The equations of motion of a certain nonlinear damped pe
Question and Solution
The equations of motion of a certain nonlinear damped pendulum are dx dt = y, dy dt = ?3 sin x ? ?y, where ? is the damp
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The equations of motion of a certain nonlinear damped pendulum are dx dt = y, dy dt = ?3 sin x ? ?y, where ? is the damping coefficient. Suppose that the initial conditions are x(0) = 0, y(0) = ?. (a) For ? = 1 4 , plot x versus t for ? = 3 and for ? = 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 ?c be the critical value of the initial velocity where the transition from one type of motion to the other occurs. Determine ?c to two decimal places. (c) Repeat part (b) for other values of ? and determine how ?c depends on ?.
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