ME 863Nonlinear Oscillations Spring 2002

ME 863

Outline of the Lecture Notes:

1. Introduction to Nonlinear Vibration

2. Linearization

  • Time-variant autonomous systems
  • Local stability of hyperbolic fixed point governed by eigenvalues (NM 3.2.1)
  • Phase portraits
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    3. Conservative Systems: Period of Free Vibration
  • Phase portraits, x' vs. x by energy methods (NM 1.2, 2.1-2..2)
  • Elliptic integral method for period (NM 2.2)
  • Regular perturbation fails: secular terms (NM 2.3.1)
  • Lindstedt's method: allow frequency to be amplitude dependent (NM 2.3.2)
  • Harmonic-balance method (NM 2.3.4)
  • Period indeed is amplitude dependent (NM 2.4)
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    4. Limit Cycles
  • Polar coordinates
  • Hopf bifurcation
  • Multiple scales introduced in analyzing Rayleigh's equation
  • Averaging: variation of constants viewpoint (NM 3.3.4)
  • Relaxation oscillations (NM 3.3.4, 3.5)
  • Friction-induced vibration: piecewise linearity and stick-slip
  • Bendixson's criterion for the nonexistence of limit cycles
  • Poincaré index: classification of equilibria, nonexistence of l.c.
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    5. Forced Vibration and Nonlinear Resonance (NM 4.1, 4.3)
  • Harmonic balance method on Duffing equation
  • Jump phenomena
  • Coexistence of steady-state responses
  • Multiple scales: resonances and stabilities
  • Primary resonance
  • Secondary resonances: subharmonic and superharmonic
  • Combination resonances
  • Averaging: same results
  • Forced systems with limit cycles
  • Resonance
  • Entrainment, quenching, and quasiperiodic responses
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    6. Floquet Theory (NM 5)
  • Time-varying systems
  • Fundamental solution matrix
  • Difference equation: Poincaré map
  • Eigenvalues and stability
  • Hill's equation
  • transition to instability coincident with periodic response
  • Mathieu's equation
  • transition curves
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    7. Internal Resonance in Multi-Degree-of-Freedom Systems (NM 6.2)
  • Swinging spring
  • Internal resonance: spring freq. near twice the pendulum freq.
  • Beating motion and exchange of energy between modes

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