1. 1940 Tacoma Narrows Bridge Collapse
(see

2. Second-Order System Dynamic Response
The general expression for a 2nd-order system is
This is a linear 2nd-order ODE, which can be rearranged as
wn (= ) is the natural frequency, which equals
for a RLC circuit and for a spring-mass-damper
system.
z is the damping ratio, which equals
for a RLC circuit and for a spring-mass-damper
system.

3. Solutions to the ODE with Step-Input Forcing
For step-input forcing, there are 3 specific solutions of the ODE because there are 3 different roots of the characteristic equation (see Appendix I). =

4. The solution is of the form
where
and
with OR
with
The two initial conditions used are and y(0)=0.

5. Underdamped Case (0<z<1 )
The solution is Eqn. 5.57:
M(w) = y(t)/KA
versus wnt

6. Critically Damped Case (z = 1 )
The solution is Eqn. 5.59:

7. Overdamped Case (z > 1 )
The solution is Eqn. 5.60:

8. Step-Input Forcing Terminologywd
ringing frequency
Figure 5.7

9. Sinusoidal-Input Forcing
For sinusoidal-input forcing, the solution typically is recast into expressions for M(w) and f(w) (see Eqns and Appendix I).
Figures 5.9 and 5.10

10. In-Class Example
Consider the RLC circuit (R = 2 W; C = 0.5 F; L = 0.5 H);
see Figure H.12, pg 513 for an example circuit diagram
Is it underdamped, critically damped, or overdamped ?
z = R/√(4L/C) = 1 → critically damped
With sine input forcing of 3sin2t:
What is its magnitude ratio ?
w/wn = w/√(1/LC)
= (2)(0.5) = 1 →
M = 1/(2zw/wn)
= 0.5 (see eqn [5.64])