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I am confused on how to use matlab to build the function and use ode for these problems. If someone could show me how to set up the code to use/run with ode, I can answer the questions/build plots from there. Thank you!
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 + u3 ,
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 + u + u3 = 0
u(0) = 0, u (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
1
1
u + 5 u + u + 5 u3 = cos ωt
u(0) = 0, u (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.