Geometrically nonlinear von Karman beam

Contents

References

Finite element model from the following reference:

Jain, S., Tiso, P., & Haller, G. (2018). Exact nonlinear model reduction for a von Kármán beam: slow-fast decomposition and spectral submanifolds. Journal of Sound and Vibration, 423, 195–211. ;https://doi.org/10.1016/J.JSV.2018.01.049;

Finite element code taken from the following package:

Jain, S., Marconi, J., Tiso P. (2020). YetAnotherFEcode (Version v1.1). Zenodo. ;http://doi.org/10.5281/zenodo.4011282;

system parameters

clear all, close all
nElements = 10;
epsilon = 5e-4;

generate model

[M,C,K,fnl,f_0,outdof] = build_model(nElements);
n = length(M);
disp(['Number of degrees of freedom = ' num2str(n)])
disp(['Phase space dimensionality = ' num2str(2*n)])

Building FE model
Assembling M,C,K matrices
Applying boundary conditions
Solving undamped eigenvalue problem
Assembling external force vector
Getting nonlinearity coefficients
Loaded tensors from storage
Total time spent on model assembly = 00:00:01
Number of degrees of freedom = 30
Phase space dimensionality = 60

Dynamical system setup

We consider the forced system

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

which can be written in the first-order form as

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

where

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

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

We assume periodic forcing of the form

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

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 5 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 3.691472e-04
modal damping ratio for 2 mode is 2.313481e-03
modal damping ratio for 3 mode is 6.479248e-03
modal damping ratio for 4 mode is 1.270560e-02
modal damping ratio for 5 mode is 2.103617e-02

 The first 10 nonzero eigenvalues are given as 
   1.0e+02 *

  -0.0000 + 0.0517i
  -0.0000 - 0.0517i
  -0.0007 + 0.3239i
  -0.0007 - 0.3239i
  -0.0059 + 0.9071i
  -0.0059 - 0.9071i
  -0.0226 + 1.7786i
  -0.0226 - 1.7786i
  -0.0620 + 2.9444i
  -0.0620 - 2.9444i

Choose Master subspace (perform resonance analysis)

S = SSM(DS);
set(S.Options, 'reltol', 0.1,'notation','multiindex')
masterModes = [1,2];
S.choose_E(masterModes);

The master subspace contains the following eigenvalues
 
lambda1 == - 0.0019078 + 5.1681i
 
lambda2 == (-0.0019078) - 5.1681i
 
No (near) outer resonances detected in the (truncated) spectrum
sigma_out = 3247
(near) inner resonance detected for the following combination of master eigenvalues:
 
2*lambda1 + 1*lambda2 == lambda1
 
3*lambda1 + 2*lambda2 == lambda1
 
4*lambda1 + 3*lambda2 == lambda1
 
5*lambda1 + 4*lambda2 == lambda1
 
1*lambda1 + 2*lambda2 == lambda2
 
2*lambda1 + 3*lambda2 == lambda2
 
3*lambda1 + 4*lambda2 == lambda2
 
4*lambda1 + 5*lambda2 == lambda2
 
sigma_in = 3247

Forced response curves using SSMs

Obtaining forced response curve in reduced-polar coordinate

order = [3,5,7]; % Approximation order

setup options

set(S.Options, 'reltol', 1,'IRtol',0.02,'notation', 'multiindex','contribNonAuto',true)
set(S.FRCOptions, 'nt', 2^7, 'nRho', 200, 'nPar', 200, 'nPsi', 100, 'rhoScale', 2 )
set(S.FRCOptions, 'method','continuation ep', 'z0', 1e-4*[1; 1]) % 'level set'
set(S.FRCOptions, 'outdof',outdof)

choose frequency range around the first natural frequency

omega0 = imag(S.E.spectrum(1));
omegaRange = omega0*[0.9 1.1];

extract forced response curve

FRC = S.extract_FRC('freq',omegaRange,order);
figFRC = gcf;

Computation of FRCs at order 3

*****************************************
Calculating FRC using SSM with master subspace: [1  2]
The master subspace contains the following eigenvalues
 
lambda1 == - 0.0019078 + 5.1681i
 
lambda2 == (-0.0019078) - 5.1681i
 
(near) outer resonance detected for the following combinations of master eigenvalues
 They are in resonance with the following eigenvalues of the slave subspace 
 
6*lambda1 + 0*lambda2 == - 0.0749307 + 32.3886i
 
7*lambda1 + 0*lambda2 == - 0.0749307 + 32.3886i
 
7*lambda1 + 1*lambda2 == - 0.0749307 + 32.3886i
 
8*lambda1 + 1*lambda2 == - 0.0749307 + 32.3886i
 
8*lambda1 + 2*lambda2 == - 0.0749307 + 32.3886i
 
0*lambda1 + 6*lambda2 == (-0.0749307) - 32.3886i
 
0*lambda1 + 7*lambda2 == (-0.0749307) - 32.3886i
 
1*lambda1 + 7*lambda2 == (-0.0749307) - 32.3886i
 
1*lambda1 + 8*lambda2 == (-0.0749307) - 32.3886i
 
2*lambda1 + 8*lambda2 == (-0.0749307) - 32.3886i
 
sigma_out = 3247
(near) inner resonance detected for the following combination of master eigenvalues:
 
2*lambda1 + 1*lambda2 == lambda1
 
3*lambda1 + 2*lambda2 == lambda1
 
4*lambda1 + 3*lambda2 == lambda1
 
5*lambda1 + 4*lambda2 == lambda1
 
1*lambda1 + 2*lambda2 == lambda2
 
2*lambda1 + 3*lambda2 == lambda2
 
3*lambda1 + 4*lambda2 == lambda2
 
4*lambda1 + 5*lambda2 == lambda2
 
sigma_in = 3247
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 = 6.15E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 8.63E-02 MB

 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                          3.73e-03  8.32e+00    0.0    0.0    0.0
   1   1  1.00e+00  6.04e-03  2.38e-07  8.32e+00    0.0    0.0    0.0
   2   1  1.00e+00  4.55e-06  9.98e-12  8.32e+00    0.0    0.0    0.0
   3   1  1.00e+00  1.95e-10  2.44e-16  8.32e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   8.3246e+00      1  EP      5.1723e+00   5.1935e-04   2.8101e+00   5.0000e-04
   10  00:00:01   8.4394e+00      2          5.0920e+00   1.4760e-04   3.1117e+00   5.0000e-04
   20  00:00:01   7.9579e+00      3          4.6710e+00   2.3186e-05   3.1377e+00   5.0000e-04
   21  00:00:01   7.9348e+00      4  EP      4.6513e+00   2.2299e-05   3.1379e+00   5.0000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:01   8.3246e+00      5  EP      5.1723e+00   5.1935e-04   2.8101e+00   5.0000e-04
    7  00:00:02   7.6003e+00      6  FP      5.2300e+00   7.8126e-04   1.2366e+00   5.0000e-04
    7  00:00:02   7.6002e+00      7  SN      5.2300e+00   7.8123e-04   1.2364e+00   5.0000e-04
   10  00:00:02   7.3743e+00      8  FP      5.2114e+00   4.0279e-04   1.7928e-01   5.0000e-04
   10  00:00:02   7.3744e+00      9  SN      5.2114e+00   4.0213e-04   1.7870e-01   5.0000e-04
   10  00:00:02   7.3734e+00     10          5.2118e+00   3.6212e-04   1.4161e-01   5.0000e-04
   20  00:00:02   7.4368e+00     11          5.2586e+00   1.2956e-04   2.5167e-02   5.0000e-04
   29  00:00:03   8.0396e+00     12  EP      5.6849e+00   2.2303e-05   3.7108e-03   5.0000e-04
  

Computation of FRCs at order 5

   *****************************************
Calculating FRC using SSM with master subspace: [1  2]

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 = 6.24E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 8.72E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 1.10E-01 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 1.55E-01 MB

 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                          1.93e-05  8.32e+00    0.0    0.0    0.0
   1   1  1.00e+00  2.18e-05  6.51e-12  8.32e+00    0.0    0.0    0.0
   2   1  1.00e+00  2.69e-10  2.64e-16  8.32e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   8.3232e+00      1  EP      5.1723e+00   5.2237e-04   2.8080e+00   5.0000e-04
   10  00:00:00   8.4395e+00      2          5.0922e+00   1.4786e-04   3.1116e+00   5.0000e-04
   20  00:00:00   7.9626e+00      3          4.6751e+00   2.3375e-05   3.1377e+00   5.0000e-04
   21  00:00:00   7.9348e+00      4  EP      4.6513e+00   2.2299e-05   3.1379e+00   5.0000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   8.3232e+00      5  EP      5.1723e+00   5.2237e-04   2.8080e+00   5.0000e-04
    7  00:00:01   7.5945e+00      6  FP      5.2288e+00   7.8634e-04   1.2242e+00   5.0000e-04
    7  00:00:01   7.5945e+00      7  SN      5.2287e+00   7.8617e-04   1.2240e+00   5.0000e-04
   10  00:00:01   7.3742e+00      8  FP      5.2112e+00   4.0513e-04   1.8088e-01   5.0000e-04
   10  00:00:01   7.3743e+00      9  SN      5.2112e+00   4.0493e-04   1.8078e-01   5.0000e-04
   10  00:00:01   7.3732e+00     10          5.2118e+00   3.6004e-04   1.3943e-01   5.0000e-04
   20  00:00:02   7.4366e+00     11          5.2584e+00   1.2980e-04   2.5225e-02   5.0000e-04
   29  00:00:02   8.0396e+00     12  EP      5.6849e+00   2.2303e-05   3.7108e-03   5.0000e-04
  

Computation of FRCs at order 7

   *****************************************
Calculating FRC using SSM with master subspace: [1  2]

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 = 6.35E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 8.83E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 1.11E-01 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 1.56E-01 MB
Manifold computation time at order 6 = 00:00:00
Estimated memory usage at order  6 = 2.01E-01 MB
Manifold computation time at order 7 = 00:00:00
Estimated memory usage at order  7 = 2.74E-01 MB

 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                          5.09e-07  8.32e+00    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:00   8.3232e+00      1  EP      5.1723e+00   5.2229e-04   2.8080e+00   5.0000e-04
   10  00:00:00   8.4395e+00      2          5.0922e+00   1.4785e-04   3.1116e+00   5.0000e-04
   20  00:00:00   7.9623e+00      3          4.6748e+00   2.3365e-05   3.1377e+00   5.0000e-04
   21  00:00:00   7.9348e+00      4  EP      4.6513e+00   2.2299e-05   3.1379e+00   5.0000e-04

 STEP      TIME        ||U||  LABEL  TYPE            om         rho1          th1          eps
    0  00:00:01   8.3232e+00      5  EP      5.1723e+00   5.2229e-04   2.8080e+00   5.0000e-04
    7  00:00:01   7.5950e+00      6  FP      5.2288e+00   7.8599e-04   1.2252e+00   5.0000e-04
    7  00:00:01   7.5949e+00      7  SN      5.2288e+00   7.8583e-04   1.2251e+00   5.0000e-04
   10  00:00:01   7.3742e+00      8  FP      5.2112e+00   4.0507e-04   1.8084e-01   5.0000e-04
   10  00:00:01   7.3743e+00      9  SN      5.2113e+00   4.0486e-04   1.8072e-01   5.0000e-04
   10  00:00:01   7.3732e+00     10          5.2118e+00   3.6009e-04   1.3947e-01   5.0000e-04
   20  00:00:02   7.4366e+00     11          5.2584e+00   1.2980e-04   2.5224e-02   5.0000e-04
   29  00:00:02   8.0396e+00     12  EP      5.6849e+00   2.2303e-05   3.7108e-03   5.0000e-04
Total time spent on FRC computation upto O(3) = 00:00:06
Total time spent on FRC computation upto O(5) = 00:00:04
Total time spent on FRC computation upto O(7) = 00:00:04

FRC in physical space:

Published with MATLAB® R2021a