Eigenfunctions for a Uniform Free Beam

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Introduction

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

Equations

We can find eigenfunctions which satisfy

[math]\partial_x^4 X_n = \lambda_n^4 X_n \,\,\, -L \leq x \leq L [/math]

plus the edge conditions of zero bending moment and shear stress

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

Solution

General solution of the differential equation for [math]\lambda \neq 0[/math] is

[math]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)\,[/math]

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 [math]\lambda = 0[/math]

There are two modes for [math]\lambda = 0[/math] which are the two rigid body motions; they are given by

[math] X_0 = \frac{1}{\sqrt{2L}} [/math]

and

[math] X_1 = \sqrt{\frac{3}{2L^3}} x [/math]

Symmetric modes

[math]C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)[/math]

By imposing boundary conditions at [math]x = L[/math] :

[math] \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} [/math]

For a nontrivial solution one gets:

[math]\tan(\lambda_n L)+\tanh(\lambda_n L)=0\,[/math]

The first three roots are :

[math]\lambda_0 L = 0, \lambda_2 L = 2.365, \lambda_4 L = 5.497\,[/math]


Symmetric natural modes can be written in normalized form as :

[math]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 [/math]

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

Anti-symmetric modes

[math]C_2 = C_4 = 0 \Rightarrow w_n(x) = C_1 \sin(\lambda_n x) + C_3 \sinh(\lambda_n x)[/math]

By imposing boundary conditions at [math]x = L[/math] :

[math] \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} [/math]

For a nontrivial solution one gets:

[math]-\tan(\lambda_n L)+\tanh(\lambda_n L)=0\,[/math]

The first three roots are :

[math]\lambda_1 L = 0, \lambda_3 L = 3.925, \lambda_5 L = 7.068\,[/math]

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

[math]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 [/math]

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

[math] \partial_x^2\left(\beta(x)\partial_x^2 \zeta\right) + \gamma(x) \partial_t^2 \zeta = p [/math]

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

[math] \partial_x^2 \zeta = 0, \,\,\partial_x^3 \zeta = 0 [/math]

at the edges of the plate.

The problem is subject to the initial conditions

[math] \zeta(x,0)=f(x) \,\! [/math]
[math] \partial_t \zeta(x,0)=g(x) [/math]

Solution for a uniform beam in eigenfunctions

If the beam is uniform the equations can be written as

[math] \beta \frac{\partial^{4}\zeta}{\partial x^{4}} + \gamma \frac{\partial^{2}\zeta}{\partial t^{2}}=0 [/math]

We can express the deflection as the series

[math] \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} [/math]

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

Then [math] A_n \,\![/math] and [math] B_n \,\![/math] can be found using orthogonality properties:

[math] 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} \,\! [/math]
[math] 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} [/math]

Note that we cannot give the plate an initial velocity that contains a rigid body motions which is why the sum starts at [math]n=2[/math] 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