Improved efficiency of vibration-based sound power computation through multi-layered radiation resistance matrix symmetry

Sound power has become an increasingly important metric in the field of acoustics. Three main approaches exist for measuring the sound power of a source. First, pressure microphone measurements within a controlled acoustic environment can be converted to power. Second, by using an acoustic intensity probe, a surface surrounding a source can be scanned to measure the sound intensity and then converted to power. Third, vibration measurements can be used in conjunction with geometric data to calculate sound power by utilizing the acoustic radiation impedance. This paper explores how to optimize the computation for this third approach.

The radiation resistance matrix provides knowledge of how the acoustic field couples with a structure vibrating in one of its fundamental modes. It is fundamental in analyzing and understanding the structural response to noise, acoustic fatigue, sound transmission, and radiation efficiency and is responsible for the radiation of sound from noise sources.^1,2

Since the early 1990s, researchers have used radiation modes to compute sound power.^3–7 This method involves using singular value decomposition (SVD) and functional analysis to obtain the eigenvectors, or radiation modes, and eigenvalues of the radiation resistance matrix from which the sound power can be computed. However, as more frequencies of interest are included, many more radiation modes are required to capture the full sound power, and the efficiencies gained by using SVD are curtailed. For this reason, alternative numerical methods are desirable to determine sound power while minimizing the computational expense.⁸ 

Several recent papers,^9–13 both computational and experimental, have verified the use of a radiation resistance matrix in conjunction with complex-valued surface velocity measurements to calculate sound power. Using the radiation resistance matrix, R, and the surface velocity vector, $ve$⁠, (consisting of all the measured element velocities arranged in vector form), the sound power, P, at a given frequency, $ω$⁠, is given as^13,14

As an example, a baffled flat plate with N elements has an $R$ matrix given as¹¹ 

with the distance matrix, $dij$⁠, given as

where $xj,1$ and $xj,2$ represent the x and y coordinates of the jth point on the scan grid, respectively, $ω$ is a given frequency, $ρ0$ is the fluid density surrounding the vibrating surface, $Ae$ is the area of a single radiator [assumed constant in Eq. (2)], c is the speed of sound in the surrounding fluid, and $k$ is the acoustic wavenumber. This radiation resistance matrix accounts for the influence of each radiator on every other radiator, which is primarily discriminated through distance. In addition to flat plates, prior research has developed the $R$ matrix to encode these acoustic radiation modes for two additional geometries, which have been tested with promising results. These geometries correspond to curved plate and cylindrical shell geometries, respectively. In each of these cases, the formulation of $R$ includes using some version of a distance matrix. Preliminary research shows that these radiation formulations may be used for some arbitrarily curved structures as well.¹³ 

At this point, it is instructive to consider the size of the matrices at play to give an idea of why the computation is considered “heavy,” which in turn will motivate the work done in this paper. For example, a 5 × 5 scan grid will yield a 25 × 25 $R$ matrix that must be calculated for each frequency of interest. In most of the experimental work done in this paper, scans typically involved at least 20 scan points in each Euclidean direction, which created much larger $R$ matrices and, thus, severely increase the computation time for each individual frequency of interest. Furthermore, a general rule of thumb when considering the size of the scan grid is to have about six scan points per wavelength to capture the sound power at the corresponding frequency.¹⁵ Therefore, denser scan grids are needed to achieve the resolution needed to examine higher frequencies, again adding to the size of the $R$ matrix. One can easily see how these calculations become computationally expensive quite quickly, especially when these computations are performed for curved plates and cylindrical shells, where the basis functions become much more complex.

Assuming a plate modeled with N radiators, a clear symmetry in the distance matrix is seen. First, it follows trivially that

from the fact that distance is calculated using a mathematical norm, which implies the classical matrix symmetry of the distance matrix. However, additional symmetries in the matrix exist given the assumption that all points of the scan grid are equally spaced in the x direction by a factor, dx, and all points are also equally spaced in the y direction by a factor, dy, not necessarily equal to dx. Indeed, this assumption yields a distance matrix having two layers of Toeplitz¹⁶ matrix symmetry. A visual description of a 5 × 5 scan grid with 25 radiators is presented in Fig. 1. In this figure, the various matrix elements are coded by color and fill pattern to easily visualize elements in the matrix that have the same value, thereby making the double-layered Toeplitz symmetry more identifiable.

The large numbers shown in Fig. 1 showcase a Toeplitz symmetry made of submatrices. However, within each large-numbered submatrix, another Toeplitz symmetry appears. Thus, the distance matrix and, by relation, the radiation resistance matrix both exhibit a double-layered Toeplitz symmetry, given the assumption of constant spacing in the x and y directions. Note that the number of scan points need not be equal in either direction. The real assumption is that the spacing of scan grid points is constant.

More importantly, due to the distance matrix's intimate connection with the radiation resistance matrix, as seen in Eq. (2), all but the $Ae$ term share this symmetry. After observation, it also becomes clear that all the unique values of the resulting radiation resistance matrix are actually present in just the first row of that matrix, as can be seen by looking at Fig. 1. Thus, the heavy computation using complex-valued basis functions (or other necessary approximations) may be performed on the first row of the radiation resistance matrix, after which a careful expansion of the first row, using the double-layered Toeplitz symmetry as a guide, yields an equivalent radiation resistance matrix for sound power computation.

The aforementioned symmetries rely on the assumption of equally spaced points in the x and y directions, respectively. The sensitivity of the output of the algorithm due to the strictness of this assumption is not explored in the scope of this paper. However, the results in Sec. 3 assure that at least for most cases, even though the scan grid spacing may not have exactly constant spacing, the method still delivers an accurate reading of sound power. This assumption could be investigated in future research to establish bounds on the accuracy that can be achieved for scan grid spacing that is not completely uniform.

As presented in Eq. (2), the $Ae$ term is an integral part of the formulation of the radiation resistance matrix. That formulation is adequate for flat plate geometry; however, when curvature is introduced, the assumption of uniform radiator areas across the plate no longer holds, as projecting a two-dimensional (2D) laser scan grid with uniform spacing onto a three-dimensional (3D) surface creates differences in radiator area. To account for these differences, Bates et al. experimented with using an $Ae$ matrix with the ijth element in the matrix corresponding to the product of the ith radiator's area with the jth radiator's area.¹⁷ This matrix was then elementwise multiplied by the radiation resistance matrix in the sound power computation before the velocity vector multiplication to weight each of the radiator areas more accurately. The results showed an improved accuracy over using just a single $Ae$ term average. More specifically, the improvement correlated positively with the curvature of the structure.

However, as discussed in their work, large oscillations in specific radiator areas were present in the matrix version of the $Ae$ term, though these were mostly localized to the seam that stitched together two scans from the laser head to get a full reading of a curved surface, something necessary when the radius of curvature of the plate is sufficiently small. Even though a matrix version of the $Ae$ term can increase the accuracy of the sound power calculation in general, for some cases, the large amplitude of noise in this matrix that can result from variations that occur along scanning seams can lead to inaccuracies.

As a result, a new approach was formulated to mitigate the problems from the seams: first, approximating the plate geometry with a perfectly spaced and seam-free scan grid and then taking the average of the matrix version of the $Ae$ term for this new smooth geometry and using this in the sound power calculation. Note that the matrix version of the $Ae$ term was calculated using the same method that Bates et al. used, taking the product of the ith radiator's area with the jth radiator's area.¹⁷ Results proved to retain accuracy and, in some cases, speed up computation time, due to elementwise matrix multiplication for large matrices being minimized. Experimental results are presented in Sec. 3.

As an example of a data set for which this approximation is optimal, a curved aluminum plate of length 0.3 m, width 0.4 m, and thickness of 1.59 mm was considered. The radius of curvature for the plate was 0.51 m, and the percent difference in average radiator area for the scanned plate was about 9.5%. Thus, in investigating the use of a single term average, the smooth geometric approximation average was used, rather than the average from the highly oscillatory matrix version of $Ae$⁠. Results from experimental data indicate that using the single term average from the geometric approximation yields a result that better matches the sound power measurement using the ISO 3741 standard.¹⁸ These results are presented in Sec. 3. Furthermore, this effectively reduced the $Ae$ matrix to a single term, and the efficiency gains based on experimental data are also presented in Sec. 3.

Exploiting this symmetry has led to meaningful efficiency gains in computation time with the calculation of sound power for curved plate and cylindrical shell structures, as shown in Table 1. The data sets were obtained using a Polytec (Baden-Württemberg, Germany) PSV-500 3D scanning laser doppler vibrometer with the experimental setup depicted in Fig. 2.

The hard Fock V coupling function has been sufficiently characterized to produce useful series representations with ten terms or fewer, which are given in an appendix of Pathak and Wang¹⁹ and in McNamara et al.²⁰ At very large curvature and low frequency, the Fock V function behaves like a sinc function, and Eq. (2) approximates the radiation resistance matrix for these cases of curved plates.

The results are very encouraging. Notice the speed up ratio increases as the radius of curvature decreases. Translating the ratios into percentages, we obtain a ∼1300% maximum efficiency gain for curved plate computations and a stunning ∼9600% maximum efficiency gain for cylindrical geometry.

Calculation is further optimized by smoothing the radiator area matrix using a normalized geometric approximation and using its average value in the calculation. This reduced the $Ae$ term in Eq. (2) to simply a constant value for all geometries. The results are shown in Table 2. Please note that these results were calculated with a machine slightly different from the one used to calculate the previous results; hence, the emphasis on speed up ratios and percentages rather than actual calculation speeds.

Regarding accuracy, the Toeplitz method and single area approximation are no worse than the traditional VBSP method. Figures 3 and 4 are representative of our data sets.

The ISO 3741 standard does not provide precision grade results below the Schroeder frequency of the reverberation chamber used, which was about 385 Hz, leading to the large discrepancy between sound power methods. It is likely that the noise floor of the chamber is masking the true energy produced from the vibrating panels below the 400 Hz one-third octave (OTO) band. Thus, it is possible that the VBSP/Toeplitz results may be a more accurate representation of the energy produced by those frequency bands. In addition, Fig. 4 shows that the single term area average method can be employed without sacrificing the accuracy of the computation. Also, note that the computation time per frequency is approximately constant.

As seen from the experimental data, allowing the assumption of constant spacing to take full advantage of the radiation resistance matrix's Toeplitz symmetry dramatically speeds up sound power computation time without significant loss of accuracy, compared to both ISO and previous VBSP methods. Approximating the surface and then using a well-informed single average area radiator term also preserves the integrity of the result and yields further speed improvement. Future areas to research include using a tensor or hypermatrix and/or multithreading to compute all frequencies simultaneously. Such improvements could continue to dramatically expand the number of applications of the VBSP method.

This work was funded by National Science Foundation Grant No. 1916696.