This course is designed for graduate students. It is aimed to teach the fundamental concepts how continuous systems vibrate. Fundamental aspects of vibrations for mathematical modeling, derivation/solution of boundary value problem, and subsequent system analysis will be covered both using analytical and approximate methods: 1) Advanced principles of dynamics (Elasticity and strain energy, generalized coordinates, Hamilton’s principle) 2) General Formulation of Natural Modes of Vibration in Continuous System (Boundary value problem, eigenvalue problem, orthogonality) 3) Natural Modes of Vibration in Continuous Systems (vibration of strings, longitudinal vibration of beams, bending vibrations of beams, vibration of membranes, expansion and enclosure theorems, Rayleigh’s quotient) 4) Natural Modes of Vibration in Continuous Systems using Approximate Methods (Rayleigh’s energy method, Rayleigh-Ritz method, Assumed modes method, Galerkin’s method) 5) Response of Undamped Continuous Systems (modal analysis, assumed modes method, Galerkin’s method)