice least prominent.
3.3.3. Component TPA: free velocity
The direct counterpart of the blocked-force method is the free-velocity TPA as depicted in Fig. 5b. In this case the
component's interfaces are left free, such that all vibrations are seen as “free displacements” u
free
2
at the interface DoFs. From
here on, equivalent forces
11
can be calculated by inverting the admittance measured at the free subsystem's interfaces,
which can be understood by comparing the free displacements with the blocked force definition of Eq. (12)
u
1
u
A
2
¼ u
free
2
"#
¼
Y
A
11
Y
A
12
Y
A
21
Y
A
22
"#
f
1
g
A
2
¼ 0
"#
⟹
u
free
2
¼ Y
A
21
f
1
f
eq
2
¼ Y
A
22
1
u
free
2
8
>
<
>
:
ð14Þ
Analogue to the blocked-force TPA, this method can be seen as a strict application of a Neumann boundary condition and is
furthermore related to Norton's equivalent source theorem for electric networks [17]. Again, imposing free boundary
Fig. 5. Component-based TPA methods: various approaches to obtain equivalent forces representing the excitation. (a) Blocked force. (b) Free velocity.
(c) Hybrid interface on test rig. (d) in situ in original assembly.
10
For numerical analysis the blocked-force concept can be particularly effective to reduce a distributed load on the active component to fewer DoFs of
the interfaces. This was recently demonstrated for transient simulation of offshore structures [100].
11
If the admittances of the subsystems are available separately, one may also apply the free velocities directly, as shown in [82,103].
M.V. van der Seijs et al. / Mechanical Systems and Signal Processing 68-69 (2016) 217–244 225
conditions can be troublesome as the vibrating system often needs to be mounted at one or more connection points to be
able to run in operation. Therefore the method is in practice expected to perform best for frequencies well above the rigid
body modes of the structure. Also note that effects such as friction and damping, which would occur at the interfaces of the
assembled system, are absent in the operational measurement on the free component.
The fact that the method makes reference to free velocities rather than free displacements has an historical motive, as the former
quantity is commonly applied in acoustical engineering in combination with acoustic pressure. For such acoustic problems, dynamic
couplingofadmittanceisnormallydiscarded.Thisisafairassumption, considering that the impedance of the radiating source is much
larger than the impedance of the receiving surrounding fluid, i.e. air . In structure-borne vibrations the source and the receiving system
often e xhibit simila r dynamics, hence explicit coupling is needed. This was real ised for a single -DoF proble m by means of a non-
dimensional coupling function in [58]. Extensions to multi-DoF systems and further discussion of this topic are found in [59–61].Asa
final note, the free-v elocity concep t has been standardised into an ISO norm for characterisation of structure borne sound sources [80].
3.3.4. Component TPA: hybrid interface
As both above-mentioned methods pose their limitations in practice, one often prefers to conduct operational tests on an
appropriate support structure. Such a coupled structure unavoidably displays its own dynamics on the interfaces, which
need to be accounted for in order to render the equivalent forces independent of any connected part. Let us therefore
imagine the active component fixed onto a test bench or test rig R as illustrated in Fig. 5c. Denoting the interface admittance
of the test rig by Y
R
22
, we obtain for the extended system of equations
u
A
1
u
A
2
u
R
2
2
6
4
3
7
5
¼
Y
A
11
Y
A
12
0
Y
A
21
Y
A
22
0
00Y
R
22
2
6
6
4
3
7
7
5
f
1
g
A
2
g
R
2
2
6
4
3
7
5
with
u
A
2
¼ u
R
2
ðdisplacement compatibilityÞ
g
A
2
¼g
R
2
ðforce equilibriumÞ
(
ð15Þ
After enforcing the compatibility and equilibrium conditions on the interface DoFs, a derivation similar to Section 3.1.2 can
be followed to find the operational interface forces g
R
2
. These forces are now dependent on the properties of both the active
component A and the test rig R. Substituting the forces back into the second row of Eq. (15), the corresponding interface
displacements u
2
are obtained as well (the superscript is dropped because of compatibility)
g
R
2
¼ Y
A
22
þY
R
22
1
Y
A
21
f
1
ð16aÞ
u
2
¼ I Y
A
22
Y
A
22
þY
R
22
1
Y
A
21
f
1
ð16bÞ
8
>
>
<
>
>
:
Eqs. (16a) and (16b) hold for any Y
R
22
, as long as no external force is acting on the test rig. The desired set of equivalent forces
equation (12) is obtained by combining Eqs. (16a) and (16b) in such way that the dynamics of the test rig Y
R
22
are eliminated
f
eq
2
¼ Y
A
22
1
Y
A
21
f
1
⟹ f
eq
2
¼ g
R
2
þ Y
A
22
1
u
2
ð17Þ
As it turns out, Eq. (17) forms the sum of contributions of both the blocked force and the free velocity experiment,
respectively, Eqs. (13) and (14). One could therefore speak of a hybrid interface condition, or Robin boundary condition in a
mathematical sense. It can indeed be verified that Eqs. (16a) and (16b) converge
12
to the blocked forces for Y
R
→0 and to free
velocities when Y
R
-1.
The hybrid interface approach combining force and motion was originally published in [78]; the displacement term was
regarded in this work as the “non-rigid test bench compensation” to the imperfect blocked forces. Although generally
applicable in theory, it should be mentioned that the method can be rather costly and time-consuming in practice, as one
needs to explicitly measure collocated forces and motion in all directions for every interface node [101].
In lack of force measurement, one may substitute the third row of (15) for g
B
2
. The so obtained approach was suggested in
[102] and takes displacement measurement only
f
eq
2
¼ Y
R
22
1
u
2
þ Y
A
22
1
u
2
ð18Þ
The price for not having to measure interface forces is that separate FRF measurements should now be conducted, to obtain
the subsystem admittances of the active component and the test rig.
3.3.5. Component TPA: in situ
Looking again at Eq. (18) we observe that the two inverted admittance FRF matrices in fact represent the dynamic
stiffness matrices of respectively component A and R for the same set of collocated interface DoFs. They can be combined by
12
A intuitive presentation of the range between the two limit cases is given in [61].
M.V. van der Seijs et al. / Mechanical Systems and Signal Processing 68-69 (2016) 217–244226
simple impedance addition
13
f
eq
2
¼ Z
R
22
þZ
A
22
u
2
¼ Z
AR
22
u
2
ð19Þ
The result of Eq. (19) is indeed indifferent to the dynamics of R or any other mounting structure that component A is
connected to. Transforming back to admittance notation, we find that the blocked forces can be calculated inversely from
the admittance of the assembly's interface or, in an overdetermined fashion, using a sufficient set of indicator points u
4
on
the receiving subsystem
f
eq
2
¼ Y
AR
22
1
u
2
or f
eq
2
¼ Y
AR
42
þ
u
4
ð20Þ
Obtained from a different derivation, the approach of Eq. (20) has first been proposed by Moorhouse and Elliott [62,63] as
the in situ method. As implied by the name, the operational measurement may even be conducted in the target assembly AB,
avoiding dismounting of any part, namely
f
eq
2
¼ Y
AB
22
1
u
2
or f
eq
2
¼ Y
AB
42
þ
u
4
ð21Þ
The equivalent forces resulting from application of Eq. (21) are nonetheless a property of component A only and thus
transferable to an assembly with another passive side.
The in situ force determination procedure is illustrated in Fig. 5d. Indeed, Eq. (21) represents the inverse of Eq. (11) yet
with an extended set of response DoFs in order to render the system (over) determined and thus invertible. As a
consequence, the method shows great resemblance with the classical matrix-inverse method of Eq. (10) the difference being
the choice for the assembled admittance instead of the passive subsystem's admittance. Similar techniques regarding matrix
conditioning (e.g. over-determination, singular value rejection) apply as well to the in situ method. Most remarkable is that
the receiving side can be chosen arbitrarily, as the equivalent forces identified by Eq. (20) or (21) are theoretically invariant
of any subsystem coupled to it. In that respect, two important conditions need to be kept in mind:
1. Operational excitations f
1
may only originate from the domain of component A. Any excitation coming from the passive
side will disturb the determination of equivalent forces.
2. Although the responses used for the matrix inversion (u
2
or u
4
) can be positioned rather arbitrarily , they are bounded to the
domain of the interface and the passive side. This relates back to the remark made after Eq. (12) and discussed in Appendix A:
vibrations at the source structure are not only caused by the interface forces but also by the source excitations directly.
Physically one could interpret the in situ method as follows: knowing the transfer functions from the interface DoFs to several
points on the passive side, Eq. (20) or (21) seeks for a set of equivalent forces f
eq
2
that, applied to the interface DoFs of the
assembled structure, generates the same responses u
4
at the passive side. Given that this response set is overdetermined, the
equivalent forces are calculated such that they minimise the sum of the squared errors (or l
2
-norm) in the extended set
14
u
4
.
Several numerical and experimental studies of in situ source characterisation have been reported [98,102,107–110].A
time-domain force reconstruction algorithm was proposed by Sturm [111,112]; see also Section 4.5. Further generalisation of
the in situ concept can be recognised in the pseudo-forces method that is discussed next.
3.3.6. Component TPA: pseudo-forces
The last member of the component-based TPA family to discuss is the pseudo-forces method as proposed by Janssens and
Verweij in the late 1990s [64,65,113]. It assumes the existence of a non-unique set of pseudo-forces acting on the outer
surface of the active component, cancelling out the effect of the operational vibrations transmitted through the interface to
the passive side. This is illustrated in Fig. 6a. If those forces are now applied in the opposite direction to the assembly with
the source shut-down, an identical response at the receiving side should be obtained, see Fig. 6c. In other words, the
pseudo-forces are supposed to represent the source excitation for responses at the passive side.
The fact that it can be regarded as a component TPA method lies in the former assumption: if there exists a set of pseudo-
forces that cancels out the operational dynamics at the interface, the responses beyond this interface shall be zero as well,
hence these forces are invariant of any structure attached to it. A similar reasoning may as well be applied to the previously
discussed equivalent force methods, yet the locations of the pseudo-forces are not bounded to the interface but extend to
the full domain of the active component (see Fig. 6).
The actual determination of the pseudo-forces is carried out slightly differently. The first step is to define a set of s
pseudo-forces f
ps
on the active component that is (a) sufficient to represent the excitation source and (b) easily accessible
for impact hammer or shaker measurement. One could think of a minimum of s¼6 forces when the interface behaviour can
be considered as rigid, or a larger number in case of more intricate connectivity. Secondly, a set of n Z s indicator response
DoFs has to be chosen on the passive side from which the operational excitation is monitored. These responses are denoted
13
This is standard practice for FEM assembly.
14
Similar ideas are used in the field of experimental substructure decoupling: the identification of the force that decouples the residual substructure
can be improved by defining an “extended interface”, adding some additional DoFs on the structure of interest distant from the interface [104–106].
M.V. van der Seijs et al. / Mechanical Systems and Signal Processing 68-69 (2016) 217–244 227
by u
4
and can be written as a result of f
1
, similar to Eq. (9)
u
4
¼ Y
AB
41
f
1
¼ Y
B
42
Y
A
22
þY
B
22
1
u
free
2;f
1
with u
free
2;f
1
9Y
A
21
f
1
ð22aÞ
A substitution is made here using the free velocity caused by the excitation f
1
, or in other words: the theoretical motion at
the interface of component A if the interfaces were left free, see also Section 3.3.3. This is by no means a quantity that needs
to be measured, but will prove useful for the derivation later on.
The next step is to determine pseudo-forces f
ps
that best recreate the operational responses at the indicator DoFs u
4
when the source excitation is shut down. Note that the assumption is similar to the equivalent forces problem stated in
Section 3.3.1, yet the pseudo-forces act on the outer surface of the active component rather than its interfaces. Similar to
Eq. (22a), the response at the indicator DoFs as a result of the pseudo-forces reads
u
4
¼ Y
AB
4ps
f
ps
¼ Y
B
42
Y
A
22
þY
B
22
1
u
free
2;f
ps
with u
free
2;f
ps
9Y
A
21
f
ps
ð22bÞ
Now, the pseudo-forces f
ps
that best describe the operational source excitations are found from solving an overdetermined
system with the response set u
4
that was measured under operation (see Fig. 6b)
f
ps
¼ Y
AB
4ps
þ
u
4
ð23Þ
It can be shown that these forces, under certain conditions, are a property of component A only. A sufficient condition is that
the free interface velocities u
free
2
as a result of the original source excitation can be fully represented by the set of pseudo-
forces determined from Eq. (23). In that case the definitions of Eqs. (22a) and (22b) may be equated and one finds that the
pseudo-forces are specific for the source component
u
free
2;f
ps
¼ u
free
2;f
1
⟹ f
ps
¼ Y
A
2ps
þ
Y
A
21
f
1
ð24Þ
Applying these pseudo-forces to the FRFs of an assembled system of interest, i.e. Y
AB
3ps
, it can be verified that the correct
receiver responses u
3
at the passive side are obtained
u
3
¼ Y
AB
3ps
f
ps
¼ Y
B
32
Y
A
22
þY
B
22
1
Y
A
2ps
Y
A
2ps
þ
Y
A
21
f
1
¼ Y
AB
31
f
1
ð25Þ
The pseudo-forces determined from Eq. (23) are indeed transferable to an assembly with another passive side. Eq. (25)
yields the responses for this new assembly, provided that the columns of the FRF matrix Y
AB
3ps
correspond to the same
pseudo-forces, i.e. excitation points at the source.
The previously discussed in situ method can be regarded as a special case of the pseudo-force method, namely for the
case where the pseudo-forces are located at the interfaces. The pseudo-forces calculated from Eq. (23) shall then equal the
equivalent (blocked) forces from Eq. (12), namely f
ps
¼ f
eq
2
. With regard to the positioning of the indicator DoFs u
4
, the same
restriction holds as for the in situ method, namely that they must be located at the passive system B or at the interface.
15
A related idea was proposed [78] to calculate pseudo-forces (or substitute forces in the original work) from equivalent
forces that were in turn obtained from test bench measurements
f
ps
¼ Y
A
2ps
þ
Y
A
22
f
eq
2
To find the receiver responses u
3
, Eq. (25) can now be applied instead of Eq. (11). This is useful if Y
AB
32
is unmeasurable,
namely when the interfaces of the assembled system AB are inaccessible for FRF measurement.
Fig. 6. The concept of the pseudo-forces component TPA method. (a) Pseudo-forces cancelling out the operational excitation at the interface. (b) Inverse
determination of the pseudo-forces under operation. (c) Application of pseudo-forces to the assembly, causing the same response.
15
This was erroneously stated in the original work [65].
M.V. van der Seijs et al. / Mechanical Systems and Signal Processing 68-69 (2016) 217–244228
3.4. Transmissibility-based TPA
The two previously discussed families of TPA have in common that they attempt to model the vibration transmission in a
physically correct sense, namely by determining as many forces and moments as required to describe the subsystem
connectivity in full. Consequently, both families explicitly require the corresponding FRFs for the interface DoFs to the
receiving response locations. It is evident that this approach ultimately reveals a wealth of information on the particular
functioning of the active component, force distribution over the interface, resonances in the structure, etc. If however the
mere purpose of a TPA is to identify the dominant path contributions in the assembled product, the efforts to set up and
conduct the full experiment can be fairly disproportionate and costly. This is especially the case when multiple incoherent
vibration sources are to be investigated.
The last family to discuss avoids the stage of explicit force determination. Instead the path contributions are determined
from so-called “transmissibilities” between sensors, possibly calculated from operational measurements. Various methods
are discussed in the following sections that share the following properties:
1. Measurements are conducted on the assembled product only, saving time to dismount the active components. In fact the
interfaces between the active and passive components are no longer of principal interest.
2. Path contributions are determined from well-chosen indicator points around the sources or connections. These indicator
points function as inputs to the TPA.
3. The result of the analysis is highly subject to the choice for these indicator points; care should therefore be taken to
include all transmission paths.
The family of transmissibility-based TPA methods indeed departs from the traditional source-transfer-receiver model that
assumes a physically meaningful set of loads, FRFs and responses. Although potentially less accurate, methodologies derived
from this concept tend to be easy to set up, versatile concerning sensor type and particularly effective for ranking
contributions from several sources. From a practical point of view, transmissibility-based TPA tries to outrun the physically
correct methods by its ability to conduct multiple cycles in shorter time. Nevertheless, under certain conditions, results of
similar (or even equivalent) accuracy can be achieved with classical and component-based TPA methods.
3.4.1. The transmissibility concept
To discuss the theoretical concepts behind transmissibility-based methods, consider an assembled system AB with two
connection points
16
as shown in Fig. 7. The active side contains a vibration mechanism that is characterised by internal
forces f
1
; the receiver responses at the passive side are denoted by p DoFs in u
3
. To monitor the vibrations transmitted across
the interfaces, n indicator DoFs u
4
are positioned around the connection points.
Let us first assume that the source excitation f
1
can be described by o forces (or independent force distributions) and that
FRFs are measurable for all of the above-mentioned DoFs. The equations for the passive-side responses then read
u
3
¼ Y
AB
31
f
1
for p receiver DoFs ð26aÞ
u
4
¼ Y
AB
41
f
1
for n indicator DoFs ð26bÞ
(
Provided that nZ o and Y
AB
41
is full rank, all excitation forces f
1
are observable from u
4
. Hence Eq. (26b) can be inverted and
substituted into Eq. (26a). The responses u
3
are now expressed in terms of the DoFs u
4
that can be measured under
operation:
u
3
¼ T
AB
34;f
1
u
4
with T
AB
34;f
1
9Y
AB
31
Y
AB
41
þ
ð27Þ
The so obtained transmissibility matrix T
AB
34;f
1
relates motion at the indicator DoFs u
4
(the inputs) to the receiver DoFs u
3
(the
outputs) for excitation forces f
1
. Interestingly, the size of the transmissibility matrix has become p n, obfuscating the o
excitations associated with the original FRFs. This raises the question which excitations are really needed to construct the
transmissibility matrix and to what extend this matrix is generally valid for the problem ðu
4
-u
3
Þ under different excitations
of the source structure.
To gain more insight in the transmissibility problem stated by Eq. (27), let us expand Eqs. (26a) and (26b) in terms of the
subsystems' admittances. As seen in previous derivations, the terms that couple the subsystems are identical for all
responses ðu
3
; u
4
Þ at the passive side
u
3
u
4
"#
¼
Y
AB
31
Y
AB
41
"#
f
1
¼
Y
B
32
Y
B
42
"#
Y
A
22
þY
B
22
1
Y
A
21
f
1
16
An example with two connection points was chosen here merely to provide better insight into some important cross-correlation properties. There is
no fundamental consequence for the generality of the methods derived.
M.V. van der Seijs et al. / Mechanical Systems and Signal Processing 68-69 (2016) 217–244 229
Recalling now the expressions for the interface force and free velocity, respectively Eqs. (4) and (14), the following
substitutions can be made:
u
3
u
4
"#
¼
Y
B
32
Y
B
42
"#
g
B
2
with
g
B
2
¼ Y
A
22
þY
B
22
1
u
free
2
for m interface forces
u
free
2
¼ Y
A
21
f
1
for m free velocities
8
>
<
>
:
ð28Þ
Hence, Eq. (28) shows that the transmission of vibrations from o forces f
1
to n responses u
4
is limited by the number of DoFs
of the interface forces/displacements m. This means that the interface acts as a bottleneck: it limits the effective rank of the
transmissibility problem to a maximum of m. Furthermore, Eq. (28) exposes two interesting properties of the
transmissibility concept:
1. Regarding the source excitation, Eq. (28) shows a direct relation between the interface forces g
B
2
of the coupled system and
the theoretical free velocities u
free
2
at the disconnected interfaces of component A. As understood from the component-
based TPA methods, various sets of forces can be defined that equivalently produce these free interface velocities u
free
2
,such
as the pseudo-forces of Eq. (22b). This is illustrated in Fig. 7b. In fact, any set of forces on the source that renders the
interface fully controllable will guaran