me304_hw4_sol - MIDDLE EAST TECHNICAL UNIVERSITY MECHANICAL...

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ME304/Spring 2005/Sections 02+03+04/HW4-Solutions Page 1 / 8 SOLUTIONS TO HOMEWORK 4 for Sections 02, 03 & 04 Prepared by Gökhan BAYAR (C-204), Atilla BAYRAM (A-148), Erdinç İ Y AY (C-209) Problem 1 Let’s obtain the necessary equations for the block diagram first. For the electrical system we can write ) ( ) ( ) ( ) ( t V dt t di L R t i t v a a a = (1) For the electromechanical transducer the following relationships can be written. ) ( 1 ) ( t V K t a a = ω ( 2 ) ) ( ) ( t i K t T a a e = (3) where T e (t) denotes the torque produced by the electrical system. For the rotational mechanical system one can write ) t ( b ) t ( J ) t ( T t i ω + ω = & (4) Note that the inertia torque T i for the rotational mechanical system is equal to the difference between the torque produced by the electrical system (T e ) and the torque used by the ideal pump denoted by T pump which is defined by ) t ( P K ) t ( T 1 p pump = ( 5 ) Then equation (4) becomes ) t ( T ) t ( b ) t ( J ) t ( T ) t ( T i t pump e = ω + ω = & (6) For the ideal pump one may also write ) ( ) ( t K t Q p = (7) MIDDLE EAST TECHNICAL UNIVERSITY MECHANICAL ENGINEERING DEPARTMENT ME 304 CONTROL SYSTEMS SPRING 2005
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ME304/Spring 2005/Sections 02+03+04/HW4-Solutions Page 2 / 8 Now consider the hydraulic system. For the flow rates one may write ) t ( Q ) t ( Q ) t ( Q 10 1 + = ( 8 ) where ) t ( x A ) t ( Q 1 1 & = ( 9 ) f 1 10 R P ) t ( Q = ( 1 0 ) Then f 1 1 R P ) t ( x A ) t ( Q + = & ( 1 1 ) Note that for the valve one can write ) t ( Q R P 10 f 1 = .
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me304_hw4_sol - MIDDLE EAST TECHNICAL UNIVERSITY MECHANICAL...

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