Template:Equations for a beam: Difference between revisions

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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
There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the [http://en.wikipedia.org/wiki/Euler_Bernoulli_beam_equation Bernoulli-Euler Beam] theory (other beam theories include the [http://en.wikipedia.org/wiki/Timoshenko_beam_theory 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
by the following
<center><math>
<center><math>
\partial_x^2\left(D\partial_x^2 \zeta\right) + \rho_i h \partial_t^2 \zeta = p
\partial_x^2\left(\beta(x)\partial_x^2 \zeta\right) + \gamma(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>\beta(x)</math> is the non dimensionalised [http://en.wikipedia.org/wiki/Flexural_rigidity flexural rigidity], and <math>\gamma </math> is non-dimensionalised linear mass density function.
<math>h</math> is the thickness of the plate, <math> p</math> is the pressure
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. Note that this equations simplifies if the plate has constant
and <math>\zeta</math> is the plate vertical displacement)
properties.   
.   


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


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
The problem is subject to the initial conditions
<math>\zeta(x,t) = e^{i\omega t} \bar{\zeta}(x) </math> from linearity. In this case the equations reduce to
<center>
 
:<math>   \zeta(x,0)=f(x) \,\! </math>
<center><math>
:<math>   \partial_t \zeta(x,0)=g(x)  </math></center>
\partial_x^2\left(D\partial_x^2 \bar{\zeta}\right)  -\omega^2 \rho_i h \bar{\zeta} = \bar{p}
</math></center>

Latest revision as of 23:39, 2 July 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(\beta(x)\partial_x^2 \zeta\right) + \gamma(x) \partial_t^2 \zeta = p }[/math]

where [math]\displaystyle{ \beta(x) }[/math] is the non dimensionalised flexural rigidity, and [math]\displaystyle{ \gamma }[/math] is non-dimensionalised linear mass density function. Note that this equations simplifies if the plate has constant properties (and that [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) .

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]
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