# Template: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

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

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

$\displaystyle{ \partial_x^2 \zeta = 0, \,\,\partial_x^3 \zeta = 0 }$

at the edges of the plate.

The problem is subject to the initial conditions

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