APL203 — Dynamics of Mechanical Systems

Course Information

Syllabus

• Part I: Newton-Euler Mechanics
  • Preliminaries: coordinate systems, vectors, tensors, tensor product; coordinate transformation.
  • Rotating frames; rotation tensor; Euler angles; angular velocity.
  • Rigid-body kinematics: five-term acceleration formula; examples.
  • Rigid-body kinetics: linear momentum; angular momentum; inertia tensor; kinetic energy.
  • Rigid-body kinetics: balance laws; governing equations; Euler's equations.
  • Examples: rigid body in free space; gyroscopes.
• Part II: Analytical Mechanics
  • Generalized coordinates; constraints; degrees of freedom.
  • Principle of virtual work in statics: virtual displacements; virtual work; constraint forces; workless constraints; Lagrange multipliers; equilibria and stability.
  • Dynamics: D'Alembert's principle; Lagrange's equations; Kane's equations; holonomic and nonholonomic systems.
  • Conservative systems; Legendre transformation; Hamiltonian mechanics; energy theorem.
• Part III: Vibrations
  • Single degree of freedom system: free, damped, forced; examples.
  • Two-degree of freedom systems: normal modes; examples.
  • Extension to multi-degree of freedom systems.
  • Introduction to nonlinear vibrations and stability analysis.

References

Primary

• Greenwood, D.T. Principles of Dynamics. Pearson Education.
• Beatty, M.F. Principles of Engineering Mechanics: Part I, II. Springer.
• Meriam, J.L. & Kraige, L.G. Engineering Mechanics: Dynamics. Wiley.

Additional

• Meirovitch, L. Methods of Analytical Dynamics. Dover Publications.
• Den Hartog, J.P. Mechanical Vibrations. Dover Publications.
• Strogatz, S.H. Nonlinear Dynamics and Chaos. CRC Press.
• Roithmayr, C.M. & Hodges, D.H. Dynamics: Theory and Application of Kane's Method. Cambridge University Press.

Grading

Audit students must write minor and major exams and maintain attendance ≥ 75% in lectures and tutorials separately. The passing grade for audit is 40%.

Quizzes, Minor, and Major

• Two quizzes: Quiz #1 before the minor exam, Quiz #2 after. Exact date, time, and place will be notified later.
• Minor and major exams scheduled based on the institute timetable.

Homework and Tutorials

• Homework assigned almost every week. These are not graded; submission is not required.
• A subset of homework problems will be discussed during tutorial sessions; solutions to remaining problems will be provided.
• Students are strongly encouraged to attempt homework independently before tutorials.

Optional Project

The project involves studying and simulating the spin reversal behavior of a rattleback. You may watch this YouTube demonstration or visit the instructor's office for a live demo.

Read the paper "Spin reversal of the rattleback: theory and experiment" by Garcia and Hubbard. Derive the equations of motion, create a MATLAB simulator, and demonstrate spin reversal.

Deliverables:

• Submit a report and MATLAB codes by 10th November 2023. Requirements:
  1. The report must include a detailed derivation of the equations of motion.
  2. Figures 2, 3, and 4 of the Garcia & Hubbard paper must be reproduced.
  3. The submitted code must generate all results and a video simulating the spin reversal.
• Presentation in the week starting 13th November 2023. Exact date, time, and place to be decided.

Attendance

• Attendance recorded for each lecture and tutorial. One grade lowered if attendance falls below 75% in either.
• In case of absence, notify the instructor by email within one week.
• Medical certificate from IIT Delhi hospital required for exam deferral or re-exam.