Course Content:

Mathematical preliminaries: Vectors; Tensors; Coordinate transformations. Newton-Euler Mechanics: Rotation; Three-dimensional Rigid-body kinematics and dynamics; Specialisation to two-dimensions; Gyroscopes. Analytical Mechanics: Virtual work; Lagrange multipliers; Lagrange’s equations; Holonomic and non-holonomic systems; Hamiltonian mechanics. Vibrations: Free, damped and forced single-degree of freedom system; Two degree of freedom system; Normal modes; Multi-degree of freedom systems; Lab demos/sessions.

Lecturewise Breakup (based on 75min per lecture)

I: Newton-Euler mechanics (15 lectures)

• Mathematical preliminaries: Coordinate systems, Vectors, Tensors, Outer product; Coordinate transformation. (2)
• Rotating frames; Rotation tensor; Euler angles; Angular velocity. (3)
• Rigid-body kinematics; Five-term acceleration formula; Examples. (3)
• Rigid-body kinetics: Linear Momentum; Angular momentum; Inertia tensor; Kinetic energy. (3)
• Rigid-body kinetics: Balance laws; Governing equations; Euler’s equations. (1)
• Examples: Rigid body in free space; Gyroscopes. (3)

II: Analytical mechanics (15 lectures)

• Generalized coordinates; Constraints; Degrees of freedom (2)
• Principal of virtual work in statics: Virtual displacements; Virtual work; Constraint forces; Workless constraints; Principal of virtual work; Lagrange multipliers; Equilibria and stability of conservative systems; Examples. (4)
• Dynamics: d’Alembert’s principal; Lagrange’s equations of motion for holonomic and nonholonomic systems; (1)
• Examples: Rigid bodies; Čaplygin’s sleigh (5)
• Conservative systems. Legendre transformation; Hamiltonian mechanics; Energy theorem; Examples (3)

III: Vibrations (10 lectures)

• Single degree of freedom system: free, damped, forced. (1)
• Convolution integral. (1)
• Two-degree of freedom systems: Normal modes; (2)
• Extension to multi-degree of freedom systems. (1)
• Examples. (2)
• Laboratory demos/sessions: (3)

References:

1. Greenwood, D. T. 1987. Principles of Dynamics 2nd edition. Pearson Education.
2. Beatty, M. F. 1986. Principles of Engineering Mechanics: Part I, II. Springer.
3. Meirovitch, L. 1986. Elements of Vibration Analysis 2nd edition. McGraw Hill Education (India).
4. Meirovitch, L. 2010 Methods of Analytical Mechanics. Dover publications.
5. Thomson, W. T. 2002. Theory of Vibrations with Applications 3rd edition. CBS publishers.
6. Hartog, D. 1985. Mechanical Vibrations. Dover publishers.
7. Lanczos, C. 1986. The Variational Principles of Mechanics 4th edition. Dover publications.
8. Sharma, I. & S. S. Gupta. 2016. Understanding Rigid Body Dynamics. (under preparation)