In the system shown in Figure P4.42, the input is the displacement y and the out
ID: 1854258 • Letter: I
Question
In the system shown in Figure P4.42, the input is the displacement y and the output is the displacement x. When x = y = 0 the springs are at their free lengths. Derive the equation of motion. Figure P4.43 shows a rack-and-pinion gear in which a damping force and a spring force act against the rack. Develop the equivalent rotational model of the system with the applied torque T as the input variable and the angular displacement theta is the output variable. Neglect any twist in the shaft. Figure P4.44 shows a drive train with a spur-gcar pair. The first shaft turns N times faster than the second shaft. Develop a model of the system including the elasticity of the second shaft. Assume the first shaft is rigid, and neglect the gear and shaft masses. The input is the applied torque . The outputs are the angles theta1 and theta3. Assuming that theta is small, derive the equations of motion of the systems shown in parts (a) and (b) of Figure P4.45. When theta = 0 the systems arc in equilibrium. Arc the systems stable, neutrally stable, or unstable? Assuming that theta is small, derive the equation of motion of the pendulum shown in Figure P4.46. The pendulum is in equilibrium when theta = 0. Is the system stable, neutrally stable, or unstable? Assuming that theta is small, derive the equation of motion of the pendulum shown in Figure P4.47. The input is y(t) and the output is theta. The equilibrium corresponds to y = theta = 0, when the springs arc at their free lengths. The rod inertia about the pivot is I. Figure P4.48 shows a quartcr-car model that includes the mass of the scats (including passengers). The constants k.3 and C3 represent the stiffness and damping in the scat supports. Derive the equations of motion of this system.Explanation / Answer
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