Operational transfer path analysis: Theory, guidelines and tire noise application

Mechanical Systems and Signal Processing 24 (2010) 1950–1962

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Operational transfer path analysis: Theory, guidelines and tire noise application D. de Klerk a,b,, A. Ossipov c a

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 Netherlands M¨ uller-BBM VAS B.V., Zwolle, The Netherlands c Goodyear Technical Center, L-7750 Colmar-Berg, Luxembourg b

a r t i c l e i n f o

abstract

Article history: Received 12 May 2010 Accepted 16 May 2010 Available online 21 May 2010

The operational transfer path analysis (OTPA) method is the subject of research in this article, which starts with a discussion on it’s theory. Here clear similarities with the MIMO technique in experimental modal analysis are found. Based on the knowledge of MIMO, one finds that input signals are allowed to be coherent to a certain extend. As coherence can be larger in OTPA in practice, the method is extended with the singular value decomposition method to reduce influences of noise. The article proceeds with a discussion on points of attention, or boundary conditions, in practical applications. An analysis on tire noise is included to illustrate the points of attention and the methods strength in, for example, vehicle TPA on tires. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Operational transfer path analysis OTPA Tire noise Experimental dynamics MIMO Cross talk cancelation CTC

1. Introduction Operational transfer path analysis (OTPA), using cross talk cancelation (CTC) and singular value decomposition (SVD), is a signal processing method which finds the linearized transfer function (TF) matrix between a set of chosen input and output channels from a measurement. The in- and output relations are determined such, that the transfer functions are linearly independent with respect to each other, hence the name CTC. The resulting transfer functions can be used in a transfer path analysis (TPA), determining a source’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 components describing the transfer functions. Indeed, in practice the numerical operations involved to determine the TF matrix between inputs and outputs often suffer from measurement noise. By rejecting smaller principal components one reduces these influences on the TF estimates. Basically OTPA find its roots in work of Bendat et al. [2,1], yet evolved from the TPA approaches which are based on FRF measurements:

 The classical TPA approach measuring interface force [19].  The matrix inversion method [22,25,15].  Corresponding author at: 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 Netherlands. E-mail addresses: [email protected], [email protected] (D. de Klerk). URL: http://www.MuellerBBM-VAS.nl (D. de Klerk).

0888-3270/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2010.05.009

D. de Klerk, A. Ossipov / Mechanical Systems and Signal Processing 24 (2010) 1950–1962

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 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 determined between defined input/reference points and chosen output point consisting of sound and/or vibration. They are determined by use of impulse hammer or shaker(s) if structural vibration is considered or by use of loudspeaker(s) for air-borne vibration. Secondly, these FRFs are combined with operational forces determined at the reference points1 to generate synthesized response signals. The synthesized output can thereafter be analyzed, determining the contribution of each propagation path. The OTPA method uses a one step approach and builds model of a structure without FRF measurements by hammer, shaker or loudspeaker. Basically, the method uses a response to response transfer function matrix,2 to represent the propagation paths of the structure. The signals all originate from a measurement of the operating system, so that 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 large reduction 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 difficult to reproduce this sound field with loudspeakers, yet OTPA uses the actual excitation source to determine the TF.

 As essentially response data is used for input, careful design of the OTPA model and basic understanding of the analyzed system are required. The goal of this paper is to obtain a better understanding of the OTPA method, highlighting its capabilities and 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 check criteria as well. In Section 4 the method is applied on a vehicle’s tire noise propagation, highlighting some interesting 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 between a chosen set of input and output quantities from a measurement. In the following discussion, these sets of input and output quantities can best be seen as degrees of freedom (DoF) describing the measured object’s excitation (inputs) and the object’s responses (output) as a linear combination of the chosen/assumed excitations. Section 2.1 describes the determination of the TF matrix, in a first step, as a least-squares approximation problem. In this view it can be seen that the OTPA method without singular value decomposition (SVD) is equivalent to the MIMO technique of finding frequency response functions (FRF) if the same input and output variables are measured. In Section 2.2 the OTPA theory is, however, enhanced with the SVD method, yielding TF matrix estimation with reduced noise influences. 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 HðjoÞxðjoÞ ¼ yðjoÞ:

ð1Þ

Here HðjoÞ is the transfer function matrix linking the vector of input DoF xðjoÞ to the vector of output DoF yðjoÞ. The dependency on frequency is denoted by ðjoÞ. In NVH problems, the measured signals are typically motions, denoted uðjoÞ, forces fðjoÞ and sound pressures pðjoÞ. The input and output vectors can thus in general be assembled from these quantities as 2 3 2 3 uy ux 6f 7 6 fy 7 x ¼ 4 x 5; y ¼ 4 5, py px ðkÞ T ux ¼ ½uð1Þ x , . . . ,ux  ;

1 2 3

ðnÞ T uy ¼ ½uð1Þ y , . . . ,uy  ,

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

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f x ¼ ½fxð1Þ , . . . ,fxðlÞ T ; ðmÞ T px ¼ ½pð1Þ x , . . . ,px  ;

f y ¼ ½fyð1Þ , . . . ,fyðoÞ T , ðpÞ T py ¼ ½pð1Þ y , . . . ,py  ,

ð2Þ

where the dependency on frequency is omitted for clarity.4 Indices k,l,m denote the number of input channels 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 restricted to choose forces as excitations only, but motion and sound pressure, which are responses from a physical point of view, may also be chosen to model the system’s behavior. In the latter cases, care should be 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 the same dimensions. In fact, usually the number of excitation channels in the input vector x will be larger than the number of DoF in the response/output vector. In vehicle analysis a typical example is to find the transfer 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 matrix H have the form  y ; kaj: ð3Þ Hij ¼ i  xj x ¼ 0 k

Typically this matrix element property is used in experimental modal analysis (EMA), where an external applied force f is applied (e.g. xj), by shaker or impulse hammer, as only input DoF (e.g. xk ¼ 0; kaj) and the resulting responses u of the system are chosen as outputs (e.g. y). These kind of transfer functions are denoted receptance frequency response function (FRF) in the literature and have the special property that their frequency peaks show the free system’s eigenfrequencies. Strictly, one could thus determine a column of the TF matrix in the OTPA method also by exciting the system with only the input DoF xj, while suppressing 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 functions element 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 one measurement only where all excitations are present at once. This determination is discussed next by first taking the transpose of Eq. (1) and writing the equation on entry level: 2 3 H11 . . . H1n 6 & ^ 7 ð4Þ ½xð1Þ , . . . ,xðmÞ 4 ^ 5 ¼ ½yð1Þ , . . . ,yðnÞ : Hm1 . . . Hmn Here m and n denote the number of in- and output DoF. Taking the transpose does not allow the determination of the TF elements though. In order to do so, notice that during an operational measurement of, for example, a vehicle run-up on a dynamometer, a set of synchronized measurement blocks will be stored to disk. In general these sets will not have the same content, as the excitations change continuously during the measurement. If one requires, or defines, the relation between the input and output DoF as being linear(ized) and constant during the total measurement, Eq. (4) should, however, hold for each individual measurement block. One could thus extend Eq. (4) writing the equation for all measurement blocks r, yielding 32 3 2 3 2 ð1Þ H11 . . . H1n y1 . . . yðnÞ xð1Þ . . . xðmÞ 1 1 76 7 6 1 6 6 ^ & ^ 7 & ^ 7 & ^ 7 ð5Þ 5¼6 54 ^ 5m: 4 4 ^ Hm1 . . . Hmn . . . yðnÞ yð1Þ . . . xðmÞ xð1Þ r r r r This formulation, or system model, now requires the transfer functions to be linearly independent with respect 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 matrix compensates for it. Here it is assumed that the experiment is performed such that the number of measurement blocks is bigger than the total amount of in- DoF, e.g. r 4 m. This approach makes Eq. (5) a solvable least-squares optimization problem with an additional residue m for the content which cannot be modelled by the (chosen) set of input DoF.5 Indeed, notice that in general the individual observations/measurement blocks will contain a distortion due to, for example, measurement noise or additional unmeasured excitation sources that are not taken into account in the model. Furthermore, the TF of the system might not be constant during the measurement either, due to nonlinear system behavior of some kind.6 Mentioned issues make a well constructed experiment and model essential, which will be discussed in more detail in Section 3. 4 5 6

It is also allowed to use any other physical quantities, as long as they have the same FFT parameter during measurement. Notice, that if different physical quantities are used one needs to perform a weighted least-squares estimate [7]. Possible mechanisms could be a system’s dependency on temperature, applied excitation amplitudes or rotation speed [4].

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To solve Eq. (5) one can first simplify its formulation writing it in a more compact way in matrices XHþ m ¼ Y:

ð6Þ

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, requiring the residue vector to lay in the null space of the input DoF, i.e. XT m ¼ 0. The TF matrix is thereafter found as H ¼ ðXT XÞ1 XT Y ¼ X þ Y,

ð7Þ

+

where matrix X is called the ‘‘pseudoinverse’’ of matrix X, readily defined as X þ 9ðXT XÞ1 XT :

ð8Þ

As the TF matrix is now determined (7), one can calculate the residual vector m, which shows the unmodelled part of Y, by substitution of (7) into (6) yielding after some manipulation

m ¼ ðIXðXT XÞ1 XT ÞY:

ð9Þ

If the OTPA method is considered purely as a least-squares calculation, it can be observed that the method is equivalent to the MIMO technique of finding FRF estimates, if the same input and output variables are chosen. Indeed, observe that XTX in (7) is in fact the averaged auto power spectrum (APS) matrix Gxx of the input signals and XT Y is the averaged cross power spectrum (CPS) matrix Gxy between the input and output signals, accept for them not being divided by the number of measurement blocks. This normalization cancels in the equation though and the OTPA method as a least-squares method is thus equal to the MIMO technique of determining FRF estimates: H ¼ ðXT XÞ1 XT Y ¼ G1 xx Gxy ,

ð10Þ

Gxx 91=rXT X,

ð11Þ

Gxy 91=rXT Y,

ð12Þ

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 signals will be partly coherent with respect to each other in MIMO techniques as well. Indeed, as they are all connected to the test structure simultaneously, their vibrations propagate to one another and are measured by all 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 a numerical point of view, with today’s 64 bit computer technology, even very high coherence in either MIMO or OTPA does not pose a problem for the calculation itself. Due to measurement noise and other disturbances coherence 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 disturbances get 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. Therefore theoretically 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 channels might well exceed 30–40% in practice. In order to reduce the influences of measurement noise in such often occurring events, the OTPA algorithm is extended with a singular value decomposition in Section 2.2 to overcome this problem. Note once more that OTPA typically does not determine receptance FRF, but transfer functions also known as transmissibilities. The transmissibility describes the isolation of a system. Amplitude peaks and drops over a frequency do therefore not necessarily refer to resonances or anti-resonances of the system.7 Care should therefore be taken in the interpretation of TFs in OTPA, e.g. it is often better to concentrate 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 talk compensation (CTC).  OTPA and MIMO both allow almost fully coherent input signals in the TF matrix estimation from a theoretical point of



view. In practice measurement noise/errors will become more and more influential on 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 m in (9) shows how much of the output signal cannot be modelled by the (chosen) input signals. In OTPA this is an useful property, as one can determine if the (chosen) input signals well model the output signals. 7

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

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2.2. OTPA enhancement with singular value decomposition The explicit determination of the transfer function matrix H by (7) can cause erroneous estimates if input signals are highly coherent in combination with measurement noise.8 Use is therefore made of singular value decomposition (SVD), to prevent poor estimates. Indeed matrix X can be expressed by a singular value decomposition as X ¼ URVT :

ð13Þ

U is an r  r unitary matrix, R is a r  m matrix with nonnegative numbers on the diagonal (as defined for a rectangular matrix) and zeros off the diagonal. VT denotes the conjugate transpose of V, an m  m unitary matrix.9 The SVD is very general in the sense that it can be applied to any r  m matrix. A standard eigenvalue decomposition, 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: XT X ¼ VRT UT URVT ¼ VðRT RÞVT ,

ð14Þ

XXT ¼ URVT VRT UT ¼ UðRT RÞUT :

ð15Þ

The right hand sides of the relations above therefore describe the eigenvalue decompositions of the left hand sides. Consequently, the squares of the non-zero singular values of X are equal to the non-zero eigenvalues of either XT X or X XT. Furthermore, the columns of U (left singular vectors) are eigenvectors of X XT and the columns of V (right singular vectors) are eigenvectors of XT X.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 R contains the singular values, which can be thought of as scalar ‘‘gain controls’’ by which each corresponding input is multiplied to give a corresponding output. A common convention is to order the values Sii in nonincreasing fashion, in which case the diagonal matrix R is uniquely determined by X. The singular value decomposition of X can be directly used in the computation of the pseudoinverse X + , yielding X þ ¼ VR1 UT ,

ð16Þ

1

where R is the inverse of R. Notice that here it is assumed that the additional rows in R were omitted in the SVD ~ using the computation making it a square matrix m  m. Substitution of (16) in (7) yields an estimate on the TF matrix H SVD method as12 ~ ¼ VR1 UT Y: H

ð17Þ

From an engineering and statistical point of view it was found in applications that smaller singular values are mainly caused by noise influences and other external disturbances [18]. They are therefore unwanted and should be rejected. Note that the least-squares fit for the analyzed measurement will be better with all singular values kept though. Yet the amount of used singular values in the TF calculation of one measurement reveals a tradeoff in the resulting fit on another, similar, measurement. Indeed, as the noise will be different in both measurements, this cross validation process reveals which of the smallest singular values are related to the noise influences [8,13,3]. Taking only a reduced set of singular values into account therefore improves the TF estimates in general. 3. Points of attention in OTPA In this section different practical considerations on the OTPA and its implementation will be discussed. Here the 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. 8

The measurement noise is amplified in the inversion term (XT X)  1. Typically the matrix sizes are chosen different, e.g. U has size r  r, R has size r  m and VT has size m  m. This will be explained shortly. 10 Determining the SVD matrices in this way is numerically unstable in practice, especially for singular values close to zero. It is explicit determination in this sense does therefore not enhance the analysis given in the previous section. Instead, special algorithms such as the (modified) Golub–Reinsch algorithm and the one-sided Jacobi orthogonalization were developed which overcome possible numerical conditioning problems [11]. 11 In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (their inner product is 0) and both of unit length (the norm of each is 1). 12 Notice that with the SVD approximation, the residual content m lays in the null space of UT. 9

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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 systems dynamics 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 be dominated from the engines combustion and thus implicitly characterizes the combustion itself. One could also think of responses measured on the rear axle differential or at the wheel spindle. Such responses will characterize either the internal gear noise excitation or the road input from the tires. Hence such signal can be well used as input signals/variables. Furthermore a connection point of the driveline to the bodywork contains 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 noise is propagated. In general one can classify locations on any structure in source, transition and response positions, as schematically represented in Fig. 1 for a vehicle. One should not combine channels of different location classes in the input/excitation vector in the OTPA algorithm. Indeed, as the signals are cross talk canceled by the algorithm, combinations yield illogical results as cause and effect are mixed. It is our experience that although the true structure excitation is not measured, building models with response data on well chosen positions as model inputs gives, for example, a good estimation of the structure’s noise propagation in automotive applications [18]. If one forgets a propagation path, coherent parts are redistributed over the other signals though. To make a correct interpretation and analysis of a structure in practice, the following considerations may be used as guidelines:

 Choosing response data measured on different sources as input signal allows to separate a measured output 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 how the source’s excitation propagate into its neighboring component(s).

 Using transition locations as OTPA inputs allow one to determine which locations transmit vibration the most. They do not indicate the origin of the source.

 Using additional responses within the structure allows to identify how the vibration is propagated into the vehicle through succeeding components. Notice that structures which are well equipped with sensors can be used for multiple OPTA analysis from a single operational data set. With smaller channel count, one can make multiple setups to determining different 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 measured responses. This can be done by observing the least-squares residual m amplitudes (9) or by comparing the measured responses Y with the synthesized ones according Y~ ¼ XVR1 UT Y:

ð18Þ

Differences in amplitudes of Y and Y~ can be evaluated using any number of singular values in R. Observed discrepancies are caused by either:

 The input and/or output signals contain additional noise content which is filtered by the OTPA algorithm.  The chosen input signals are not the only sources which contribute to the response signal. Receiver Transition Point

Excitation Source Fig. 1. It is essential in OTPA to define positions on the system which one classifies in source locations, transition points and response positions. In the OTPA calculation, one should not combine channels of different location classes in the input vector, as cause and effect are mixed faulty.

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Table 1 Measurement details. Vehicle properties Type Tires

Volkswagen Golf 5 Six different types varying from slick’s to off road profiles and summer to winter tires

Dynamometer properties Surface Smooth or rough road Transmission Two wheel drive coastdown OTPA setup Sources

Transitions Receiver

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 in. Microphones at left and right front tires at their lead, side and trail position (Le,Si,Tr) about 2 m apart and aside from the wheel. Air-borne and structure-borne paths are denoted with (AB) or (SB), respectively None Driver’s ear (1x) (MIC)

Measurement settings Sampling rate Block size Measurement time

16 000 Hz 8192 samples 100 s

Acceleration sensors were placed at the wheel connection to the vehicle and the microphones for lead, trail and side positions.

 The system might behave nonlinear during the chosen set of measurement blocks. The TF matrix will represent 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 into account in the TF matrix estimation. If the TF matrix changes considerably this could indicate nonlinear system 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 as possible. Larger variation minimizes the coherence between the chosen input channels and results in a better conditioning, i.e. higher values of the lowest singular values, of the input matrix. As such, the noise influence is 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 measurement. Indeed, during the vehicle run-up excitation sources change continuously in amplitude and direction. It was noticed in [10] that care should be taken, as higher frequencies are excited by less engine orders, hence less variation might be expected at higher frequencies. Typically such problems can be traced by a significant drop of the responses’ multiple coherence and a large residue (9). Yet in general higher frequencies are dominated by air-borne noise, also see the result in the Section 4. When these signals are also measured, 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 10–15 dB or less should be sufficient to reduce influences of noise. However, depending on the level of external noise, number of input channels and coherence between the input channels, this criteria can only be used as a rule of thumb. It was found, for example, that larger number of input channels as well as high coherence between excitation signals yield a higher relevant threshold. It means, typically, that the condition number should be around a factor 100 from 12 input channels upwards. 4. OTPA application on tire noise This section discusses a tire noise analysis at Goodyear’s test facility in Luxembourg with specifications listed in Table 1. Here we used a 3D acceleration sensor at each wheel hub. Furthermore we used three microphones for each tire at approximately 2 m aside of the wheel. Indeed, experience learned that these positions are suited to analyze the air-borne path up to about 1200 Hz, as later seen in Fig. 2. More microphones are needed for the analysis of higher frequency ranges 13

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

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dB (A) OTPA synthesis versus measured response

Source: Tire type 1

60

50

50

40

40

Front_Driver P syn_ AB_LF_Side AB_LF_Lead AB_LF_Trail AB_RF_Side AB_RF_Lead AB_RF_Trail SB_LF_x SB_LF_y SB_LF_z SB_RF_x SB_RF_y SB_RF_z

30 20 10 0 200

300

400

500

600 700 1/min

800

900 1000 1100

Sound Pressure

Sound Pressure

dB (A)

1957

30 20 10 Measured Synthesis Airborn StrucBorn

0 -10 0

100 200 300 400 500 600 700 800 900 1000 Hz

Fig. 2. (a) The response signal, e.g. the microphone at the driver’s ear, separated in individual path contributions using the OTPA method and resynthesis of the resulting time signal. (b) Averaged auto power spectra of the synthesized path contributions from the complete vehicle coastdown compared to the originally measured response.

though, due to the increasing complexity of the sound field.14 The analysis at hand was setup such, that both air-borne and structure-borne vibration is analyzed with six different tire combinations, all on both rough and smooth road surface. In Section 4.1 one tire combination, with different tire types left and right, is analyzed on the dynamometer’s smooth road surface. Thereafter all 12 measurement will be compared to analyze how the OTPA results change 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 synthesized output 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 Eq. (18) allows to synthesize the output channels in Y~ with the originally measured inputs X. Fig. 2a shows this comparison for an arbitrary tire set. Note that here the overall sound level is depicted during the coastdown to get a good overview. The figure shows that the total synthesized sound level at the driver’s ear matches very well with the original measured signal, e.g. the red curves overlap. This indicates, that on an overall level, the OTPA model is well able to describe the receiving sound pressure at the driver with the six structure-borne and six air-borne channels. The other curves in blue and gray show the individual path contributions from each of the 12 input/source channels. Observe that depending on the vehicle speed, individual path dominance vary. The structure-borne excitations in z-direction clearly have the largest overall impact on the driver’s sound level and are therefore most worthwhile optimizing. It can be observed though that the left and right excitation not always yield the same contribution, which is due to the two different tires left and right.15 From Fig. 2a with its overall sound levels one does not have an indication what frequency range should be tackled/ optimized. In a second step one therefore proceeds to compare the frequency content of the synthesized and measured outputs. In order to do so, all auto power spectrum (APS) measurement blocks of the complete coastdown are averaged. This gives an overall indication of the model’s fit in frequency over the complete run. As seen in Fig. 2b, the model fits the measurement well up to a frequency of about 1000 Hz. 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. Therefore one can conclude that the dominant air-borne propagation path cannot be properly measured above 1500 Hz, spatially, with six 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 about 700 Hz. At higher frequencies, air-borne vibration becomes the dominant one for the tested vehicle. As the highest amplitudes occur below 700 Hz, the structure-borne dominance on the overall level, see Fig. 2a, is well explained. Should higher ‘‘whissle’’/tonal kind of noises be a psycho-acoustical target, this analysis shows that air-borne contributions are most worthwhile investigating. By now we have an indication that structure-borne vibration is dominant on an overall level, their dominance occurs at frequencies lower that 700 Hz and from Fig. 2a one knows that the z-direction path is most dominant. In a third step, Figs. 3(SB) and 3(AB) show the individual air-borne and structure-borne contributions (again on average). From these figures one can concluded that the right tire has a more dominant impact on the driver. Furthermore the figures show that the dominance amongst channels change as a function of frequency, which is probably caused by different 14 Note that in case one measures a vehicle with different number of microphones, these measurement’s transfer functions cannot be directly compared as coherent parts between microphones is distributed differently. 15 In measurements where the same tires were used at each side, the contributions show much better ‘‘symmetry’’. 16 This result shows that the OTPA method can be used for these kind of indications as well.

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Sound Pressure [dB (A)]

Sound Pressure [dB (A)]

Sound Pressure [dB (A)]

Sound Pressure [dB (A)]

50 30 SB SB_LF_x SB_LF_y SB_LF_z SB_RF_x SB_RF_y SB_RF_z

10 SB SB_LF SB_RF

-10 -30

10

Hz Sound Pressure [dB (A)]

25

63 Hz

160

AB AB_LF_Side AB_LF_Lead AB_LF_Trail AB_RF_Side AB_RF_Lead AB_RF_Trail

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

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Fig. 4. (a) Causal effect of the dominant SB-RF-Z node, separated in excitation, vibration propagation and the resulting response at the driver’s ear. (b) Singular values when both SB and AB noise paths are used in the TF matrix determination with the two smallest singular values showing an out of bound character.

vehicle modes or excitation, yet to be investigated. In the final step of this individual analysis, the so called ‘‘causal effect’’ is therefore studied. An example is shown in Fig. 4a for the most dominant excitation node SB_RF_Z. The figure is marked with three individual frequency lines from which interesting observations can be made: line 1 At about 100 Hz, in average, the response signal has the highest value over the complete frequency range. From the excitation and transfer function it can be concluded, that both have high amplitudes. Measures for optimization clearly require work on both aspects. line 2 At about 240 Hz, in average, the excitation has a very high amplitude, whereas the vehicle transfer function does not. 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 Measures for optimization require work on the vehicle. In this case a more detailed OTPA would be useful, where sensors are 17

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

D. de Klerk, A. Ossipov / Mechanical Systems and Signal Processing 24 (2010) 1950–1962

FRF determined by Impulse Hammer

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Fig. 5. (a) The FRF measured with an impulse hammer show a resonance at 410 Hz with the highest amplitude for the z-direction excitation. (b) Comparison of the OTPA TF with the measured FRF using an impulse hammer. The amplitudes of the TF from the OTPA analysis does not comply to the correct y-axis values for the sake of clarity, yet the frequency axis does.

placed at transition points of the individual suspension parts. From that measurement 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 the dominant path only. One saves time as only one (or a few) path(s) need to be examined, from the vehicle as it is. Further investigation of line 3 was indeed conducted with an impulse hammer measurement, which yields Figs. 5a and b. Left all three structure-borne FRF from the left tire are displayed. It can be observed that a resonance is indeed situated around 410 Hz, confirming the OTPA analysis outcome. On the right, the FRF (e.g. force as input) in z-direction is compared with the TF (e.g. displacement as input) from the OTPA measurement. Observe that at most FRF resonances shows great similarity with the transfer functions as well. In summary the OTPA well shows the engineer it’s optimization options for this tire noise problem, or alternatively which FRFs need to be determined/examined. Interesting to note is that for the different tires left 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, psychoacoustic analysis on the time data like tonality and roughness and as discussed in [5] dominance of excitation sources in a frequency band specific vehicle speed. These analysis results were obtained by rejecting the two smallest singular values, Fig. 4b showing all of the singular values of this measurement. Notice that the two smallest, as well as the two/three largest stand out of the middle ones. 4.2. Analysis of all measurements In this section a parameter study is conducted to determine how stable the TF matrix determination is with different setups. First Figs. 6a and b show the analysis results when the air-borne channels are left in the TF matrix calculation. Clearly in the frequency range where structure-borne noise is dominant no significant differences exist, showing dominant paths are not affected by small (yet coherent) ones. Also at higher frequencies, where air-borne noise is dominant, no large differences exist, showing that low coherent paths do not pose a problem. Yet in the frequency range 750–1000 Hz differences get into existence as coherence between the air-borne and structure-borne channels reach up 40%. This causes a redistribution of 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. Fig. 7 shows how 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 capable of identifying stable results, and that an indication of all measurements simultaneously in the TF matrix calculation yields some average estimates. Notice furthermore, in Fig. 8, that the air-borne path matches well on rough and smooth road at frequencies higher than 100 Hz. Up to about 100 Hz, high differences are found though. Whereas the structure-borne path does not 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 case when the noise on the smooth road is unidirectional, whereas the noise on the rough road is omnidirectional. Indeed, the excitation level, Fig. 8b, is similar up to 60 Hz and then deviates more and more towards 1000 Hz between both road types.

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D. de Klerk, A. Ossipov / Mechanical Systems and Signal Processing 24 (2010) 1950–1962

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Fig. 6. (a) Difference between the contribution of the structure-borne excitation when the air-borne excitation is or is not taken into account in the OTPA calculation. (b) The dominant transfer function of the structure-borne path does not change when the air-borne channels are deactivated in the OTPA algorithm up to 650 Hz, where SB noise is dominant. At higher frequencies, the AB path get dominant and difference is apparent.

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5. Summary and conclusion Operational TPA finds it’s roots in the work of Bendat [1,2], yet evolved from FRF methods in TPA. Its main 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 finding FRFs. Yet, the OTPA allows for any kind of signal, like sound pressure and acceleration, for the definition of the OTPA model. As this kind of signals are responses from a physical perspective, this article introduces guidelines for a proper definition. Indeed, one can group locations on a system in source, transition and response nodes. Furthermore, by the construction of the algorithm, cross talk between input signals is accounted for.

D. de Klerk, A. Ossipov / Mechanical Systems and Signal Processing 24 (2010) 1950–1962

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Fig. 8. (a) The transfer functions calculated from all measurements at ones are similar for structure-borne noise on rough and smooth road conditions. However, between 30 and 150 Hz clear differences are apparent for the air-borne path. (b) The excitation level from the air-borne paths is similar up to 60 Hz, yet deviates at higher frequencies.

The SVD algorithm is required for an accurate transfer function (TF) matrix estimation as coherence between input signals can be quite large. This does not pose a problem from a theoretical point, yet measurement noise get’s an ever increasing effect on the TF estimate. Indeed, it was found that measurement noise reveals itself in the smallest singular values [18]. This article includes a tire noise analysis which shows, for example, that the OTPA method can determine structureborne and air-borne dominance easily. Furthermore a parameter study is conducted to examine the effect of forgetting paths, changes of the TF matrix with different tyre types and road surfaces. The study shows that only in case signals are forgotten which are coherent and in the same order of magnitude may not be neglected. The structure-borne TF’s do not depend on the tyre type and road condition much, showing that the OTPA is independent on the operational condition. Yet the parameter study also reveals that the air-borne path does show a large difference up to about 150 Hz due to the difference in directionality of the tire on smooth and rough road. References [1] J. Bendat, System identification from multiple input/output data, Journal of Sound and Vibration 49 (3) (1976) 293–308. [2] J. Bendat, A. Piersol, Engineering Applications of Correlation and Spectral Analysis, John Wiley and Sons, New York, 1980. [3] D. de Klerk, Determining the significant number of singular values in experimental dynamic substructuring, in: Proceedings of the 27th International Modal Analysis Conference, Orlando, FL. Society for Experimental Mechanics, Bethel, CT, 2009. [4] D. de Klerk, Dynamic response characterization of complex systems through operational identification and dynamic substructuring, Ph.D. Thesis, Delft University of Technology, 2009. [5] D. de Klerk, M. Lohrmann, M. Quickert, W. Foken, Application of operational transfer path analysis on a classic car, NAG/DAGA 2009, 2009. [6] D. de Klerk, D. Rixen, Component transfer path analysis method with compensation for test bench dynamics, Mechanical Systems & Signal Processing, 2010, doi:10.1016/j.ymssp.2010.01.006. ¨ L. Meester, Probability and Statistics for the 21st Century, Delft University of Technology, Delft, The [7] F. Dekking, C. Kraaikamp, H. Lopuhaa, Netherlands, 2004. [8] P.A. Devijver, J. Kittler, Pattern Recognition: A Statistical Approach, Prentice-Hall, London, 1982. [9] D. Ewins, W. Liu, Transmissibility properties of mdof systems, in: Proceedings of 16th IMAC, Santa Barbara, CA, 1998, pp. 847–854. [10] P. Gajdatsy, K. Janssens, L. Gielen, P. Mas, H.V. der Auweraer, Critical assessment of operational path analysis: mathematical problems of transmissibility estimation, in: Proceedings of Acoustics’ 2008, 2008, pp. 5461–5466. [11] G. Golub, W. Kahan, Calculating the singular values and pseudo-inverse of a matrix, Journal of the Society for Industrial and Applied Mathematics: Series B, Numerical Analysis 2 (2) (1965) 205–224. [12] K. Janssens, P. Gajdatsy, H.V. der Auweraer, Operational path analysis: a critical review, in: Proceedings of the International Conference on Noise & Vibration Engineering (ISMA). Katholieke Universiteit Leuven, Leuven, BE, 2008, pp. 3657–3672. [13] R. Kohavi, A study of cross-validation and bootstrap for accuracy estimation and model selection, in: International Joint Conference on Artificial Intelligence (IJCAI), Morgan Kaufmann, 1995, pp. 1137–1143. [14] M. Lee, J.S. Bolton, S. Suh, A procedure for estimating the combustion noise transfer matrix of a diesel engine, in: 15th International Congress on Sound and Vibration, 2008, pp. 974–981. [15] M. Lohrmann, T. Hohenberger, Operational transfer path analysis: comparison with conventional methods, DAGA 2008. [16] F. Magrans, Path analysis, in: Proceedings of the NAG/DAGA Conference, 2009, pp. 768–771. [17] N. Maia, J. Silva, A. Ribeiro, The transmissibility concept in multi-degree-of-freedom systems, Mechanical Systems & Signal Processing 15 (1) (2001) 129–137. [18] K.Noumura J. Yoshida Method of Transfer Path Analysis for Interior Vehicle Sound by Actual Measurement, Honda R&D Tech Rev, L0353A, 2006, vol. 18, No. 1, pp. 136-141, Japan.

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[19] J. Plunt, Strategy for transfer path analysis (TPA) applied to vibro-acoustic systems at medium and high frequencies, in: Proceedings of the 23rd International Conference on Noise & Vibration Engineering (ISMA). Katholieke Universiteit Leuven, Leuven, BE, pp. 139–158. [20] J. Plunt, Finding and fixing vehicle NVH problems with transfer path analysis, Sound and Vibration 39 (11) (2005) 12–17. [21] A. Ribeiro, J. Silva, N. Maia, M. Fontul, Transmissibility matrix in harmonic and random processes, Journal of Shock and Vibration 11 (5–6) (2004) 563–571. [22] J. Starkey, G. Merril, On the ill-conditioned nature of indirect force measurement techniques, International Journal of Analytical and Experimental Modal Analysis 4 (3) (1989) 103–108. [23] P. Varoto, K. McConnell, Single point vs. multi point acceleration transmissibility concepts in vibration testing, in: Proceedings of the 16th IMAC, Santa Barbara, CA, 1998, pp. 83–90. [24] J. Verheij, Multipath sound transfer from resiliently mounted shipboard machinery, Ph.D. Dissertation, Technische Physische Dienst TNO-TH, Delft, 1986. [25] J. Verheij, Experimental procedures for quantifying sound paths to the interior of road vehicles, in: Proceedings of 2nd International Conference on Vehicle Comfort, Bologna, Italy, October 14–16 1992, pp. 483–491.