User:Eml4500.f08.jamama.jan/Meeting29

Initial Conditions  (2 conditions) at t = 0, prescribe:

Condition 1: $$ u(x, t=0) = \bar{u}(x) $$
 * $$ \uparrow $$
 * known function displacement

Condition 2: $$ \frac{\partial u}{\partial t}(x, t=0) = \dot {u}(x, t=0) = \bar{v}(x)$$
 * $$ \uparrow $$
 * known function velocity

PVW (continuous) of dynamics of elastic bar

PDE (Partial Differential Equation):

$$ \frac{\partial}{\partial x}[(EA)\frac{\partial u}{\partial x}] + f = m \ddot u \; \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (1)$$
 * $$ \uparrow $$

Discrete EOM (Equation of Motion)
 * $$\downarrow $$

$$ -\mathbf{M\ddot d} + \mathbf{F} = \mathbf{Kd} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad \qquad \qquad (2)$$


 * $$\Downarrow $$

$$ \mathbf{M\ddot d} + \mathbf{Kd} = \mathbf{F} $$
 * $$ \nwarrow $$
 * Multiple Degree of Freedom System (MDOF)

Single Degree of Freedom System (SDOF)

$$ \Rightarrow \ \mathbf{m\ddot d} + \mathbf{kd} = \mathbf{f} $$

Derive (2) from (1)

$$ \int \limits_0^L ( \frac{\partial}{\partial x}[EA \frac{\partial u}{\partial x}] + f - m \ddot {u} ) dx = 0,$$ $$ \qquad $$   for all possible  $$\mathbf{w(x)} \qquad \qquad \qquad (3)$$
 * $$ \uparrow $$
 * weighting function

$$ (1) \Rightarrow (3) $$  >>  trivial

$$ (3) \Rightarrow (1) $$  >>  not trivial

(3) rewritten as : $$ \int w(x) g(x) dx, \qquad $$ for all $$ \mathbf{w(x)}$$

Since (3) holds for all $$ \mathbf{w(x)}$$, select $$ \mathbf{w(x)} = \mathbf{g(x)}$$, then (3) becomes $$\int \underbrace{g^2(x)}_{\ge 0} dx = 0 \Rightarrow g(x) = 0 $$

{| class="toccolours collapsible collapsed" style="width:100%" ! Matlab Code For Electric Pylon

Matlab Code


results=[]; for i=1:n_elem lm=lmm(i,:); location = [x(node_connect(1,i)),y(node_connect(1,i)); x(node_connect(2,i)),y(node_connect(2,i))]; results = [results; PlaneTrussResults(E,A,location,d(lm))]; end results