Chapter 28, Problem 46P

The following ordinary differential equation describes the motion of a damped spring-mass system (Fig. P28.46):

$m\frac{d^{2}x}{d{t}^2}+a\left|\frac{dx}{dt}\right|\frac{dx}{dt}+b{x}^3=0$

where $x=$ displacement form the equilibrium position, $t=$ time, $m=1\text{ kg}$ mass, and $a=5\text{ n}{\left(\text{m/s}\right)}^2$. The damping term is nonlinear and represents air damping.

The spring is cubic spring and is also nonlinear with $b=5{\text{ N/m}}^3$.

The initial conditions are

Initial velocity $\frac{dx}{dt}=0.5\text{ m/s}$

Initial displacement $x=1\text{ m}$

Solve this equation using a numerical method over the time period $0\le t\le 8\text{ s}$. 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;

$m\frac{d^{2}x}{d{t}^2}+2\frac{dx}{dt}+5x=0$

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

$\frac{d^{2}x}{d{t}^2}+2\frac{dx}{dt}+b{x}^3=0$

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

$m\frac{d^{2}x}{d{t}^2}+a\left|\frac{dx}{dt}\right|\frac{dx}{dt}+5x=0$1

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

$m\frac{d^{2}x}{d{t}^2}+a\left|\frac{dx}{dt}\right|\frac{dx}{dt}+b{x}^3=0$1

FIGURE P28.46