Application of decomposition-based technique in NVH source contribution analysis

Transcription

1 Application of decomposition-based technique in NVH source contribution analysis D. Tcherniak, A.P. Schuhmacher Brüel & jær, Sound & Vibration Measurement A/S Skodsborgvej 37, D-285, Nærum, Denmark Abstract Calculation of contributions from different noise sources is an important part of a vehicle NVH evaluation process. Typically, the contributions calculations are based on measured transfer functions and estimated source strengths. The main disadvantage of this technique is that it requires measurements of transfer functions, which is an awkward time consuming process. The technique is also error-prone due to the approach the source strengths are estimated. The current study focuses on another approach to the problem. It is based on the decomposition of the received sound rather than the synthesis of the contributions, and therefore requires only operational data. The technique is based on the estimation of the transmissibility matrix. It can be shown that for a given excitation set, the transmissibility matrix does not depend on applied loads, and therefore is a property of the system under test. It can also be shown that the transmissibility matrix can be estimated from several sets of operating measurements. Due to the method assumptions, it does not calculate exact contributions but provides results that can help NVH engineer to quantify the main noise sources of the vehicle. In the paper we apply the method to data synthesized in well controlled test environments which however resemble a real vehicle. A number of special cases typical for automotive NVH are being highlighted. 1 Introduction The main goal of Transfer Path Analysis (TPA) is estimation of contributions of different sound sources approaching a receiver by different paths. Automotive industry is a main customer of TPA, where the analysis plays important role helping to improve the interior and exterior sound of a vehicle. During more than 15 years of research, a number of TPA methods were developed [1]. Most of the classical TPA methods are based on the synthesis of contributions: the contributions are calculated as products of source (or path) strength and path sensitivity [2]. By their nature, these methods are error-prone since they require estimation of source strengths (which cannot be measured directly) and time-consuming due to the measurements of path sensitivities. There is another family of methods which is based on decomposition rather then synthesis. These methods attempt to decompose the received sound (or vibration) using one or another decomposition criteria. One of the examples is the Multiple Coherence method which splits the received sound or vibration assuming the contributing sources being uncorrelated. Another recently published method [3, 4] uses so-called transmissibility functions to estimate the contributions. As defined in [5], transmissibility is a ratio between two responses measured at different degrees-offreedom (DOFs). Though simple for Single Input Multiple Output (SIMO) systems, transmissibilities are not so obvious for Multiple Input Multiple Output (MIMO) systems which are the case in TPA applications. The concept of transmissibilities was generalized for MIMO systems by Ribeiro et al [6, 7, 8]. It is worthy to note that studying of transmissibilities gained more and more attention in the recent years: e.g. there are applications of transmissibility functions to Operational Modal Analysis [9, 1]; new

2 properties of the transmissibility functions in application to mechanical systems were found and described by Maia et al. [11]. There are some known attempts of using transmissibility functions for TPA. E.g. an analog of the transmissibility method (often called pressure-to-pressure method) is often used in automotive NVH programs as a simple technique to investigate contributions from single localized sources, e.g. exhaust orifice. As was found in [6], the transmissibility matrix can be estimated from operating data only, and does not require time consuming measurements of transfer functions. This makes the method very attractive for using for TPA [3, 4, 1]. Simply speaking, the method requires simultaneous recording of the operating signals from reference transducers placed in a vicinity of the selected sources and receiver signals. Sets of signals corresponding to different operating conditions should be recorded. Based on these measurements, a transmissibility matrix between references and receivers is being calculated. sing the matrix and the measurement set for a given operating condition, the contributions from the sources to the receivers can be estimated. Typically, when starting using a new method in real-life cases, a number of questions concerning the applicability of the method arise. For example, How the correlation between the sources could affect the method results? What is the best technique to collect operating data for the transmissibility matrix calculation? What are the influences of the sources which are not accounted for? How sensitive is the method to simultaneous strong and weak sources? How to deal with big distributed sources? The paper is trying to answer these questions focusing on the vehicle air-borne contributions. The results provided in the paper are based on artificially synthesized input data which suppose to resemble the noise from real vehicle sources (using transfer functions measured on a real vehicle and operating data synthesized using Vehicle NVH Simulator [12]). The paper is organized as following: in Section 2 the theoretical description of the transmissibility method is given; it also concerns with the practical application of the method to TPA; Section 3 describes the numerical test system used for the method testing, and in Section 4 the method is being applied to different test cases. 2 Transmissibility method and its application to TPA 2.1 Transmissibilities for a multiple-input system In a linear system, a response relates to the causing excitation by x = H f (1) N res N s where x C is a vector of responses measured at N res DOFs, f C is a vector of excitation acting at N s N s DOFs, and res N H C is a matrix of frequency transfer functions (FRFs) between all excitation and response DOFs. Expression (1) is written in frequency domain. Excitation and response can be thought of as vectors of phase-assigned autospectra (PAS) w.r.t. some (and same) reference signal. This means that all elements of vector f are correlated. If this is not the case, the excitations should be decorrelated using for example Singular Value Decomposition (SVD), i.e. split into mutually uncorrelated (principal) components. Then expression (1) is valid for each principal component. All further derivations are based on the assumption that the excitations and responses are correlated and can be presented as PAS w.r.t. some reference.

3 Following Ribeiro et al.[6], let us assume that the excitation may be applied to the system only at a given subset A of DOFs: x = H f A, (2) and the response vector x can be split into two sub-vectors corresponding to two disjoint subsets of DOFs and : x x =. (3) x Then one can define transmissibility matrix T A which relates x and x under the assumptions that the excitation is applied at DOFs from subset A: sing (2), the following is valid: and substitution to (4) yields to x = T x. (4) x x = H A = H A A f f A A + A = HA H A, (5) T, (6) where + denotes matrix pseudoinverse. sing (6), one can obtain the transmissibility matrix if all transfer functions are known. It is important to note that since H A and H A are properties of the system and do not depend on the excitation, the transmissibility matrix also does not depend on the excitation (though, this is only valid under the condition that the excitation are acting at the subset A). Let us consider different realizations of excitation (1) (2) ( M ) (1) (2) ( M ) f A : f (1) A, f (2) A... f ( M ) A. Corresponding responses are: x, x... x and x, x... x. According to (4), the responses relate via transmissibility matrix, so one can re-write expression (4) as follows or Solving (8) for T A, one obtains (1) (2) ( M ) (1) (2) ( M ) [ x... x ] T [ x x... x ] x = (7) A A X = T X. (8) + A = X X T. (9) Expression (9) shows that the transmissibility matrix can be obtained from operational measurements only (under condition that the excitation is applied at the same subset A and matrix X is invertible). 2.2 Application of transmissibility method to TPA problems In the original works of Ribeiro et al. [6,7], the main application of the method was the estimation of the vibrations of a mechanical structures at the points where the sensors could not be mounted (subscripts and historically originate from this application: stands for known vibration, for unknown vibrations). Later Nomura et al. [3,4] and Van der Auweraer et al. [1] suggested using the method for TPA. Let us call subset receivers (e.g. driver ears or pass-by microphones), and subset references. The references are related to some sources, assuming that each reference picks most of its energy from the

4 corresponding source. Then the response at the j-th receiver under the i-th operating condition can be written as follows: ( i) x j = T = = N ref k= 1 N A ref k= 1 T C j1 x A ( i) jk ( i) 1 jk + T x ( i) k A j 2 x ( i) T A j x ( i) N ref (1) (i) Frequency domain function C jk can be considered as a contribution from the k-th source to the j-th receiver under the i-th operating condition. However, as it was mentioned in [1], one has to be careful (i) interpreting C jk since this is not the same contribution which one used to obtain following classical TPA. While the classical contribution is a product of the source strength and path sensitivity; the value of (i) C jk refers to the strength of the reference signal in the overall signal perceived by the receiver. However, under a number of conditions, the value of can be used in practice. (i) C jk is reasonably close to classical contribution, and therefore It is also important to note that the classical TPA approach splits the contribution into two meaningful parts: source strength part and path sensitivity part; both are helpful in understanding the nature of received signal (e.g. is it a path or a source responsible for the contribution?). The transmissibility approach also splits the contribution into two parts (cf. (1)) but these parts do not hold any practical meaning and cannot be used for making any design decisions. Expression (1) can be re-written in time domain. In this case the transmissibility function can be converted to FIR-filters, and multiplication should be replaced by convolution. This will allow one to generate contributions from the sources in time domain, listen to them for evaluation and proceed with frequency/sound quality analysis if needed. 3 Test system 3.1 Model system In order to understand the practical aspects of the applicability of transmissibility method and answer the questions listed in Introduction, a number of numerical experiments were conducted. A simple model system was introduced (Figure 1). The main idea behind the model was to create simple fully controllable environments to test and validate the method for a variety of source signals. The system was supposed to generate signals similar to those perceived by the reference and receiver sensors under real-life operating conditions. The model consists of three point sources, three corresponding reference sensors and one receiver sensor. A number of selected time domain signals s(t) = {s 1 (t), s 2 (t), } T are mixed together using constant matrix ; the mix is instantaneous: q ( t) = s( t). (11) Signals q(t) = {q 1 (t), q 2 (t), q 3 (t)} T excite the model at source points 1,2,3; the model response at corresponding reference positions p ref (t) = {p 1 (t), p 2 (t), p 3 (t) } T and at the receiver position p res (t) are computed as a convolutive mix: p p ref res ( t) = H( t) q( t), (12) ( t)

5 Figure 1: Model system under considerations where H(t) is a matrix of FIR-filters generated from the FRF matrix H(f). To ensure the simple test system resembles properties of a real vehicle, we used FRFs measured on a real car. A Volvo S6 served as a test object; the measurements were conducted using Volume Velocity Source B& Type 4295/4299 and a number of B& microphones Type The three point sources were placed: under the engine (source 1), under the gear-box (source 2) and in front of the intake (source 3). The reference microphones were placed few centimeters way from the sources. The receiver microphone was located about 1 m way from the left side of the vehicle. 3.2 Common steps of the numerical experiments Most of the numerical experiments which are described in the following sections were performed following these steps: 1. Source signal vector s(t) was generated for a chosen test scenario. Note that though we used measured FRFs (Pa/(m 3 /s) units), the signals s(t) and q(t) are synthetic and dimensionless. Therefore all further results do not hold any quantitative information and presented without units. ( m) ref ( ) p m res 2. sing expressions (11) and (12), p ( t) and ( t), m = 1..M were computed for M different operating conditions; This step models the measurements which had to be conducted if the method was used in practice. 3. Phase-assign spectra w.r.t. a chosen PAS-reference signal was computed: p p ( m) ref ( m) res ( t) x ( t) x 4. Matrices X and X were generated, and the transmissibility matrix T A was computed using (9); 5. As the first validation step, the computed matrix T A was compared with the exact transmissibility matrix calculated via FRFs using (6); (m) 6. For a selected operating measurement, the contributions C j of all sources were calculated using (1) and compared with exact contribution obtained from the model when the other two sources are switched off ; ( m) ( m) 7. The sum of contributions was compared with measured x. (m) (13)

6 4 Application of the method to different test cases In this chapter the method is applied to different test conditions in attempt to answer the questions listed in the Introduction. 4.1 Influence of the correlation between the sources As it was mentioned in section 2.1, the method assumes the excitation sources are correlated. If the sources are uncorrelated, an obvious approach would be to decorrelate the excitation to a number of uncorrelated (principal) components and apply the method to each of them. In practice this could be done by computing the cross-spectra matrix G pp for all references and responses: and applying Singular Value Decomposition (SVD) to it: p ref ( t) FFT p( t) = G pp (14) pres ( t) pp H G = S V, (15) where, V are the matrices containing left and right singular vectors, S is a diagonal matrix of singular values, and * H denotes conjugate transpose. The original cross-spectra matrix G pp can be presented as a sum pp = G pp = H G u s v ; (16) the summation can be truncated to the significant singular values only. For all principal components held in the summation, PAS function x can be calculated according to its definition, ij G ji x ij = G ii, (17) and be used to generate the X and X matrices. Here, actually, one can utilize extra information obtained from principal components, by populating X and X in the following way (compare to (7)): G (1) (1) (2) (2) [ x x... x x...] / = = 1 = 2 = 1 = 2 ji X. (18) To confirm the approach, a simple numerical experiment was conducted. Two sets of source signals were generated, s cor (t) with all signals correlated and s unc (t) with uncorrelated signals. In order to check if the resulting mix is well separated by the transmissibility method, the signals contained energy in different but overlapping frequency bands, Figure x x 1-5 c 11 c 11 1 c 22 1 c 22 c 33 c 33.8 c 12.8 c 12 c 23 c Figure 2. Magnitude of cross-spectra computed for: s cor (t) (left) and s unc (t) (right). In both cases, signal frequency bands are s 1 : Hz; s 2 : 65-95Hz; s 3 : 45-75Hz.

7 Figure 3 compares transmissibility functions obtained according to (9) with the exact functions, computed using expression (6). To simplify the plot, only one element of T A, namely T 11 is plotted. The figure shows a good match between computed and exact functions in the frequency range Hz, i.e. the frequency range of the excitation. For uncorrelated sources, if SVD is not applied, the computed function differs from the exact curve on the frequency lines where the excitation bands overlap, e.g Hz and Hz (Figure 3b). This can be specially seen on the error plot (bottom). Introduction of SVD (15)-(18) solves the problem, and almost perfect match between the functions can be observed, Figure 3c. 4.2 Selection of operating conditions to estimate the transmissibility matrix Following the method, the contributions are being estimated based on transmissibility matrix, which is calculated based on operating measurements only, (9). As it was mentioned before, this is the main advantage of the method. In order to use expression (9), the X matrix must be invertible. Matrix X has a number of rows equal to number of sources (and references) N s, and the number of columns M corresponds to number of different operating conditions used during the test (7),(8). The X matrix will be a full-rank matrix if it contains at least N s independent columns. In this case the right pseudoinverse can be + found: X X = I. How to achieve the independency of the columns in X in practice? As it follows from the method assumptions, the excitations should be applied at the same subset of DOFs (subset A) during all M operating tests. However, the level of excitation applied to different DOFs can vary. In the case of a car, where many sources of excitation act simultaneously (engine, transmission, tyres, etc.), the excitation level depends on engine RPM or speed of the vehicle. As it was mentioned in [1], engine or wheels RPM and their harmonics excite the system, and the level of excitation varies depending on the RPM. Therefore the obvious choice is to use the two test scenarios which are standard in automotive NVH: constant speed tests and run-up tests. a) b) c) Figure 3. Transmissibility function T 11, phase (top), magnitude (middle) and relative error, % of magnitude db values. Blue computed using (6), red computed using (9): a) Correlated signals, no SVD; b) ncorrelated signals, no SVD bad agreement on the frequencies where the excitation bands overlap; c) ncorrelated signals, with SVD.

8 In order to compare different excitation scenarios, the source time data s(t) resembling the data typically observed during real tests were generated. We used B& PLSE NVH Vehicle Simulator Type 3644 [12] to synthesize the data: s 1 (t) resembles typical engine sound, s 2 (t) transmission, s 3 (t) intake. For the constant speed test scenario, 7 engine regimes were used: 25, 35, 37, 4, 42, 45, 5 RPM. The gear box was set to the second gear. Engine RPM was slightly and slowly varying around the above listed mean values. Time data characteristics: length 22 s, sampling frequency 496 Hz. For the run-up tests, two data sets were generated: slow run-up from 1 to 6 RPM in 22 s and fast run-up, same RPM bounds in 12 s. In case of stationary operating conditions, each RPM regime forms a column in the X and X matrices (18). Figure 4 a-c shows how the number and the selection of the stationary operating regimes affect the quality of the T A matrix estimation. All 22 s of data were used (43 averages for 16 lines FFT, 5% overlap, 1 Hz resolution). se of all 7 operating data sets gives the best results (Figure 4a) but reducing the number of data sets to three (corresponding to the lowest (25), middle (4) and highest (5) RPM) does not significantly affect the estimation quality (Figure 4b). However when utilizing three data sets from the higher RPM range (42, 45 and 5 RPM data sets), the result quality drops (Figure 4c). Figure 4 d-f shows the effect of averaging on the quality of the T A estimation. All 22 seconds of the recorded data that correspond to 43 averages give the best result (Figure 4d); decreasing the length of the recorded block and consequently the number of averages leads to bigger estimation errors (Figure 4e,f). In case of run-up, the time data are split by chunks; each chunk, after it is subjected to Fourier transform and averaging, forms a column in the X and X matrices. Results are shown on Figure 5. Figure 5a,b compares slow (a) and fast (b) runups. In both cases the chunks were 3 s long that correspond to 5 FFT averages. In the case of the slow runup X and X contained 8 columns (with.6 s overlap between chunks introduced). In the case of fast runup the matrices contained only 3 columns. As one can see the run-up speed does not seriously affect the quality of T A estimation. Figure 5c,d compares result for the different chunk length: short chunks of 1.5 s with only 2 FFT averages but generating 23 columns in X and X (Figure 5c) and 6 s long chunks (11 averages, 3 columns, Figure 5d). Similar to the constant speed a) d) b) e) c) f) , Figure 4. Relative estimation error for log(t 11 ), %. Constant speed operating conditions: a) All 7 regimes used: 25, 35, 37, 4, 42, 45, 5 RPM; b) Three regimes used: 25, 4, 5 RPM; c) Three higher RPM regimes used: 42, 45, 5 RPM. d) All recorded data length (22 s) used for PAS calculations (43 averages). Same as a), note different y- axis scaling; e) 7.3 s of data used (13 averages); f) 3.7 s of data used (6 averages).

9 a) 5 c) b) 5 d) Figure 5. Relative estimation error for log(t 11 ), %. Runup operating conditions: a) Slow runup (from 1 to 6 RPM in 22 s), chunk length 3 s; b) Fast runup (1-6 RPM in 12 s), same chunk length; c) Slow runup, chunk length 1.5 s; d) Slow runup, chunk length 6 s. case, by increasing the number of FFT averages one can improve quality of the T A matrix estimation. Based on these observations, one can conclude that generally constant speed operating measurements (though more time consuming) give better results than runup measurements. Interesting to note that the number of FFT averages affect the T A matrix estimation quality in much higher degree than the number of operating conditions (i.e. the number of columns in the X and X matrices). 4.3 Influence of the sources which are not accounted for One of the assumptions of the considered method is that all active excitation sources are accounted for by reference sensors. However, this condition is difficult to satisfy in practice. A modern car has hundreds of noise and vibration sources, and it is not feasible to characterize all of them by setting up references. In this section we will try to understand how the sources which are not accounted for will influence the estimation of the transmissibility matrix and the contributions. The not-accounted for source is modeled by source #3; modifying the coefficients of the matrix, one can control its strength q 3. Signal p 3 (t) picked by the reference sensor #3 (Figure 1) was ignored, i.e. removed from the reference vector p ref and the corresponding row in X was also taken out. The results of matrix T A estimation and resulting contributions are shown on Figure 6. The calculations were performed for the constant speed operating conditions as described in the previous section. The strength of the not accounted for source q 3 was gradually increased from very weak (-4 db from its original value used in the previous section) to modest (-2dB), and further to strong (-1 db). Figure 6a shows that the error of the T A estimation increases with the increase of the q 3 strength. The same is happening with computed contributions (Figure 6b) which are getting more and more overestimated. The obvious reason of the error in the T A estimation is that the method believes the signal picked by the receiver sensor is originated from the sources accounted for by the reference sensors. The bigger contribution of the not accounted for source into the reference and/or receiver sensors, the bigger error can be expected. This is demonstrated by Figure 7. The blue line is the contribution of the source #3. In both cases, whether this source is weak or strong, its contribution to the first reference signal p 1 is relatively small (Figure 7, a,b), at least much smaller compare to the contribution of the source #1 which is

10 a) b) Figure 6. a) Relative estimation error of the log(t 11 ), %. Legend: Thick magenta line estimation error when q 3 is accounted for, dotted line weak q 3, dark green line modest q 3, blue line strong q 3 ; b) (7) Contribution C 11. Legend: Thick magenta line exact contribution, dotted line weak q 3, dark green line modest q 3, blue line strong q 3. the closest to the reference. Same can be observed for other reference signals. This means that matrix X is only slightly affected by the third source. This is almost always the case since, according to the method, the reference sensors are placed close to the sources. However, for the receiver signal, the situation is different (Figure 7, c,d). In case of weak q 3, the source #2 is the main contributor to the receiver signal (green line). But with increase of q 3 (Figure 7d), source #3 becomes the main contributor (compare the blue and green lines), so matrix X becomes strongly affected. (m) This results in overestimation of the T A matrix and further in the overestimation of contributions C ij. Concluding, one can note that a source can be neglected if (i) it is weak compared to the sources which are accounted for, or (ii) it is located far away from the receiver sensors (i.e. contributes relatively little to the received signal). a) 1 5 b) c) d) Figure 7. Exact contributions of q 1, q 2, q 3 into: (a,b) first reference signal p 1 ; (c,d) receiver signal p rec. (a,c) weak q 3 ; (b,d) strong q 3. Colors: black total signal; red contribution of q 1 ; green contribution of q 2 ; blue contribution of q 3. All plots are for constant speed (5 RPM) operating data.

11 4.4 Influence of weak and strong sources One of the drawbacks of the classical TPA methods is their weakness in separation of the noise coming from the sources with significant difference in strength. This section models the situation when one source is much stronger than the others. In our test model (Figure 1), the first and second sources are located close to each other (modeling vehicle engine and transmission) while the third source (the intake) is quite away from the first two. Since the goal of the experiment is to see the effect of different strength of the sources that are close to each other, we selected the source #1 as a test source: Its strength q 1 was varied, and the effect on the estimation quality of T A and contributions were observed. As it was noticed in a number of previous experiments, the best estimation results are obtained when the strength of contributions from different sources has approximately same level. The relative errors of the T 11, T 12, T 13 estimations for such a case are shown in blue on Figure 8 (same operating condition as described in the previous section). Green and red curves on Figures 8 correspond to the estimation errors when the strength of q 1 was increased by 2dB and 3dB respectively. As it can be seen, the estimation error growths significantly for T 12, T 13 but less for T 11 (note different y- axis scaling), which is the transmissibility function corresponding to source #1. The contributions computed using the T A matrix are plotted on Figure 9. As one can see, with the increase of q 1 (from (a) to (c)), the precision of contributions decreases, especially for the weak sources #2 and #3 (generally, they are overestimated). However, the estimation of the strongest source contribution C 11, is still quite good. Since the strongest source defines the total response (the bottom plot), the estimation of the total is good for all test cases. Concluding the section, one can notice that the method is quite robust in case of neighboring weak and strong sources. It provides satisfactory estimation of contributions from weak sources and relatively good estimation for strong sources and the total response. 4.5 Effect of using several references in case of distributed noise sources Engineers working in automotive NVH used to deal with distributed noise sources, i.e. relatively big volumes with boundaries radiating sound. Examples of such sources are engine, transmission, muffler, wheels, etc. A common approach for dealing with distributed sources is to use a number of measurement points surrounding distributed sources. E.g. the Source Substitution Method models such sources by placing a number of point sources on the radiating surface (typically used for modeling engine). Multiple Coherence Method uses several reference sensors to characterize a single distributed source (typical example is a vehicle tyre). This section investigates how the multiple sensors approach is suited for the Transmissibility Matrix Method. The test model had to be modified: there are only two sources left, source #1 and #3 (modeling engine and intake respectively). As before, source #3 is assumed to be a point source and being characterized by a single reference picking up signal p 3. In contrast, the source #1 is considered to be distributed, and there are two references set to characterize it; they pick signals p 1 and p 2. Obviously, if the Figure 8. Estimation error for T 11 (left, note the different y-axis scaling), T 12 (middle) and T 13 (right) for different strength of q 1 : blue original strength; green 1 times of original; red 32 times of original.

12 a) b) c) 1 5 Engine 1 5 Engine 1 5 Engine Transmission Transmission Transmission Intake Intake Intake Total Total Total Figure 9. Contributions C 11 to C 13 and total (top to bottom). a) Original strength of q 1 ; b) q 1 times 1; c) q 1 times 32; Colors: thick magenta computed, thin black exact. reference sensors located far away from each other, they pick relatively different signals, and the considerations from the previous sections are valid. Then the contribution from the distributed source can be computed as a sum of two single sources corresponding to the first and second references. Let s consider a limit case when the two references are located very close to each other and effectively pick the same signal. If the method can deal with this limit case, and does not require any modifications, it can be successfully used for estimation of contributions from big distributed sources. Considering how the X matrix is populated (7)-(8) or (18), one can notice that if p 1 and p 2 measure the same signals, the first two rows in the matrix will be identical. Then the X matrix is not a full-rank matrix + anymore, and the equality X X = I does not hold. It can be shown [13, p. 21] that two corresponding columns of the pseudoinverse are equal. Just for the illustration of this property, let s print out the X and + X matrices for a random frequency line: X i i i i i i ; i 3 + = i X i = i i i i i i i i i Imagine, only one reference sensor was used instead of two located next to each other; then the X matrix and its pseudoinverse, would look as follows: i i i + X = 1 ; X = i i i i i i i i i One can notice that the first column of the second pseudoinverse is exactly the sum of the two first columns of the original pseudoinverse. Similarly, the transmissibility matrix T A calculated according to (9) will have two identical columns which values are exactly halves of the values if only one reference sensor was used. Obviously, resulting contributions C 11 and C 12 are equal, and their sum is equal to the single contribution in case of only one reference sensor was used. It is important to note that since the X matrix is in this case always ill-conditioned, some regularizations techniques shall be applied to ensure valid pseudoinverse.

13 Finishing the section, we can conclude that using several references in order to characterize a big distributed source does not contradict with the principles of the method and shall not be avoided. 5 Conclusion and further research The paper presents the theoretical background of the Transmissibility Matrix Method and concerns its application to air-borne TPA. A simple synthetic numerical test model was used for method validation. A number of practical issues typical for automotive applications were considered such as method applicability for correlated and uncorrelated sources, effect of neglected sources, effect of neighboring weak and strong sources, dealing with big distributed sources. Different types of operating test regimes were compared to select one providing the best estimation of contributions. As a further research, the additional validation of the method can be suggested, especially on a real vehicle, when the resulting contributions can be compared with the results of classical TPA method such a Source Substitution Method. Application of the method to vehicle pass-by test and as a data pre-processing tool for Vehicle NVH Simulator can also be interesting. Acknowledgements The authors would like to thank aare Brandt Petersen for his help in understanding properties of the Moore-Penrose pseudoinverse of rank deficient matrices. References [1] H. Van der Auweraer, P. Mas, S. Dom, A. Vecchio,. Janssens and P. Van de Ponseele, Transfer Path Analysis in the Critical Path of Vehicle Refinement: The Role of Fast, Hybrid and Operational Path Analysis, SAE , Proceedings SAE Noise and Vibration Conference, St.Charles (IL), SA, 27 May [2] H. Van der Auweraer,. Wyckaert, W. Hendricx, P. Van Der Linden, Noise and vibration transfer path analysis, Lecture series - van areman Institute for fluid dynamics, van areman Institute, Rhode-Saint-Genèse, Belgium (1979). [3] ousuke Noumura, Junji Yoshida, Method of Transfer Path Analysis for Vehicle Interior Sound with No Excitation Experiment, F26D183, Proceedings of FISITA World Automotive Congress, Yokohama, Japan, 26 October [4] ousuke Noumura, Junji Yoshida, Method of Transfer Path Analysis for Interior Vehicle Sound by Actual Measurement Data, 26541, Proceedings of JSAE Annual Congress, Yokohama, Japan 26 May 24. [5] D. J. Ewins, Modal Testing: Theory, Practice and Application, Research Studies Press Ltd, Badlock, Hertfordshire, England (2). [6] A. M. R. Ribeiro, N. M. M. Maia, J. M. M. Silva, Experimental Evaluation of the Transmissibility Matrix, Proceedings of IMAC, issimmee (FL), SA, 1999 February. [7] A. M. R. Ribeiro, J. M. M. Silva, N. M. M. Maia, On Generalization of the Transmissibility Concept, Mechanical Systems and Signal Processing, Vol. 14, No. 1 (2), pp [8] N. M. M. Maia, J. M. M. Silva, A. M. R. Ribeiro, The Transmissibility Concept in Multi-Degree-of- Freedom Systems, Mechanical Systems and Signal Processing, Vol. 15, No. 1 (21), pp

14 [9] P. Guillaume, C. Devriendt, G. De Sitter, An Operational Modal Analysis Approach Based on Parametrically Identified Multivariable Transmissibilities, Proceedings of IOMAC, Copenhagen, Denmark, 27 April-May. [1] C. Devriendt, P. Guillaume, S. Vanlanduit, G. De Sitter, The Sequel on the Identification of Modal parameters from Transmissibility Measurements, Proceedings of IOMAC, Copenhagen, Denmark, 27 April-May. [11] N. M. M. Maia, R. A. B. Almeida, A. P. V. rgueira, nderstanding Transmissibility Properties, Proceedings of IMAC, Orlando (FL), SA, 28 February 4-7. [12] R. Williams, M. Allman-ward, B. Ginn, D. Tcherniak, An efficient approach to include interactivity in the vehicle NVH evaluation process, Proceedings of ISMA, Leuven, Belgium, 26 September [13]. B. Petersen, M. S. Pedersen, The Matrix Cookbook,