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Computer Project 1. Nonlinear Springs Goal: Investigate the behavior of nonlinear springs. Tools needed: ode45, plot Description: For certain...

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Computer Project 1. Nonlinear Springs Goal: Investigate the behavior of nonlinear springs. Tools needed: ode45 , plot Description: For certain (nonlinear) spring-mass systems, the spring force is not given by Hooke’s Law but instead satisfies F spring = ku + ±u 3 , where k > 0 is the spring constant and ± is small but may be positive or negative and represents the “strength” of the spring ( ± = 0 gives Hooke’s Law). The spring is called a hard spring if ± > 0 and a soft spring if ± < 0. Questions: Suppose a nonlinear spring-mass system satisfies the initial value problem ( u 00 + u + ±u 3 = 0 u (0) = 0 , u 0 (0) = 1 Use ode45 and plot to answer the following: 1. Let ± = 0 . 0 , 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 , 1 . 0 and plot the solutions of the above initial value problem for 0 t 20. Estimate the amplitude of the spring. Experiment with various ± > 0. What appears to happen to the amplitude as ± → ∞ ? Let μ + denote the first time the mass reaches equilibrium after t = 0. Estimate μ + when ± = 0 . 0 , 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 , 1 . 0. What appears to happen to μ + as ± → ∞ ? 2. Let ± = - 0 . 1 , - 0 . 2 , - 0 . 3 , - 0 . 4 and plot the solutions of the above initial value problem for 0 t 20. Estimate the amplitude of the spring. Experiment with various ± < 0. What appears to happen to the amplitude as ± → -∞ ? Let μ - denote the first time the mass reaches equilibrium after t = 0. Estimate μ - when ± = - 0 . 1 , - 0 . 2 , - 0 . 3 , - 0 . 4. What appears to happen to μ - as ± → -∞ ? 3. Given that a certain nonlinear hard spring satisfies the initial value problem ( u 00 + 1 5 u 0 + ± u + 1 5 u 3 ) = cos ωt u (0) = 0 , u 0 (0) = 0 plot the solution u ( t ) over the interval 0 t 60 for ω = 0 . 5 , 0 . 7 , 1 . 0 , 1 . 3 , 2 . 0. Continue to experiment with various values of ω , where 0 . 5 ω 2 . 0, to find a value ω * for which | u ( t ) | is largest over the interval 40 t 60.
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