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Unformatted text preview: Nonlinear Dynamic Systems Homework 3 1. Figure 1 is a graphic of a MEMS microphone. Due to the geometric nonlinearity of stretching, the free vibration of the center plate will contain a cubic nonlinear term. The governing equation of motion for the first mode of vibration is given by Eq. 1. Use multiple scales to obtain an expression for the transient motion of the center plate with the following parameters: ζ = 0 . 05 ω = 1, β = ω . Compare your multiple scales results with simulation for the following initial condition: x (0) = 1 . 5, ˙ x (0) = 0. Comment on how well the multiple scales approach predicts the transient behavior. ¨ x + 2 ζω ˙ x + ω 2 x + βx 3 = 0 . (1) Figure 1: Graphic of a dual back plate MEMS microphone. Stretching of the center plate causes a cubic nonlinearity. 2. In order to characterize the behavior of a system near the bifurcation point, the dynamics of the system are often reduced via center manifold reduction and the use of the normal forms. Use center manifold reduction and normal forms to reduce the dynamics of Eq. (2) to a single dimensional expression for the stability. Use this expression to construct a bifurcation diagram.expression for the stability....
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- Spring '08
- Chaos Theory, Manifold, Fundamental physics concepts, Linear system, Nonlinear system, Dynamical systems