2. Nonlinear Systems. Part I 1.Nonlinear Mechanics 2.Competition Phenomena 3.Nonlinear Electrical Phenomena 4.Chemical and Other Oscillators

2.1. Nonlinear Mechanics 1.The Simple Pendulum 2.The Eardrum 3.Nonlinear Damping 4.Nonlinear Lattice Dynamics

The Simple Pendulum

Lagrangian approach : The file MF01.nb / MF01.mws attempts, but failed, to solve this analytically.MF01.nbMF01.mws Example 2-1: Parametric Excitation: Example 2-1.nb02-1.nb

The Eardrum Harmonic force: After an initial transient period, respond only to  Forced simple harmonic oscillator

Eardrum is asymmetrically loaded Helmholtz’s eardrum equation, 1895

Nonlinear Damping For an ellipsoidal object moving in a fluid without creating turbulence Equation of motion for an object moving near the earth’s surface Stoke’s Law of Resistence: n =1 Newton’s Law of Resistence : n = 2

Motion of a military shell in air 1.n ~ 1 for v  24 m/s or 86 km/h 2.n ~ 2 for 24 m/s < v < v s ( ~300 m/s ) 3.For v ≥ v s, there is a “bump” in the |F drag | vs |v| curve 4.n ~ 1 above ~600 m/s.

Nonlinear Air Drag on a Sphere: MF02.nbMF02.mws Drag and Lift on a Golf Ball: MF03.nbMF03.mws

Nonlinear Lattice Dynamics E. Fermi, J.Pasta, S. Ulam: FPU problem N  64 (eardrum problem )

If springs are harmonic (   0 ), motion is linear combinations of normal modes. which conserve energies. A system of harmonic springs is NOT ergodic. ( can’t reach thermal equilibrium) FPU showed that their problem was also non-ergodic. This is known as the FPU anomaly. If  >  C, there is a transition into chaos.  equi-partition of energy, if not actual ergodicity.

Toda lattice

For small r, (FPU problem)