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Question: 8 figure 1 a rocking mechanism 1 figure 1 illustrates...

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8 Figure 1: A rocking mechanism1. Figure 1 illustrates a rocking mechanism consisting of a point mass, a linear spring, and a rigid rod. The point mass has mass m and undergoes an absolute displacement x(t). The point mass slides on a fixed guide via lubricating fluid that has a constant damping coefficient c. The point mass is also connected to the tip of the rigid rod via a linear spring with constant stiffness k. The rigid rod is hinged at the bottom end, so the rod behaves like an inverted pendulum undergoing angular displacement θ(t). The rod has length l, mass m2, and mass moment inertia Γ about the hinge point. In addition, an input torque M(t) is applied to the rod at the hinged end. Also, the gravitational acceleration is g. Let us assume that the angular motion is small. In this case, sin θ θ and cos θ 1 . Answer the following questions (a) Draw a free-body diagram of the point mass and applies Newtons second law to obtain the following equation of motion. (b) Draw a free-body diagram of the rigid rod and applies Newtons second law (specifically the moment equation) to obtain the following equation of motion. m2gl 2 (c) Let us define the following 4 state variables Use the four state variables defined above to write the equations of motion (1) and (2 into a state equation. (d) If the acceleration of the point mass and the spring force Fk are the output variables of interest. Derive the output equations based on the state variables defined in (3).

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