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=Problem 7.2 Static and Dynamic Finite Element Modelling and Analysis for Vibrating Membrane=

Given: Problem defined on [[media:fe1.s11.mtg40.djvu|Mtg 40 (c)]]
Strong Form

Essential boundary conditions,

Find
1. Find the static solution $$ \quad U_{s}^h $$ to $$ \quad 10^{-6} $$ accuracy at x = 0.

2. Dynamic (Transient) Solution

2a. Free Vibration Analysis

2b. Plot $$ u^h(0,t) \ vs \ t \quad  for \ t \ \epsilon \ [0,2T_1]$$  and produce a movie of the vibrating membrane for symmetric

2c. Plot $$ u^h(0,t) \ vs \ t \quad  for \ t \ \epsilon \ [0,2T_1]$$  and produce a movie of the vibrating membrane for non-symmetric

i. Weak Form
From the PDE the weak form can be written as: Now from the theory of Integration by parts, we can write

Hence Eq. 2.1 can be now written as

Now we can write $$\displaystyle d\Omega $$ as $$\displaystyle d\Omega ={{\Gamma }_{g}}\cup {{\Gamma }_{h}}$$ , where $$\displaystyle {{\Gamma }_{g}}$$ = essential boundary condition. $$\displaystyle {{\Gamma }_{h}}$$ = natural boundary condition.

So by applying the Gauss theorem on the first term we obtain,

We select $$\displaystyle w $$ such that $$\displaystyle w=0 $$ on $$\displaystyle {{\Gamma }_{g}}$$

This Equation is of the form:- where,

ii. Finding approximate solution
Approximated solution $$ u^{h} \left ( x \right ) $$ and $$ w^{h} \left ( x \right ) $$ :

where $$ \left ( c_{i} \right ) $$ and $$ \left ( d_{j} \right ) $$ are constants and $$ \left ( b_{i} \right ) $$ and $$ \left (b_{j}\right ) $$ are the LIBF. We will use these LIBF as a shape function satisfying the CBS.

2DLIBF are given by

$$L_{i,m}$$ is the Lagrange basis function along x and $$L_{i,m}$$ is the Lagrange basis function along y given by

The LIBF be defined by a function

where,

Capacitance matrix

Conductivity matrix

Force matrix

1. Static solution
We have strong form of vibrating membrane equation (Eq. 2.1) where T = 4, $$f\left( x \right){\text{  =  1}}$$,  $$\quad $$   $$\quad \rho = 3 $$,  $$\quad $$ $$\qquad\frac{\partial^2u}{\partial t^2}=0$$

we can also write for 2D

$$\frac{{{\partial }^{2}}u}{\partial x_{i}^{2}}= \frac{\partial^2u}{\partial x_{1}^2} + \frac{\partial^2u}{\partial x_{2}^2}$$

If we rearrange Eq 2.1 for static case conditions, we get

2. Dynamic solution
$$ \quad \rho \ = \ 3$$

2a. Free Vibration Analysis
The equation is,

Where,

$$ \quad \lambda - Eigenvalues $$

$$\quad \mathbf{\phi} - Eigenvectors $$

We calculate eigenvalues and then put them into the equation below to find frequencies.

The fundamental frequency is $$\omega _{1}=\lambda_{1}^{1/2}$$, and  $$f_{1}=\frac{\omega _{1}}{2\Pi }$$. The period T is $$T_{1}=\frac{1}{f_{1}}$$

2b. Plotting displacement and the movie of the vibrating membrane for symmetric
Initial conditions:

2c. Plotting displacement and the movie of the vibrating membrane for non-symmetric
Initial conditions: