## Homework 7

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## Homework 7

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- Pages:
- 2
- School:
- Massachusetts Institute of Technology
- Course:
- 2 003j - Dynamics and Control I

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Massachusetts Institute of Technology Department of Mechanical Engineering 2 003J 1 053J Dynamics Control I Fall 2007 Homework 7 Issued Nov 2 2007 Due Nov 9 2007 Problem 7 1 Derivation of the equation of the motion for a rolling half disk Half disk is rolling without slipping on the plane surface i Derive the equation of the motion Keep all nonlinear terms and do not linearize ii Linearize the nonlinear motion in case of small angle oscillation Hint use small angle approximations such as sin iii Solve the linearized equation of motion obtained in ii analytically with following initial conditions 0 0 0 0 g Radius r r r CM 4r 1 and I CM mr 2 mr 2 3 2 Cite as 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 Problem 7 2 Generate simulation codes for motion for rolling half disk Generate functions to simulate the trajectory of for rolling half disk based on following instructions Simulation time is 10 seconds Set r 1m i Use the nonlinearized equation of motion obtained in problem 7 1 i Use ode45 for simulation Function name and m file name should be RockerRK your kerberos name and upload it to 2 003 MIT Server site You also submit the hardcopy of your code with appropriate comments Function has following structure function t theta RockerRK your kerberos name theta0 t time matrix N 1 theta angle matrix N 1 theta0 initial condition matrix 1 2 ii Use analytical solution obtained in 7 1 iii Trajectory can be obtained by simply evaluating the analytical solution as a function of time Function name and m file name should be RockerAN your kerberos name and upload it to 2 003 MIT Server site You also submit the hardcopy of your code with appropriate comments Function has following structure function t theta RockerAN your kerberos name theta0 t time matrix N 1 theta angle matrix N 1 theta0 initial condition matrix 1 2 Problem 7 3 Trajectory of t for both small and large angle oscillations For the initial conditions given below simulate the nonlinear motion and the linearized motion for rolling half disk up to 10 seconds use results of Problem 7 2 Compare these results by plotting both of them in the same figure with the appropriate legends Are they identical Otherwise explain why not You should hand in hardcopy of the plots i Small angle oscillation 0 5 o and 0 0 ii Large angle oscillation 0 30o and 0 0 Cite as 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

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