# ps10 - 1.053/2.003 Dynamics and Control I Fall 2007 Problem...

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F ( t ) k Rack, m 1 Pinion, m 2 + b a Viscous damping, c 1.053/2.003 Dynamics and Control I Fall 2007 Problem Set 10 Out: Tuesday, December 4 th , 2007 Due: Wednesday, December 12 th , 2007 Free and forced system response Problem 1. Rack and pinion Consider the rack and the pinion system shown in Fig.1 below as in Problem 1 of Pset 8. The axis of the pinion is fixed in frictionless bearings. A massless rocket is attached to the circular massless pulley of radius a at a point along its edge as shown in the figure. It exerts a thrust F(t) which remains tangential to the pulley at all times. Assume that the pinion can be modeled as a uniform cylinder of mass m 2 and radius b and that the friction between the rack and the horizontal surface can be modeled as viscous damping having a dashpot constant c . a. Find the equilibrium of the system, and linearize the equations of motion about it. b. Reduce the system to the canonical form with equivalent mass, spring, dashpot, and generalized force. Identify the values of parameters ω n and ζ . c. Determine the conditions on the values of m 1 , m 2 , k and c for which the system is under-damped, critically damped and over-damped. d. For F(t) = F o cos( ω t) , determine the particular solution to the ODE. Determine the value of the driving frequency at which the resonance occurs (when ζ = 0). Figure by MIT OpenCourseWare. Figure 1. Rack and pinion Cite as: Sanjay Sarma, Nicholas Makris, Yahya Modarres-Sadeghi, and Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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## This note was uploaded on 11/29/2011 for the course CIVIL 1.018j taught by Professor Markusbuehler during the Fall '08 term at MIT.

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ps10 - 1.053/2.003 Dynamics and Control I Fall 2007 Problem...

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