The following ordinary differential equation describes the motion of a damped spring-mess system (Figure):

where x = displacement from the equilibrium position, t = time, m = 1 kg mass, and α = 5 N/(m/s)². The damping term is nonlinear and represents air damping.

The spring is a cubic spring and is also nonlinear with b = 5 N/m³. The Initial conditions are

Initial velocity             dx/dt = 0.5 m/s

Initial displacement     x = 1 m

Solve this equation using a numerical method over the time period 0 ≤ t ≤ 8 s. Plot the displacement and velocity versus time and plot the phase-plane portrait (velocity versus displacement) for all the following cases:

(a) A similar linear equation

(b) The nonlinear equation with only a nonlinear spring term

(c) The nonlinear equation with only a nonlinear damping term

(d) The full nonlinear equation where both the damping and spring terms are nonlinear

Transcribed Image Text:

dx dx + br' = 0 +a di? di