Dynamics of Structures and Finite Elements

Introduction to structural dynamics. Vibrations of single DOF systems: free vibrations of undamped and damped system, response to harmonic excitation, response to general loading. Undamped vibrations of multiple DOF systems: free vibrations, natural frequencies and natural modes, response to general loading, modal analysis. Damped vibrations of multiple DOF systems, modal damping assumption (Rayleigh damping). Elements of analytical dynamics, variational formulation in dynamics: virtual work principle, kinematic constraints, generalized coordinates, Hamilton’s principle and Lagrange’s equation. Continuous systems: rods, beams, thin plates. Application of Hamilton’s principle. Examples: beams Euler-Bernoulli and Timoshenko, beams on elastic foundation, effect of axial forces on beam’s vibration, vibration of rotating beams, stability of rotating shafts, tubes with flowing fluid. Order reduction techniques, approximation of continuous systems: Rayleigh’s-Ritz method, assumed modes method, Galerkin’s method. Structural connections, direct stiffness method, consistent and lumped formulations, condensation. Direct time integration methods for dynamic response: basic formula for time integration.