hw_8 - ME163 Mechanical Vibrations(Winter 2009 HW-8 Due...

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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 o± 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. 2. 2DOF Oscillator. Consider the system in Figure 2. Suppose k 1 = 10 4 N/m, k 2 = k 3 = 2 k 1 , and that m 1 = m 2 = 20 kg (there is no gravity). (a) Derive equations of motion. Write down the mass and sti±ness matrix. (b) Consider f ( t ) = 0. Calculate the frequencies ω 1 , ω 2 without using a computer. Also, calculate mode shapes and draw them. (c) Now, consider
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hw_8 - ME163 Mechanical Vibrations(Winter 2009 HW-8 Due...

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