ME 863 Nonlinear Oscillations Spring 2008

Outline of the Lecture Notes:

1. Introduction to Nonlinear Vibration

2. Linearization and local stability

  • Time-variant autonomous systems
  • Local stability of hyperbolic (real parts of eigenvalues are nonzero) fixed point governed by eigenvalues (Rand 1.1; NM 3.2.1)
  • Phase portraits (Rand 1.1)

    • 3. Conservative Systems: Period of Free Vibration

  •  Phase portraits, x' vs. x by energy methods (NM 1.2, 2.1-2..2; Rand 2.0)
  • Elliptic integral method for period (NM 2.2; Rand 2.2)
  • Regular perturbation fails: secular terms (NM 2.3.1)
  • Lindstedt's method: allow frequency to be amplitude dependent (NM 2.3.2; Rand 2.1)
  • Harmonic-balance method (NM 2.3.4)
  • Period indeed is amplitude dependent (NM 2.4; Rand 2.1-2)

    • 4. Limit Cycles

  •  Polar coordinates (notes)
  • Hopf bifurcation (Rand 3.2)
  • Multiple scales (two-variable expansion) introduced in analyzing Rayleigh's equation (notes)
  • Averaging: variation of constants viewpoint (Rand 3.1; NM 3.3.4)
  • Relaxation oscillations (Rand 3.4; NM 3.3.4, 3.5)
  • Friction-induced vibration: piecewise linearity and stick-slip (notes)
  • Bendixson's criterion for the nonexistence of limit cycles (notes)
  • Poincaré index: classification of equilibria, nonexistence of limit cycles (notes).

    • 5. Forced Vibration and Nonlinear Resonance (NM 4.1, 4.3; Rand 4, 5)

  •  Harmonic balance method on Duffing equation
  • Jump phenomena
  • Coexistence of steady-state responses
  • Multiple scales (two-variable expansion): resonances and stabilities
  • Primary resonance
  • Secondary resonances: subharmonic and superharmonic (NM)
  • Combination resonances (NM)
  • Averaging: same results
  • Forced systems with limit cycles
  • Resonance
  • Entrainment, quenching, and quasiperiodic responses

    • 6. Floquet Theory (NM 5; Rand 6)

  • 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

    • 7.  Multi-Degree-of-Freedom Systems (NM 6.2; Rand 9-10)

  • Swinging spring (NM 6.2)
  • Internal resonance: spring freq. near twice the pendulum freq. (NM 6.2)
  • Beating motion and exchange of energy between modes (NM 6.2)
  •  Phase locking in coupled oscillators (Rand 9).  
  •  Traveling waves (Rand 10).

    • Books on Reserve at the Engineering Library:

    Nonlinear Oscillations
    A. H. Nayfeh and D. T. Mook
    Wiley-Interscience, 1979

    Introduction to Perturbation Techniques
    A. H. Nayfeh
    Wiley, 1980