Lagrangian mechanics of dynamical systems with an emphasis on vibrations; constraints and generalized coordinates; motion in accelerating frames; conservation laws and invariance of the Lagrangian; particle motion in one dimension, the two-body problem, and central-force motion; free and forced vibration of linearized single-degree-of-freedom and multi-degree-of-freedom discrete systems; weakly nonlinear vibrations; parametric resonance; introduction to Hamiltonian dynamics; rigid-body motions. Prerequisite: TAM 212; MATH 225 or MATH 415; and MATH 285. 4 undergraduate or graduate hours.
1. Foundations of Lagrangian mechanics: Coordinate systems and transformation equations, Generalized coordinates and degrees of freedom, Generalized velocities and virtual displacements, Configuration constraints and redundant coordinates, Work and kinetic energy, Generalized forces, Moving frames
2. Fundamentals of rigid body mechanics: Moments and ellipsoids of inertia, Dynamically equivalent bodies, Angular velocities, Euler angles, Kinetic Energy
3. d’Alembert’s principle of virtual work and Euler-Lagrange’s equation of motion for unconstrained and constrained systems of particles and rigid bodies, Lagrange multipliers, Automated formulation of configuration constraints
4. Conservative systems, Potential energy and relation to generalized forces, Euler-Lagrange’s equations in terms of Lagrangians
5. Ignorable coordinates and conservation laws of Lagrangian mechanics: Energy-time, Momentum-translation, Angular momentum-rotation
6. Small oscillations about positions of equilibrium: Matrix representations of equations of motion, Free and forced vibrations, Normal coordinates and modal analysis, Eigenvalue and eigenvector decompositions of symmetric matrices
7. Small oscillations about steady-state motion, Elimination of ignorable coordinates using Routhians, Free oscillations, Stability
8. Forces of constraint: Inverse dynamics, Frictional contact
9. Foundations of Hamiltonian mechanics: Generalized momenta, Legendre transformations, Hamilton’s canonical equations, Derivation of Hamilton’s principle of least action
ME: MechSE or technical elective.