Raymond J. Nagem

Dynamic Analysis of Large Space Structures Using Transfer Matrices and Joint Coupling Matrices

Abstract Linear dynamic analysis of lattice structures using transfer matrices and joint coupling matrices is presented. A lattice structure is defined as a network of one-dimensional members that are connected by joints. Two examples are considered to illustrate how transfer matrices and joint coupling matrices may be used to compute natural frequencies of vibration. These two examples indicate that the transfer matrix and joint coupling matrix analysis is numerically accurate over a wide range of frequencies and becomes increasingly efficient, compared to the finite element method, as the frequency increases. Some suggestions for further improvements in computational efficiency and some comments about applicability to numerical analysis of wave propagation problems are given.

Generalized Helmholtz-Kirchhoff Model for Two-Dimensional Distributed Vortex Motion ∗

The two-dimensional Navier–Stokes equations are rewritten as a system of coupled nonlinear ordinary differential equations. These equations describe the evolution of the moments of an expansion of the vorticity with respect to Hermite functions and of the centers of vorticity concentrations. We prove the convergence of this expansion and show that in the zero viscosity and zero core size limit we formally recover the Helmholtz–Kirchhoff model for the evolution of point vortices. The present expansion systematically incorporates the effects of both viscosity and finite vortex core size. We also show that a low-order truncation of our expansion leads to the representation of the flow as a system of interacting Gaussian (i.e., Oseen) vortices, which previous experimental work has shown to be an accurate approximation to many important physical flows [P. Meunier, S. Le Dizes, and T. Leweke, C. R. Phys., 6 (2005), pp. 431–450].

Vorticity dynamics and sound generation in two-dimensional fluid flow

An approximate solution to the two-dimensional incompressible fluid equations is constructed by expanding the vorticity field in a series of derivatives of a Gaussian vortex. The expansion is used to analyze the motion of a corotating Gaussian vortex pair, and the spatial rotation frequency of the vortex pair is derived directly from the fluid vorticity equation. The resulting rotation frequency includes the effects of finite vortex core size and viscosity and reduces, in the appropriate limit, to the rotation frequency of the Kirchhoff point vortex theory. The expansion is then used in the low Mach number Lighthill equation to derive the far-field acoustic pressure generated by the Gaussian vortex pair. This pressure amplitude is compared with that of a previous fully numerical simulation in which the Reynolds number is large and the vortex core size is significant compared to the vortex separation. The present analytic result for the far-field acoustic pressure is shown to be substantially more accurate than previous theoretical predictions. The given example suggests that the vorticity expansion is a useful tool for the prediction of sound generated by a general distributed vorticity field.

Curle’s equation and acoustic scattering by a sphere

Recent papers have initiated interesting comparisons between aeroacoustic theory and the results of acoustic scattering problems. In this paper, we consider some aspects of these comparisons for acoustic scattering by a sphere. We give a derivation of Curles equation for a specific class of linear acoustic scattering problems, and, in response to previous claims to the contrary, give an explicit confirmation of Curles equation for plane wave scattering by a stationary rigid sphere of arbitrary size in an inviscid fluid. We construct the complete solution for scattering by a rigid sphere in a viscous fluid, and show that the neglect of viscous terms in Curies equation yields an incomplete prediction of the far field dipole pressure. We also consider the null field solution of the sphere scattering problem, and give its extension to the vorticity modes associated with viscosity. Finally, we construct a solution for an elastic sphere in a viscous fluid, and show that the rigid sphere/null field solution is recovered from the limit of infinite longitudinal and shear wave speeds in the elastic solid.

Influence of viscosity on the diffraction of sound by a circular aperture in a plane screen

The linearized equations of viscous fluid flow are used to analyze the diffraction of a time-harmonic acoustic plane wave by a circular aperture in a rigid plane screen. Arbitrary aperture size and arbitrary angle of incidence are considered. Sets of dual integral equations are derived for the diffracted velocity and pressure fields, and are solved by analytic reduction to sets of linear algebraic equations. In the case of normal incidence, numerical results are presented for the fluid velocity in the aperture and the power absorption due to viscous dissipation. The theoretical results for power absorption are compared to previously obtained results from high amplitude acoustic experiments in air. The conditions under which the dissipation predicted by linear theory becomes significantare quantified in terms of the fluid viscosity and sound speed, the acoustic frequency, and the aperture radius.

Control of a One-Dimensional Distributed Structure Based on Wave Propagation Analysis∗

ABSTRACT Wave propagation in a one-dimensional structure is analyzed. Based on this wave propagation analysis, a control system for the structure is developed. The purpose of the control system is to improve the dynamic response of the structure to an external force. It is shown that the control system can completely isolate a section of the structure from disturbances caused by the external force, or it can add damping to the structure. The effects of two types of errors on the control system are considered. The example given here illustrates the potential importance of control systems based on wave propagation concepts and raises many issues that are involved in the development of such control systems.

Effect of viscosity on acoustic diffraction by a circular disk

A complete solution is obtained for the diffraction of a time-harmonic acoustic plane wave by a circular disk in a viscous fluid. Arbitrary disk radius size and arbitrary angle of incidence are considered. The linearized equations of viscous flow and the no-slip condition on the rigid disk are used to derive sets of dual integral equations for the fluid velocity and pressure. The dual integral equations are solved by analytic reduction to sets of linear algebraic equations. An asymptotic approximation for the far-field scattered pressure is given, and this approximation is compared to results of previous inviscid acoustic analyses. It is shown that our results for the force on the disk and the far-field scattered pressure are consistent with the prediction of the theory of aerodynamic sound. Numerical results are presented for the fluid velocity field in the case of tangential incidence. The velocity field near the disk is shown to contain vortices that are swept along the disk with the passage of the inc...

The general solution of the three-dimensional acoustic equation and of Maxwell's equations in the infinite domain in terms of the asymptotic solution in the wave zone

The general solution of the three‐dimensional scalar wave equation (or acoustic equation) and of Maxwell’s equations in the infinite spatial domain is given in terms of the asymptotic forms for large times in the future and in the past, or, equivalently, in terms of the fields in the wave zone. One is therby able to obtain the exact solutions from arbitrary solutions in the wave zone. It is shown that the exact fields computed from an arbitrary wave zone solution always satisfy an initial value problem, and that, therefore, they are always physical. In contrast to earlier derivations of related results which required the use of Radon transforms and the introduction of somewhat sophisticated geometrical concepts, the derivations are simple and use only elementary properties of the Fourier transform.

Acoustic diffraction by a half-plane in a viscous fluid medium

We consider the diffraction of a time-harmonic acoustic plane wave by a rigid half-plane in a viscous fluid medium. The linearized equations of viscous fluid flow and the no-slip condition on the half-plane are used to derive a pair of disjoint Wiener-Hopf equations for the fluid stresses and velocities. The Wiener-Hopf equations are solved in conjunction with a requirement that the stresses are integrable near the edge of the half-plane. Specific wave components of the scattered velocity field are given analytically. A Padé approximation to the Wiener-Hopf kernel function is used to derive numerical results that show the effect of viscosity on the velocity field in the immediate vicinity of the edge of the half-plane.

Vortex pairs and dipoles

Point vortices have been extensively studied in vortex dynamics. The generalization to higher singularities, starting with vortex dipoles, is not so well understood.We obtain a family of equations of motion for inviscid vortex dipoles and discuss limitations of the concept. We then investigate viscous vortex dipoles, using two different formulations to obtain their propagation velocity. We also derive an integro-differential for the motion of a viscous vortex dipole parallel to a straight boundary.