Chapter 7.2, Problem 23P

The equations of motion of a certain nonlinear damped pendulum are

$\frac{dx}{dt}=y,\frac{dy}{dt}=-3\sin x-\gamma y$,

where $\gamma$ is the damping coefficient. Suppose that the initial conditions are

$x\left(0\right)=0$, $y\left(0\right)=v$.

(a) For $\gamma =\frac{1}{4}$, plot $x$ versus $t$ for $v=3$ and for $v=6$. Explain these plots in terms of the motions of the pendulum that they represent. Also explain how they relate to the corresponding graphs in Problem $21$ of Section $7.1$.

(b) Let $v_c$ be the critical value of the initial velocity where the transition from one type of motion to the other occurs. Determine $v_c$ to two decimal places.

(c) Repeat part $\left(\text{b}\right)$ for other values of $\gamma$ and determine how $v_c$ depends on $\gamma$.