In this paper we investigate a class of combined discrete-continuous mechanical systems consisting of a continuous elastic structure and a finite number of concentrated masses, elastic supports, and linear oscillators of arbitrary dimension. After the motion equations for such combined systems are derived, they are formulated as an abstract evolution equation on an appropriately defined Hilbert space. Our main objective is to ascertain conditions under which the combined systems have classical normal modes. Using the sesquilinear form approach, we show that unless some matching conditions are satisfied, the combined systems cannot have normal modes even if Kelvin-Voigt damping is considered.
On the Existence of Normal Modes of Damped Discrete-Continuous Systems
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Banks, H. T., Luo, Z., Bergman, L. A., and Inman, D. J. (December 1, 1998). "On the Existence of Normal Modes of Damped Discrete-Continuous Systems." ASME. J. Appl. Mech. December 1998; 65(4): 980–989. https://doi.org/10.1115/1.2791942
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