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Component replacement transfer path analysis

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In-situ Transfer Path Analysis (TPA) is a diagnostic method used to analyse the propagation of noise and vibration through complex built-up structures. Its defining feature is the independent characterisation of an assembly's vibratory source in terms of its blocked force, an invariant property that is unchanged by the dynamics of neighbouring components. This invariance enables downstream structural modifications to be made to an assembly, without affecting the source's operational characteristics. Modifications made upstream of the defined source-receiver interface, however, are prohibited, as they would lead to a change in the blocked force. Note that the source-receiver interface is somewhat arbitrary, and typically chosen for convenience rather than to satisfy some physical distinction (e.g. resilient mounts are often included as part of a source definition). To this end, in the present paper we are interested in computing the modification of a 'source' given the replacement of one of its constituent components (e.g. installing new resilient mounts).
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Component replacement transfer path analysis
J.W.R. Meggitt 1, A.S. Elliott 1, A.T. Moorhouse 1, A. Jalibert 2, G. Franks 3
1Acoustics Research Centre, University of Salford,
Greater Manchester, M5 4WT, England
e-mail: j.w.r.meggitt1@salford.ac.uk
2Bentley Motors Ltd., Pyms Lane,
Crewe, CW1 3PL, England
3Bruel & Kjaer Sound & Vibration Engineering Services,
Millbrook, Bedfordshire, MK45 2YT , England
Abstract
In-situ Transfer Path Analysis (TPA) is a diagnostic method used to analyse the propagation of noise and
vibration through complex built-up structures. Its defining feature is the independent characterisation of
an assembly’s vibratory source in terms of its blocked force, an invariant property that is unchanged by
the dynamics of neighbouring components. This invariance enables downstream structural modifications
to be made to an assembly, without affecting the source’s operational characteristics. Modifications made
upstream of the defined source-receiver interface, however, are prohibited, as they would lead to a change
in the blocked force. Note that the source-receiver interface is somewhat arbitrary, and typically chosen for
convenience rather than to satisfy some physical distinction (e.g. resilient mounts are often included as part
of a source definition). To this end, in the present paper we are interested in computing the modification of a
‘source’ given the replacement of one of its constituent components (e.g. installing new resilient mounts).
1 Introduction
Transfer path analysis (TPA) is a diagnostic method used for analysing the propagation of noise and vibration
in complex built-up structures, for example, ships, vehicles, trains, etc. It has become an essential tool
in the development and refinement of structures whose vibro-acoustic response is of interest. There exist
many varieties of TPA, differing in their implementation and interpretation [1]. In the present paper we are
concerned with the variant known widely as in-situ TPA (also blocked force TPA) [2].
In an in-situ TPA the active components of an assembly (i.e. the vibration sources) are each characterised
by their blocked force, i.e. the force required to constrain their interface such that their velocity (also dis-
placement/acceleration) is zero. The blocked force independently describes the operational activity of a
vibration source; it does not depend on what the source is connected to. This is advantageous as it means
that the blocked force can be used in conjunction with structural modification techniques, where the receiver
structure is modified in some way.
Once characterised, the contribution of each active component to an operational response is determined using
transfer functions measured between the source-receiver interface and the chosen target degree of freedom
(e.g. sound pressure level at the driver position). Based on their relative contributions an engineer is able to
identify troublesome vibration sources and make appropriate design changes.
Application of in-situ TPA first requires the definition of a source-receiver interface; this interface implicitly
defines what is considered as the ‘source’ of vibration. The source-receiver interface is somewhat arbitrary,
and is typically chosen for convenience rather than to satisfy some physical distinction; as a practical exam-
ple, vibration isolators are often included as part of the source definition (i.e. the source-receiver interface
is defined as being that between the isolator and the receiver, as opposed to the source and the isolator). Al-
though this interface is perfectly admissible, it does place a limitation on what structural modifications can
be investigated. Although downstream modifications may be applied without affecting the blocked force,
upstream modifications (i.e. those above the source-receiver interface) are prohibited as these will lead to a
change in the blocked force.
Given the often arbitrary positioning of the source-receiver interface, we are interested in extending the
possibility of structural modification upstream, such that a constituent source sub-component (e.g. a resilient
coupling that has been included in the source definition) can be modified or replaced. In the present paper we
will employ transmissibility theory to modify the blocked force and forward transfer functions obtained from
an initial assembly according to a known component replacement. The resulting equations make possible
the upstream structural modification of a structure.
The remainder of this paper will be organised as follows. Section 2 will begin by briefly introducing in-situ
TPA before section 3 describes the proposed Component Replacement TPA method; sections 3.1 and 3.2
will present the modified blocked force and forward transfer function equations. A numerical validation will
be provided in section 4 before section 5 presents a demonstrative experimental application. Finally, section
6 will draw some concluding remarks.
2 In-situ Transfer Path Analysis
In-situ Transfer Path Analysis aims to identify the dominant sources of vibration that contribute to a partic-
ular response (e.g. vehicle cabin sound pressure level) by first characterising their operational activity, and
then the transfer paths through which they contribute. In-situ TPA differs from classical TPA in that the
source activity is characterised independently using a blocked force, as opposed to a contact force (which is
dependent on the dynamics of the receiver structure).
In an in-situ TPA the blocked force is obtained using the inverse relation [3],
¯
fS
c=YC
bc1vb(1)
where, with reference to figure 1a: vbis an operation velocity measured on the coupled assembly at the in-
dicator DoFs (degrees of freedom) b;YC
bc is the mobility matrix of the coupled assembly, measured between
the indicator and (defined) interface DoFs, band c, respectively; and ¯
fS
cis the sought after blocked force.
Implementation of equation 1 requires a two part measurement procedure. In part 1, the source is turned off
and the mobility matrix YC
bc is measured. In part 2, the source is operated and the velocity vbmeasured at
the indicator DoFs.
Once the blocked force has been obtained it may be used to predict the operational response at some reference
DoF in the assembly. This forward response prediction is given by,
pr
vr
=
HC
rc
YC
rc
¯
fS
c(2)
where: prand vrare the operational pressure and velocity predictions; HC
rc and YC
rc are the vibro-acoustic
transfer function and transfer mobility, respectively, between the interface and reference DoFs, cand r; and
¯
fS
cis the acquired blocked force from equation 1. The blocked force is an independent source property, and
so it can also be used to predict the operational response in an assembly different to the one in which it was
characterised, provided that its coupled transfer functions are known.
S
o
R
c
¯
fS
c
a
r
b
(a)
S
o
a
c1
I
c2
R
b
r
(b)
Figure 1: Diagrams of general assemblies; (a) - Source-Receiver assembly, (b) - Source-Mount-Receiver
assembly.
Together, equations 1 and 2 constitute what is known as in-situ/blocked force TPA. In the more general case
that HC
rc (or YC
rc) is predicted using dynamic sub-structuring, this is typically referred to as component-based
TPA [1] (also known as Virtual Acoustic Prototyping [4])).
3 Component Replacement Transfer Path Analysis
Often when performing an in-situ TPA, access to the preferred source-receiver interface is limited and the
required measurements cannot be undertaken. In this case it is possible to redefine the source-receiver
interface elsewhere, typically further downstream (i.e. into the receiver structure). By doing so we redefine
the source and receiver to include/exclude the appropriate ‘components’. Take for example the source-
mount-receiver assembly in figure 1b. Although the interface c1may be the preferred choice, as it is the
natural source interface, the interface c2may equally be considered. In this case, the source definition is set
to include the mount, which is now excluded from the receiver definition.
Although its definition is somewhat arbitrary, if we wish to investigate the effect of replacing (or modifying)
a component it must be located downstream of the defined source-receiver interface (i.e. within the receiver
structure). Only then may standard dynamic sub-structuring/structural modification techniques be used. Any
modifications made upstream of the source-receiver interface will affect the source definition and in turn
cause a change in the blocked force. For example, suppose the mount in figure 1b were replaced by some
other coupling; the blocked force at interface c2would be modified accordingly.
When performing an in-situ TPA, one must also consider the forward transfer function between the defined
source-receiver interface and the chosen target DoFs. Clearly, any upstream modification will have an influ-
ence on this also.
In the present paper we are interested in the replacement (or modification) of an upstream component Iin
the source definition, and predicting its effect on the blocked force ¯
fSI
c2, the forward transfer function YC
rc2,
and the target response prediction pr.
The proposed Component Replacement TPA requires first that a primary interface c1is defined. Any com-
ponent modifications must take place downstream of this primary interface. The blocked force, defined (or
obtained) at some secondary interface c2, is then related to the primary interface blocked force ¯
fS
c1through a
blocked force transmissibility, taking into account the properties of the initial assembly. The primary inter-
face blocked force will be independent of any modifications made upstream of the secondary interface c2, so
long as they remain downstream of the primary interface. By using a second blocked force transmissibility,
based on a modified assembly, a modified blocked force at the secondary interface can be obtained. A similar
procedure is used to obtain the modified transfer function matrix.
The modified blocked force and forward transfer function equations will be presented in sections 3.1 and 3.2,
respectively. Detailed derivations are omitted for brevity. Section 4 will then present a numerical validation
of the proposed Component Replacement TPA before section 5 demonstrates an experimental application.
3.1 Modified Blocked Force
Consider the assembly shown in figure 1 whose source is excited externally (or internally) at some DoF a
(or o). Note that this excitation can be represented by an equivalent excitation at the primary interface c1,
corresponding to a negative blocked force [5],
¯
fS
c1
0
0
=
ZSI
c1c1ZI
c1c20
ZI
c2c1ZIR
c2c2ZR
c2r
0 ZR
rc2ZR
rr
vc1
vc2
vr
.(3)
By applying an appropriate blocking force at the secondary interface c2, we are able to enforce the constraint
vc2=0(and consequently vb=0),
¯
fS
c1
¯
fSI
c2
0
=
ZSI
c1c1ZI
c1c20
ZI
c2c1ZIR
c2c2ZR
c2R
0 ZR
Rc2ZR
RR
vc1
0
0
.(4)
From the top row of equation 4 we obtain,
vc1=ZSI
c1c11¯
fS
c1(5)
which upon substitution into the second row yields,
¯
fSI
c2=ZI
c2c1ZSI
c1c11¯
fS
c1.(6)
Equation 6 relates the primary and secondary interface blocked forces (due to an excitation at a), through
what may be interpreted as a blocked force transmissibility,
¯
Ta
c2c1=ZI
c2c1ZSI
c1c11(7)
where the over-bar ¯is used to denote a blocked force transmissibility (as opposed to a velocity transmis-
sibility, for example). Note that the primary interface blocked force, ¯
fS
c1, is by definition independent of the
components Iand R, whilst the secondary interface blocked force, ¯
fSI
c2, is independent of only the receiver
component R.
By considering the inverse of the above we can formulate an equivalent expression that relates the secondary
interface blocked force to that of the primary,
¯
fS
c1=ZSI
c1c1ZI
c2c11fSI
c2.(8)
Suppose we obtain (experimentally) a secondary interface blocked force in an initial assembly with the
component I1installed, ¯
fSI1
c2. From equation 8, we can obtain the primary interface blocked force ¯
fS
c1. Then,
using equation 6 for a modified assembly with component I2installed, we can obtain the blocked force ¯
fSI2
c2.
All together we have that,
¯
fSI2
c2=ZI2
c2c1ZSI2
c1c11ZSI1
c1c1ZI1
c2c11¯
fSI1
c2(9)
where the superscripts 1 and 2 are used to denote the initial and modified assemblies, respectively. Equation
9 provides an exact modification of the blocked force ¯
fSI
c2, due to the component replacement/modification
I1I2.
Substituting the coupled SI impedance for the summed component impedances, ZS I
c1c1=ZS
c1c1+ZI
c1c1,
equation 9 may be rewritten as,
¯
fSI2
c2=ZI2
c2c1ZS
c1c1+ZI2
c1c11ZS
c1c1+ZI1
c1c1ZI1
c2c11¯
fSI1
c2(10)
where it is noted that the source impedance is unchanged between the two assemblies. By simply adding and
subtracting the initial coupling component point impedance within the left most matrix inversion,
¯
fSI2
c2=ZI2
c2c1ZS
c1c1+ZI2
c1c1+ZI1
c1c1ZI1
c1c11ZS
c1c1+ZI1
c1c1ZI1
c2c11¯
fSI1
c2
we arrive at,
¯
fSI2
c2=ZI2
c2c1ZSI1
c1c1+ZI2
c1c1ZI1
c1c11ZSI1
c1c1ZI1
c2c11¯
fSI1
c2.(11)
Equation 11 provides an exact modification of the blocked force, requiring the impedance characteristics of
the coupling component Iand source S, the latter of which may be obtained from experiment through the
free source mobility. As we are considering the replacement/modification of the coupling component, it is
reasonable to assume that its properties are available. These may be obtained numerically, for example by
FE modelling, or experimentally, for example by the in-situ approach [6] or sub-structure decoupling [7].
3.1.1 Special Case - Resilient Coupling
In the special case that the coupling component being replaced constitutes some form of isolation (e.g. a
resilient coupling), we can assume an impedance mismatch between the source and coupling component.
Mathematically we can express this assumption in the following form,
ZSI
c1c1=ZS
c1c1+κΛI(12)
where: ZI
c1c1=κΛIis the coupling component point impedance, κis some small numerical constant, and
ΛIis an appropriate unscaled matrix. Assuming that κΛIis sufficiently small, a (first order) Taylor series
expansion can be used to approximate the inverse of the coupled impedance ZS I
c1c1,
ZSI
c1c11
ZS
c1c11
κZS
c1c11ΛIZS
c1c11.(13)
Substitution of equation 13 into equation 9, after some manipulations, yields,
¯
fSI2
c2ZI2
c2c1I+YS
c1c1ZI1
c1c1ZI2
c1c1ZI1
c2c11¯
fSI1
c2.(14)
Equation 14 constitutes a first order approximation of the modified blocked force ¯
fSI2
c2, due to the component
replacement I1I2. In the case that an identical mount is used (I1=I2) the above yields, ¯
fSI2
c2=¯
fSI1
c2, as
expected. Although simplified, in its current form equation 14 still requires the free source mobility YS
c1c1
(its inversion, however, is avoided). Again, this may be obtained from experiment or via numerical modeling.
Noting that the free source mobility in equation 14 is multiplied by the small quantity that is the component
impedance difference ZI1
c1c1ZI2
c1c1, it is argued that, provided that the mounts are not too dissimilar, a
reasonable approximation can be obtained from the zeroth order term alone,
¯
fSI2
c2ZI2
c2c1ZI1
c2c11¯
fSI1
c2.(15)
Equation 15 constitutes a zeroth order approximation to the modified blocked force ¯
fSI2
c2, due to the compo-
nent replacement I1I2. Equation 15 requires only the transfer impedance of the two coupling compo-
nents, and thus avoids the need to perform any additional measurements over and above those required for a
standard in-situ TPA.
In summary, equations 9, 14, and 15, provide, respectively, an exact, first order, and zeroth order approxima-
tion to the modified blocked force ¯
fSI
c2due to the component replacement I1I2.
3.2 Modified Forward Transfer Function
Equations 9, 14, and 15, provide relations to modify the secondary interface blocked force ¯
fSI
c2given the
component replacement I1I2. To make an in-situ TPA response prediction in this modified assembly,
the modified blocked force must be accompanied by an appropriately modified forward transfer function
YC2
rc2(i.e. taking into account the component replacement). In this section we will present an appropriate
modification of the initial forward transfer function YC1
rc2to accompany the modified blocked force ¯
fSI2
c2.
We begin by recalling the invariant properties of the transmissibility [8]. It can been shown that the velocity
transmissibility between the interface DoFs c2and the remote receiver DoF r, due to an applied force at c2,
Tc2
rc2, is independent of the forcing applied. Consequently, the same transmissibility Tc2
rc2will be obtained
whether or not the SI assembly is attached to the receiver R. As such, we have that,
YC
rc2YC
c2c21=YR
rc2YR
c2c21=Tc2
rc2.(16)
Making use of this invariance, a second equation can be established for a new assembly whose coupling
component has been replaced/modified,
YC2
rc2YC2
c2c21=YR
rc2YR
c2c21.(17)
The left hand sides of equations 16 and 17 can now be equated (introducing the superscript C1for the initial
assembly). Post-multiplication by YC2
c2c2, after some further manipulations, yields,
YC2
rc2=YC1
rc2YC1
c2c21ZC1
c2c2+ZSI2
c2c2ZSI1
c2c21.(18)
Equation 18 provides an exact modification of the forward transfer function YC1
rc2, due to the component
replacement/modification I1I2.
3.2.1 Special Case - Resilient Coupling
After some manipulation (including a first order Taylor expansion for right most matrix inversion) we arrive
at the simplified equation,
YC2
rc2YC1
rc2IZSI2
c2c2ZSI1
c2c2YC1
c2c2.(19)
Equation 19 constitutes a first order approximation of the modified forward transfer function YC2
rc2, due to
the component replacement I1I2.
It is important to note that the impedance terms ZSI2
c2c2and ZSI2
c2c2are properties of the SI assembly, and
include dynamic contributions from both the source and the coupling component. As such, these are not
directly available from experiment.
We are interested in simplifying the impedance difference ZSI2
c2c2ZSI1
c2c2such that it can be expressed in
terms of available quantities. Under the present assumptions (i.e. an impedance mismatch) the source will
have a high impedance compared to the coupling component. As such, the coupling component will be
approximately blocked at the primary interface, and ZSI
c2c2ZI
c2c2.
Finally, from equation 19 a zeroth order approximation can be identified as simply,
YC2
rc2YC1
rc2(20)
i.e. in the case of a high impedance mismatch the component replacement has no effect on the forward
transfer function.
In summary, equations 18, 19, and 20, provide, respectively, an exact, first order, and zeroth order approxi-
mation to the modified forward transfer function YC2
rc2due to the component replacement I1I2.
4 Numerical Case Study
The purpose of this numerical example is to demonstrate the exact and approximate modifications of the
blocked force and forward transfer function, due to an upstream component replacement.
Source Coupling Receiver
c1c2
a r
Figure 2: Diagram of numerical simulation; two steel beams coupled via a third resilient beam.
The study considered is shown diagrammatically in figure 2; two free-free beams (source and receiver) are
coupled via a third (resilient) beam element. As above, we identify the primary and secondary interfaces as
c1and c2, respectively. We are interested in predicting the blocked force at, and the forward transfer function
from, the secondary interface c2based on the modification of some initial assembly. This modification will
be the replacement of the resilient coupling by another of different geometry and material properties. Results
will be compared against those obtained directly from the modified assembly. The geometry and material
properties of each beam element are given in table 1.
Table 1: Beam element properties - llength, EYoung’s modulus, ρdensity, and ηloss factor. All beam
elements are given the a thickness hof 0.01 m and a width wof 0.1 m.
Element l[m] E[GPa] ρ[kg/m3]η
Source 1 200 7800 0.05
Receiver 0.5 200 7800 0.1
Coupling (init) 0.2 2 2000 0.1
Coupling (mod) 0.3 0.2 4000 0.1
The initial blocked force ¯
fSI1
c2C2is obtained using the inverse relation of equation 1 based on the mobility
of the coupled SI R (source-isolator-receiver) assembly. The initial forward transfer function YC1
rc2C1×2
is obtained from the same assembly. These are shown in grey in figures 3 and 4, respectively. Two plots are
shown in each figure, corresponding to the blocked force (top plot) and the blocked moment (bottom plot) in
figure 3, and their associated transfer functions in figure 4. The same procedure is followed for the modified
assembly, and the true blocked force and transfer function are obtained. These are shown in orange in figures
3 and 4, respectively.
Comparison of the initial and (true) modified blocked force in figure 3 demonstrates the large affect a mod-
ification of the coupling element can have on the blocked force at the secondary interface. Comparison of
the initial and (true) modified forward transfer function in figure 4 suggests this modification has less of an
effect on the forward transfer function. This is to be expected given the resilient nature of the coupling.
Also shown in figure 3 are the exact (yellow), first order (purple), and zeroth order (green) blocked force
modifications. As expected, the exact modification is in near perfect agreement with the true blocked force.
The first order approximation can be seen to provide a good estimation of the blocked force across most of
the frequency range, although there are some notable discrepancies (see for example 150 Hz). The zeroth
order approximation provides a reasonable approximation given its simplicity.
102103
103
101
(a)
Force [N]
Initial True Exact 1st order 0th order
102103
104
102
(b)
Frequency [Hz]
Moment [Nm]
Figure 3: Modified blocked force at the interface c2obtained using the Component-Replacement TPA ap-
proach and its approximations; top: blocked force, bottom: blocked moment. Grey plot corresponds to the
blocked force obtained for the initial assembly. Remaining plots correspond to the new assembly with a
mount replacement.
102103
105
104
103
102
Mobility [ms1/N]
Initial True Exact 1st order
102103
104
103
102
101
Frequency [Hz]
Mobility [ms1/Nm]
Figure 4: Modified forward transfer function between the interface and reference DoFs c2and robtained us-
ing the Component-Replacement TPA approach and its approximations; top: force-velocity mobility, bottom:
moment-velocity mobility. Grey plot corresponds to the transfer function obtained for the initial assembly.
Remaining plots correspond to the new assembly with a mount replacement.
Shown in figure 4 are the exact (yellow), and first order (purple) transfer function modifications (the zeroth
order approximation corresponds to the unmodified transfer function, shown in grey). Again, the exact
modification is in near perfect agreement with the true transfer function. The first order approximation
can be seen to provide a noticeable improvement over the unmodified transfer function (i.e. zeroth order
approximation).
These results demonstrate the validity and application of the modified blocked force and forward transfer
function equations. In the following section an experimental case study will be presented.
5 Industrial Case Study
Presented in this section is a demonstrative example of the proposed Component Replacement TPA method
applied to a luxury vehicle. Note that the predictions presented here could not be independently validated
due to practical constraints; this example is intended therefore to serve as a demonstrative application only,
as opposed to a validation of the method, which was provided by section 4 through a numerical example.
In the present example we are interested in the structure-borne contribution of a gearbox to the acoustic
response inside the cabin. An in-situ TPA was performed on the vehicle including engine, gearbox and
suspension contributions.
100 1000
Frequency [Hz]
Stiness
Mount 1
Mount 2
Figure 5: Dynamic stiffness of initial and replacement mount.
Due to restricted access, the gearbox interface was defined below its (nominally identical) resilient supports.
Consequently, the blocked forces obtained include the resilient element as part of the source. Using the
proposed Component Replacement TPA approach, we are able to modify the acquired blocked force to
account for a replacement of the resilient elements with another type. From this we are able to predict and
auralise the modified contribution of the gearbox due to its new (virtual) resilient supports.
To apply a mount replacement the original and replacement supports must first be characterised so as to
obtain their independent transfer impedances ZI1
c1c2and ZI2
c1c2. This characterisation was performed using the
in-situ method [6, 9] taking into account the x,yand ztranslational DoFs. The vertical transfer stiffness for
the two mount types are shown in figure 5. When performing the mount replacement it was further assumed
that the x,yand zDoFs were uncoupled, such that ZI
c1c2C3×3was a diagonal matrix, containing the
transfer impedance of each DoF along the diagonal (this assumption is not a requirement of the method).
The modified blocked force was then estimated using the zeroth order approximation. In line with the zeroth
order approximation the forward transfer function as assumed to be unaffected by the mount replacement.
Shown in figure 6 are the gearbox contributions, in both the time and frequency domain, predicted using the
original in-situ TPA data (in blue), and a modified blocked force obtained using the Component Replacement
TPA method (in orange). The virtual mount replacement clearly decreases the transmitted vibration, partic-
ularly at low and high frequencies. Although we are unable to provide a validation of this result it appears
sensible, given that the replacement mount is considerably softer than the initial mount in these ranges.
0 1 2 3 4 5 6 7 8 9 10 11 12 13
0.5
0
0.5
Time [s]
Sound Level
Original TPA
Mount replacement TPA
100 1000
105
103
101
101
103
Frequency [Hz]
Sound Level
Figure 6: Time (top) and frequency(bottom) domain response of vehicle run up due to gearbox contribution.
In blue is the original in-situ TPA prediction, and in orange is the prediction based on a mount replacement.
Note that for confidentiality reasons scales are normalized to an unspecified value.
6 Conclusions
In the present paper we propose a novel Component-Replacement TPA (CR-TPA) methodology which en-
ables the upstream modification of an assembly, e.g. the replacement of resilient mounts in a source def-
inition. The method is based on the modification of blocked forces and forward transfer functions using
transmissibility functions that characterise the initial and modified assemblies. The necessary blocked force
transmissibilities may be approximated to various degrees, and used to investigate the effect of different
structural modifications. Equations are presented for exact, first, and zeroth order approximations. First and
zeroth order approximations are valid only in the presence of a resilient coupling. A key result is that for a
zeroth order approximation a straight forward modification of the blocked force is obtained, requiring only
the transfer impedance of the initial and modified coupling elements. This information is easily obtainable
from experimental measurements or numerical modeling.
The CR-TPA method is validated numerically using a coupled beam simulation. It is shown that the blocked
force, and to a lesser extent the forward transfer function, is affected by a change in the material properties
of a coupling element, and that the CR-TPA method is able to predict the resulting changes exactly, if the
appropriate FRFs are available. When, as is often the case, the full set of FRFs are not available the method
provides an approximate prediction based on a subset of FRFs that are available from measurement with
good accuracy. An industrial case study is presented using real experimental data to demonstrate a practical
application of the proposed method.
Acknowledgements
This work was funded through the EPSRC Research Grant EP/P005489/1 Design by Science.
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... Pre-multiplication of Eq. (13) by the inverse mobility matrix Y C c 2 c 1 −1 then yields, (18) , which required the free source mobility, Eq. (19) requires the coupled impedance of the initial SI assembly, Z SI 1 c 1 c 1 . This is available experimentally if excitations can be performed at both the primary and secondary interface. ...
... Although simplified, in its current form Eq. (31) still requires the free source mobility Y S c 1 c 1 . However, it avoids the need for a double matrix inversion, as in Eq. (18) . Matrix inversions are notoriously sensitive to experimental error and inconsistencies. ...
... As such, providing there is a sufficient impedance mismatch, Eq. (31) may provide a more reliable estimate than Eq. (18) in the presence of experimental uncertainty. ...
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