EXTRACT_BACKBONE

Extract backbone
setup
loop over orders
compute autonomous SSM coefficients
compute backbone
Backbone curves in Physical Coordinates
plotting

function BB = extract_backbone(obj, modes, omegaRange, order, varargin)

Extract backbone

This function extracts the Backbone Curves in polar coordinates. For two-dimensional SSMs, we use the normal form of paramaterization, where we choose the following form of autonomous reduced dynamics as

$\mathbf{R}_{0}(\mathbf{p})=\left[\begin{array}{c}\lambda p\\\bar{\lambda}\bar{p}\end{array}\right]+\sum_{j=1}^{M}\left[\begin{array}{c}\gamma_{j}p^{j+1}\bar{p}^{j}\\\bar{\gamma}_{j}p^{j}\bar{p}^{j+1}\end{array}\right],$

Subsitution of $p=\rho e^{\mathrm{i}\theta}$ and $\bar{p}=\rho e^{-\mathrm{i}\theta}$ to $\dot{\mathbf{p}}=\mathbf{R}_0(\mathbf{p})$ yields

$\dot{\rho}=a(\rho), \dot{\theta}=b(\rho)$

where

$a(\rho)=\sum_{j=1}^{M}\Re(\gamma_{j})\rho^{2j+1}+\rho\Re(\lambda),$

$b(\rho)=\sum_{j=1}^{M}\Im(\gamma_{j})\rho^{2j+1}+\rho\Im(\lambda).$

It follows that the backbone curves in polar coordinates is given by $\Omega=\frac{b(\rho)}{\rho}$ .

The range of rho is determined by quadratic approximation of backbone curve if varargin is empty. Otherwise, it is specified via varargin.

f1 = figure('Name','Norm');
if isnumeric(obj.FRCOptions.outdof)
    f2 = figure('Name',['Amplitude at DOFs ' num2str(obj.FRCOptions.outdof(:)')]);
else
    f2 = figure('Name','Amplitude at DOFs');
end
figs = [f1, f2];
colors = get(0,'defaultaxescolororder');


assert(numel(modes)==2,'The analytic backbone computation can only be performed for a two-dimensional SSM/LSM')
% get options
[nt, nRho, nOmega, rhoScale, outdof, saveIC]  = ...
    deal(obj.FRCOptions.nt, obj.FRCOptions.nRho, ...
    obj.FRCOptions.nPar, obj.FRCOptions.rhoScale, ...
    obj.FRCOptions.outdof, obj.FRCOptions.saveIC);

setup

startBB = tic;
obj.choose_E(modes);
lambda = obj.E.spectrum(1);

% some checks
assert(~isreal(lambda),'The eigenvalues associated to the modal subspace must be complex for analytic backbone computation')
omega0 = abs(imag(lambda));
assert(prod([omega0-omegaRange(1),omega0-omegaRange(end)])<0,'The supplied omegaRange must contain the natural frequency associated to the modes')

loop over orders

norders = numel(order);

compute autonomous SSM coefficients

[W0,R0] = obj.compute_whisker(order(end));
gamma = compute_gamma(R0); BB = cell(norders,1);
for k=1:norders

compute backbone

    gidx  = round((order(k)+0.5)/2)-1;
    if numel(varargin)==0
        rho = compute_rho_grid(omegaRange,nOmega,rhoScale,gamma(1:gidx),lambda,nRho);
    else
        rhomax = varargin{1};
        rho = linspace(0.001*rhomax,rhomax,nRho);
    end
    [~,b] = frc_ab(rho, 0, gamma(1:gidx), lambda);
    omega = b./rho;
    idx = [find(omega<omegaRange(1)) find(omega>omegaRange(2))];
    rho(idx) = []; omega(idx) = [];

Backbone curves in Physical Coordinates

    stability = true(size(rho)); psi = zeros(size(rho)); epsilon = 0;
    BC = compute_output_polar2D(rho,psi,stability,epsilon,omega,W0(1:order(k)),[],1,nt, saveIC, outdof);
    BB{k} = BC;

plotting

    plot_FRC(BC,outdof,order(k),'freq','lines',figs,colors(k,:));

end
if norders==1
    BB = BB{1};
end
totalComputationTime = toc(startBB);
disp(['Total time spent on backbone curve computation = ' datestr(datenum(0,0,0,0,0,totalComputationTime),'HH:MM:SS')])

end

function rho = compute_rho_grid(omegaRange,nOmega,rhoScale,gamma,lambda,nRho)

omega = linspace(omegaRange(1),omegaRange(end), nOmega);

Explicit quadratic approximation of the backbone curve

$\rho_{backbone} = \sqrt{\frac{\Omega-\Im(\lambda)}{\Im(\gamma_1)}}$

rho_bb = real(sqrt((omega - imag(lambda))/imag(gamma(1))));
rhomax = rhoScale * max(rho_bb);
rho = (rhomax/nRho) * (1:nRho);

end

EXTRACT_BACKBONE

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