Finite Element Solution of Time Dependent Partial Differential Equation in Structural Dynamics

Abstract

This thesis investigates the Finite Element Method (FEM) solution of time dependent Par-tial Differential Equations (PDEs) in structural dynamics, focusing on the transient free vibration of a simply supported elastic cable governed by the one-dimensional wave equa-tion. The primary objective is to numerically model the dynamic behaviour of the structure under various initial conditions and rigorously validate the numerical accuracy, stability, and convergence properties of the selected time integration schemes. The study analysed two distinct vibration modes across three problem configurations: 1. First Mode Vibration (Problems 2 and 3) The cable was initiated in its first mode shape, f(x, 0) = sin(pi*x) and released from rest with a wave speed of c = 2m / s This configuration was analyzed using two explicit time step sizes, k = 0.0495 and k = 0.05 to investigate the critical stability condition. 2. Second Mode Vibration (Problem 1) The string was excited in its second mode shape using prescribed initial displacement and velocity conditions, f(x, 0) = sin(2pi*x) partial f partial t (x, 0) = 2pi * sin(2pi*x) . The wave speed for this case was c = 1m / s