Self Excited Oscillators

Contents

Self Excited Oscillators

We reproduce results for the example of a 4DOF coupled oscillator system with parametric excitation as given in Szabelski, K. & Warminki, J. Vibration of a Non-Linear Self-Excited System with Two Degrees of Freedomunder External and Parametric Excitation. Nonlinear Dynamics 1997 14:114,23–36 (1 1997). https://doi.org/10.1023/A:1008227315259

The negative linear damping with coefficient $c$ leads to exponentially growing amplitudes, the effect of which is balanced with positive nonlinear damping with coefficient $\gamma$ which starts contributing at significant oscillation amplitudes. Furthermore a nonlinear spring is coupled to the first oscillator. They are connected via a spring which induces parametric excitation via a time-dependent stiffness coefficient.

clear all; close all; clc

Generate model

[M,C,K,fnl,fext] = build_model();

Dynamical system setup

We consider the forced and parametrically excited system

$\mathbf{M}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}+\mathbf{f}(\mathbf{x},\dot{\mathbf{x}})=\epsilon\mathbf{g}(\mathbf{x},\dot{\mathbf{x}},\Omega t),$

which can be written in the first-order form as

$\mathbf{B}\dot{\mathbf{z}}=\mathbf{Az}+\mathbf{F}(\mathbf{z})+\epsilon\mathbf{G}(\mathbf{z},\phi)$

$\dot{\mathbf{\phi}}=\mathbf{\Omega}$

where

$\mathbf{z}=\left[\begin{array}{c}\mathbf{x}\\\dot{\mathbf{x}}\end{array}\right],\quad\mathbf{A}=\left[\begin{array}{cc}-\mathbf{K} & \mathbf{0}\\\mathbf{0} & \mathbf{M}\end{array}\right],\mathbf{B}=\left[\begin{array}{cc}\mathbf{C} & \mathbf{M}\\\mathbf{M} & \mathbf{0}\end{array}\right],\quad\quad\mathbf{F}(\mathbf{z})=\left[\begin{array}{c}\mathbf{-\mathbf{f}(\mathbf{x},\dot{\mathbf{x}})}\\\mathbf{0}\end{array}\right],\quad\mathbf{G}(\mathbf{z},\mathbf{\phi})=\left[\begin{array}{c}\mathbf{g}(\mathbf{x, \dot{x}},\phi)\\\mathbf{0}\end{array}\right]$

% Dynamical System
order = 2;
DS = DynamicalSystem(order);
set(DS,'order',2)
set(DS,'M',M,'C',C,'K',K,'fnl',fnl);
set(DS.Options,'Emax',5,'Nmax',10,'notation','multiindex')

Add forcing

The dynamical system is forced externally and parametrically with $\mathbf{g(x},\Omega t) = 4 \mu \left[\begin{array}{cc} k & -k \\ -k & k \end{array}\right] \cos(2\Omega t) \mathbf{x} +\left[\begin{array}{cc} q \cos (\Omega t ) \\ 0\end{array}\right]$

DS.add_forcing(fext,1e-1);

Linear Modal Analysis

% Analyse spectrum
[V,D,W] = DS.linear_spectral_analysis();
% Choose Master subspace (perform resonance analysis)

S = SSM(DS);
set(S.Options, 'reltol', 0.5,'notation','multiindex')

%Choose Master subspace
masterModes = [3,4];
S.choose_E(masterModes);

 The first 4 nonzero eigenvalues are given as 
   0.0037 + 0.5463i
   0.0037 - 0.5463i
   0.0013 + 2.5887i
   0.0013 - 2.5887i

(near) outer resonance detected for the following combination of master eigenvalues
     1     1
     1     1

These are in resonance with the follwing eigenvalues of the slave subspace
   0.0037 + 0.5463i
   0.0037 - 0.5463i

sigma_out = 2
sigma_in = 1

Forced response curves using SSMs

Obtaining forced response curve in reduced-polar coordinate

order = 5; % Approximation order

setup options

outdof = [1,2];
set(S.Options, 'reltol', 0.5,'IRtol',0.02,'notation', 'multiindex','contribNonAuto',true)
set(S.FRCOptions, 'nt', 2^7, 'nRho', 80, 'nPar',600, 'nPsi', 80, 'rhoScale', 4 )
set(S.FRCOptions, 'method','level set') % 'continuation ep'
set(S.FRCOptions, 'outdof',outdof)
set(S.FRCOptions,'coordinates','cartesian')

choose frequency range around the master mode frequency

omega0 = imag(S.E.spectrum(1));
OmegaRange =omega0*[0.95 1.1];

Extract forced response curve

startFRCSSM = tic;
FRC = S.extract_FRC('freq',OmegaRange,order);
figFRC = gcf;
timings.FRCSSM = toc(startFRCSSM)

*****************************************
Calculating FRC using SSM with master subspace: [3  4]
(near) outer resonance detected for the following combination of master eigenvalues
     1     1
     1     1

These are in resonance with the follwing eigenvalues of the slave subspace
   0.0037 + 0.5463i
   0.0037 - 0.5463i

sigma_out = 2
sigma_in = 1
Due to (near) outer resonance, the exisitence of the manifold is questionable and the underlying computation may suffer.
Attempting manifold computation
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 7.78E-03 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 9.10E-03 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 1.13E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 1.40E-02 MB
gamma = 
  -0.0022 + 0.0038i
   0.0000 - 0.0000i

Total time spent on FRC computation upto O(5) = 00:00:53

timings = 

  struct with fields:

    FRCSSM: 54.3701

Verification: Collocation using coco

Dankowicz, H., & Schilder, F. (2013). Recipes for Continuation, SIAM Philadelphia. https://doi.org/10.1137/1.9781611972573

nCycles = 10;

coco = cocoWrapper(DS, nCycles, outdof);
set(coco,'initialGuess','forward')
set(coco.Options, 'NAdapt', 1);
set(coco.Options,'NTST', 70,'PtMX',350);

figure(figFRC)
hold on
startcoco = tic;
bd3 = coco.extract_FRC(OmegaRange);
timings.cocoFRC = toc(startcoco)

 Run='FRC0.1': Continue primary family of periodic orbits.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          2.03e-02  5.35e+00    0.0    0.0    0.0
   1   1  3.59e-01  9.75e-01  1.30e-02  5.24e+00    0.0    0.0    0.0
   2   1  5.14e-01  6.26e-01  6.33e-03  5.16e+00    0.0    0.1    0.0
   3   1  9.78e-01  3.04e-01  1.37e-04  5.10e+00    0.0    0.1    0.0
   4   1  1.00e+00  6.62e-03  4.22e-11  5.10e+00    0.0    0.2    0.0
   5   1  1.00e+00  7.63e-09  3.03e-16  5.10e+00    0.1    0.2    0.0

 STEP      TIME        ||U||  LABEL  TYPE         omega    po.period          eps         amp1         amp2
    0  00:00:00   5.0981e+00      1  EP      2.4593e+00   2.5549e+00   1.0000e-01   2.3797e-02   1.1582e-02
   10  00:00:03   6.8076e+00      2          2.5558e+00   2.4584e+00   1.0000e-01   1.3156e-01   5.7213e-02
   20  00:00:05   9.8632e+00      3          2.5683e+00   2.4464e+00   1.0000e-01   2.9035e-01   1.2452e-01
   30  00:00:07   1.2868e+01      4          2.5736e+00   2.4414e+00   1.0000e-01   4.7607e-01   2.0294e-01
   40  00:00:09   1.5329e+01      5          2.5784e+00   2.4368e+00   1.0000e-01   6.9846e-01   2.9611e-01
   50  00:00:10   1.9891e+01      6          2.5839e+00   2.4317e+00   1.0000e-01   9.2886e-01   3.9175e-01
   60  00:00:12   2.4170e+01      7          2.5902e+00   2.4258e+00   1.0000e-01   1.1444e+00   4.7916e-01
   70  00:00:14   2.7873e+01      8          2.5972e+00   2.4192e+00   1.0000e-01   1.3316e+00   5.5463e-01
   80  00:00:15   3.0690e+01      9          2.6043e+00   2.4126e+00   1.0000e-01   1.4760e+00   6.1102e-01
   90  00:00:17   3.2368e+01     10          2.6105e+00   2.4069e+00   1.0000e-01   1.5633e+00   6.4542e-01
  100  00:00:18   3.2714e+01     11          2.6149e+00   2.4028e+00   1.0000e-01   1.5839e+00   6.5424e-01
  110  00:00:20   3.1777e+01     12          2.6171e+00   2.4008e+00   1.0000e-01   1.5398e+00   6.3760e-01
  115  00:00:21   3.0950e+01     13  FP      2.6173e+00   2.4006e+00   1.0000e-01   1.4991e+00   6.2161e-01
  115  00:00:21   3.0950e+01     14  SN      2.6173e+00   2.4006e+00   1.0000e-01   1.4991e+00   6.2161e-01
  120  00:00:22   2.9750e+01     15          2.6171e+00   2.4009e+00   1.0000e-01   1.4399e+00   5.9819e-01
  130  00:00:24   2.6803e+01     16          2.6153e+00   2.4025e+00   1.0000e-01   1.2926e+00   5.3974e-01
  140  00:00:26   2.3213e+01     17          2.6123e+00   2.4052e+00   1.0000e-01   1.1119e+00   4.6702e-01
  150  00:00:27   1.9177e+01     18          2.6088e+00   2.4084e+00   1.0000e-01   9.0652e-01   3.8235e-01
  160  00:00:29   1.4885e+01     19          2.6054e+00   2.4116e+00   1.0000e-01   6.8559e-01   2.9040e-01
  170  00:00:30   1.0379e+01     20          2.6024e+00   2.4143e+00   1.0000e-01   4.5684e-01   1.9379e-01
  180  00:00:31   6.4423e+00     21          2.6000e+00   2.4166e+00   1.0000e-01   2.2275e-01   9.3949e-02
  190  00:00:33   5.7212e+00     22          2.5960e+00   2.4204e+00   1.0000e-01   1.6157e-01   6.7334e-02
  200  00:00:35   6.3551e+00     23          2.5903e+00   2.4256e+00   1.0000e-01   2.2928e-01   9.5979e-02
  210  00:00:36   1.0871e+01     24  FP      2.5856e+00   2.4301e+00   1.0000e-01   5.0832e-01   2.1385e-01
  210  00:00:37   1.0871e+01     25  SN      2.5856e+00   2.4301e+00   1.0000e-01   5.0833e-01   2.1385e-01
  210  00:00:37   1.1051e+01     26          2.5856e+00   2.4301e+00   1.0000e-01   5.1893e-01   2.1831e-01
  220  00:00:38   1.5516e+01     27          2.5876e+00   2.4282e+00   1.0000e-01   7.7540e-01   3.2554e-01
  230  00:00:40   1.9842e+01     28          2.5922e+00   2.4239e+00   1.0000e-01   1.0162e+00   4.2489e-01
  240  00:00:41   2.5565e+01     29          2.5983e+00   2.4182e+00   1.0000e-01   1.2155e+00   5.0585e-01
  250  00:00:43   2.8278e+01     30          2.6046e+00   2.4124e+00   1.0000e-01   1.3548e+00   5.6157e-01
  260  00:00:45   2.9460e+01     31          2.6098e+00   2.4075e+00   1.0000e-01   1.4173e+00   5.8693e-01
  270  00:00:46   2.8754e+01     32          2.6126e+00   2.4050e+00   1.0000e-01   1.3862e+00   5.7519e-01
  275  00:00:47   2.7817e+01     33  FP      2.6129e+00   2.4047e+00   1.0000e-01   1.3404e+00   5.5721e-01
  275  00:00:47   2.7817e+01     34  SN      2.6129e+00   2.4047e+00   1.0000e-01   1.3404e+00   5.5721e-01
  280  00:00:48   2.6349e+01     35          2.6126e+00   2.4050e+00   1.0000e-01   1.2672e+00   5.2828e-01
  290  00:00:50   2.2673e+01     36          2.6103e+00   2.4070e+00   1.0000e-01   1.0816e+00   4.5386e-01
  300  00:00:51   1.8244e+01     37          2.6071e+00   2.4100e+00   1.0000e-01   8.5793e-01   3.6183e-01
  310  00:00:53   1.3576e+01     38          2.6040e+00   2.4129e+00   1.0000e-01   6.1717e-01   2.6142e-01
  320  00:00:54   8.7229e+00     39          2.6019e+00   2.4149e+00   1.0000e-01   3.6917e-01   1.5666e-01
  326  00:00:56   6.4940e+00     40  FP      2.6014e+00   2.4153e+00   1.0000e-01   2.2757e-01   9.6146e-02
  326  00:00:56   6.4937e+00     41  SN      2.6014e+00   2.4153e+00   1.0000e-01   2.2755e-01   9.6137e-02
  330  00:00:56   5.5522e+00     42          2.6022e+00   2.4146e+00   1.0000e-01   1.3495e-01   5.6308e-02
  340  00:00:58   5.2251e+00     43          2.6097e+00   2.4076e+00   1.0000e-01   8.8372e-02   3.6242e-02
  350  00:00:59   5.0528e+00     44  EP      2.7002e+00   2.3270e+00   1.0000e-01   2.3382e-02   8.7140e-03

timings = 

  struct with fields:

     FRCSSM: 54.3701
    cocoFRC: 62.2742

Published with MATLAB® R2023a