1d wave equation - physicsforums.com

1D Wave EquationThe derivation and solution for modeling a fixed-free cantilever

beam

Nick BrunoMAT 661

Spring 2010

If time… Extension to voltage in polyvinylidene fluoride (PVDF) strip

IntroductionThe lateral response of a vibrating cantilever beam is studied using the

wave equation by:

Deriving a fourth order wave equation considering a free body diagram.

Solving the equation using the technique of separation of variables with a Fourier expansion.

Finding orthogonal eigenfunctions and producing eigenvalues using root finding.

Applying initial conditions which are used to determine coefficients which yield a complete system time response.

2Bruno

Introduction

3Bruno

PZT - Lead zirconate titanate (MFC)

Derivation

A one dimensional beam will be considered with an applied distributed vertical load as seen below:

4Bruno

Derivation

Summing all the forces in the z direction and moments about the origin “O” results in the following equations:

5Bruno

We also know:

Derivation

Making substitutions with the latter yields:

6Bruno

Which simplifies to a fourth order PDE for a uniform beam…

orwhere

Free vibration

Separation of VariablesLetting the response equal the product of independent

functions, separating the variables, and setting them equal to a positive separation constant yields the following:

7Bruno

Application of Boundary Conditions

The boundary conditions for a cantilever beam are defined by:

8Bruno

No Displacement @ x =0 No Slope @ x =0 No Bending

Moment @ x = LNo Shear Force @ x = L

These translate to the following for our eigenvalue problem:

We will use these

Application of Boundary Conditions

This results in a system of equations that can be used to determine the eigenvalues and a relation between coefficients C1 and C2

9Bruno

In order for a non-trivial solution to be found

so equating zero with the determinant of the 2X2 matrix seen will yield eigenvalues for the system

Application of Boundary Conditions

10Bruno

Application of Boundary Conditions

The relation between C1 and C2 can be found as follows:

11Bruno

Application of Initial ConditionsRemember from before…

12Bruno

The chosen initial conditions for the beam are as follows:

whereδ= 0.004 [m] = Initial displacement at the tipL = 0.107 [m] = Length of the beamthus,

Orthogonality?Before solving for An we must first check orthogonality of the

eigenfunctions.

13Bruno

Check

Example Solution

Use inner products

No weight function!

Assuming convergence

Summary of Data

14Bruno

Summary of Data

15Bruno

Summary of Data

16Bruno

Summary of Data

17Bruno

Summary of Data

18Bruno

Summary of Data

19Bruno

u

ut utt

Summary of Data

20Bruno

Summary of Data

21Bruno

V * %e

Comparable

Sodano et al. 2003