problem set 3

1Problem Set 3 MEEM 3700 1

MEEM 3700MEEM 3700Mechanical VibrationsMechanical Vibrations

Mohan D. Rao Chuck Van Karsen

Mechanical Engineering-Engineering MechanicsMichigan Technological University

Copyright 2003

Problem Set 3 MEEM 3700 2

Set up the differential equation of motion for the double pendulum shown inthe figure below, using the coordinates x1 and x2 and assuming small amplitudes.Find the natural frequencies, the ratios of amplitudes, and the locations of nodes for the two modes of vibration when m1 = m2 = m and l2 = l2 = l.

Problem 1Problem 1


Problem 2Problem 2Determine the natural frequencies and normal modes of the torsional system shownin the figure below for kt2 = 2 kt1 and J2 = 2 J1.


Problem 3Problem 3A machine tool, having a mass of m = 1000 kg and a mass moment of inertia of J0 =300 kg m2, is supported on elastic supports as shown in the figure below. Ifthe stiffnesses of the supports are given by k1 = 3000 N/mm, and k2 = 2000 N/mm,the supports are located at l1 = 0.5 m and l2 = 0.8 m, find the natural frequencies and mode shapes of the machine tool.


Problem 4Problem 4An electric train made up of two cars, each weighing 50,000 lb, is connected by couplings of stiffness equal to 16,000 lb/in., as shown in the figure below. Determine the natural frequency of the system.


Problem 5Problem 5Determine the natural frequencies of the system shown in the figure below by assuming that the rope passing over the cylinder does not slip.


Problem 6Problem 6The two identical cylinders of mass m, radius r, and centroidal mass moment of inertia I = mr2/2 shown in the accompanying figure are attached to each other and to rigid supports by springs having the stiffnesses shown. Assuming that the cylinders roll without slipping, determine (a) the differential equations of motion of the system in terms of the generalized coordinates 1 and 2 shown in the figure, and (b) the systems natural frequencies and modal matrix [u] if m = 2 lbsec2/in., r = 6 in., and k = 300 lb/in.


Problem 7Problem 7The system show in the accompanying figure is subjected to a sinusoidal excitation force F0 sin t. Determine (a) the differential equations of motion of the system, (b) the steady state response amplitudes X1 and X2 , and (c) the natural circular frequencies of the system. For parts b and c let m1 = m2 = m and k1 = k2 = k.


Problem 8Problem 8The gear and shaft system shown in the accompanying figure is subjected to the moment M = M0 sin t which acts on the center one of the three gears. The centroidal mass moment of inertia of each gear and the stiffness of each shaft is as shown in the figure. The mass of the shafts is negligible with respect to that of the gears. Determine (a) the differential equations of motion of the system and (b) the amplitudes of the steady state response of the system.


Problem 9Problem 9A centrifugal pump, having an imbalance of me, is supported on a rigid foundation of mass m2 through isolator springs of stiffness k1, as shown in the figure below. Ifthe soil stiffness and damping are k2 and c2, find the displacements of the pump and the foundation for the following data: mg = 0.5 lb, e = 6 in., m1g = 800 lb, k1 = 2000 lb/in., m2g = 2000 lb, k2 = 1000 lb/in., c2 = 200 lb-sec/in., and speed of pump = 1200 rpm.


Problem 10Problem 10Use the free body diagram method to derive the differential equations governing the motion of the systems show in the figure below using the indicated generalized coordinates. Make linearizing assumptions and write the resulting equations in matrix form.


Problem 11Problem 11The system of the figure below represents a simplified model of a vehicle suspension system and a passenger in the vehicle. The seat is modeled as a spring and viscous damper in parallel. For the suspension system shown, plot the acceleration amplitude of the passenger as a function of vehicle speed.

problem set 3

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