Difference between revisions of "Template:Equations for a beam"

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For a [http://en.wikipedia.org/wiki/Euler_Bernoulli_beam_equation Bernoulli-Euler Beam] on the surface of the water, the equation of motion is given
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For a [http://en.wikipedia.org/wiki/Euler_Bernoulli_beam_equation Bernoulli-Euler Beam], the equation of motion is given
 
by the following
 
by the following
 
<center><math>
 
<center><math>
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</math></center>
 
</math></center>
 
at the edges of the plate.
 
at the edges of the plate.
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If we assume that the pressure is of the form <math>p(x,t) = e^{i\omega t} \bar{p}(x)</math> then it follows that
 
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<math>\zeta(x,t) = e^{i\omega t} \bar{\zeta}(x) </math> from linearity. In this case the equations reduce to
 
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<center><math>
 
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\partial_x^2\left(D\partial_x^2 \bar{\zeta}\right)  -\omega^2 \rho_i h \bar{\zeta} = \bar{p}
 
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</math></center>
 

Revision as of 21:22, 24 March 2009

For a Bernoulli-Euler Beam, the equation of motion is given by the following

[math]\displaystyle{ \partial_x^2\left(D\partial_x^2 \zeta\right) + \rho_i h \partial_t^2 \zeta = p }[/math]

where [math]\displaystyle{ D }[/math] is the flexural rigidity, [math]\displaystyle{ \rho_i }[/math] is the density of the plate, [math]\displaystyle{ h }[/math] is the thickness of the plate, [math]\displaystyle{ p }[/math] is the pressure and [math]\displaystyle{ \zeta }[/math] is the plate vertical displacement. Note that this equations simplifies if the plate has constant properties.

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

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

at the edges of the plate.

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