User:Eml4500.f08.a-team.melvin/Lecture 28



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

$$\Sigma$$ Fx = $$ \frac{\delta N}{\delta x} $$ dx + h.o.t. + f(x,t) dx - m(x) ,math> \ddot{u}, dx neglect h.o.t.

(h.o.t. = higher order terms)

Recall Taylor series expansion:

f(x+dx) = f(x) + $$ \frac{df(x)}{dx} $$ dx + $$ \underbrace{\frac{1}{2} \frac{d^2f(x)}{dx^2} dx^2 + ...}_{h.o.t.} $$

Eq. (1) $$ \Rightarrow \frac{\delta N}{\delta x} + f = m \ddot{u} $$ (2)

Equation of motion (EOM)

$$ N(x,t) = A(x) \underbrace{\sigma (x,t)}_{E(x) \underbrace{\epsilon (x,t)}_{\frac{\delta u}{\delta x} (x,t)}} $$ (3)

Constitutive relation

(3) in (2) yields

$$ \frac{\delta}{\delta x} [A(x)E(x) \frac{\delta u}{\delta x}] + f(x,t) = m(x) \underbrace{\ddot{u}}_{\frac{\delta^2 u}{\delta t^2}} $$

Need 2 b.c.s (2nd order derivative w.r.t. x)

2 initial conditions (2nd order derivative w.r.t. t)

(initial displacement, initial velocity)



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

$$ \Rightarrow \frac{\delta u}{\delta x} (L,t) = \frac{F(t)}{A(L)E(L)} $$



1) u(0,t) = 0

2)

$$ \underbrace{N(L,t)}_{A(t)\underbrace{\sigma (L,t)}_{E(L) \underbrace{\epsilon (L,t)}_{\frac{\delta u}{\delta x} (L,t)}}} = F(t)$$

Initial condition at t=0, prescribe

u(x,t=0) = $$\overline{u}$$(x) known function (displacement)

$$ \frac{\delta u}{\delta t} $$ (x,t=0) = $$ \dot{u} $$ (x,t=0) = $$ \overline{\nu} $$ (x)

known function velocity