sol02 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of...

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± ± MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.004 Dynamics and Control II Fall 2007 Problem Set #2 Solution Posted: Friday, Sept. 21, ’07 1. In class, we showed in two different ways that the torque constant of a DC motor equals the back–EMF constant, K m = K v . Verify from the definitions of these constants, K m i = T and K v ω = v e , respectively, that the units associated with these constants are consistent as well. Answer: From the definition of K m and K v , [ N · m ] [ J ] [ K m ]= = , [ A ] [ A ] and [ V ] [ W/A ] [ W · s ] [ J ] [ K v = = = . [1 /s ] [1 /s ] [ A ] [ A ] Therefore they have the same units. 2. Rework Problem 2 of Problem Set #1 (the motor–shaft system of Lecture 2 with non–zero initial condition) by using the Laplace transform. You should arrive at the same step response as you did in Problem Set #1. Answer: The equation of motion is b T 0 u ( t ) + ω = . dt J J Using the Laplace transform, we can write as b T 0 s Ω( s ) ω 0 + Ω( s )= . J Js and it can be solve as T 0 1 Ω( s + ω 0 sJ s + b/J ω 0 T 0 = + s + b/J sJ ( s + b/J ) ω 0 T 0 1 1 = + . s + b/J b s s + b/J 1
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± ± ² ³ ² ³ ´ By taking the inverse Laplace transform, we end up with ω ( t )= ω 0 e t/τ + T 0 1 e t/τ , b where τ = J/b . The result is identical to the solution we derived in the lecture.
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sol02 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of...

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