Nonlinear vibrations

Nonlinear effects

1. Duffing oscillator
2. Potential well escape (static/dynamic)
3. Self-oscillation (limiting cycles)
4. Bifurcations (various):
   • changing phase diagram (Hopf)
   • jumping between attractors (Lorenz)
   • appearing/disappearing limiting cycle (VDP)
   • stochastic
5. Stochastic (sys. with 3 and more DOFs like stochastic pendulim, Lorenz attractor)
6. Autoparametric resonance (modes coupling due to nonlinear terms)
7. Sub/super-harmonic resonances (duffing)
8. Instabilities (Mathieu equation, autoparametric resonances)

Seminar plan

1st part

1. Начать с Pi-теоремы размерности, которая превращает физическую задачу в чисто математическую
2. показать, что в книге сказано про straightforward approach
3. показать в Wikipedia нет зависимости от амплитуды
4. Duffing (amplitude-frequency interaction, subharmonic resonance)
5. Autoparametric resonance

2nd part

1. talk about nonlinear effects (self oscillations, limiting cycle, multivibrator)
2. open original paper and show the generator and equations
3. demonstrate notebook with derivation of limiting cycle for small amplitudes
4. explain the difference
5. show numerical examples
6. open Nayfeh for demonstrating strong nonlinear case
7. explain this derivation in my notebook
8. show intersting thing in the original paper

Interesting article

W.L. Keith, R.H. Rand, Dynamics of a system exhibiting the global bifurcation of a limit cycle at infinity, International Journal of Non-Linear Mechanics, Volume 20, Issue 4, 1985, https://doi.org/10.1016/0020-7462(85)90040-X

References

1. Nayfeh - Nonlinear oscillations
2. Nayfeh - Perturbation methods
3. Richard H. Rand - Lecture notes on nonlinear vibrations
4. Strogatz - Nonlinear dynamics and chaos
5. Tondl - Nonlinear vibrations
6. Tondl - Autoparametric resonance in mechanical systems