If the restoring force of a nonlinear spring satisfies t
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If the restoring force of a nonlinear spring satisfies the relation Fs(?x) = ?k?x ? ?(?x) 3 , where k > 0, then the diff
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If the restoring force of a nonlinear spring satisfies the relation Fs(?x) = ?k?x ? ?(?x) 3 , where k > 0, then the differential equation for the displacement x(t) of the mass from its equilibrium position satisfies the differential equation (see Problem 17, Section 4.1) mx?? + ?x? + kx + ?x3 = 0. Assume that the initial conditions are x(0) = 0, x? (0) = 1. (a) Find x(t) when ? = 0 and also determine the amplitude and period of the motion. (b) Let ? = 0.1. Plot a numerical approximation to the solution. Does the motion appear to be periodic? Estimate the amplitude and period. (c) Repeat part (c) for ? = 0.2 and ? = 0.3. (d) Plot your estimated values of the amplitude A and the period T versus ?. Describe the way in which A and T, respectively, depend on ?. (e) Repeat parts (c), (d), and (e) for negative values of ?
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