Sean F. Wu

On reconstruction of acoustic pressure fields using the helmholtz equation least squares method

This paper presents analyses and implementation of the reconstruction of acoustic pressure fields radiated from a general, three-dimensional complex vibrating structure using the Helmholtz equation least-squares (HELS) method. The structure under consideration emulates a full-size four-cylinder engine. To simulate sound radiation from a vibrating structure, harmonic excitations are assumed to act on arbitrarily selected surfaces. The resulting vibration responses are solved by the commercial FEM (finite element method) software I-DEAS. Once the normal component of the surface velocity distribution is determined, the surface acoustic pressures are calculated using standard boundary element method (BEM) codes. The radiated acoustic pressures over several planar surfaces at certain distances from the source are calculated by the Helmholtz integral formulation. These field pressures are taken as the input to the HELS formulation to reconstruct acoustic pressures on the entire source surface, as well as in the field. The reconstructed acoustic pressures thus obtained are then compared with benchmark values. Numerical results demonstrate that good agreements can be obtained with relatively few expansion functions. The HELS method is shown to be very effective in the low-to-mid frequency regime, and can potentially become a powerful noise diagnostic tool.

Reconstruction of transient acoustic radiation from a sphere

Transient near-field acoustical holography (NAH) formulation is derived from the Helmholtz equation least squares (HELS) method to reconstruct acoustic radiation from a spherical surface subject to transient excitations in a free field. To facilitate derivations of temporal solutions, we make use of the Laplace transform and expansion in terms of the spherical Hankel functions and spherical harmonics, with their coefficients settled by solving a system of equations obtained by matching an assumed-form solution to the measured acoustic pressure. To derive a general form of solution for a temporal kernel, we replace the spherical Hankel functions and their derivatives by polynomials, recast infinite integrals in the inverse Laplace transform as contour integrals in a complex s-plane, and evaluate it via the residue theorem. The transient acoustic quantities anywhere including the source surface are then obtained by convoluting the temporal kernels with respect to the measured acoustic pressure. Numerical examples of reconstructing transient acoustic fields from explosively expanding, impulsively accelerating, and partially accelerating spheres, and that from a sphere subject to an arbitrarily time-dependent excitation are depicted. To illustrate the effectiveness of HELS-based transient NAH formulations, all input data are collected along an arbitrarily selected line segment and used to reconstruct transient acoustic quantities everywhere.

Noise Transmission Through a Vehicle Side Window Due to Turbulent Boundary Layer Excitation

This paper presents results of an investigation on noise transmission through an aluminum panel clamped to a greenhouse vehicle model subject to random acoustics, random vibration, and turbulent boundary layer excitations. Experiments on random acoustics and random vibration excitations were carried out in a reverberation chamber, and those on turbulent boundary layer excitation were conducted in the wind tunnel at the Chrysler Technology Center. The transmitted noise spectra were also calculated using a single computer program VibroAcoustic Payload Environment Prediction System (VAPEPS) based on Statistic Energy Analysis (SEA). The acoustic absorption coefficient (AAC) and the damping loss factor (DLF) for the vehicle were determined based on experimental data. Results showed that the largest differences between the measured and calculated sound pressure levels in any frequency band above 500 Hz were less than 2.5 dB for random acoustics excitation, 5.0 dB for random vibration excitation, and 5 dB for turbulent boundary layer excitation. In spite of the presence of differences in individual frequency bands, the calculated total sound pressure levels compared well with the measured ones. The differences between the calculated and measured total sound pressure levels were 0.7 dB for random acoustics excitation, 0.4 dB for random vibration excitation, and 1.8 dB for turbulent boundary layer excitation.

Reconstructing the vibro-acoustic quantities on a highly non-spherical surface using the Helmholtz equation least squares method

This paper presents helpful guidelines and strategies for reconstructing the vibro-acoustic quantities on a highly non-spherical surface by using the Helmholtz equation least squares (HELS). This study highlights that a computationally simple code based on the spherical wave functions can produce an accurate reconstruction of the acoustic pressure and normal surface velocity on planar surfaces. The key is to select the optimal origin of the coordinate system behind the planar surface, choose a target structural wavelength to be reconstructed, set an appropriate stand-off distance and microphone spacing, use a hybrid regularization scheme to determine the optimal number of the expansion functions, etc. The reconstructed vibro-acoustic quantities are validated rigorously via experiments by comparing the reconstructed normal surface velocity spectra and distributions with the benchmark data obtained by scanning a laser vibrometer over the plate surface. Results confirm that following the proposed guidelines and strategies can ensure the accuracy in reconstructing the normal surface velocity up to the target structural wavelength, and produce much more satisfactory results than a straight application of the original HELS formulations. Experiment validations on a baffled, square plate were conducted inside a fully anechoic chamber.

Visualization of vehicle interior sound field using nearfield acoustical holography based on the helmholtz-equation least-squares (HELS) method

This paper examines the application of Nearfield Acoustical Holography based on the Helmholtz Equation Least-Squares (HELS) method for visualization of the sound-pressure field in the passenger compartment of a vehicle with hard interior surfaces. In the HELS method, sound pressure is determined by an expansion of a set of orthogonal basis functions that are solutions to the Helmholtz equation. Coefficients associated with these basis functions were determined by a collocation method with errors minimized by the least-squares method. The number of expansion terms in the method was optimized to ensure a convergence of the solution anywhere in the interior region of the vehicle, including the boundary surfaces. The reconstructed sound pressures were compared with benchmark sound pressures calculated by the boundary element method (BEM). Results demonstrated that the HELS method permitted reconstruction of the sound-pressure field with much greater computational efficiency than the BEM procedure. Accuracy of reconstruction was relatively insensitive to the choice of the locations for input data. For hard interior surfaces, a small box-shaped surface on which input data were collected was shown to be sufficient to provide good reconstruction results. Such a configuration could be implemented relatively easily in practice. Reconstruction of sound-pressure fields at resonance frequencies of the vehicle interior cavity was also validated.

Simulation of Vehicle Pass-by Noise Radiation

This paper presents an extended Kirchhoff integral formulation for predicting sound radiation from an arbitrarily shaped vibrating structure moving along an infinite baffle. In deriving this formulation, the effect of sound reflection from the baffle is taken into account by using the image source theory. Moreover, the effect of source convection motion and that of motion-induced fluid-structure interaction at the interface on the resulting acoustic pressure field are considered. The formulation thus derived is used to calculate sound radiation from a simplified vehicle model cruising along a solid ground at constant speeds. Since analytical and benchmark numerical solutions for an arbitrarily shaped vibrating object in motion are not available, validations of numerical results are made with respect to those of point source. Next, sound radiation from a full-size vehicle is simulated. For simplicity, the vehicle is assumed to be made of shell-type structure and excited by harmonic forces acting on its four tires. Vibration responses subject to these excitations are calculated using finite element method (FEM) with HyperMesh® version 2.0 as pre- and post-processors. Once the normal component of the surface velocity is specified, the radiated acoustic pressure fields are determined using boundary element method (BEM). Numerical results show that the effect of source convection motion enhances sound radiation in the forward direction, but reduces that in the rearward direction. Changes in the resulting sound pressure fields become obvious when the Mach number exceeds 0.1, or equivalently, when a vehicle cruises at 70 mph or higher.

An alternative formulation for predicting sound radiation from a vibrating object

An alternative formulation is derived for predicting acoustic radiation from a vibrating object in an unbounded fluid medium. The radiated acoustic pressure is shown to be expressible as a surface integral of the particle velocity, which is determinable by using a nonintrusive laser velocimeter. This approach is in contrast with the Kirchhoff integral formulation which requires the knowledge of both the normal component of the surface velocity and the surface acoustic pressure prior to predicting the radiated acoustic pressure. Solutions thus obtained are unique. Moreover, the efficiency of numerical computations is high because the surface integration can be readily implemented numerically by using standard Gaussian quadratures. This alternative formulation may be desirable to analyze the acoustic and vibration responses of a lightweight, a flexible, or a structure under an adverse environment for which a nonintrusive laser measurement technique must be used. Validations of this alternative formulation are demonstrated both analytically and numerically for vibrating spheres and right circular cylinders.

Panel acoustic contribution analysis

Formulations are derived to analyze the relative panel acoustic contributions of a vibrating structure. The essence of this analysis is to correlate the acoustic power flow from each panel to the radiated acoustic pressure at any field point. The acoustic power is obtained by integrating the normal component of the surface acoustic intensity, which is the product of the surface acoustic pressure and normal surface velocity reconstructed by using the Helmholtz equation least squares based nearfield acoustical holography, over each panel. The significance of this methodology is that it enables one to analyze and rank relative acoustic contributions of individual panels of a complex vibrating structure to acoustic radiation anywhere in the field based on a single set of the acoustic pressures measured in the near field. Moreover, this approach is valid for both interior and exterior regions. Examples of using this method to analyze and rank the relative acoustic contributions of a scaled vehicle cabin are demonstrated.

An explicit integral formulation for transient acoustic radiation

An explicit integral formulation is derived for predicting transient acoustic radiation. The radiated acoustic pressure is shown to be expressible as a sum of integrals over simple and doublet sources and their couplings induced by the normal and tangential components of the particle velocity, which can be measured by a nonintrusive laser velocimeter. Such an integral formulation is computationally advantageous in that solutions thus obtained are unique, and the efficiency of numerical computations is high. This is because the radiated acoustic pressure is expressed as an explicit integral of known quantities, which can be readily implemented numerically by using the Gaussian quadratures. For arbitrary time-dependent excitations, the radiated acoustic pressure can be written as a sum of convolution integrals of impulse response functions over the time history of the particle velocity specified on the control surface. Validations of this integral formulation are demonstrated both analytically and numerical...

Nonuniqueness of solutions to extended Kirchhoff integral formulations

This paper examines the nonuniqueness of solutions to an extended Kirchhoff integral formulation at certain discrete frequencies corresponding to the eigenfrequencies of the related interior boundary value problem. In particular, effects of surface motion and interaction between turbulence and vibrating surface in motion on nonuniqueness difficulties are investigated. It is shown that surface motion affects nonuniqueness difficulties in two ways: (1) it shifts the eigenfrequencies of the related interior boundary value problem and (2) it excites more eigenfrequencies at which nonuniqueness difficulties occur than the corresponding stationary case. The interaction between turbulence and vibrating surface in motion further shifts the eigenfrequencies. Although changes in these eigenfrequencies are small at low Mach numbers, they increase with the Mach number. It is also demonstrated that although an extended Kirchhoff integral equation fails to yield a unique solution at certain discrete frequencies, a combined extended surface Kirchhoff integral equation and an extended interior Kirchhoff integral equation always yield a unique solution for all frequencies. This is because there is only one set of solutions that satisfy both an extended surface Kirchhoff integral equation and an extended interior Kirchhoff integral equation.