Technion – Israel Institute of Technology
Department of Mechanical Engineering

The objectives of this course are to introduce and develop an understanding of nonlinear vibrations of (deterministic) dynamical systems that govern the domain and applications of mechanical engineering. The topics of the course include free, self-excited and forced (external/parametric) vibration of multiple rigidbody and continuum systems that are near to (or far from) primary or secondary (ultrasubharmonic) resonances, and that undergo internal or combination resonances. The methodology incorporates both analytical (asymptotic expansions/solvability) and numerical techniques (algebraic/integration), local (orbital) stability of nonlinear system response and an introduction to bifurcation theory.


  • nonlinear modeling (overview):
    1. single/multi-degree-of-freedom dynamical systems with single/multiple inputs.
    2. evolution equations obtained from continuous systems.
    3. existence of nonlinear resonances (primary/secondary/internal/combination/sub-combination).
  • asymptotic techniques:
    1. regular perturbations (polynomial series vs. harmonic/spectral balance)
    2. singular perturbation methods (Lindstedt-Poincare’, renormalization, averaging)
    3. multiple-scales (ordinary and partial differential equations).
  • equilibrium stability vs. orbital stability of periodic (ultrasubharmonic) and quasiperiodic (beat) solutions.
  • introduction to local bifurcation theory [divergence (saddle-node/pitchfork) and flutter (Hopf)].
  • numerical techniques [integration, sampling (Poincare’ maps/spectral), bifurcation continuation].

Applications include external/parametric excitation of resonant engineering systems, self-excited oscillatory systems (fluid-structure interaction, nonlinear control, friction-induced vibration), boundary/field excitation of continuous large scale, micro- and nano-systems (string/rod/beam, buckling/whirling).


Hinch JE, Perturbation Methods, 1991.
Jordan, DW and Smith, P, Nonlinear Ordinary Differential Equations, 1988.

  • Kevorkian, J and Cole, JD, Multiple Scale and Singular Perturbation Methods, 1995.
    Khalil, KH, Nonlinear Systems, 1996.
  • Nayfeh, AH and Mook, DT, Nonlinear Oscillations, 1979.
    Thomsen, JJ, Vibration and Stability, 2003.
  • selected papers


home work problems (3 problem sets = 60%), final project/term paper (40%).

Time & Place

Winter semester 2009
Wednesday 1430-1730
Lady Davis Bldg. Rm. 441, Technion
Oded Gottlieb