User:Eml4500.f08.delta 6.lopez/HW5

Lecture Notes - Oct. 31


$$ \sum{F_{x}} = 0 = -N(x, t) + N(x + dx, t) + f(x, t)dx - m(x)\ddot{u} $$

$$ = \frac{dN}{dx}(x, t)dx + h.o.t. + f(x, t)dx + m(x)\ddot{u} $$

In the above equation, we can neglect the h.o.t. (higher order terms). If we recall Taylor Series Expansion:

$$ f(x+dx) = f(x) + \frac{df(x)}{dx}dx + \frac{1}{2}\frac{d^{2}f(x)}{dx^{2}} + ... $$

The term with the 1/2 coefficient and everything after that term are know as higher order terms. Equation (1) now turns into:

$$ \frac{dN}{dx} + f = m\ddot{u}\; \; \; \; \; \;\; \; \;\; \; \;Eq. (2) $$

This equation is also know as the equation of motion.

$$ N(x, t) = A(x)\sigma (x, t)\; \; \; \; \; \; \; \; \; \; \; \; Eq. (3) $$

$$ \sigma (x, t) = E(x)\varepsilon (x, t) $$

$$ \varepsilon (x, t) = \frac{du}{dx}(x, t) $$

The equation above is known as the Constitutive Relation. Now, if we substitute Equation (3) into Equation (2):

$$ \frac{d}{dx}[N(x, t) = A(x)\sigma (x, t)] + f(x, t) = m(x)\ddot{u} $$

This is know as the Partial Differential Equation of Motion. We need two boundary conditions, since this is a second order derivative with respect to x. These two conditions are the initial displacements and initial velocities.

where $$ u(0, t) = 0 = u(L, t) $$



$$ N(L, t) = A(L)\sigma (L, t) $$

$$ \sigma (L, t) = E(L)\varepsilon (L, t) $$

$$ \varepsilon (L, t) = \frac{du}{dx}(L, t) $$

$$ \frac{du}{dx}(L, t) = \frac{F(t)}{A(L)E(L)} $$

Homework
If we set w1 = 1, then w2...w6 = 0

$$ \mathbf{w}^{T} = \begin{bmatrix} 1 & 0& 0 &  0& 0 & 0 \end{bmatrix} $$

$$ \mathbf{w}*(\mathbf{k}\mathbf{q}- \mathbf{p}) $$

$$ = (1)[\sum{k_{1j}}q_j- p_1] + (0)[\sum{k_{2j}}q_j- p_2]+ (0)[\sum{k_{3j}}q_j- p_3]+ (0)[\sum{k_{4j}}q_j- p_4]+ (0)[\sum{k_{5j}}q_j- p_5]+ (0)[\sum{k_{6j}}q_j- p_6] $$

$$ =\sum{k_{1j}}q_j- p_1 $$

For all other values of w (w2...w6), the trend remains the same, and the answers can be found below.

w2 = 1, w1,3,4,5,6 = 0 $$ =\sum{k_{2j}}q_j- p_2 $$

w3 = 1, w1,2,4,5,6 = 0 $$ =\sum{k_{3j}}q_j- p_3 $$

w4 = 1, w1,2,3,5,6 = 0 $$ =\sum{k_{4j}}q_j- p_4 $$

w5 = 1, w1,2,3,4,6 = 0 $$ =\sum{k_{5j}}q_j- p_5 $$

w6 = 1, w1,2,3,4,5 = 0 $$ =\sum{k_{6j}}q_j- p_6 $$