Template:Equations for a beam

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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]