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### Test1_Q-F02

Course: ME 3015, Fall 2008
School: Georgia Tech
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Word Count: 690

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___________________________(Please Name Print) Georgia Institute of Technology Woodruff School of Mechanical Engineering ME3015 System Dynamics and Control First mid-term: October 1, 2002 You need to show all the work clearly. Only answers or partial steps are not acceptable. Transfer functions must be written in standard form: G(s) =N(s)/D(s) where N(s) and ...D(s) are polynomials of &quot;s&quot; in...

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___________________________(Please Name Print) Georgia Institute of Technology Woodruff School of Mechanical Engineering ME3015 System Dynamics and Control First mid-term: October 1, 2002 You need to show all the work clearly. Only answers or partial steps are not acceptable. Transfer functions must be written in standard form: G(s) =N(s)/D(s) where N(s) and ...D(s) are polynomials of "s" in descending order. All problems are equally weighed. Answer three of the four problems. Closed book/ closed notes, one-page formula sheet (with no examples) must be turned in with the test. Check problems to be graded: Problem 1:_____ Problem 2:_____ Problem 3:_____ Problem 4:_____. [1] (33.33%) Consider the following mechanical system, where v(t) is the velocity of the mass and vi(t) is an input velocity. vi(t) b m v(t) k (a) Draw a free-body diagram, and clearly label the velocities and the forces for each of the elements. (b) Write the physical law (equation) governing each of the elements (m, b, and k) in terms of velocity and force variables. (c) Derive the transfer function relating the velocity of the mass to the input velocity. (d) Given that the equation of motion of a mechanical system is given by dx d 2x dx m 2 + b + kx = b i dt dt dt 2 D where x(t = 0) = 0 ; x(t = 0) = 0 ; and b > 4mk , find x(t) if the input to the system xi(t) is a unit step 1(t). What is the steady state value of x? [2] (33.33%) Consider the electrical circuit given below, where the operational amplifier has a gain in the order of K=108 and a very high input resistance R g ; Z1 consists of a capacitor and a resistor in series; and Zf consists of a capacitor and a resistor in parallel. Zf R1 R2 Z1 A B Z1 + K Eo(s) R C Zf Cf Ei(s) Rf (a) Find the complex impedances, Z1 and Zf. (b) Derive the expressions for EA(s) and EB(s) in terms of Eo(s) and Ei(s). (c) The value of the output voltage is finite such that E B ( s ) - E B ( s ) = transfer for function the electrical system, G ( s ) = Eo ( s) 0 . Derive the K Eo (s ) . (Due to the limited time in test, you Ei ( s) may express your solution in terms of Z1 and Zf.) (d) Given that the transfer function of an electrical system has the following form: E o ( s ) b2 s 2 + b1 s + bo G (s) = = Ei ( s ) a 2 s 2 + a1 s + ao where the coefficients are constant, what is the steady state value of the output voltage eo(t) if ei(t) is a unit step input? [3] (33.33%) Consider the following schematics of a rack and pinion mechanism which converts the driving rotational motion into translational motion. In the figure below, the viscous frictions are modeled as ideal dampers bp and bt in the rotational and translational subsystems respectively; and f are the input torque and the loading force respectively. f x pinion of radius r and moment of inertia Jp rack/table of mass m bp bt k (a) Complete the following free-body diagram, and clearly label the displacements and the forces acting on the rotational and translational subsystems. (b) Write the physical law or laws (equations) governing each of the elements including the transformer. (c) Derive...

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