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Chapter 1 gives the aims and scopes of the book. Chapter 2 is the literature review. Chapter 3 develops an alternative approach to predict natural frequencies of whirling rods. Nonlinear steady equations of whirling rods are formulated. An original numerical method is applied to solve for rod shape profiles under whirling motion. Classical receptance approach is then applied to solve for the natural frequencies from the deformed geometry of the whirling rod. Conditions for existence and uniqueness of solutions are derived. Chapter 4 investigates in-plane vibration of annuli. Two different model equations that govern rotational motions of annuli are scrutinized. Solutions of the equations assume small oscillations of vibration being superimposed on the steady state of the annulus while it is rotating. Exact and approximate solutions are obtained, where the approximate solutions are acquired by ignoring the Coriolis effect. A proposed numerical scheme is implemented to solve the governing equations coupled with radial and circumferential displacements. Chapter 5 provides critiques on current work. Finally, Chapter 6 is the summary.