Example 9: Subharmonic response of a Duffing oscillator

MMS example on a Duffing oscillator subject to hard forcing triggering a subharmonic response. This configuration was studied by Nayfeh and Mook in Nonlinear Oscillations (1995), sections 4.1.2 and 4.1.4.

System description

Nonlinear system.

Illustration of a forced Duffing oscillator. Subharmonic oscillations (of order 3) are triggered provided \(F\) is large and \(\omega_0 \approx 1/3 \omega\).

The system’s equation is

\[\ddot{x} + c \dot{x} + \gamma \dot{x}^{3} + \omega_{0}^{2} x = F \cos(\omega t),\]

where

  • \(x\) is the oscillator’s coordinate,

  • \(t\) is the time,

  • \(\dot{(\bullet)} = \mathrm{d}(\bullet)/\mathrm{d}t\) is a time derivative,

  • \(c\) is the linear viscous damping coefficient,

  • \(\omega_0\) is the oscillator’s natural frequency,

  • \(\gamma\) is the nonlinear coefficient,

  • \(F\) is the forcing amplitude,

  • \(\omega\) is the forcing frequency.

A parametric response around 3 times the oscillator’s frequency is sought so the frequency is set to

\[\omega = 3\omega_0 + \epsilon \sigma\]

where

  • \(\epsilon\) is a small parameter involved in the MMS,

  • \(\sigma\) is the detuning.

The parameters are then scaled to indicate how weak they are:

  • \(c = \epsilon \tilde{c}\) indicates that damping is weak,

  • \(\gamma = \epsilon \tilde{\gamma}\) indicates that nonlinear damping is weak.

Note that the forcing is not scaled, i.e. \(F\) appears at leading order. This is called hard forcing.

Code description

The script below allows to

  • Construct the dynamical system.

  • Apply the MMS to the system,

  • Evaluate the MMS results at steady state,

  • Compute the forced response and the backbone curve,

 1# -*- coding: utf-8 -*-
 2
 3#%% Imports and initialisation
 4from sympy import symbols, Function
 5from sympy.physics.vector.printing import init_vprinting, vlatex
 6init_vprinting(use_latex=True, forecolor='White') # Initialise latex printing 
 7from oscilate import MMS
 8
 9# Parameters and variables
10omega0, F, c = symbols(r'\omega_0, F, c', real=True,positive=True)
11gamma        = symbols(r'\gamma',real=True)
12t            = symbols('t')
13x            = Function(r'x', real=True)(t)
14
15# Dynamical system
16Eq  = x.diff(t,2) + omega0**2*x + gamma*x**3 + c*x.diff(t) # Equation (unforced)
17dyn = MMS.Dynamical_system(t, x, Eq, omega0, F=F)
18
19# Initialisation of the MMS sytem
20eps            = symbols(r"\epsilon", real=True, positive=True) # Small parameter epsilon
21Ne             = 1      # Order of the expansions
22omega_ref      = omega0 # Reference frequency
23ratio_omegaMMS = 3      # Look for a subharmonic resonance of order 3
24
25param_to_scale = (gamma, F, c)
26scaling        = (1    , 0, 1)
27param_scaled, sub_scaling = MMS.scale_parameters(param_to_scale, scaling, eps)
28
29mms = MMS.Multiple_scales_system(dyn, eps, Ne, omega_ref, sub_scaling, ratio_omegaMMS=ratio_omegaMMS)
30
31# Application of the MMS
32mms.apply_MMS(rewrite_polar="all")
33
34# Evaluation at steady state
35ss = MMS.Steady_state(mms)
36
37# Solve the evolution equations for a given dof
38solve_dof = 0 # dof to solve for
39ss.solve_bbc(solve_dof=solve_dof, c=param_scaled[-1])
40ss.solve_forced(solve_dof=solve_dof)
41
42# %%