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Problems 39 and 40 illustrate two types of resonance in a mass-spring-dashpot system with given external force F(t) and with the initial conditions x (0) = x'(0) = 0. 39. Suppose that m = 1, k = 9, c = 0, and F(t) = 6cos 31. Use the inverse transform given in Eq. (16) to derive the solution x(1) = t sin 3t. Construct a figure that illustrates the resonance that occurs. 40. Suppose that m = 1, k = 9.04, c = 0.4, and F(t) = 6e-i/5 cos 3t. Derive the solution x(1) = te/5 sin 3t. Show that the maximum value of the amplitude function A(t) = te-t/5 is A(5) = 5/e. Thus (as indicated in Fig. 7.3.5) the oscillations of the mass increase in am- plitude during the first 5 s before being damped out as t + +0o. X= + te -s 10n *=- te -5 FIGURE 7.3.5. The graph of the damped oscillation in Problem 40.

Problems 39 and 40 illustrate two types of resonance in a mass-spring-dashpot system with given external force F(t) and with the initial conditions x (0) = x'(0) = 0. 39. Suppose that m = 1, k = 9, c = 0, and F(t) = 6cos 31. Use the inverse transform given in Eq. (16) to derive the solution x(1) = t sin 3t. Construct a figure that illustrates the resonance that occurs. 40. Suppose that m = 1, k = 9.04, c = 0.4, and F(t) = 6e-i/5 cos 3t. Derive the solution x(1) = te/5 sin 3t. Show that the maximum value of the amplitude function A(t) = te-t/5 is A(5) = 5/e. Thus (as indicated in Fig. 7.3.5) the oscillations of the mass increase in am- plitude during the first 5 s before being damped out as t + +0o. X= + te -s 10n *=- te -5 FIGURE 7.3.5. The graph of the damped oscillation in Problem 40.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Problems 39 and 40 illustrate two types of resonance in a mass-spring-dashpot system with given external force F(t) and with the initial conditions x (0) = x'(0) = 0. 39. Suppose that m = 1, k = 9, c = 0, and F(t) = 6cos 31. Use the inverse transform given in Eq. (16) to derive the solution x(1) = t sin 3t. Construct a figure that illustrates the resonance that occurs. 40. Suppose that m = 1, k = 9.04, c = 0.4, and F(t) = 6e-i/5 cos 3t. Derive the solution x(1) = te/5 sin 3t. Show that the maximum value of the amplitude function A(t) = te-t/5 is A(5) = 5/e. Thus (as indicated in Fig. 7.3.5) the oscillations of the mass increase in am- plitude during the first 5 s before being damped out as t + +0o. X= + te -s 10n *=- te -5 FIGURE 7.3.5. The graph of the damped oscillation in Problem 40.
Transcribed Image Text:Problems 39 and 40 illustrate two types of resonance in a mass-spring-dashpot system with given external force F(t) and with the initial conditions x (0) = x'(0) = 0. 39. Suppose that m = 1, k = 9, c = 0, and F(t) = 6cos 31. Use the inverse transform given in Eq. (16) to derive the solution x(1) = t sin 3t. Construct a figure that illustrates the resonance that occurs. 40. Suppose that m = 1, k = 9.04, c = 0.4, and F(t) = 6e-i/5 cos 3t. Derive the solution x(1) = te/5 sin 3t. Show that the maximum value of the amplitude function A(t) = te-t/5 is A(5) = 5/e. Thus (as indicated in Fig. 7.3.5) the oscillations of the mass increase in am- plitude during the first 5 s before being damped out as t + +0o. X= + te -s 10n *=- te -5 FIGURE 7.3.5. The graph of the damped oscillation in Problem 40.
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