2.004: MODELING, DYNAMICS, & CONTROL II Spring Term 2003 PLEASE ALSO NOTE THAT ALL PRELAB EXERCISE ARE DUE AT THE START (WITHIN 10 MINUTE) OF THE LAB SESSION, NO LATE WORK IS ACCEPTED. Pre-Lab Exercise for Experiment 6 Consider a second order mechanical system. A steel rod of mass m is driven by a transducer with force f. The position of the rod, x, is monitored by a positional sensor. The coupling between the rod and the force transducer also exerts a damping force, dxFd =γ dt . The damping force opposes the motion of the rod and is proportional to the velocity of the rod with a proportional constant, γ . The steel rod is further coupled to a spring with coefficient k. f =γ(dx/dt)d Positional m f k x Spring Force Sensor Transducer (a) Show that the equation of motion has the form: a&y&(t) + by&(t) + cy(t) = f (t) Express coefficients a, b, c, in terms of physical parameters. 1(b) Show that the transfer function can be expressed as: Y (s) 1 = F (s) as 2 + bs + c F(s) Y( s) a s + b s + c2 1 (c) Using the following parameters: m = 0.85kg Nk = 500 m Nsγ= 14 m Where are the poles of this system? 2(d) Sketch the open loop root-locus of this system. (e) Consider implementing proportional gain feedback, sketch the root-locus of this system. 3(f) Consider implementing differential feedback. Where would you put the zero? Sketch the root-locus of this system.
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