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4. Figure 4 illustrates a torsional rebound mechanism. A T-shape lever is pivoted at its bottom with a mass moment of inertia I. Moreover, the lever is connected to a linear spring (with spring constant k) and a viscous damper (with damping coefficient c) at one end. In addition, the distance between the spring/damper to the center line is b. The lever is subjected to a torque M(t). The T-shape lever lies in a horizontal plane, so we do not need to consider the gravity. In analyzing this problem, let us use the angular displacement θ as our dependent variable. Also, the spring is undeformed, when θ 0 as shown in Fig. 4. Answer the following questions(a) Draw a free-body diagram and derive the ordinary differential equation governing the (b) Identify the natural frequency wn and viscous damping factor < from the ordinary dif (c) If the moment M applied is a constant, what is the long-term response of the angular (d) An unknown moment M is applied to the rebound mechanism, and the rebound mecha- angular displacement 0 ferential equation you derived in part (a) displacement θ. nism responds with the angular displacement 0(t) shown in Fig. 5. Based on this angular displacement, answer the following questions. i. Is the response a step response or impulse response. Justify your answer ii. Estimate the natural frequency wn of the rebound mechanism iii. How would you find the viscous damping factor < from the response? Please briefly explain your strategy, but you dont need to find the numerical value. iv. Estimate the initial conditions of the response. M(t) Figure 5: Time response of the rebound mechanism Figure 4: A rebound mecha- nism

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