3. Stabilizing a rocket to a vertical position during the early stages of a launch is...

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3. Stabilizing a rocket to a vertical position during the early stages of a launch is essentially the same task as stabilizing an inverted pendulum. Think of how you move your hand if trying to keep a pole on your hand upright. A simple 2-dimensional version of this problem is an inverted pendulum mounted on a moving cart with mass Me as shown below. The sphere has mass M and diameter d and is connected to the cart by a massless bar with zero friction at the coupling pin. There is zero rolling resistance at the cart wheels and the input to the system is the force Fi which is used to keep as close to zero as possible. From tables, Jcg of the sphere is M.d/10. ME M 3.1 The following modeling equations have been formulated for this system. Note, for now assume F; is an unspecified input. Verify that the system is nonlinear and that the unknowns are y, x, 0, z and Fx. (1) Ms(g)Lsine + M*Lcose +J = 0 (2) x z + Lsine Sum of moments at connection pin horizontal position x of sphere cg (3) y L Lcos vertical position y of sphere cg (4) Mc + Fi - Fx = 0 d'Alembert equation for cart position z (5) Fx + M5x = 0 d'Alembert equation for horizontal displacement x of sphere 3.2 Verify that the substitution of the 2nd derivatives of (2) and (3) into (1) gives (1) (ML + 1) + M,Lcos (0) - MsgLsin(0) = 0 and using (5) and the 2nd derivative of (2), equation (4) becomes (4) (Mc+Ms) + MLcos (0) - M,L sin(0) + F = 0 3. Stabilizing a rocket to a vertical position during the early stages of a launch is essentially the same task as stabilizing an inverted pendulum. Think of how you move your hand if trying to keep a pole on your hand upright. A simple 2-dimensional version of this problem is an inverted pendulum mounted on a moving cart with mass Me as shown below. The sphere has mass M and diameter d and is connected to the cart by a massless bar with zero friction at the coupling pin. There is zero rolling resistance at the cart wheels and the input to the system is the force Fi which is used to keep as close to zero as possible. From tables, Jcg of the sphere is M.d/10. ME M 3.1 The following modeling equations have been formulated for this system. Note, for now assume F; is an unspecified input. Verify that the system is nonlinear and that the unknowns are y, x, 0, z and Fx. (1) Ms(g)Lsine + M*Lcose +J = 0 (2) x z + Lsine Sum of moments at connection pin horizontal position x of sphere cg (3) y L Lcos vertical position y of sphere cg (4) Mc + Fi - Fx = 0 d'Alembert equation for cart position z (5) Fx + M5x = 0 d'Alembert equation for horizontal displacement x of sphere 3.2 Verify that the substitution of the 2nd derivatives of (2) and (3) into (1) gives (1) (ML + 1) + M,Lcos (0) - MsgLsin(0) = 0 and using (5) and the 2nd derivative of (2), equation (4) becomes (4) (Mc+Ms) + MLcos (0) - M,L sin(0) + F = 0