CHALMERS UNIVERSITY OF TECHNOLOGY

School of Mechanical and Vehicular Engineering

Department of Mechanics

Graduate Course: Analytical Mechanics

Period 2, Academic Year 2000/2001

Overview

An advanced course in classical mechanics has long been a time-honoured part of education of the graduate and postgraduate students enrolled in mechanical engineering, physics, or applied mathematics curricula.

The given course content consists of a combination of topics in advanced dynamics and the elegant variational approach to formulating problems in mechanics. Special attention is paid to the calculus of variations, stability of motion and optimal control theory. The course will provide knowledge about utilization of principles of dynamics together with variational methods for solving actual problems in different technical areas including robotics, mechatronics and biomedical engineering.

Teacher: Professor Viktor Berbyuk

Credits: 4 credit points

Program: The course is given in the second period of the academic year 2000-2001. It is planned to consist of about 20 lectures of two hours each (2x45 min) and the home solving assigned problems. The preliminary pace is two-three lectures per week. Exact schedule will be decided at the first lecture in order to suit everyone. The course program consists of the following chapters: principles of dynamics, Lagrangian mechanics, Hamiltonian systems, calculus of variations, stability of motion, dynamics of rotating bodies, actual problems of dynamics, control and optimisation of robotic, biorobotic and other advanced dynamical systems.

Examination: Final grade will be given after passed oral exam. Completed assigned problems will be a pre-requisite for the exam.

Recommended Literature: The main part of the course will follow the book by Josef S. Török, Analytical Mechanics with an Introduction to Dynamical Systems, John Wiley & Sons, 2000. The department of mechanics will order the book. Those who are interested in buying it this way are requested to tell Gunilla Ekman (031-7721515, gunek@mec.chalmers.se) before October 10, 2000. An extend list of literature will be given on a request.

Pre-Requisites: Knowledge of the level of postgraduate or senior graduate students enrolled in mechanical engineering, physics or applied mathematics curricula.

First Lecture: Oct. 25, 2000, at 13.00, Seminar Room of department of mechanics, Hörsalsvägen 5.

Contact: Viktor Berbyuk

Department of Mechanics, Chalmers University of Technology

412 96 Göteborg, Sweden

Tel: +46-31-7721516, Fax: +46-31-7723477, E-mail: berbyuk@mec.chalmers.se

www.mec.chalmers.se/~berbyuk

Contents

October 26; 30

Principles of Dynamics
Kinematics, Kinetics, Energy, Virtual Work

November 02; 06

Lagrangian Dynamics
Holonomic Systems, Lagrange’s Equations, Conservative Systems,

Lagrangian Systems, Dissipative Systems, Nonholonomic Systems,
Integrals of Motion, Steady Motion

November 09; 13

Calculus of Variations
Extrema of Functions and Functionals, Euler-Lagrange Equations,

The Variational Operator, Variational Problems with Constraints,

Hamilton’s Principle

November 16

Optimal Control

The Basic Optimal Control Problem

Time-Optimal Control, Bang-Bang Control, Energy-Optimal Control

CORRECTION: See below!

November 23; 27

Hamiltonian Systems

Legendre Transformations, Hamilton’s Equations,
Canonical Transformations, Hamilton-Jacobi Equation,
Conservative Systems

November 30
Stability Theory
The Variational Equations, Stability of Linear Systems,

Lyapunov’s Indirect Method, Lyapunov’s Direct Method

December 04; 07

Dynamics, Control and Optimisation Problems of Advanced Mechanical Systems
Under,-Fully,-and Over Actuated Systems,
Semi-Passively Actuated Systems,
Inverse Dynamics-Based Optimisation,
Direct Dynamics-Based Optimisation,
Semi-Inverse Dynamics-Based Optimisation
Applications
Manipulator Robots, Bipedal Locomotion Robots,
Human Locomotor Apparatus with Lower Limb Prostheses.

Recommended Literature

1. Török J.S., Analytical Mechanics with an Introduction to Dynamical Systems, John Wiley & Sons, 2000.

2. Burghes D. N. and A. M. Downs, Modern Introduction to Classical Mechanics & Control, Ellis Horwood Limited, 1975.

3. Krasovskii N.N., Stability of Motion, Stanford University Press, 1963.

4. Goldstein H., Classical Mechanics, Second Edition, Addison-Wesley Publishing Company, Inc., 1980.

5. Gantmacher F., Lectures in Analytical Mechanics, Translated from Russian, Moscow, Mir Publishers, 1975.

6. Lanczos C., The Variational Principles of Mechanics, 4^th ed., Toronto, University of Toronto Press, 1970.

7. Arnold V.I., Mathematical Methods of Classical Mechanics, New York, Springer-Verlag, 1978.

8. Berbyuk V.E., "The use of first integrals in problems of synthesizing optimal control systems", J. Appl. Maths Mechs, 1986, Vol.50, No.1, pp.11-16.

9. Berbyuk V.E., "Using the first integrals to estimate limiting possibilities of optimal control systems", J. Appl. Maths Mechs, 1992, Vol.56, No.5, pp.747-753.

10. Berbyuk V.E., "Synthesizing suboptimal control in nonlinear dynamical systems", J. Appl. Maths Mechs, 1993, Vol.57, No.1, pp.29-35.

11. Berbyuk V., Peterson B. and Nishchenko N., "Linear viscoelastic actuator-based control system of a bipedal walking robot", In MECHATRONICS'98, Proceedings of the 6th UK Mechatronics Forum International Conference, Editors J. Adolfsson and J. Karlsen, Skövde, Sweden, 9-11 September 1998, Elsevier, 1998, pp.379-384.

12. Berbyuk V., Boström A., Lytwyn B. and Peterson B., "Optimization of control laws of the bipedal locomotion systems", In Advances in Computational Multibody Dynamics, Jorge A.C. Ambrósio and Werner O. Schiehlen (Eds.), IDMEC/IST, Lisbon, Portugal, September 20-23, 1999, pp.713-728.

13. Berbyuk V., "Dynamic simulation of a human gait and design problems of the lower limb prostheses", Proceedings of the 12^th International Biomechanics Seminar, Ed. C. Högfors, Chalmers, Gothenburg, Sweden, September 10-11, 1999, Vol. XII, pp.1-20.

14. Berbyuk V., Boström A. and Peterson B., ”Modelling and design of robotic systems having spring-damper actuators”. In: Mechatronics 2000, 6-8 September 2000, Atlanta, USA, Elsevier.

Schedule of the Course

Lectures: Monday and Thursday

Time: 13:15 – 15:00

Place: Seminar Room of Department of mechanics, Chalmers, Hörsalsvägen 5

October: 26; 30

November: 02; 06; 09; 13; 16; 23; 27; 30

December: 04; 07

Exam: December 11 – December 18, 2000

Assigned Problems

Variant # 1

Problems ##: 1.102; 1.192; 2.37; 2.88; 3.6 (a); 3.16; 5.11; 5.36; 6.21; 6.34.

Variant # 2

Problems ##: 1.105; 1.193; 2.17; 2.89; 3.6 (b); 3.18; 5.8; 5.34 (a); 6.22; 6.35.

Variant # 3

Problems ##: 1.131; 1.194; 2.24; 2.90; 3.6 (c); 3.19; 5.13(b); 5.35; 6.42; 6.36.

Variant # 4

Problems ##: 3.6 (d); 3.20; 6.41; 6.43.

CORRECTION

November 30

Hamiltonian Systems

Canonical Transformations, Hamilton-Jacobi Equation,
Conservative Systems

December 04
Stability Theory
The Variational Equations, Stability of Linear Systems,

Lyapunov’s Indirect Method, Lyapunov’s Direct Method

December 07

Dynamics, Control and Optimisation Problems of Advanced Mechanical Systems
Under,-Fully,-and Over Actuated Systems,
Semi-Passively Actuated Systems,
Inverse Dynamics-Based Optimisation,
Direct Dynamics-Based Optimisation,
Semi-Inverse Dynamics-Based Optimisation
Applications
Manipulator Robots, Bipedal Locomotion Robots,
Human Locomotor Apparatus with Lower Limb Prostheses.

_______________________________________________________________

Last modified: December 04, 2000 by Viktor Berbyuk: berbyuk@mec.chalmers.se