Physics 319 Classical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 22 G. A. Krafft Jefferson Lab

General Solution Place following boundary conditions on solution Then get Phase delayed oscillations with amplitude that goes from one degree of freedom to the other and back again

Beats In normal mode view have oscillators at slightly different frequencies. Beating is constructive and destructive interference

Lagrangian Version of Two Carts Euler-Lagrange equations of motion are identical to before

Double Pendulum Oscillations Double Pendulum Lagrangian Small Oscillation (Linearization) Approximation (0,0) l1 l2

Equations of Motion The Euler-Lagrange equations of motion are written in the matrix form as Same solution method as before

Equal Mass and Length Case When m1 = m2 and l1 = l2

Normal Modes First Normal Mode Symmetric Displacements Note: amplitudes different in the normal mode!

Symmetric Displacement

Second Normal Mode Other normal mode Antisymmetric normal mode

Antisymmetric Displacement

n Coupled Bodies Use Lagrangian description to analyze general case of n coupled bodies. The general displacement is described by a column vector Evaluate the kinetic energy Defines the M matrix as before

General Potential Matrix For small oscillations about an equilibrium point, the dominant contribution for the potential follows from a Taylor expansion of the potential energy function Euler-Lagrange equations yield, for each index i This is the familiar matrix equation

Solution Method Find n normal mode frequencies by solving the secular or characteristic equation Find n normal mode vectors Write general solution as the sum over the normal modes. Expansion coefficients determined by the initial conditions

Three Coupled Oscillators Solve the problem of three coupled oscillators Kinetic energy easy Gravitation potential easy

Equations of Motion Spring potential energy Dynamical matrices

Secular Equation First normal mode totally symmetric

Second normal mode, antisymmetric like Third normal mode, also antisymmetric like

From Taylor

Resolution into Normal Modes Given the time displacement of the total system and that you know the normal mode eigenvectors, one can easily determine the normal mode amplitude and phase. Simply perform the sum using the eigenvector of the mode