User:Eml4500.f08.team.dwyer.jd/HW5 Nov 3 (Meeting 29)

The Principle of Virtual Work Continued
The initial conditions for the above elastic bar are described as follows:
 * At t=0, prescribe $$u(x,t=0)=\bar{u}(x)$$
 * $$\bar{u}(x)$$ is a known function of displacement
 * Also at t=0, $$\frac{\partial u}{\partial x}(x,t=o)=\dot{u}(x,t=0)=\bar{v}(x)$$
 * $$\bar{v}(x)$$ is a known function of velocity

Equation (1)  $$\frac{\partial }{\partial x}[A(x)E(x)\frac{\partial u}{\partial x}]+f(x,t)=m(x)\ddot{u}$$

Knowing that the Discrete Equation of Motion is as follows:  $$-\mathbf{k}*\mathbf{d}+\mathbf{F}=\mathbf{M}*\mathbf{\ddot{d}}$$

Noticing that there is a one to one correspondence between the two equations, for example $$\mathbf{M}*\mathbf{\ddot{d}}$$ is similar to $$m(x)\ddot{u}$$, the above Discrete EOM can be rewritten as:

Equation (2)  $$\mathbf{M}*\mathbf{\ddot{d}}+\mathbf{k}*\mathbf{d}=\mathbf{F}$$

Equation (2) is a multiple degree of freedom equation (MDOF).

Single Degree of Freedom Example


A single degree of freedom (SDOF) situation is shown above. The equation of motion is: Equation (3): $$m\ddot{d}+kd=F$$(3)


 * 2 initial conditions are needed since it is a 2nd order derivative.

Deriving Equation (2) from Equation (1)

 * To derive Equation (2) from Equation (1) the following integral is used.

Equation (4)

$$ \int_{x=0}^{x=L} w(x)\left({\frac{\partial}{\partial x}\left[(EA)\frac{\partial u}{\partial x}\right]+f-m\ddot{u}}\right)dx=0 $$

Let $$ \left({\frac{\partial}{\partial x}\left[(EA)\frac{\partial u}{\partial x}\right]+f-m\ddot{u}}\right)=g(x) $$

Equation 4 then becomes $$ \int_{x=0}^{x=L} w(x)g(x)dx=0 $$ for all $$ w\left(x\right) $$

Since Equation 4 holds for all $$ w\left(x\right) $$, select $$ w\left(x\right)=g\left(x\right) $$,

then Equation 4 becomes $$ \int_{x=0}^{x=L}g^2dx=0 $$ ⇨ $$ g\left(x\right)=0 $$

Rectangular Truss Zero Eigenvalues Plot Revisited
The purpose of this exercise was to plot the displacements corresponding to the zero eigenvalues of the stiffness matrix for the rectangular truss presented below.



In the above diagram a = b = 1 m, and for case (a) E = 2 and A = 3.