University
of Pennsylvania
School of Engineering and Applied Science
Department of Mechanical Engineering
MEAM 535: Dynamics
Fall 2002
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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
Instructors
Professor Vijay Kumar
Office: 111 Towne
Phone: 898-8241
Email: kumar@cis.upenn.edu
Web site: http://www.cis.upenn.edu/~kumar
Dr. Herbert Tanner
Office: GRASP Lab, Room 311C
Phone: 898 - 8741
Email: tanner@grasp.cis.upenn.edu
Web site: http://www.cis.upenn.edu/~tanner/
Teaching Assistant
Fan Zhang (Michael)
Office: GRASP Lab, Room 325C
Phone:
Email: zhangf@grasp.cis.upenn.edu
Office Hours
Vijay Kumar, kumar@cis.upenn.edu
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).
Herbert Tanner,
Fan Zhang, Mon 4:30
– 6:00 and Wed 4:30 – 6:00, GRASP Lab Room 325C
Texts
Bulk Pack
[BP] MEAM 535 Fall 2001 V. Kumar. University of
Pennsylvania.
An electronic version of the bulk pack is available here.
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,
- [KL] Dynamics:
Theory and Applications, T. R. Kane and D. A. Levinson,
- [RR] Analytical
Dynamics of Discrete Systems, R. M. Rosenberg, Plenum Press,
- [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,
- [HB] Analytical
Dynamics, Haim Baruh, WCB/McGraw-Hill,
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,
- Principles of Mechanics by J. L. Synge and B. A. Griffith
- Analytical Dynamics of Discrete Systems R. M. Rosenberg, Plenum Press,
- Rational Mechanics by C. W. Kilmister and J. E. Reeve
- Methods of Applied Mathematics, Francis B. Hildebrand, Prentice-Hall, 1965/Dover Publications
1992.
A. Introduction
- Lesser, Chapter 1
- Goldstein, Chapter 1
- Abraham and Shaw, Chapters 1 and 2
- Review: Particle Dynamics
B. Rigid Body
Kinematics
- Preliminaries - V. Kumar
- Kinematics - V. Kumar
- Constraints and Degrees of Freedom - V.
Kumar
C. Rigid Body Dynamics and Analytical
Mechanics
- Basics: Dynamics of a System of
Particles
- Mass Distribution and Inertia Tensor
- Principle of Virtual Work
- Lagrange’s equations and D’Alembert’s principle - V. Kumar
D. Stability of
Dynamical Systems
1.
Stability of Dynamic Systems - V. Kumar and G. K. Ananthasuresh
2. Vidyasagar, Lyapunov
Stability
E. Simulation
Basics of Numerical
Integration - V. Kumar
F. Homework Problems
Sources of material for the bulkpack include
- Dynamics: Theory and Applications, T. R. Kane and D. A. Levinson,
- The Analysis of Complex Nonlinear Mechanical Systems: A Computer
Algebra Assisted Approach, M. Lesser, World Scientific Series on Nonlinear
Science, 1995.
- Classical Mechanics, H. Goldstein, Addison-Wesley,
- Principle of Dynamics, Donald T. Greenwood, Prentice Hall,
- Analytical Dynamics of Discrete Systems, R. M. Rosenberg, Plenum
Press,
- Nonlinear Systems Analysis, Prentice-Hall,
- Dynamics: The Geometry of Behavior, R. H. Abraham and C. D. Shaw,
Addison Wesley, 1992.
E-Bulk Pack
NB: You must be registered for the course before accessing these notes.
Course Information
Course
Description
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:
- Rigid body kinematics: the description of the motion of systems of
rigid bodies.
- 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).
- Dynamical systems: The study of systems governed by ordinary
differential equations including the trajectory of the system, stability,
and periodicity.
- 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.
- 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.
Students are
expected to have a basic background in physics and must be familiar with
Time and Location
Tuesdays
and Thursdays
Towne 315. (See http://www.upenn.edu/fm/map.html
to locate
Tentative Schedule
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Week |
Date |
Topic |
Work |
Remarks |
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1 |
5-Sep |
Introduction (history, connection to geometry) |
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2 |
10-Sep |
Rigid body kinematics, angular velocity |
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12-Sep |
Differentiation of vectors |
HW 1 |
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3 |
17-Sep |
Differentiation (cont'd), frame dependence |
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19-Sep |
Velocity and acceleration |
HW 2 |
HT |
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4 |
24-Sep |
Holonomic and nonholonomic constraints |
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26-Sep |
Configuration space, constraints |
HW 3 |
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30-Sep |
Review Session 1 |
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5 |
1-Oct |
Generalized coordinates, partial velocities |
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HT |
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3-Oct |
Configuration space, generalized coord., speeds |
HW 4 |
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6 |
8-Oct |
Kinetics of particles, momentum, energy |
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10-Oct |
System of particles |
HW 5 |
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14-Oct |
Review Session 2 |
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7 |
15-Oct |
Kinetic energy, angular momentum, inertia dyadic |
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17-Oct |
Inertia Dyadic, eigen values and eigenvectors |
HW 6 |
HT |
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8 |
22-Oct |
Principle of virtual work |
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24-Oct |
Partial velocities, generalized forces |
HW 7 |
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28-Oct |
Review Session 3 |
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9 |
29-Oct |
D' Alembert's principle |
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31-Oct |
Applications to nonholonomic systems |
HW 8 |
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10 |
5-Nov |
Constraint forces |
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7-Nov |
Newton-Euler Equations of Motion |
HW 9 |
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11 |
12-Nov |
Lagrange's equations |
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14-Nov |
Nonholonomic systems |
HW 10 |
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18-Nov |
Review Session 4 |
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12 |
19-Nov |
Examples |
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21-Nov |
Numerical methods and simulation |
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13 |
26-Nov |
Integrals of motion |
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28-Nov |
Thanksgiving |
HW 11 |
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14 |
3-Dec |
Stability of dynamical systems (cont'd) |
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HT |
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5-Dec |
Make-up, problems, etc. |
HW 12 |
HT |
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9-Dec |
Tentative Date for Final Exam |
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Material Covered in Class -
Weekly Updates
Weekly homework
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Week |
Date |
Topic |
Covered material |
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1 |
Sep 5 |
Introduction, history, geometric mechanics. |
A.1, A.3 (pages 1-6). |
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2 |
Sep 20 |
Potential Fields, transformation and differentiation of vectors, angular velocity. |
B.1, B.2 (2.1, 2.3). |
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2 |
Sep 12 |
B.2 (2.2.4, 2.4-2.5). |
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3 |
Sep 17 |
B.2 (2.4-2.7). |
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4 |
Sep 24 |
B.3 (3.1-3.3). |
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4 |
Sep 26 |
B.3 (3.4), B.4. |
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5 |
Oct 1 |
B.3 (3.5). |
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5 |
Oct 3 |
A.2 (Sections 1.1-1.3), A.4 |
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6 |
Oct 8 |
Review of basic concepts for HW3, Dynamics of Systems of Particles, Example: Rolling Cone (Maple program) |
C.1 (4.1-4.5), C.2 (Section 3.1) |
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6 |
Oct 10 |
System of particles: Angular momentum and kinetic energy |
C.2 (Section 3.2-3.8) |
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7 |
Oct 15 |
Inertia Dyadic, angular momentum and kinetic energy for a rigid body |
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7 |
Oct 17 |
C.2 |
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8 |
Oct 22 |
C.3 |
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8 |
Oct 24 |
C.3 |
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9 |
Oct 29 |
Midterm |
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9 |
Oct 31 |
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10 |
Nov 5 |
D'Alembert's Principle, Kane's Equations, Newton-Euler Equations |
(C.4: 8.1-8.4, 8.8) |
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10 |
Nov 7 |
Kane-Lagrange Equations (Two examples: Spherical and BalancingPendulum) |
(C.4: 8.5-8.7) |
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11 |
Nov 12 |
Kane-Lagrange Equations for Nonholonomic Systems |
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11 |
Nov 14 |
Lagrange's Equations for Systems with Constraints |
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12 |
Nov 19 |
Integrals of Motion and |
(Goldstein, Chapter 8: pages 339-356) |
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12 |
Nov 21 |
Introduction to the Calculus of Variations |
(Goldstein, Chapter 2) |
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13 |
Nov 26 |
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14 |
Dec 3 |
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14 |
Dec 5 |
Problems Discussion |
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15 |
Dec 9 |
Final Exam Assigned |
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15 |
Dec 10 |
Final Exam Due (Please hand in
to Michael before |
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16 |
Dec 20 |
Project Due |
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Assignments/Homeworks
- Homework assignments will be announced on a weekly basis and posted
on the web site.
- 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.
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Homeworks |
Problems |
Due dates |
Solutions |
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HW 1 |
9/12 (Thursday) |
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HW 2 |
9/19 (Thursday) |
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HW 3 |
10/3 (Thursday) |
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HW 4 |
10/10 (Thursday) |
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HW 5 |
10/17 (Thursday) |
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HW 6 |
10/24 (Thursday) |
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HW 7 |
11/5 (Tuesday) |
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HW 8 |
11/12 (Tuesday) |
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HW 9 |
11/21 (Thursday) |
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HW 10 |
12/3 (Tuesday) |
Problems for HW 2 (Solution)
Problem 1 , 2 (parts (a)-(c) only) , 3.
Problems for HW 3 (Solution)
Problem 2 (parts (d) and (e)) , 6, 7, 9, 11(b), 12.
Problems
for HW 4 (Solution)
(a). Problem 25
(b). When your car accelerates forward, the body of the car rotates. In which
direction, and why?
(c). Greenwood,
3.1, 3.10, 3.13.
Problems for HW 5 (Solution)
(Sample Maple program for
the rolling disk problem, “diskOnPlane.mws”)
(a). Solve Problem 2 using Maple. Calculate the velocity and accleration of the point of
contact in the inertial frame A: (a)
without assuming rolling; (b) imposing the rolling constraint.
(b). Show that if the center of mass is used as a reference point, the rate of
change of the angular momentum
for a system of particles in an inertial
frame is equal to the net external moment about the reference point.
(c). Problem 19
Problems
for HW6 (Solution)
Problems 14, 15, 16
Solution
to the midterm exam
Problems
for HW 7 (Solution)
(a). See the problem here.
(b). Problems 20, 21.
Problems
for HW 8 (Solution)
(a).
In Problem 7(a) in the bulkpack, the two disks roll
on the plane, A, and the system has three independent speeds. Consider an
arbitrary oriented external force acting at point S*.
(i) Use
the expression for the velocity of S* to identify the nonholonomic
velocity partials.
(ii) Derive the conditions for
static equilibrium.
(b). In Problem 9(b), derive the condition(s) for static equilibrium if the
particle is subject to an external force. How does this condition change if the
nonholomic constraint is removed?
(c). In Problem 2, assume q2=0 and the disk is rolling. The disk has two
independent speeds. Let the generalized speeds be given by u1 and u2, where u1
is the forward speed of the disk, and u2 is the vertical component of the
angular velocity. Given an external force F acting at the center of the disk,
and a torque, T, about the vertical axis, find the nonholonomic
active generalized forces.
(d)
Problem 28.
Problems for HW 9
(Solution)
(See solution for problem
32 by Maple)
Problem
31, 32, 33.
Problem for HW 10
(Solution)
Problem 34 (Please write Hamilton
Problem
35, 37, 39.
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.
Grading Policy
The tentative
grading policy is as follows:
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Homeworks |
25% |
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Midterm |
25% |
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Final |
35% |
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Project |
15% |
Students taking this course
generally have diverse backgrounds. This will be taken into account when
grading.
Resources
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.
Notes/Handouts
See bulkpack.
Introduction
(pdf file, 53Kb)
Dynamics:
the Geometry of Behavior (pdf file, 734Kb)
Potential
Fields (pdf file, 661Kb)
Kinematics
Part І (pdf file, 194Kb)
Kinematics
Part II (pdf file, 138Kb)
Degrees
of Freedom (pdf file, 51Kb)
Constraints
(pdf file, 92Kb)
Dynamics
of Particles (pdf file, 186 Kb)
The
Inertia Dyadic (pdf file, 80 Kb)
Virtual
Work (pdf file, 199 Kb)
D’Alembert’s
Principle (pdf file, 30 Kb)
Newton-Euler
Equations (pdf file, 30 Kb)
Kane’s
Equations for NH Systems (pdf file, 140 Kb)
Kane’s
Equations (pdf file, 107 Kb)
Stability
of Dynamic Systems (pdf file, 124Kb)
PDF files for all slides
are available here
Software Tutorials
HELP
Questions
concerning the following topics can be sent to the corresponding email
addresses
- Course administration, personal questions or general assistance
- Homework and assistance with the course
- Web site problems/comments (meam535@seas.upenn.edu)
Links
Seminar series that will have talks in the area of
dynamics and related topics.
Links to
Research Laboratories
Please send us
links that you'd like to see here.
Other Useful
Links
Maintained by meam535@seas.upenn.edu