midterm_solution

# midterm_solution - ME201 Advanced Dynamics(Fall 2007...

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Unformatted text preview: ME201 Advanced Dynamics (Fall 2007) Midterm Solution ( 100 pts ) Consider the system shown in the figure. Planar pendulum (no gravity) of mass m is free to rotate around the point O by 360 ◦ : The mass is connected to point O by a massless inextensible bar of length l . The mass is also attached to a massless linear spring of constant stiffness k and equilibrium length seq. The spring is attached to the slider B that is free to slide on the bar A: The spring always stays parallel to the (0; h ) axis. There is no friction. The system will be studied in cylindrical coordinates r-the radius measured from O and θ-the angle measured from the vertical axis passing through O in the counterclockwise direction, as shown in the figure. Figure 1: Phase space: m = 10 ,L = 1 ,h = 5 ,s = 0 . 1 ,k = 1 A. 10 pts Derive the Newton’s equation of motion for the mass m for both the radial direction r and the tangential direction θ : Recall that the radial acceleration and tangential acceleration are given by a r = ¨ r- r ˙ θ 2 , a θ = r ¨ θ + 2˙ r ˙ θ Newton’s equations of motion are: m ¨ a r = m ‡ ¨ r- r ˙ θ 2 · =- T + k ( h- r cos θ- s eq )cos θ m ¨ a θ = m ‡ r ¨ θ + 2˙ r ˙ θ · =- k ( h- r cos θ- s eq )sin θ Where T is tension in the rod. B. 10 pts Rewrite the equations as a system of first order ordinary differential equations....
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midterm_solution - ME201 Advanced Dynamics(Fall 2007...

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