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AIA-DAGA 2013 Merano Approximate description of structure-borne sound sources with continuous line-shaped contact interfaces Sebastian Mathiowetz1 , Hannes A. Bonhoﬀ Institute of Fluid Mechanics and Engineering Acoustics, Technische Universität Berlin, Germany 1 Email: [email protected] Introduction A majority of research concerning the characterization of structure-borne sound sources focuses on sources with multiple discrete contact points. Large eﬀorts have been made in ﬁnding simpliﬁed methods for source characterization and the calculation of the power injected into adjacent structures. However, there is a lack of studies that investigate the applicability of such methods to source structures with continuous line-shaped contact interfaces by means of discretization. In this work the accuracy of the single equivalent approximation [1] and the interface mobility method [2] to be expected with regard to power transmission at continuous line-shaped contact interfaces is examined. Both methods have their advantages with respect to simplicity or physical insight, making each approach inherently better suited for diﬀerent use. The study is based on simulated as well as experimentally collected data from two sources with generic line-shaped contact interfaces connected continuously to plate-like receiver structures. The study is conﬁned to the vertical component of motion. of the interface and with the maximum frequency to be resolved, the problem is generally more severe than in the case of point-connected contact interfaces. Additionally, the identiﬁcation of dominating contributions to the power transmission is hindered as only one single power spectrum is obtained. Particularly for practitioners like consultant engineers or manufacturers, a matrix description is hardly to manage and a description in terms of singular values is favored. Recently, the approach of single equivalent approximation was presented [1]. Here, the concept of eﬀective mobilities [4] was generalized by applying a spatial averaging over the contact points to both the eﬀective mobilities and the activity of the source. Source and receiver mobility matrices are represented by singular equivalent source and receiver mobilities YS,eq and YR,eq , respectively, whereas the source activity can be given as the sum of squared free velocity magnitudes at the N 2 |v contact points n=1 SF,n | . The calculation of the transmitted power thereby is rigorously simpliﬁed and given by Theory The assessment of the structure-borne sound transmission from an active source into a passive receiver connected continuously to each other generally requires an integral formulation. An integral formulation, however, would assume that integral descriptions of mobilities and free velocities are available which is usually not the case in practice. To circumvent this problem, the contact interface can be discretized. Discretizing the interface at N sampling points and introducing source and receiver mobility matrices YS and YR as well as the source activity in terms of a free velocity vector vSF leads to the well known matrix formulation for the transmitted power, as shown in [3], for instance, W = 1 T T ∗ −1 ∗ Re(vSF (YS +YR )−T YR (YS∗ +YR ) vSF ). (1) 2 In Eq. (1), superscripts −1 indicate a matrix inversion and ∗ denotes the complex conjugate. Provided that the spatial Nyquist criterion is fulﬁlled, that is the distance between sampling points has to be smaller than half the minimum wavelength of interest, the complex power at the discretized interface equals that of the continuous interface. The matrix formulation can be problematic as it involves matrix inversions. As the size of mobility matrices of discretized interfaces increases with the size Weq N Re(YR,eq ) 1 . = |vSF,n |2 2 n=1 |YR,eq + YS,eq | (2) In order to derive the eﬀective mobilities, assumptions about the force ratios between the contact points have to be made. At low frequencies where the source is supposed to be acting in bouncing mode, force ratios with unit magnitude and a zero phase diﬀerence can be assumed [4], [5]. At higher frequencies where the structure is controlled by resonant behavior, force ratios with unit magnitude and random phase distribution are likely to be applicable [4], [5]. Thus, the approximate character of the method is due to the spatial averaging of data over the contact points as well as due to the applied force ratio assumption. The method is tested for both force ratio assumptions in this study. The interface mobility method [2] oﬀers a complementary approach where the vibrational behavior at the interface is split up into a sum of orders p. Each order can be associated with a certain component of motion as it reveals the phase diﬀerence between two adjacent contact points such that Δφ = 2π · p/N . For p = 0 all contact points move in phase, so it can be associated with the bouncing mode of the source, for instance. Interface mobilities of source and receiver are derived 2227 AIA-DAGA 2013 Merano as the Fourier coeﬃcients of the respective ordinary mobility matrices. From a two-dimensional Fourier transformation of mobility matrices, equal- and crossorder interface mobilities are derived [2]. By neglecting the latter, only equal-order interface mobilities Ŷp−p,S and Ŷp−p,R remain. Along with the Fourier transform of the free velocity, v̂p,SF , a singular value description is obtained for each power order Wp and the complete power is retained by the sum of all power orders: installations, source and receiver mobilities show the same order of magnitude, i.e. matched conditions. N/2−1 WIF = WIF,p (3) p=−N/2 with WIF,p = 1 |v̂p,SF |2 Re(Ŷp−p,R ) . 2 |Ŷp−p,S + Ŷp−p,R |2 Cross-order interface mobilities account for a variation of ordinary point mobilities along the main diagonal of the mobility matrix and for variations of transfer mobilities with an equidistant ”spacing” in the mobility matrix, e.g. Y12 , Y23 , Y31 in a three-point interface [2]. With regard to continuous line-shaped contact interfaces, cross-order interface mobilities account for a non-circular interface contour, an interrupted interface contour, as well as for discontinuities of source and receiver structures. Their importance with regard to continuous line-shaped contact interfaces is tested in this study. Test setup In this work, structure-borne sound power transmission from two source-receiver installations is examined. First, a laboratory source structure with a circular interface (r = 0.16 m) that is driven by a shaker with pink noise is considered (see Fig. 1 (a), top). Initially, the source is assumed to be connected to a rectangular simply supported chipboard plate (h = 8 mm) at a nonsymmetric, oﬀ-diagonal position. Source and receiver mobilities as well as the free velocities were measured in the dissembled state at N = 24 sampling points (see Fig. 1 (a), bottom). This allows the vibrational behavior at the interface to be discretized up to about 5, 000 Hz. Subsequently, the circular source is assumed to be connected to a corresponding inﬁnite plate. In a second setup, the power transmission from a fan unit source is considered (see Fig. 1 (b), top). The ﬂangelike footing structure of the source is made from three thin bent metal sheets that are arranged perpendicular to each other. The source mobility and free velocity were measured at N = 27 sampling points (see Fig. 1 (b), bottom). In conjunction with a thin inﬁnite steel plate (h = 10 mm) assumed as receiver, the interface can be discretized up to about 3, 500 Hz. However, due to some yet unsolved problems in the higher frequency region, results are analyzed up to 1, 000 Hz and calculated from N = 15 sampling points. Mobilities of the inﬁnite plate receivers were derived by calculation. In both Figure 1: (a) Laboratory source with circular interface, (b) Fan unit source with ﬂange-like rectangular interface •: sampling points Results Laboratory source with circular interface In order to study the accuracy of the approximate methods with respect to the circular interface source (see Fig. 1 (a)), the transmitted power was calculated by means of Eqs. (2) and (3) and compared with the transmitted power calculated by Eq. (1). For the case that the source is connected to a simply supported plate, results are shown in Fig. 2. In the mass-like region, that is up to about 90 Hz, the single equivalent approximation with zero phase assumption Weq,zero is in good agreement with the matrix calculation. Above, the deviations become larger. From 160 Hz onwards, the single equivalent approximation with random phase assumption Weq,rand yields an overall agreement of ±5 dB or less. The interface mobility method WIF yields a good agreement throughout the frequency range, but deviations of about 7 dB can be observed at two distinct frequencies. In summary, for both methods the largest deviations are observed in the transition region between mass-controlled and resonance-controlled frequency region of the source, that is from 100 Hz to 250 Hz. Here, pronounced variations of point and transfer mobilities appear due to the onset of resonances and anti-resonances in the source structure that are not accounted for when mobilities are averaged over the interface or when cross-order interface mobilities are neglected. 2228 AIA-DAGA 2013 Merano 10 10 10 W / WMatrix(dB) (b) -6 Weq, zero WIF Weq, rand (a) 10 10 W (Nm/s) 10 WMatrix -7 -8 10 10 10 -9 63 125 250 500 f (Hz) 1k 2k 10 4k (b) 15 10 W / WMatrix(dB) W (Nm/s) (a) 10-5 5 0 -5 -10 -15 63 125 250 500 f (Hz) 1k 2k -3 WMatrix Weq, zero -4 WIF Weq, rand -5 -6 -7 -8 125 250 500 f (Hz) 1k 2k 63 125 250 500 f (Hz) 1k 2k 15 10 5 0 -5 -10 -15 4k 63 4k Figure 2: (a) Transmitted power from circular source on simply supported plate, (b) Normalized transmitted power Figure 4: (a) Transmitted power from circular source on inﬁnite plate, (b) Normalized transmitted power The vibrational behavior at the interface can be analyzed in more detail by using the interface mobility method. In Fig. 3, the ﬁrst three power orders WIF,0 , WIF,1 and WIF,2 are compared with the sum of all power orders WIF for the frequency range from 50 Hz to 1, 000 Hz. Here, corresponding positive and negative power orders have been summed up, that is W1 = W−1 + W1 and W2 = W−2 + W2 . contact points becomes important. Accordingly, the structural wavelength matches half the circumference of the source in this case. From about 250 Hz onwards, the motion components occur less distinctive. 10 W (Nm/s) 10 10 10 10 10 WIF -5 -6 WIF, 0 WIF, 1 WIF, 2 -7 -8 -9 -10 10 2 10 3 f (Hz) Figure 3: Power orders for circular source on simply supported plate The zero order that represents the bouncing motion component where all sampling points move in phase is seen to be dominant mainly in the mass-like region of the source. The ﬁrst orders describe motion components where adjacent sampling points exhibit a phase diﬀerence of Δφ = ±2π · 1/24. Here, the structural wavelength matches the circumference of the source. Positive and negative ﬁrst order can superimpose and result in a rocking motion. The ﬁrst order motion is seen to be most dominant where the source behaves mass-like and in the transition region between mass-like and resonancecontrolled frequency region of the source. Above, the second order motion component that describes a phase diﬀerence of Δφ = ±2π · 2/24 between two adjacent When the circular source is assumed to be connected to a corresponding inﬁnite plate for which the receiver mobilities were derived by calculation, the deviations shown in Fig. 2 (b) are expected to be reduced. As the interface is rotationally symmetric, there is no variation in point and transfer mobility of the receiver when moving along the interface with a constant distance between the sampling points. In this case, cross-order interface mobilities of the receiver are equal to zero. Equivalently, spatial averaging of mobilities over the contact points does not introduce errors. Results are shown in Fig. 4. It can be seen that the deviations in the transition region between mass-controlled and resonancecontrolled frequency region of the source remain, even if reduced. The zero phase assumption, Weq,zero , leads to large errors when applied to frequencies above the masslike region. As the inﬁnite plate mobility is independent of location, the vibrational behavior at the interface and thus the force distribution can be assumed to be predominantly determined by the source [1]. Fan unit source In this section, the installation of a fan unit source (see Fig. 1 (b)) connected to an inﬁnite steel plate receiver is considered. In Fig. 5, the transmitted power calculated by means of Eqs. (2) and (3) is compared with the transmitted power calculated by Eq. (1). Below the main excitation frequency of 50 Hz, the deviation of the single equivalent approximation is largest. The zero phase assumption Weq,zero yields deviations in a range of ±5 dB throughout the rest of the spectrum. The random phase assumption Weq,rand yields larger deviations in the higher frequency region. The interface 2229 AIA-DAGA 2013 Merano mobility method overestimates the transmitted power at most parts of the spectrum with a maximum deviation of about 8 dB at 63 Hz. W (Nm/s) (a) 102 10 10 10 10 W / WMatrix(dB) (b) WMatrix 0 Weq, rand Weq, zero -2 WIF -4 -6 63 125 250 f (Hz) 500 1k 63 125 250 f (Hz) 500 1k 15 10 5 0 -5 -10 -15 Figure 5: (a) Transmitted power from fan unit source on inﬁnite plate, (b) Normalized transmitted power In Fig. 6, the ﬁrst three power orders are compared with the sum of all power orders WIF for the frequency range from 40 Hz to 200 Hz. Except for the main excitation frequency of the electric motor at 50 Hz and its harmonics, the single power orders cannot be identiﬁed as dominant in certain frequency regions. This might be due to the structure of the source whose dynamic behavior is less distinctive with regard to resonances and mode shapes compared to that of the circular source. On the other hand, the frequency region between the main excitation frequency and its harmonics can be seen as a more or less random ”noise ﬂoor” to the peaks in the spectrum, thus it is of minor importance with regard to power transmission. 10 10 W (Nm/s) interfaces was tested in the example of two sourcereceiver installations. In summary, both methods reveal a similar degree of accuracy on average. The input data have been collected by discretizing source and receiver structures which results in large mobility matrices. The calculation of reference values involves matrix inversions, thus the shown deviations between approximate methods and reference values might also result from uncertainties in the reference values to some extent. With regard to prediction, the approach of the single equivalent approximation is generally more appealing as it allows the transmission process to be described by singular values of source and receiver. The interface mobility method, in contrast, allows to obtain some physical insight in the transmission process, but requires larger data sets to be analyzed. The results presented are based on small datasets compared to the variety of conditions that can be found in practice. The study should be extended to other generic interface geometries in order to yield results on a more general basis. 10 10 10 10 10 2 WIF 0 WIF, 0 -2 WIF, 1 -4 WIF, 2 -6 -8 -10 10 f (Hz) 2 Figure 6: Power orders for fan unit source on inﬁnite plate Concluding remarks Acknowledgments The authors would like to thank Dr. Andreas Mayr from the University of Applied Sciences Rosenheim, Germany, for collaborative discussions on the work presented in this paper. The ﬁnancial support received from the German Federal Ministry of Economics and Technology - BMWi through grant 03SX305G is gratefully acknowledged. References [1] Mayr, A.R., Gibbs, B.M.: Single equivalent approximation for multiple contact structure-borne sound sources in buildings. Acta Acustica united with Acustica 98 (2012), 402-410 [2] Bonhoﬀ, H.A., Eslami, A.: Interface mobilities for characterization of structure-borne sound sources with multi-point coupling. Acta Acustica united with Acustica 98 (2012), 384-391 [3] Bonhoﬀ, H.A.: The inﬂuence and signiﬁcance of cross-order terms in interface mobilities for structureborne sound source characterization (Ph.D.). Universitätsverlag der TU Berlin, Berlin, 2010 [4] Petersson, B.A.T., Plunt, J.: On eﬀective mobilities in the prediction of structure-borne sound transmission between a source structure and a receiver structure, part 1: Theoretical background and basic experimental studies. Journal of Sound and Vibration 82 (1982), 517-529 [5] Fulford, R.A., Gibbs, B.M.: Structure-borne sound power and source characterization of structure-borne sound sources in multi-point-connected systems, part 1: Case studies for assumed force distributions. Journal of Sound and Vibration 204 (1997) 659-677 In this paper, the applicability of the single equivalent approximation as well as the interface mobility method with regard to continuous line-shaped contact 2230