Unit Checks

There has been a lot of problems with my units. I came up with the strain equation back in March, but I realized that the units weren't matching up and my strain was at least to the 5th power.

Looking back at the Transverse Vibration of Beam notes I realized that

w(x,t) = SUM[ W_i(x)* N_i(t)]

where

W_i = ith natural mode

N_i = time dependent function

I already have W_i(x), so I need to solve for N_i(t).

Two equations I can use are:

N_i(0) = INTEGRAL(0,L) of p*A(x)*W_i(x)*w_o(x) dx

d N_i(0) = INTEGRAL(0,L) of p*A(x)*W_i(x)*w_o(x) dx

I need w(x,t), when t=0 it becomes w_o(x)

NEED: initial deflection of beam under distributed load of intensity f_o => use the static deflection curve.

ASSUMPTION: initial velocity of beam assumed zero

d w_o(x)=0

The static deflection curve can be found by double integrating the equation

d^2 y/ dx^2 = -Px/EI + PL/EI

If P=f and with the I.C. being y(0)=0, y'(0)=0 (fixed bottom)

y(x) = -f/EI * (x^3/3 + x^2*L/4)

I will email my professors with this.

Goals:

• Apply this information to find my N_i(x) equation
• Check my units from previous applications (and on MATLAB files)