Difference between revisions of "Template:Equations for a beam"
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\partial_x^2\left(D(x)\partial_x^2 \zeta\right) + m(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 = EI</math> is the flexural rigidity (<math>E</math> is the | + | where <math>D = EI</math> is the [http://en.wikipedia.org/wiki/Flexural_rigidity flexural rigidity] (<math>E</math> is the [http://en.wikipedia.org/wiki/Young_modulus Young's modulus], <math>I</math> is the |
| − | moment of inertia), <math>m </math> is the mass per unit length. | + | [http://en.wikipedia.org/wiki/Moment_of_inertia moment of inertia]), <math>m </math> is the mass per unit length. |
<math>h</math> is the thickness of the plate), <math> p</math> is the pressure | <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 | ||
Revision as of 09:35, 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
where [math]\displaystyle{ D = EI }[/math] is the flexural rigidity ([math]\displaystyle{ E }[/math] is the Young's modulus, [math]\displaystyle{ I }[/math] is the moment of inertia), [math]\displaystyle{ m }[/math] is the mass per unit length. [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).
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]
