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  3. problem 3 mechanical system modeling the hanging crane structure supporting...

Question: problem 3 mechanical system modeling the hanging crane structure supporting...

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Problem 3. Mechanical System Modeling: The hanging crane structure supporting the Space Shuttle Atlantis, along with its simple schematic representation are shown below, where M is the mass of the cart, m is the mass of the payload, L is the length of the massless rigid connector, x(t) is the cart displacement, Fb(t) =-bi(t) is the friction force, φ(t) is the connector angle with respect to the vertical, and u(t) is the force applied to the cartx(t) Fb(t) u(t) ф(t) (a) Write the equations of motion describing the motion of the cart and the payload. (Hint: Consider the reaction force between the cart and the payload along the connector. Write separate equations of motion for the cart and the payload, and eliminate the reaction force from your equations in order to obtain your final equations of motion) (b) Assume ф 0, to linearize your equations. Also assume no friction (i.e., b 0). Find the transfer function fron cart velocity u(t)- (t) to the connector angle φ(t), i.e., (c) Assume the cart starts moving at a constant speed, i.e., v(t) is a unit step function. Using the above transfer function, find the resulting connector angle p(t). Show that in this case, the payload will oscillate with a frequency wo VE where g is gravity. (d) Find the transfer function from the applied force to the cart position, i.e., (e) Show that if a constant force is applied to the cart (i.e., if u(t) a unit step function), ts velocity will increase without bound as too

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