Small Vibrations Concepts: Equilibrium position in multidimensional space Oscillations around that position If coordinate system is stationary Equations of motion: Equilibrium is the point where:

Stable Equilibria In one dimension, an equilibrium position is stable if: In two dimensions, you must add: Etc.

Motion near Equilibrium

Solution Trial solution: n solutions for w 2 each with a [C] Solution is superposition of normal modes

Eigenvalue/vector Solution If q’s are orthogonal, then: Redefine coordinates: Eigenvalue equation

Normal Coordinates Each -p 2 is an eigenvalue associated with a eigenvector Normalize such that:

Normal Coordinates (continued)

Forced Oscillations where Equation of Forced Harmonic Oscillator

Forced Damped Oscillations? Equation of Forced, Damped Harmonic Oscillator If Possibly unrealistic assumption

Perturbations Perturbation PotentialSolved Potential if Find approximate solution if V p is small. And near an equilibrium point of V 0 Expand V p around equilibrium point:

Effect of First Derivatives Normal Coordinates: Equation of motion: Shift in equilibrium point:

Second Derivatives Normal Coordinates: Equation of motion: 00 Diagonal terms change w. Off-diagonal mix modes.

Second Derivatives (continued) Look for mode close to unperturbed mode 1:

Second Derivatives (solution)

Recalculate Frequency (Second Order – or – 1.5 Order)

Assumptions Off diagonal terms Diagonal terms Other modes – repeat! Degenerate (or nearly degenerate) modes?

Degenerate Modes Assume:and