Consider the mechanical system shown on the right. A mass m slides on a surface with...

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Consider the mechanical system shown on the right. A mass m slides on a surface with viscous friction b, and a spring with stiffness k is connected between the mass and a wall on the left. The coordinate x Io No gravity, g = 0. denotes the horizontal position of the mass, and x = 0 is the rest position of the spring. The mass is connected to the end of a thin rod via a rigid, inextensible T-bar. The thin rod of length L pivots about point 0, and has moment of inertia lo about the pivot. The angle 0 denotes the counter-clockwise angle of the rod from the vertical; you may assume that the spring is at rest when 0 = 0 (that is that x = 0 when 0 = 0). A horizontal, rightward-pointing input force u(t) is applied at the midpoint of the rod (at length L/2 down the rod). The T-bar connection between the mass and the u(t) k end of the rod simply makes sure that any force transferred between the mass and the rod is purely horizontal. The left side of the T-bar slides up and down in a frictionless groove in the mass m, and the right side is connected to a frictionless pivot on the end of the rod (at length L down the rod). m viscous friction, b Derive the equation of motion (EOM) for this dynamic system. Assume that there is no gravity, but do not make any small-angle approximations. Consider the mechanical system shown on the right. A mass m slides on a surface with viscous friction b, and a spring with stiffness k is connected between the mass and a wall on the left. The coordinate x Io No gravity, g = 0. denotes the horizontal position of the mass, and x = 0 is the rest position of the spring. The mass is connected to the end of a thin rod via a rigid, inextensible T-bar. The thin rod of length L pivots about point 0, and has moment of inertia lo about the pivot. The angle 0 denotes the counter-clockwise angle of the rod from the vertical; you may assume that the spring is at rest when 0 = 0 (that is that x = 0 when 0 = 0). A horizontal, rightward-pointing input force u(t) is applied at the midpoint of the rod (at length L/2 down the rod). The T-bar connection between the mass and the u(t) k end of the rod simply makes sure that any force transferred between the mass and the rod is purely horizontal. The left side of the T-bar slides up and down in a frictionless groove in the mass m, and the right side is connected to a frictionless pivot on the end of the rod (at length L down the rod). m viscous friction, b Derive the equation of motion (EOM) for this dynamic system. Assume that there is no gravity, but do not make any small-angle approximations.

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Solution Solution a In addition to the coordinate 0 identified in the original figure we a... View the full answer