Stationary Pipe Conveying Fluid

Contents

We consider a geometrically nonlinear cantilever pipe conveying fluid [1]. We set the flow velocity to be zero and then construct a 2D stable SSM around the stable straight configuration. More details about the construction can be found in [2]

[1] Paidoussis, M. P. (1998). Fluid-structure interactions: slender structures and axial flow (Vol. 1). Academic press.

[2] M. Li, H. Yan, L. Wang: Nonlinear model reduction for a cantilevered pipe conveying fluid: A system with asymmetric damping and stiffness matrices. MSSP. 2023. https://doi.org/10.1016/j.ymssp.2022.109993

clearvars
close all

Example setup

The $N$-degree of freedom dynamical system is of the form

$\mathbf{M\ddot{q}} + \mathbf{C\dot{q}} + \mathbf{Kq} + \mathbf{f}(\mathbf{q},\mathbf{\dot{q}}) = \mathbf{0}$

where $\mathbf{f}=\mathcal{O}(|\mathbf{q}|^2,|\mathbf{\dot{q}}|^2,|\mathbf{q}||\mathbf{\dot{q}}|)$ represents the nonlinearities and $\mathbf{M}$, $\mathbf{C}$, and $\mathbf{K}$ are the $n\times n$ mass, stiffness, and damping matrices, respectively. Here, both the damping and stiffness matrices are not symmetric.

clear all;
nmodes = 4;
fcload = 1;
flowspeed = 0.0;
miu = 0;
Gamma=0;
alpha=0.001;
beta=0.2;
[M, C, K, fnl, fext] = build_model(nmodes,flowspeed,beta,miu,Gamma,alpha,'clamped-free','nonlinear_damp');
n = size(M,1);    % mechanical dofs (axial def, transverse def, angle)
[F, lambda] = functionFromTensors(M, C, K, fnl);
lambda = sort(lambda);

Getting linear and nonlinearity coefficients
Loaded coefficients from storage

Create dynamical system

order = 2;
DS = DynamicalSystem(order);
set(DS,'M',M,'C',C,'K',K,'fnl',fnl);
set(DS.Options,'Emax',10,'Nmax',10,'notation','multiindex')
set(DS.Options,'RayleighDamping',false)
[V,D,W] = DS.linear_spectral_analysis();

 The first 8 nonzero eigenvalues are given as 
   1.0e+02 *

  -0.0001 + 0.0352i
  -0.0001 - 0.0352i
  -0.0024 + 0.2203i
  -0.0024 - 0.2203i
  -0.0190 + 0.6167i
  -0.0190 - 0.6167i
  -0.0731 + 1.2068i
  -0.0731 - 1.2068i

Choose Master subspace (perform resonance analysis)

We take the first pair of modes as the master subspace and compute the corresponding 2D stable SSM

S = SSM(DS);
set(S.Options, 'reltol', 1,'notation','multiindex');
order = 7;
resonant_modes = [1,2];
S.choose_E(resonant_modes);
[W0,R0] = S.compute_whisker(order);

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

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i

sigma_out = 1182
(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 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i

sigma_in = 1182
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 = 1.74E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 1.99E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 2.39E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 2.91E-02 MB
Manifold computation time at order 6 = 00:00:00
Estimated memory usage at order  6 = 3.62E-02 MB
Manifold computation time at order 7 = 00:00:00
Estimated memory usage at order  7 = 4.51E-02 MB

Reduced dynamics in symbolic form

lamdMaster = DS.spectrum.Lambda(resonant_modes);
options = struct();
options.isauto = true;
options.isdamped = true;
options.numDigits = 4;
y = reduced_dynamics_symbolic(lamdMaster,R0,options)

 
y =
 
0.001971*rho_1^7 + 0.0008166*rho_1^5 + 0.001503*rho_1^3 - 0.006181*rho_1
                1.114*rho_1^6 + 0.7655*rho_1^4 + 0.05768*rho_1^2 + 3.516
 

Convergence of backbone curve

syms rho_1 positive
[coeffs,powers]=coeffs(y(2));
tmp = simplify(log(powers));
exp_idx = double(tmp./log(rho_1));

rhosamp = 0:0.01:0.12;
orders = [3 5 7];
plot_backbone_curves(double(coeffs),exp_idx,rhosamp,orders)

Transient response prediction

We apply a static force at the free end and then release the beam to observe the free vibration of the pipe

phiend = zeros(n,1);
lamda  = zeros(n,1);
lamda(1)=1.8751040687119611664453082410782141625701117335311;
lamda(2)=4.6940911329741745764363917780198120493898967375458;
lamda(3)=7.8547574382376125648610085827645704578485419292300;
lamda(4)=10.995540734875466990667349107854702939612972774652;
lamda(5)=14.137168391046470580917046812551772068603076792975;
lamda(6)=17.278759532088236333543928414375822085934519635550;
lamda(7)=20.420352251041250994415811947947837046137288894544;
lamda(8)=23.561944901806443501520253240198075517031265990051;
lamda(9)=26.703537555518298805445478738088174009227496691444;
lamda(10)=29.845130209102817263788730907430405063369916050237;
lamda(11)=32.986722862692838446168314183853683000367036041646;
lamda(12)=36.128315516282621834281162204648446323089539352502;
for k=1:n
   x    = 1;
   phin = cos(lamda(k)*x)-cosh(lamda(k)*x)-(cos(lamda(k))+cosh(lamda(k)))/(sin(lamda(k))+sinh(lamda(k)))*...
        (sin(lamda(k)*x)-sinh(lamda(k)*x));%clamped-free
   phiend(k) = phin;
end
loads = 1;
loadvector = loads.*phiend;
IC = getStaticResponse(K, M, F, loadvector);

Forward simulation with given IC

tf = 100;
nsteps = 2000;
outdof = [1 2 n+1];
redtic = tic;
traj = transient_traj_on_auto_ssm(DS, resonant_modes, W0, R0, tf, nsteps, outdof, IC);
timered = toc(redtic)
fulltic = tic;
[tInt,zInt] = time_integration_transient(DS,0,...
    'nSteps', nsteps, 'nCycles', ceil(tf/2/pi), ...
    'integrationMethod','ode45','outdof',outdof,'init',IC);
timefull = toc(fulltic)

figure;
subplot(2,1,1)
plot(traj.time,traj.phy(:,1),'r-','LineWidth',1); hold on
plot(tInt,zInt(:,1),'b--','LineWidth',1);
ylabel('$q_1$','Interpreter',"latex",'FontSize',14);
set(gca,'FontSize',14); grid on
xlim([0,min(traj.time(end),tInt(end))]);
subplot(2,1,2)
plot(traj.time,traj.phy(:,2),'r-','LineWidth',1); hold on
plot(tInt,zInt(:,2),'b--','LineWidth',1);
xlabel('$\tau$','Interpreter',"latex",'FontSize',14);
ylabel('$q_2$','Interpreter',"latex",'FontSize',14);
set(gca,'FontSize',14); grid on
legend('SSM','Full')
xlim([0,min(traj.time(end),tInt(end))]);

rhosamp = [0:0.01:0.5];
plotdofs = [1 2 3];
plot_2D_auto_SSM(W0,rhosamp,outdof(plotdofs));
view([30 30])
hold on
plot3(traj.phy(:,plotdofs(1)),traj.phy(:,plotdofs(2)),traj.phy(:,plotdofs(3)),'r-');
plot3(zInt(:,plotdofs(1)),zInt(:,plotdofs(2)),zInt(:,plotdofs(3)),'b--');

timered =

    0.7018


timefull =

   19.5664

Forced response - periodic orbit

% add Forcing
kappas = [-1; 1];
coeffs = [fext fext]/2;
epsilon = 0.0006;
DS.add_forcing(coeffs, kappas, epsilon);
set(DS.Options,'BaseExcitation',true);
% set up obervables
% disp at the free end
obsfun = @(x,mapx,nmodes) mapx*x(1:nmodes,:);
obs = @(x) obsfun(x,phiend',n);

% set up FRC options
freqrange = [0.99 1.01]*imag(D(1));
set(S.FRCOptions, 'nCycle',500, 'initialSolver', 'fsolve');
set(S.contOptions, 'PtMX', 300, 'h_max', 0.1);
set(S.FRCOptions, 'omegaSampStyle', 'cocoBD');
set(S.FRCOptions, 'outdof',obs,'method','continuation ep','p0',[]);
S.extract_FRC('freq',freqrange,[3,5,7]);
figfrc = gcf;

*****************************************
Calculating FRC using SSM with master subspace: [1  2]
(near) outer resonance detected for the following combination of master eigenvalues
     6     0
     7     0
     7     1
     8     1
     8     2
     0     6
     0     7
     1     7
     1     8
     2     8

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i

sigma_out = 1182
(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 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i

sigma_in = 1182
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 = 1.54E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 1.78E-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='freqSubint1.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          4.65e-13  5.00e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   4.9994e+00      1  EP      3.5160e+00   2.5252e-01   2.6656e-01   6.0000e-04
   10  00:00:00   5.1313e+00      2          3.4990e+00   9.5678e-02   9.5544e-01   6.0000e-04
   13  00:00:00   5.1719e+00      3  EP      3.4808e+00   4.9107e-02   1.1204e+00   6.0000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   4.9994e+00      4  EP      3.5160e+00   2.5252e-01   2.6656e-01   6.0000e-04
   10  00:00:00   5.0364e+00      5          3.5220e+00   2.9387e-01  -4.3806e-01   6.0000e-04
   20  00:00:00   5.2454e+00      6          3.5253e+00   1.9228e-01  -1.1369e+00   6.0000e-04
   28  00:00:01   5.5522e+00      7  EP      3.5512e+00   5.1525e-02  -1.6732e+00   6.0000e-04
*****************************************
Calculating FRC using SSM with master subspace: [1  2]
(near) outer resonance detected for the following combination of master eigenvalues
     6     0
     7     0
     7     1
     8     1
     8     2
     0     6
     0     7
     1     7
     1     8
     2     8

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i

sigma_out = 1182
(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 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i

sigma_in = 1182
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 = 1.64E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 1.88E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 2.29E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 2.81E-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='freqSubint1.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          9.41e-12  5.02e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   5.0166e+00      1  EP      3.5160e+00   2.2725e-01   4.1138e-01   6.0000e-04
   10  00:00:00   5.1677e+00      2          3.4853e+00   5.6028e-02   1.0963e+00   6.0000e-04
   11  00:00:00   5.1719e+00      3  EP      3.4808e+00   4.9101e-02   1.1205e+00   6.0000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   5.0166e+00      4  EP      3.5160e+00   2.2725e-01   4.1138e-01   6.0000e-04
   10  00:00:00   5.0233e+00      5          3.5274e+00   2.9916e-01  -2.9084e-01   6.0000e-04
   13  00:00:00   5.0478e+00      6  FP      3.5278e+00   2.9426e-01  -4.5599e-01   6.0000e-04
   13  00:00:00   5.0478e+00      7  SN      3.5278e+00   2.9426e-01  -4.5601e-01   6.0000e-04
   20  00:00:00   5.1901e+00      8          3.5261e+00   2.2325e-01  -9.9277e-01   6.0000e-04
   21  00:00:00   5.2029e+00      9  SN      3.5261e+00   2.1624e-01  -1.0273e+00   6.0000e-04
   21  00:00:00   5.2029e+00     10  FP      3.5261e+00   2.1624e-01  -1.0273e+00   6.0000e-04
   30  00:00:00   5.5521e+00     11  EP      3.5512e+00   5.1533e-02  -1.6732e+00   6.0000e-04
*****************************************
Calculating FRC using SSM with master subspace: [1  2]
(near) outer resonance detected for the following combination of master eigenvalues
     6     0
     7     0
     7     1
     8     1
     8     2
     0     6
     0     7
     1     7
     1     8
     2     8

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i

sigma_out = 1182
(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 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i

sigma_in = 1182
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 = 1.74E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 1.99E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 2.39E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 2.91E-02 MB
Manifold computation time at order 6 = 00:00:00
Estimated memory usage at order  6 = 3.62E-02 MB
Manifold computation time at order 7 = 00:00:00
Estimated memory usage at order  7 = 4.51E-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='freqSubint1.ep': Continue equilibria along primary branch.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          3.83e-12  5.02e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   5.0176e+00      1  EP      3.5160e+00   2.2597e-01   4.1806e-01   6.0000e-04
   10  00:00:00   5.1689e+00      2          3.4842e+00   5.4179e-02   1.1027e+00   6.0000e-04
   11  00:00:00   5.1719e+00      3  EP      3.4808e+00   4.9101e-02   1.1205e+00   6.0000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   5.0176e+00      4  EP      3.5160e+00   2.2597e-01   4.1806e-01   6.0000e-04
   10  00:00:00   5.0236e+00      5          3.5282e+00   2.9940e-01  -2.8399e-01   6.0000e-04
   13  00:00:00   5.0446e+00      6  FP      3.5286e+00   2.9576e-01  -4.3092e-01   6.0000e-04
   13  00:00:00   5.0446e+00      7  SN      3.5286e+00   2.9576e-01  -4.3093e-01   6.0000e-04
   20  00:00:00   5.1879e+00      8          3.5262e+00   2.2462e-01  -9.8603e-01   6.0000e-04
   21  00:00:00   5.2098e+00      9  FP      3.5262e+00   2.1257e-01  -1.0450e+00   6.0000e-04
   21  00:00:00   5.2098e+00     10  SN      3.5262e+00   2.1257e-01  -1.0450e+00   6.0000e-04
   30  00:00:00   5.5499e+00     11          3.5507e+00   5.2273e-02  -1.6706e+00   6.0000e-04
   31  00:00:00   5.5521e+00     12  EP      3.5512e+00   5.1533e-02  -1.6732e+00   6.0000e-04
Total time spent on FRC computation upto O(3) = 00:00:02
Total time spent on FRC computation upto O(5) = 00:00:01
Total time spent on FRC computation upto O(7) = 00:00:01

adding backbone curve

BB = S.extract_backbone(resonant_modes, freqrange, order);
numPts = numel([BB.Omega]);
numOutdof = numel([BB.Aout])/numPts;
Aout = reshape([BB.Aout], [numOutdof, numPts]);
Aout = Aout';
figure(figfrc); hold on
plot([BB.Omega],Aout(:,1),'k-','DisplayName','backbone');
% Validation using collocation method from COCO

nCycles = 500;
coco = cocoWrapper(DS, nCycles, obs);
set(coco.Options, 'PtMX', 1000, 'NTST',10, 'dir_name', 'bd_ld_v0p01');
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)
% Exercies

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

These are in resonance with the follwing eigenvalues of the slave subspace
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 +22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i
  -0.2428 -22.0332i

sigma_out = 1182
(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 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 + 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i
  -0.0062 - 3.5160i

sigma_in = 1182
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 = 1.74E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 1.99E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 2.39E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 2.91E-02 MB
Manifold computation time at order 6 = 00:00:00
Estimated memory usage at order  6 = 3.62E-02 MB
Manifold computation time at order 7 = 00:00:00
Estimated memory usage at order  7 = 4.51E-02 MB
gamma = 
   0.0015 + 0.0577i
   0.0008 + 0.7655i
   0.0020 + 1.1141i

Total time spent on backbone curve computation = 00:00:00

 Run='bd_ld_v0p01.FRC0.0006': Continue primary family of periodic orbits.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          2.55e-03  5.55e+00    0.0    0.0    0.0
   1   1  6.44e-01  3.79e-01  9.44e-04  5.55e+00    0.0    0.0    0.0
   2   1  1.00e+00  1.33e-01  7.47e-05  5.56e+00    0.0    0.1    0.0
   3   1  1.00e+00  4.88e-04  1.30e-09  5.56e+00    0.0    0.1    0.0
   4   1  1.00e+00  2.57e-08  1.36e-16  5.56e+00    0.0    0.1    0.0

 STEP      TIME        ||U||  LABEL  TYPE         omega    po.period          eps         amp1
    0  00:00:00   5.5604e+00      1  EP      3.4808e+00   1.8051e+00   6.0000e-04   4.5345e-02
   10  00:00:01   5.7238e+00      2          3.5084e+00   1.7909e+00   6.0000e-04   1.4375e-01
   20  00:00:02   5.9329e+00      3          3.5180e+00   1.7860e+00   6.0000e-04   2.1958e-01
   30  00:00:03   6.0747e+00      4          3.5251e+00   1.7824e+00   6.0000e-04   2.5915e-01
   40  00:00:04   6.0739e+00      5  FP      3.5273e+00   1.7813e+00   6.0000e-04   2.5809e-01
   40  00:00:04   6.0666e+00      6          3.5273e+00   1.7813e+00   6.0000e-04   2.5555e-01
   50  00:00:05   5.9158e+00      7          3.5261e+00   1.7819e+00   6.0000e-04   2.1145e-01
   53  00:00:05   5.8741e+00      8  FP      3.5260e+00   1.7820e+00   6.0000e-04   1.9708e-01
   60  00:00:06   5.7184e+00      9          3.5284e+00   1.7807e+00   6.0000e-04   1.3143e-01
   69  00:00:07   5.6278e+00     10  EP      3.5512e+00   1.7693e+00   6.0000e-04   4.7136e-02

timings = 

  struct with fields:

    cocoFRCbd_ld: 7.8049

Published with MATLAB® R2023a