hw_8_solu - ME163 Mechanical Vibrations(Winter 2009 HW-8...

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ME163 Mechanical Vibrations (Winter 2009) HW-8 Due Wednesday 4.04.09 7pm (in the box or to a TA) Answers on web by 9pm, midterm the day after 1. 2DOF Modeling. Consider the system shown in Figure 1 which is a crude model of a coal cart with a scoop attached to its side. The cart has a mass M and is propelled along a horizontal track by a force f . The scoop hangs off the side and is actuated by a torque T (the scoop has a total mass m and moment of inertia I about the center of mass C ). Obtain the equations of motion using x 1 and θ as coordinates and f and T as the inputs. Consider gravity. T M f g m C ϑ a y x 1 x 2 Figure 1: Coal cart with attached scoop. One choice of coordinates is x 1 and θ . The free body diagrams are shown in Figure 2. F x and F y are the reaction forces in the x and y directions due to the physical connection between f F M M F R R g 1 2 x y T x F F y mg C Figure 2: Free body diagrams the pendulum and the mass M . R 1 and R 2 are the vertical reaction forces due to the wheels. Summing forces on M in the horizontal and vertical directions gives M ¨ x 1 = F x + f ME163 Mechanical Vibrations 1 HW8
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M ¨ y 1 = F y + R 1 + R 2 - Mg For mass m , summing forces in the horizontal and vertical directions gives m ¨ x 2 = - F x m ¨ y 2 = - F y - mg Summing moments about the center of mass gives I ¨ θ = aF x cos θ + aF y sin θ + T. Using the previous equations to eliminate F x and F y we obtain M ¨ x 1 + m ¨ x 2 = f (1) I ¨ θ + m ¨ x 2 a cos θ + ma y 2 + g ) sin θ = T (2) The displacements are related as follows: x 2 = x 1 + a sin θ y 2 = - a cos θ Thus ¨ x 2 = ¨ x 1 - a ˙ θ 2 sin θ + a ¨ θ cos θ ¨ y 2 = a ˙ θ 2 cos θ + a ¨ θ sin θ Substituting ¨ x 2 and ¨ y 2 in (1) and (2) we obtain ( I + ma 2 ) ¨ θ + ma ¨ x 1 cos θ + mga sin θ = T ( M + m x 1 - ma ˙ θ 2 sin θ + ma ¨ θ cos θ = f.
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