1940 Tacoma Narrows Bridge Collapse (see www.enm.bris.ac.uk/research/nonlinear/tacoma/tacoma.html)

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.

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). =

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

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

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

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

Step-Input Forcing Terminology wd ringing frequency Figure 5.7

Sinusoidal-Input Forcing For sinusoidal-input forcing, the solution typically is recast into expressions for M(w) and f(w) (see Eqns. 5.62-5.64 and Appendix I). Figures 5.9 and 5.10

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])