Euler Bernoulli beam with cubic spring and damper

% Generate model
clear all;
nElements = 25;
kappa = 6; % cubic spring
gamma = -0.02; % cubic damping
[M,C,K,fnl,f_0] = build_model(kappa, gamma, nElements);
n = length(M);
% Dynamical system setup

DSorder = 2;
DS = DynamicalSystem(DSorder);
set(DS,'M',M,'C',C,'K',K,'fnl',fnl);
set(DS.Options,'Emax',3,'Nmax',10,'notation','multiindex')

We assume periodic forcing of the form

$\mathbf{f}^{ext}(\phi) = \mathbf{f}_0\cos(\phi)=\frac{\mathbf{f}_0}{2}e^{i\phi} + \frac{\mathbf{f}_0}{2}e^{-i\phi}$

epsilon = 0.01;

Fourier coefficients of Forcing

kappas = [-1; 1];
coeffs = [f_0 f_0]/2;
DS.add_forcing(coeffs, kappas,epsilon);
% Linear Modal analysis and SSM setup

[V,D,W] = DS.linear_spectral_analysis();

Due to high-dimensionality, we compute only the first 3 eigenvalues with the smallest magnitude. These would also be used to compute the spectral quotients
Assuming a proportional damping hypthesis with symmetric matrices
modal damping ratio for 1 mode is 8.839900e-04
modal damping ratio for 2 mode is 5.485350e-03
modal damping ratio for 3 mode is 1.535575e-02

 The first 6 nonzero eigenvalues are given as 
   1.0e+02 *

  -0.0001 + 0.0700i
  -0.0001 - 0.0700i
  -0.0024 + 0.4387i
  -0.0024 - 0.4387i
  -0.0189 + 1.2283i
  -0.0189 - 1.2283i

Choose Master subspace (perform resonance analysis)

S = SSM(DS);
set(S.Options, 'reltol', 0.5,'notation','multiindex')
masterModes = [1,2];
S.choose_E(masterModes);
% Forced response curves using SSMs
% Obtaining *forced response curve* in reduced coordinate

order = 5; % SSM approximation order
outdof = n-1; %[n-1 n 2*n-1 2*n]; % degree of freedom at which output is displayed
freqRange = [6.96 7.04];
epsRange = [0.01 1]*epsilon;
optdof = outdof;
mFreq  = 1;
set(S.contOptions, 'h0',1e-3,'h_min',1e-3,'h_max',0.005, 'PtMX', 250);  % continuation setting
set(S.FRCOptions, 'initialSolver', 'fsolve');   % solver for initial solution
set(S.FRCOptions, 'coordinates','cartesian');
set(S.FRSOptions, 'calFRS', true);
set(S.Options, 'contribNonAuto', false, 'COMPtype', 'first');

(near) outer resonance detected for the following combination of master eigenvalues
     6     0
     7     1
     8     2
     0     6
     1     7
     2     8

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.2406 +43.8707i
  -0.2406 +43.8707i
  -0.2406 +43.8707i
  -0.2406 -43.8707i
  -0.2406 -43.8707i
  -0.2406 -43.8707i

sigma_out = 304
(near) inner resonance detected for the following combination of master eigenvalues
     2     1
     3     2
     4     3
     5     4
     1     2
     2     3
     3     4
     4     5

These are in resonance with the follwing eigenvalues of the master subspace
  -0.0062 + 7.0005i
  -0.0062 + 7.0005i
  -0.0062 + 7.0005i
  -0.0062 + 7.0005i
  -0.0062 - 7.0005i
  -0.0062 - 7.0005i
  -0.0062 - 7.0005i
  -0.0062 - 7.0005i

sigma_in = 304

We calcuate the frequency response surface.

(1) two-dimensional continuation based computation

set(S.contOptions,'atlasdim',2, 'R', 0.01,'R_max',10,'R_min',1e-4,'almax',9.5,'PtMX',2000);
parRange = {freqRange,epsRange};
p0 = [6.97 0.5*epsilon]';
z0 = [1e-3 0]';
set(S.FRSOptions, 'method', 'continuation');
startfrs = tic;
S.extract_FRS('frs_cont',masterModes,order,mFreq,parRange,outdof,optdof,[],[],{p0,z0});
timings.FRS = toc(startfrs);

(near) outer resonance detected for the following combination of master eigenvalues
     6     0
     7     1
     8     2
     0     6
     1     7
     2     8

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.2406 +43.8707i
  -0.2406 +43.8707i
  -0.2406 +43.8707i
  -0.2406 -43.8707i
  -0.2406 -43.8707i
  -0.2406 -43.8707i

sigma_out = 304
(near) inner resonance detected for the following combination of master eigenvalues
     2     1
     3     2
     4     3
     5     4
     1     2
     2     3
     3     4
     4     5

These are in resonance with the follwing eigenvalues of the master subspace
  -0.0062 + 7.0005i
  -0.0062 + 7.0005i
  -0.0062 + 7.0005i
  -0.0062 + 7.0005i
  -0.0062 - 7.0005i
  -0.0062 - 7.0005i
  -0.0062 - 7.0005i
  -0.0062 - 7.0005i

sigma_in = 304
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 = 6.33E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 9.40E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 1.36E-01 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 2.05E-01 MB

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.


 Run='frs_cont.FRSep': Continue equilibria along FRS.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          1.59e-11  9.86e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om          eps         Rez1         Imz1          amp        l2z49      linfz49         rho1
    0  00:00:00   9.8571e+00      1  EP      6.9700e+00   5.0000e-03   1.6162e-03  -3.1090e-04   6.7148e-03   6.7148e-03   9.4959e-03   1.6458e-03
    5  00:00:01   9.8457e+00      2  EP      6.9619e+00   1.0000e-02   2.5798e-03  -3.6210e-04   1.0628e-02   1.0628e-02   1.5030e-02   2.6050e-03
    6  00:00:02   9.8429e+00      3  EP      6.9600e+00   2.4214e-03   6.0169e-04  -9.1317e-05   2.4830e-03   2.4830e-03   3.5114e-03   6.0858e-04
   10  00:00:03   9.8529e+00      4          6.9670e+00   1.5938e-03   4.7505e-04  -8.7999e-05   1.9712e-03   1.9712e-03   2.7875e-03   4.8313e-04
   11  00:00:03   9.8684e+00      5  EP      6.9780e+00   1.0000e-02   4.1373e-03  -7.5304e-04   1.7155e-02   1.7155e-02   2.4261e-02   4.2052e-03
   20  00:00:05   9.8869e+00      6          6.9911e+00   5.2768e-03   4.2631e-03  -1.5488e-03   1.8503e-02   1.8503e-02   2.6167e-02   4.5357e-03
   24  00:00:06   9.8987e+00      7  EP      6.9994e+00   1.0000e-04   2.7879e-05  -1.6175e-04   6.6967e-04   6.6967e-04   9.4701e-04   1.6414e-04
   29  00:00:08   9.8681e+00      8  EP      6.9778e+00   9.9796e-05   4.2090e-05  -1.1916e-05   1.7847e-04   1.7847e-04   2.5240e-04   4.3744e-05
   30  00:00:08   9.8642e+00      9          6.9750e+00   6.9884e-04   2.6686e-04  -6.4503e-05   1.1201e-03   1.1201e-03   1.5840e-03   2.7454e-04
   31  00:00:08   9.8752e+00     10  EP      6.9828e+00   1.0000e-04   5.1984e-05  -1.8226e-05   2.2475e-04   2.2475e-04   3.1783e-04   5.5086e-05
   32  00:00:09   9.8540e+00     11  EP      6.9678e+00   1.0001e-04   3.0999e-05  -6.3632e-06   1.2911e-04   1.2911e-04   1.8259e-04   3.1646e-05
   40  00:00:11   9.8898e+00     12          6.9931e+00   3.3985e-03   3.0865e-03  -1.7720e-03   1.4519e-02   1.4519e-02   2.0533e-02   3.5590e-03
   42  00:00:11   9.8863e+00     13  EP      6.9906e+00   1.0000e-04   7.5010e-05  -4.7134e-05   3.6144e-04   3.6144e-04   5.1113e-04   8.8589e-05

   .
   . 
   .
   
 1987  00:15:32   9.9373e+00    494  EP      7.0267e+00   9.9802e-05  -3.7382e-05  -8.8366e-06   1.5672e-04   1.5672e-04   2.2162e-04   3.8412e-05
 1989  00:15:40   9.9373e+00    495  EP      7.0267e+00   9.9802e-05  -3.7382e-05  -8.8366e-06   1.5672e-04   1.5672e-04   2.2162e-04   3.8412e-05
 1990  00:15:40   9.9375e+00    496          7.0268e+00   1.5287e-04  -5.6880e-05  -1.3371e-05   2.3839e-04   2.3839e-04   3.3712e-04   5.8430e-05
 1991  00:15:49   9.9373e+00    497  EP      7.0267e+00   9.9802e-05  -3.7382e-05  -8.8366e-06   1.5672e-04   1.5672e-04   2.2162e-04   3.8412e-05
 1993  00:15:57   9.9373e+00    498  EP      7.0267e+00   9.9802e-05  -3.7382e-05  -8.8366e-06   1.5672e-04   1.5672e-04   2.2162e-04   3.8412e-05
 1995  00:16:05   9.9373e+00    499  EP      7.0267e+00   9.9802e-05  -3.7382e-05  -8.8366e-06   1.5672e-04   1.5672e-04   2.2162e-04   3.8412e-05
 1997  00:16:13   9.9373e+00    500  EP      7.0267e+00   9.9802e-05  -3.7382e-05  -8.8366e-06   1.5672e-04   1.5672e-04   2.2162e-04   3.8412e-05
 1999  00:16:21   9.9373e+00    501  EP      7.0267e+00   9.9802e-05  -3.7382e-05  -8.8366e-06   1.5672e-04   1.5672e-04   2.2162e-04   3.8412e-05
 2000  00:16:22   9.9375e+00    502  EP      7.0268e+00   1.5287e-04  -5.6880e-05  -1.3371e-05   2.3839e-04   2.3839e-04   3.3712e-04   5.8430e-05

runid = 'frs_cont.FRSep';
figure;
plot_atlas_2d(runid, 1, 2, 5);
xlim(freqRange); ylim(epsRange);
xlabel('$\Omega$','FontSize',16,'Interpreter',"latex");
ylabel('$\epsilon$','FontSize',16,'Interpreter',"latex");
zlabel('$A_{\mathcal{L}_2}$','FontSize',16,'Interpreter',"latex");

(2) analytic prediction

set(S.FRSOptions, 'method', 'level set');
tic
set(S.FRSOptions, 'rhoMax', 0.02, 'meshDens', 100);
S.extract_FRS('frs_ana',masterModes,order,1,parRange,outdof,optdof,[],[]);
toc

(near) outer resonance detected for the following combination of master eigenvalues
     6     0
     7     1
     8     2
     0     6
     1     7
     2     8

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.2406 +43.8707i
  -0.2406 +43.8707i
  -0.2406 +43.8707i
  -0.2406 -43.8707i
  -0.2406 -43.8707i
  -0.2406 -43.8707i

sigma_out = 304
(near) inner resonance detected for the following combination of master eigenvalues
     2     1
     3     2
     4     3
     5     4
     1     2
     2     3
     3     4
     4     5

These are in resonance with the follwing eigenvalues of the master subspace
  -0.0062 + 7.0005i
  -0.0062 + 7.0005i
  -0.0062 + 7.0005i
  -0.0062 + 7.0005i
  -0.0062 - 7.0005i
  -0.0062 - 7.0005i
  -0.0062 - 7.0005i
  -0.0062 - 7.0005i

sigma_in = 304
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 = 6.33E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 9.40E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 1.36E-01 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 2.05E-01 MB
gamma = 
   1.0e+03 *

   0.1018 + 0.0891i
   0.6709 - 2.7512i

the number of faces is 42538
Elapsed time is 4.069162 seconds.

Published with MATLAB® R2023a