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The Mathieu analysis is also used to investigate the stability of the forced periodic vibration of systems with nonlinear stiffness.. However, for weakly nonlinear systems, the main engineering interest. A solutions manual is being prepared and it is hoped that this will be available shortly after the book is published. Details. This book presents a collection of exact solutions for vibration of strings, beams (bars), and plates, which are very common both in engineering and natural science.. It focuses on free vibrations for analysis and design of structures and machines and presents the exact vibration solutions for strings, membranes, beams, and. [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] Progress in Civil, Architectural and Hydraulic Engineering IV – Kim. Harik I.E., Liu X., Balakrishnan N. An analytical solution to free vibration of rectangular plates. Journal of Sound and Vibration, 1992, 153(1):51–62. Hwang S.S. 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