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1940 Tacoma Narrows Bridge Collapse

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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. ) a 2 / ( 1 =

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 The two initial conditions used are and y(0)=0.

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

6 Underdamped Case (z < 1 )
The solution is Eqn. 5.57:

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

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

9 Step-Input Forcing Terminology
wd ringing frequency Figure 5.7

10 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

11 In-Class Example Consider the RLC circuit (R = 2 W; C = 0.5 F; L = 0.5 H).


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