ME625A
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Applied Dynamics and Vibrations
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Credits:
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3L-0T-0L-0D (9 Credits)
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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)
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Mathematical preliminaries: Coordinate systems, Vectors, Tensors, Outer product; Coordinate transformation. (2)
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Rotating frames; Rotation tensor; Euler angles; Angular velocity. (3)
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Rigid-body kinematics; Five-term acceleration formula; Examples. (3)
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Rigid-body kinetics: Linear Momentum; Angular momentum; Inertia tensor; Kinetic energy. (3)
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Rigid-body kinetics: Balance laws; Governing equations; Euler’s equations. (1)
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Examples: Rigid body in free space; Gyroscopes. (3)
II: Analytical mechanics (15 lectures)
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Generalized coordinates; Constraints; Degrees of freedom (2)
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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)
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Dynamics: d’Alembert’s principal; Lagrange’s equations of motion for holonomic and nonholonomic systems; (1)
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Examples: Rigid bodies; Čaplygin’s sleigh (5)
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Conservative systems. Legendre transformation; Hamiltonian mechanics; Energy theorem; Examples (3)
III: Vibrations (10 lectures)
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Single degree of freedom system: free, damped, forced. (1)
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Convolution integral. (1)
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Two-degree of freedom systems: Normal modes; (2)
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Extension to multi-degree of freedom systems. (1)
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Examples. (2)
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Laboratory demos/sessions: (3)
References:
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Greenwood, D. T. 1987. Principles of Dynamics 2nd edition. Pearson Education.
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Beatty, M. F. 1986. Principles of Engineering Mechanics: Part I, II. Springer.
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Meirovitch, L. 1986. Elements of Vibration Analysis 2nd edition. McGraw Hill Education (India).
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Meirovitch, L. 2010 Methods of Analytical Mechanics. Dover publications.
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Thomson, W. T. 2002. Theory of Vibrations with Applications 3rd edition. CBS publishers.
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Hartog, D. 1985. Mechanical Vibrations. Dover publishers.
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Lanczos, C. 1986. The Variational Principles of Mechanics 4th edition. Dover publications.
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Sharma, I. & S. S. Gupta. 2016. Understanding Rigid Body Dynamics. (under preparation)
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