Prismatic Beam under parametric excitation

Setup Dynamical System
Linear Modal Analysis
Stability Diagram from Reduced Dynamics
Verification: Collocation using coco

We consider a clamped-pinned beam.

Nayfeh [1] and Li [2] inverstigated the forced response of such a system under external harmonic response. Specifically, modal expansion (with linear modes) is used to transfer PDEs to a set of ODEs

$\ddot{u}_n+\omega_n^2u_n=-2c_n\dot{u}_n+\nu\sum_{m,p,q}\alpha_{nmpq}u_mu_pu_q+\epsilon\left(f_n\cos\lambda t\right), n=1,\cdots,$

This set of ODEs has to be adjusted for the case where the harmonic excitation $p_a(t)$ occurs in an axial direction and direct transverse excitation is applied as $p_t(x,t)$ , where $x$ are dimensionless coordinates along the beam . Then the equations of the modal coordinates in transverse direction read

$\omega_j^2 u_j + \ddot{u}_j + 2c_j\dot{u}_j= \nu\sum_{i,k,s} \alpha_{jiks} u_i u_k u_s + \epsilon \bigg(f_j(t) + \sum_i p_a(t) u_i a_{ji}\bigg)$

where the coefficients $a_{ji}$ are defined in terms of the spatial eigenmodes as

$a_{ji} := \int_0^l \psi_j \ \psi''_i dx$

and

$f_j = \int_0^l \psi_j p(x,t) dx$

Here no mode is ecited externall so $f_j = 0$ for all $j$ and $p_a(t) = \mu \cos(\Omega t)$ . The system is forced around the principal resonance of the first mode so $\Omega \approx 2\omega_1$

[1] Nayfeh, A. H., Mook, D. T., & Sridhar, S. (1974). Nonlinear analysis of the forced response of structural elements. The Journal of the Acoustical Society of America, 55(2), 281-291.

[2] M. Li, S. Jain, and G. Haller. Nonlinear analysis of forced mechanical systems with internal resonance using spectralsubmanifolds–part I: Periodic response and forced response curve. arXiv preprint arXiv:2106.05162, 2021

Setup Dynamical System

clear all;

rLsq = 1e-4;
cs  = [100,200,300,400];
n = 10;               % number of modes

ii = 1;
for c = cs

[mass,damp,stiff,fnl,fext] = build_model_parametric(c,rLsq,n);

% Create
order = 2;
DS = DynamicalSystem(order);
set(DS,'M',mass,'C',damp,'K',stiff,'fnl',fnl);
set(DS.Options,'Emax',5,'Nmax',10,'notation','multiindex')
% Forcing
DS.add_forcing(fext);

Getting nonlinearity coefficients
Loaded coefficients from storage
Loaded coefficients from storage

Linear Modal Analysis

% Analyse spectrum
[V,D,W_evec] = DS.linear_spectral_analysis();

% Choose Master subspace (perform resonance analysis)

% Set up SSM object
S = SSM(DS);
set(S.Options, 'reltol', 0.3,'notation','multiindex')

%Choose Master subspace
resModes = [1,2];
S.choose_E(resModes);

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 1.874028e-03
modal damping ratio for 2 mode is 5.783382e-04
modal damping ratio for 3 mode is 2.772339e-04
modal damping ratio for 4 mode is 1.621083e-04
modal damping ratio for 5 mode is 1.062379e-04

 The first 10 nonzero eigenvalues are given as 
  -0.0100 + 5.3361i
  -0.0100 - 5.3361i
  -0.0100 +17.2909i
  -0.0100 -17.2909i
  -0.0100 +36.0706i
  -0.0100 -36.0706i
  -0.0100 +61.6872i
  -0.0100 -61.6872i
  -0.0100 +94.1284i
  -0.0100 -94.1284i

sigma_out = 1
sigma_in = 1

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 3.748056e-03
modal damping ratio for 2 mode is 1.156676e-03
modal damping ratio for 3 mode is 5.544678e-04
modal damping ratio for 4 mode is 3.242166e-04
modal damping ratio for 5 mode is 2.124758e-04

 The first 10 nonzero eigenvalues are given as 
  -0.0200 + 5.3361i
  -0.0200 - 5.3361i
  -0.0200 +17.2909i
  -0.0200 -17.2909i
  -0.0200 +36.0706i
  -0.0200 -36.0706i
  -0.0200 +61.6872i
  -0.0200 -61.6872i
  -0.0200 +94.1284i
  -0.0200 -94.1284i

sigma_out = 1
sigma_in = 1

Stability Diagram from Reduced Dynamics

We extract the stability diagram using continuation of bifurcations. By extending the dynamical system

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

to an autonomous system of variables $(\mathbf{z}, \tau) \in \mathbf{R}^N \times S^1$ the trivial fixed point $\mathbf{z} =\mathbf{0}$ of the paremtrically excited system can be interpreted as the periodic orbit $(\mathbf{z}, \tau ) = (\mathbf{0}, t \ \textrm{mod} \ 2\pi )$ . Any change of the stability behaviour of this periodic orbit is then given by some bifurcation. At the stability boundary of the principal resonance with $\Omega \approx 2 \omega_0$ nontrivial periodic orbits with response period $T = 4 \pi / \Omega$ emerge. If continuation of $2 \pi / \Omega$ periodic orbits is used then these bifurcations show up as period doubling ('PD') bifurcations. Initially continuing $4 \pi / \Omega$ periodic orbits leads to a saddle node ('SN') bifurcation. The function extract_Stability_Diagram allows to chose between these two options for constructing the stability diagram. Settings

set(S.contOptions,'PtMX',40,'bi_direct',true)
set(S.FRCOptions,'branchSwitch',true)
PlotSD = false;
order  = 5;

% Initial condition and parameter range


omega0 = imag(S.E.spectrum(1));
p0 = [2*omega0,0.01]; % Initial condition

OmegaRange =[10.3,11];
epRange = [0,1];

% Computation

startSDSSM = tic;
SD = S.extract_Stability_Diagram(resModes, order, OmegaRange,epRange,'amp', p0,'PD',PlotSD);
timings(ii).SDSSM = toc(startSDSSM);

Damping Parameter: c = 100

sigma_out = 1
sigma_in = 1
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 5.72E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 6.26E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 7.27E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 8.53E-02 MB

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          0.00e+00  1.07e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE           eps    po.period
    0  00:00:00   1.0705e+01      1  EP      1.0000e-02   5.8874e-01
    1  00:00:00   1.0705e+01      2  EP      3.4694e-18   5.8874e-01

 STEP      TIME        ||U||  LABEL  TYPE           eps    po.period
    0  00:00:01   1.0705e+01      3  EP      1.0000e-02   5.8874e-01
    1  00:00:01   1.0705e+01      4  PD      5.3576e-02   5.8874e-01
    5  00:00:01   1.0798e+01      5  EP      1.0000e+00   5.8874e-01

 Run='ROM_family_bif1': Continue bifurcations from point 4 in run 'ROM_detect_bif'.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          5.59e-09  1.67e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:00   1.6688e+01      1  EP      1.0672e+01   5.8874e-01   5.3576e-02
   10  00:00:04   1.6628e+01      2          1.0624e+01   5.9139e-01   1.3856e-01
   20  00:00:08   1.6459e+01      3          1.0479e+01   5.9959e-01   5.1992e-01
   26  00:00:11   1.6276e+01      4  EP      1.0300e+01   6.1002e-01   9.9843e-01

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:12   1.6688e+01      5  EP      1.0672e+01   5.8874e-01   5.3576e-02
   10  00:00:15   1.6750e+01      6          1.0720e+01   5.8612e-01   1.3856e-01
   20  00:00:19   1.6950e+01      7          1.0865e+01   5.7828e-01   5.1993e-01
   25  00:00:22   1.7152e+01      8  EP      1.1000e+01   5.7120e-01   8.7979e-01
Total time spent on Stability Diagram computation = 00:00:27

Damping Parameter: c = 200

sigma_out = 1
sigma_in = 1
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 5.72E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 6.26E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 7.27E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 8.53E-02 MB

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          8.88e-16  1.07e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE           eps    po.period
    0  00:00:00   1.0705e+01      1  EP      1.0000e-02   5.8875e-01
    1  00:00:00   1.0705e+01      2  EP      3.4694e-18   5.8875e-01

 STEP      TIME        ||U||  LABEL  TYPE           eps    po.period
    0  00:00:00   1.0705e+01      3  EP      1.0000e-02   5.8875e-01
    2  00:00:00   1.0706e+01      4  PD      1.0715e-01   5.8875e-01
    5  00:00:00   1.0798e+01      5  EP      1.0000e+00   5.8875e-01

 Run='ROM_family_bif1': Continue bifurcations from point 4 in run 'ROM_detect_bif'.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          1.23e-08  1.67e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:00   1.6688e+01      1  EP      1.0672e+01   5.8875e-01   1.0715e-01
   10  00:00:04   1.6576e+01      2          1.0581e+01   5.9382e-01   2.6675e-01
   20  00:00:08   1.6364e+01      3          1.0389e+01   6.0477e-01   7.6491e-01
   23  00:00:09   1.6277e+01      4  EP      1.0300e+01   6.1002e-01   1.0026e+00

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:10   1.6688e+01      5  EP      1.0672e+01   5.8875e-01   1.0715e-01
   10  00:00:13   1.6808e+01      6          1.0763e+01   5.8376e-01   2.6675e-01
   20  00:00:17   1.7084e+01      7          1.0955e+01   5.7355e-01   7.6493e-01
   22  00:00:18   1.7153e+01      8  EP      1.1000e+01   5.7120e-01   8.8481e-01
Total time spent on Stability Diagram computation = 00:00:20

Damping Parameter: c = 300

sigma_out = 1
sigma_in = 1
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 5.72E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 6.26E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 7.27E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 8.53E-02 MB

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          8.88e-16  1.07e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE           eps    po.period
    0  00:00:00   1.0704e+01      1  EP      1.0000e-02   5.8875e-01
    1  00:00:00   1.0704e+01      2  EP      3.4694e-18   5.8875e-01

 STEP      TIME        ||U||  LABEL  TYPE           eps    po.period
    0  00:00:00   1.0704e+01      3  EP      1.0000e-02   5.8875e-01
    2  00:00:00   1.0707e+01      4  PD      1.6072e-01   5.8875e-01
    5  00:00:00   1.0797e+01      5  EP      1.0000e+00   5.8875e-01

 Run='ROM_family_bif1': Continue bifurcations from point 4 in run 'ROM_detect_bif'.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          1.12e-08  1.67e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:00   1.6689e+01      1  EP      1.0672e+01   5.8875e-01   1.6072e-01
   10  00:00:04   1.6534e+01      2          1.0545e+01   5.9586e-01   3.7699e-01
   20  00:00:08   1.6289e+01      3          1.0312e+01   6.0930e-01   9.7729e-01
   21  00:00:08   1.6278e+01      4  EP      1.0300e+01   6.1002e-01   1.0095e+00

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:09   1.6689e+01      5  EP      1.0672e+01   5.8875e-01   1.6072e-01
   10  00:00:12   1.6859e+01      6          1.0799e+01   5.8181e-01   3.7699e-01
   20  00:00:17   1.7154e+01      7  EP      1.1000e+01   5.7120e-01   8.9312e-01
Total time spent on Stability Diagram computation = 00:00:18

Damping Parameter: c = 400

sigma_out = 1
sigma_in = 1
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 5.72E-02 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 6.26E-02 MB
Manifold computation time at order 4 = 00:00:00
Estimated memory usage at order  4 = 7.27E-02 MB
Manifold computation time at order 5 = 00:00:00
Estimated memory usage at order  5 = 8.53E-02 MB

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          0.00e+00  1.07e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE           eps    po.period
    0  00:00:00   1.0704e+01      1  EP      1.0000e-02   5.8876e-01
    1  00:00:00   1.0704e+01      2  EP      3.4694e-18   5.8876e-01

 STEP      TIME        ||U||  LABEL  TYPE           eps    po.period
    0  00:00:00   1.0704e+01      3  EP      1.0000e-02   5.8876e-01
    2  00:00:00   1.0709e+01      4  PD      2.1430e-01   5.8876e-01
    5  00:00:00   1.0797e+01      5  EP      1.0000e+00   5.8876e-01

 Run='ROM_family_bif1': Continue bifurcations from point 4 in run 'ROM_detect_bif'.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          7.35e-09  1.67e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:00   1.6690e+01      1  EP      1.0672e+01   5.8876e-01   2.1430e-01
   10  00:00:04   1.6500e+01      2          1.0514e+01   5.9759e-01   4.7368e-01
   19  00:00:07   1.6279e+01      3  EP      1.0300e+01   6.1002e-01   1.0190e+00

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:08   1.6690e+01      4  EP      1.0672e+01   5.8876e-01   2.1430e-01
   10  00:00:11   1.6902e+01      5          1.0830e+01   5.8019e-01   4.7369e-01
   18  00:00:15   1.7155e+01      6  EP      1.1000e+01   5.7120e-01   9.0463e-01
Total time spent on Stability Diagram computation = 00:00:16

Verification: Collocation using coco

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

nCycles = 10;
coco_sd = cocoWrapper(DS, nCycles, []);
set(coco_sd.Options, 'PtMX',70, 'bi_direct',true);
set(coco_sd,'branchSwitch',true)

startcoco = tic;
SD_full = coco_sd.extract_Stability_Diagram(OmegaRange,epRange,'amp',p0,'PD',PlotSD);
timings(ii).cocoSD = toc(startcoco);


name = strcat('SD_damp',num2str(c),'n',num2str(n));
save(name, 'SD','SD_full')

ii = ii+1;

Damping Parameter: c = 100

 Run='full_detect_bif': Continue primary family of periodic orbits.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          0.00e+00  1.51e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE           eps           om
    0  00:00:00   1.5104e+01      1  EP      1.0000e-02   1.0672e+01
    1  00:00:00   1.5104e+01      2  EP      3.4694e-18   1.0672e+01

 STEP      TIME        ||U||  LABEL  TYPE           eps           om
    0  00:00:01   1.5104e+01      3  EP      1.0000e-02   1.0672e+01
    1  00:00:01   1.5104e+01      4  PD      5.3576e-02   1.0672e+01
    5  00:00:02   1.5170e+01      5  EP      1.0000e+00   1.0672e+01

 Run='full_family_bif_1': Continue bifurcations from point 4 in run 'full_detect_bif'.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          2.00e-09  1.67e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:00   1.6683e+01      1  EP      1.0672e+01   5.8874e-01   5.3576e-02
   10  00:00:04   1.5888e+01      2          1.0614e+01   5.9196e-01   1.6375e-01
   20  00:00:08   1.5681e+01      3          1.0473e+01   5.9994e-01   5.3188e-01
   24  00:00:11   1.5474e+01      4  EP      1.0300e+01   6.1002e-01   9.8404e-01

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:11   1.6683e+01      5  EP      1.0672e+01   5.8874e-01   5.3576e-02
   10  00:00:14   1.8313e+01      6          1.0686e+01   5.8798e-01   6.5272e-02
   20  00:00:18   2.1160e+01      7          1.0702e+01   5.8712e-01   9.5665e-02
   30  00:00:21   2.4639e+01      8          1.0723e+01   5.8595e-01   1.4685e-01
   40  00:00:26   2.8467e+01      9          1.0769e+01   5.8343e-01   2.6708e-01
   50  00:00:31   3.1148e+01     10          1.0926e+01   5.7509e-01   6.8987e-01
   55  00:00:33   3.1565e+01     11  EP      1.1000e+01   5.7120e-01   8.9590e-01

Damping Parameter: c = 200

 Run='full_detect_bif': Continue primary family of periodic orbits.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          8.88e-16  1.51e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE           eps           om
    0  00:00:00   1.5104e+01      1  EP      1.0000e-02   1.0672e+01
    1  00:00:00   1.5104e+01      2  EP      3.4694e-18   1.0672e+01

 STEP      TIME        ||U||  LABEL  TYPE           eps           om
    0  00:00:00   1.5104e+01      3  EP      1.0000e-02   1.0672e+01
    2  00:00:01   1.5105e+01      4  PD      1.0715e-01   1.0672e+01
    5  00:00:02   1.5170e+01      5  EP      1.0000e+00   1.0672e+01

 Run='full_family_bif_1': Continue bifurcations from point 4 in run 'full_detect_bif'.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          8.14e-10  1.67e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:00   1.6678e+01      1  EP      1.0672e+01   5.8875e-01   1.0715e-01
   10  00:00:04   1.5835e+01      2          1.0571e+01   5.9438e-01   2.9001e-01
   20  00:00:08   1.5578e+01      3          1.0387e+01   6.0491e-01   7.6235e-01
   22  00:00:10   1.5476e+01      4  EP      1.0300e+01   6.1002e-01   9.8814e-01

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:10   1.6678e+01      5  EP      1.0672e+01   5.8875e-01   1.0715e-01
   10  00:00:13   1.8323e+01      6          1.0700e+01   5.8721e-01   1.3073e-01
   20  00:00:17   2.1182e+01      7          1.0731e+01   5.8550e-01   1.9197e-01
   30  00:00:20   2.4674e+01      8          1.0774e+01   5.8317e-01   2.9487e-01
   40  00:00:25   2.8525e+01      9          1.0863e+01   5.7838e-01   5.2867e-01
   47  00:00:29   3.0637e+01     10  EP      1.1000e+01   5.7120e-01   9.0103e-01

Damping Parameter: c = 300

 Run='full_detect_bif': Continue primary family of periodic orbits.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          8.88e-16  1.51e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE           eps           om
    0  00:00:00   1.5104e+01      1  EP      1.0000e-02   1.0672e+01
    1  00:00:00   1.5104e+01      2  EP      3.4694e-18   1.0672e+01

 STEP      TIME        ||U||  LABEL  TYPE           eps           om
    0  00:00:00   1.5104e+01      3  EP      1.0000e-02   1.0672e+01
    2  00:00:02   1.5106e+01      4  PD      1.6073e-01   1.0672e+01
    5  00:00:02   1.5170e+01      5  EP      1.0000e+00   1.0672e+01

 Run='full_family_bif_1': Continue bifurcations from point 4 in run 'full_detect_bif'.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          6.86e-09  1.67e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:00   1.6673e+01      1  EP      1.0672e+01   5.8875e-01   1.6073e-01
   10  00:00:04   1.5800e+01      2          1.0539e+01   5.9618e-01   3.8848e-01
   20  00:00:09   1.5503e+01      3          1.0320e+01   6.0881e-01   9.4242e-01
   21  00:00:09   1.5479e+01      4  EP      1.0300e+01   6.1002e-01   9.9493e-01

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:10   1.6673e+01      5  EP      1.0672e+01   5.8875e-01   1.6073e-01
   10  00:00:13   1.8332e+01      6          1.0714e+01   5.8645e-01   1.9637e-01
   20  00:00:17   2.1203e+01      7          1.0761e+01   5.8388e-01   2.8889e-01
   30  00:00:21   2.4709e+01      8          1.0825e+01   5.8042e-01   4.4390e-01
   40  00:00:25   2.8558e+01      9          1.0953e+01   5.7366e-01   7.8087e-01
   43  00:00:27   2.9349e+01     10  EP      1.1000e+01   5.7120e-01   9.0953e-01

Damping Parameter: c = 400

 Run='full_detect_bif': Continue primary family of periodic orbits.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          0.00e+00  1.51e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE           eps           om
    0  00:00:00   1.5104e+01      1  EP      1.0000e-02   1.0672e+01
    1  00:00:00   1.5104e+01      2  EP      3.4694e-18   1.0672e+01

 STEP      TIME        ||U||  LABEL  TYPE           eps           om
    0  00:00:00   1.5104e+01      3  EP      1.0000e-02   1.0672e+01
    2  00:00:02   1.5107e+01      4  PD      2.1431e-01   1.0672e+01
    5  00:00:02   1.5170e+01      5  EP      1.0000e+00   1.0672e+01

 Run='full_family_bif_1': Continue bifurcations from point 4 in run 'full_detect_bif'.

    STEP   DAMPING               NORMS              COMPUTATION TIMES
  IT SIT     GAMMA     ||d||     ||f||     ||U||   F(x)  DF(x)  SOLVE
   0                          4.72e-09  1.67e+01    0.0    0.0    0.0

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:00   1.6668e+01      1  EP      1.0672e+01   5.8876e-01   2.1431e-01
   10  00:00:04   1.5770e+01      2          1.0511e+01   5.9778e-01   4.7860e-01
   19  00:00:09   1.5483e+01      3  EP      1.0300e+01   6.1002e-01   1.0044e+00

 STEP      TIME        ||U||  LABEL  TYPE            om    po.period          eps
    0  00:00:09   1.6668e+01      4  EP      1.0672e+01   5.8876e-01   2.1431e-01
   10  00:00:12   1.8342e+01      5          1.0728e+01   5.8569e-01   2.6220e-01
   20  00:00:16   2.1226e+01      6          1.0791e+01   5.8226e-01   3.8642e-01
   30  00:00:20   2.4743e+01      7          1.0876e+01   5.7770e-01   5.9366e-01
   39  00:00:25   2.7938e+01      8  EP      1.1000e+01   5.7120e-01   9.2128e-01

end

Plot for paper

timings
PBplotSD(n,cs); % Plot SD for various damping parameters

Prismatic Beam under parametric excitation

Contents

Setup Dynamical System

Linear Modal Analysis

Stability Diagram from Reduced Dynamics

Damping Parameter: c = 100

Damping Parameter: c = 200

Damping Parameter: c = 300

Damping Parameter: c = 400

Verification: Collocation using coco

Damping Parameter: c = 100

Damping Parameter: c = 200

Damping Parameter: c = 300

Damping Parameter: c = 400