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 Fspring = ku + \u000fu3 , where k > 0 is the spring constant and \u000f is small but may be positive or negative and represents the \"strength\" of the spring (\u000f = 0 gives Hooke's Law). The spring is called a hard spring if \u000f > 0 and a soft spring if \u000f < 0. Questions: Suppose a nonlinear spring-mass system satisfies the initial value problem ( u00 + u + \u000fu3 = 0 u(0) = 0, u0 (0) = 1 Use ode45 and plot to answer the following: 1. Let \u000f = 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 \u000f > 0. What appears to happen to the amplitude as \u000f ? Let + denote the first time the mass reaches equilibrium after t = 0. Estimate + when \u000f = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. What appears to happen to + as \u000f ? 2. Let \u000f = 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 \u000f < 0. What appears to happen to the amplitude as \u000f ? Let denote the first time the mass reaches equilibrium after t = 0. Estimate when \u000f = 0.1, 0.2, 0.3, 0.4. What appears to happen to as \u000f ? 3. Given that a certain nonlinear hard spring satisfies the initial value problem ( \u0001 u00 + 51 u0 + u + 51 u3 = cos t u(0) = 0, u0 (0) = 0 plot the solution u(t) over the interval 0 t 60 for = 0.5, 0.7, 1.0, 1.3, 2.0. Continue to experiment with various values of , where 0.5 2.0, to find a value for which |u(t)| is largest over the interval 40 t 60