Consider the ball-and-beam apparatus shown below (not drawn to scale). This is one of a few...

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Consider the ball-and-beam apparatus shown below (not drawn to scale). This is one of a few classical test cases for control systems that have been designed to maintain balance. In the diagram below, the ball (red circle) rolls along the beam (AB) without slipping. One end of the beam (A) rotates about a pin that is attached to ground; the other end of the beam (B) is attached to a linkage (BC). A servo rotates a disc (blue circle) about point D relative to ground; the pin at point C is attached to this disc. A X(t) D When the servo rotates counterclockwise a small amount (blue arrow), the beam pivots about point A and causes the ball to roll to the left. As shown in the diagram below, an angular displacement of the servo by 0 (t) causes the beam to tilt to an angle of a(t). L x(t) B Folt) (a) [2 marks] Derive the following equation of motion for the ball: 5g *(t) = sin a(t) Some reminders to help get you started: The moment of inertia of a solid sphere is J = m, where m and r are, respectively, the mass and radius of the sphere. You will need two equations of motion: one in the x(t) direction (from your free-body diagram) and one from taking moments about the center of mass of the ball: FfrictionB = JB (t). "Rolling without slipping" means that x(t) = TB0B (t). (b) [2 marks] Linearize the equation of motion from part (a) about the point 0 = 0 (i.e., where the beam is approximately horizontal) and find the transfer function SX(s)/80 (s). You can use the approximate relation L sin a(t) = r sin 0(t), where r is the length of the linkage BC. (c) [2 marks] Linearize the equation of motion from part (a) about the point 0 = 15 (i.e., where the beam is slightly inclined, as illustrated in the last drawing on the previous page) and find the transfer function 8X(s)/80 (s). You can again use the approximate relation L sin a(t) = r sin 0 (t). Consider the ball-and-beam apparatus shown below (not drawn to scale). This is one of a few classical test cases for control systems that have been designed to maintain balance. In the diagram below, the ball (red circle) rolls along the beam (AB) without slipping. One end of the beam (A) rotates about a pin that is attached to ground; the other end of the beam (B) is attached to a linkage (BC). A servo rotates a disc (blue circle) about point D relative to ground; the pin at point C is attached to this disc. A X(t) D When the servo rotates counterclockwise a small amount (blue arrow), the beam pivots about point A and causes the ball to roll to the left. As shown in the diagram below, an angular displacement of the servo by 0 (t) causes the beam to tilt to an angle of a(t). L x(t) B Folt) (a) [2 marks] Derive the following equation of motion for the ball: 5g *(t) = sin a(t) Some reminders to help get you started: The moment of inertia of a solid sphere is J = m, where m and r are, respectively, the mass and radius of the sphere. You will need two equations of motion: one in the x(t) direction (from your free-body diagram) and one from taking moments about the center of mass of the ball: FfrictionB = JB (t). "Rolling without slipping" means that x(t) = TB0B (t). (b) [2 marks] Linearize the equation of motion from part (a) about the point 0 = 0 (i.e., where the beam is approximately horizontal) and find the transfer function SX(s)/80 (s). You can use the approximate relation L sin a(t) = r sin 0(t), where r is the length of the linkage BC. (c) [2 marks] Linearize the equation of motion from part (a) about the point 0 = 15 (i.e., where the beam is slightly inclined, as illustrated in the last drawing on the previous page) and find the transfer function 8X(s)/80 (s). You can again use the approximate relation L sin a(t) = r sin 0 (t).