Application of the Transmissibility Concept in Transfer Path Analysis

Transcription

1 Alication of the Transmissibility Concet in Transfer Path Analysis P. Gajdatsy,, K. Janssens, Wim Desmet, H. Van der Auweraer LMS International NV Interleuvenlaan 8, 00 Leuven, Belgium K. U. Leuven, Deartment of Mechanical Engineering Celestijnenlaan 00B, 00 Leuven, Belgium Abstract In recent years, there has been a renewed interest in develoing faster and simler transfer ath analysis (TPA) methods. A dominant class of these new aroaches, often referred to as Oerational Path Analyses (OPA), is designed to achieve this goal by using only oerational data in conjunction with the alication of the transmissibility concet. Desite the reduction in measurement time and comlexity, these suffer from a number of limitations, such as roblems related to the estimation of transmissibility, or the unreliability of the results due to cross couling between ath inuts, etc, which makes them rone to errors. Some of these only aly to one secific method while others are common to all transmissibility based aroaches. The goal of this aer is to identify and describe these limitations and oint out the otential dangers of alying such methods without taking these into account. Introduction The concet of transmissibility has been around for many years [-], but has recently gained new interest, esecially in the field of structural health monitoring and oerational modal analysis []. Furthermore, it has also found its way into other areas, such as noise ath analysis. The link between some new ad-hoc aroaches to noise ath analysis and the transmissibility concet was demonstrated in []. The motivation for the research into new noise ath analysis techniques is that the classical Transfer Path Analysis (TPA) method, desite having become a well-established and reliable technique for tackling NVH roblems, still remains a time consuming and comlex rocedure. The use of transmissibilities aarently offers a ossibility for a significant reduction in measurement time and comlexity. Among these new TPA methods, two grous can be distinguished. Earlier ublications [, 7] describe an aroach which corresonds to the use of transmissibilities to relace the frequency resonse functions (FRFs) to describe the transfer aths [, 8]. As oosed to this, a recently ublished method [9] uses them in an indirect way, for the in-situ estimation of the frequency resonse functions of the system, thus eliminating the need for exerimental FRF measurements. In general, all of these methods are referred to as Oerational Transfer Path Analysis (OTPA), or simly Oerational Path Analysis (OPA) methods, since just the oerational data in itself is sufficient for the analysis. But a fundamental question arises, namely, as to what extent the notions of Transfer Paths and Transfer Path Analysis are still alicable to the results obtained this way. The goal of this aer is to give an answer to this question by identifying the limitations ertaining to both tyes of aroaches. In order to achieve this, first a general system model is introduced which is then followed by an overview of the basic ideas of the transmissibility concet. Then the transmissibility based OTPA methods are discussed in detail and finally, the identified limitations are analyzed through both analytical considerations and numerical simulations. Partial results of this study have already been ublished at various conferences [0-]. Aears in MSSP vol. (7) - ISMA00 secial issue 909

2 90 PROCEEDINGS OF ISMA00 INCLUDING USD00 Transfer Path Analysis (TPA). Basic TPA formulation The transfer ath model used throughout the following discussions is the classical source-ath-receiver TPA model which dates back to the '80s [-]. It is first briefly reviewed and the key ideas and its limitations are discussed. The basic assumtion of this TPA model is that the global system can be divided into an active and a assive art, the former containing the sources, the latter the transmission aths and receiver oints. A schematic reresentation is shown in Fig.. At the interface between the two arts, structural or acoustic loads can be defined deending on the tye of couling. These excitations are then roagated to the receiver oints through the transfer aths. Generally, the loads are force (F i ) or volume velocity (Q j ), e.g. an engine mount or the air intake orifice resectively, and the receiver resonses are sound ressure ( target ) or acceleration (e.g. the sound at the driver s ear or steering wheel vibration). The transfer aths are reresented by their corresonding Frequency Resonse Functions (FRF), which are often called Noise Transfer Functions (NTF s). Each of them describes the relationshi between one inut degree-of-freedom (DoF) and one resonse DoF. There is also a second grou of FRF s in the model, denoted by H ij in Fig. and usually referred to as local FRF s. These FRF s exress the relationshi between the resonses at the inut DoF s (a n, i ) and also at additional resonse locations, which are called overdetermination oints (a k ). This model resumes a causal load-resonse relationshi and that all the FRF s are system characteristics of the global system. Active Subsystem Q j F F F n i NTF j a a H ij a n t a k Passive Subsystem Figure : General TPA model Based on these considerations, the resonse at the receiver location t can therefore be exressed as: ( ) t n i NTF ( ).F ( ) i i r j NTF ( ).Q ( ) where F i (i = n) denotes the interface forces due to structural loads, Q j (j = r) the volume velocities due to acoustic loads, and NTF i and NTF j the corresonding noise transfer functions. j j ()

3 TRANSFER PATH ANALYSIS AND SOURCE IDENTIFICATION 9 Further, a assive side resonse a i can be written as where, Hij reresents the corresonding local FRF s. a i FjH ij () j All of these quantities deend on the dynamics of the actual system. Therefore, if the system is modified and its dynamic behavior is changed then these inut quantities have to be identified again. An alternative solution would be to directly characterize the excitation source of the system [9], but this is beyond the scoe of classical TPA. Frequently, the contributions from structural loads or the contributions from acoustic loads are searately investigated. The former case is generally referred to as Transfer Path Analysis, while the latter as Acoustic Source Quantification or also as Panel Contribution Analysis in case the acoustic sources are radiating anels [7-8]. The underlying general rinciles however remain the same. For the sake of simlicity, the discussions in the following chaters will be restricted to a single case in which the inuts are structural loads and the target is a single ressure resonse.. Practical limitations of classical TPA TPA is a owerful methodology to assess dominant noise source contributions. The most common way to visualize the results of a TPA analysis is the so called artial ath contribution (PPC) lot, such as the one shown in Fig.. In such a lot each row shows the artial contribution of a single ath which is the load multilied by its corresonding NTF to the target resonse as a function of for a given engine order. However, such lot should be handled with care as it only dislays the magnitude information but not the hase relationshi between the different aths. For examle, the black rectangle shows a region where transfer aths and seem to have a high contribution yet the total contribution remains low. This is due to a canceling hase relationshi between the transfer aths, comensating each other. The hase information therefore should always be taken into account during an analysis. To facilitate this, the results are often also lotted on Bode or vector diagrams. 7 Paths Figure : A artial ath contribution (PPC) lot The test rocedure to build a conventional TPA model tyically requires two basic stes, first the identification of the oerational loads during in-oeration tests (e.g. run-u, run-down, etc.) e.g. on the road or on a chassis dyno; and secondly the estimation of the NTF s from exerimental tests (e.g. hammer imact test, shaker test, etc.). The rocedure is similar for both structural and acoustical loading cases, but the ractical imlementation is governed by the nature of the signals and loads.

4 9 PROCEEDINGS OF ISMA00 INCLUDING USD00 As for the second ste, a number of reliable and well-known methods exist to measure and estimate both acoustic and structural NTF s. For examle, recirocity techniques are commonly used, exciting at the acoustic resonse location and measuring corresonding accelerations at the load location [9]. However, with regard to the estimation of the oerational loads, they remain the main factor in the accuracy of the analysis [0-]. Presently, there are three classical measurement methods in use. The most straightforward aroach is to measure the forces directly by using dedicated measuring devices such as e.g. load cells for structural forces. But such direct measurements are not feasible in the majority of the cases because load cells require sace and well-defined suort surfaces. This either distorts the original mounting conditions or simly makes mounting imossible. The second, and most widely used, aroach is the mount stiffness method which can be used in cases when the active and assive system comonents are connected through flexible mounts. This aroach calculates the transmitted force from the relative oerational dislacement across a given mount and the corresonding known mount stiffness rofile. (a ai ( ) a i ( )) F i ( ) K i ( ). () F i is the force in the mount, K i is the mount stiffness rofile and a ai and a i are the active and assive side mount accelerations. The mount stiffness method is a fast method, but its main drawback is that accurate mount stiffness data are seldom available and as most mounts are non-linear, they deend on the load conditions (re-loads) and excitation amlitudes. The third aroach is the inverse force identification method which estimates the oerational forces from a large set of (mostly local) oerational indicator accelerations and the corresonding FRF matrix. For a set of n oerational forces F nx, m oerational acceleration indicator resonses, a mx, are measured on the assive side, along with the corresonding FRF matrix H mxn between all inuts and all indicator resonses. The oerational forces then can be estimated by multilying a mx with the seudo-inverse of H mxn : {F( )} [H( )].{a( )} This equation is calculated searately for each frequency. The number of indicator resonses (m) must exceed the number of forces (n), tyically by a factor as a rule of thumb, in order to minimize illconditioning roblems during the seudo-inverse calculation. The main drawback of this aroach is the need to erform a large number of FRF measurements to build the full H mxn matrix, which is time consuming. Moreover, the active art has to be removed for the FRF measurements, further increasing the duration of the measurement [-]. Desite these limitations this method is used most often in the NVH field, and therefore in this article it is referred to as the classical TPA aroach. For the sake of comleteness, it must be mentioned that there are some other aroaches, such as combining numerical models with measurement data [], the use of a different aroach to FRF measurement [] or the use of Fast TPA aroaches for assessing global subsystem resonses [,8], but these are not yet as widesread as the classical methods described reviously. () Transmissibility The common feature in most of the recently roosed, so-called oerational TPA methods is that they attemt to avoid the time consuming stage of FRF measurements by relying on the in-oeration measurement of system transmissibilities. Transmissibility exresses the relationshi between two resonses in a system. Let a a and a b denote these two resonses, F i the single force acting on the system and H ai, H bi the corresonding transfer functions from H ij. The transmissibility can be formulated then, as: T ab a a ( ) Hai ( ) F i ( ) Hai ( ) () a ( ) H ( ) F ( ) H ( ) b bi i bi

5 TRANSFER PATH ANALYSIS AND SOURCE IDENTIFICATION 9 Basically, transmissibility is the ratio of two FRF s in a single inut case. Naturally, this formulation can also be extended to MIMO systems [-]. In order to gain more insight into the meaning of the revious result, the transfer functions are rewritten in a ole-zero form in Eq. (), as it was formulated by Guillaume et al in []. Here D() reresents the denominator olynomial whose roots are the system oles and N xx () the numerator olynomial containing the zeros of the transfer function. An imortant fact is that while N xx () is secific to each transfer function, D() is common to every member of H ij. Therefore the transmissibility reduces to Eq. (7). H ai Nai ( ) Nbi ( ) ( ) and Hbi ( ) () D( ) D( ) T ab Nai ( ) (7) N ( ) This result has a number of imortant consequences. First of all it can be seen that the eaks in the magnitude of the transmissibility functions do not coincide with the eaks in the FRF s since the oles of one transfer function become the zeros of the transmissibility T ab. Another imortant observation is that the transmissibility deends on the location of the forces []. In case an additional force is acting on the system the transmissibilities will change since the corresonding FRF s will be changed. One way to acquire the transmissibility functions of a system is to measure them exerimentally before the structure is laced in oeration, which is the common ractice in structural health monitoring alications. However, in other cases, such as OMA and OPA, they need to be estimated from oerational data. bi Oerational Transfer Path Analyses In order to overcome the test-time limitations of the traditional TPA aroach and to seed u the analysis rocess, a number of solutions and imrovements have been roosed as mentioned in Section, making different trade-offs regarding seed, accuracy, detail of analysis and causality [-8]. During the following discussions the classical TPA aroach is used as a reference, in site of its shortcomings. The reasons for this choice are twofold: firstly because it is the most well-known aroach in the NVH field and secondly because the new methods in general claim to be equivalent to it. In some of these methods the transmissibility concet is directly alied to the TPA model, by relacing the transfer functions with transmissibilities. The goal of these methods is to use only oerational data to derive TPA-like results without the need for all the additional measurements characteristic of classical TPA aroaches. This is achieved by using a different TPA-like model in which the target resonse is formulated using oerational resonses measured at the load locations instead of the loads themselves. Since the mathematical formulation is similar to that of TPA, there is often confusion about the terminology, which unfortunately leads to an incorrect use of the method and interretation of the results. These methods are discussed in Section.. Another tye of transmissibility based TPA aroach, described in Section., uses the concet in an indirect way. The system descrition remains the classical source-ath-receiver TPA model, but the transfer functions are estimated from the transmissibilities [9, ]. Although this avoids most of the itfalls of the revious grou it still has some limitations.. OPA methods based on direct alication of transmissibility The aroaches belonging to this grou are referred to as OPA or OPA method. The basic idea of the OPA method was described in [] as an ad-hoc measurement method and was then further elaborated by

6 9 PROCEEDINGS OF ISMA00 INCLUDING USD00 [7], however, without linking this aroach to the underlying transmissibility formulation. In [], this aroach was shown to be an imlicit alication of MIMO transmissibility theory. Using the notation of Fig. the mathematical model for a ressure target can be exressed as: ( ) t n i T ( ).a i i ( ) r j T ( ). where a i (i = n) are the body side mount accelerations reresenting the structural aths, j (j = r) oerational ressures reresenting the acoustic sources and, finally, T i and T j the transmissibilities between the acceleration and ressure inuts at the load locations and the target resonse. For the sake of clarity the () term, denoting the frequency deendence, is omitted in the following. The transmissibilities can be estimated from the oerational inut and resonse data using an H estimator: t n i T.a i i r j T T j j t t j j ( ) T {. a, } {a, }' T {a, } {a, }' - - {a, } {a, }' S S {a, }' (0) The sign denotes the comlex conjugate transose of the vectors, the horizontal line the average, the < > sign a row vector, the { } a column vector, [ ] a matrix, and S ta and S aa stand for the averaged cross- and autoower matrix resectively. At first sight this aroach looks romising since there is no need for time consuming FRF measurements. All that is required is some oerational data and the results can be immediately calculated. Desite the fact that this aroach claims to be similar to the classical TPA aroach, there are a number of fundamental differences: first of all, the OPA model is not causal, as oosed to the classical TPA model. Instead of a load-resonse relationshi, it is based on a resonseresonse relationshi, which means that while in TPA one can draw a conclusion as to what effect a certain load has on the total resonse, in OPA one can only talk, with a few excetions, about a similarity, or in other words, a "co-existence" relationshi between the target and the inut resonses. Secondly, the transmissibilities are not system characteristics but deend on the osition and number of forces acting on the system, as it was mentioned in Section, and thirdly, transmissibilities are not equivalent to the NTF s. However, there might be some situations where such a method can still be useful for troubleshooting. Earlier studies [0-] showed that the necessary conditions for this are i) the forces acting on the structure should have a low correlation with each other in order to have good transmissibility and ath contribution estimates, ii) cross couling between aths should be small, meaning that the effect of a given force must be the most rominent in the corresonding inut resonse and negligible in all other inut resonses and iii) all aths must be taken into account. All these limitations are thoroughly elaborated through a set of examles in Section. ta aa (8) (9). OPA method based on indirect alication of transmissibility Desite the fact that this recently ublished method [9] also calls itself an OPA aroach, it is fundamentally different from the revious one. In this method the transmissibilities are used indirectly, only to estimate the NTF s, but not to relace them and the classical TPA model is reserved. This aroach avoids most of the roblems related to the direct transmissibility methods and the results can actually be interreted in the same way as in the traditional TPA. Still it has a few limitations on its own. The most obvious one is that the method requires a large number of incoherent forces acting on the active art and a number of known forces acting at the connection oints during oeration. This requirement limits the alicability of the method since in most ractical situations only a few, and even then often coherent sources are resent on the active art (e.g. engine imbalance), and as for the known forces on the

7 TRANSFER PATH ANALYSIS AND SOURCE IDENTIFICATION 9 assive art, it is usually quite comlicated, if not imossible, to excite the connection oints during oeration, let alone at the exact same DOF as the oerational forces. Limitations of the direct OPA methods In the following, the required conditions for the OPA method, mentioned in Section., are discussed: low coherence between inut forces low cross couling between inut resonses all active aths included in the analysis These conditions need to be fulfilled to obtain hysically meaningful results through the direct OPA methods. In order to assess their effect from a ractical oint of view, a measurement based test case is used. An extensive dataset is synthesized from an existing and well-validated exerimental TPA model originating from an engine noise TPA dataset of a -cylider car in run-u conditions (from 000 to 000 rm). It contains structural aths: engine mounts and subframe mounts measured in three orthogonal directions, lus ressure target at the driver s location. A high quality analysis was erformed and the model was validated for consistency. The use of such reresentative dataset is referred over both ure simulation data, which can not include all real-life critical factors, and over ure exerimental data, which is hard to control to allow consistent comarisons. The resulting dataset includes: i) oerational mount forces F x for orders 0. u to 0, ii) body FRF matrix H x and iii) NTF x between forces and ressure target. Based on this data, a number of numerical simulations are carried out to re-synthesize the load and resonse data, such as the oerational assive-, sometimes called body side accelerations and ressure resonses, for the searate assessment of each condition. The results are shown in PPC lots. The analysis only treats structural transfer aths. A related investigation regarding acoustic sources is discussed in [8]. Traditionally the similarity between synthesized and measured target ressure is used as an indication of the quality of the synthesis. In the case of classical TPA this is a valid aroach, but when it comes to OPA this has to be discarded and the individual ath contributions must be comared. This is due to the fact that the considered OPA methods could be called backward-forward calculations. The target signal is used already during the estimation of the transmissibilities, as seen in Eq., therefore it is not surrising that the ath contributions sum u to the measured target in most cases. In the following stes the error on the individual ath contributions is used to indicate the quality and reliability of the direct OPA method.. Low coherence for good transmissibility estimates The first critical element of the OPA method is the ill-conditioning roblem related to estimating transmissibilities from oerational data. As described in Section. they are tyically estimated by using an H estimator, well-known from classical Least-Squares (LS) estimation: { T} S ta S aa - () The basic condition for erforming this oeration is the invertibility of the averaged autoower matrix which is equal to having a full rank matrix. This requires that in the build-u or averaging of the ower sectra, a number of different conditions at least equal to the number of aths are realized at each frequency. This might be obtained by combining tests under different conditions. For examle, during an engine run-u, each frequency will, with increasing, be excited by different orders, each of these causing a different hase relation between the engine mount resonses. Averaging over the range or in different driving conditions with varying torques also decorrelates the autoower matrix. In most S aa

8 9 PROCEEDINGS OF ISMA00 INCLUDING USD00 ractical cases, however, this is only satisfied in the high frequency range. At lower frequencies the inut vibrations are largely coherent because of the strong modal behavior of the structure, making the autoower matrix rank deficient. In such cases, Singular Value Decomosition (SVD) can be alied to get an aroximate seudo-inverse solution. However, this may lead to an incorrect estimation of the transmissibilities, giving rise to errors in the OPA calculations. By examining tyical oerational run-u signals (Fig. ), one can clearly observe that these are dominated by the engine orders and that besides them there are hardly any other henomena resent. The useful information at a certain frequency is therefore limited by the number of orders resent at that frequency. In order to examine this effect, the original inut forces of the reference TPA dataset are relaced by incoherent signals and the cross-couling between the aths is set to zero, thus excluding the other critical elements of OPA. Then multile scenarios are tested while varying the number of transfer aths Tacho (T) rm Pa Hz FRLE:S (CH) Figure : Colorma of a tyical run-u ressure target signal First, a reduced TPA system with small number of aths was used with at least twice as many orders in the signal than the number of aths and no couling between aths. This reduction to a smaller system served two uroses, first it is easier to comare the results with less aths, secondly this way it could be ensured that for most of the frequency range the number of orders significantly exceeds the number of aths. The OPA method gives similar results as TPA as it is shown in Fig. for order contributions. By a close examination of the two figures, small differences can still be discovered desite all the efforts to make the inut forces incoherent. The reason for this lies in the rocess of averaging. The concet of incoherence only makes sense if a very large number of samles is available. Taking a limited number of averages (smaller than the number of henomena or covering only a set of test conditions smaller than the dimension of the dataset) results in weak decorrelation for the individual contributions and hence illconditioning of the autoower matrix (Eq. ()) and introduces errors in the estimated transmissibilities. In case the number of aths is close to or higher than the number of orders or if the measurement data does not contain enough variation, the averaged autoower matrix S aa becomes rank deficient and only an aroximate solution can be found using a seudo inverse. But as this is only an aroximation, the OPA results diverge more and more from the TPA results. Fig. shows the PPC lots of the TPA and OPA method for a ath system with incoherent inut forces and no cross-couling between the aths. The number of orders (0) no longer exceeds strongly the number of aths (), leading to transmissibility estimation errors and deviating ath contribution results.

9 TRANSFER PATH ANALYSIS AND SOURCE IDENTIFICATION 97 Synthesized resonse OPA resonse 7 Paths Figure : First transmissibility estimation examle, TPA (left) and OPA (right) PPC lots. Paths Synthesized resonse OPA resonse Figure : Second transmissibility estimation examle, TPA (left) and OPA (right) PPC lots. 7 0 The last figure in this section (Fig. ) shows that, for a given number of orders, the average error of the OPA results increases with the number of aths resent in the system. It has been suggested that including different measurement conditions (e.g. using run-u and run-down data from different gear ositions) might alleviate this roblem. As mentioned in Section, earlier studies [, ] have shown that the transmissibilities, as oosed to the NTF's, intrinsically deend on the loading configuration. Changing the amlitude or the hase of the loads will have no effect on the transmissibilities but introducing a new load (e.g. a different airborne source, extra imact tests on the assive side) changes the transmissibilities. Averaging under such circumstances will no longer give valid results therefore combining different loading conditions must be done with roer circumsection. Average OPA error Number of aths Figure : OPA errors increase as more aths are included in the analysis for a given dataset.

10 98 PROCEEDINGS OF ISMA00 INCLUDING USD00 So far, an ideal situation has been resented in which all the inut forces are incoherent. However, in a real test case, the forces are always artially correlated due to the modal behavior of the system, which means that to a significant extent they carry the same information, thus reducing the rank of the autoower matrix and increasing the error of the estimation. Therefore, in such a real-life alication with correlated inuts, the average error can be exected to be much higher than the ideal examle shown here.. Cross-couling between the ath inuts Cross-couling between ath inuts means that the body side acceleration at a certain ath inut deends not only on the force acting at that oint but also on all the other forces. This is clearly exressed in Eq. (), where H ij denotes the assive side transfer functions between the ath inuts, keeing with the notation of Fig. a F H () i j For the simulations, the smaller dataset with only structural aths is used. There is one single force acting at each ath inut which are mutually incoherent to minimize the effects of ill-conditioned transmissibility estimation. Two examles are given in the following to illustrate the different roblems that stem from cross couling. The first examle below shows the roblem of including a false ath inut, meaning that there is actually no significant force acting at a given oint on the subsystem, yet it is still considered as a ath inut and so the acceleration is measured at that oint. To simulate this effect, the force acting on ath is set to zero. Since the ath contributions in the TPA model are exressed as a roduct of the inut force and NTF, ath has no artial contribution as shown on the left PPC lot in Fig. 7: whereas, the OPA artial contributions deend on the degree of cross couling between the aths. It follows from Eq. () that even with no force acting at ath, OPA might still show an imortant artial contribution deending on the H ij assive side transfer functions. This effect is clearly visible on the right hand side lot in Fig. 7. where ath shows a quite high contribution at some s desite the excitation being zero at that ath inut. The OPA aroach thus leads to a false interretation in this case. j ji Synthesized resonse OPA resonse 7 Paths Figure 7: The effect of cross couling on OPA in case of a false inut ath In the second examle the force at ath is restored to its original value and cross couling is resent in the simulated system (Fig. 8). The ath contributions of OPA are fundamentally different even though the summed contributions still show good agreement. For examle, whereas ath is insignificant around 00 in the reference set, it becomes very imortant in the OPA analysis in the same region.

11 TRANSFER PATH ANALYSIS AND SOURCE IDENTIFICATION 99 Synthesized resonse OPA resonse 7 Paths Figure 8: The effect of cross couling on OPA. Eq. () can hel to understand the underlying mechanism that causes this error in OPA. As it is shown in that formula, the acceleration at a given ath inut is the result of all the forces acting on the subsystem. Therefore it is ossible that a high ath inut acceleration level is caused not by the force acting at the same oint but by the cross couling to another inut force. To rovide a visual aid, the artial acceleration values F i *H ij for ath a 00 are shown in Fig. 9. The bar grah reveals that the high vibration level is caused by the force acting at ath and not by the force at ath. acceleration [m/s ] inut force location Figure 9: Partial contributions of inut forces to the ath inut acceleration at ath at 00 Rm for the examle shown in Fig 8.. Errors due to missing aths in the analysis The last set of examles deals with the effect of missing a ath inut in the analysis. A missing ath means that a load is acting on the body at a given lace but no ath inut (e.g. an acceleration signal) is measured at that oint. Although this load also contributes to the target ressure it is not taken exlicitly into account in the analysis. In case of the TPA mount stiffness method, missing a ath results in an incorrect estimation of the total ressure. Whether this is an over- or underestimation deends on the hase relationshi between the individual ath contributions. Both ways, the error can be recognized and at the same time the artial ath contributions will still remain valid since the NTF s do not change with the number and osition of the excitations. Fig. 0 and illustrate this for a ath system, the first dislays the artial ressure contribution lots, the latter the comarison of the original and the synthesized target.

12 90 PROCEEDINGS OF ISMA00 INCLUDING USD00 Synthesized resonse Synthesized resonse 7 Paths Paths missing ath Figure 0: Effect of missing ath in a classical TPA mount stiffness analysis: original case, (left) and ath missing, (right). 80 Comarison of target resonses Figure : Effect of missing ath in a classical TPA mount stiffness analysis: summed TPA contribution comared (dashed line) against the original target resonse (solid line). For OPA the effect is more varied. The reason is that it deends on the degree of correlation between the missing aths and the ones measured in the analysis. To get a better view on this, it is useful to analyze the two extremes, namely what haens if the missing inuts are fully correlated with the other inuts and what if they are uncorrelated. In Eq. (0) it was shown that the transmissibilities are the roduct of a crossower and an autoower matrix. The crossower matrix contains information from every ath, even from the missing ones because it includes the target ressure. In the first case, since the contribution from the missing aths is correlated with the others, the energy of this contribution will be simly sread across the rest. This means that the sum of the artial contributions will be equal to the target resonse, thus the error will go unnoticed. The results for the simulation are shown in Fig.. Although the total OPA synthesis in Fig. shows a good agreement with the target ressure, the estimated artial contributions are incorrect. In the second case, when the missing ath s contribution is uncorrelated with the rest, it doesn t show u in the results because it disaears during the H estimation rocess. Theoretically, just as in TPA, the individual contributions remain the same and the summed contribution will be different from the target resonse, showing the effect of missing some aths. However, in ractice the results are not so clear which is due to the nature of the H estimation. The signals must be long enough to have a high number of averages. If this is not the case, then a certain amount of the uncorrelated signal s energy will show u as error in the other aths. The results of the simulation for the ath system in Fig. dislay this behavior. The discreancy between the original target and the summed OPA contributions (Fig ) does indicate that some aths are missing in the analysis. However, the number of averages is insufficient to comletely eliminate the effect of these aths and this results in small errors in the estimation of the ath contributions.

13 TRANSFER PATH ANALYSIS AND SOURCE IDENTIFICATION 9 Synthesized resonse OPA resonse 7 Paths missing ath missing ath Figure : Effect of missing aths in a OPA analysis in case of coherent inuts: reference TPA synthesis (left) and OPA results (right). 80 Comarison of target resonses Figure : Effect of missing aths in a OPA analysis in case of coherent inuts: summed OPA contribution (dotted line) comared against the target resonse (solid line). Synthesized resonse OPA resonse Paths missing ath Figure : Effect of missing aths in a OPA analysis in case of incoherent inuts: reference TPA synthesis (left) and OPA results (right). 0

14 9 PROCEEDINGS OF ISMA00 INCLUDING USD00 80 Comarison of target resonses Figure : Effect of missing aths in an OPA analysis in case of coherent inuts: summed OPA contribution (dotted line) comared against the target resonse (solid line).. Validation of results In classical TPA the comarison of the synthesized and the measured target is a generally acceted tool for assessing the validity of the results. However, the data rocessing flow of the OPA method from analysis to validation actually corresonds to a backward-forward calculation: using the same data twice. This means that for OPA, validation by synthesis does not make sense, since the fit of the synthesized target to the original target will in most cases be very good, regardless of the amount of error on the individual contributions. This is clearly illustrated both by Fig. in Section. and also by Fig. in this section. In the first figure, the synthesis fit is excellent desite the fact that a significant ath was neglected during the measurements. The second one, Fig., shows the synthesis fit of the OPA calculation from Fig. in Section.: the dashed line reresenting the OPA synthesis result, the solid gray line the original target. The agreement is very good, so based on this result one might conclude that the synthesis is reliable. However, the comarison of the left and right side lots in Fig. reveals that in this case OPA gives unreliable artial contribution results. Since in a real situation there is often no oortunity for such a comarison, but only the OPA results are available, one is left in doubt about the quality of the artial contributions even when a good synthesized target is achieved. Therefore, using the fit of the synthesis for the validation of the results is not recommended for the OPA methods, other quality indicators will have to be found. 90 Comarison of target resonses Figure : Comarison of the summed OPA contribution (dashed line) to the reference synthesis result (solid gray line).

15 TRANSFER PATH ANALYSIS AND SOURCE IDENTIFICATION 9. The inverse roblem of transmissibility estimation Finally it should be mentioned that using an inverse method for the estimation of transmissibilities has a number of consequences. One of them is the limitation due to the coherence between the ath inuts as described before, in Section.. A detailed discussion is beyond the scoe of this aer, but the following aragrahs give a brief descrition of two ossible ways to alleviate the effects of coherent ath inuts. First, it is ossible to use various regularization methods, such as the L-curve method with singular value truncation or Tikhonov regularization in order to find the best aroximate solution. However, in the author s exerience, in the context of industrial OPA datasets these aroaches often yield different results. This, combined with the fact that the fit of the synthesis to the measured target can no longer be used as a validation tool, leaves the user with a number of ossible good solutions without any guidelines to judge their validity. A second ossibility is to use different arametric aroaches to estimate the transmissibilities, such as the one roosed by Verboven in [7] for oerational modal analysis. Methods like this might be able to give more reliable estimates even at resonance frequencies, where the ath inuts become highly correlated, and might even require less data than the standard H aroach resently used by the OPA method. Another consequence of using an inverse method is the need for roer balancing between different tyes of inut. When, for examle, both ressure and acceleration data is used simultaneously, the numerical values of the first one are tyically in the range of ~00 Pa while acceleration levels are generally around 0 - ~0 - m/s. Combining these in the same autoower matrix results in a quite ill-conditioned matrix, moreover, the singular values corresonding to the different inut tyes are searated into two grous according to the magnitude of the inut data. Using the regularization methods mentioned before would mean that the effect of the weaker inuts, that would be the accelerations in this examle, is either comletely ignored or at least severely reduced. The solution is to find some kind of balancing scheme between these two grous, e.g. multilication of the accelerations values with a constant so that their magnitude is comarable to the ressure inuts, etc. However, as all these schemes introduce an artificial weighting in favor of one or the other inuts, the choice of a balancing method is far from straightforward. Conclusions The objective of this aer is to understand the limitations of the transmissibility based, so called Oerational TPA or OPA methods. Two grous can be distinguished among these methods. In the first one, transmissibilities are directly alied in the TPA rocess, relacing the noise transfer functions, which results in a resonse-resonse tye aroach instead of the classical load-resonse model. The second grou of methods kees the classical TPA model but uses the transmissibilities to estimate the noise transfer functions instead of measuring them. With regard to the limitations of the first grou, it is shown that a number of conditions must be fulfilled in order to get meaningful results. First of all, the inut forces should be incoherent. If this is not the case, the transmissibility estimation will suffer from illconditioning, resulting in incorrect estimates. Secondly, all structures exhibit a certain amount of cross couling between the ath inuts. Deending on the degree of the couling, the estimated artial contributions will be unreliable. The greatest danger here is that this error will in most cases go unnoticed because these OPA aroaches are inherently backward forward calculations, therefore the sum of the contributions will match the measured target no matter how much error the artial contributions contain. Finally, it is ossible to neglect an imortant ath during measurements. In this case, again deending on the degree of correlation between the load at the unmeasured inut and the rest of the ath inuts, the energy of the unmeasured ath inut will be sread over the other aths, causing an error that is hard to recognize. Aarently, these methods have quite a few limitations besides their advantages and these limit the scoe of their alication and reliability in real-life situations. Particular care has to be taken when interreting OPA results. The second tye OPA aroaches are more romising, but the required measurement conditions might be too strict for a generic alication. In conclusion, it can be said that there is still a need for a fast but reliable TPA method. Oerational Transfer Path Analysis methods are a

16 9 PROCEEDINGS OF ISMA00 INCLUDING USD00 first ste in this direction, but the inherent limitations severely restrict the range of alications, and demand great care with the interretation of results. Further research is therefore required for finding more suitable aroaches with a better seed/accuracy trade-off. In this asect novel methods based on arametric load models show a lot of romise. 7 Acknowledgements The resented research was conducted as art of the Smart Structures Marie Curie Research Training Network (MRTN-CT-00-09). Péter Gajdátsy, has been hosted in this roject as an early stage researcher first at LMS International and resently at the Katholieke Universiteit Leuven. The financial suort of the Euroean Commission is gratefully acknowledged. References [] D. Ewins and W. Liu, Transmissibility roerties of MDOF systems, Proc. th IMAC Conf., Santa Barbara, CA, US, 998, [] N.M.M. Maia, J.M.M. Silva and A.M.R. Ribeiro, The transmissibility concet in multi-degree-offreedom systems, Mechanical Systems & Signal Processing, Vol., No., Jan. 00, [] A.M.R. Riberio, J.M.M. Silva and N.M.M. Maia, On the generalization of the transmissibility concet, Mechanical Systems and Signal Processing, 000, Vol., No.,. 9- [] C. Devriendt, P. Guillaume, Identification of modal arameters from transmissibility measurements, Journal of Sound and Vibration, Vol., No. -, 8 July 008,. - [] H. Van der Auweraer, P. Mas, S. Dom, A. Vecchio, K. Janssens and P. Van de Ponseele, Transfer Path Analysis in the critical ath of vehicle refinement: the role of fast, hybrid and oerational ath nalysis, Proc. SAE Noise and Vibration Conf., May -7, 007, aer no [] K. Noumura, J. Yoshida, Method of transfer ath analysis for vehicle interior sound with no excitation exeriment, Proc. FISITA 00, F00D8, Yokohama, Jaan, Oct. 00. [7] M Lohrmann, Oerational transfer ath analysis: comarison with conventional methods, ICSV Conf., -0 July, 008, Daejeon, Korea. [8] H. Van der Auweraer, P. Mas, B. Peeters, K. Janssens, A. Vecchio, Modal and Path contribution Models from In-Oeration Data: Review and New Aroaches, Shock and Vibration Vol., No. -, 008,. 0- [9] G. de Sitter, C. Devriendt, P. Guillaume,, Oerational Transfer Path Analysis, MSSP, Article in Press, htt://dx.doi.org/0.0/j.ymss [0] P. Gajdatsy, K. Janssens, L. Gielen, P. Mas, H. Van der Auweraer, Critical assessment of Oerational Path Analysis: effect of couling between ath inuts, Acoustics 08 Conf., June 9 July 0, 008, Paris, France. [] P. Gajdatsy, K. Janssens, L. Gielen, P. Mas, H. Van der Auweraer, Critical assessment of Oerational Path Analysis: mathematical roblems of transmissibility estimation, Acoustics 08 Conf., June 9 July 0, 008, Paris, France. [] P. Gajdatsy, K. Janssens, L. Gielen, P. Mas, H. Van der Auweraer, Critical assessment of Oerational Path Analysis: effect of neglected aths, ICSV Conf., -0 July, 008, Daejeon, Korea. [] K. Janssens, P. Gajdatsy, H. Van der Auweraer, Oerational ath analysis: a critical review, ISMA Conf., -7 Setember, 008, Leuven, Belgium. [] J. Verheij, Multiath sound transfer from resiliently mounted shiboard machinery, PhD Dissertation, Technische Physische Dienst TNO-TH, Delft, 98.

17 TRANSFER PATH ANALYSIS AND SOURCE IDENTIFICATION 9 [] J. Verheij, Exerimental rocedures for quantifying sound aths to the interior of road vehicles, Proc. nd Int. Conf. on Vehicle Comfort, Bologna, Italy, Oct.-, 99, [] F.X. Magrans, Method of measuring transmission aths, J. Sound Vib., Vol 7, No.,. -0 (98) [7] J. Verheij, Hoebrichts, Thomson, Acoustical source strength characterisation for heavy road vehicles engines in connection with ass-by noise, rd ICSV Conf., June 99, Montreal (CND). [8] P. van der Linden, J. Schnur, T. Schomburg, Quantifying the noise emission of engine oilsums, valve covers, etc. using artificial excitation, th ICSV Conf., Adelaide (AU), December 997. [9] P. van der Linden, J. Fun, Using mechanical-acoustic recirocity for diagnosis of structure borne sound in vehicles, SAE Noise & Vibration Conference & Exosition, May 99, Traverse City, MI, USA, SAE aer no. 90. [0] J. Starkey and G. Merril, On the ill-conditioned nature of indirect force measurement techniques, Int. J. Analyt. Ex. Modal Anal., Vol., No., 989, [] P. van der Linden and H. Floetke, Comaring inverse force identification and the mount stiffness force identification methods for noise contribution analysis, Proc. 00 ISMA Conf., Leuven, Belgium, Set. 00. [] P. Mas, P. Sas and K. Wyckaert, Indirect force identification based uon imedance matrix inversion: a study on statistical and deterministical accuracy, Proc. 99 ISMA Conf., Leuven, Belgium, Set. -, 99. [] M. Blau, Indirect measurement of multile excitation force sectra by FRF matrix inversion: influence of errors in statistical estimates of FRFs and resonse sectra, Acta Acustica united with Acustica, Volume 8, Number, July/August 999,. -79 () [] P. van der Linden, F. Gerard, K. Michiels, H. Van der Auweraer, D. Storer, Body in white anel noise assessment through satial and modal contribution analysis, Proc. ISMA-, Leuven, Belgium, Set. 000,.-8. [] W. Biermayer, F. Brandl, R. Hoeldrich, A. Sontacchi, S. Brandl, H. H. Priebsch, Efficient Transfer Path Analysis for Vehicle Sound Engineering, JSAE 008 Annual Congress, - May, 008, Pacifico Yokohama, Jaan [] P. Guillaume, C. Devriendt, G. De Sitter, An oerational modal analysis aroach based on arametrically identified multivariable transmissibilities, MSSP, Article in Press, htt://dx.doi.org/0.0/j.ymss [7] P. Verboven, Frequency domain system identification for modal analysis. PhD Thesis, Deartment of Mechanical Engineering, Vrije Universiteit Brussel, Belgium, avrg.vub.ac.be, 00. [8] D. Tcherniak, A.P. Schuhmacher, Alication of Transmissibility Matrix Method to NVH Source Contribution Analysis, IMAC XVII, 9- February 009, Orlando, Florida, USA [9] D. de Klerk and D.J. Rixen, Comonent transfer ath analysis method with comensation for test bench dynamics, Mechanical Systems and Signal Processing, 00 (in ress, available online), doi:0.0/j.ymss

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