A stability analysis of a nonlinear plate clamped to an infinite baffle in mean flow is given. The effect of structural nonlinearities induced by in-plane forces and shearing forces due to stretching of plate bending motion, and that of viscous damping are taken into account in the derivation of the plate equation. The plate flexural displacement is obtained by modal expansion based on Galerkin's method. The critical mean flow speeds at which local instabilities may occur are determined by Routh algorithm. The mechanisms that trigger the local instabilities are uncovered. The effect of structural nonlinearities, and that of plate aspect and plate length/thickness ratios on local instabilities are examined. Numerical examples of the transition from stable to locally unstable vibration, as the mean flow speed exceeds the critical values, are demonstrated. The results show that while the overall amplitude of the plate flexural displacement may be bounded when the mean flow speed exceeds the critical values, plate vibration may be locally unstable, jumping from one equilibrium position to another. Furthermore, the jumping may be random, and the plate vibration may seem chaotic. The results also show that viscous damping may stabilize plate flexural vibration and settle the plate in one of its equilibria.
Formulations are derived for predicting sound radiation from two semi-infinite dissimilar plates subject to a line force excitation at the joint in the presence of mean flow using the Wiener–Hopf technique and Fourier transformations. The acoustic pressure is solved in the frequency-wave-number domain first in terms of the decomposition factors and parameters associated with displacements, slopes, bending moments, and shear forces at the joint of two plates. The decomposition factors are evaluated via contour integrations, and the parameters are determined by a set of simultaneous equations derived from boundary conditions and the finiteness requirement imposed at the joint. It is shown that the set of equations reduces to a 2×2 matrix for a welded joint, a 3×3 matrix for a hinged joint, and a 4×4 matrix for a free–free joint, i.e., two plates being mechanically unconnected. The frequency-domain acoustic pressure is subsequently obtained by taking an inverse Fourier transformation and evaluated by using the residue theory and contour integrals along the branch cut. Asymptotic behaviors of the plate flexural displacement and radiated acoustic pressure in the frequency-wave-number domain are obtained. Effects of mean flow on resulting sound radiation are examined.
Asymptotic formulations are derived for describing far-field acoustic radiation from two semi-infinite dissimilar plates subject to an harmonic line force excitation at the joint in the presence of mean flow. Analysis shows that mean flow affects the acoustic pressure in three ways: (1) It enhances the wave number and amplitude of an acoustic wave propagating in the upstream direction, while it suppresses those of an acoustic wave propagating in the downstream direction; (2) it reduces the decay rate of the upstream propagating wave, while it increases that of the downstream propagating wave as they propagate in the plate normal direction; and (3) it rotates the radiation beam angles toward the downstream direction. The effects of mean flow are obvious when the excitation frequency is above the plate coincidence frequency, but decays significantly at low frequencies. The condition at the joint of two plates does not change the characteristics of the radiation pattern, but merely the amplitude of the radiated acoustic pressure. The looser the joint, for example, two plates being mechanically unconnected, the higher the amplitude of the resulting acoustic pressure.
Asymptotic solutions to the dynamic and acoustic responses of a fluid-loaded infinite plate subject to a time-dependent line force excitation in mean flow are obtained. Results show that mean flow has a significant effect on the plate traveling waves, the modification factor being α2(1∓M)8 when the excitation frequency is above the plate coincidence frequency, where α is the fluid-loading factor, M is the mean flow Mach number, and minus and plus signs indicate upstream and downstream propagating waves, respectively. At the coincidence frequency, the amplitudes of traveling waves are modified by mean flow by (c0+c1M+c2M2), where cj, j=0, 1, and 2 are α-dependent constants. For light fluid loading, for example, an air/steel interface, α≊6.5×10−4, the leading term of the traveling wave is proportional to α−4/3M2. For heavy fluid loading, for example, a water/steel interface, α≊0.133, the leading term is proportional to α−2/3M. Below the coincidence frequency, the effect of mean flow on plate traveling waves is insignificant. Mean flow also changes the radiated acoustic pressures. In particular, the amplitudes of the leaky waves are modified by mean flow, to the leading order of a small value of α, by (√Ω±M)2[4Ω√(Ω±M)2−1]−1, where Ω is the ratio of the excitation frequency to the plate coincidence frequency. The effect of mean flow on the cylindrical waves depends on Ω. When Ω≳1, the peak amplitudes of the cylindrical waves are modified by mean flow by √(1−Ω−1)±2Ω−1/2M−M2, and the beam angles are rotated by θb=sin−1(M∓Ω−1/2). When Ω<1, the effect of mean flow on the cylindrical waves is greatly decreased. Mean flow modifies the wave numbers of the traveling waves and acoustic waves as well. As a result, the waves propagating in the upstream direction are squeezed and those in the downstream direction are relaxed in space.
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