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100644 index 0000000..e291e46 --- /dev/null +++ b/examples/AxialMovingBeam/AxialMovingBeam.m @@ -0,0 +1,271 @@ +%% +% 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'); + +%% +% *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. + +DS = DynamicalSystem(); +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(); +%% +% *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); +%% +% 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) +%% Linear Damping +% Setup Dynamical System + +n = 10; +[mass,damp,gyro,stiff,fnl,fext] = build_model(n,'linear_damp'); + +% Create model +DS = DynamicalSystem(); +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(); +%% +% *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) +%% 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 \ No newline at end of file diff --git a/examples/AxialMovingBeam/AxialMovingBeamBook.mlx b/examples/AxialMovingBeam/AxialMovingBeamBook.mlx new file mode 100644 index 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+fext = zeros(n,1); +cubic_coeff = zeros(n,n,n,n); +for i=1:n + damp(i,i) = alpha*(i*pi)^4; + stiff(i,i) = kf^2*(i*pi)^4-(gamma^2-1)*(i*pi)^2; + fext(i) = (1-(-1)^i)/(i*pi); + for j=1:n + if j~=i + gyro(i,j) = 4*gamma*i*j/(i^2-j^2)*(1-(-1)^(i+j)); + end + cubic_coeff(i,j,j,i) = i^2*j^2; + end +end + +disp('the first four eigenvalues for undamped system'); +lamd = eigs([zeros(n) eye(n);-mass\stiff -mass\(gyro)],4,'smallestabs') + +if numel(varargin)>0 && strcmp(varargin{1}, 'nonlinear_damp') + % visoelastic damping + tmp = sptensor(cubic_coeff); + subs1 = tmp.subs; + subs2 = subs1; + subs2(:,3) = subs1(:,3)+n; + subs = [subs1; subs2]; + vals = [0.25*k1^2*pi^4*tmp.vals; 0.5*alpha*k1^2/kf^2*pi^4*tmp.vals]; + f3 = sptensor(subs, vals, [n,2*n,2*n,2*n]); +else + % viscoelastic damping but ignore nonlinear part + f3 = 0.25*k1^2*pi^4*sptensor(cubic_coeff); +end +fnl = {[],f3}; + +end \ No newline at end of file diff --git a/examples/AxialMovingBeam/cal_parameters.m b/examples/AxialMovingBeam/cal_parameters.m new file mode 100644 index 0000000..d2ac7bc --- /dev/null +++ b/examples/AxialMovingBeam/cal_parameters.m @@ -0,0 +1,64 @@ +function [alpha, omega, nu] = cal_parameters(N,l) + +syms x +% N = 2; % number of modes +% l = 2; + +% construct modal functions +phis = []; +dphis = []; +ddphis = []; +alphal = zeros(N,1); +alphal(1:4) = [3.927 7.069 10.210 13.352]'; +alphal(5:N) = (4*(5:N)'+1)*pi/4; +alpha = alphal/l; +R = sin(alphal)./sinh(alphal); +E = (0.5*l*(1-R.^2)+(R.^2.*sinh(2*alphal)-sin(2*alphal))/(4*alpha)).^(-0.5); +for n=1:N + phin = E(n)*(sin(alpha(n)*x)-R(n)*sinh(alpha(n)*x)); + dphin = diff(phin,x); + ddphin = diff(dphin,x); + phis = [phis; phin]; + dphis = [dphis; dphin]; + ddphis = [ddphis; ddphin]; + normphi = int(phin^2, x, 0, l); + fprintf('L2 norm of phi%d is %d\n', n, normphi); +end + +% compute nonlinear coeffficients +alpha1 = zeros(N); +alpha2 = zeros(N); +alpha = zeros(N,N,N,N); + +for i=1:N + for j=1:N + alpha1(i,j) = int(phis(i)*ddphis(j),x,0,l); + alpha2(i,j) = int(dphis(i)*dphis(j),x,0,l); + end +end + + +for n=1:N + for m=1:N + for p=1:N + for q=1:N + alpha(n,m,p,q) = alpha1(n,q)*alpha2(m,p); + end + end + end +end + +% compute natural frequencies +omega = zeros(N,1); +for i=1:N + int1 = int(phis(i)^2,x,0,l); + int2 = int(diff(phis(i),x,4)*phis(i),x,0,l); + omega(i) = sqrt(int2/int1); +end + +nu = 1/(2*l); + +end + + + \ No newline at end of file diff --git a/examples/BenchamrkSSM1stOrder/build_model.m b/examples/BenchamrkSSM1stOrder/build_model.m new file mode 100644 index 0000000..adb11df --- /dev/null +++ b/examples/BenchamrkSSM1stOrder/build_model.m @@ -0,0 +1,17 @@ +function [A,B,F] = build_model() + +A = [-1 0;0 -sqrt(24)]; +B = eye(2); + +F2 = sptensor([2 2 2]); +F3 = sptensor([2,2,2,2]); +F4 = sptensor([2,2,2,2,2]); +F5 = sptensor([2,2,2,2,2,2]); + +F2(2,1,1) = 1; +F3(2,1,1,1) = 1; +F4(2,1,1,1,1) = 1; +F5(2,1,1,1,1,1) = 1; + +F = {F2,F3,F4,F5}; +end \ No newline at end of file diff --git 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any(orderk==orders) + plot(y,rhosamp,'LineWidth',1.5,'DisplayName',['SSM-$\mathcal{O}(',num2str(orderk),')$']); + end +end +hl = legend('show','Location', 'Best'); +set(hl, 'Interpreter','latex') +legend boxoff +set(gca,'FontSize',14); +set(gca,'LineWidth',1.2); +grid on +xlabel('$\omega$','Interpreter',"latex",'FontSize',16); +ylabel('$\rho$','Interpreter',"latex",'FontSize',16); +end \ No newline at end of file diff --git a/examples/Lorenz1stOrder/build_model.m b/examples/Lorenz1stOrder/build_model.m new file mode 100644 index 0000000..5d12860 --- /dev/null +++ b/examples/Lorenz1stOrder/build_model.m @@ -0,0 +1,10 @@ +function [A,B,F] = build_model(sigma,rho,beta) +% linear part +A = [-sigma sigma 0;rho -1 0;0 0 -beta]; +B = eye(3); +% quadratic part +F2 = sptensor([3 3 3]); +F2(2,1,3) = -1; % -xz term +F2(3,1,2) = 1; % xy term +F = {F2}; +end \ No newline at end of file diff --git a/examples/Lorenz1stOrder/demo.mlx b/examples/Lorenz1stOrder/demo.mlx new file mode 100644 index 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