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Unformatted text preview: Nonlinear Dynamic Systems Homework 4 Due: 28Mar05 1. Figure 1 shows a turning process that can be described by a delay differential equation. Assume Eq. 1 will adequately describe this process m ¨ y + c ˙ y + ky =- bK n [ h o + y ( t )- y ( t- τ )] . (1) You are asked to determine the stability boundaries and to plot a stability chart for the following range of rpm’s 1,000 – 30,000 [rpm]. You must show your work! You should also use the following parameters K n = 2 × 10 8 [N/m 2 ], m = 0 . 5 [Kg], c = 1 [Ns/m], k = 1 × 10 6 [N/m], h o = 0 . 001 [m]. Figure 1: Schematic of a turning process with regeneration that can be described with delay differential equations. 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