User:EML4500.f08.FEABBQ.Hamdan/HW-6

Notes from November-03-08


 * Redo Matlab Rectangular Trusses

Continuing the derivation:

Initial conditions: At t=0, prescribed

$$\left(x,t=0 \right) =\bar{U}\left(x \right)$$ Known Function Displacement $$\frac{\partial }{\partial t}\left(x,t=0 \right)=\dot{U }\left(x,t=0 \right)=\bar{V}\left(x \right)$$Known Function Velocity

$$\frac{\partial }{\partial x}\left[\left(EA \right)\frac{\partial }{\partial x} \right]+f=m\ddot{U}$$ Eq.(1)
 * PVW (continuous) of the dynamics of elastic bar
 * PDE

Discrete EOM $$\Leftarrow $$ Equation of motion. $$-Kd+F=M\ddot{d}$$

$$\frac{\partial }{\partial x}\left[\left(EA \right)\frac{\partial }{\partial x} \right]\Rightarrow -Kd$$

$$f \Rightarrow F$$

$$m\ddot{U} \Rightarrow M\ddot{d} $$

$$M\ddot{d}+Kd=F$$    Eq.(2)

Goal is to derive Eq.(2) from Eq.(1):
 * MDOF $$\Rightarrow$$ Multiple Degree of Freedom system
 * SDOF $$\Rightarrow$$ Single Degree of Freedom system

Motivation a SDOF spring mass system shown below where : $$M\ddot{d}+Kd=F$$



$$\int_{0}^{x=L}{W\left(x \right)\left\{\frac{\partial }{\partial x}\left[EA\frac{\partial U}{\partial x} \right]+f-m\ddot{U} \right\}}dx=0$$ Eq.(3)    For all posible $$W(x)$$

As seen
 * Eq.(1)$$\Rightarrow$$ Eq.(3) is trivial


 * Eq.(3)$$\Rightarrow$$ Eq.(1) is not trivial

Eq.(3) can be rewritten as $$\int W(x)g(x)dx=0$$

Since Eq.(3) holds for all $$W(x)$$, select $$W(x)=g(x)$$, then Eq.(3) becomes:

$$\int g^{2}dx=0 $$ where $$g^2\geq 0$$  \Rightarrow $$g(x)=0$$