General Information

MEAM 535 is a graduate level course in rigid body dynamics and dynamical systems. This course is open to all engineering graduate students. If you are an undergraduate student, you must talk to the instructor before registering for the course. A more detailed description of the course is available here.

Announcements

1) Final exam scheduled for Thursday, December 18^th, 11 am – 1 pm, DRLB A7.

2) Rolling disk project description is here. Solutions to HW#5 & alternate solutions to P25 are posted.

Instructors

Office Hours

Professor Vijay Kumar
Office: 111 Towne
Phone: 898-8241
Email: kumar@cis.upenn.edu
Web site: http://www.cis.upenn.edu/~kumar/

Office Hours: Office hours are scheduled on a weekly basis and are available here. If you want to see me at another time, please make an appointment by e-mailing me or by contacting Ms. Emily Hoover (phone: 215.898.5771, email address: ehoover@seas.upenn.edu).

Dr. Peng Song
Office: GRASP Lab, Levine Hall L402
Phone: 898 - 5814
Email: pengs@grasp.cis.upenn.edu
Web site: http://www.cis.upenn.edu/~pengs/

Office Hours: Wed. 9:00 – 10:30 am

Location: Small conference room outside of Levine 403 (GRASP Lab)

Teaching Assistant

Ani Hsieh
Office: GRASP Lab, Levine Hall L402
Phone: 898 - 5814
Email: mya@grasp.cis.upenn.edu

Office Hours: Tues. 4:30 – 6:00 pm

Location: Small conference room outside of Levine 403 (GRASP Lab)

Course Information

MEAM 535 deals with advanced concepts in dynamics. The course will emphasize the tools of analytical mechanics with the main goal of developing mathematical models that describe the dynamics of systems of rigid bodies and continuous systems. The course will also address the formulation of equations of motion for complicated mechanical systems and methods for solving these equations. In particular, the course will include:

1. Rigid body kinematics: the description of the motion of systems of rigid bodies.

2. Rigid body kinetics: the study of the forces that cause motion and the relationship between the forces and the motion. This relationship is generally described by equations of motion. The main focus will be on analytical mechanics, a set of principles that allow us to write the equations of motion using analytical methods (as opposed to graphical or numerical methods).

3. Dynamical systems: The study of systems governed by ordinary differential equations including the trajectory of the system, stability, and periodicity.

4. Energy methods and integrals of the equations of motion: The description of dynamical systems with simplified models in which the order of differential equations is reduced by exploiting conservation laws.

5. Simulation: the basic techniques for numerical integration, solving differential equations, and for visualization of results.

The course will include applications to multibody systems, and in particular, robots and spatial mechanisms. See tentative schedule for a list of topics covered. Also see the websites for material covered in previous years.

o MEAM 535: Spring 1997

o MEAM 535: Spring 1998

o MEAM 535: Fall 1998

o MEAM 535: Fall 1999

o MEAM 535: Fall 2000

o MEAM 535: Fall 2001

o MEAM 535: Fall 2002

Students are expected to have a basic background in physics and must be familiar with Newton's laws and their application to particles in two and three dimensions. We will assume that everybody is familiar with matrices and determinants, and has had a basic course in ordinary differential equations (equivalent of MEAM 240 and 241 at Penn). Students must also know how to manipulate, multiply and differentiate vectors.

Time and Location

Mondays and Wednesdays 10:30am-12:00pm

Towne 303. (See http://www.upenn.edu/fm/map.html to locate Towne Building)

Tentative Schedule

Week	Date	Topic	Work
1	3-Sep	Introduction: history, connection to geometry	HW 0
2	8-Sep	Transformation and differential of vectors
	10-Sep	Matlab, numerical integration, simulation**	HW 1
3	15-Sep	Velocity and acceleration
	17-Sep	Velocity and acceleration	HW 2
4	22-Sep	Degree of Freedom, Constraints **
	24-Sep	Holonomic and nonholonomic constraints	HW 3
5	29-Sep	Generalized coordinates, partial velocities
	1-Oct	Generalized speeds, partial velocities
6	6-Oct	Principle of virtual work	HW 4
	8-Oct	Principle of virtual work
7	13-Oct	Fall Break
	15-Oct	Kinetics of particles, momentum, energy**	HW 5
	17-Oct	Help session** (to be scheduled)
8	20-Oct	Midterm I**
	22-Oct	Kinetic energy, angular momentum, inertia dyadic**	HW 6
9	27-Oct	Friction, inequalities in mechanics**
	29-Oct	Dynamics with complementarity constraints**	HW 7
10	3-Nov	Constraint forces
	5-Nov	Principle of virtual work for nonholonomic systems	HW 8
11	10-Nov	D'Alembert Equations
	12-Nov	Lagrange's equations for nonholonomic systems	HW 9
12	17-Nov	Conservative systems
	19-Nov	Variations on the theme: Lagrange's equations
13	24-Nov	Stability of dynamical systems
	26-Nov	Stability of dynamical systems	HW 10
14	1-Dec
	3-Dec	Reading Day - Help with problems	HW 11

	*	needs to be rescheduled
	**	taught by Peng Song

Texts

E-Bulk Pack

[BP] MEAM 535 Fall 2001 V. Kumar. University of Pennsylvania.

You have to be registered in the course to get access to this website.

There is no prescribed text for this class. Several suggested references are provided here.

References

· [DG] Principle of Dynamics, Donald T. Greenwood, Prentice Hall, Englewood Cliffs, New Jersey, 1988. ISBN 0-13-709981-9.

· [KL] Dynamics: Theory and Applications, T. R. Kane and D. A. Levinson, McGraw Hill, New York, 1985. ISBN 0-07-037846-0.

· [RR] Analytical Dynamics of Discrete Systems, R. M. Rosenberg, Plenum Press, New York, 1977. ISBN 0-306-31014-7.

· [ML] The Analysis of Complex Nonlinear Mechanical Systems: A Computer Algebra Assisted Approach, M. Lesser, World Scientific Series on Nonlinear Science, 1995. ISBN 981-02-2209-2.

· [HG] Classical Mechanics, H. Goldstein, Addison-Wesley, Reading, 1989.

· [HB] Analytical Dynamics, Haim Baruh, WCB/McGraw-Hill, Boston, ISBN 0-07-365977-0. 1999.

Other references

· Dynamics: The Geometry of Behavior, R. H. Abraham and C. D. Shaw, Addison Wesley, 1992.

· A Treatise on the Analytical Dynamics of Particles & Rigid Bodies by E. T. Whittaker.

· A Treatise on Analytical Dynamics by L. A. Pars.

· Calculus of Variations with Applications to Physics and Engineering, Robert Weinstock, McGraw-Hill, 1952/Dover 1974.

· Variational Principles of Mechanics by C. Lanczos, Dover.

· Principles of Mechanics by J. L. Synge and B. A. Griffith

· Analytical Dynamics of Discrete Systems R. M. Rosenberg, Plenum Press, New York, 1977. ISBN 0-306-31014-7.

· Rational Mechanics by C. W. Kilmister and J. E. Reeve

· Methods of Applied Mathematics, Francis B. Hildebrand, Prentice-Hall, 1965/Dover Publications 1992.

Assignments/Homeworks

· Homework assignments will be announced on a weekly basis and posted on the web site. Reading assignments and homework problem sets will be announced through the web (see Homeworks and Solutions). Problem sets will be assigned at least one week before they are due. Only selected problems from each set will be graded. You may consult your colleagues or talk to the instructors before doing the homework problems. Solutions to all problems will be kept in the library. No late homework will be accepted without special permission from one of the instructors.

· There will be in-class, 1.5 hour midterm.

· There will be one take-home, 2 hour final exam.

· Note: Reload the page so you are viewing the most current updates.

For this week’s homework and solution, click here.

Projects

The project is an opportunity to (a) apply basic concepts learned in the class to problem and application areas; or (b) to learn new concepts in dynamics that are not covered in the syllabus. Examples of projects include:

· Dynamics of continuous systems with applications to robotics or computer graphics;

· Potential field based navigation methods for autonomous agents (vehicles, synthetic figures in animations, robots)

· Multibody dynamics: Dynamics of closed chain systems

· Electronic throttle control, cruise control and the dynamics of automatic transmission vehicles

· Dynamics of biological networks: Modeling, analysis and simulation of biomolecular inter and intra cellular networks.

Suggestions for Projects:

· Rolling Disk Project

Grading Policy

The tentative grading policy is as follows:

Homeworks	25%
Midterm	25%
Final	35%
Project	15%

Students taking this course generally have diverse backgrounds. This will be taken into account when grading.

Weekly Updates

NOTE: A few pages require the use of Adobe Acrobat Reader in order to view the material. You may download a FREE copy from the Adobe web site by clicking on the Adobe icon above.

Material Covered in Class

Week	Date	Topic	Covered material
1	Sep 3	Introduction: history, connection to geometry	A.1, A.3 (pages 15-26)
2	Sep 8	Transformation and differential of vectors	B.1, B.5 (Chapter 1)
	Sep 10	Matlab, numerical integration, simulation	Matlab Tutorial
3	Sep 15	Velocity and acceleration	B.2, B.5 (Chapter 2)
	Sep 17	Velocity and acceleration (continued)	B.2, B.5 (Chapter 2)
4	Sep 22	Velocity and acceleration (continued)	B.2, B.5 (Chapter2)
	Sep 24	Constraints and Degrees of Freedom	B.3, B.4
5	Sep 29	Constraints, Holonomic and Nonholonomic Constraints	B.3, B.4
	Oct 1	Constraints, Holonomic and Nonholonomic Constraints (continued)	B.3, B.4
6	Oct 6	Principle of Virtual Work	C.3
	Oct 8	Principle of Virtual Work (continued)	C.3
7	Oct 13	Fall Break
	Oct 15	Complete Principle of Virtual Work (continued)	C.3
8	Oct 20	Midterm (in class)
	Oct 22	Review of Particle Dynamics	C.1, C.3
9	Oct 27	System of Particles	C.1, C.3
	Oct 29	Kinetic energy, angular momentum, inertia dyadic	C.2
10	Nov 3	Kinetic energy, angular momentum, inertia dyadic
	Nov 5	Kinetic energy, angular momentum, inertia dyadic
11	Nov 10	D'Alembert Equations	C.4
	Nov 12	D'Alembert Equations	C.4
12	Nov 17	Kane Lagrange's equations for holonomic and conservative systems	C.4
	Nov 19	Kane Lagrange's equations for nonholonomic systems
13	Nov 24	Stability of Dynamical Systems	D.1
	Nov 26	Stability of Dynamical Systems	D.2
14	Dec 1
	Dec 2

Notes: Materials listed under “Covered material” refers to sections in the Bulk Pack. Additional materials will be noted accordingly.

Homeworks & Solutions

Homework	Due Date	Solution
HW 0	Wed. 9/10 (Optional)	HW 0 Solution
Matlab Exercise 1 & Problem 1	Wed. 9/17	HW1 Solution Matlab Exercise sample code: Example1, Example2
HW 2	Mon. 9/29	HW 2 Solution
HW 3	Wed. 10/8	HW3 Solution
HW 4	Wed. 10/15	HW4 Solution
Matlab Exercise 2	Wed 10/29	get_R_r.m, get_qdot_from_u.m
HW 5	Wed. 11/5	HW5 Solution, Alternate Solution to P25
HW 6	Wed 11/12	HW6 Solution
HW 7	Wed 11/19	HW7 Solution
HW 8	Mon 12/1	HW8 Solution
HW 9	Wed 12/10

Notes/Handouts

Week	Date	Notes/Handouts	Additional Material
1	Sep 3	Class Slides
2	Sep 8
	Sep 10	Class Slides	examples.m, main_m_file.m, test_function.m, demo_msd.m, diff_msd.m
3	Sep 15	Class Slides (Kinematics Part I)
	Sep 17
4	Sep 22	Kinematics Part II
	Sep 24	Constraints and Degrees of Freedom
5	Sep 29	Rolling Disk Kinematics Constraints and Degrees of Freedom
	Oct 1	Constraints and Degrees of Freedom
6	Oct 6	Principle of Virtual Work
	Oct 8	Principle of Virtual Work
7	Oct 13	Fall Break
	Oct 15	HW#4 Questions and Matlab Exercise 2.
8	Oct 20	MidTerm
	Oct 22	Dynamics of Particles
9	Oct 27	Dynamics of Particles
	Oct 29	System of Particles
10	Nov 3	Kinetic Energy, Inertia Dyadic
	Nov 5	Kinetic Energy, Inertia Dyadic
11	Nov 10	D'Alembert Equations
	Nov 12	D'Alembert Equations
12	Nov 17	Kane Lagrange's equations
	Nov 19	Kane Lagrange’s equations
13	Nov 24	Stability of Dynamical Systems
	Nov 26	Stability of Dynamical Systems
14	Dec 1	Rolling disk project
	Dec 3
15	Dec 8

Resources

HELP

Questions concerning the following topics can be sent to the corresponding email addresses:

· Course administration, personal questions or general assistance: kumar@cis.upenn.edu

· Homework and assistance with the course: mya@grasp.cis.upenn.edu

· Web site problems/comments: meam535@seas.upenn.edu

Software Tutorials

· Matlab