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

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by the following
 
by the following
 
<center><math>
 
<center><math>
\partial_x^2\left(D(x)\partial_x^2 \zeta\right) + \rho_i h(x) \partial_t^2 \zeta = p
+
\partial_x^2\left(D(x)\partial_x^2 \zeta\right) + m(x) \partial_t^2 \zeta = p
 
</math></center>
 
</math></center>
where <math>D</math> is the flexural rigidity, <math>\rho_i</math> is the density of the plate,
+
where <math>D = EI</math> is the flexural rigidity (<math>E</math> is the Youngs modulus, <math>I</math> is the
<math>h</math> is the thickness of the plate, <math> p</math> is the pressure
+
moment of inertia), <math>m</math> is the mass (<math>\rho_i</math> is the density of the plate,
 +
<math>h</math> is the thickness of the plate), <math> p</math> is the pressure
 
and <math>\zeta</math> is the plate vertical displacement. Note that this equations simplifies if the plate has constant
 
and <math>\zeta</math> is the plate vertical displacement. Note that this equations simplifies if the plate has constant
 
properties.   
 
properties.   
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<center>
 
<center>
 
:<math>  \zeta(x,0)=f(x) \,\! </math>
 
:<math>  \zeta(x,0)=f(x) \,\! </math>
:<math>  \frac{\partial \zeta(x,0)}{\partial t}=g(x)  </math></center>
+
:<math>  \partial_t \zeta(x,0)=g(x)  </math></center>

Revision as of 08:38, 7 April 2009

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

where [math]\displaystyle{ D = EI }[/math] is the flexural rigidity ([math]\displaystyle{ E }[/math] is the Youngs modulus, [math]\displaystyle{ I }[/math] is the moment of inertia), [math]\displaystyle{ m }[/math] is the mass ([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.

The problem is subject to the initial conditions

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