Gary H. Koopmann

A method for computing acoustic fields based on the principle of wave superposition

A method for computing the acoustic fields of arbitrarily shaped radiators is described that uses the principle of wave superposition. The superposition integral, which is shown to be equivalent to the Helmholtz integral, is based on the idea that the combined fields of an array of sources interior to a radiator can be made to reproduce a velocity prescribed on the surface of the radiator. The strengths of the sources that produce this condition can, in turn, be used to compute the corresponding surface pressures. The results of several numerical experiments are presented that demonstrate the simplicity of the method. Also, the advantages that the superposition method has over the more commonly used boundary‐element methods are discussed. These include simplicity of generating the matrix elements used in the numerical formulation and improved accuracy and speed, the latter two being due to the avoidance of uniqueness and singularity problems inherent in the boundary‐element formulation.

Material tailoring of structures to achieve a minimum radiation condition

A strategy is developed for designing structures that radiate sound inefficiently in light fluids. The problem is broken into two steps. First, given a frequency and overall geometry of the structure, a surface velocity distribution is found that produces a minimum radiation condition. This particular velocity distribution is referred to as the ‘‘weak radiator’’ velocity profile. Second, a distribution of Young’s modulus and density distribution is found for the structure such that it exhibits the weak radiator velocity profile as one of its mode shapes. In the first step, a finite element adaptation of the integral wave equation is combined with the Lagrange multiplier theorem to obtain a surface velocity distribution that minimizes the radiated sound power. In the second step, extensive use of structural finite element modeling as well as linear programming techniques is made. The result is a weak radiator structure. When compared to a structure with uniform material properties, the weak radiator struct...

A boundary element method for acoustic radiation valid for all wavenumbers

A boundary element method for solving the exterior acoustic radiation problem that is valid for all wavenumbers has been developed. The new formulation derives from the earlier work of Burton and Miller [Proc. R. Soc. London Ser. A 323, 201–210 (1971)], which uses a linear combination of the Helmholtz integral equation and its gradient to overcome the uniqueness problem. A main feature of this formulation is the absence of integral singularities. By restricting the field points of the kernels of the integrals to interior regions well away from radiating boundaries, the usual complication of dealing with singular kernels is circumvented. The new formulation has been named CHI, for coupled Helmholtz integrals. A corresponding computer program, CHI 4.35, can be used to compute the surface acoustic pressure on an arbitrary body for a given specified surface normal velocity. Results generated with CHI 4.35 are presented for the spherical monopole and dipole, along with results for parametric studies on the acc...

Numerical errors associated with the method of superposition for computing acoustic fields

The method of “wave superposition” is based on the idea that an acoustic radiator can be approximately represented by the sum of the fields due to a finite number of interior point sources. The accuracy of this representation depends upon how well the velocity boundary condition on the surface of the body is approximated. The ultimate objective of this study, then, is to provide some guidelines for improving the accuracy of the surface velocity reconstruction and, consequently, the accuracy of the superposition solutions. In general, this is dependent upon the particular surface velocity distribution to be reconstructed, as well as other formulation factors such as the acoustic wave number, the number and locations of the surface nodes, and the number and locations of the point sources. Velocity interpolation functions are introduced as a means of quantifying the dependence of reconstruction errors on the acoustic wave number and the placement of the surface nodes and point sources. Numerical experiments ...

A numerical solution for the general radiation problem based on the combined methods of superposition and singular‐value decomposition

In the method of wave superposition, the acoustic field, due to a complex radiator, is expressed in terms of a Fredholm integral equation of the first kind called the ‘‘superposition integral equation.’’ In general, Fredholm integral equations of the first kind are ill‐posed and therefore difficult to solve numerically. In this paper, it will be shown that a simple collocation procedure, when combined with the singular‐value decomposition, can yield accurate results for the numerical solution of the superposition integral. As an example of the application of the method, the acoustic radiation from a circular cylinder will be analyzed using this numerical procedure and compared to the exact solution. It is shown that, for this problem, the accuracy of the numerical solution can be judged by evaluating how well the superposition solution approximates the specified boundary condition on the surface of the radiator. An example is also given of a problem which has no exact solution. In this situation, it is suggested, without proof, that the accuracy of the numerical solution can be judged in a similar manner by evaluating the error in the superposition solution’s satisfaction of the boundary condition.

Method for computing the sound power of machines based on the Helmholtz integral

A computational method is presented for assessing the sound power characteristics of machines. The method, which is based upon a Helmholtz integral formulation, requires a knowledge of the geometry and the modal characteristics of a machine’s vibrating surfaces so that the pressure on the surface can be computed. Closed form integration of the associated surface integrals is carried out in a piecewise manner over planar surface elements in the shape of rectangles or triangles. This choice of element allows the grid geometries associated with the acoustic power computations to be made identical to those used in existing structural modal analysis methods. In this way, the sound power characteristics of a given machine can be computed in terms of the sound power radiated by each of the structural modes comprising the overall response. The structural modes can then be ranked in order of their radiation efficiency for purposes of noise control treatments. The accuracy of the method is demonstrated by calculati...

A general optimization strategy for sound power minimization

A general approach for minimizing radiated acoustic power of a baffled plate excited by broad band harmonic excitation is given. The steps involve a finite element discretization for expressing acoustic power and vibration analysis, analytical design sensitivity analysis, and the use of gradient-based optimization algorithms. Acoustic power expressions are derived from the Rayleigh integral for plates. A general methodology is developed for computing design sensitivities using analytical expressions. Results show that analytical sensitivity analysis is important from both computational time and accuracy considerations. Applications of the optimization strategy to rectangular plates and an engine cover plate are presented. Thicknesses are chosen as design variables.

Design and performance of a high-force piezoelectric inchworm motor

A linear inchworm motor was developed for applications in adaptive, conformable structures for flow control. The device is compact (82 X 57 X 13 mm), and capable of unlimited displacement and high force actuation (150 N). The static holding force is 350 N. Four piezoceramic stack elements (two for clamping and two for extension) are integrated into the actuator, which is cut from a single block of titanium alloy. Actuation is in the form of a steel shaft pushed through a precision tolerance hole in the device. Unlimited displacements are achieved by repetitively advancing and clamping the steel shaft. Although each step is only on the order of 10 microns, a step rate of 100 Hz results in a speed of 1 mm/s. Since the input voltage can readily control the step size, positioning on the sub-micron level is possible.

Experimental and Numerical Analysis of Bimodal Acoustic Agglomeration

The interaction between fly ash particles (first mode) and sorbent particles (second mode) in coal combustion processes is studied under the influence of a low frequency, high intensity acoustic field. The effect of bimodal acoustic agglomeration is evaluated in a numerical sensitivity analysis on parameters such as residence time in the combustion chamber and mass loading of the particle modes. An Acoustic Agglomeration Simulation Model (AASM) developed by Song at the Pennsylvania State University is used for these numerical studies. Experimental examinations carried out in a down-fired combustor show evidence of bimodal agglomeration and enhanced particle interaction under the influence of a low frequency (44 Hz), high intensity (160 dB) sound field. The results of the experiments are compared to the equivalent numerical studies and good agreement can be shown between the two sets of data.

A Design Method for Minimizing the Sound Power Radiated from Plates by Adding Optimally Sized, Discrete Masses

In this paper, a novel method for minimizing the sound power radiated from a structure is presented. The method involves placing strategically sized masses at specific locations on the structure’s surface. The minimization procedure modifies the shapes of the resonant modes of the structure in the frequency range of interest such that they are forced to radiate sound inefficiently. Because of this, they are referred to as “weak radiator” mode shapes. The method uses an optimization procedure that directly minimizes the radiated sound power from the surface of a plate in an infinite baffle. The procedure can be carried out for a single frequency or over a range of frequencies. Analytical sensitivities of sound power with respect to the design variables are developed and used in the optimization algorithm. Results on various test cases show sound power reductions of 10 dB or more even when several resonances are included in the frequency band. An acoustic intensity probe is used to experimentally verify the results for one test case. The experiment confirms the sound power reductions predicted by the optimization program.