Axially Moving Beam

Nonlinear Damping
Linear Damping
Comparison of FRC with/without nonlinear damping

We consider an axially moving beam under 1:3 internal resonance between the first two bending modes. The equation of motion is given by

$\ddot{\mathbf{u}}+(\mathbf{C}+\mathbf{G})\dot{\mathbf{u}}+\mathbf{f}(\mathbf{u},\dot{\mathbf{u}})=\epsilon\Omega^2\mathbf{g}\cos\Omega t$

where $\mathbf{G}^\top=-\mathbf{G}$ is a gyroscopic matrix. With viscoelastic material model, the system has nonlinear damping. In addition, the beam is subject to base excitation such that the forcing amplitude is a function of forcing frequency.

In this notebook, we will calculate the forced response curve of the system with/without the nonlinear damping taken into consideration to explore the effects of nonlinear damping

Nonlinear Damping

Setup Dynamical System

clear all;
n = 10;
[mass,damp,gyro,stiff,fnl,fext] = build_model(n,'nonlinear_damp');

the first four eigenvalues for undamped system

lamd =

  -0.0000 + 3.1954i
  -0.0000 - 3.1954i
  -0.0000 + 9.5862i
  -0.0000 - 9.5862i

Create model

Given the assembled damping matrix $\mathbf{C}+\mathbf{G}$ is not Rayleigh damping anymore, one should set the corresponding Options in DS to be false.
The BaseExcitation option in DS should be set true to account for the $\Omega$ -dependent forcing amplitude.

order = 2
DS = DynamicalSystem(order);
set(DS,'M',mass,'C',damp+gyro,'K',stiff,'fnl',fnl);
set(DS.Options,'Emax',6,'Nmax',10,'notation','multiindex');
set(DS.Options,'RayleighDamping',false,'BaseExcitation',true);

% Forcing
h = 1.5e-4; % h characterizes the vibration amplitude of base excitation. It plays the role of epsilon here
kappas = [-1; 1];
coeffs = [fext fext]/2;
DS.add_forcing(coeffs, kappas, h);
% Linear Modal analysis

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

 The first 6 nonzero eigenvalues are given as 
  -0.0180 + 3.1954i
  -0.0180 - 3.1954i
  -0.2721 + 9.5828i
  -0.2721 - 9.5828i
  -1.3571 +19.6071i
  -1.3571 -19.6071i

Choose Master subspace

Due to the 1:3 internal resonance, we take the first two complex conjugate pairs of modes as the spectral subspace to SSM. So we hvae resonant_modes = [1 2 3 4].

S = SSM(DS);
set(S.Options, 'reltol', 1,'notation','multiindex');
resonant_modes = [1 2 3 4];
order = 3;
outdof = [1 2];

Primary resonance of the first mode

We consider the case that $\Omega\approx\omega_1$ . Although the second mode is not excited externally ( $f_2=\langle \phi_2,f^\mathrm{ext}\rangle=0$ ), the response of the second mode is nontrivial due to the modal interactions.

freqrange = [0.98 1.04]*imag(D(1));
set(S.FRCOptions, 'nCycle',500, 'initialSolver', 'fsolve');
set(S.contOptions, 'PtMX', 300, 'h0', 0.1, 'h_max', 0.2, 'h_min', 1e-3);
set(S.FRCOptions, 'coordinates', 'polar');

We first compute the FRC with O(3,5,7) expansion of SSM and then check the convergence of FRC with increasing orders.

% O(3)
start = tic;
FRC_ND_O3 = S.SSM_isol2ep('isol-nd-3',resonant_modes, order, [1 3], 'freq', freqrange,outdof);
timings.FRC_ND_O3 = toc(start);
% O(5)
sol = ep_read_solution('isol-nd-3.ep',1);
start = tic;
FRC_ND_O5 = S.SSM_isol2ep('isol-nd-5',resonant_modes, order+2, [1 3],...
    'freq', freqrange,outdof,{sol.p,sol.x});
timings.FRC_ND_O5 = toc(start);
fig_ssm = gcf;
% O(7)
start = tic;
FRC_ND_O7 = S.SSM_isol2ep('isol-nd-7',resonant_modes, order+4, [1 3],...
    'freq', freqrange,outdof,{sol.p,sol.x});
timings.FRC_ND_O7 = toc(start);

(near) outer resonance detected for the following combination of master eigenvalues
     0     0     2     0
     1     0     2     0
     0     0     3     1
     .
     .
     .
     5     2     0     3
     6     0     0     4

These are in resonance with the follwing eigenvalues of the slave subspace
  -1.3571 +19.6071i
  -1.3571 +19.6071i
  -1.3571 +19.6071i
  .
  .
  .
  -1.3571 -19.6071i
  -1.3571 -19.6071i
  -1.3571 -19.6071i

sigma_out = 75
(near) inner resonance detected for the following combination of master eigenvalues
     0     2     1     0
     1     0     1     1
     2     1     0     0
     .
     .
     6     0     0     3
     2     7     1     0
     4     6     0     0

These are in resonance with the follwing eigenvalues of the master subspace
  -0.0180 + 3.1954i
  -0.0180 + 3.1954i
  -0.0180 + 3.1954i
  .
  .
  -0.2721 - 9.5828i
  -0.2721 - 9.5828i
  -0.2721 - 9.5828i

Order 3 SSM Computation

sigma_in = 75
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 = 3.63E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 6.73E-02 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='isol-nd-3.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          4.37e-16  1.15e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   1.1478e+01      1  EP      3.1954e+00   2.1592e-03   2.5798e-06   4.1005e+00   6.2323e+00   1.5000e-04
   10  00:00:00   1.3116e+01      2          3.1473e+00   1.1492e-03   3.4299e-07   4.3980e+00   7.5342e+00   1.5000e-04
   13  00:00:00   1.3485e+01      3  EP      3.1315e+00   9.3460e-04   1.7140e-07   4.4563e+00   7.8265e+00   1.5000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   1.1478e+01      4  EP      3.1954e+00   2.1592e-03   2.5798e-06   4.1005e+00   6.2323e+00   1.5000e-04
   10  00:00:00   9.9061e+00      5          3.2333e+00   3.0311e-03   6.9840e-06   3.7729e+00   4.9372e+00   1.5000e-04
   20  00:00:01   8.3616e+00      6          3.2645e+00   3.6281e-03   1.1061e-05   3.3939e+00   3.5752e+00   1.5000e-04
   30  00:00:01   7.0729e+00      7  FP      3.2762e+00   3.7209e-03   1.1387e-05   2.9981e+00   2.3003e+00   1.5000e-04
   30  00:00:01   7.0729e+00      8  SN      3.2762e+00   3.7209e-03   1.1387e-05   2.9981e+00   2.3003e+00   1.5000e-04
   30  00:00:01   7.0035e+00      9          3.2761e+00   3.7082e-03   1.1263e-05   2.9734e+00   2.2250e+00   1.5000e-04
   40  00:00:01   5.9593e+00     10          3.2628e+00   3.0886e-03   6.6607e-06   2.5146e+00   8.8743e-01   1.5000e-04
   50  00:00:01   5.4889e+00     11  SN      3.2486e+00   1.9500e-03   1.6982e-06   2.0947e+00  -3.5064e-01   1.5000e-04
   50  00:00:01   5.4889e+00     12  FP      3.2486e+00   1.9500e-03   1.6982e-06   2.0947e+00  -3.5065e-01   1.5000e-04
   50  00:00:01   5.4804e+00     13          3.2488e+00   1.8476e-03   1.4384e-06   2.0635e+00  -4.5269e-01   1.5000e-04
   60  00:00:01   5.8787e+00     14          3.3050e+00   6.8295e-04   5.2288e-08   1.7381e+00  -1.8263e+00   1.5000e-04
   62  00:00:01   5.9804e+00     15  EP      3.3232e+00   5.8904e-04   3.0413e-08   1.7133e+00  -1.9757e+00   1.5000e-04

the forcing frequency 3.1315e+00 is nearly resonant with the eigenvalue -1.7969e-02 + i3.1954e+00
the forcing frequency 3.1324e+00 is nearly resonant with the eigenvalue -1.7969e-02 + i3.1954e+00
the forcing frequency 3.1402e+00 is nearly resonant with the eigenvalue -1.7969e-02 + i3.1954e+00
.
.
the forcing frequency 3.3217e+00 is nearly resonant with the eigenvalue -1.7969e-02 + i3.1954e+00
the forcing frequency 3.3232e+00 is nearly resonant with the eigenvalue -1.7969e-02 + i3.1954e+00

Order 5 SSM Computation


sigma_in = 75
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 = 3.85E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 6.95E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 1.34E-01 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 2.67E-01 MB

Equation solved, inaccuracy possible.

The vector of function values is near zero, as measured by the value
of the function tolerance. However, the last step was ineffective.


 Run='isol-nd-5.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          2.51e-13  1.15e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   1.1467e+01      1  EP      3.1954e+00   2.1663e-03   2.6319e-06   4.0982e+00   6.2242e+00   1.5000e-04
   10  00:00:00   1.3105e+01      2          3.1476e+00   1.1554e-03   3.5008e-07   4.3963e+00   7.5260e+00   1.5000e-04
   13  00:00:00   1.3484e+01      3  EP      3.1315e+00   9.3472e-04   1.7179e-07   4.4563e+00   7.8262e+00   1.5000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   1.1467e+01      4  EP      3.1954e+00   2.1663e-03   2.6319e-06   4.0982e+00   6.2242e+00   1.5000e-04
   10  00:00:00   9.8949e+00      5          3.2325e+00   3.0391e-03   7.1899e-06   3.7689e+00   4.9295e+00   1.5000e-04
   20  00:00:00   8.3489e+00      6          3.2625e+00   3.6323e-03   1.1489e-05   3.3879e+00   3.5680e+00   1.5000e-04
   30  00:00:00   7.0499e+00      7  SN      3.2735e+00   3.7143e-03   1.1769e-05   2.9878e+00   2.2821e+00   1.5000e-04
   30  00:00:00   7.0499e+00      8  FP      3.2735e+00   3.7143e-03   1.1769e-05   2.9878e+00   2.2821e+00   1.5000e-04
   30  00:00:00   6.9910e+00      9          3.2734e+00   3.7029e-03   1.1652e-05   2.9668e+00   2.2180e+00   1.5000e-04
   40  00:00:00   5.9514e+00     10          3.2614e+00   3.0779e-03   6.7577e-06   2.5098e+00   8.7979e-01   1.5000e-04
   50  00:00:00   5.4902e+00     11  SN      3.2484e+00   1.9641e-03   1.7518e-06   2.0991e+00  -3.3600e-01   1.5000e-04
   50  00:00:00   5.4902e+00     12  FP      3.2484e+00   1.9641e-03   1.7517e-06   2.0991e+00  -3.3600e-01   1.5000e-04
   50  00:00:00   5.4796e+00     13          3.2486e+00   1.8388e-03   1.4285e-06   2.0608e+00  -4.6107e-01   1.5000e-04
   60  00:00:00   5.8842e+00     14          3.3059e+00   6.7742e-04   5.0824e-08   1.7367e+00  -1.8350e+00   1.5000e-04
   62  00:00:01   5.9805e+00     15  EP      3.3232e+00   5.8903e-04   3.0435e-08   1.7133e+00  -1.9758e+00   1.5000e-04

the forcing frequency 3.1315e+00 is nearly resonant with the eigenvalue -1.7969e-02 + i3.1954e+00
the forcing frequency 3.1329e+00 is nearly resonant with the eigenvalue -1.7969e-02 + i3.1954e+00
the forcing frequency 3.1407e+00 is nearly resonant with the eigenvalue -1.7969e-02 + i3.1954e+00
.
.

Order 7 SSM Computation


sigma_in = 75
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 = 4.25E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 7.34E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 1.38E-01 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 2.71E-01 MB
Manifold computation time at order 6 = 00:00:00
Estimated memory usage at order  6 = 4.66E-01 MB
Manifold computation time at order 7 = 00:00:00
Estimated memory usage at order  7 = 8.31E-01 MB

Equation solved, inaccuracy possible.

The vector of function values is near zero, as measured by the value
of the function tolerance. However, the last step was ineffective.


 Run='isol-nd-7.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          2.38e-13  1.15e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   1.1467e+01      1  EP      3.1954e+00   2.1661e-03   2.6305e-06   4.0982e+00   6.2244e+00   1.5000e-04
   10  00:00:00   1.3106e+01      2          3.1476e+00   1.1552e-03   3.4993e-07   4.3964e+00   7.5262e+00   1.5000e-04
   13  00:00:00   1.3484e+01      3  EP      3.1315e+00   9.3472e-04   1.7179e-07   4.4563e+00   7.8262e+00   1.5000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   1.1467e+01      4  EP      3.1954e+00   2.1661e-03   2.6305e-06   4.0982e+00   6.2244e+00   1.5000e-04
   10  00:00:00   9.8951e+00      5          3.2326e+00   3.0390e-03   7.1791e-06   3.7690e+00   4.9297e+00   1.5000e-04
   20  00:00:00   8.3493e+00      6          3.2626e+00   3.6323e-03   1.1455e-05   3.3881e+00   3.5681e+00   1.5000e-04
   30  00:00:00   7.0523e+00      7  SN      3.2736e+00   3.7149e-03   1.1736e-05   2.9887e+00   2.2844e+00   1.5000e-04
   30  00:00:00   7.0523e+00      8  FP      3.2736e+00   3.7149e-03   1.1736e-05   2.9887e+00   2.2844e+00   1.5000e-04
   30  00:00:00   6.9914e+00      9          3.2736e+00   3.7032e-03   1.1616e-05   2.9670e+00   2.2182e+00   1.5000e-04
   40  00:00:00   5.9516e+00     10          3.2614e+00   3.0781e-03   6.7493e-06   2.5099e+00   8.7997e-01   1.5000e-04
   50  00:00:00   5.4902e+00     11  SN      3.2484e+00   1.9638e-03   1.7504e-06   2.0990e+00  -3.3633e-01   1.5000e-04
   50  00:00:00   5.4902e+00     12  FP      3.2484e+00   1.9637e-03   1.7504e-06   2.0990e+00  -3.3634e-01   1.5000e-04
   50  00:00:00   5.4797e+00     13          3.2486e+00   1.8390e-03   1.4287e-06   2.0609e+00  -4.6087e-01   1.5000e-04
   60  00:00:00   5.8841e+00     14          3.3059e+00   6.7755e-04   5.0859e-08   1.7367e+00  -1.8348e+00   1.5000e-04
   62  00:00:00   5.9805e+00     15  EP      3.3232e+00   5.8903e-04   3.0435e-08   1.7133e+00  -1.9758e+00   1.5000e-04

the forcing frequency 3.1315e+00 is nearly resonant with the eigenvalue -1.7969e-02 + i3.1954e+00
the forcing frequency 3.1329e+00 is nearly resonant with the eigenvalue -1.7969e-02 + i3.1954e+00
the forcing frequency 3.1406e+00 is nearly resonant with the eigenvalue -1.7969e-02 + i3.1954e+00

Plot FRC at different orders in the same figure to observe the convergence

FRCs = {FRC_ND_O3,FRC_ND_O5,FRC_ND_O7};
thm = struct();
thm.SN = {'LineStyle', 'none', 'LineWidth', 2, ...
  'Color', 'cyan', 'Marker', 'o', 'MarkerSize', 8, 'MarkerEdgeColor', ...
  'cyan', 'MarkerFaceColor', 'white'};
thm.HB = {'LineStyle', 'none', 'LineWidth', 2, ...
  'Color', 'black', 'Marker', 's', 'MarkerSize', 8, 'MarkerEdgeColor', ...
  'black', 'MarkerFaceColor', 'white'};
color = {'r','k','b','m'};
figure(20);
ax1 = gca;
for k=1:3
    FRC = FRCs{k};
    SNidx = FRC.SNidx;
    HBidx = FRC.HBidx;
    FRC.st = double(FRC.st);
    FRC.st(HBidx) = nan;
    FRC.st(SNidx) = nan;
    % color
    ST = cell(2,1);
    ST{1} = {[color{k},'--'],'LineWidth',1.5}; % unstable
    ST{2} = {[color{k},'-'],'LineWidth',1.5};  % stable
    legs = ['SSM-$\mathcal{O}(',num2str(2*k+1),')$-unstable'];
    legu = ['SSM-$\mathcal{O}(',num2str(2*k+1),')$-stable'];
    hold(ax1,'on');
    plot_stab_lines(FRC.om,FRC.Aout_frc(:,1),FRC.st,ST,legs,legu);
    SNfig = plot(FRC.om(SNidx),FRC.Aout_frc(SNidx,1),thm.SN{:});
    set(get(get(SNfig,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
    HBfig = plot(FRC.om(HBidx),FRC.Aout_frc(HBidx,1),thm.HB{:});
    set(get(get(HBfig,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
    xlabel('$\Omega$','Interpreter','latex');
    ylabel('$||u_1||_{\infty}$','Interpreter','latex');
    set(gca,'FontSize',14);
    grid on; axis tight;
end
% Validation using collocation method from COCO

figure(fig_ssm); hold on
nCycles = 500;
coco = cocoWrapper(DS, nCycles, outdof);
set(coco.Options, 'PtMX', 1000, 'NTST',20, 'dir_name', 'bd_nd');
set(coco.Options, 'NAdapt', 0, 'h_max', 200, 'MaxRes', 1);
coco.initialGuess = 'linear';
start = tic;
bd_nd = coco.extract_FRC(freqrange);
timings.cocoFRCbd_nd = toc(start)

 Run='bd_nd.FRC0.00015': Continue primary family of periodic orbits.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          2.90e-04  5.26e+00    0.0    0.0    0.0
   1   1  1.00e+00  4.21e-02  2.64e-05  5.26e+00    0.0    0.1    0.0
   2   1  1.00e+00  3.17e-03  3.55e-07  5.26e+00    0.0    0.2    0.0
   3   1  1.00e+00  4.42e-05  6.25e-11  5.26e+00    0.0    0.2    0.0
   4   1  1.00e+00  7.80e-09  8.88e-16  5.26e+00    0.0    0.3    0.0

 STEP      TIME        ||U||  LABEL  TYPE         omega    po.period          eps         amp1         amp2
    0  00:00:01   5.2599e+00      1  EP      3.1315e+00   2.0064e+00   1.5000e-04   1.8748e-03   1.9789e-04
   10  00:00:05   5.3338e+00      2          3.2294e+00   1.9456e+00   1.5000e-04   5.8545e-03   6.3780e-04
   20  00:00:10   5.3635e+00      3          3.2661e+00   1.9238e+00   1.5000e-04   7.2225e-03   7.8958e-04
   26  00:00:15   5.3682e+00      4  SN      3.2720e+00   1.9203e+00   1.5000e-04   7.2282e-03   7.9137e-04
   26  00:00:15   5.3682e+00      5  FP      3.2720e+00   1.9203e+00   1.5000e-04   7.2282e-03   7.9137e-04
   30  00:00:17   5.3649e+00      6          3.2682e+00   1.9225e+00   1.5000e-04   6.7607e-03   7.4076e-04
   40  00:00:21   5.3483e+00      7          3.2491e+00   1.9338e+00   1.5000e-04   4.4414e-03   4.8801e-04
   42  00:00:25   5.3472e+00      8  FP      3.2480e+00   1.9345e+00   1.5000e-04   3.8834e-03   4.2738e-04
   42  00:00:25   5.3472e+00      9  SN      3.2480e+00   1.9345e+00   1.5000e-04   3.8832e-03   4.2735e-04
   50  00:00:28   5.3727e+00     10          3.2809e+00   1.9151e+00   1.5000e-04   1.7174e-03   1.9240e-04
   54  00:00:30   5.4072e+00     11  EP      3.3232e+00   1.8907e+00   1.5000e-04   1.1458e-03   1.3070e-04

timings = 

  struct with fields:

       FRC_ND_O3: 6.2155
       FRC_ND_O5: 3.1502
       FRC_ND_O7: 3.2323
    cocoFRCbd_nd: 31.8611

Linear Damping

Setup Dynamical System

n = 10;
[mass,damp,gyro,stiff,fnl,fext] = build_model(n,'linear_damp');

% Create model
order = 2
DS = DynamicalSystem(order);
set(DS,'M',mass,'C',damp+gyro,'K',stiff,'fnl',fnl);
set(DS.Options,'Emax',6,'Nmax',10,'notation','multiindex');
set(DS.Options,'RayleighDamping',false,'BaseExcitation',true);
DS.add_forcing(coeffs, kappas, h);
% Linear Modal analysis

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

the first four eigenvalues for undamped system

lamd =

  -0.0000 + 3.1954i
  -0.0000 - 3.1954i
  -0.0000 + 9.5862i
  -0.0000 - 9.5862i


 The first 6 nonzero eigenvalues are given as 
  -0.0180 + 3.1954i
  -0.0180 - 3.1954i
  -0.2721 + 9.5828i
  -0.2721 - 9.5828i
  -1.3571 +19.6071i
  -1.3571 -19.6071i

Choose Master subspace

S = SSM(DS);
set(S.Options, 'reltol', 1,'notation','multiindex');
resonant_modes = [1 2 3 4];
order = 5;
outdof = [1 2];
% Primary resonance of the first mode

freqrange = [0.98 1.04]*imag(D(1));
set(S.FRCOptions, 'nCycle',500, 'initialSolver', 'fsolve');
set(S.contOptions, 'PtMX', 300, 'h0', 0.1, 'h_max', 0.2, 'h_min', 1e-3);
set(S.FRCOptions, 'coordinates', 'polar');

sol = ep_read_solution('isol-nd-3.ep',1);
start = tic;
FRC_LD_O5 = S.SSM_isol2ep('isol-ld-5',resonant_modes, order, [1 3],...
    'freq', freqrange,outdof,{sol.p,sol.x});
timings.FRC_LD_O5 = toc(start);
% validation using collocation method from COCO

nCycles = 500;
coco = cocoWrapper(DS, nCycles, outdof);
set(coco.Options, 'PtMX', 1000, 'NTST',20, 'dir_name', 'bd_ld');
set(coco.Options, 'NAdapt', 0, 'h_max', 200, 'MaxRes', 1);
coco.initialGuess = 'linear';
start = tic;
bd_ld = coco.extract_FRC(freqrange);
timings.cocoFRCbd_ld = toc(start)

(near) outer resonance detected for the following combination of master eigenvalues
     0     0     2     0
     1     0     2     0
     0     0     3     1
     . 
     .
     4     4     0     2
     5     2     0     3
     6     0     0     4

These are in resonance with the follwing eigenvalues of the slave subspace
  -1.3571 +19.6071i
  -1.3571 +19.6071i
  -1.3571 +19.6071i
  . 
  . 
  -1.3571 -19.6071i
  -1.3571 -19.6071i
  -1.3571 -19.6071i

sigma_out = 75
(near) inner resonance detected for the following combination of master eigenvalues
     0     2     1     0
     1     0     1     1
     2     1     0     0
     . 
     .
     6     0     0     3
     2     7     1     0
     4     6     0     0

These are in resonance with the follwing eigenvalues of the master subspace
  -0.0180 + 3.1954i
  -0.0180 + 3.1954i
  -0.0180 + 3.1954i
   . 
   .
  -0.2721 - 9.5828i
  -0.2721 - 9.5828i
  -0.2721 - 9.5828i

Order 5 SSM Computation

sigma_in = 75
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 = 3.18E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 6.27E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 1.27E-01 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 2.60E-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='isol-ld-5.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          8.42e-16  1.15e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   1.1463e+01      1  EP      3.1954e+00   2.1730e-03   2.6567e-06   4.1120e+00   6.2114e+00   1.5000e-04
   10  00:00:00   1.3097e+01      2          3.1458e+00   1.1279e-03   3.2286e-07   4.4058e+00   7.5141e+00   1.5000e-04
   12  00:00:00   1.3425e+01      3  EP      3.1315e+00   9.3493e-04   1.7173e-07   4.4573e+00   7.7748e+00   1.5000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1         rho2          th1          th2          eps
    0  00:00:00   1.1463e+01      4  EP      3.1954e+00   2.1730e-03   2.6567e-06   4.1120e+00   6.2114e+00   1.5000e-04
   10  00:00:00   9.8985e+00      5          3.2357e+00   3.1334e-03   7.8609e-06   3.7899e+00   4.9150e+00   1.5000e-04
   20  00:00:00   8.3650e+00      6          3.2708e+00   3.8522e-03   1.3465e-05   3.4171e+00   3.5513e+00   1.5000e-04
   30  00:00:00   7.1056e+00      7  FP      3.2844e+00   4.0076e-03   1.4408e-05   3.0299e+00   2.2973e+00   1.5000e-04
   30  00:00:00   7.1056e+00      8  SN      3.2844e+00   4.0076e-03   1.4408e-05   3.0299e+00   2.2973e+00   1.5000e-04
   30  00:00:00   7.0169e+00      9          3.2843e+00   3.9914e-03   1.4218e-05   2.9981e+00   2.2008e+00   1.5000e-04
   40  00:00:00   5.9733e+00     10          3.2672e+00   3.2849e-03   8.1210e-06   2.5332e+00   8.6532e-01   1.5000e-04
   50  00:00:00   5.4937e+00     11  FP      3.2486e+00   1.9522e-03   1.7168e-06   2.0846e+00  -4.3764e-01   1.5000e-04
   50  00:00:00   5.4937e+00     12  SN      3.2486e+00   1.9521e-03   1.7168e-06   2.0846e+00  -4.3765e-01   1.5000e-04
   50  00:00:00   5.4914e+00     13          3.2486e+00   1.9156e-03   1.6195e-06   2.0740e+00  -4.7197e-01   1.5000e-04
   60  00:00:00   5.8894e+00     14          3.3011e+00   7.0769e-04   5.9444e-08   1.7442e+00  -1.8447e+00   1.5000e-04
   62  00:00:00   6.0174e+00     15  EP      3.3232e+00   5.8906e-04   3.0403e-08   1.7131e+00  -2.0313e+00   1.5000e-04

the forcing frequency 3.1315e+00 is nearly resonant with the eigenvalue -1.7969e-02 + i3.1954e+00
the forcing frequency 3.1386e+00 is nearly resonant with the eigenvalue -1.7969e-02 + i3.1954e+00
the forcing frequency 3.1458e+00 is nearly resonant with the eigenvalue -1.7969e-02 + i3.1954e+00
...

 Run='bd_ld.FRC0.00015': Continue primary family of periodic orbits.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          2.90e-04  5.26e+00    0.0    0.0    0.0
   1   1  1.00e+00  4.21e-02  2.64e-05  5.26e+00    0.0    0.0    0.0
   2   1  1.00e+00  3.17e-03  3.55e-07  5.26e+00    0.0    0.1    0.0
   3   1  1.00e+00  4.42e-05  6.19e-11  5.26e+00    0.0    0.1    0.0
   4   1  1.00e+00  7.74e-09  8.88e-16  5.26e+00    0.0    0.1    0.0

 STEP      TIME        ||U||  LABEL  TYPE         omega    po.period          eps         amp1         amp2
    0  00:00:00   5.2599e+00      1  EP      3.1315e+00   2.0064e+00   1.5000e-04   1.8752e-03   1.9793e-04
   10  00:00:03   5.3386e+00      2          3.2354e+00   1.9420e+00   1.5000e-04   6.1597e-03   6.7089e-04
   20  00:00:06   5.3722e+00      3          3.2764e+00   1.9177e+00   1.5000e-04   7.7132e-03   8.4203e-04
   26  00:00:10   5.3769e+00      4  SN      3.2822e+00   1.9143e+00   1.5000e-04   7.7583e-03   8.4859e-04
   26  00:00:10   5.3769e+00      5  FP      3.2822e+00   1.9143e+00   1.5000e-04   7.7583e-03   8.4859e-04
   30  00:00:12   5.3724e+00      6          3.2772e+00   1.9172e+00   1.5000e-04   7.2829e-03   7.9727e-04
   40  00:00:14   5.3520e+00      7          3.2534e+00   1.9313e+00   1.5000e-04   5.1319e-03   5.6288e-04
   44  00:00:18   5.3474e+00      8  FP      3.2482e+00   1.9343e+00   1.5000e-04   3.8587e-03   4.2476e-04
   44  00:00:18   5.3474e+00      9  SN      3.2482e+00   1.9343e+00   1.5000e-04   3.8585e-03   4.2474e-04
   50  00:00:20   5.3628e+00     10          3.2685e+00   1.9223e+00   1.5000e-04   2.0450e-03   2.2805e-04
   56  00:00:21   5.4072e+00     11  EP      3.3232e+00   1.8907e+00   1.5000e-04   1.1459e-03   1.3070e-04

timings = 

  struct with fields:

       FRC_ND_O3: 6.2155
       FRC_ND_O5: 3.1502
       FRC_ND_O7: 3.2323
    cocoFRCbd_nd: 31.8611
       FRC_LD_O5: 2.3849
    cocoFRCbd_ld: 24.0015

Comparison of FRC with/without nonlinear damping

% plot SSM results
FRCs = {FRC_LD_O5,FRC_ND_O5};
legs = {'LD-SSM-$\mathcal{O}(5)$-unstable','ND-SSM-$\mathcal{O}(5)$-unstable'};
legu = {'LD-SSM-$\mathcal{O}(5)$-stable','ND-SSM-$\mathcal{O}(5)$-stable'};
color = {'r','b'};
fig25 = figure;
fig26 = figure;
for k=1:2
    FRC = FRCs{k};
    SNidx = FRC.SNidx;
    HBidx = FRC.HBidx;
    FRC.st = double(FRC.st);
    FRC.st(HBidx) = nan;
    FRC.st(SNidx) = nan;
    % color
    ST = cell(2,1);
    ST{1} = {[color{k},'--'],'LineWidth',1.5}; % unstable
    ST{2} = {[color{k},'-'],'LineWidth',1.5};  % stable
    figure(fig25); hold on
    plot_stab_lines(FRC.om,FRC.Aout_frc(:,1),FRC.st,ST,legs{k},legu{k});
    SNfig = plot(FRC.om(SNidx),FRC.Aout_frc(SNidx,1),thm.SN{:});
    set(get(get(SNfig,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
    xlabel('$\Omega$','Interpreter','latex');
    ylabel('$||u_1||_{\infty}$','Interpreter','latex');
    set(gca,'FontSize',14);
    grid on, axis tight;
    legend boxoff;
    figure(fig26); hold on
    plot_stab_lines(FRC.om,FRC.Aout_frc(:,2),FRC.st,ST,legs{k},legu{k});
    SNfig = plot(FRC.om(SNidx),FRC.Aout_frc(SNidx,2),thm.SN{:});
    set(get(get(SNfig,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
    xlabel('$\Omega$','Interpreter','latex');
    ylabel('$||u_2||_{\infty}$','Interpreter','latex');
    set(gca,'FontSize',14);
    grid on; axis tight;
    legend boxoff
end

% load coco solution
legs = {'LD-Collocation-unstable','ND-Collocation-unstable'};
legu = {'LD-Collocation-stable','ND-Collocation-stable'};

bd = bd_nd{:};
ndom   = coco_bd_col(bd,'omega');
ndamp1 = coco_bd_col(bd, 'amp1');
ndamp2 = coco_bd_col(bd, 'amp2');
ndst   = coco_bd_col(bd, 'eigs');
ndst = all(abs(ndst)<1,1);
logs  = [1 2 3 4:3:numel(ndst)-4 numel(ndst)-1 numel(ndst)];
ndamp1 = ndamp1(logs);
ndamp2 = ndamp2(logs);
ndom = ndom(logs);
ndst = ndst(logs);
figure(fig25); hold on
plot(ndom(ndst), ndamp1(ndst), 'ro', 'MarkerSize', 6, 'LineWidth', 1,...
    'DisplayName', 'ND-Collocation-stable');
plot(ndom(~ndst), ndamp1(~ndst), 'ms', 'MarkerSize', 6, 'LineWidth', 1,...
    'DisplayName', 'ND-Collocation-unstable');
figure(fig26); hold on
plot(ndom(ndst), ndamp2(ndst), 'ro', 'MarkerSize', 6, 'LineWidth', 1,...
    'DisplayName', 'ND-Collocation-stable');
plot(ndom(~ndst), ndamp2(~ndst), 'ms', 'MarkerSize', 6, 'LineWidth', 1,...
    'DisplayName', 'ND-Collocation-unstable');

bd = bd_ld{:};
ldom   = coco_bd_col(bd,'omega');
ldamp1 = coco_bd_col(bd, 'amp1');
ldamp2 = coco_bd_col(bd, 'amp2');
ldst   = coco_bd_col(bd, 'eigs');
ldst = all(abs(ldst)<1,1);
logs  = [1 2 3 4:3:numel(ldst)-4 numel(ldst)-1 numel(ldst)];
ldamp1 = ldamp1(logs);
ldamp2 = ldamp2(logs);
ldom = ldom(logs);
ldst = ldst(logs);
figure(fig25); hold on
plot(ldom(ldst), ldamp1(ldst), 'kd', 'MarkerSize', 6, 'LineWidth', 1,...
    'DisplayName', 'LD-Collocation-stable');
plot(ldom(~ldst), ldamp1(~ldst), 'gv', 'MarkerSize', 6, 'LineWidth', 1,...
    'DisplayName', 'LD-Collocation-unstable');
figure(fig26); hold on
plot(ldom(ldst), ldamp2(ldst), 'kd', 'MarkerSize', 6, 'LineWidth', 1,...
    'DisplayName', 'LD-Collocation-stable');
plot(ldom(~ldst), ldamp2(~ldst), 'gv', 'MarkerSize', 6, 'LineWidth', 1,...
    'DisplayName', 'LD-Collocation-unstable');

timings

timings = 

  struct with fields:

       FRC_ND_O3: 6.2155
       FRC_ND_O5: 3.1502
       FRC_ND_O7: 3.2323
    cocoFRCbd_nd: 31.8611
       FRC_LD_O5: 2.3849
    cocoFRCbd_ld: 24.0015

Axially Moving Beam

Contents

Nonlinear Damping

Order 3 SSM Computation

Order 5 SSM Computation

Order 7 SSM Computation

Linear Damping

Order 5 SSM Computation

Comparison of FRC with/without nonlinear damping