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Component based OTPA: An alternative to component TPA

without impact measurements

M. Haeussler 1, S. W. B. Klaassen 1, D. de Klerk 1, T. Rumpel 2

1Vibes Technology B.V. ,

Molengraaffsingel 14, 2629JD Delft, The Netherlands,

e-mail: mhaeussler@vibestechnology.com

2Mercedes-Benz AG ,

70546 Stuttgart, Germany

Abstract

Component based TPA allows to obtain a vibration source-description that is independent of the receiver.

Typically, blocked forces are determined on a component testbench to predict vibration levels on a receiver.

Therefore, impact measurements must be performed. Operational TPA (OTPA) is typically used as a tool

for path ranking of different noise sources in a troubleshooting phase. OTPA has the advantage of not

requiring laborious impact campaigns but lacks the possibility to predict receiver noise levels from testbench

measurements. In this paper, we derive an OTPA method offering this “testbench to vehicle” capability. This

novel, component-based OTPA method will be compared to the blocked force TPA method in a numerical

test case. The sensitivities of both methods to typically encountered measurement uncertainties are shown.

This is achieved by introducing different random errors in a Monte-Carlo simulation. The resulting variance

and bias error in the predicted receiver responses are used to evaluate the error sensitivity of both methods.

1 Introduction, motivation & outline of the paper

Transfer path analysis (TPA) has established in industry as a tool for noise vibration harshness (NVH) engi-

neering. A broad review and comparison of methods in a uniﬁed notation can be found in [2]. In general, a

TPA studies subcomponents which actively excite a larger assembly and thereby cause noise and vibrations.

As one of the ﬁrst applications, Verheij described the transmission of vibrations, from a ship engine to the

hull, by interface forces transmitted over the rubber isolators [3]. In 1982 this was mainly driven by the

desire to make military ships more stealthy. Nowadays, TPA is commonly applied in NVH engineering of

vehicles [4, 5]. Classically, TPA has been used as trouble-shooting tool, using interface forces to understand

the transmission of vibrations from the source to the receiver. A current trend is to use approaches which

describe the source independently from a speciﬁc receiver, e.g. via blocked forces [6–8]. A popular method

for obtaining the blocked forces is the in-situ method [9], which will also be used in this paper. It yields

results comparable to classical TPA, with some improvements if also rotational degrees of freedom (DoF)

are included in the source description (see [8]).

The reason for industry adaption of component based TPA is best explained with an example. Imagine a

vehicle manufacturer (OEM) which is producing nvcar models, say 30, see table 1. These vehicles typically

contain multiple, potentially noisy mechanical components. For example rear axle differentials (RADs).

There are in total ncdifferent variants of this source component. The different RADs are required for

different maximum speeds and torques of the individual vehicle conﬁgurations. However, on average only

ncv of all nvvehicle’s can be conﬁgured with one speciﬁc RAD.

Table 1: Exemplary numbers of vehicle models and variants of a single component at an OEM.

Number of vehicle models nv30

Number of components nc25

Vehicle models conﬁgured with one component ncv 10

The OEM wants to make sure that each potential vehicle conﬁguration meets the high customer demands for

NVH comfort. The number of tests depends on the approach chosen:

•Classical TPA & OTPA

With classical methods it is required to test each conﬁguration separately in the vehicle. That means,

the OEM would require to test each component in all vehicles which can be conﬁgured with this

component. The total number of physical tests required in the vehicle prototypes would be:

nc×ncv = 250.

These are typically done late in the development cycle, in the validation phase, where it is hard to

change the isolation concept.

•Component based TPA

With component based TPA methods it is possible to test each RAD on a testbench to obtain a valid

description of that source component. This can be combined with measured or simulated transfer

functions of each vehicle to obtain a description of the receiver. In total this would mean,

nv+nc= 55,

physical tests on the component testbench and in the vehicle prototypes.

From this example it is clear that component based TPA methods can reduce the number of physical tests.

Another advantage of component based TPA, is that potential NVH problems can be identiﬁed at an earlier

development stage, where it is still possible to take action by changing the design.

In this paper, the advantage of component based TPA methods is combined with the ease of OTPA, which

works without measured transfer functions. This method will be called component based OTPA (cOTPA).

Therefore, the concept underlying component based TPA and OTPA will be brieﬂy explained in section 2.

From this, the formulation of cOTPA will be derived in section 3. In section 4, a numerical example is used

validate both methods. Additionally, the sensitivities of both methods to typically encountered measurement

uncertainties are studied and compared. The results are discussed in section 5.

2 Transfer path analysis - Theory

In this section the principle of component based TPA using blocked forces and the used notation is explained.

From this derivation the principle of operational TPA (OTPA) is introduced. This will give the theoretical

basis for the new method, called component based OTPA, which is introduced in section 3.

2.1 Component based TPA with blocked forces

The general principle of component based TPA is described in ﬁgure 1. An assembly AB contains a vibration

source A, which is subject to internal loads fA

1. The exact mechanisms creating the internal forces fA

1might

be unknown or cumbersome to model. It is therefore desirable for an NVH engineer to ﬁnd another more

practical, yet complete description of the source. Component TPA methods are a solution to this.

The following explanation treats the underlying concepts of component TPA, using so called blocked forces

as a source description. We focus on an intuitive explanation and therefore reduce mathematical detail (see

[2] for a full derivation). The fact that blocked forces can be used as a source description can be derived from

the three depictions in ﬁgure 1:

Figure 1: The general source receiver problem and the equivalent modeling of vibrations on the receiver side

by blocked forces fbl

2.

1. Noise cancellation

Consider the following thought experiment: The blocked forces fbl

2are deﬁned as the forces which, when

applied as an external load at the interface between source and receiver, would block all vibration at the

interface. The motion on the interface of assembly AB would thus be zero:

uAB

2=YAB

21 fA

1+YAB

22 fbl

2

!

=0.(1)

where superscript (⋆)AB indicates that the quantity is a property of the coupled system, source Aand receiver

B. Subscript (⋆)21 indicates that the FRF matrix describes the vibration transfer from the internal source

DoF, subscript (⋆)1, to the interface DoF, subscript (⋆)2.

If assembly AB has no motion on the interface and there is no other vibration source on the receiver B, then

the sound and vibration at all other points of the receiver would be zero:

pAB

3=YAB

31 fA

1+YAB

32 fbl

2

!

=0,(2)

where subscript (⋆)3indicates points on the receiver, for example some sound pressure pAB

3. Thus, the

blocked forces act as a noise cancellation on the receiver. This is the theoretical basis for the blocked force

concept (or in fact all component based TPA concepts, see [2]).

2. Force superposition

Of course, the discussion so far was just a thought experiment, as artiﬁcially applying the blocked forces at

the interface DoF is usually not possible. In normal operation, the source’s internal forces fA

1are transferred

to vibrations uAB

3or sound pressures pAB

3in the receiver, via the frequency response function (FRF) matrix

YAB

31 :

pAB

3=YAB

31 fA

1,(3)

However, in NVH we are considering small vibration amplitudes and model the assembly AB as a linear

time-invariant system Therefore, it is allowed to add and subtract the effect of the blocked forces from the

original problem in equation (3) without modifying the outcome (superposition principle):

pAB

3=YAB

31 fA

1+

=0

z }| {

YAB

32 fbl

2−YAB

32 fbl

2.(4)

Figure 2: In-Situ determination of blocked forces.

3. Equivalent source description

Considering the noise cancellation from equation (2), one ﬁnds that:

pAB

3=−YAB

32 fbl

2.(5)

This means that all vibrations on the receiver side of assembly AB can be described with the transfer function

of the coupled system and the blocked forces of the source. Note, that here is the difference to classical TPA.

There, the interface forces are used as a source description and the FRF of the uncoupled receiver (YB),

with the source component removed, is used for propagation. Notice that the derivation did not specify

which particular receiver structure Bis used. The blocked forces are thus a valid source description for any

receiver B. Also, note that the blocked forces are a property of the source alone.

A thorough derivation of the concept, as well as different methods for obtaining the blocked forces in prac-

tice, are described in [2]. A theoretical comparison of these methods is given in [10]. An important assump-

tion for the derivation of the blocked force concept is that the internal source excitation fA

1is independent

of the source mounting, i.e. the receiver B. This is (to the author’s experience) a good assumption for cli-

mate compressors, electric motors, rear axle differentials, and many other components. However, it typically

requires some studies to apply the concept in an industrial context. For example, one needs to investigate

how many degrees of freedom need to be used for describing the source-receiver interface [11], and what the

most practical approach for obtaining the blocked forces of the component is.

2.2 In-Situ determination of blocked forces

A popular method for determining the blocked forces in practice is the in-situ method [9]. For identifying the

blocked forces, the source component could be attached to any receiver, like the the vehicle Bor a component

test-bench R, see ﬁgure 2. The receiver is equipped with indicator sensors, denoted as u4, which have to

be at or downstream of the interface (see ﬁgure 2). As discussed in the previous section, when artiﬁcially

applying the blocked forces fbl

2to the interface, they would cancel all vibration on the receiver:

0!

=YAB

41 fA

1

| {z }

=uAB

4

+YAB

42 fbl

20!

=YAR

41 fA

1

| {z }

=uAR

4

+YAR

42 fbl

2,(6)

The responses in the indicator sensors u4can be recorded for different operational conditions of the source.

The blocked forces for each operational condition can then be computed by:

fbl

2=−YAB

42 +uAB

4,fbl

2=−YAR

42 +uAR

4,(7)

where (⋆)+indicates a least-squares pseudo inverse. A pseudo-inverse has to be used if the system of

equations is over-determined, i.e. the vector u4contains more channels than the actual number of blocked

forces to be computed in fbl

2. This over-determination is generally recommended [1]. The pseudo-inverse

can either be built with least-squares, or with a regularized inverse to suppress the detrimental effects of

measurement noise [1]. Notice that the blocked forces can be determined either on a testbench or in the

vehicle, equation (7). These can be used for predicting the sound and vibration caused in the vehicle, see

equation (5).

2.3 Operational TPA (OTPA)

NVH problems are often encountered late in the development process. Finding a solution is then time-

critical, while initially it is often not even clear what source or path is the critical contributor to the noise

problem. Determining the blocked forces of the potential noise source(s) would be too laborious since the

FRFs and operational signals need to be measured (see equation (7)).

It would be faster to perform only operational measurements, and ﬁnd the critical noise sources and transfer

paths from them, as the impact-hammer or shaker tests are typically the most time-consuming part of the

measurement campaign. This is the application ﬁeld of operational TPA, introducedd by Nomura [12].

In the following, the equations for OTPA will be derived from the principle of blocked force component

TPA. Writing out the equations for predicting a sound pressure pAB

3from the individual contributions in the

operational measurements uAB

4yields:

pAB

3=YAB

32 fbl

2=YAB

32 YAB

42 +

| {z }

=:TAB

34

uAB

4.(8)

It can be seen that the receiver response pAB

3can be directly predicted from the operational signals in the

indicator sensors uAB

4with the so called transmissibility matrix TAB

34 :

pAB

3=TAB

34 uAB

4.(9)

In equation (8) this matrix is written as a result of the FRF matrices. In OTPA one estimates this trans-

missibility matrix from a large set of recorded operational measurements. The relationship in equation (9)

holds for any operational condition of the source(s). Stacking together all recorded operational conditions as

columns of the matrices PAB

3and UAB

4, one can write:

PAB

3=TAB

34 UAB

4.(10)

Provided that the system of equations has full rank, one can estimate the transmissibility matrix:

TAB

34 =PAB

3UAB

4+.(11)

With the transmissiblility matrix and the recorded signals, one can perform a path ranking of the individual

paths and noise sources, via equation (8). This gives a good estimate for the dominant paths in the operational

conditions that exhibit the NVH issue.

One assumption of OTPA is that all relevant modes of the structure are excited in the captured signal in UAB

4

[2]. Typically, many operational conditions are included in UAB

4, such as run-ups with different loads of

all components under investigation. Additionally, one can increase the number of independent excitations

in UAB

4by performing a couple of impacts with an uninstrumented hammer on the source, as suggested in

[13]. All important paths must be included in the signals of the indicator sensors uAB

4, since the method

yields unrealistic results otherwise [14]. However, if there are too many DoF in uAB

4, measurement noise

can be strongly ampliﬁed [15]. This can be attenuated by principal component analysis [12, 16].

3 Component based OTPA

In this section the formulation of a new OTPA method is derived. The authors think is best described by

the name component based OTPA. With it, two advantages of component based TPA and OTPA shall be

Figure 3: Principle signals required for the new component based OTPA method

combined, namely:

1. The component TPA ability to predict sound and vibration levels in the receiver pAB

3with operational

data from a component testbench.

2. The OTPA time-saving by omitting the laborious FRF tests.

.

OTPA gives the ability to predict receiver responses pAB

3from signals at the indicator sensors uAB

4see (9).

The indicator sensor signals uAB

4can also be predicted with the in-situ determined blocked forces from a

testbench, see (7):

uAB

4=YAB

42 fbl

2=YAB

42 YAR

42 +

| {z }

=:TAR/AB

44

uAR

4.(12)

This shows that there is a transmissibility matrix TAR/AB

44 from the TB to the vehicle:

uAB

4=TAR/AB

44 uAR

4(13)

Following the same idea as in OTPA, we try to ﬁnd this transmissibility matrix from a large number of

operational measurements:

TAB/AR

44 =UAB

4UAR

4+(14)

Note that in equation (14), we are implicitly making the assumption that the source is running in exactly

the same operational condition for every column of UAB

4and the corresponding column of UAR

4. Provided

that this is the case, one can ﬁnd the transmissibility from indicator sensor signals on the testbench to the

indicator sensor signals on the receiver, see ﬁgure 3.

A forward prediction to the receiver DoF pAB

3can then be done with the OTPA formula from (9):

pAB

3=TAB

34 uAB

4=TAB

34 TAB/AR

44

| {z }

=:TAB/AR

34

uAR

4.(15)

Note that the transmissibility TAB/AR

34 could also be directly determined from the operational measurements

at only the receiver DoF pAB

3and the indicator sensors uAR

4. This would even further simplify the method,

as then no indicator sensors in the receiver assembly uAB

4would be required. However, practical tests would

need to show if this is a viable option.

Figure 4: Signals for the numerical example

4 Numerical Example

In this section, blocked force TPA is compared to the new cOTPA method. Therefore, a numerical test case is

created, where the individual substructures are described by linear mass, spring, damper systems. It consists

of a source component A, a testrig Rand a receiver B, which contain 4,7and 9DoF respectively. The

number of DoF which are chosen as internal, interface, indicator and response DoF are listed in table 2.

Table 2: Number of internal, interface, indicator and response DoF in numerical example.

Internal source DoF fA

12

Interface DoF fbl

22

Indicator DoF uAB

4,uAR

43

Response DoF pAB

32

The system FRFs are computed from the stiffness, damping and mass matrices of the numerical example

(see upper left part of ﬁgure 4). The internal force of the source fA

1(b)is constructed for different blocks

bin the frequency domain. There are nBand nRblocks, representing the source operational forces on

the ﬁnal assembly AB and component testbench AR respectively. In the following examples it is chosen:

nB=nR= 15. For the proof of concept, it is assumed that the internal force fA

1(b)is constant in all blocks,

i.e. the source is running in exactly the same operational condition on the testbench and receiver assembly.

The resulting vibrations on the testbench and receiver can be computed from the system matrices and fA

1(b),

see the middle left part of ﬁgure 4.

The resulting vibration responses, see lower right part of ﬁgure 4, are the input for the comparison of BF-TPA

and cTPA. They are computed as follows:

•BF TPA: Compute the transmissibility matrix TAR/AB

44 from the FRF matrices as in (12). Compute

the blocked forces fbl

2as in the right side of (7), and predict the receiver responses pAB

3with (5).

•cOTPA: Compute the transmissibility matrix TAR/AB

44 from the frequency blocks of the operational

signals as in (14). Predict the receiver responses pAB

3with (15).

The ﬁnal receiver response in one channel and a resulting transmissibility matrix entry is plotted in the upper

part of ﬁgure 5. It can be observed that both methods yield the same prediction on the receiver, and that the

computed transmissibility is equal. The reference solution is computed as:

pAB

3(bR) = YAB

31 fA

1(bR),(16)

as this would be the response if the source was exciting the ﬁnal receiver with the same excitation as on the

testbench.

4.1 Sensitivity to measurement errors: BF TPA vs. cOTPA

Additionally to the proof of concept, three cases of measurement imperfections, which are expected to appear

in practice, are investigated with the numerical example (see also ﬁgure 4). These are described in the

following:

•Case 1: Amplitude Variance in the source exitation fA

1. This is introduced by scaling the nominal

source internal force ˆ

fA

1with a random scalar

fA

1(b) = ˆ

fA

1(1 + X),with: X∼ N µ= 0, σ2= 1,(17)

where Nis the standard normal distribution.

•Case 2: Phase variance in the source exitation fA

1. This is introduced by changing the phase of the

nominal source internal force ˆ

fA

1:

fA

1(b) = ˆ

fA

1eiX ,with: X∼ N µ= 0, σ2=π

42.(18)

•Case 3: Random noise on test bench signals uAR

4and receiver signals pAB

3,uAB

4.

The results of these three cases of measurement imperfections are shown in ﬁgure 5.

4.2 Uncertainty Quantiﬁcation: BF TPA vs. cOTPA

In order to understand the effects of the measurement imperfections on both methods better, the number of

blocks is increased to nB=nR= 150. The resulting nRpredictions in one channel of pAB

3are then used to

compute the mean value of the predictions. Additionally, the variance σis computed and plotted in a band,

shown in ﬁgure 6.

5 Discussion of results

A new method called component based OTPA (cOTPA) has been introduced in this contribution. It was

derived from the concept of blocked force (BF) TPA. In a proof of concept, it could be shown that cOTPA

yields equivalent and exact results if the underlying assumptions are valid, as can be seen in the upper part of

ﬁgure 5. In case of an amplitude variance of the internal source strength, the predicted mean values for the

receiver signal pAB

3are equal for both methods (Case 1, in ﬁgure 5 and 6). Also the standard deviation in the

prediction is comparable. Interestingly, the individual entries in the transmissibilities TAR/AB

44 determined

with cOPTA are multiple orders of magnitude too high. This is presumably due to the underdetermined

system of equations which is solved when using equation (14) with the signals from the numerical exam-

ple. Both UAB

4and UAR

4have 3 DoF, but only 2 principal components, since they are both caused by an

internal force with only two DoF (see table 2). A phase variance of the internal source strength (Case 2, in

ﬁgure 5 and 6), yields different results. The predicted mean value for the receiver signal pAB

3is too low for

cOTPA, whereas BF-TPA yields on average correct solutions. This is by no means surprising, since one of

the underlying assumptions of cOTPA is that the source is running in exactly the same operational condition

Proof of concept

Case 1: Amplitude Variance fA

1

Case 2: Phase Variance fA

1

Case 3: Output Sensor Noise

Figure 5: Results for a noise free proof of concept and cases of expected measurement imperfections. In the

left column one channel of pAB

3is shown. The reference and the predictions with BF TPA and cOTPA are

compared. The mean value of the amplitude over all nBand nRblocks is shown. The right column shows

the estimated transmissibilities.

Case 1: Amplitude Variance fA

1

Case 2: Phase Variance fA

1

Case 3: Output Sensor Noise

Figure 6: Results of a Monte Carlo simulation. Mean Value and variance of BF-TPA and cOTPA predictions

for the receiver response pAB

3.

(including the phase reference) on the testbench and the receiver. This is not a requirement for BF-TPA, as

the differences in phase are implicitly accounted for via the FRF matrices. This result can be seen particu-

larly clear in the Monte Carlo Simulation. It emphasizes the requirement for an accurate measurement of the

source’s operational phase for deriving the cOTPA transmissibility matrices from only operational measure-

ments (e.g. via a triggered rpm channel). For noise polluted signals (Case 2, in ﬁgure 5 and 6), both methods

yield a similar scatter in the predicted mean values of the receiver signal pAB

3. However, the predicted mean

value for cOTPA is lower as the reference and the BF-TPA solution.

6 Summary

The newly derived component OTPA method comes with the advantage that it is possible to predict in-vehicle

responses with testbench measurements very efﬁciently, due to the fact that no FRFs need to be measured.

This is a very relevant advantage for industries with many product variants. It requires testing at least one

source component in the ﬁnal vehicle variant and on the testbench, under different operational conditions,

while controlling and recording the operational states of the component very accurately. Compared to com-

ponent based TPA methods, like the BF TPA, the method will most likely have less predictive value, when

it comes to deriving acoustic design improvements, as the measurements cannot be easily transferred to a

new, virtually modiﬁed receiver design. This is reason why the authors see most application potential for

the method in end of line component testing. There, the required operational measurements for deriving the

transmissibilities can be done with one exemplar of a nominally equivalent source component in the vehicle

and on the end of line testbench. These transmissibility matrices can then be used to automatically check

the expected vibration levels for each exemplar of the source component that leaves the factory. The method

seems to be particularly interesting for components which are very easy to control in their operational con-

dition (e.g. e-motors or servo motors). If there is not enough variance in the excitation modes with the

source’s operational conditions, one could enhance the measurement basis with some impacts on the source

component, similar as it is suggested in [13] for classical OTPA. However, this would most likely make the

method very similar, if not equivalent to the pseudo force method [2].

The authors also want to disclose, why the contribution is not showing any real operational data. Initially, the

project intention was to investigate the potential for an "impact-testing free" method for predicting vehicle

noise and vibration from testbench operational measurements only. If this was possible, this would have

given the chance to post-process big data silos already available for rear axle differentials. As it was shown

in the mathematical derivation and the numerical example, it is crucial to have a clear phase reference for

cOTPA. As this information was not available in the recorded data, the project was discontinued after the

method development and numerical proof of concept phase. Still the authors want to invite colleagues in the

ﬁeld of NVH engineering, to test, improve and collaborate on advancing the method.

Nomenclature:

FRF frequency response function DoF degree of freedom

TPA transfer path analysis OTPA operational TPA

NVH noise vibration and harshness cOTPA component OTPA

OEM original equipment manufacturer RAD rear axle differential

BF blocked force

u,fmeasured accelerations / forces Yadmittance FRF matrix

(⋆)Aquantity pertaining to source A (⋆)Bquantity pertaining to receiver B

(⋆)Rquantity pertaining to test-bench R (⋆)AB coupled quantity of Aand B

(⋆)1internal DoF of source A (⋆)2interface DoF at source connection points

(⋆)3receiver point on B (⋆)4indicator sensor point.

ˆ

(⋆)noise or pertubation free signal

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