The Wave Propagation Method for the Analysis of Boundary Stabilization in Vibrating Structures
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In the modern vibration control of flexible space structures and flexible robots, various boundary feedback schemes have been employed to cause energy dissipation and damping, thereby achieving stabilization. The mathematical analysis of eigenspectrum of vibration is usually carried out by classical separation of variables and by solving the transcendental equations. This involves rather lengthy and tedious work due to the complexity and the numerous boundary conditions. A different approach, developed by Keller and Rubinow, uses ideas from wave propagation to obtain asymptotic estimates of eigenvalues for multidimensional scattering problems. This approach is powerful and yields accurate eigenvalue estimates even at a relatively low frequency range [Ann. Physics, 9 (1960), pp. 24-75]. In this paper, we take advantage of this wave approach to study one-dimensional vibration problems with boundary damping. We decompose vibration waves into incident, reflected (including transmitted) and evanescent waves. Based on their amplitude and reflection coefficients we are able to derive eigenfrequency estimates and compute numerical values. This wave propagation method greatly simplifies the asymptotic estimation procedures and reproduces earlier results in [SIAM J. Appl. Math., 47 (1987), pp. 751-780], [Lecture Notes in Pure and Applied Mathematics, Vol. 108, Marcel Dekker, New York, 1987], [SIAM J. Appl., Math., 49 (1989), pp. 1665-1693]. It also yields new insights into various structural control problems such as feedback with tipped mass , nonrobustness of feedback delays [SIAM J. Control Optim., 24 (1986), pp. 152-156], [SIAM J. Control Optim., 26 (1988), pp. 697-713], noncollocated sensors and actuators, and special medium-low frequency structural damping patterns.
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