1. The figure below shows two pendulums suspended from pivots with viscous friction, b, and connected...

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1. The figure below shows two pendulums suspended from pivots with viscous friction, b, and connected at their midpoints by a linear spring, k; the angles (radians) are relatively small so that sin 0 and sin02 02. At equilibrium, the pendulums are vertical and there is no force in the spring. Assume that each pendulum can be represented by a spherical mass M with diameter d at the end of a massless bar of length L. The input to the system is the force f(t). Denote gravity by g. a. (10%) Draw a free body diagram for the components in this system. Note, your diagram should show 3-components with all displacements, forces, and torques labeled on the diagram. Be sure to include d'Alembert forces and torques. b. (10%) Write the individual component equations resulting from the summations of torques and/or forces. c. (10%) Since we have assumed small angles, your equations should be linear. Laplace transform the equations assuming all initial conditions are zero. Since the input force f(t) has not been specified, use F(s) for its Laplace transform. d. (15%) Working with your equations in part b, define state variables for this system and write the state variable derivative equations in matrix format assuming 02 is the output of interest. 0(t) f(t) 82(t) www k M L/2 M 1. The figure below shows two pendulums suspended from pivots with viscous friction, b, and connected at their midpoints by a linear spring, k; the angles (radians) are relatively small so that sin 0 and sin02 02. At equilibrium, the pendulums are vertical and there is no force in the spring. Assume that each pendulum can be represented by a spherical mass M with diameter d at the end of a massless bar of length L. The input to the system is the force f(t). Denote gravity by g. a. (10%) Draw a free body diagram for the components in this system. Note, your diagram should show 3-components with all displacements, forces, and torques labeled on the diagram. Be sure to include d'Alembert forces and torques. b. (10%) Write the individual component equations resulting from the summations of torques and/or forces. c. (10%) Since we have assumed small angles, your equations should be linear. Laplace transform the equations assuming all initial conditions are zero. Since the input force f(t) has not been specified, use F(s) for its Laplace transform. d. (15%) Working with your equations in part b, define state variables for this system and write the state variable derivative equations in matrix format assuming 02 is the output of interest. 0(t) f(t) 82(t) www k M L/2 M