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

From WikiWaves
Jump to navigationJump to search
(changing notation)
 
(7 intermediate revisions by 2 users not shown)
Line 1: Line 1:
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] (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).  
+
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
 
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).  
Line 15: Line 15:
 
</math></center>
 
</math></center>
 
at the edges of the plate.
 
at the edges of the plate.
 +
 +
The problem is subject to the initial conditions
 +
<center>
 +
:<math>  \zeta(x,0)=f(x) \,\! </math>
 +
:<math>  \partial_t \zeta(x,0)=g(x)  </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]