Interpolation of transfer functions for damped vibrating systems
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This thesis presents methods for interpolating transfer functions of damped vibrating systems. Primary applications lie in the design and control of damped structures. The interpolations reduce the number of frequencies at which the transfer function must be computed or measured. The transfer functions are assumed to have impulse responses that are real-valued and causal, so a method is developed for constructing interpolations that implicitly satisfy these conditions. The method is applied to a particular choice of basis function that corresponds to a Fourier series in the time domain. Numerical results indicate that satisfaction of the causality condition increases the accuracy of the interpolation. A detailed investigation is made into interpolations for viscously damped systems, whose transfer functions are linear combinations of basis functions derived from the complex-valued eigenpairs of the system. Since the estimation of all eigenpairs is computationally expensive, a method is developed to estimate only those eigenpairs that significantly contribute to the transfer function in the specified frequency band. The method uses eigenvalues of the corresponding undamped system, which are much easier to compute, as starting guesses in an iterative algorithm. One advantage of the method is the assurance that it finds all eigenvalues in a specified region of the complex plane.
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