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lecture_15 - MIT OpenCourseWare http/ocw.mit.edu 2.004...

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MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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i n K 9 ± ± G J B J 9 ± ± f r i c t i o n a l d r a g p l a t e s Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 15 1 Reading: Class Handout: Modeling Part 1: Energy and Power Flow in Linear Systems Sec. 3. Class Handout: Modeling Part 2: Summary of One-Port Primitive Elements Nise: Secs. 2.4 and 2.6. 1 Rotational Systems (continued) Example 1 The diagram shows a mechanical tachometer that uses frictional drag plates to create a torque proportional to angular velocity difference. The angular velocity is indicated by the displacement θ of a torsional spring. Find the transfer function relating the displacement of the indicator θ to the input angular velocity Ω in θ ( s ) H ( s ) = Ω in ( s ) and show that for a constant input angular velocity the steady-state indicated speed θ ss Ω i . Solution: There are two distinct angular velocities, and the system graph is: 1 copyright c D.Rowell 2008 15–1
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J B K 9 = 0 i n 9 ± ± a c r o s s - v a r i a b l e s o u r c e Using impedances, redraw the graph combining the inertia and the spring into a single impedance Z 2 : Z Z 9 = 0 i n 9 ± ± J 9 ± ± 1 2 T Z = 1 1 B 1 Z = 2 1 J s s K = 1 / J K s / K + 1 / J s Then Z 2 Ω J = Ω in ( s ) Z 1 + Z 2 s/ ( Js 2 + K ) = Ω in ( s ) 1 /B + s/ ( Js 2 + K ) Bs = Ω in ( s ) . Js 2 + Bs + K But the angular displacement θ ( s ) = Ω J ( s ) /s so that B H ( s ) = Js 2 + Bs + K If the input velocity is a step function at t = 0, the Ω in ( s ) = Ω /s and the steady-state indicated value will be Ω B θ ss = lim θ ( t ) = lim ( s ) = lim sH ( s ) = Ω t →∞ s 0 s 0 s K show that
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