Content uploaded by Joshua W.R. Meggitt

Author content

All content in this area was uploaded by Joshua W.R. Meggitt on Sep 09, 2020

Content may be subject to copyright.

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

bc−1vb(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

c1c1−1¯

fS

c1(5)

which upon substitution into the second row yields,

¯

fSI

c2=−ZI

c2c1ZSI

c1c1−1¯

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

c1c1−1(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

c2c1−1fSI

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

c1c1−1ZSI1

c1c1ZI1

c2c1−1¯

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

I1→I2.

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

c1c1−1ZS

c1c1+ZI1

c1c1ZI1

c2c1−1¯

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

c1c1−ZI1

c1c1−1ZS

c1c1+ZI1

c1c1ZI1

c2c1−1¯

fSI1

c2

we arrive at,

¯

fSI2

c2=ZI2

c2c1ZSI1

c1c1+ZI2

c1c1−ZI1

c1c1−1ZSI1

c1c1ZI1

c2c1−1¯

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

c1c1−1

≈ZS

c1c1−1

−κZS

c1c1−1ΛIZS

c1c1−1.(13)

Substitution of equation 13 into equation 9, after some manipulations, yields,

¯

fSI2

c2≈ZI2

c2c1I+YS

c1c1ZI1

c1c1−ZI2

c1c1ZI1

c2c1−1¯

fSI1

c2.(14)

Equation 14 constitutes a ﬁrst order approximation of the modiﬁed blocked force ¯

fSI2

c2, due to the component

replacement I1→I2. 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

c1c1−ZI2

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

c2≈ZI2

c2c1ZI1

c2c1−1¯

fSI1

c2.(15)

Equation 15 constitutes a zeroth order approximation to the modiﬁed blocked force ¯

fSI2

c2, due to the compo-

nent replacement I1→I2. 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 I1→I2.

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 I1→I2. 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

c2c2−1=YR

rc2YR

c2c2−1=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

c2c2−1=YR

rc2YR

c2c2−1.(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

c2c2−1ZC1

c2c2+ZSI2

c2c2−ZSI1

c2c2−1.(18)

Equation 18 provides an exact modiﬁcation of the forward transfer function YC1

rc2, due to the component

replacement/modiﬁcation I1→I2.

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

rc2≈YC1

rc2I−ZSI2

c2c2−ZSI1

c2c2YC1

c2c2.(19)

Equation 19 constitutes a ﬁrst order approximation of the modiﬁed forward transfer function YC2

rc2, due to

the component replacement I1→I2.

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

c2c2−ZSI1

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

c2c2≈ZI

c2c2.

Finally, from equation 19 a zeroth order approximation can be identiﬁed as simply,

YC2

rc2≈YC1

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 I1→I2.

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

c2∈C2is 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

rc2∈C1×2

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

10−3

10−1

(a)

Force [N]

Initial True Exact 1st order 0th order

102103

10−4

10−2

(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

10−5

10−4

10−3

10−2

(a)

Mobility [ms−1/N]

Initial True Exact 1st order

102103

10−4

10−3

10−2

10−1

(b)

Frequency [Hz]

Mobility [ms−1/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]

Stiness

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

c1c2∈C3×3was 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

10−5

10−3

10−1

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.

References

[1] M. Van Der Seijs, D. De Klerk, and D. Rixen, “General framework for transfer path analysis: History,

theory and classiﬁcation of techniques,” Mechanical Systems and Signal Processing, vol. 68-69, pp.

217–244, 2016.

[2] A. S. Elliott, A. T. Moorhouse, T. Huntley, and S. Tate, “In-situ source path contribution analysis of

structure borne road noise,” Journal of Sound and Vibration, vol. 332, no. 24, pp. 6276–6295, 2013.

[3] A. Moorhouse, A. Elliott, and T. Evans, “In situ measurement of the blocked force of structure-borne

sound sources,” Journal of Sound and Vibration, vol. 325, no. 4-5, pp. 679–685, 2009.

[4] J. Meggitt, A. Elliott, and A. Moorhouse, “Virtual assemblies and their use in the prediction of vibro-

acoustic responses,” in Proceedings of the Institute of Acoustics, vol. 38. Warwickshire: Institute of

Acoustics, 2016, pp. 165–172.

[5] Y. Bobrovnitskii, “A theorem on representation of the ﬁeld of forced vibrations of a composite elastic

system,” Akusticheskij Zhurnal, vol. 47, no. 5, pp. 586–590, 2001.

[6] J. Meggitt, A. Elliott, A. Moorhouse, and H. Lai, “In situ determination of dynamic stiffness for resilient

elements,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical En-

gineering Science, vol. 230, no. 6, pp. 986–993, 2015.

[7] M. Haeussler, S. Klaassen, and D. Rixen, “Experimental twelve degree of freedom rubber isolator mod-

els for use in substructuring assemblies,” Journal of Sound and Vibration, vol. 474, pp. 9–11, 2020.

[8] N. M. M. Maia, A. P. V. Urgueira, and R. A. B. Almei, “Whys and Wherefores of Transmissibility,” in

Vibration Analysis and Control - New Trends and Developments. Intechopen, 2011, ch. 10.

[9] J. Meggitt, “On in-situ methodologies for the characterisation and simulation of vibro-acoustic assem-

blies,” Ph.D. dissertation, University of Salford, 2017.