Operational Transfer Path Analysis: Theory, Guidelines...

Operational Transfer Path Analysis: Theory, Guidelines &Tire Noise Application

D. de Klerk 1,2, A. Ossipov 3

1 Delft University of Technology,Faculty of Mechanics, Maritime and Material Engineering,Department of Precision and Microsystems Engineering, Engineering Dynamics,Mekelweg 2, 2628 CD Delft, The Netherlandse-mail: [email protected]

2 Muller-BBM VibroAkustik Systeme B.V.,Menno van Coehoornsingel 9a,8011XA, Zwolle, The Netherlandse-mail: [email protected]

3 Goodyear Innovation Center,TVM Vibration Mechanics Group,L-7750 Colmar-Berg, Luxembourgemail: Alexander [email protected]

AbstractThe Operational Transfer Path Analysis (OTPA) method is subject of research in this article, which startswith a discussion on it’s theory. Here clear similarities with the MIMO technique in Experimental ModalAnalysis are found. Based on the knowledge of MIMO, one finds that input signals are allowed to becoherent to a certain extend. As coherence can be larger in OTPA in practice, the method is extendedwith the Singular Value Decomposition method to reduce influences of noise. The article proceeds with adiscussion on points of attention, or boundary conditions, in practical applications. An analysis on tire noiseis included to illustrate the points of attention and the methods strength in, for example, vehicle TPA on tires.

1 Introduction

Operational Transfer Path Analysis (OTPA), using Cross Talk Cancelation (CTC) and Singular Value De-composition (SVD), is a signal processing method which finds the linearized Transfer Function (TF) matrixbetween a set of chosen input and output channels from a measurement. The in- and output relations aredetermined such, that the Transfer Functions are linearly independent with respect to each other, hence thename CTC. The resulting Transfer Functions can be used in a Transfer Path Analysis (TPA), determining asource’s propagation of noise and the resulting content in the response signal.

The OTPA uses the Singular Value Decomposition (SVD) algorithm to find independent principal compo-nents describing the Transfer Functions. Indeed, in practice the numerical operations involved to determinethe TF matrix between inputs and outputs often suffer from measurement noise. By rejecting smaller princi-pal components one reduces these influences on the TF estimates. Basically OTPA find its roots in work ofBendat et al [2, 1], yet evolved from the TPA approaches which are based on FRF measurements:

• The classical TPA approach measuring interface force [19].

Appears in MSSP vol. 24(7) - ISMA2010 special issue 3961

• The Matrix Inversion method [22, 25, 15].

• The Mount Stiffness method [24, 20].

• The Gear Noise Propagation (GNP) or Component TPA method [4, 6].

These other methods basically consist of two steps. First, Frequency Response Functions (FRF) are de-termined between defined input / reference points and chosen output point consisting of sound and / orvibration. They are determined by use of impulse hammer or shaker(s) if structural vibration is consideredor by use of loudspeaker(s) for air-borne vibration. Secondly, these FRFs are combined with operationalforces determined at the reference points1 to generate synthesized response signals. The synthesized out-put can thereafter be analyzed, determining the contribution of each propagation path. The OTPA methoduses a one step approach and builds model of a structure without FRF measurements by hammer, shaker orloudspeaker. Basically, the method uses a response to response Transfer Function matrix2, to represent thepropagation paths of the structure. The signals all originate from a measurement of the operating system, sothat implicitly the operating excitations are used to determine the transfer paths.3

Compared to the FRF approaches one can make following remarks:

• the OTPA method is very easy and fast to setup as it uses only an operational measurement. A largereduction in analysis time can therefore be achieved compared with FRF approaches.

• As the method uses operational data, operational influences are accounted for.

• Especially air-borne noise has a spatially distributed sound field on the excitation source. It is dif-ficult to reproduce this sound field with loudspeakers, yet OTPA uses the actual excitation source todetermine the TF.

• As essentially response data is used for input, careful design of the OTPA model and basic understand-ing of the analyzed system are required.

The goal of this paper is to obtain a better understanding of the OTPA method, highlighting its capabilitiesand point of attention in it’s application. Furthermore, this paper includes an OTPA analysis on tire noise,showing the method’s strength and illustrating points of attention.

The paper starts with a general framework for the OTPA method in section 2. Based on the framework,section 3 presents points of attention in OTPA applications, which by themselves can serve as quality checkcriteria as well. In section 4 the method is applied on a vehicle’s tire noise propagation, highlighting someinteresting features. The paper is concluded with a summary in section 5.

2 Operational Transfer Path Analysis Method Framework

The OTPA method, based on CTC / PCA, tries to find the (linearized) Transfer Function (TF) matrix betweena chosen set of input and output quantities from a measurement. In the following discussion, these sets ofinput and output quantities can best be seen as Degrees of Freedom (DoF) describing the measured object’sexcitation (inputs) and the object’s responses (output) as a linear combination of the chosen / assumed ex-citations. Subsection 2.1 describes the determination of the TF matrix, in a first step, as a least-squaresapproximation problem. In this view it can be seen that the OTPA method without Singular Value Decom-position (SVD) is equivalent to the MIMO technique of finding Frequency Response Functions (FRF) if thesame input and output variables are measured. In subsection 2.2 the OTPA theory is however enhanced withthe SVD method, yielding TF matrix estimation with reduced noise influences.

1Each method determines these forces in a different way.2also known as Transmissibility matrix if acceleration sensors are used.3Other work on the study to obtain the MIMO transmissibiliy matrix from response measurements are found in [9, 23, 17, 21]

3962 PROCEEDINGS OF ISMA2010 INCLUDING USD2010

2.1 OTPA Theory with Least-Squares Algorithm

Consider an arbitrary linear(ized) system model described by a set of input and output DoF, represented as

H(jω)x(jω) = y(jω). (1)

Here H(jω) is the Transfer Function matrix linking the vector of input DoF x(jω) to the vector of outputDoF y(jω). The dependency on frequency is denoted by (jω). In NVH problems, the measured signals aretypically motions, denoted u(jω), forces f(jω) and sound pressures p(jω). The input and output vectorscan thus in general be assembled from these quantities as

x =

ux

fx

px

; y =

uy

fy

py

(2)

ux = [u(1)x . . . u(k)

x ]T ; uy = [u(1)y . . . u(n)

y ]T

fx = [f (1)x . . . f (l)

x ]T ; fy = [f (1)y . . . f (o)

y ]T

px = [p(1)x . . . p(m)

x ]T ; py = [p(1)y . . . p(p)

y ]T ,

where the dependency on frequency is omitted for clarity.4 Indices k, l,m denote the number of input chan-nels for the different quantities and n, o, p the number of output channels for different quantities respectively.Notice that it is up to the engineer to define the input and output sets from the measured data. He is not re-stricted to choose forces as excitations only, but motion and sound pressure, which are responses from aphysical point of view, may also be chosen to model the system’s behavior. In the latter cases, care shouldbe taken, which will be discussed in more detail in section 3.

Not all physical quantities have to be present in each set defined in (2), neither do the vectors have to havethe same dimensions. In fact, usually the number of excitation channels in the input vector x will be largerthan the number of DoF in the response / output vector. In vehicle analysis a typical example is to find theTransfer Functions between motions measured on the driveline and the sound pressure at the driver’s ear.

By the construction of (1), the elements of the TF matrixH have the form

Hij =yi

xj

∣∣∣∣xk=0

; k 6= j. (3)

Typically this matrix element property is used in Experimental Modal Analysis (EMA), where an externalapplied force f is applied (e.g. xj), by shaker or impulse hammer, as only input DoF (e.g. xk = 0; k 6= j)and the resulting responses u of the system are chosen as outputs (e.g. y). These kind of Transfer Functionsare denoted receptance Frequency Response Function (FRF) in the literature and have the special propertythat their frequency peaks show the free system’s eigenfrequencies. Strictly, one could thus determine acolumn of the TF matrix in the OTPA method also by exciting the system with only the input DoF xj , whilesuppressing all other input excitations. In practice this is very hard to achieve as inputs are not only forces,but also motions, sound pressures or any kind of quantities. The determination of the transfer functionselement wise will therefore often lead to very difficult, impractical and often impossible experimental setups.Analysis as such will therefore require a big expense in time and resources.

To overcome this disadvantage, the OTPA method tries to determine all elements of the TF matrix from onemeasurement only where all excitations are present at once. This determination is discussed next by firsttaking the transpose of equation (1) and writing the equation on entry level:

[x(1) . . . x(m)

] H11 . . . H1n...

. . ....

Hm1 . . . Hmn

=

[y(1) . . . y(n)

]. (4)

4It is also allowed to use any other physical quantities, as long as they have the same FFT parameter during measurement.

TYRE/ROAD NOISE AND EXPERIMENTAL VALIDATION 3963

Here m and n denote the number of in- and output DoF. Taking the transpose does not allow the determi-nation of the TF elements though. In order to do so, notice that during an operational measurement of, forexample, a vehicle run-up on a dynamometer, a set of synchronized measurement blocks will be stored todisk. In general these sets will not have the same content, as the excitations change continuously duringthe measurement. If one requires, or defines, the relation between the input and output DoF as being lin-ear(ized) and constant during the total measurement, equation (4) should however hold for each individualmeasurement block. One could thus extend equation (4) writing the equation for all measurement blocks r,yielding: x

(1)1 . . . x

(m)1

.... . .

...x

(1)r . . . x

(m)r

H11 . . . H1n

.... . .

...Hm1 . . . Hmn

=

y(1)1 . . . y

(n)1

.... . .

...y

(1)r . . . y

(n)r

− µ. (5)

This formulation, or system model, now requires the Transfer Functions to be linearly independent withrespect to each other, hence the name Cross Talk Cancelation. Indeed, although the input quantities might(and most often will) be coherent with respect to each other, the calculation of the Transfer Function matrixcompensates for it.

Here it is assumed that the experiment is performed such that the number of measurement blocks is biggerthan the total amount of in- DoF, e.g. r > m. This approach makes equation (5) a solvable least-squaresoptimization problem with an additional residue µ for the content which can not be modelled by the (cho-sen) set of input DoF.5 Indeed, notice that in general the individual observations / measurement blocks willcontain a distortion due to, for example, measurement noise or additional unmeasured excitation sources thatare not taken into account in the model. Furthermore, the TF of the system might not be constant duringthe measurement either, due to nonlinear system behavior of some kind.6 Mentioned issues make a wellconstructed experiment and model essential, which will be discussed in more detail in section 3.

To solve equation (5) one can first simplify its formulation writing it in a more compact way in matrices

XH + µ = Y . (6)

The calculation needs to be performed for each individual frequency line of the FFT spectrum. Solving (6)for each frequency is now performed, in explicit sense, by pre-multiplying the equation by XT , requiringthe residue vector to lay in the null space of the input DoF, i.e. XTµ = 0 . The TF matrix is thereafter foundas

H =(XTX

)−1XTY = X+Y , (7)

where matrixX+ is called the “pseudoinverse” of matrixX , readily defined as

X+ ,(XTX

)−1XT . (8)

As the TF matrix is now determined (7), one can calculate the residual vectorµ, which shows the unmodelledpart of Y , by substitution of (7) into (6) yielding after some manipulation:

µ =(I −X (

XTX)−1

XT)Y . (9)

If the OTPA method is considered purely as a least-squares calculation, it can be observed that the methodis equivalent to the MIMO technique of finding FRF estimates, if the same input and output variables are

5Notice, that if different physical quantities are used one needs to perform a weighted least squares estimate [7].6Possible mechanisms could be a system’s dependency on temperature, applied excitation amplitudes or rotation speed [4].


chosen. Indeed, observe thatXTX in (7) is in fact the averaged Auto Power Spectrum (APS) matrixGxx ofthe input signals andXTY is the averaged Cross Power Spectrum (CPS) matrixGxy between the input andoutput signals, accept for them not being divided by the number of measurement blocks. This normalizationcancels in the equation though and the OTPA method as a least-squares method is thus equal to the MIMOtechnique of determining FRF estimates:

H =(XTX

)−1XTY = G−1

xxGxy (10)

Gxx , 1/rXTX (11)

Gxy , 1/rXTY , (12)

if only shaker input forces would be used for excitation.

Note that coherence between input signals is often seen as an issue in OTPA [16, 12]. Yet, shaker signalswill be partly coherent with respect to each other in MIMO techniques as well. Indeed, as they are allconnected to the test structure simultaneously, their vibrations propagate to one another and are measured byall force sensors. Furthermore a limited number of measurement blocks is used for the MIMO calculation,rendering some remaining correlation, and thus coherence, among the excitation signals as well. From anumerical point of view, with today’s 64 bit computer technology, even very high coherence in either MIMOor OTPA does not pose a problem for the calculation itself. Due to measurement noise and other disturbancescoherence between the shaker forces should not exceed 30 - 40% for MIMO analysis in practice though.Indeed, higher values might well lead to inaccurate FRF estimates as the small measurement disturbancesget more influence in (10) and more averages might be required. Important to realize though, is that (10)compensates for remaining coherence between (shaker) inputs by the off-diagonal terms in Gxx. Thereforetheoretically exact FRFs (or TF for OTPA) are calculated, even if the input signals are almost fully coherent.

The reasoning above also holds for the OTPA calculation (7), although coherence among the input channelsmight well exceed 30-40% in practice. In order to reduce the influences of measurement noise in such oftenoccurring events, the OTPA algorithm is extended with a Singular Value Decomposition in subsection 2.2 toovercome this problem. Note once more that OTPA typically doesn’t determine receptance FRF, but TransferFunctions also known as Transmissibilities. The transmissibility describes the isolation of a system. Ampli-tude peaks and drops over a frequency do therefore not necessarily refer to resonances or anti-resonances ofthe system.7 Care should therefore be taken in the interpretation of TFs in OTPA, e.g. it is often better toconcentrate on path contributions. This issue will be addressed in more detail in section 3.

To summarize, this subsection showed:

• The OTPA algorithm has the same basis as the MIMO method in EMA and can be seen as a least-squares estimate.

• The OTPA algorithm compensates for cross talk between input channels, hence the name Cross TalkCompensation (CTC).

• OTPA and MIMO both allow almost fully coherent input signals in the TF matrix estimation from atheoretical point of view. In practice measurement noise / errors will become more and more influentialon the TF estimation with increasing coherence though. Therefore coherence should be below 30-40%in MIMO applications and an SVD technique is required for the OTPA method.

• The least-squares residue µ in (9) shows how much of the output signal can not be modelled by the(chosen) input signals. In OTPA this is a useful property, as one can determine if the (chosen) inputsignals well model the output signals.

7At a resonance frequency of a system, the transmissibility actually represents the ratio between the modal amplitudes of twopoints.


2.2 OTPA enhancement with Singular Value Decomposition

The explicit determination of the Transfer Function matrix H by (7) can cause erroneous estimates if inputsignals are highly coherent in combination with measurement noise.8 Use is therefore made of SingularValue Decomposition (SVD), to prevent poor estimates. Indeed matrix X can be expressed by a SingularValue Decomposition as

X = UΣV T . (13)

U is an r × r unitary matrix, Σ is a r ×m matrix with nonnegative numbers on the diagonal (as definedfor a rectangular matrix) and zeros off the diagonal. V T denotes the conjugate transpose of V , an m ×munitary matrix.9

The SVD is very general in the sense that it can be applied to any r × m matrix. A standard eigenvaluedecomposition, on the other hand, can only be applied to certain classes of square matrices. Nevertheless,analytically, the SVD can be determined by an eigenvalue decomposition by the following relations:

XTX = V ΣTUTUΣV T = V(ΣTΣ

)V T (14)

XXT = UΣV TV ΣTUT = U(ΣTΣ

)UT (15)

The right hand sides of the relations above therefore describe the eigenvalue decompositions of the left handsides. Consequently, the squares of the non-zero singular values of X are equal to the non-zero eigenvaluesof either XTX or XXT . Furthermore, the columns of U (left singular vectors) are eigenvectors of XXT

and the columns of V (right singular vectors) are eigenvectors of XTX .10 A physical way to interpret (13)would be to see matrix V as a set of orthonormal “input” or “analyzing” basis vector directions for X .11

The matrix U contains a set of orthonormal “output” basis vector directions for X and matrix Σ containsthe singular values, which can be thought of as scalar “gain controls” by which each corresponding input ismultiplied to give a corresponding output. A common convention is to order the values Σii in non-increasingfashion, in which case the diagonal matrixΣ is uniquely determined byX .

The Singular Value Decomposition of X can be directly used in the computation of the pseudoinverse X+,yielding

X+ = V Σ−1UT , (16)

whereΣ−1 is the inverse ofΣ. Notice that here it is assumed that the additional rows inΣ were omitted inthe SVD computation making it a square matrix m×m. Substitution of (16) in (7) yields an estimate on theTF matrix H using the SVD method as12

H = V Σ−1UTY . (17)

From an engineering and statistical point of view it was found in applications that smaller singular valuesare mainly caused by noise influences and other external disturbances [18]. They are therefore unwantedand should be rejected. Note that the least-squares fit for the analyzed measurement will be better with allsingular values kept though. Yet the amount of used singular values in the TF calculation of one measurementreveals a tradeoff in the resulting fit on another, similar, measurement. Indeed, as the noise will be differentin both measurements, this cross validation process reveals which of the smallest singular values are relatedto the noise influences[8, 13, 3]. Taking only a reduced set of singular values into account therefore improvesthe TF estimates in general.

8The measurement noise is amplified in the inversion term (XTX)−1.9Typically the matrix sizes are chosen different, e.g. U has size r × r,Σ has size r ×m and V T has size m×m. This will be

explained shortly.10Determining the SVD matrices in this way is numerically unstable in practice, especially for singular values close to zero.

It’s explicit determination in this sense does therefore not enhance the analysis given in the previous section. Instead, specialalgorithms such as the (modified) Golub-Reinsch algorithm and the one-sided Jacobi orthogonalization were developed whichovercome possible numerical conditioning problems [11].

11In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (their inner product is 0) and bothof unit length (the norm of each is 1).

12Notice that with the SVD approximation, the residual content µ lays in the null space of UT .


3 Points of Attention in OTPA

In this section different practical considerations on the OTPA and its implementation will be discussed. Herethe following issues are brought forth:

• OTPA model design: Source, Transition and Response locations.

• Quality of OTPA model using Least-Squares Residue.

• Variation in the Structure’s Excitation.

Notice furthermore that coherence between input / excitation signals, which is often seen as an issue as well,was addressed in the previous section.

3.1 OTPA model design: Source, Transition and Response Locations

Essential for an accurate OTPA system model is a model definition which implicitly represent the systemsdynamics best, yet uses the measured operational response signals only. Vehicle (driveline) components are,for example, often decoupled from other parts. Measured responses on the engine, for example, will bedominated from the engines combustion and thus implicitly characterizes the combustion itself. One couldalso think of responses measured on the rear axle differential or at the wheel spindle. Such responses willcharacterize either the internal gear noise excitation or the road input from the tires. Hence such signal canbe well used as input signals / variables. Furthermore a connection point of the driveline to the bodyworkcontains a combination of the engine, tire, and gear noise excitation. Therefore such kind of locations cannot be used for source characterization, yet will tell the engineer at which bodywork connection most noiseis propagated.

In general one can classify locations on any structure in Source, Transition and Response positions, asschematically represented in figure 1 for a vehicle. One should not combine channels of different locationclasses in the input / excitation vector in the OTPA algorithm. Indeed, as the signals are cross talk canceledby the algorithm, combinations yield illogical results as cause and effect are mixed. It is our experience

ExcitationSource

TransitionPoint

Receiver

Figure 1: It is essential in OTPA to define positions on the system which one classifies in Source locations, Transitionpoints and Response positions. In the OTPA calculation, one should not combine channels of different location classesin the input vector, as cause and effect are mixed faulty.

that although the true structure excitation is not measured, building models with response data on wellchosen positions as model inputs gives, for example, a good estimation of the structure’s noise propagationin automotive applications [18]. If one forgets a propagation path, coherent parts are redistributed over theother signals though. To make a correct interpretation and analysis of a structure in practice, the followingconsiderations may be used as guidelines:


• Choosing response data measured on different sources as input signal allows to separate a measuredoutput responses, at for example the driver’s ear, in the different source contents.

• Choosing several responses on one excitation source as input variables allows the identification of howthe source’s excitation propagate into its neighboring component(s).

• Using Transition locations as OTPA inputs allows one to determine which locations transmit vibrationmost. The do not indicate the origin of the source.

• Using additional responses within the structure allows to identify how the vibration is propagated intothe vehicle through succeeding components.

Notice that structures which are well equipped with sensors can be used for multiple OPTA analysis froma single operational data set. With smaller channel count, one can make multiple setups to determiningdifferent items from the guidelines.

3.2 Quality of the OTPA model

One way to test the quality of the OTPA model is to verify if the synthesized responses are similar to the mea-sured responses. This can be done by observing the least-squares residual µ amplitudes (9) or by comparingthe measured responses Y with the synthesized ones according:

Y = XV Σ−1UTY . (18)

Differences in amplitudes of Y and Y can be evaluated using any number of singular values inΣ. Observeddiscrepancies are caused by either:

• The input and / or output signals contain additional noise content which is filtered by the OTPA algo-rithm.

• The chosen input signals are not the only sources which contribute to the response signal.

• The system might behave nonlinear during the chosen set of measurement blocks. The TF matrix willrepresent the average transfer functions during the measurement.

One way to verify if the system behaves nonlinear is to vary the number of measurement blocks taken intoaccount in the TF matrix estimation. If the TF matrix changes considerably this could indicate nonlinearsystem behavior.

3.3 Variation in the Structure’s Excitation

It is important to vary the input quantities of matrix X during the operational measurement as much aspossible. Larger variation minimizes the coherence between the chosen input channels and results in a betterconditioning, i.e. higher values of the lowest singular values, of the input matrix. As such, the noise influenceis minimized, yielding accurate TF matrix estimates.13

In this view a vehicle run-up is, for example, a better operational measurement than a constant speed mea-surement. Indeed, during the vehicle run-up excitation sources change continuously in amplitude and di-rection. It was noticed in [10] that care should be taken, as higher frequencies are excited by less engineorders, hence less variation might be expected at higher frequencies. Typically such problems can be tracedby a significant drop of the responses’ multiple coherence and a large residue (9). Yet in general higher

13See discussion in section 2.1 on the influences of noise in the TF matrix estimation.


frequencies are dominated by air-borne noise, also see the result in the section 4. When these signals are alsomeasured, engine order dominance shows not to be a problem.

The question which often rises is what level of variation is required for an accurate OTPA analysis? In [14],where diesel combustion is analyzed in an OTPA fashion, it is suggested that a condition number of 10 to15 dB or less should be sufficient to reduce influences of noise. However, depending on the level of externalnoise, number of input channels and coherence between the input channels, this criteria can only be used asa rule of thumb. It was found, for example, that larger number of input channels as well as high coherencebetween excitation signals yield a higher relevant threshold. It means, typically, that the condition numbershould be around a factor 100 from 12 input channels upwards. If more variation is required, one can alsocombine multiple measurements, e.g. repeated measurement runs. One can also combine different kind ofmeasurements, like vehicle run-ups and coastdowns, shaker tests and impact hammer tests.

4 OTPA Application on Tire Noise

This section discusses a tire noise analysis at Goodyear’s test facility in Luxembourg with specificationslisted in table 1. Here we used a 3D acceleration sensor at each wheel hub. Furthermore we used 3 mi-crophones for each tire at approximately 2 meters aside of the wheel. Indeed, experience learned that thesepositions are suited to analyze the air-borne path up to about 1200Hz, as later seen in figure 2. More micro-phones are needed for the analysis of higher frequency ranges though, due to the increasing complexity ofthe sound field.14 The analysis at hand was setup such, that both air-borne and structure-borne vibration isanalyzed with 6 different tire combinations, all on both rough and smooth road surface.

In subsection 4.1 one tire combination, with different tire types left and right, is analyzed on the dynamome-ter’s smooth road surface. Thereafter all 12 measurement will be compared to analyze how the OTPA resultschange with different configurations.

4.1 Analysis of a Single Measurement

After the OTPA model is calculated according section 2, one first starts with a comparison of synthesizedoutput channels and their actually measured ones. Indeed, after the Transfer Functions are determined from(17) in a least-squares sense, using them as FIR filters in equation (18) allows to synthesize the outputchannels in Y with the originally measured inputsX . Figure 2.a shows this comparison for an arbitrary tireset. Note that here the overall sound level is depicted during the coastdown to get a good overview. Thefigure shows that the total synthesized sound level at the driver’s ear matches very well with the originalmeasured signal, e.g. the red curves overlap. This indicates, that on an overall level, the OTPA model iswell able to describe the receiving sound pressure at the driver with the 6 structure-borne and 6 air-bornechannels. The other curves in blue and gray show the individual path contributions from each of the 12input / source channels. Observe that depending on the vehicle speed, individual path dominance vary. Thestructure-borne excitations in z-direction clearly have the largest overall impact on the driver’s sound leveland are therefore most worthwhile optimizing. It can be observed though that the left and right excitationnot always yield the same contribution, which is due to the 2 different tires left and right.15

From figure 2.a with its overall sound levels one doesn’t have an indication what frequency range shouldbe tackled / optimized. In a second step one therefore proceeds to compare the frequency content of thesynthesized and measured outputs. In order to do so, all Auto Power Spectrum (APS) measurement blocksof the complete coastdown are averaged. This gives an overall indication of the model’s fit in frequencyover the complete run. As seen in the figure 2.b, the model fits the measurement well up to a frequency of

14Note that in case one measures a vehicle with different number of microphones, these measurement’s Transfer Functions can’tbe directly compared as coherent parts between microphones is distributed differently.

15In measurements where the same tires were used at each side, the contributions show much better “symmetry”.


about 1000Hz. At higher frequencies, not shown here for clarity, differences get into play above 1500 Hz.Evaluation of the coherence between the microphones above 1500 hz also reveals a clear drop. Thereforeone can conclude that the dominant air-borne propagation path can’t be properly measured above 1500 Hz,spatially, with 6 microphones next to the vehicle .16 Interesting to note is the different air-borne and structure-borne dominance over frequency. On average structure-borne vibration dominates up to frequencies of about700 Hz. At higher frequencies, air-borne vibration becomes the dominant one for the tested vehicle. As thehighest amplitudes occur below 700 Hz, the structure-borne dominance on the overall level, see figure 2.a,is well explained. Should higher “whissle” / tonal kind of noises be a psycho-acoustical target, this analysisshows that air-borne contributions are most worthwhile investigating. By now we have an indication that

Vehicle PropertiesType Volkswagen Golf 5Tires 6 different types varying from slick’s to off road profiles and summer to winter tires

Dynamometer PropertiesSurface Smooth or Rough roadTransmission 2 wheel drive coastdown

OTPA SetupSources 3D ICP acc. Sensor at left and right Front Tire (FL / FR) hubs in global x,y,z

directions (X,Y,Z). These global vehicle directions are X: longitudinal,Y: lateral, Z: vertical. 1/2 Inch Microphones at left and right Front Tiresat their Lead, Side and Trail position (Le,Si,Tr) about 2 meters apart andaside from the wheel.Airborne and Structure borne paths are denoted with (AB) or (SB) respectively.

Transitions NoneReceiver Driver’s ear (1x) (MIC)

Measurement SettingsSampling rate 16000 HzBlock size 8192 samplesMeasurement time 100 seconds

Table 1: Measurement details. Acceleration sensors were placed at the wheel connection to the vehicle andthe microphones for lead, trail and side positions.

structure-borne vibration is dominant on an overall level, their dominance occurs at frequencies lower that700 Hz and from figure 2.a one knows that the z-direction path is most dominant. In a third step, figures3.sb and 3.ab show the individual air-borne and structure-borne contributions (again on average). From thesefigures one can concluded that the right tire has a more dominant impact on the driver. Furthermore thefigures show that the dominance amongst channels change as a function of frequency, which is probablycaused by different vehicle modes or excitation, yet to be investigated. In the final step of this individualanalysis, the so called “causal effect” is therefore studied. An example is shown in figure 4.a for the mostdominant excitation node SB_RF_Z. The figure is marked with 3 individual frequency lines from whichinteresting observations can be made:

line 1 At about 100 Hz, in average, the response signal has the highest value over the complete frequencyrange. From the excitation and Transfer Function it can be concluded, that both have high amplitudes.Measures for optimization clearly require work on both aspects.

16This result shows that the OTPA method can be used for these kind of indications as well.


200 300 400 500 600 700 800 900 1000 11001/min

0

10

20

60dB(A) Source: Tire type 1

30

40

50

Front_DriverP_synAB_LF_SideAB_LF_LeadAB_LF_TrailAB_RF_SideAB_RF_LeadAB_RF_TrailSB_LF_xSB_LF_ySB_LF_zSB_RF_xSB_RF_ySB_RF_z

0 100 200 300 400 500 600 700 800 900 1000Hz

-10

0

10

20

30

40

50dB(A)

MeasuredSynthesisAirbornStrucBorn

OTPA synthesis versus measured response

Sou

nd P

ress

ure

Sou

nd P

ress

ure

Figure 2: (a) The response signal, e.g. the microphone at the driver’s ear, separated in individual path contributionsusing the OTPA method and resynthesis of the resulting time signal. (b) Averaged Auto Power Spectra of the synthesizedpath contributions from the complete vehicle coastdown compared to the originally measured response.

Sound Pressure [dB(A)] Sound Pressure [dB(A)] Sound Pressure [dB(A)]

Sound Pressure [dB(A)] Sound Pressure [dB(A)] Sound Pressure [dB(A)] Sound Pressure [dB(A)]

-30

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Figure 3: (SB) Contribution of the individual structure-borne paths in the response signal at the driver’s ear inleft / right comparison, x,y,z at left tire, x,y,z at right tire and all represented in 1/3 octaves. (AB) Contribution of theindividual air-borne paths in the response signal at the driver’s ear in left / right comparison, side, lead, trail at lefttire, side, lead, trail at right tire and all represented in 1/3 octaves.

line 2 At about 240 Hz, in average, the excitation has a very high amplitude, whereas the vehicle transferfunction doesn’t. The “resonance” peak in the excitation is known to originate from a tire cavity mode.Measures for optimization require in this case work on the tire.

line 3 At about 410 Hz, in average the vehicle’s transfer function shows a clear “resonance” peak.17 Mea-sures for optimization require work on the vehicle. In this case a more detailed OTPA would be useful,where sensors are placed at transition points of the individual suspension parts. From that measure-ment more detailed optimization measures can be conducted as the real mode shape is not known yet.Alternatively a targeted FRF measurement, by impulse hammer for example, can be conducted for thedominant path only. One saves time as only one (or a few) path(s) need to be examined, from thevehicle as it is.

Further investigation of line 3 was indeed conducted with an impulse hammer measurement, which yieldsfigures 5.a and 5.b. Left all three structure-borne FRF from the left tire are displayed. It can be observed

17As discussed in 2.1, higher amplitudes do not necessarily indicate a structure resonance.


that a resonance is indeed situated around 410 Hz, confirming the OTPA analysis outcome. On the right, theFRF (e.g. force as input) in z-direction is compared with the TF (e.g. displacement as input) from the OTPAmeasurement. Observe that at most FRF resonances shows great similarity with the Transfer Functions aswell. In summary the OTPA well shows the engineer it’s optimization options for this tire noise problem, oralternatively which FRFs need to be determined / examined. Interesting to note is that for the different tiresleft and right can also be found in the OTPA analysis. Here the right tire shows to be loudest.

Other analysis tools useful, but not discussed in this paper, are for example analysis on individual tire orders,psycho-acoustic analysis on the time data like tonality and roughness and as discussed in [5] dominance ofexcitation sources in a frequency band specific vehicle speed.

These analysis results were obtained by rejecting the two smallest singular values, figure 4.b showing all ofthe singular values of this measurement. Notice that the two smallest, as well as the two / three largest standout of the middle ones.

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Figure 4: (a) Causal Effect of the dominant SB −RF −Z node, separated in excitation, vibration propagation andthe resulting response at the driver’s ear. (b) Singular Values when both SB and AB noise paths are used in the TFmatrix determination with the two smallest singular values showing an out of bound character.

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Figure 5: (a) The FRF measured with an impulse hammer show a resonance at 410 Hz with the highest amplitudefor the z-direction excitation. (b) Comparison of the OTPA TF with the measured FRF using an impulse hammer. Theamplitudes of the TF from the OTPA analysis doesn’t comply to the correct y-axis values for the sake of clarity, yet thefrequency axis does.


4.2 Analysis of all Measurements

In this section a parameter study is conducted to determine how stable the TF matrix determination is withdifferent setups. First figure 6.a and 6.b show the analysis results when the Air-borne channels are leftin the TF matrix calculation. Clearly in the frequency range where Structure-borne noise is dominant nosignificant differences exist, showing dominant paths are not affected by small (yet coherent) ones. Also athigher frequencies, where Air-borne noise is dominant, no large differences exist, showing that low coherentpaths do not pose a problem. Yet in the frequency range 750 - 1000 Hz differences get into existence ascoherence between the Air-borne and Structure-borne channels reach up 40%. This causes a redistributionof the energy, and stresses the importance of a proper OTPA model.

Secondly a parameter study is conducted with different tires and rough / soft road conditions. Figure 7 showshow the TF matrix change. From the figure it can be seen that the transfer functions stay quite constant,although the corresponding excitation changed quite a bit. This shows that the OTPA method is capableof identifying stable results, and that an indication of all measurements simultaneously in the TF matrixcalculation yields some average estimates. Notice furthermore, in figure 8, that the Air-borne path matches

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With AB Source - TF FL-Z to DriverWith AB Source - TF FR-Z to DriverNo AB source - TF FL-Z to DriverNo AB source - TF FR-Z to Driver

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Figure 6: (a) Difference between the contribution of the Structure-Borne excitation when the Air-borne excitation isor is not taken into account in the OTPA calculation. (b) The dominant Transfer Function of the Structure-Borne pathdoesn’t change when the Air-borne channels are deactivated in the OTPA algorithm up to 650 Hz, where SB noise isdominant. At higher frequencies, the AB path get dominant and difference is apparent.

well on rough and smooth road at frequencies higher than 100Hz. Up to about 100Hz, high differences arefound though. Whereas the Structure-borne path doesn’t really change over the complete frequency range,this example shows the Air-borne path behaves different on smooth and rough road. This could be the casewhen the noise on the smooth road is unidirectional, whereas the noise on the rough road is omnidirectional.Indeed, the excitation level, figure 8.b, is similar up to 60 Hz and then deviates more and more towards 1000Hz between both road types.


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5x individual tyresTF from combined tyres



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Figure 7: The Transfer Functions of the dominant path are quite similar for the different tyre types, especially for theStructure-borne noise; somewhat larger differences are found for the Air-borne path.

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Figure 8: (a) The Transfer Functions calculated from all measurements at ones are similar for Structure-borne noiseon Rough and Smooth Road conditions. However, between 30 - 150 Hz clear differences are apparent for the Air-bornepath. (b) The excitation level from the Air-borne paths is similar up to 60 Hz, yet deviates at higher frequencies.

5 Summary & Conclusion

Operational TPA finds it’s roots in the work of Bendat [1, 2], yet evolved from FRF methods in TPA. Itsmain benefits are being faster than FRF based TPA and being operational.

This article shows that the OTPA algorithm, without SVD, is equivalent to the MIMO technique of findingFRFs. Yet, the OTPA allows for any kind of signal, like sound pressure and acceleration, for the definitionof the OTPA model. As this kind of signals are responses from a physical perspective, this article intro-duces guidelines for a proper definition. Indeed, one can group locations on a system in source, transitionand response nodes. Furthermore, by the construction of the algorithm, cross talk between input signals isaccounted for.

The SVD algorithm is required for an accurate Transfer Function (TF) matrix estimation as coherence be-tween input signals can be quite large. This does not pose a problem from a theoretical point, yet measure-


ment noise get’s an ever increasing effect on the TF estimate. Indeed, it was found that measurement noisereveals itself in the smallest singular values [18].

This article includes a tire noise analysis which shows, for example, that the OTPA method can determineStructure-borne and Air-borne dominance easily. Furthermore a parameter study is conducted to examinethe effect of forgetting paths, changes of the TF matrix with different tyre types and road surfaces. The studyshows that only in case signals are forgotten which are coherent and in the same order of magnitude may notbe neglected. The structure-borne TF’s do not depend on the tyre type and road condition much, showingthat the OTPA is independent on the operational condition. Yet the parameter study also reveals that theAir-borne path does show a large difference up to about 150 Hz due to the difference in directionality of thetire on smooth and rough road.

References

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[2] J.S. Bendat and A.G. Piersol. Engineering Applications of Correlation and Spectral Analysis. John Willey andSons, 1980.

[3] D. de Klerk. Determining the Significant Number of Singular Values in Experimental Dynamic Substructuring.In Proceedings of the Twentyseventh International Modal Analysis Conference, Orlando, FL, Bethel, CT, 2009.Society for Experimental Mechanics.

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[7] F.M. Dekking, C. Kraaikamp, H.P. Lopuhaa, and L.E. Meester. Probability and Statistics for the 21st Century.Delft University of Technology, Delft, The Netherlands, 2004.

[8] P. A. Devijver and J. Kittler. Pattern Recognition: A Statistical Approach. Prentice-Hall, London, 1982.

[9] D. Ewins and W. Liu. Transmissibility properties of mdof systems. In Proc. 16th IMAC, pages 847–854, SantaBarbara (CA), 1998.

[10] P. Gajdatsy, K. Janssens, L. Gielen, P. Mas, and H. Van der Auweraer. Critical Assessment of Operational PathAnalysis: Mathematical Problems of Transmissibility Estimation. In Proceedings of Acoustics’ 2008, pages5461–5466, 2008.

[11] G. Golub and W. Kahan. Calculating the Singular Values and Pseudo-Inverse of a Matrix. Journal of the Societyfor Industrial and Applied Mathematics: Series B, Numerical Analysis, 2(2):205–224, 1965.

[12] K. Janssens, P. Gajdatsy, and H. Van der Auweraer. Operational Path Analysis: a Critical Review. In Proceedingsof the International Conference on Noise & Vibration Engineering (ISMA), pages 3657–3672, Leuven, BE, 2008.Katholieke Universiteit Leuven.

[13] Ron Kohavi. A Study of Cross-Validation and Bootstrap for Accuracy Estimation and Model Selection. InInternational Joint Conference on Artificial Intelligence (IJCAI), pages 1137–1143. Morgan Kaufmann, 1995.

[14] M. Lee, J. Stuart Bolton, and S. Suh. A Procedure for Estimating the Combustion Noise Transfer Matrix of aDiesel Engine. In 15th International Congress on Sound and Vibration, pages 974–981, 2008.

[15] M. Lohrmann and T. Hohenberger. Operational Transfer Path Analysis: Comparison with Conventional Methods.DAGA 2008, 2008.

[16] F.X. Magrans. Path Analysis. In Proceedings of the NAG/DAGA Conference, pages 768–771, 2009.

[17] N.M.M. Maia, J.M.M. Silva, and A.M.R. Ribeiro. The transmissibility concept in multi-degree-of-freedom sys-tems. Mechanical Systems & Signal Processing, 15(1):129–137, 2001.


[18] K. Noumura and J. Yoshida. Method of Transfer Path Analysis for Interior Vehicle Sound by Actual Measure-ment. Society of Automotive Engineers of Japan, Inc, May 2006.

[19] J. Plunt. Strategy for Transfer Path Analysis (TPA) applied to Vibro-Acoustic Systems at Medium and HighFrequencies. In Proceedings of the Twentythird International Conference on Noise & Vibration Engineering(ISMA), pages 139–158, Leuven, BE, 1998. Katholieke Universiteit Leuven.

[20] J. Plunt. Finding and Fixing Vehicle NVH Problems with Transfer Path Analysis. Sound and Vibration, 11,2005.

[21] A.M.R. Ribeiro, J.M.M. Silva, N.M.M. Maia, and M. Fontul. Transmissibility matrix in harmonic and randomprocesses. Journal of Shock and Vibration, 11(5-6):563–571, 2004.

[22] J. Starkey and G. Merril. On the ill-conditioned nature of indirect force measurement techniques. Int. J. Analyt.Experim. Modal Anal., 4(3):103–108, 1989.

[23] P. Varoto and K. McConnell. Single point vs. multi point acceleration transmissibility concepts in vibrationtesting. In Proc. 16th IMAC, pages 83–90, Santa Barbara (CA), 1998.

[24] J. Verheij. Multipath sound transfer from resiliently mounted shipboard machinery. Phd dissertation, TechnischePhysische Dienst TNO-TH, Delft, 1986.

[25] J. Verheij. Experimental procedures for quantifying sound paths to the interior of road vehicles. In Proc. 2nd Int.Conference on Vehicle Comfort, Bologna, Italy, pages 483–491, Oct.14-16 1992.


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