Waves on a Variable Beam
We consider here the equations for a non-uniform free beam. We begin by presenting the theory for a uniform beam.
An example of the motion for a non-uniform beam is demonstrated below:
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
where is the non dimensionalised flexural rigidity, and is non-dimensionalised linear mass density function. Note that this equations simplifies if the plate has constant properties (and that is the thickness of the plate, is the pressure and 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).
at the edges of the plate.
The problem is subject to the initial conditions
Solution for a uniform beam in eigenfunctions with no pressure
If the beam is uniform the equations can be written as
We can express the deflection as the series
where are the Eigenfunctions for a Uniform Free Beam and where are the eigenfunctions.
Then and can be found using orthogonality properties:
Note that we cannot give the plate an initial velocity that contains a rigid body motions which is why the sum starts at for time derivative.
where is known as the energy functional, and is some constraint.
In the case of a uniform beam with zero transverse load, our energy functional is (Lanczos 1949):
If we choose
Then minimizing the Rayleigh Quotient can be expressed as a variational problem
which will in turn, also solve our eigenvalue problem.
We can essentially replace the variational problem of finding a that extremises to finding a set of constants which extremise , through using the Rayleigh-Ritz method.
for all .
In our example of non-uniform beams, the assumption is made that we are able to approximate the eigenfunctions for a non-uniform beam as a linear combination of the eigenfunctions for a linear beam:
Solution for a Non-uniform free beam in eigenfunctions
In the case of a non-uniform free beam, the Euler-Bernoulli beam equation becomes:
Using separation of variables, where , we obtain a corresponding eigenfunction problem (with eigenvalues denoted by ):
which leads us to the following variational expression:
We can approximate this using Rayleigh-Ritz and obtain:
Then extremising as follows
for all , allows us to obtain:
for all (here denotes both the Lagrange multiplier and eigenvalues of a non-uniform beam)
Converting to Matrix Form
If we define
then for all :
Let us create the matrix , with elements , the matrix , with elements , the sparse matrix with terms on the diagonal, and the vector . We can consequently express the above sum in the following way:
So we can solve for in any of the problems below to obtain coefficients :
Rather than tediously evaluating an integral for each element of the matrices and , we can break up the integral into subintervals with weights using Composite Simpson's Rule:
So we can express as follows:
and weights are defined as: .
We can extend this concept to the the full matrix if we form:
For we use an identical approach:
Using the MATLAB program eig.m, we can solve the eigenvalue problem
using the either of the commands
[Ai_values,nonuniform_eigvals] = eig(M^(-1)*K)
[Ai_values,nonuniform_eigvals] = eig(K,M)
Where the diagonal of nonuniform_eigvals denotes the eigenvalues for the nonuniform beam (), and each column of the matrix Ai_values represents the coefficients corresponding to :
We can obtain the matrix by taking Xhat_mat= Ai_valuesTX_mat.
Non-uniform beam revisited
We have already solved the eigenfunction problem. We now turn our attention to the time component arising from separation of variables:
which has solutions of the form
Introducing the initial conditions
Then using the the first of these initial conditions we obtain:
multiplying by and integrating allows us to obtain:
Then using the second initial condition, and applying the same technique yields
where and are the two rigid modes of the non-uniform beam. Note that and do not exist.
A program to solve for a variable beam can be found here variable_beam.m
This program requires