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Unformatted text preview: Nonlinear Dynamic Systems Homework 1 1. A particle of mass m is constrained to travel along the path shown in Figure 1, which is described by the following function y ( x ) = 5 x 2 + 10 x 4 , (1) where x is defined as the horizontal location of the particle. Develop the equation of motion for the system, find the fixed points for the system, classify the stability of each fixed point, and sketch the neighboring trajectories. Explain how the stable and unstable fixed point(s) match or do not match your intuition?10.80.60.40.2 0.2 0.4 0.6 0.8 11 1 2 3 4 5 x [m] y(x) [m] Track Shape For Multiple Potential Wells Figure 1: Graph of the curved path y ( x ) in a vertical gravitational field. Prob. 1 Soln. ~ r cm = x ˆ i + y ( x ) ˆ j , (2) 1 ˙ ~ r cm = ˙ x ˆ i + ˙ y ( x ) ˆ j , . . . but . . . dy dt = ∂y ∂x ˙ x = y x ˙ x , (3) T = 1 2 m ˙ ~ r cm · ˙ ~ r cm = 1 2 m ( ˙ x 2 + ( y x ˙ x ) 2 ) , (4) V = mgy ( x ) = mg ( 5 x 2 + 10 x 4 ) , (5) Using Lagrange’s equation.. d dt ∂T ∂ ˙ x ∂T ∂x + ∂V ∂x = 0 (6) the resulting equation of motion becomes..... m ¨ x (1 + y 2 x ) + my x y xx ˙ x 2 + mgy x = 0 , (7) Fixed points are found from the state space form of the above equation f ( ~x ) = 0, which results in f 2 ( ~x ) = mgy x = mg ( 10 x f + 40 x 3 f ) = mgx f ( 10 + 40 x 2 f ) = 0 (8) with fixed points at ~x f = (0 , 0) , (0 , ± 1 2 ). Stability of each fixed point is found from eigenvalues of Jacobian matrix evaluated at the fixed point. Df ( ~x )  ~x = ~x f = " ∂f 1 ∂x 1 ∂f 1 ∂x 1 ∂f 2 ∂x 2 ∂f 2 ∂x 2 # ~x = ~x f (9) Results are that (0 , 0) is an unstable fixed point (i.e. at lease one < ( λ ’s) > 0). Stable fixed points are found at (0 , ± 1 2 ) since all eigenvalues satisfy < ( λ ’s) < 0. The phase space portriat is shown in Figure 2. 210.5 0.5 1321 1 2 3 x 1 x 2 Phase Space: Particle on a Double Potential Well Curve Figure 2: Phase space portrait of a particle following a curve with a double potential well. 3 2. Figure 3 is a schematic of a double pendulum undergoing parametric excitation (i.e. time periodic movement of the pivot point). The first pendulum is subjected to a harmonic displacement, of amplitude A and frequency Ω, oriented along an angle ( α ) which is at an incline with the horizontal.) which is at an incline with the horizontal....
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This note was uploaded on 05/19/2011 for the course EML 4220 taught by Professor Chen during the Spring '08 term at University of Florida.
 Spring '08
 CHEN
 Strain

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