Eigenfunctions for a Uniform Free Beam

Introduction

We show here how to find the eigenfunction for a beam with free edge conditions.

Equations

We can find eigenfunctions which satisfy

$\partial_x^4 X_n = \lambda_n^4 X_n \,\,\, -L \leq x \leq L$

plus the edge conditions of zero bending moment and shear stress

$\begin{matrix} \partial_x^3 X_n= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L, \end{matrix}$
$\begin{matrix} \partial_x^2 X_n = 0 \;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L. \end{matrix}$

Solution

General solution of the differential equation for $\lambda \neq 0$ is

$X_n(x) = C_1 \sin(\lambda_n x) + C_2 \cos(\lambda_n x) + C_3 \sinh(\lambda_n x) + C_4 \cosh(\lambda_n x)\,$

Due to symmetry of the problem, dry natural vibrations of a free beam can be split into two different sets of modes, symmetric (even) modes and skew-symmetric (odd) modes.

Modes for $\lambda = 0$

There are two modes for $\lambda = 0$ which are the two rigid body motions; they are given by

$X_0 = \frac{1}{\sqrt{2L}}$

and

$X_1 = \sqrt{\frac{3}{2L^3}} x$

Symmetric modes

$C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)$

By imposing boundary conditions at $x = L$ :

$\begin{bmatrix} - \cos(\lambda_n L)&\cosh(\lambda_n L)\\ \sin(\lambda_n L)&\sinh(\lambda_n L)\\ \end{bmatrix} \begin{bmatrix} C_2\\ C_4\\ \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ \end{bmatrix}$

For a nontrivial solution one gets:

$\tan(\lambda_n L)+\tanh(\lambda_n L)=0\,$

The first three roots are :

$\lambda_0 L = 0, \lambda_2 L = 2.365, \lambda_4 L = 5.497\,$

Symmetric natural modes can be written in normalized form as :

$X_{2n}(x) = \frac{1}{\sqrt{2L}}\left( \frac{\cos(\lambda_{2n} x)}{\cos(\lambda_{2n} L)}+\frac{\cosh(\lambda_{2n} x)}{\cosh(\lambda_{2n} L)} \right ) \,\,\,n\geq 1$

where the The symmetric modes have been normalised so that their inner products equal the Kronecker delta.

Anti-symmetric modes

$C_2 = C_4 = 0 \Rightarrow w_n(x) = C_1 \sin(\lambda_n x) + C_3 \sinh(\lambda_n x)$

By imposing boundary conditions at $x = L$ :

$\begin{bmatrix} - \sin(\lambda_n L)&\sinh(\lambda_n L)\\ -\cos(\lambda_n L)&\cosh(\lambda_n L)\\ \end{bmatrix} \begin{bmatrix} C_1\\ C_3\\ \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ \end{bmatrix}$

For a nontrivial solution one gets:

$-\tan(\lambda_n L)+\tanh(\lambda_n L)=0\,$

The first three roots are :

$\lambda_1 L = 0, \lambda_3 L = 3.925, \lambda_5 L = 7.068\,$

Anti-symmetric natural modes can be written in normalized form as :

$X_{2n+1}(x) = \frac{1}{\sqrt{2L}}\left( \frac{\sin(\lambda_{2n+1} x)}{\sin(\lambda_{2n+1} L)}+\frac{\sinh(\lambda_{2n+1} x)}{\sinh(\lambda_{2n+1} L)} \right ) \,\,\,n\geq 1$

where the eigenfunctions have been chosen so that their inner products equal the Kronecker delta.

Equations for a beam

There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the Bernoulli-Euler Beam theory (other beam theories include the Timoshenko Beam theory and Reddy-Bickford Beam theory where shear deformation of higher order is considered). For a Bernoulli-Euler Beam, the equation of motion is given by the following

$\partial_x^2\left(\beta(x)\partial_x^2 \zeta\right) + \gamma(x) \partial_t^2 \zeta = p$

where $\beta(x)$ is the non dimensionalised flexural rigidity, and $\gamma$ is non-dimensionalised linear mass density function. Note that this equations simplifies if the plate has constant properties (and that $h$ is the thickness of the plate, $p$ is the pressure and $\zeta$ is the plate vertical displacement) .

The edges of the plate can satisfy a range of boundary conditions. The natural boundary condition (i.e. free-edge boundary conditions).

$\partial_x^2 \zeta = 0, \,\,\partial_x^3 \zeta = 0$

at the edges of the plate.

The problem is subject to the initial conditions

$\zeta(x,0)=f(x) \,\!$
$\partial_t \zeta(x,0)=g(x)$

Solution for a uniform beam in eigenfunctions

If the beam is uniform the equations can be written as

$\beta \frac{\partial^{4}\zeta}{\partial x^{4}} + \gamma \frac{\partial^{2}\zeta}{\partial t^{2}}=0$

We can express the deflection as the series

$\zeta(x,t)=\sum_{n=0}^{\infty} A_n X_n(x) \cos(k_n t) + \sum_{n=2}^{\infty}B_n X_n(x) \frac{\sin(k_n t)}{k_n}$

where $X_n$ are the Eigenfunctions for a Uniform Free Beam and $k_m = \lambda^2_n \sqrt{\beta/\gamma}$ where $\lambda_n$ are the eigenfunctions.

Then $A_n \,\!$ and $B_n \,\!$ can be found using orthogonality properties:

$A_n=\frac{\int_{-L}^{L}f(x)X_n(x)\mathrm{d}x}{\int_{-L}^{L}X_n(x)X_n(x)\mathrm{d}x} \,\!$
$B_n=\frac{\int_{-L}^{L}g(x)X_n(x)\mathrm{d}x}{\int_{-L}^{L}X_n(x)X_n(x)\mathrm{d}x}$

Note that we cannot give the plate an initial velocity that contains a rigid body motions which is why the sum starts at $n=2$ for time derivative.

Matlab Code

A program to calculate the eigenvalues can be found here beam_ev.m

A program to calculate the eigenvectors can be found here beam_em.m