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.vvVabsi n d i c a t e s t h a t w ea s s u m e v > vbaabFsi n d i c a t e s t h a t w e a s s u m e t h e c u r r e n tf l o w i n t h e d i r e c t i o n o f t h e a r r o wo rt h a t t h e s o u r c e a c t s t o m o v e n o d e ai n t h e p o s i t i v e r e f e r e n c e d i r e c t i o n i n am e c h a n i c a l s y s t e m� Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 161 Reading: • Class Handout - Modeling Part 3: Two-Port Energy Transducing Elements 1 Arrow Conventions on Ideal Sources To this point we have simply told you to draw the arrows on source elements (a) In the direction of the assumed across-variable drop for across-variable sources (voltage and velocities), (b) in the direction of the assumed through-variable direction for through-variable sources (currents and forces) When we draw a branch on a graph we make the assumption that P > 0, that is that power is flowing into the element. There are in fact two arrows implicit on each branch: one representing the assumed across-variable drop, and a second representing the assumed through-variable direction. If P > 0 (power is flowing into the element), the two arrows are in the same direction. For example, consider a capacitor 1copyright cD.Rowell 2008 16–1S y s t e mIsa c r o s s - v a r i a b l ed r o pt h r o u g h - v a r i a b l ed i r e c t i o nRd r o p+-iRd r o piZZZv = 0sV 12Fs123L o o p 1 : v + v - V = 0Z1sZ2L o o p 2 : v - v = 0Z3Z2CvbvaiCd r o pvCabiCvCabv o l t a g e d r o pa n d c u r r e n ti n t h e s a m ed i r e c t i o nv o l t a g e d r o pa n d c u r r e n ti n t h e o p p o s i t ed i r e c t i o nP = vcic > 0 when either P = vcic < 0 when either 1. vc > 0 and ic > 0,or 1. vc > 0 and ic < 0,or On 2. vc < 0 and ic < 0. 2. vc < 0 and ic > 0. passive elements, with the assumption P > 0 we can combine the two arrows into one because they point in the same direction. For sources, the assumption is the opposite - that is we assume that the source is supplying energy/power to the system, P < 0. There are two arrows (pointing in opposite directions) associated with any source – one for the across-variable, the second for the through-variable. In modeling sources, we choose to show the arrow that will normally be used to solve the system; (a) An across-variable source will usually be included in a loop-equation, therefore the convention is to show the arrow associated with the across-variable drop. (b) A through-variable source will be included in a node-equation, therefore the convention is to show the arrow representing the direction of the assumed through variable. 16–2JWd r i v em o t o rTsv+-ie l e c t r i c a lr o t a t i o n a la d c s e r v o m o t o r i sa t r a n s d u c e r b e t w e e n t h e ee l e c t r i c a l a n d r o t a t i o n a le n e r g y d o m a i n sllvv1122F12Fa l e v e r i s a t r a n s d u c e r b e t w e e nt w o t r a n s l a t i o n a l d o m a i n s ( f o rs m a l l d i s p l a c e m e n t s )sF ( a )v = 0Z1Z2a t n o d e ( a ) : F - F - F = 0Z1Z2sa c r o s s - v a r i a b l ed r o po p p o s i t e t od i r e c t i o n o f Fs2 Energy Transduction – Two-Port Elements Reading: Class Handout - Modeling Part 3: Two-Port Energy Transducing Ele-ments Many systems involve two or more energy domains, for example a system containing a dc motor or there may be a scaling of the across- and through-variables within a single domain, for example a mechanical lever The following two pages show some examples of two-port elements. 16–3T+-viWA r e a AvFQPQQTWWPPR e s e r v o i ri = - T1K_ _v( b ) E l e c t r i c a l m o t o r / g e n e r a t o rV = rW( a ) R a c k a n d p i n i o nF = A P( d ) F l u i d p i s t o n - c y l i n d e rT = D P( c ) R o t a r y p o s i t i v e d i s p l a c e m e n t p u m pW = - Q1D_rTvFvFWWP i n i o nR a c kv = - Q1A_v = K WvF = - T1r_WFvFvTWv = - rWF = T1r_r+-iFm a g n e tc o i lEvE = K vi = - F1K_ _( b ) S l i d e r - c r a n k( d ) M o v i n g - c o i l l o u d s p e a k e r16–4FvFv1122l1l2+-+-vviiN t u r n s N t u r n s121122v = v211_NN = N / N21F o r a . c . i n p u t s( e ) E l e c t r i c a l t r a n …
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