final - 1.053J/2.003J Dynamics and Control I Fall 2007...

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Unformatted text preview: 1.053J/2.003J Dynamics and Control I Fall 2007 Final Exam 18 th December, 2007 Important Notes: 1. You are allowed to use three letter-size sheets (two-sides each) of notes. 2. There are five (5) problems on the exam, and seven pages in total (including this page). Please make sure you are not missing any pages. 3. Each problem is worth 20 points. 4. You have three hours to solve the exam. GOOD LUCK! 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]. 1 Formulae for the moments of inertia I of some two-dimensional bodies about their center of mass: Disk with mass m and radius R : I = 1 mR 2 2 R Thin rod with mass m and length l : 1 I = ml 2 12 l Rectangular block with mass m , length l, and width w : I = 1 2 2 m ( l + w ) 12 l w 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 Problem 1 v q 1 P l O Q mg B A L (initial length) C q 2 Figure for Problem 1 A robotic arm shown in the figure above consists of link A and link B. Link A rotates at constant angular speed q # 1 = (with respect to an inertial frame), driven by a motor at point O. Link A is also telescoping, i.e ., increasing in length at a constant rate v from an initial length L . Link B is driven by a motor at point Q and rotates at constant angular speed q # 2 = with respect to link A. Link B can be taken as a rigid thin rod of constant length l . Note that gravity acts, the mass of link B is m , and the center of mass of link B is point C. The motor at point Q must clearly apply a torque (a moment couple, in fact) to link B to make it move the way it does. We seek to calculate this torque for motor and bearing sizing purposes. The sub-parts below break this up into two steps....
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final - 1.053J/2.003J Dynamics and Control I Fall 2007...

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