1.053/2.003 Dyn mi dCancsll2007 aFaI ltonro m S teleP bro 9 Out: Tuesday,November27th,2007 Due: Monday,December3rd,2007 Eq ds btai i itnearza on an,b i lra, i liu ityli1. m 1leP broAdisc ofmassm andradiusr issandwichedbetweena horizontalsurfaceanda slenderrodofmassm whichcanslideinandout ofa horizontalgroovewithout h lt tou sfrictionasshowninFigure1. Thedisc ro wills page ip withrespect toboth thehorizontalbottom aswellastherod. Therodisconnectedtoa springofa springconstant k asshowninthefigure,andtheotherendofthespringis attachedtotheinertialframe. AssumesomelengthL0 oftheun-stretchedspring (althoughthat willturnout tobeirrelevant totheproblem). Therodispulledout ofthegroovesuchthat thespringisstretchedbysomele danngth thenr eleased. a. Derive the equation ofmotion ofthe system using the Lagrangian approach. b. Findtheequilibria ofthesystem. c. Linearize the equation ofmotion about the equilibrium pointsand examinethestabilityoftheequilibria. Figure1 Cite as: Sanjay Sarma, Nicholas Makris, Yahya Modarres-Sadeghi, and Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].2. m 2leP broAdisc ofmassm andradiusr rollswithout slippingonaninclinewhichisat angle� withrespect tothehorizontalasshowninFigure2 below. Aspringofa springconstant k isconnectedtoa fixedsupport anda pivot at thecenterofthe disc asshowninthefigure. Notethat gravityacts. Initiallythedisc isheldsuch that thespringisun-stretched. Thedisc isthensuddenlyreleased. It startsrolling without slippingontheincline. a. Derivetheequationofmotionofthesystem. b. Findtheequilibria ofthesystem. c. Linearize the equation ofmotion about the equilibrium pointsand examinethestabilityoftheequilibria. igurF e2 Cite as: Sanjay Sarma, Nicholas Makris, Yahya Modarres-Sadeghi, and Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].3. m 3leP broConsiderthesystem showninthefollowingfigureasinProblem 2 ofExam 2. You mayusetheequationofmotionfrom thesolutionspostedonline. a. Findtheequilibria ofthesystem. b. Linearizetheequationofmotionabout theequilibrium q=0 andexamine lony).thestabilityofthisequilibrium (m, l k C k g M, R igurF e3 Cite as: Sanjay Sarma, Nicholas Makris, Yahya Modarres-Sadeghi, and Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].4. m 4leP broInthelinkageshowninFigure4below,thediskBandsliderCeachhavea massm. Thelinkseachhavea lengthlandareofnegligiblemass. Thespringhasa stiffnesskandunstretchedlength2l. Thesliderispusheddownbya smalldistancesuchthat thespringiscompressed,andthesystem isreleased. Notethat gravityacts. Alsonoticethat point Aispinnedsoit doesnot move. a. Findtheequationofmotion. b. Findtheequilibria ofthesystem. c. Linearize the equation ofmotion about the equilibrium pointsand examinethestabilityoftheequilibria. g igurF e4 Cite as: Sanjay Sarma, Nicholas Makris, Yahya Modarres-Sadeghi, and Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month
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