Single Degree of Freedom System:
Degrees of freedom, undamped system, springs in parallel, in series. Newton’s laws of motion, free body diagrams. D’Alembert’s principle, solution of the differential equation of motion, frequency and period, amplitude of motion. Damped Single degree of freedom system – viscous damping, equation of motion, critically damped system, over damped system, under damped system, and logarithmic decrement. Response of single degree of freedom system to harmonic loading – undamped harmonic excitation, damped harmonic excitation, evaluation of damping at resonance, bandwidth method (Half power) to evaluate damping, response to support motion, force transmitted to the foundation, seismic instruments.
Response to General Dynamic Loading
Impulsive loading and Duhamel’s integral,numerical evaluation of Duhamel’s integral, un-damped system, numerical evaluation of Duhamel’s integral, damped system. Fourier analysis and response in frequency domain – Fourier analysis, Fourier coefficient for piece-wise liner functions, exponential form of Fourier series, discrete Fourier analysis, fast Fourier transform.
Generalised Co-ordinates and Rayleigh’s method
Principle of virtual work, generalized single degree of freedom system (rigid body and distributed elasticity), Raylegh’s method. Multistory Shear Building. Free vibration – natural frequencies and normal modes, Zero modes of vibration. Forced motion – modal superposition method – response of a shear building to base motion. Damped motion of shear building – equations of motions – uncoupled damped equation – conditions for uncoupling. Damping.
Discretization of Continuous Systems
Longitudinal Vibration of a uniform rod. Transverse vibration of a pre-tensioned cable. Free transverse vibration of uniform beams – Rotary inertia and shear effects – The effect of axial loading. Orthogonality of normal modes. Undamped forced vibration of beams by mode superposition.
Dynamic Analysis of Beams
Stiffness matrix, mass matrix (lumped and consistent); equations of motion for the discretiesed beam in matrix form and its solutions.