Domain

Explanation

Phenomenon

• Vibration as a naturally-occurring phenomenon, as process towards attaining equilibrium with environmental conditions, as dissipation of resultant energies or forces
• Introductory Physics
• Fun with vibrations
• Acoustics and Vibration Animations
• Waves and vibrations
• Wave Equation
• Eric Weisstein’s World of Mathematics

Concepts

• Newton’s laws: 1^st inertia, 2^nd momentum, 3^rd action-reaction
• Momentum: conservation, force-moment, angular momentum
• Work, energy: conservation, dissipation
• Types: potential energy, kinetic energy, mechanical/chemical/biological
• Dynamics: 1) particles (SDOF), 2) rigid bodies (SDOF+continuity), 3) flexible bodies (MDOF+continuity)
• Systems: 1) discrete, 2) continuous
• Vibrations: 1) excitations, 2) self-induced, 3) about equilibrium points (stable)
• Physical phenomenon: vibration modes, frequency, phase, periodic/non-periodic, random/arbitrary, (un)forced

SDOF

• Angular oscillation, periods, frequency, single variable
• Coulomb damping
• Modeled after Meirovitch as well as Hart
• System response to: 1) initial excitation, 2) harmonic & periodic, 3) non-periodic
• Sensitivity analysis of response
• Normal mode method

2DOF

• Modal analysis: orthogonality, beat phenomenon, convolution sum
• Rayleigh damping
• Sensitivity analysis of response

MDOF

• Eigenvalue problem
• Coefficients of mass, stiffness & damping
• Modal orthogonality, decomposition & analysis

Analytical dynamics

• DOF & generalized coordinates
• Principle of virtual work
• Principle of D’Alembert
• Extended Hamilton’s Principle
• Lagrange’s equations
• Exact solution to model as eigenvalue problem using wave equation

Approximate dynamics

• Holzer’s method for torsional vibration
• Myklestad’s method for bending vibration
• Rayleigh’s principle
• Galerkin’s method
• Collocation method (numerical family: Newmark & Wilson part)

Nonlinear oscillations/response

• Nonlinearity occurs w.r.t. oscillations (excitations), material properties, mass (rocket fuels), geometry (deformations) & uncertainties
• Stability: w.r.t. equilibrium (critical) points, limit cycles, perturbation technique & stiffness criterion
• Lindstedt’s method
• Van der Pol oscillator