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Final Project in course, Computational Physics 118094

Numerical Analysis of PDEs: A Finite Difference Method for the Wave Equation, Damped Wave Equation and Free Vibration of Beam

Introduction

There are many numerical methods for solving partial differential equations. Finite Difference method will be used to solve PDEs with given initial condition. In order to simplify the calculations, the following assumptions are made:

The following cases will be discussed:

Wave Equation

It is convenient to rescale equations to reduce to a problem with dimensionless parameters.

Damped Wave equation

Real strings are subject to a damping force which accounts for dissipation of energy in the form of heat. As a first approximation, the damping force is proportional to the velocity of the string, so the wave equation is modified to become

It is obvious that if b=0 we have standard wave equation.

Free Vibration of Beam (Modeling of CNT Dynamics)

Although CNTs (Carbon Nano-Tubes) can have diameter only several times larger than the length of a bond between carbon atoms, continuum models have been found to describe their mechanical behavior very well under many circumstances. Dynamics of CNT can be described with the following equation:

The derivatives of u with respect to x have physical meaning.

This equation is derived assuming that displacements are small and that sections of the beam normal to the central axis in the unload state remain normal during bending: assumptions are usually valid for small deformations of long, thin beam. It is convenient to rescale our governing equation to reduce to dimensionless form. We scale axial coordinate with the length of the beam, time with natural time scale of vibration, and capture deflection with some characteristic displacement.

It is common when dealing with vibration behavior of nanotubes, to neglect the tension term due to the fact that for a small deformation the tension is much smaller than flexural rigidity.

Finite Difference Method will be used later to solve this equation.