├── Utilities ├── cosAndSin.m ├── blkmat.m ├── catmat.m ├── split_harmonics.m ├── hbm_excitation_forces.m ├── combine_harmonics.m ├── unpackfreq.m ├── packfreq.m ├── hbm_excitation.m ├── unpackdof.m ├── phasor2ampl.m ├── packdof.m ├── aft_jacobian_fft.m ├── linear_jacobian.m ├── hbm_scaling.m ├── hbm_floquet.m ├── aft_jacobian_ifft.m ├── aft_jacobian.m ├── hbm_derivatives.m └── hbm_objective.m ├── .gitignore ├── Test ├── test_obj.m ├── test_excite.m ├── test_odefun.m ├── test_params.m ├── test_model.m ├── test_model_allnl.m ├── hbm_solve_test.m ├── fft_test.m ├── hbm_amp_test.m ├── hbm_frf_test.m └── hbm_bb_test.m ├── CITATION.cff ├── .gitattributes ├── Setup ├── setupLin.m ├── setupNonlin.m ├── setupHarm.m └── setuphbm.m ├── Standard ├── hbm_output.m ├── hbm_states.m ├── get_time_series.m ├── hbm_balance.m └── hbm_nonlinear.m ├── Generalised ├── hbm_output3d.m ├── hbm_states3d.m ├── get_time_series3d.m ├── hbm_balance3d.m └── hbm_nonlinear3d.m ├── Functions ├── hbm_bb_plot.m ├── hbm_solve.m ├── hbm_res.m ├── hbm_amp_plot.m ├── hbm_frf_plot.m ├── hbm_amp.m ├── hbm_bb.m └── hbm_frf.m ├── README.md └── LICENSE /Utilities/cosAndSin.m: -------------------------------------------------------------------------------- 1 | function [cl, sl] = cosAndSin(ph) 2 | cl = cos(ph); sl = sin(ph); -------------------------------------------------------------------------------- /Utilities/blkmat.m: -------------------------------------------------------------------------------- 1 | function C = blkmat(A) 2 | B = num2cell(A,[1,2]); 3 | C = blkdiag(B{:}); -------------------------------------------------------------------------------- /Utilities/catmat.m: -------------------------------------------------------------------------------- 1 | function C = catmat(A,dim) 2 | B = num2cell(A,[1,2]); 3 | C = cat(dim,B{:}); -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | *.MATLABDriveTag 2 | *.bak 3 | *.asv 4 | *.m~ 5 | *.DS_Store 6 | desktop.ini 7 | thumbs.db -------------------------------------------------------------------------------- /Test/test_obj.m: -------------------------------------------------------------------------------- 1 | function G = test_obj(X,U,F,hbm,problem,w0) 2 | P = problem.P; 3 | H = X(2,P.iDof)./U(2,P.iInput); 4 | % G = -(angle(H) + pi/2).^2; 5 | G = abs(H); -------------------------------------------------------------------------------- /Utilities/split_harmonics.m: -------------------------------------------------------------------------------- 1 | function Xfft2 = split_harmonics(Xfft,harm) 2 | for i = 1:length(harm.group) 3 | Xfft2{i} = Xfft(harm.group{i}.iFreq,harm.group{i}.iDof); 4 | end -------------------------------------------------------------------------------- /CITATION.cff: -------------------------------------------------------------------------------- 1 | cff-version: 1.2.0 2 | message: "If you use this library, please cite it as below." 3 | authors: 4 | - family-names: Haslam 5 | given-names: Alexander 6 | title: HarmLAB 7 | url: https://github.com/alxhslm/HarmLAB 8 | -------------------------------------------------------------------------------- /Utilities/hbm_excitation_forces.m: -------------------------------------------------------------------------------- 1 | function frf = hbm_excitation_forces(problem,frf) 2 | for i = 1:length(frf) 3 | frf(i).Fe = (problem.Ku*frf(i).U.' + problem.Cu*(1i*frf(i).W.*frf(i).U).' + problem.Mu*(frf(i).W.^2.*frf(i).U).').'; 4 | end -------------------------------------------------------------------------------- /Utilities/combine_harmonics.m: -------------------------------------------------------------------------------- 1 | function Xfft2 = combine_harmonics(Xfft,harm) 2 | NDof = sum([harm.group{:}.NDof]); 3 | Xfft2 = zeros(harm.NFreq,NDof); 4 | for i = 1:length(harm.group) 5 | Xfft2(harm.group{i}.iFreq,harm.group{i}.iDof) = Xfft{i}; 6 | end -------------------------------------------------------------------------------- /Utilities/unpackfreq.m: -------------------------------------------------------------------------------- 1 | function Xfull = unpackfreq(X,NHarm,iSub) 2 | Xfull = zeros(2*NHarm(1)+1,2*NHarm(2)+1,size(X,2)); 3 | for i = 1:length(iSub) 4 | Xfull(iSub(i,1),iSub(i,2),:) = X(i,:); 5 | end 6 | Xfull = 0.5*(Xfull + conj(flip(flip(Xfull,1),2))); -------------------------------------------------------------------------------- /Test/test_excite.m: -------------------------------------------------------------------------------- 1 | function U = test_excite(hbm,problem,w0) 2 | P = problem.P; 3 | U = zeros(hbm.harm.NFreq,1); 4 | U(1) = P.f0; 5 | U(2) = P.f; 6 | if hbm.harm.NHarm(2)>0 7 | ii = hbm.harm.kHarm(:,1) == 0 & hbm.harm.kHarm(:,2) == 1; 8 | U(ii) = P.f2; 9 | end -------------------------------------------------------------------------------- /Utilities/packfreq.m: -------------------------------------------------------------------------------- 1 | function X = packfreq(Xfull,iSub) 2 | X = zeros(length(iSub),size(Xfull,3)); 3 | 4 | for i = 1:length(iSub) 5 | X(i,:) = Xfull(iSub(i,1),iSub(i,2),:); 6 | end 7 | 8 | %now double everything as we want single-sided coefficients 9 | X(2:end,:) = 2*X(2:end,:); %except 0Hz 10 | -------------------------------------------------------------------------------- /Utilities/hbm_excitation.m: -------------------------------------------------------------------------------- 1 | function Fe = hbm_excitation(hbm,problem,w0,U) 2 | w = hbm.harm.kHarm*(repmat(hbm.harm.rFreqBase',1,size(w0,2)) .* w0); 3 | w = permute(w,[1 3 2]); 4 | Wu = repmat(1i*w,1,size(U,2),1); 5 | Udot = U.*Wu; 6 | Uddot = Udot.*Wu; 7 | 8 | Fe = mtransposex(mtimesx(problem.Ku,mtransposex(U)) + mtimesx(problem.Cu,mtransposex(Udot)) + mtimesx(problem.Mu,mtransposex(Uddot))); -------------------------------------------------------------------------------- /Utilities/unpackdof.m: -------------------------------------------------------------------------------- 1 | function Xc = unpackdof(X,NHarm,NDof,iRetain) 2 | NComp = prod(2*NHarm+1); 3 | if nargin < 4 4 | iRetain = 1:(NDof*NComp); 5 | end 6 | Xfull = zeros(NDof*NComp,size(X,2)); 7 | Xfull(iRetain,:) = X; 8 | 9 | if ~isempty(Xfull) 10 | X = permute(reshape(Xfull,NDof,NComp,[]),[2 1 3]); 11 | else 12 | X = zeros(NComp,NDof,size(X,2)); 13 | end 14 | Xc = [X(1,:,:); X(2:2:end,:,:) + 1i*X(3:2:end,:,:)]; -------------------------------------------------------------------------------- /Utilities/phasor2ampl.m: -------------------------------------------------------------------------------- 1 | function [amp,x] = phasor2ampl(X,Nfft) 2 | if ~iscell(X) 3 | X = {X}; 4 | end 5 | 6 | if nargin < 2 7 | Nfft = 1000; 8 | end 9 | 10 | NHarm = size(X{1},1)-1; 11 | 12 | for i = 1:length(X) 13 | x{i} = permute(freq2time(X{i},NHarm,Nfft),[2 3 1]); 14 | x{i} = cat(3,x{i},x{i}(:,:,1)); 15 | amp{i} = max(x{i},[],3) - min(x{i},[],3); 16 | end 17 | 18 | if length(x) == 1 19 | x = x{1}; 20 | amp = amp{1}; 21 | end -------------------------------------------------------------------------------- /.gitattributes: -------------------------------------------------------------------------------- 1 | *.mlx -crlf -diff -merge 2 | *.mat -crlf -diff -merge 3 | *.fig -crlf -diff -merge 4 | *.p -crlf -diff -merge 5 | *.slx -crlf -diff -merge 6 | *.mdl -crlf -diff -merge 7 | *.mdlp -crlf -diff -merge 8 | *.slxp -crlf -diff -merge 9 | *.sldd -crlf -diff -merge 10 | *.mexa64 -crlf -diff -merge 11 | *.mexw64 -crlf -diff -merge 12 | *.mexmaci64 -crlf -diff -merge 13 | *.xlsx -crlf -diff -merge 14 | *.docx -crlf -diff -merge 15 | *.pdf -crlf -diff -merge 16 | *.jpg -crlf -diff -merge 17 | *.png -crlf -diff -merge -------------------------------------------------------------------------------- /Utilities/packdof.m: -------------------------------------------------------------------------------- 1 | function X = packdof(Xc,iRetain) 2 | %order is 3 | % X = [X0; @ 0 4 | % real(X1) @ 1*w0 5 | % imag(X1) @ 1*w0 6 | % real(X2) @ 2*w0 7 | % imag(X2) @ 2*w0 8 | % ... 9 | % real(XN) @ NHarm*w0 10 | % imag(XN)] @ NHarm*w0 11 | 12 | X = mtransposex([real(Xc(2:end,:,:)) imag(Xc(2:end,:,:))]); 13 | X0 = permute(Xc(1,:,:),[2 3 1]); 14 | 15 | X = [X0; reshape(X,[],size(X,3))]; 16 | 17 | if nargin < 2 18 | iRetain = 1:size(X,1); 19 | end 20 | X = X(iRetain,:); 21 | -------------------------------------------------------------------------------- /Setup/setupLin.m: -------------------------------------------------------------------------------- 1 | function lin = setupLin(harm,problem) 2 | %% state-jacobian 3 | [lin.Ak,lin.Ac,lin.Am,lin.Ax] = linear_jacobian(harm,problem.K,problem.C,problem.M,problem.G); 4 | 5 | %% input-jacobian 6 | [lin.Bk,lin.Bc,lin.Bm,lin.Bx] = linear_jacobian(harm,problem.Ku,problem.Cu,problem.Mu); 7 | 8 | %% constant terms 9 | lin.b = [problem.F0; 10 | zeros(problem.NDof*(harm.NComp-1),1)]; 11 | 12 | %% floquet multipliers 13 | [floquet.D1xdot,floquet.D1xddot] = linear_jacobian(harm,problem.C,problem.M,0*problem.M); 14 | floquet.D1Gxdot = linear_jacobian(harm,problem.G,0*problem.M,0*problem.M); 15 | floquet.D2 = kron(eye(harm.NComp),problem.M); 16 | 17 | lin.floquet = floquet; -------------------------------------------------------------------------------- /Utilities/aft_jacobian_fft.m: -------------------------------------------------------------------------------- 1 | function Ju = aft_jacobian_fft(harm) 2 | Nfft = harm.Nfft; 3 | NComp = harm.NComp; 4 | 5 | theta1 = 2*pi/Nfft(1)*(0:(Nfft(1)-1)); 6 | theta2 = 2*pi/Nfft(2)*(0:(Nfft(2)-1)); 7 | [theta1,theta2] = ndgrid(theta1,theta2); 8 | theta0 = [theta1(:)'; theta2(:)']; 9 | 10 | theta1 = permute(theta0(1,:),[1 3 2]); 11 | theta2 = permute(theta0(2,:),[1 3 2]); 12 | 13 | ju = zeros(NComp,1,Nfft(1)*Nfft(2)); 14 | ju(1,1,:) = 1/(Nfft(1)*Nfft(2)); 15 | for l = 1:(harm.NFreq-1) 16 | ph = harm.kHarm(l+1,1)*theta1 + harm.kHarm(l+1,2)*theta2; 17 | cl = cos(ph); sl = sin(ph); 18 | ju(2*l,:,:) = 2*cl/(Nfft(1)*Nfft(2)); 19 | ju(2*l+1,:,:) = -2*sl/(Nfft(1)*Nfft(2)); 20 | end 21 | Ju = ju; -------------------------------------------------------------------------------- /Standard/hbm_output.m: -------------------------------------------------------------------------------- 1 | function o = hbm_output(hbm,problem,w,u,x) 2 | NInput = problem.NInput; 3 | NDof = problem.NDof; 4 | NHarm = hbm.harm.NHarm; 5 | 6 | ii = find(NHarm ~= 0); 7 | r = hbm.harm.rFreqRatio(ii); 8 | w0 = w * r + hbm.harm.wFreq0(ii); 9 | 10 | if size(x,1) == hbm.harm.NFreq && size(x,2) == problem.NDof 11 | X = x; 12 | U = u; 13 | elseif isvector(x) && size(x,1) == hbm.harm.NComp*problem.NDof 14 | X = unpackdof(x,NFreq-1,NDof); 15 | U = unpackdof(u,NFreq-1,NInput); 16 | end 17 | 18 | %work out the time domain 19 | States = hbm_states(w0,X,U,hbm); 20 | 21 | %push through the nl system 22 | o = feval(problem.model,'output',States,hbm,problem); 23 | 24 | %finally convert into a fourier series 25 | O = hbm.nonlin.FFT*o.'; 26 | 27 | if ndims(x) < 2 28 | o = packdof(O); 29 | else 30 | o = O; 31 | end -------------------------------------------------------------------------------- /Generalised/hbm_output3d.m: -------------------------------------------------------------------------------- 1 | function o = hbm_output3d(hbm,problem,w,u,x) 2 | if hbm.options.bUseStandardHBM 3 | o = hbm_output(hbm,problem,w,u,x); 4 | return; 5 | end 6 | 7 | NInput = problem.NInput; 8 | NDof = problem.NDof; 9 | 10 | r = hbm.harm.rFreqRatio; 11 | w0 = w .* r + hbm.harm.wFreq0; 12 | 13 | if size(x,1) == hbm.harm.NFreq && size(x,2) == problem.NDof 14 | X = x; 15 | U = u; 16 | elseif isvector(x) && size(x,1) == hbm.harm.NComp*problem.NDof 17 | X = unpackdof(x,NFreq-1,NDof); 18 | U = unpackdof(u,NFreq-1,NInput); 19 | end 20 | 21 | %work out the time domain 22 | States = hbm_states3d(w0,X,U,hbm); 23 | 24 | %push through the nl system 25 | o = feval(problem.model,'output',States,hbm,problem); 26 | 27 | %finally convert into a fourier series 28 | O = hbm.nonlin.FFT*o.'; 29 | 30 | if size(x,1) == hbm.harm.NFreq && size(x,2) == problem.NDof 31 | o = O; 32 | else 33 | o = packdof(O); 34 | end -------------------------------------------------------------------------------- /Test/test_odefun.m: -------------------------------------------------------------------------------- 1 | function Mydot = test_odefun(t,y,w0,U,hbm,problem) 2 | NInput = size(U,2); 3 | w = hbm.harm.kHarm*w0(:); 4 | Wu = repmat(1i*w,1,NInput); 5 | 6 | ph = repmat(permute(exp(1i*w*t), [3 2 1]),NInput,1); 7 | amp = repmat(permute(U, [2 3 1]),1,length(t)); 8 | ampdot = repmat(permute(U.*Wu, [2 3 1]),1,length(t)); 9 | ampddot = repmat(permute(U.*(Wu.^2),[2 3 1]),1,length(t)); 10 | u = sum(real(amp .* ph) ,3); 11 | udot = sum(real(ampdot .* ph) ,3); 12 | uddot = sum(real(ampddot .* ph) ,3); 13 | 14 | %extract x and xdot 15 | NDof = problem.NDof; 16 | x = y(1:NDof,:); 17 | xdot = y(NDof+1:end,:); 18 | 19 | Fe = problem.Ku*u + problem.Cu*udot + problem.Mu*uddot; 20 | Fl = -problem.C*xdot - problem.K*x; 21 | 22 | States = struct('t',t,'x',x,'xdot',xdot,'u',u,'udot',udot,'uddot',uddot,'w0',w0); 23 | Fnl = -test_model('nl' ,States,hbm,problem); 24 | 25 | Ftot = Fl + Fnl + Fe; 26 | 27 | %finally assemble the xdot vector 28 | Mydot = [xdot; 29 | Ftot]; -------------------------------------------------------------------------------- /Standard/hbm_states.m: -------------------------------------------------------------------------------- 1 | function States = hbm_states(w0,X,U,hbm) 2 | ii = find(hbm.harm.NHarm ~= 0); 3 | 4 | NFreq = hbm.harm.NFreq; 5 | Nfft = hbm.harm.Nfft(ii); 6 | kHarm = hbm.harm.kHarm(:,ii); 7 | wBase = hbm.harm.rFreqBase(ii)*w0; 8 | 9 | %unpack the inputs 10 | w = kHarm*wBase; 11 | States.t = (0:Nfft-1)/Nfft*2*pi/wBase; 12 | 13 | %compute the fourier coefficients of the derivatives 14 | Wx = repmat(1i*w,1,size(X,2)); 15 | Xdot = X.*Wx; 16 | Xddot = Xdot.*Wx; 17 | 18 | %precompute the external inputs 19 | Wu = repmat(1i*w,1,size(U,2)); 20 | Udot = U.*Wu; 21 | Uddot = Udot.*Wu; 22 | 23 | States.w0 = w0; 24 | States.wBase = wBase; 25 | 26 | %create the time series from the fourier series 27 | States.x = real(hbm.nonlin.IFFT*X).'; 28 | States.xdot = real(hbm.nonlin.IFFT*Xdot).'; 29 | States.xddot = real(hbm.nonlin.IFFT*Xddot).'; 30 | 31 | %create the vector of inputs 32 | States.u = real(hbm.nonlin.IFFT*U).'; 33 | States.udot = real(hbm.nonlin.IFFT*Udot).'; 34 | States.uddot = real(hbm.nonlin.IFFT*Uddot).'; 35 | -------------------------------------------------------------------------------- /Generalised/hbm_states3d.m: -------------------------------------------------------------------------------- 1 | function States = hbm_states3d(w0,X,U,hbm) 2 | Nfft = hbm.harm.Nfft; 3 | kHarm = hbm.harm.kHarm; 4 | wBase = hbm.harm.rFreqBase.*w0; 5 | 6 | %unpack the inputs 7 | w = kHarm*wBase'; 8 | t1 = (0:Nfft(1)-1)/Nfft(1)*2*pi/(wBase(1)+eps); 9 | t2 = (0:Nfft(2)-1)/Nfft(2)*2*pi/(wBase(2)+eps); 10 | [t1,t2] = ndgrid(t1,t2); 11 | States.t = [t1(:) t2(:)].'; 12 | 13 | %compute the fourier coefficients of the derivatives 14 | Wx = repmat(1i*w,1,size(X,2)); 15 | Xdot = X.*Wx; 16 | Xddot = Xdot.*Wx; 17 | 18 | %precompute the external inputs 19 | Wu = repmat(1i*w,1,size(U,2)); 20 | Udot = U.*Wu; 21 | Uddot = Udot.*Wu; 22 | 23 | States.w0 = w0; 24 | States.wBase = wBase; 25 | 26 | %create the time series from the fourier series 27 | States.x = real(hbm.nonlin.IFFT*X).'; 28 | States.xdot = real(hbm.nonlin.IFFT*Xdot).'; 29 | States.xddot = real(hbm.nonlin.IFFT*Xddot).'; 30 | 31 | %create the vector of inputs 32 | States.u = real(hbm.nonlin.IFFT*U).'; 33 | States.udot = real(hbm.nonlin.IFFT*Udot).'; 34 | States.uddot = real(hbm.nonlin.IFFT*Uddot).'; 35 | -------------------------------------------------------------------------------- /Test/test_params.m: -------------------------------------------------------------------------------- 1 | function problem = test_params 2 | rng(5); 3 | P.k1 = 10+rand(1)*0; 4 | P.k2 = 10+rand(1)*0; 5 | P.k3 = 5+rand(1)*0; 6 | P.m1 = 1; 7 | P.m2 = 2; 8 | P.c1 = 0.05; 9 | P.c2 = 0.05; 10 | P.c3 = 0.05; 11 | 12 | P.cnl = 0*[0 0 0.005]'; 13 | P.knl = [0 0 0.004]'; 14 | P.n = 3; 15 | 16 | P.R = [1 0; 17 | -1 1; 18 | 0 1]; 19 | 20 | %forcing 21 | P.f0 = 0; 22 | P.f = 2*rand(1)*exp(1i*pi*rand(1)); 23 | P.f2 = 0.5*rand(1)*exp(1i*pi*rand(1)); 24 | 25 | P.iDof = 1; 26 | P.iInput = 1; 27 | 28 | problem.K = P.k1*[1 0; 29 | 0 0]; 30 | problem.K = problem.K + P.k2*[1 -1; 31 | -1 1]; 32 | problem.K = problem.K + P.k3*[0 0; 33 | 0 1]; 34 | problem.C = P.c1*[1 0; 35 | 0 0]; 36 | problem.C = problem.C + P.c2*[1 -1; 37 | -1 1]; 38 | problem.C = problem.C + P.c3*[0 0; 39 | 0 1]; 40 | problem.M = diag([P.m1 P.m2]); 41 | 42 | problem.F0 = zeros(2,1); 43 | 44 | problem.Mu = zeros(2,1); 45 | problem.Cu = zeros(2,1); 46 | problem.Ku = [0;1]; 47 | 48 | problem.model = @test_model; 49 | problem.excite = @test_excite; 50 | 51 | problem.P = P; -------------------------------------------------------------------------------- /Utilities/linear_jacobian.m: -------------------------------------------------------------------------------- 1 | function [Bk,Bc,Bm,Bx] = linear_jacobian(harm,Ku,Cu,Mu,Gu) 2 | if nargin < 5 3 | Gu = 0*Cu; 4 | end 5 | 6 | %terms dependent on each frequency 7 | Ak{1} = Ku; 8 | for j = 2:harm.NFreq 9 | Ak{j} = blkdiag(Ku,Ku); 10 | end 11 | 12 | for k = 1:2 13 | Ac{k}{1} = 0*Cu; 14 | Am{k}{1} = 0*Mu; 15 | for j = 2:harm.NFreq 16 | Ac{k}{j} = harm.rFreqBase(k)*harm.kHarm(j,k)*antiblkdiag(-Cu,Cu); 17 | Am{k}{j} = -(harm.rFreqBase(k)*harm.kHarm(j,k))^2*blkdiag(Mu,Mu); 18 | end 19 | end 20 | 21 | %cross-terms 22 | Ax{1} = 0*Mu; 23 | for j = 2:harm.NFreq 24 | Ax{j} = -2*prod(harm.kHarm(j,:).*harm.rFreqBase)*blkdiag(Mu,Mu); 25 | end 26 | 27 | %add on gyro terms 28 | for j = 2:harm.NFreq 29 | Am{1}{j} = Am{1}{j} + harm.rFreqBase(1)*harm.kHarm(j,1)*antiblkdiag(-Gu,Gu); 30 | Ax{j} = Ax{j} + harm.rFreqBase(2)*harm.kHarm(j,2)*antiblkdiag(-Gu,Gu); 31 | end 32 | 33 | Bk = blkdiag(Ak{:}); 34 | for k = 1:2 35 | Bc{k} = blkdiag(Ac{k}{:}); 36 | Bm{k} = blkdiag(Am{k}{:}); 37 | end 38 | Bx = blkdiag(Ax{:}); 39 | 40 | Bk = sparse(Bk); 41 | for k = 1:2 42 | Bm{k} = sparse(Bm{k}); 43 | Bc{k} = sparse(Bc{k}); 44 | end 45 | Bx = sparse(Bx); 46 | -------------------------------------------------------------------------------- /Standard/get_time_series.m: -------------------------------------------------------------------------------- 1 | function varargout = get_time_series(hbm,w0,X,tspan) 2 | NHarm = hbm.harm.NHarm(1); 3 | Nfft = hbm.harm.Nfft(1); 4 | 5 | %unpack the inputs 6 | w = (0:NHarm)'*w0; 7 | t = (0:Nfft-1)'/Nfft*2*pi/w0; 8 | 9 | %compute the fourier coefficients of the derivatives 10 | Wx = repmat(1i*w,1,size(X,2)); 11 | 12 | %create the time series from the fourier series 13 | x = real(hbm.nonlin.IFFT*X); 14 | if nargout > 2 15 | Xdot = X.*Wx; 16 | xdot = real(hbm.nonlin.IFFT*Xdot); 17 | if nargout > 3 18 | Xddot = Xdot.*Wx; 19 | xddot = real(hbm.nonlin.IFFT*Xddot); 20 | end 21 | end 22 | 23 | if nargin < 5 || isempty(tspan) %empty 24 | ti = linspace(0,2*pi/min(w0),Nfft); 25 | elseif length(tspan) < 2 %duration 26 | ti = linspace(0,tspan,Nfft); 27 | else 28 | ti = tspan; 29 | end 30 | 31 | for i = 1:size(x,2) 32 | xi(:,i) = interp1(t,x(:,i),ti); 33 | if nargout > 2 34 | xdoti(:,i) = interp1(t,xdot(:,i),ti); 35 | if nargout > 3 36 | xddoti(:,i) = interp1(t,xddot(:,i),ti); 37 | end 38 | end 39 | end 40 | 41 | varargout{1} = ti; 42 | varargout{2} = xi; 43 | if nargout >2 44 | varargout{3} = xdoti; 45 | if nargout > 3 46 | varargout{4} = xddoti; 47 | end 48 | end -------------------------------------------------------------------------------- /Utilities/hbm_scaling.m: -------------------------------------------------------------------------------- 1 | function problem = hbm_scaling(problem,hbm,results) 2 | X = results.X; 3 | w = results.w; 4 | A = results.A; 5 | 6 | tol = hbm.scaling.tol; 7 | switch hbm.scaling.method 8 | case 'max' 9 | xdc = max(abs(X(1,:)),tol); 10 | xmax = max(max(abs(X(2:end,:)),[],1),tol); 11 | xharm = repmat(xmax,hbm.harm.NFreq-1,1); 12 | case 'abs' 13 | xdc = max(abs(X(1,:)),tol); 14 | xharm = max(abs(X(2:end,:)),tol); 15 | end 16 | Xscale = [xdc; xharm*(1+1i)]; 17 | problem.Xscale = packdof(Xscale); 18 | problem.Zscale = problem.Xscale; 19 | problem.Fscale = ones(length(problem.Xscale),1); 20 | 21 | switch problem.type 22 | case 'frf' 23 | problem.wscale = w; 24 | problem.Zscale(end+1) = w; 25 | case 'resonance' 26 | problem.wscale = w; 27 | problem.Zscale(end+1) = w; 28 | case 'bb' 29 | problem.wscale = w; 30 | problem.Zscale(end+1) = w; 31 | 32 | %amplitude is the continuation variable 33 | problem.Ascale = A; 34 | problem.Zscale(end+1) = A; 35 | %need extra constraint 36 | problem.Fscale(end+1) = 1; 37 | case 'amp' 38 | problem.Ascale = A; 39 | problem.Zscale(end+1) = A; 40 | case 'solve' 41 | end 42 | 43 | problem.Jscale = (1./problem.Fscale(:))*problem.Zscale(:)'; -------------------------------------------------------------------------------- /Utilities/hbm_floquet.m: -------------------------------------------------------------------------------- 1 | function sol = hbm_floquet(hbm,problem,sol) 2 | if length(sol) > 1 3 | [sol.L] = deal(0); 4 | for i = 1:length(sol) 5 | sol(i) = hbm_floquet(hbm,problem,sol(i)); 6 | end 7 | return; 8 | end 9 | 10 | if size(sol.X,1) == hbm.harm.NFreq && size(sol.X,2) == problem.NDof 11 | x = packdof(sol.X); 12 | u = packdof(sol.U); 13 | else 14 | x = sol.X; 15 | u = sol.U; 16 | end 17 | 18 | w = sol.w; 19 | 20 | if any(isnan(x) | isinf(x)) 21 | lambda = NaN(hbm.harm.NComp*problem.NDof,1); 22 | else 23 | [A,B] = floquetMatrices(hbm,problem,w,u,x); 24 | lambda = eig(A,B,'vector'); 25 | [~,iSort] = sort(abs(imag(lambda))); 26 | lambda = lambda(iSort); 27 | end 28 | 29 | sol.L = lambda; 30 | 31 | function [A,B] = floquetMatrices(hbm,problem,w,u,x) 32 | hbm.bIncludeNL = 1; 33 | NPts = size(u,2); 34 | 35 | A = zeros(2*size(x,1),2*size(x,1),NPts); 36 | B = zeros(2*size(x,1),2*size(x,1),NPts); 37 | for i = 1:NPts 38 | D0 = hbm_balance3d('floquet0',hbm,problem,w(i),u(:,i),x(:,i)); 39 | D1 = hbm_balance3d('floquet1',hbm,problem,w(i),u(:,i),x(:,i)); 40 | D2 = hbm_balance3d('floquet2',hbm,problem,w(i),u(:,i),x(:,i)); 41 | 42 | I = eye(hbm.harm.NComp*problem.NDof); 43 | Z = zeros(hbm.harm.NComp*problem.NDof); 44 | 45 | B1 = [D1 D0; 46 | -I Z]; 47 | B2 = [D2 Z; 48 | Z I]; 49 | 50 | A(:,:,i) = B1; 51 | B(:,:,i) = -B2; 52 | end -------------------------------------------------------------------------------- /Test/test_model.m: -------------------------------------------------------------------------------- 1 | function varargout = test_model(part,States,hbm,problem) 2 | if ~iscell(part) 3 | part = {part}; 4 | end 5 | 6 | P = problem.P; 7 | NPts = size(States.t,2); 8 | x = States.x; 9 | xdot = States.xdot; 10 | 11 | varargout = {}; 12 | 13 | for i = 1:length(part) 14 | switch part{i} 15 | %% Nonlin 16 | case {'nl','output'} 17 | fel = P.knl .* sgn_power(P.R*x,P.n); 18 | fdamp = P.cnl .* (P.R*xdot).^3; 19 | Fnl = P.R' * (fel + fdamp); 20 | 21 | varargout{end+1} = Fnl; 22 | case 'nl_x' 23 | [~,d] = sgn_power(P.R*x,P.n); 24 | Knl = zeros(problem.NDof); 25 | for j = 1:size(P.R,1) 26 | Knl = Knl + mtimesx(P.knl(j) .* permute(d(j,:),[1 3 2]),P.R(j,:)'*P.R(j,:)); 27 | end 28 | varargout{end+1} = Knl; 29 | case 'nl_xdot' 30 | Cnl = zeros(problem.NDof); 31 | d = 3 * P.cnl .* (P.R*xdot).^2; 32 | for j = 1:size(P.R,1) 33 | Cnl = Cnl + mtimesx(P.cnl(j) .* permute(d(j,:),[1 3 2]),P.R(j,:)'*P.R(j,:)); 34 | end 35 | varargout{end+1} = Cnl; 36 | case 'nl_u' 37 | varargout{end+1} = zeros(2,1,NPts); 38 | case 'nl_udot' 39 | varargout{end+1} = zeros(2,1,NPts); 40 | otherwise 41 | varargout{end+1} = []; 42 | end 43 | end -------------------------------------------------------------------------------- /Test/test_model_allnl.m: -------------------------------------------------------------------------------- 1 | function varargout = test_model(part,States,hbm,problem) 2 | 3 | P = problem.P; 4 | 5 | P = problem.P; 6 | NPts = size(States.t,2); 7 | x = States.x; 8 | xdot = States.xdot; 9 | u = States.u; 10 | udot = States.udot; 11 | 12 | varargout = {}; 13 | 14 | for i = 1:length(part) 15 | switch part 16 | case {'nonlin','output'} 17 | Fe = problem.Ku*u + problem.Cu*udot + problem.Mu*uddot; 18 | Fl = -problem.C*xdot - problem.K*x - problem.M*xddot; 19 | x1 = x(1,:); x2 = x(2,:); 20 | f = sgn_power(x1-x2,P.n); 21 | Fnl = [-1;1]*P.knl * f - P.cnl * xdot.^3; 22 | varargout{end+1} = Fl + Fnl + Fe; 23 | case 'nl_x' 24 | x1 = permute(x(1,:),[1 3 2]); 25 | x2 = permute(x(2,:),[1 3 2]); 26 | [~,d] = sgn_power(x1-x2,P.n); 27 | k = P.knl * d; 28 | 29 | Knl = -[k -k; -k k]; 30 | 31 | varargout{end+1} = Knl; 32 | case 'nl_xdot' 33 | xdot1 = permute(xdot(1,:),[1 3 2]); 34 | xdot2 = permute(xdot(2,:),[1 3 2]); 35 | c1 = P.cnl * 3 * xdot1.^2; 36 | c2 = P.cnl * 3 * xdot2.^2; 37 | 38 | Cnl = -[c1,0*c1;0*c2,c2]; 39 | varargout{end+1} = Cnl; 40 | case {'nl_u','nl_udot'} 41 | varargout{end+1} = zeros(2,1,NPts); 42 | case 'freqderiv' 43 | varargout{end+1} = []; 44 | end 45 | end -------------------------------------------------------------------------------- /Utilities/aft_jacobian_ifft.m: -------------------------------------------------------------------------------- 1 | function [Ju,Judot,Juddot] = aft_jacobian_ifft(harm) 2 | Nfft = harm.Nfft; 3 | NComp = harm.NComp; 4 | 5 | theta1 = 2*pi/Nfft(1)*(0:(Nfft(1)-1)); 6 | theta2 = 2*pi/Nfft(2)*(0:(Nfft(2)-1)); 7 | [theta1,theta2] = ndgrid(theta1,theta2); 8 | theta0 = [theta1(:)'; theta2(:)']; 9 | 10 | Ju = zeros(1,NComp,Nfft(1)*Nfft(2)); 11 | Ju(1,1,:) = 1; 12 | for l = 1:(harm.NFreq-1) 13 | ph = permute(harm.kHarm(l+1,:)*theta0,[1 3 2]); 14 | cl = cos(ph); sl = sin(ph); 15 | Ju(1,2*l,:) = cl; 16 | Ju(1,2*l+1,:) = -sl; 17 | end 18 | 19 | Judot = cell(1,2); 20 | for n = 1:2 21 | Judot{n} = zeros(1,NComp,Nfft(1)*Nfft(2)); 22 | for l = 1:(harm.NFreq-1) 23 | ph = permute(harm.kHarm(l+1,:)*theta0,[1 3 2]); 24 | cl = cos(ph); sl = sin(ph); 25 | Judot{n}(1,2*l,:) = -harm.kHarm(l+1,n)*sl; 26 | Judot{n}(1,2*l+1,:) = -harm.kHarm(l+1,n)*cl; 27 | end 28 | end 29 | 30 | Juddot = cell(1,3); 31 | for n = 1:2 32 | Juddot{n} = zeros(1,NComp,Nfft(1)*Nfft(2)); 33 | for l = 1:(harm.NFreq-1) 34 | ph = permute(harm.kHarm(l+1,:)*theta0,[1 3 2]); 35 | cl = cos(ph); sl = sin(ph); 36 | Juddot{n}(1,2*l,:) = -harm.kHarm(l+1,n)^2*cl; 37 | Juddot{n}(1,2*l+1,:) = harm.kHarm(l+1,n)^2*sl; 38 | end 39 | end 40 | 41 | n = 3; 42 | Juddot{n} = zeros(1,NComp,Nfft(1)*Nfft(2)); 43 | for l = 1:(harm.NFreq-1) 44 | ph = permute(harm.kHarm(l+1,:)*theta0,[1 3 2]); 45 | cl = cos(ph); sl = sin(ph); 46 | Juddot{n}(1,2*l,:) = -prod(harm.kHarm(l+1,:))*cl; 47 | Juddot{n}(1,2*l+1,:) = prod(harm.kHarm(l+1,:))*sl; 48 | end -------------------------------------------------------------------------------- /Test/hbm_solve_test.m: -------------------------------------------------------------------------------- 1 | function hbm_solve_test(b3d) 2 | problem = test_params; 3 | 4 | hbm.harm.rFreqRatio = 1; 5 | hbm.harm.NHarm = 2; 6 | hbm.harm.Nfft = 32; 7 | hbm.harm.iHarmPlot = 2; 8 | 9 | if nargin < 1 10 | b3d = 1; 11 | end 12 | 13 | if b3d 14 | hbm.harm.rFreqRatio(end+1) = 1.376; 15 | hbm.harm.NHarm(end+1) = 2; 16 | hbm.harm.Nfft(end+1) = 16; 17 | hbm.harm.iHarmPlot(end+1) = 3; 18 | end 19 | 20 | omega = sqrt(eig(problem.K,problem.M)); 21 | w0 = 4; 22 | A = 100; 23 | 24 | [hbm,problem] = setuphbm(hbm,problem); 25 | NDof = problem.NDof; 26 | 27 | tic; 28 | sol1 = hbm_solve(hbm,problem,w0,A,[]); 29 | [t1,x1,xdot1] = get_time_series3d(hbm,w0,sol1.X); 30 | tRun(1) = toc; 31 | 32 | tic; 33 | y0 = [x1(1,:) xdot1(1,:)]; 34 | fun = @(t,y)test_odefun(t,y,w0*hbm.harm.rFreqRatio,sol1.U,hbm,problem); 35 | M = blkdiag(eye(NDof),problem.M); 36 | options = odeset('Mass',M,'Vectorized','on','MassSingular','yes','RelTol',1E-12,'AbsTol',1E-12); 37 | [t2,y] = ode15s(fun,[t1(1) t1(end)],y0,options); 38 | tRun(2) = toc; 39 | 40 | x2 = y(:,1:NDof); 41 | xdot2 = y(:,NDof+(1:NDof)); 42 | 43 | figure 44 | subplot(2,2,1) 45 | plot(t1,x1(:,1:NDof)','-',t2,x2(:,1:NDof)','o-'); 46 | ylabel('x'); 47 | 48 | subplot(2,2,3) 49 | plot(t1,xdot1(:,1:NDof)','-',t2,xdot2(:,1:NDof)','o-'); 50 | ylabel('xdot'); 51 | xlabel('Time (s)'); 52 | 53 | names = {'HBM','ODE'}; 54 | 55 | subplot(1,2,2) 56 | plot(x1(:,1:NDof),xdot1(:,1:NDof)); 57 | hold on 58 | plot(x2(:,1:NDof),xdot2(:,1:NDof),'o-'); 59 | xlabel('x'); 60 | ylabel('xdot'); 61 | leg = {}; 62 | for i = 1:length(tRun) 63 | leg = [leg cellsprintf([names{i} '(x%d)'],num2cell(1:NDof))]; 64 | end 65 | legend(leg) 66 | 67 | for i = 1:length(tRun) 68 | fprintf('%s: %f s\n',names{i},tRun(i)); 69 | end -------------------------------------------------------------------------------- /Test/fft_test.m: -------------------------------------------------------------------------------- 1 | function fft_test 2 | 3 | NHarm = [4 4]; 4 | Nfft = [128 128]; 5 | iSub = [2 7; 6 | 4 1 7 | 2 4 8 | 3 1]; 9 | for i = 1:10000 10 | X = rand(4,1) + rand(4,1)*1i; 11 | tic; 12 | x = freq2time3d(X,NHarm,iSub,Nfft,1); 13 | t(i) = toc; 14 | tic; 15 | x = freq2time3d(X,NHarm,iSub,Nfft,0); 16 | t2(i) = toc; 17 | end 18 | mean(t) 19 | mean(t2) 20 | 21 | function x = freq2time3d(X,NHarm,iSub,Nfft,b) 22 | %NB, scaling so IFFT = Expanding Fourier Coefficients 23 | 24 | %first regenerate the -ve frequency components 25 | Xfft = unpackfreq(X,NHarm,iSub); 26 | 27 | if b 28 | %now augment for Nfft frequencies 29 | iz = floor(Nfft/2)+1; 30 | Xfft2 = zeros(Nfft(1),Nfft(2),size(X,3)); 31 | Xfft2(iz(1)+(-NHarm(1):NHarm(1)),iz(2)+(-NHarm(2):NHarm(2)),:) = Xfft; 32 | Xfft = Xfft2; 33 | end 34 | 35 | %finally compute the ifft 36 | Xfft = ifftshift(Xfft,1); 37 | Xfft = ifft(Xfft,Nfft(1),1)*Nfft(1); 38 | if NHarm(2)>0 39 | Xfft = ifftshift(Xfft,2); 40 | x = ifft(Xfft,Nfft(2),2)*Nfft(2); 41 | else 42 | x = Xfft; 43 | end 44 | 45 | %the answer should now be real to machine precision 46 | x = real(x); 47 | 48 | %wrap hypertime 49 | x = reshape(x,prod(Nfft),[]); 50 | 51 | 52 | 53 | function x = freq2time(X,NHarm,Nfft,b) 54 | 55 | %first regenerate the -ve frequency components 56 | X = cat(1,0*X(2:end,:,:), X); 57 | Xfft = 0.5*(X + conj(flip(X,1))); 58 | 59 | if b 60 | %now augment for Nfft frequencies 61 | iz = floor(Nfft/2)+1; 62 | sz = size(X); 63 | Xfft2 = zeros([Nfft,sz(2:end)]); 64 | Xfft2(iz+(-NHarm:NHarm),:,:) = Xfft; 65 | Xfft = Xfft2; 66 | end 67 | 68 | %finally compute the ifft 69 | Xfft = ifftshift(Xfft,1); 70 | x = ifft(Xfft,Nfft,1) * Nfft; 71 | 72 | %the answer should now be real to machine precision 73 | x = real(x); -------------------------------------------------------------------------------- /Test/hbm_amp_test.m: -------------------------------------------------------------------------------- 1 | function hbm_amp_test(b3d) 2 | problem = test_params; 3 | 4 | hbm.harm.rFreqRatio = 1; 5 | hbm.harm.NHarm = 2; 6 | hbm.harm.Nfft = 32; 7 | hbm.harm.iHarmPlot = 2; 8 | 9 | if nargin < 1 10 | b3d = 1; 11 | end 12 | 13 | if b3d 14 | hbm.harm.rFreqRatio(end+1) = 1.376; 15 | hbm.harm.NHarm(end+1) = 2; 16 | hbm.harm.Nfft(end+1) = 16; 17 | hbm.harm.iHarmPlot(end+1) = 3; 18 | end 19 | 20 | hbm.dependence.x = true; 21 | hbm.dependence.xdot = true; 22 | hbm.dependence.w = false; 23 | 24 | hbm.scaling.tol = 1; 25 | 26 | hbm.cont.step0 = 1E-2; 27 | hbm.cont.max_step = 1E-1; 28 | hbm.cont.min_step = 1E-6; 29 | 30 | hbm.cont.method = 'predcorr'; 31 | 32 | [hbm,problem] = setuphbm(hbm,problem); 33 | 34 | omega = sqrt(eig(problem.K,problem.M)); 35 | w0 = omega(1); 36 | A0 = 1; 37 | AEnd = 5; 38 | 39 | S = {}; 40 | 41 | tic; 42 | hbm.cont.method = 'none'; 43 | sol = hbm_amp(hbm,problem,w0,A0,[],AEnd,[]); 44 | S{end+1} = storeResults(sol,toc,'none'); 45 | 46 | tic; 47 | hbm.cont.method = 'predcorr'; 48 | hbm.cont.predcorr.corrector = 'arclength'; 49 | sol = hbm_amp(hbm,problem,w0,A0,[],AEnd,[]); 50 | S{end+1} = storeResults(sol,toc,'arclength'); 51 | 52 | tic; 53 | hbm.cont.method = 'predcorr'; 54 | hbm.cont.predcorr.corrector = 'pseudo'; 55 | sol = hbm_amp(hbm,problem,w0,A0,[],AEnd,[]); 56 | S{end+1} = storeResults(sol,toc,'pseudo'); 57 | 58 | figure 59 | for j = 1:problem.NDof 60 | ax_mag(j) = subplot(2,problem.NDof,j); 61 | hold on 62 | for i = 1:length(S) 63 | h(i) = plot(ax_mag(j),S{i}.A,abs(S{i}.X(j,:))); 64 | end 65 | ylabel(ax_mag(j),sprintf('|X_%d| (mag)',j)); 66 | end 67 | 68 | for j = 1:problem.NDof 69 | ax_ph(j) = subplot(2,problem.NDof,problem.NDof+j); 70 | hold on 71 | for i = 1:length(S) 72 | hph(i,j) = plot(ax_ph(j),S{i}.A,unwrap(angle(S{i}.X(j,:)),[],2)); 73 | end 74 | xlabel(ax_ph(j),'\omega (rads)'); 75 | ylabel(ax_ph(j),sprintf('\\angle X_%d (deg)',j)); 76 | end 77 | linkaxes([ax_mag ax_ph],'x') 78 | 79 | for i = 1:length(S) 80 | leg{i} = S{i}.name; 81 | end 82 | legend(ax_ph(end),hph(:,end),leg) 83 | 84 | for i = 1:length(S) 85 | fprintf('%10s : %0.2f s\n',S{i}.name,S{i}.t) 86 | end 87 | 88 | function S = storeResults(sol,t,name) 89 | X = cat(3,sol.X); 90 | 91 | S.X = permute(X(2,:,:),[2 3 1]); 92 | S.w = cat(2,sol.w); 93 | S.A = cat(2,sol.A); 94 | S.t = t; 95 | S.name = name; -------------------------------------------------------------------------------- /Functions/hbm_bb_plot.m: -------------------------------------------------------------------------------- 1 | function varargout = hbm_bb_plot(command,hbm,problem,results) 2 | persistent figBB axBB hBB hWaitbar A H W 3 | if hbm.cont.bUpdate 4 | switch command 5 | case 'init' 6 | if ~isempty(figBB) && ishandle(figBB) 7 | close(figBB) 8 | hBB = []; 9 | end 10 | if ~isempty(hWaitbar) && ishandle(hWaitbar) 11 | close(hWaitbar) 12 | hWaitbar = []; 13 | end 14 | 15 | A = results.A; 16 | W = results.w; 17 | H = abs(results.H); 18 | [figBB,axBB,hBB] = createFig(hbm,problem,A,H,W); 19 | hWaitbar = waitbar(0, 'Amplitude Range'); 20 | if nargout > 0 21 | varargout{1} = figBB; 22 | if nargout > 1 23 | varargout{2} = axBB; 24 | end 25 | end 26 | 27 | case {'data','err'} 28 | if ~ishandle(figBB) 29 | [figBB,axBB,hBB] = createFig(hbm,problem,A,H,W); 30 | end 31 | if strcmp(command,'data') 32 | A(end+1) = results.A; 33 | H(end+1) = abs(results.H); 34 | W(end+1) = results.w; 35 | [A,isort] = sort(A); 36 | H = H(isort); W = W(isort); 37 | set(hBB(1),'xdata',A,'ydata',W); 38 | set(hBB(2),'xdata',A,'ydata',H); 39 | 40 | %update our progress 41 | if ~ishandle(hWaitbar) 42 | hWaitbar = waitbar(0, 'Amplitude Range'); 43 | end 44 | waitbar((results.A-problem.A0)/(problem.AEnd - problem.A0),hWaitbar); 45 | else 46 | plot(axBB(1),results.A,results.w,'r.') 47 | plot(axBB(2),results.A,abs(results.H),'r.') 48 | end 49 | drawnow 50 | case 'close' 51 | close(hWaitbar) 52 | close(figBB) 53 | hWaitbar = []; 54 | figBB = []; 55 | hBB = []; 56 | axBB = []; 57 | end 58 | end 59 | 60 | 61 | function [fig,axBB,hBB] = createFig(hbm,problem,A,H,w) 62 | fig = figure('Name',['BB: ' problem.name]); 63 | 64 | axBB(1) = subplot(2,1,1); 65 | hBB(1) = plot(axBB(1),A,w,'g.-'); 66 | hold on 67 | ylabel('Frequency (rad/s)') 68 | xlabel('Amplitude') 69 | xlim(axBB(1),[problem.AMin problem.AMax]); 70 | 71 | axBB(2) = subplot(2,1,2); 72 | hBB(2) = plot(axBB(2),A,H,'g.-'); 73 | hold on 74 | ylabel('Peak amp') 75 | xlabel('Amplitude') 76 | xlim(axBB(2),[problem.AMin problem.AMax]); -------------------------------------------------------------------------------- /Utilities/aft_jacobian.m: -------------------------------------------------------------------------------- 1 | function [Ju,Judot] = aft_jacobian(harm,NOutput,NInput) 2 | Nfft = harm.Nfft; 3 | NComp = harm.NComp; 4 | 5 | theta1 = 2*pi/Nfft(1)*(0:(Nfft(1)-1)); 6 | theta2 = 2*pi/Nfft(2)*(0:(Nfft(2)-1)); 7 | [theta1,theta2] = ndgrid(theta1,theta2); 8 | theta0 = [theta1(:)'; theta2(:)']; 9 | 10 | theta1 = permute(theta0(1,:),[1 3 2]); 11 | theta2 = permute(theta0(2,:),[1 3 2]); 12 | 13 | ju = zeros(NComp,NComp,Nfft(1)*Nfft(2)); 14 | ju(1,1,:) = 1/(Nfft(1)*Nfft(2)); 15 | for l = 1:(harm.NFreq-1) 16 | ph = harm.kHarm(l+1,1)*theta1 + harm.kHarm(l+1,2)*theta2; 17 | cl = cos(ph); sl = sin(ph); 18 | ju(1,2*l,:) = cl/(Nfft(1)*Nfft(2)); 19 | ju(1,2*l+1,:) = -sl/(Nfft(1)*Nfft(2)); 20 | end 21 | for k = 1:(harm.NFreq-1) 22 | ph = harm.kHarm(k+1,1)*theta1 + harm.kHarm(k+1,2)*theta2; 23 | ck = cos(ph); sk = sin(ph); 24 | ju(2*k,1,:) = 2*ck/(Nfft(1)*Nfft(2)); 25 | ju(2*k+1,1,:) = -2*sk/(Nfft(1)*Nfft(2)); 26 | for l = 1:(harm.NFreq-1) 27 | ph = harm.kHarm(l+1,1)*theta1 + harm.kHarm(l+1,2)*theta2; 28 | cl = cos(ph); sl = sin(ph); 29 | ju(2*k,2*l,:) = 2*cl.*ck/(Nfft(1)*Nfft(2)); 30 | ju(2*k,2*l+1,:) = -2*sl.*ck/(Nfft(1)*Nfft(2)); 31 | ju(2*k+1,2*l,:) = -2*cl.*sk/(Nfft(1)*Nfft(2)); 32 | ju(2*k+1,2*l+1,:) = 2*sl.*sk/(Nfft(1)*Nfft(2)); 33 | end 34 | end 35 | Ju = resize_jacobian(ju,NInput,NOutput); 36 | 37 | Judot = cell(1,2); 38 | for n = 1:2 39 | judot = zeros(NComp,NComp,Nfft(1)*Nfft(2)); 40 | for l = 1:(harm.NFreq-1) 41 | ph = harm.kHarm(l+1,1)*theta1 + harm.kHarm(l+1,2)*theta2; 42 | cl = cos(ph); sl = sin(ph); 43 | judot(1,2*l,:) = -harm.kHarm(l+1,n)*sl/(Nfft(1)*Nfft(2)); 44 | judot(1,2*l+1,:) = -harm.kHarm(l+1,n)*cl/(Nfft(1)*Nfft(2)); 45 | end 46 | for k = 1:(harm.NFreq-1) 47 | ph = harm.kHarm(k+1,1)*theta1 + harm.kHarm(k+1,2)*theta2; 48 | ck = cos(ph); sk = sin(ph); 49 | judot(2*k,1,:) = 0; 50 | judot(2*k+1,1,:) = 0; 51 | for l = 1:(harm.NFreq-1) 52 | ph = harm.kHarm(l+1,1)*theta1 + harm.kHarm(l+1,2)*theta2; 53 | cl = cos(ph); sl = sin(ph); 54 | judot(2*k,2*l,:) = -2*harm.kHarm(l+1,n).*sl.*ck/(Nfft(1)*Nfft(2)); 55 | judot(2*k,2*l+1,:) = -2*harm.kHarm(l+1,n).*cl.*ck/(Nfft(1)*Nfft(2)); 56 | judot(2*k+1,2*l,:) = 2*harm.kHarm(l+1,n).*sl.*sk/(Nfft(1)*Nfft(2)); 57 | judot(2*k+1,2*l+1,:) = 2*harm.kHarm(l+1,n).*cl.*sk/(Nfft(1)*Nfft(2)); 58 | end 59 | end 60 | Judot{n} = resize_jacobian(judot,NInput,NOutput); 61 | end 62 | 63 | function Ju = resize_jacobian(ju,NInput,NOutput) 64 | Ju = zeros(NOutput*size(ju,1),NInput*size(ju,2),size(ju,3)); 65 | for i = 1:size(ju,3) 66 | Ju(:,:,i) = kron(ju(:,:,i),ones(NOutput,NInput)); 67 | end -------------------------------------------------------------------------------- /Setup/setupNonlin.m: -------------------------------------------------------------------------------- 1 | function nonlin = setupNonlin(harm,problem) 2 | Nfft = harm.Nfft; 3 | 4 | %% fft matrices 5 | theta1 = 2*pi/Nfft(1)*(0:(Nfft(1)-1)); 6 | theta2 = 2*pi/Nfft(2)*(0:(Nfft(2)-1)); 7 | [theta1,theta2] = ndgrid(theta1,theta2); 8 | theta0 = [theta1(:)'; theta2(:)']; 9 | 10 | nonlin.FFT = zeros(harm.NFreq,Nfft(1)*Nfft(2)); 11 | nonlin.FFT(1,:) = 1/(Nfft(1)*Nfft(2)); 12 | for k = 2:harm.NFreq 13 | nonlin.FFT(k,:) = 2/(Nfft(1)*Nfft(2))*exp(-1i*harm.kHarm(k,:)*theta0); 14 | end 15 | 16 | nonlin.IFFT = zeros(Nfft(1)*Nfft(2),harm.NFreq); 17 | nonlin.IFFT(:,1) = 1; 18 | for k = 2:harm.NFreq 19 | nonlin.IFFT(:,k) = exp(1i*harm.kHarm(k,:)*theta0); 20 | end 21 | 22 | %% non-linear derivative matrices 23 | %work out the constituent jacobians first 24 | aft = get_aft_jacobians(harm); 25 | 26 | %now make the specifis matrices we will need 27 | nonlin.hbm = get_hbm_matrices(problem,harm,aft); 28 | 29 | function aft = get_aft_jacobians(harm) 30 | aft.fft.J = aft_jacobian_fft(harm); 31 | [aft.ifft.J,aft.ifft.Jdot,aft.ifft.Jddot] = aft_jacobian_ifft(harm); 32 | 33 | %total jacobian 34 | aft.J = mtimesx(aft.fft.J,aft.ifft.J); 35 | for i = 1:2 36 | aft.Jdot{i} = mtimesx(aft.fft.J,harm.rFreqBase(i)*aft.ifft.Jdot{i}); 37 | aft.Jddot{i} = mtimesx(aft.fft.J,harm.rFreqBase(i)^2*aft.ifft.Jddot{i}); 38 | end 39 | aft.Jddot{3} = mtimesx(aft.fft.J,prod(harm.rFreqBase)*aft.ifft.Jddot{3}); 40 | 41 | function hbm = get_hbm_matrices(problem,harm,aft) 42 | 43 | %hbm 44 | hbm.Jfft = resize_jacobian(repmat(aft.fft.J,1,size(aft.fft.J,1)),problem.NDof,problem.NDof); 45 | 46 | hbm.Jifft = resize_jacobian(repmat(aft.ifft.J,size(aft.ifft.J,2),1),problem.NDof,problem.NDof); 47 | 48 | for i = 1:2 49 | hbm.Jdotifft{i} = resize_jacobian(repmat(harm.rFreqBase(i)*aft.ifft.Jdot{i},size(aft.ifft.J,2),1),problem.NDof,problem.NDof); 50 | end 51 | 52 | %% States 53 | hbm.Jx = resize_jacobian(aft.J,problem.NDof,problem.NDof); 54 | hbm.Jxdot = resize_jacobian(aft.Jdot,problem.NDof,problem.NDof); 55 | hbm.Jxddot = resize_jacobian(aft.Jddot,problem.NDof,problem.NDof); 56 | 57 | %% Inputs 58 | hbm.Ju = resize_jacobian(aft.J,problem.NDof,problem.NInput); 59 | hbm.Judot = resize_jacobian(aft.Jdot,problem.NDof,problem.NInput); 60 | hbm.Juddot = resize_jacobian(aft.Jddot,problem.NDof,problem.NInput); 61 | 62 | 63 | %indices for creating the jacobian 64 | hbm.ijacobx = repmat((1:problem.NDof)',harm.NComp,1); 65 | hbm.ijacobu = repmat((1:problem.NInput)',harm.NComp,1); 66 | 67 | function Ju = resize_jacobian(ju,NOutput,NInput) 68 | if ~iscell(ju) 69 | ju = {ju}; 70 | end 71 | for k = 1:length(ju) 72 | Ju{k} = zeros(NOutput*size(ju{k},1),NInput*size(ju{k},2),size(ju{k},3)); 73 | for i = 1:size(ju{k},3) 74 | Ju{k}(:,:,i) = kron(ju{k}(:,:,i),ones(NOutput,NInput)); 75 | end 76 | end 77 | if length(Ju) == 1 78 | Ju = Ju{1}; 79 | end -------------------------------------------------------------------------------- /Setup/setupHarm.m: -------------------------------------------------------------------------------- 1 | function harm = setupHarm(harm) 2 | 3 | harm = setupHarmonics(harm); 4 | 5 | %default missing fields 6 | f = {'Nfft','rFreqRatio','rFreqBase','wFreq0'}; 7 | d = {max(8*harm.NHarm,1),0*harm.NHarm+1,0*harm.NHarm+1,0*harm.NHarm}; 8 | for i = 1:length(f) 9 | if ~isfield(harm,f{i}) 10 | harm.(f{i}) = d{i}; 11 | warning('Field %s is missing from the options. Defaulting to %s',f{i},mat2str(d{i})) 12 | end 13 | end 14 | 15 | %check all the sizes match 16 | if length(harm.rFreqRatio) ~= length(harm.NHarm) 17 | error('The number of frequencies should match the number of elements in the NHarm vector') 18 | end 19 | 20 | if length(harm.rFreqRatio) ~= length(harm.Nfft) 21 | error('The number of frequencies should match the number of elements in the Nfft vector') 22 | end 23 | 24 | %now add dummy second harmonic for single harmonic problems 25 | if length(harm.rFreqRatio) == 1 26 | harm.rFreqRatio(2) = 1; 27 | harm.NHarm(2) = 0; 28 | harm.Nfft(2) = 1; 29 | harm.kHarm(:,2) = 0; 30 | end 31 | 32 | %deal with any sub-harmonics, changing base frequencies accordingly 33 | harm = detect_subharmonics(harm); 34 | 35 | harm.kHarm = reorder_harmonics(unique(harm.kHarm,'rows','stable')); 36 | 37 | %find the subscripts for using FFT/IFFT 38 | harm.NFreq = size(harm.kHarm,1); 39 | harm.NComp = (2*(harm.NFreq-1)+1); 40 | 41 | function group = setupHarmonics(group) 42 | if isfield(group,'kHarm') 43 | group.NHarm = max(abs(group.kHarm),[],1); 44 | elseif isfield(group,'NHarm') 45 | if length(group.NHarm) == 1 46 | group.kHarm = (0:group.NHarm)'; 47 | elseif group.NHarm(1) == 0 48 | k2 = 0:group.NHarm(2); 49 | k1 = 0*k2; 50 | group.kHarm = [k1(:) k2(:)]; 51 | elseif group.NHarm(2) == 0 52 | k1 = 0:group.NHarm(1); 53 | k2 = 0*k1; 54 | group.kHarm = [k1(:) k2(:)]; 55 | else 56 | k1 = -group.NHarm(1):group.NHarm(1); 57 | k2 = -group.NHarm(2):group.NHarm(2); 58 | [K1,K2] = ndgrid(k1,k2); 59 | keep = (K1+K2)>0 | ((K1+K2)== 0 & K1>=0); 60 | group.kHarm = [K1(keep) K2(keep)]; 61 | end 62 | else 63 | error('Need either kHarm or NHarm'); 64 | end 65 | 66 | function harm = detect_subharmonics(harm) 67 | ratio = harm.NHarm*0 + 1; 68 | 69 | [~,fac] = rat(harm.kHarm); 70 | 71 | for k = 1:2 72 | for j = 1:size(harm.kHarm,1) 73 | ratio(k) = lcm(fac(j,k),ratio(k)); 74 | end 75 | end 76 | 77 | harm.kHarm = harm.kHarm .* repmat(ratio,size(harm.kHarm,1),1); 78 | 79 | harm.rFreqBase = harm.rFreqBase./ratio; 80 | harm.NHarm = harm.NHarm.*ratio; 81 | harm.Nfft = harm.Nfft.*ratio; 82 | 83 | function kHarm = reorder_harmonics(kHarm) 84 | %find 0th harmonic and put it first 85 | 86 | zero = kHarm(:,1) == 0 & kHarm(:,2) == 0; 87 | first1 = kHarm(:,1) == 1 & kHarm(:,2) == 0; 88 | first2 = kHarm(:,1) == 0 & kHarm(:,2) == 1; 89 | rest = ~zero & ~first1 & ~first2; 90 | ii = [find(zero); find(first1); find(first2); find(rest)]; 91 | kHarm = kHarm(ii,:); -------------------------------------------------------------------------------- /Standard/hbm_balance.m: -------------------------------------------------------------------------------- 1 | function varargout = hbm_balance(command,hbm,problem,w,u,x) 2 | NDofTot = hbm.harm.NComp*problem.NDof; 3 | 4 | ii = find(hbm.harm.NHarm ~= 0); 5 | r = hbm.harm.rFreqRatio(ii); 6 | w0 = w * r + hbm.harm.wFreq0(ii); 7 | 8 | A = (hbm.lin.Ak + w0*hbm.lin.Ac{ii} + w0^2*hbm.lin.Am{ii}); 9 | B = (hbm.lin.Bk + w0*hbm.lin.Bc{ii} + w0^2*hbm.lin.Bm{ii}); 10 | dAdw = (hbm.lin.Ac{ii} + 2*w0*hbm.lin.Am{ii}); 11 | dBdw = (hbm.lin.Bc{ii} + 2*w0*hbm.lin.Bm{ii}); 12 | 13 | switch command 14 | case 'func' %F, used by hbm_frf & hbm_bb 15 | cl = B*u - hbm.lin.b - A*x; 16 | if hbm.bIncludeNL 17 | cnl = hbm_nonlinear('func',hbm,problem,w0,x,u); 18 | else 19 | cnl = 0*cl; 20 | end 21 | c = (cl - cnl); 22 | varargout{1} = c; 23 | case 'jacob' %dF_dX, used by hbm_frf & hbm_bb 24 | Jl = -A; 25 | if hbm.bIncludeNL 26 | [Jx,Jxdot,Jxddot] = hbm_nonlinear({'jacobX','jacobXdot','jacobXddot'},hbm,problem,w0,x,u); 27 | Jnl = Jx + w0*Jxdot + w0^2*Jxddot; 28 | else 29 | Jnl = 0*Jl; 30 | end 31 | J = (Jl - Jnl); 32 | varargout{1} = J; 33 | case 'derivW' %dF_dw, used by hbm_frf & hbm_bb 34 | Dl = dBdw*u - dAdw*x; 35 | if hbm.bIncludeNL 36 | if hbm.dependence.xdot || hbm.dependence.w 37 | [Jxdot,Jxddot,Judot,Juddot,Dw] = hbm_nonlinear({'jacobXdot','jacobXddot','jacobUdot','jacobUddot','derivW'},hbm,problem,w0,x,u); 38 | Dxdot = r*Jxdot*x; 39 | Dxddot = 2*r*w0*Jxddot*x; 40 | Dudot = r*Judot*u; 41 | Duddot = 2*r*w0*Juddot*u; 42 | Dnl = Dxdot + Dxddot + Dudot + Duddot + Dw; 43 | else 44 | Dnl = 0*Dl; 45 | end 46 | else 47 | Dnl = 0*Dl; 48 | end 49 | D = (Dl - Dnl); 50 | varargout{1} = D; 51 | case 'derivA' %dF_dA, used in hbm_bb 52 | Dl = B*u; 53 | if hbm.bIncludeNL 54 | [Ju,Judot,Juddot] = hbm_nonlinear({'jacobU','jacobUdot','jacobUddot'},hbm,problem,w0,x,u); 55 | Dnl = (Ju + w0*Judot + w0^2*Juddot)*u; 56 | else 57 | Dnl = 0*Dl; 58 | end 59 | D = (Dl - Dnl); 60 | varargout{1} = D; 61 | case 'floquet0' 62 | D0 = -hbm_balance('jacob',hbm,problem,w0,u,x); 63 | varargout{1} = D0; 64 | case 'floquet1' 65 | D1l = hbm.lin.floquet.D1xdot + 2*hbm.lin.floquet.D1xddot{ii}*w0; 66 | if hbm.bIncludeNL 67 | [D1xdot,D1xddot] = hbm_nonlinear({'floquet1xdot','floquet1xddot'},hbm,problem,w0,x,u); 68 | D1nl = D1xdot + 2*D1xddot*w0; 69 | else 70 | D1nl = 0*D1l; 71 | end 72 | varargout{1} = (D1l - D1nl); 73 | case 'floquet2' 74 | D2l = hbm.lin.floquet.D2; 75 | if hbm.bIncludeNL 76 | D2nl = hbm_nonlinear({'floquet2'},hbm,problem,w0,x,u); 77 | else 78 | D2nl = 0*D2l; 79 | end 80 | varargout{1} = (D2l - D2nl); 81 | end -------------------------------------------------------------------------------- /Generalised/get_time_series3d.m: -------------------------------------------------------------------------------- 1 | function varargout = get_time_series3d(hbm,w,X,tspan) 2 | if hbm.options.bUseStandardHBM 3 | [varargout{1:nargout}] = get_time_series(hbm,problem,w,u,X); 4 | return; 5 | end 6 | 7 | NHarm = hbm.harm.NHarm; 8 | Nfft = hbm.harm.Nfft; 9 | kHarm = hbm.harm.kHarm; 10 | w0 = w * hbm.harm.rFreqRatio + hbm.harm.wFreq0; 11 | 12 | %unpack the inputs 13 | w = kHarm(:,1)*w0(1) + kHarm(:,2)*w0(2); 14 | t1 = (0:Nfft(1)-1)/Nfft(1)*2*pi/w0(1); 15 | t2 = (0:Nfft(2)-1)/Nfft(2)*2*pi/w0(2); 16 | [t1,t2] = ndgrid(t1,t2); 17 | %compute the fourier coefficients of the derivatives 18 | Wx = repmat(1i*w,1,size(X,2)); 19 | 20 | %create the time series from the fourier series 21 | x = real(hbm.nonlin.IFFT*X); 22 | if nargout > 2 23 | Xdot = X.*Wx; 24 | xdot = real(hbm.nonlin.IFFT*Xdot); 25 | if nargout > 3 26 | Xddot = Xdot.*Wx; 27 | xddot = real(hbm.nonlin.IFFT*Xddot); 28 | end 29 | end 30 | 31 | x = reshape(x,size(t1,1),size(t1,2),[]); 32 | if nargout > 2 33 | xdot = reshape(xdot,size(t1,1),size(t1,2),[]); 34 | if nargout > 3 35 | xddot = reshape(xddot,size(t1,1),size(t1,2),[]); 36 | end 37 | end 38 | 39 | if nargin < 5 || isempty(tspan) %empty 40 | ti = linspace(0,2*pi/min(w0),max(Nfft)); 41 | elseif length(tspan) < 2 %duration 42 | ti = linspace(0,tspan,max(Nfft)); 43 | else 44 | ti = tspan; 45 | end 46 | 47 | if ti(end) > 2*pi/w0(1)% && hbm.harm.NHarm(1)>0 48 | N = ceil(ti(end)/t1(end,1)); 49 | t0 = t1; 50 | for i = 2:N 51 | t1 = [t1; t0+2*pi*(i-1)/w0(1)]; 52 | end 53 | x = repmat(x,N,1); 54 | t2 = repmat(t2,N,1); 55 | if nargout > 2 56 | xdot = repmat(xdot,N,1); 57 | if nargout > 3 58 | xddot = repmat(xddot,N,1); 59 | end 60 | end 61 | end 62 | if ti(end) > 2*pi/w0(2)% && hbm.harm.NHarm(2)>0 63 | N = ceil(ti(end)/t2(1,end)); 64 | t0 = t2; 65 | for i = 2:N 66 | t2 = [t2 t0+2*pi*(i-1)/w0(2)]; 67 | end 68 | x = repmat(x,1,N); 69 | t1 = repmat(t1,1,N); 70 | if nargout > 2 71 | xdot = repmat(xdot,1,N); 72 | if nargout > 3 73 | xddot = repmat(xddot,1,N); 74 | end 75 | end 76 | end 77 | 78 | if Nfft(2) > 1 79 | for i = 1:size(x,3) 80 | xi(:,i) = interp2(t2,t1,x(:,:,i),ti,ti); 81 | if nargout > 2 82 | xdoti(:,i) = interp2(t2,t1,xdot(:,:,i),ti,ti); 83 | if nargout > 3 84 | xddoti(:,i) = interp2(t2,t1,xddot(:,:,i),ti,ti); 85 | end 86 | end 87 | end 88 | else 89 | for i = 1:size(x,3) 90 | xi(:,i) = interp1(t1,x(:,:,i),ti); 91 | if nargout > 2 92 | xdoti(:,i) = interp1(t1,xdot(:,:,i),ti); 93 | if nargout > 3 94 | xddoti(:,i) = interp1(t1,xddot(:,:,i),ti); 95 | end 96 | end 97 | end 98 | end 99 | 100 | varargout{1} = ti; 101 | varargout{2} = xi; 102 | 103 | if nargout >2 104 | varargout{3} = xdoti; 105 | if nargout > 3 106 | varargout{4} = xddoti; 107 | end 108 | end 109 | 110 | -------------------------------------------------------------------------------- /Utilities/hbm_derivatives.m: -------------------------------------------------------------------------------- 1 | function varargout = hbm_derivatives(fun,var,States,hbm,problem) 2 | NPts = size(States.t,2); 3 | 4 | if ~iscell(var) 5 | var = {var}; 6 | end 7 | 8 | for i = 1:length(var) 9 | switch var{i}(1) 10 | case 'w' 11 | %for frequency derivatives, it's a bit more complicated as we have to rescale the time vector as well 12 | df_dw = feval(problem.model,[fun '_w'],States,hbm,problem); 13 | if isempty(df_dw) 14 | h = 1E-6; 15 | NFreq = length(States.w0); 16 | 17 | if hbm.dependence.w 18 | for k = 1:NFreq 19 | States2 = States; 20 | States2.w0(k) = States.w0(k) + h; 21 | States2.t(k,:) = States.w0(k)*States2.t(k,:)/States2.w0(k); 22 | States2.f = feval(problem.model,fun,States2,hbm,problem); 23 | df_dw{k} = (States2.f-States.f)./h; 24 | end 25 | else 26 | df_dw = repmat({zeros(size(f,1),NPts)},1,NFreq); 27 | end 28 | end 29 | for k = 1:length(States.w0) 30 | df_dw{k} = df_dw{k}.'; 31 | end 32 | varargout{i} = df_dw; 33 | case 'x' 34 | %for everything else, we just peturb each element in turn 35 | df_dx = feval(problem.model,[fun '_' var{i}],States,hbm,problem); 36 | if isempty(df_dx) 37 | if hbm.dependence.(var{i}) 38 | h = 1E-10; 39 | df_dx = zeros(size(States.f,1),problem.NDof,NPts); 40 | for j = 1:problem.NDof 41 | States2 = States; 42 | States2.(var{i})(j,:) = States.(var{i})(j,:) + h; 43 | States2.f = feval(problem.model,fun,States2,hbm,problem); 44 | df_dx(:,j,:) = permute((States2.f-States.f)./h,[1 3 2]); 45 | end 46 | else 47 | df_dx = zeros(size(States.f,1),problem.NDof,NPts); 48 | end 49 | end 50 | varargout{i} = df_dx; 51 | case 'u' 52 | %for everything else, we just peturb each element in turn 53 | df_du = feval(problem.model,[fun '_' var{i}],States,hbm,problem); 54 | if isempty(df_du) 55 | if hbm.dependence.(var{i}) 56 | h = 1E-10; 57 | df_du = zeros(size(States.f,1),problem.NInput,NPts); 58 | for j = 1:problem.NInput 59 | States2 = States; 60 | States2.(var{i})(j,:) = States.(var{i})(j,:) + h; 61 | States2.f = feval(problem.model,fun,States2,hbm,problem); 62 | df_du(:,j,:) = permute((States2.f-States.f)./h,[1 3 2]); 63 | end 64 | else 65 | df_du = zeros(size(States.f,1),problem.NInput,NPts); 66 | end 67 | end 68 | varargout{i} = df_du; 69 | otherwise 70 | error('Unrecognized input') 71 | end 72 | end -------------------------------------------------------------------------------- /Functions/hbm_solve.m: -------------------------------------------------------------------------------- 1 | function sol = hbm_solve(hbm,problem,w,A,X0) 2 | problem.type = 'solve'; 3 | 4 | if nargin < 5 || isempty(X0) 5 | X0 = zeros(hbm.harm.NFreq,problem.NDof); 6 | end 7 | if size(X0,1) == hbm.harm.NFreq && size(X0,2) == problem.NDof 8 | x0 = packdof(X0); 9 | elseif isvector(X0) && length(X0) == hbm.harm.NComp*problem.NDof 10 | x0 = X0; 11 | X0 = unpackdof(x0,hbm.harm.NHarm,problem.NDof); 12 | else 13 | error('Wrong size for X0'); 14 | end 15 | 16 | %setup the problem for IPOPT 17 | w0 = w*hbm.harm.rFreqRatio + hbm.harm.wFreq0; 18 | W = hbm.harm.kHarm*(hbm.harm.rFreqBase.*w0)'; 19 | 20 | %we have an initial guess vector 21 | U = A*feval(problem.excite,hbm,problem,w0); 22 | u = packdof(U); 23 | 24 | hbm.bIncludeNL = true; 25 | 26 | init.X = X0; 27 | init.w = w; 28 | init.A = A; 29 | 30 | if isfield(problem,'xscale') 31 | xscale = [problem.xscale'; repmat(problem.xscale',hbm.harm.NFreq-1,1)*(1+1i)]; 32 | problem.Zscale = packdof(xscale); 33 | problem.Fscale = problem.Zscale*0+1; 34 | problem.Jscale = (1./problem.Fscale(:))*problem.Zscale(:)'; 35 | else 36 | problem = hbm_scaling(problem,hbm,init); 37 | end 38 | 39 | %and actually solve it 40 | hbm.max_iter = 3; 41 | bSuccess = false; 42 | 43 | constr_tol = 1E-6; 44 | maxit = 20; 45 | 46 | z0 = x0; 47 | Z0 = z0./problem.Zscale; 48 | 49 | attempts = 0; 50 | while ~bSuccess && attempts < hbm.max_iter 51 | switch hbm.options.solver 52 | case 'fsolve' 53 | fun_constr = @(x)hbm_constraints(x,hbm,problem,w,u); 54 | options = optimoptions('fsolve','Display','iter','SpecifyObjectiveGradient',true,'FunctionTolerance',constr_tol,'MaxIterations',maxit); 55 | [Z,~,EXITFLAG,OUTPUT] = fsolve(fun_constr,Z0,options); 56 | bSuccess = EXITFLAG == 1; 57 | iter = OUTPUT.iterations + 1; 58 | case 'ipopt' 59 | options.jacobian = @hbm_jacobian; 60 | options.jacobianstructure = hbm.sparsity; 61 | options.print_level = 5; 62 | options.max_iter = maxit; 63 | options.constr_viol_tol = constr_tol; 64 | [Z, info] = fipopt('',Z0,@hbm_constraints,options,hbm,problem,w,u); 65 | bSuccess = any(info.status == [0 1]); 66 | iter = info.iter; 67 | end 68 | Z0 = Z + 1E-8*rand(length(Z),1); 69 | attempts = attempts + 1; 70 | end 71 | 72 | if ~bSuccess 73 | Z = Z + NaN; 74 | end 75 | z = Z.*problem.Zscale; 76 | 77 | X = unpackdof(z,hbm.harm.NFreq-1,problem.NDof); 78 | 79 | sol.w = w; 80 | sol.W = W; 81 | sol.A = A; 82 | sol.X = X; 83 | sol.U = U; 84 | sol.F = hbm_output3d(hbm,problem,w,sol.U,sol.X); 85 | 86 | % floquet multipliers 87 | sol = hbm_floquet(hbm,problem,sol); 88 | 89 | %excitation forces 90 | sol = hbm_excitation_forces(problem,sol); 91 | 92 | sol.it = iter; 93 | 94 | function [c,J] = hbm_constraints(Z,hbm,problem,w,u) 95 | %unpack the inputs 96 | x = Z.*problem.Zscale; 97 | c = hbm_balance3d('func',hbm,problem,w,u,x); 98 | c = c ./ problem.Fscale; 99 | if nargout > 1 100 | J = hbm_jacobian(Z,hbm,problem,w,u); 101 | end 102 | 103 | function J = hbm_jacobian(Z,hbm,problem,w,u) 104 | x = Z.*problem.Zscale; 105 | J = hbm_balance3d('jacob',hbm,problem,w,u,x); 106 | J = J .* problem.Jscale; -------------------------------------------------------------------------------- /Test/hbm_frf_test.m: -------------------------------------------------------------------------------- 1 | function hbm_frf_test(b3d) 2 | problem = test_params; 3 | 4 | hbm.harm.rFreqRatio = 1; 5 | hbm.harm.NHarm = 2; 6 | hbm.harm.Nfft = 32; 7 | hbm.harm.iHarmPlot = 2; 8 | 9 | if nargin < 1 10 | b3d = 1; 11 | end 12 | 13 | if b3d 14 | hbm.harm.rFreqRatio(end+1) = 1.376; 15 | hbm.harm.NHarm(end+1) = 2; 16 | hbm.harm.Nfft(end+1) = 16; 17 | hbm.harm.iHarmPlot(end+1) = 3; 18 | end 19 | 20 | hbm.dependence.x = true; 21 | hbm.dependence.xdot = true; 22 | hbm.dependence.w = false; 23 | 24 | hbm.scaling.tol = 1; 25 | 26 | hbm.cont.step0 = 1E-2; 27 | hbm.cont.max_step = 1E-1; 28 | hbm.cont.min_step = 1E-6; 29 | 30 | hbm.cont.method = 'predcorr'; 31 | 32 | [hbm,problem] = setuphbm(hbm,problem); 33 | 34 | omega = sqrt(eig(problem.K,problem.M)); 35 | w0 = max(min(omega)-2,0.5); 36 | wEnd = max(omega)+2; 37 | A = 1; 38 | 39 | S = {}; 40 | 41 | tic; 42 | hbm.cont.method = 'none'; 43 | sol = hbm_frf(hbm,problem,A,w0,[],wEnd,[]); 44 | S{end+1} = storeResults(sol,toc,'none'); 45 | 46 | tic; 47 | hbm.cont.method = 'predcorr'; 48 | hbm.cont.predcorr.corrector = 'arclength'; 49 | sol = hbm_frf(hbm,problem,A,w0,[],wEnd,[]); 50 | S{end+1} = storeResults(sol,toc,'arclength'); 51 | 52 | tic; 53 | hbm.cont.method = 'predcorr'; 54 | hbm.cont.predcorr.corrector = 'pseudo'; 55 | sol = hbm_frf(hbm,problem,A,w0,[],wEnd,[]); 56 | S{end+1} = storeResults(sol,toc,'pseudo'); 57 | 58 | tic; 59 | S{end+1} = ode_frf(w0,wEnd,A,hbm,problem); 60 | 61 | figure 62 | hold on 63 | for j = 1:problem.NDof 64 | ax_mag(j) = subplot(2,problem.NDof,j); 65 | hold on 66 | for i = 1:length(S) 67 | hmag(i,j) = plot(ax_mag(j),S{i}.w,abs(S{i}.X(j,:))); 68 | end 69 | ylabel(ax_mag(j),sprintf('|X_%d| (mag)',j)); 70 | end 71 | 72 | for j = 1:problem.NDof 73 | ax_ph(j) = subplot(2,problem.NDof,problem.NDof+j); 74 | hold on 75 | for i = 1:length(S) 76 | hph(i,j) = plot(ax_ph(j),S{i}.w,unwrap(angle(S{i}.X(j,:)),[],2)); 77 | end 78 | xlabel(ax_ph(j),'\omega (rads)'); 79 | ylabel(ax_ph(j),sprintf('\\angle X_%d (deg)',j)); 80 | end 81 | linkaxes([ax_mag ax_ph],'x') 82 | 83 | for i = 1:length(S) 84 | leg{i} = S{i}.name; 85 | end 86 | legend(ax_ph(end),hph(:,end),leg) 87 | 88 | for i = 1:length(S) 89 | fprintf('%10s : %0.2f s\n',S{i}.name,S{i}.t) 90 | end 91 | 92 | function S = ode_frf(w0,wEnd,A,hbm,problem) 93 | y = zeros(1,2*problem.NDof); 94 | ws = linspace(w0,wEnd,20); 95 | NCycle = 50; 96 | Nfft = 101; 97 | for i = 1:length(ws) 98 | y0 = y(end,:); 99 | w0 = hbm.harm.rFreqRatio * ws(i); 100 | U = A*test_excite(hbm,problem,w0); 101 | T = 2*pi/ws(i); 102 | [t,y] = ode45(@(t,y)test_odefun(t,y,w0,U,hbm,problem),[0 NCycle*T],y0); 103 | 104 | ii = find(t>(t(end)-T),1); 105 | t = t(ii:end) - t(ii); 106 | y = y(ii:end,:); 107 | 108 | tReg = linspace(0,T,Nfft)'; 109 | Fs = Nfft/T; 110 | yReg = interp1(t,y,tReg,'linear','extrap'); 111 | 112 | Yfft = 2*fft(yReg,[],1)/Nfft; 113 | Wfft = (0:(Nfft-1))'/Nfft * 2*pi* Fs; 114 | 115 | X(:,i) = Yfft(2,1:problem.NDof); 116 | w(i) = Wfft(2); 117 | end 118 | 119 | S.X = X; 120 | S.w = w; 121 | S.t = toc; 122 | S.name = 'ode'; 123 | 124 | function S = storeResults(sol,t,name) 125 | X = cat(3,sol.X); 126 | 127 | S.X = permute(X(2,:,:),[2 3 1]); 128 | S.w = cat(2,sol.w); 129 | S.t = t; 130 | S.name = name; -------------------------------------------------------------------------------- /Generalised/hbm_balance3d.m: -------------------------------------------------------------------------------- 1 | function varargout = hbm_balance3d(command,hbm,problem,w,u,x) 2 | if hbm.options.bUseStandardHBM 3 | [varargout{1:nargout}] = hbm_balance(command,hbm,problem,w,u,x); 4 | return; 5 | end 6 | NDofTot = hbm.harm.NComp*problem.NDof; 7 | 8 | r = hbm.harm.rFreqRatio; 9 | w0 = w .* r + hbm.harm.wFreq0; 10 | 11 | A = hbm.lin.Ak + prod(w0)*hbm.lin.Ax; 12 | B = hbm.lin.Bk + prod(w0)*hbm.lin.Bx; 13 | dAdw = (r(1)*w0(2) + r(2)*w0(1))*hbm.lin.Ax; 14 | dBdw = (r(1)*w0(2) + r(2)*w0(1))*hbm.lin.Bx; 15 | for k = 1:2 16 | A = A + (w0(k)*hbm.lin.Ac{k} + w0(k)^2*hbm.lin.Am{k}); 17 | B = B + (w0(k)*hbm.lin.Bc{k} + w0(k)^2*hbm.lin.Bm{k}); 18 | 19 | dAdw = dAdw + r(k)*(hbm.lin.Ac{k} + 2*w0(k)*hbm.lin.Am{k}); 20 | dBdw = dBdw + r(k)*(hbm.lin.Bc{k} + 2*w0(k)*hbm.lin.Bm{k}); 21 | end 22 | 23 | switch command 24 | case 'func' %F, used by hbm_frf & hbm_bb 25 | cl = B*u - hbm.lin.b - A*x; 26 | if hbm.bIncludeNL 27 | cnl = hbm_nonlinear3d('func',hbm,problem,w0,x,u); 28 | else 29 | cnl = 0*cl; 30 | end 31 | c = (cl - cnl); 32 | varargout{1} = c; 33 | case 'jacob' %dF_dX, used by hbm_frf & hbm_bb 34 | Jl = -A; 35 | if hbm.bIncludeNL 36 | [Jx,Jxdot,Jxddot] = hbm_nonlinear3d({'jacobX','jacobXdot','jacobXddot'},hbm,problem,w0,x,u); 37 | Jnl = Jx + w0(1)*Jxdot{1} + w0(2)*Jxdot{2} + w0(1)^2*Jxddot{1} + w0(2)^2*Jxddot{2} + prod(w0)*Jxddot{3}; 38 | else 39 | Jnl = 0*Jl; 40 | end 41 | J = (Jl - Jnl); 42 | varargout{1} = J; 43 | case 'derivW' %dF_dw, used by hbm_frf & hbm_bb 44 | Dl = dBdw*u - dAdw*x; 45 | cl = B*u - hbm.lin.b - A*x; 46 | if hbm.bIncludeNL 47 | if hbm.dependence.xdot || hbm.dependence.w 48 | [Jxdot,Jxddot,Judot,Juddot,Dw] = hbm_nonlinear3d({'jacobXdot','jacobXddot','jacobUdot','jacobUddot','derivW'},hbm,problem,w0,x,u); 49 | Dxdot = (r(1)*Jxdot{1} + r(2)*Jxdot{2})*x; 50 | Dxddot = (2*r(1)*w0(1)*Jxddot{1} + 2*r(2)*w0(2)*Jxddot{2} + (r(1)*w0(2) + r(2)*w0(1))*Jxddot{3})*x; 51 | Dudot = (r(1)*Judot{1} + r(2)*Judot{2})*u; 52 | Duddot = (2*r(1)*w0(1)*Juddot{1} + 2*r(2)*w0(2)*Juddot{2} + (r(1)*w0(2) + r(2)*w0(1))*Juddot{3})*u; 53 | Dw = r(1)*Dw{1} + r(2)*Dw{2}; 54 | Dnl = Dxdot + Dxddot + Dudot + Duddot + Dw; 55 | else 56 | Dnl = 0*Dl; 57 | end 58 | else 59 | Dnl = 0*Dl; 60 | end 61 | D = (Dl - Dnl); 62 | varargout{1} = D; 63 | case 'derivA' %dF_dA, used in hbm_bb 64 | Dl = B*u; 65 | if hbm.bIncludeNL 66 | [Ju,Judot,Juddot] = hbm_nonlinear3d({'jacobU','jacobUdot','jacobUddot'},hbm,problem,w0,x,u); 67 | Dnl = (Ju + w0(1)*Judot{1} + w0(2)*Judot{2} +w0(1)^2*Juddot{1} + w0(2)^2*Juddot{2} + prod(w0)*Juddot{3})*u; 68 | else 69 | Dnl = 0*Dl; 70 | end 71 | D = (Dl - Dnl); 72 | varargout{1} = D; 73 | case 'floquet0' 74 | D0 = -hbm_balance3d('jacob',hbm,problem,w,u,x); 75 | varargout{1} = D0; 76 | case 'floquet1' 77 | D1l = hbm.lin.floquet.D1xdot + hbm.lin.floquet.D1Gxdot*w0(1) + 2*(w0(1)*hbm.lin.floquet.D1xddot{1} + w0(2)*hbm.lin.floquet.D1xddot{2}); 78 | if hbm.bIncludeNL 79 | [D1xdot,D1xddot] = hbm_nonlinear3d({'floquet1xdot','floquet1xddot'},hbm,problem,w0,x,u); 80 | D1nl = D1xdot + 2*w0(1)*D1xddot{1} + 2*w0(2)*D1xddot{2}; 81 | else 82 | D1nl = 0*D1l; 83 | end 84 | varargout{1} = (D1l - D1nl); 85 | case 'floquet2' 86 | D2l = hbm.lin.floquet.D2; 87 | if hbm.bIncludeNL 88 | D2nl = hbm_nonlinear3d({'floquet2'},hbm,problem,w0,x,u); 89 | else 90 | D2nl = 0*D2l; 91 | end 92 | varargout{1} = (D2l - D2nl); 93 | end -------------------------------------------------------------------------------- /Test/hbm_bb_test.m: -------------------------------------------------------------------------------- 1 | function hbm_bb_test(b3d) 2 | problem = test_params; 3 | 4 | P = problem.P; 5 | 6 | hbm.harm.rFreqRatio = 1; 7 | hbm.harm.NHarm = 2; 8 | hbm.harm.Nfft = 32; 9 | hbm.harm.iHarmPlot = 2; 10 | 11 | hbm.dependence.x = true; 12 | hbm.dependence.xdot = true; 13 | hbm.dependence.w = false; 14 | 15 | hbm.scaling.tol = 1; 16 | 17 | hbm.cont.step0 = 1E-2; 18 | hbm.cont.max_step = 1E-1; 19 | hbm.cont.min_step = 1E-6; 20 | 21 | if nargin < 1 22 | b3d = 1; 23 | end 24 | 25 | if b3d 26 | hbm.harm.rFreqRatio(end+1) = 1.376; 27 | hbm.harm.NHarm(end+1) = 2; 28 | hbm.harm.Nfft(end+1) = 16; 29 | hbm.harm.iHarmPlot(end+1) = 3; 30 | end 31 | 32 | hbm.cont.method = 'predcorr'; 33 | hbm.cont.predcorr.corrector = 'arclength'; 34 | 35 | problem.res.iOutput = P.iDof; 36 | problem.res.iInput = P.iInput; 37 | problem.res.iHarm = 2; 38 | problem.res.sign = 1; 39 | problem.res.output = 'x'; 40 | problem.res.input = 'u'; 41 | 42 | [hbm,problem] = setuphbm(hbm,problem); 43 | 44 | iRes = []; 45 | iRes(end+1) = find(hbm.harm.kHarm(:,1) == 1 & hbm.harm.kHarm(:,2) == 0); 46 | iRes(end+1) = find(hbm.harm.kHarm(:,1) == 0 & hbm.harm.kHarm(:,2) == 1); 47 | NRes = length(iRes); 48 | 49 | iMode = 1; 50 | omega = sqrt(eig(problem.K,problem.M)); 51 | w0 = max(omega(iMode)-2,0.5); 52 | wEnd = omega(iMode)+2; 53 | 54 | figure 55 | for j = 1:NRes 56 | ax(1,j) = subplot(2,NRes,j); 57 | hold on 58 | if j == 1 59 | ylabel('|X| (mag)'); 60 | end 61 | title(sprintf('iHarm %d',iRes(j))) 62 | 63 | ax(2,j) = subplot(2,NRes,NRes+j); 64 | hold on 65 | if j == 1 66 | ylabel('\angle X (deg)'); 67 | end 68 | 69 | xlabel('\omega (rads)'); 70 | end 71 | linkaxes(ax(:),'x') 72 | xlim([w0,wEnd]); 73 | 74 | As = [1 2 3 4]; 75 | 76 | root = fileparts(which(mfilename)); 77 | for i = 1:length(As) 78 | file = fullfile(root,sprintf('FRF_A = %0.1f.mat',As(i))); 79 | if ~isfile(file) 80 | sol = hbm_frf(hbm,problem,As(i),w0,[],wEnd,[]); 81 | results.X = cat(3,sol.X); 82 | results.U = cat(3,sol.U); 83 | results.w = cat(2,sol.w); 84 | save(file,'-struct','results'); 85 | else 86 | results = load(file); 87 | end 88 | 89 | x = results.X; 90 | u = results.U; 91 | w = results.w; 92 | 93 | for j = 1:NRes 94 | X = permute((x(iRes(j),:,:)),[2 3 1]); 95 | U = permute((u(iRes(j),:,:)),[2 3 1]); 96 | H = X(P.iDof,:)./U(P.iInput,:); 97 | plot(ax(1,j),w,abs(H)); 98 | plot(ax(2,j),w,unwrap(angle(H),[],2)); 99 | 100 | file = fullfile(root,sprintf('RES_A = %0.1f_%d.mat',As(i),j)); 101 | if ~isfile(file) 102 | problem.res.iHarm = iRes(j); 103 | [hbm,problem] = setuphbm(hbm,problem); 104 | [~,ii] = max(H); 105 | res = hbm_res(hbm,problem,w(ii),As(i),x(:,:,ii)); 106 | save(file,'-struct','res'); 107 | else 108 | res = load(file); 109 | end 110 | 111 | Xres{j}(:,:,i) = res.X; 112 | wres{j}(i) = res.w; 113 | plot(ax(1,j),res.w,abs(res.H),'o') 114 | plot(ax(2,j),res.w,angle(res.H),'o') 115 | end 116 | drawnow 117 | end 118 | 119 | bb = hbm; 120 | bb.cont.method = 'none'; 121 | 122 | for j = 1:NRes 123 | A0 = As(1); w0 = wres{j}(1); X0 = Xres{j}(:,:,1); 124 | Aend = As(end); wEnd = wres{j}(end); XEnd = Xres{j}(:,:,end); 125 | 126 | problem.res.iHarm = iRes(j); 127 | [hbm,problem] = setuphbm(hbm,problem); 128 | 129 | file = fullfile(root,sprintf('BB_%d.mat',j)); 130 | if ~isfile(file) 131 | sol = hbm_bb(bb,problem,A0,w0,X0,Aend,wEnd,XEnd); 132 | results.X = cat(3,sol.X); 133 | results.U = cat(3,sol.U); 134 | results.w = cat(2,sol.w); 135 | results.H = cat(2,sol.H); 136 | save(file,'-struct','results'); 137 | else 138 | results = load(file); 139 | end 140 | 141 | for k = 1:NRes 142 | X = squeeze(results.X(iRes(k),P.iDof,:)); 143 | U = squeeze(results.U(iRes(k),P.iInput,:)); 144 | H = X./U; 145 | plot(ax(1,k),results.w,abs(H)); 146 | plot(ax(2,k),results.w,unwrap(angle(H))); 147 | end 148 | end -------------------------------------------------------------------------------- /Functions/hbm_res.m: -------------------------------------------------------------------------------- 1 | function sol = hbm_res(hbm,problem,w,A,X0) 2 | problem.type = 'resonance'; 3 | 4 | if nargin < 5 || isempty(X0) 5 | X0 = zeros(hbm.harm.NFreq,problem.NDof); 6 | end 7 | if size(X0,1) == hbm.harm.NFreq && size(X0,2) == problem.NDof 8 | x0 = packdof(X0); 9 | elseif isvector(X0) && length(X0) == hbm.harm.NComp*problem.NDof 10 | x0 = X0; 11 | X0 = unpackdof(x0,hbm.harm.NHarm,problem.NDof); 12 | else 13 | error('Wrong size for X0'); 14 | end 15 | 16 | %setup the problem for IPOPT 17 | hbm.bIncludeNL = true; 18 | 19 | %first solve @ w0 20 | sol = hbm_solve(hbm,problem,w,A,X0); 21 | X0 = sol.X; 22 | z0 = packdof(X0); 23 | 24 | init.X = X0; 25 | init.w = w; 26 | init.A = A; 27 | 28 | if isfield(problem,'xscale') 29 | xscale = [problem.xscale'; repmat(problem.xscale',hbm.harm.NFreq-1,1)*(1+1i)]; 30 | problem.Xscale = packdof(xscale)*sqrt(length(xscale)); 31 | problem.wscale = w; 32 | problem.Fscale = problem.Xscale*0+1; 33 | problem.Zscale = [problem.Xscale; problem.wscale]; 34 | problem.Jscale = (1./problem.Fscale(:))*problem.Zscale(:)'; 35 | else 36 | problem = hbm_scaling(problem,hbm,init); 37 | end 38 | 39 | %and actually solve it 40 | hbm.max_iter = 4; 41 | bSuccess = false; 42 | 43 | constr_tol = 1E-6; 44 | opt_tol = 1E-6; 45 | maxit = 20; 46 | 47 | Z0 = [z0; w]./problem.Zscale; 48 | 49 | zlb = [z0-Inf;problem.res.wMin./problem.wscale]; 50 | zub = [z0+Inf;problem.res.wMax./problem.wscale]; 51 | 52 | attempts = 0; 53 | while ~bSuccess && attempts < hbm.max_iter 54 | switch hbm.options.solver 55 | case 'fsolve' 56 | fun_obj = @(x)hbm_fsolve_obj(x,hbm,problem,A); 57 | fun_constr = @(x)hbm_fsolve_constr(x,hbm,problem,A); 58 | options = optimoptions('fmincon','SpecifyObjectiveGradient',true,'SpecifyConstraintGradient',true,'Display','iter',... 59 | 'OptimalityTolerance',opt_tol,'ConstraintTolerance',constr_tol,'MaxIterations',maxit); 60 | [Z,~,EXITFLAG,OUTPUT] = fmincon(fun_obj,Z0,[],[],[],[],zlb,zub,fun_constr,options); 61 | bSuccess = EXITFLAG == 1; 62 | iter = OUTPUT.iterations + 1; 63 | case 'ipopt' 64 | options.jacobianstructure = [hbm.sparsity ones(problem.NDof*hbm.harm.NComp,1)]; 65 | options.jacobian = @hbm_jacobian; 66 | options.gradient = @hbm_grad; 67 | options.print_level = 5; 68 | options.max_iter = maxit; 69 | options.tol = opt_tol; 70 | options.constr_viol_tol = constr_tol; 71 | options.lb = zlb; 72 | options.ub = zub; 73 | [Z, info] = fipopt(@hbm_obj,Z0,@hbm_constr,options,hbm,problem,A); 74 | bSuccess = any(info.status == [0 1]); 75 | iter = info.iter; 76 | end 77 | Z0 = Z+rand(length(Z),1)*1E-8; 78 | attempts = attempts + 1; 79 | end 80 | if ~bSuccess 81 | Z = Z + NaN; 82 | end 83 | z = Z.*problem.Zscale; 84 | 85 | w = z(end); 86 | w0 = w*hbm.harm.rFreqRatio + hbm.harm.wFreq0; 87 | W = hbm.harm.kHarm*(hbm.harm.rFreqBase.*w0)'; 88 | 89 | U = A*feval(problem.excite,hbm,problem,w0); 90 | u = packdof(U); 91 | 92 | x = z(1:end-1); 93 | X = unpackdof(x,hbm.harm.NFreq-1,problem.NDof); 94 | 95 | sol.w = w; 96 | sol.W = W; 97 | sol.A = A; 98 | sol.X = X; 99 | sol.U = U; 100 | sol.F = hbm_output3d(hbm,problem,w,sol.U,sol.X); 101 | 102 | %floquet multipliers & objective 103 | sol.H = hbm_objective('complex',hbm,problem,w,z(1:end-1),u); 104 | sol = hbm_floquet(hbm,problem,sol); 105 | sol = hbm_excitation_forces(problem,sol); 106 | 107 | sol.it = iter; 108 | 109 | function obj = hbm_obj(Z,hbm,problem,A) 110 | x = Z(1:end-1).*problem.Xscale; 111 | w = Z(end).*problem.wscale; 112 | w0 = w * hbm.harm.rFreqRatio + hbm.harm.wFreq0; 113 | 114 | U = A*feval(problem.excite,hbm,problem,w0); 115 | u = packdof(U); 116 | 117 | H = hbm_objective('func',hbm,problem,w,x,u); 118 | obj = - problem.res.sign * H; 119 | 120 | function G = hbm_grad(Z,hbm,problem,A) 121 | x = Z(1:end-1).*problem.Xscale; 122 | w = Z(end).*problem.wscale; 123 | w0 = w * hbm.harm.rFreqRatio + hbm.harm.wFreq0; 124 | 125 | U = A*feval(problem.excite,hbm,problem,w0); 126 | u = packdof(U); 127 | 128 | [Dx, Dw] = hbm_objective({'jacobX','derivW'},hbm,problem,w,x,u); 129 | G = -problem.res.sign*[Dx Dw]; 130 | 131 | G = G.*problem.Zscale(:)'; 132 | 133 | function c = hbm_constr(Z,hbm,problem,A) 134 | x = Z(1:end-1).*problem.Xscale; 135 | w = Z(end).*problem.wscale; 136 | w0 = w * hbm.harm.rFreqRatio + hbm.harm.wFreq0; 137 | 138 | U = A*feval(problem.excite,hbm,problem,w0); 139 | u = packdof(U); 140 | 141 | c = hbm_balance3d('func',hbm,problem,w,u,x); 142 | 143 | function J = hbm_jacobian(Z,hbm,problem,A) 144 | x = Z(1:end-1).*problem.Xscale; 145 | w = Z(end).*problem.wscale; 146 | w0 = w * hbm.harm.rFreqRatio + hbm.harm.wFreq0; 147 | 148 | U = A*feval(problem.excite,hbm,problem,w0); 149 | u = packdof(U); 150 | 151 | Jx_nl = hbm_balance3d('jacob',hbm,problem,w,u,x); 152 | Dw_nl = hbm_balance3d('derivW',hbm,problem,w,u,x); 153 | 154 | J = [Jx_nl Dw_nl]; 155 | J = J .* problem.Jscale; 156 | 157 | function [obj,G] = hbm_fsolve_obj(Z,hbm,problem,A) 158 | obj = hbm_obj(Z,hbm,problem,A); 159 | if nargout > 1 160 | G = hbm_grad(Z,hbm,problem,A)'; 161 | end 162 | 163 | function [c,ceq,Jc,Jceq] = hbm_fsolve_constr(Z,hbm,problem,A) 164 | ceq = hbm_constr(Z,hbm,problem,A); 165 | c = []; 166 | % c = ceq(end); 167 | % ceq = ceq(1:end-1); 168 | if nargout > 2 169 | Jceq = hbm_jacobian(Z,hbm,problem,A); 170 | Jc = zeros(0,length(Z)); 171 | 172 | Jc = Jc'; 173 | Jceq = Jceq'; 174 | % Jc = Jceq(end,:)'; 175 | % Jceq = Jceq(1:end-1,:)'; 176 | end -------------------------------------------------------------------------------- /Utilities/hbm_objective.m: -------------------------------------------------------------------------------- 1 | function varargout = hbm_objective(part,hbm,problem,w,x,u) 2 | 3 | NOutput = problem.res.NOutput; 4 | NInput = problem.res.NInput; 5 | 6 | r = hbm.harm.rFreqRatio; 7 | w0 = w .* r + hbm.harm.wFreq0; 8 | 9 | if problem.res.iHarm > 1 10 | iRe = (1:NOutput)' + 2*(problem.res.iHarm - 2)*NOutput + NOutput; 11 | iIm = (1:NOutput)' + 2*(problem.res.iHarm - 2)*NOutput + 2*NOutput; 12 | 13 | jRe = (1:NInput)' + 2*(problem.res.iHarm - 2)*NInput + NInput; 14 | jIm = (1:NInput)' + 2*(problem.res.iHarm - 2)*NInput + 2*NInput; 15 | else 16 | iRe = (1:NOutput)'; 17 | iIm = []; 18 | 19 | jRe = (1:NInput)'; 20 | jIm = []; 21 | end 22 | 23 | kk = hbm.harm.kHarm*(hbm.harm.rFreqBase.*hbm.harm.rFreqRatio)'; 24 | D = [0 -1; 25 | 1 0]; 26 | D = blkdiag(0,kron(diag(kk(2:end)),D)); 27 | 28 | %% output 29 | Wx = kron(D,eye(problem.NDof)); 30 | switch problem.res.output 31 | case 'none' 32 | Fnl = 0*x; 33 | case 'fnl' 34 | Fnl = hbm_nonlinear3d({'func'},hbm,problem,w0,x,u); 35 | case 'x' 36 | Fnl = x; 37 | case 'xdot' 38 | Fnl = w*Wx*x; 39 | case 'xddot' 40 | Fnl = w^2*Wx*Wx*x; 41 | end 42 | 43 | Fb = Fnl(iRe); 44 | if ~isempty(iIm) 45 | Fb = Fb + 1i*Fnl(iIm); 46 | end 47 | Fb = problem.res.ROutput*Fb; 48 | 49 | %% input 50 | Wu = kron(D,eye(problem.NInput)); 51 | switch problem.res.input 52 | case 'unity' 53 | Fex = 0*u+1; 54 | case 'fe' 55 | Ju = prod(w0)*hbm.lin.Bx + hbm.lin.Bk; 56 | for k = 1:2 57 | Ju = Ju + (w0(k)*hbm.lin.Bc{k} + w0(k)^2*hbm.lin.Bm{k}); 58 | end 59 | Fex = Ju*u; 60 | case 'u' 61 | Fex = u; 62 | case 'udot' 63 | Fex = w * Wu*u; 64 | case 'uddot' 65 | Fex = w^2 * Wu*Wu*u; 66 | end 67 | 68 | Fe = Fex(jRe); 69 | if ~isempty(jIm) 70 | Fe = Fe + 1i*Fex(jIm); 71 | end 72 | Fe = problem.res.RInput*Fe; 73 | 74 | if ~iscell(part) 75 | part = {part}; 76 | end 77 | varargout = cell(1,length(part)); 78 | 79 | for i = 1:length(part) 80 | switch part{i} 81 | case 'complex' 82 | H = Fb./Fe; 83 | varargout{i} = H; 84 | case 'func' 85 | H = abs(Fb./Fe); 86 | varargout{i} = H; 87 | case 'jacobX' 88 | 89 | %nl 90 | switch problem.res.output 91 | case 'none' 92 | Jx = 0*Wx; 93 | case 'fnl' 94 | Jx = hbm_nonlinear3d({'jacobX'},hbm,problem,w0,x,u); 95 | case 'x' 96 | Jx = eye(problem.NDof*hbm.harm.NComp); 97 | case 'xdot' 98 | Jx = Wx; 99 | case 'xddot' 100 | Jx = Wx*Wx; 101 | end 102 | 103 | if ~isempty(iIm) 104 | dFbdx = (real(Fb)*Jx(iRe,:) + imag(Fb)*Jx(iIm,:))./(abs(Fb) + eps); 105 | else 106 | dFbdx = (real(Fb)*Jx(iRe,:))./(abs(Fb) + eps); 107 | end 108 | dFbdx = problem.res.ROutput*dFbdx; 109 | 110 | %excitation 111 | dFedx = 0; 112 | 113 | %put it all together 114 | dHdx = (dFbdx.*abs(Fe) - abs(Fb).*dFedx)./abs(Fe).^2; 115 | 116 | varargout{i} = dHdx; 117 | case 'derivW' 118 | 119 | %nl 120 | switch problem.res.output 121 | case 'none' 122 | Dw_nl = 0*x; 123 | case 'fnl' 124 | Dw = hbm_nonlinear3d({'derivW'},hbm,problem,w0,x,u); 125 | if ~iscell(Dw) 126 | Dw = {Dw,0*Dw}; 127 | end 128 | Dw_nl = 0*Dw{1}; 129 | for k = 1:2 130 | Dw_nl = Dw_nl + r(k)*Dw{k}; 131 | end 132 | case 'x' 133 | Dw_nl = zeros(problem.NDof*hbm.harm.NComp,1); 134 | case 'xdot' 135 | Dw_nl = (Wx*x); 136 | case 'xddot' 137 | Dw_nl = 2*w*(Wx*Wx*x); 138 | end 139 | if ~isempty(iIm) 140 | dFbdw = (real(Fb)*Dw_nl(iRe) + imag(Fb)*Dw_nl(iIm))./(abs(Fb) + eps); 141 | else 142 | dFbdw = (real(Fb)*Dw_nl(iRe))./(abs(Fb) + eps); 143 | end 144 | dFbdw = problem.res.ROutput*dFbdw; 145 | 146 | %excitation 147 | switch problem.res.input 148 | case 'unity' 149 | Dw_u = 0*u; 150 | case 'fe' 151 | Dw_u = (r(1)*w0(2) + r(2)*w0(1))*hbm.lin.Bx*u; 152 | for k = 1:2 153 | Dw_u = Dw_u + r(k)*(hbm.lin.Bc{k} + 2*w0(k)*hbm.lin.Bm{k})*u; 154 | end 155 | case 'u' 156 | Dw_u = zeros(problem.NInput*hbm.harm.NComp,1); 157 | case 'udot' 158 | Dw_u = (Wu*u); 159 | case 'uddot' 160 | Dw_u = 2*w*(Wu*Wu*u); 161 | end 162 | if ~isempty(iIm) 163 | dFedw = (real(Fe)*Dw_u(jRe) + imag(Fe)*Dw_u(jIm))./abs(Fe); 164 | else 165 | dFedw = (real(Fe)*Dw_u(jRe))./abs(Fe); 166 | end 167 | dFedw = problem.res.RInput*dFedw; 168 | 169 | dHdw = (dFbdw.*abs(Fe) - abs(Fb).*dFedw)./abs(Fe).^2; 170 | 171 | varargout{i} = dHdw; 172 | end 173 | end -------------------------------------------------------------------------------- /Standard/hbm_nonlinear.m: -------------------------------------------------------------------------------- 1 | function varargout = hbm_nonlinear(command,hbm,problem,w0,Xp,Up) 2 | NInput = problem.NInput; 3 | NDof = problem.NDof; 4 | 5 | ii = find(hbm.harm.NHarm ~= 0); 6 | 7 | NFreq = hbm.harm.NFreq; 8 | NComp = hbm.harm.NComp; 9 | NDofTot = NComp*NDof; 10 | NInputTot = NComp*NInput; 11 | 12 | %work out the time domain 13 | X = unpackdof(Xp,NFreq-1,NDof); 14 | U = unpackdof(Up,NFreq-1,NInput); 15 | 16 | %get the time series 17 | States = hbm_states(w0,X,U,hbm); 18 | 19 | %push through the nl system 20 | States.f = feval(problem.model,'nl' ,States,hbm,problem); 21 | 22 | %finally convert into the time domain 23 | F = hbm.nonlin.FFT*States.f.'; 24 | Fp = packdof(F); 25 | 26 | if ~iscell(command) 27 | command = {command}; 28 | end 29 | 30 | varargout = cell(1,length(command)); 31 | 32 | ijacobx = hbm.nonlin.hbm.ijacobx; 33 | ijacobu = hbm.nonlin.hbm.ijacobu; 34 | 35 | for o = 1:length(command) 36 | switch command{o} 37 | case 'func' 38 | %all done 39 | varargout{o} = Fp; 40 | case 'derivW' %df_dw 41 | if ~hbm.dependence.w 42 | Dw = zeros(NDofTot,1); 43 | else 44 | dfhbm_dw = hbm_derivatives('nl' ,'w',States,hbm,problem); 45 | if isfield(problem,'derivW') 46 | Dw = feval(problem.derivW,dfhbm_dw,hbm,problem); 47 | else 48 | Dw = packdof(hbm.nonlin.FFT*dfhbm_dw{ii}); 49 | end 50 | end 51 | varargout{o} = Dw; 52 | case 'jacobX' %df_dX = Jx*X 53 | if ~hbm.dependence.x 54 | Jx = zeros(NDofTot,NDofTot); 55 | else 56 | States.df_dx = hbm_derivatives('nl','x',States,hbm,problem); 57 | 58 | if isfield(problem,'jacobX') 59 | Jx = feval(problem.jacobX,States,hbm,problem); 60 | else 61 | Jx = sum(hbm.nonlin.hbm.Jx.*States.df_dx(ijacobx,ijacobx,:),3); 62 | end 63 | end 64 | varargout{o} = Jx; 65 | case 'jacobU' %df_dU = Ju*U 66 | if ~hbm.dependence.u 67 | Ju = zeros(NDofTot,NInputTot); 68 | else 69 | States.df_du = hbm_derivatives('nl' ,'u',States,hbm,problem); 70 | 71 | if isfield(problem,'jacobU') 72 | Ju = feval(problem.jacobU,States,problem); 73 | else 74 | Ju = sum(hbm.nonlin.hbm.Ju.*States.df_du(ijacobx,ijacobu,:),3); 75 | end 76 | end 77 | varargout{o} = Ju; 78 | case 'jacobXdot' %df_dX = w*Jxdot*X 79 | if ~hbm.dependence.xdot 80 | Jxdot = zeros(DofTot,NDofTot); 81 | else 82 | States.df_dxdot = hbm_derivatives('nl' ,'xdot',States,hbm,problem); 83 | 84 | if isfield(problem,'jacobXdot') 85 | Jxdot = feval(problem.jacobXdot,States,hbm,problem); 86 | else 87 | Jxdot = sum(hbm.nonlin.hbm.Jxdot{ii}.*States.df_dxdot(ijacobx,ijacobx,:),3); 88 | end 89 | end 90 | varargout{o} = Jxdot; 91 | case 'jacobUdot' %df_dU = w*Judot*U 92 | if ~hbm.dependence.udot 93 | Judot = zeros(NDofTot,NInputTot); 94 | else 95 | States.df_dudot = hbm_derivatives('nl' ,'udot',States,hbm,problem); 96 | 97 | if isfield(problem,'jacobUdot') 98 | Judot = feval(problem.jacobUdot,States,hbm,problem); 99 | else 100 | Judot = sum(hbm.nonlin.hbm.Judot{ii}.*States.df_dudot(ijacobx,ijacobu,:),3); 101 | end 102 | end 103 | varargout{o} = Judot; 104 | case 'jacobXddot' %df_dX = w^2*Jxddot*X 105 | if ~hbm.dependence.xddot 106 | Jxddot = zeros(NDofTot,NDofTot); 107 | else 108 | States.df_dxddot = hbm_derivatives('nl' ,'xddot',States,hbm,problem); 109 | 110 | if isfield(problem,'jacobXddot') 111 | Jxddot = feval(problem.jacobXddot,States,hbm,problem); 112 | else 113 | Jxddot = sum(hbm.nonlin.hbm.Jxddot{ii}.*States.df_dxddot(ijacobx,ijacobx,:),3); 114 | end 115 | end 116 | varargout{o} = Jxddot; 117 | case 'jacobUddot' %df_dU = w^2*Juddot*U 118 | if ~hbm.dependence.uddot 119 | Juddot = zeros(NDofTot,NInputTot); 120 | else 121 | States.df_duddot = hbm_derivatives('nl' ,'uddot',States,hbm,problem); 122 | 123 | if isfield(problem,'jacobUddot') 124 | Juddot = feval(problem.jacobUddot,States,hbm,problem); 125 | else 126 | Juddot = sum(hbm.nonlin.hbm.Juddot{ii}.*States.df_duddot(ijacobx,ijacobu,:),3); 127 | end 128 | end 129 | varargout{o} = Juddot; 130 | case 'floquet1xdot' 131 | if ~hbm.dependence.xdot 132 | D1 = zeros(NDofTot,NDofTot); 133 | else 134 | States.df_dxdot = hbm_derivatives('nl','xdot',States,hbm,problem); 135 | 136 | if isfield(problem,'floquet1xdot') 137 | D1 = feval(problem.floquet1xdot,States,hbm,problem); 138 | else 139 | D1 = sum(hbm.nonlin.hbm.Jx.*States.df_dxdot(ijacobx,ijacobx,:),3); 140 | end 141 | end 142 | varargout{o} = D1; 143 | case 'floquet1xddot' 144 | if ~hbm.dependence.xddot 145 | D1dd = zeros(NDofTot,NDofTot); 146 | else 147 | States.df_dxddot = hbm_derivatives('nl' ,'xddot',States,hbm,problem); 148 | 149 | if isfield(problem,'floquet1xddot') 150 | D1dd = feval(problem.floquet1xddot,States,hbm,problem); 151 | else 152 | D1dd = sum(hbm.nonlin.hbm.Jxdot{ii}.*States.df_dxddot(ijacobx,ijacobx,:),3); 153 | end 154 | end 155 | varargout{o} = D1dd; 156 | case 'floquet2' 157 | if ~hbm.dependence.xddot 158 | D2 = zeros(NDofTot,NDofTot); 159 | else 160 | States.df_dxddot = hbm_derivatives('nl' ,'xddot',States,hbm,problem); 161 | 162 | if isfield(problem,'floquet2') 163 | D2 = feval(problem.floquet2,States,hbm,problem); 164 | else 165 | D2 = sum(hbm.nonlin.hbm.Jx.*States.df_dxddot(ijacobx,ijacobx,:),3); 166 | end 167 | end 168 | varargout{o} = D2; 169 | end 170 | end -------------------------------------------------------------------------------- /Setup/setuphbm.m: -------------------------------------------------------------------------------- 1 | function [hbm,problem] = setuphbm(hbm,problem) 2 | 3 | %% Harmonics 4 | if ~isfield(hbm,'harm') 5 | hbm.harm = struct(); 6 | end 7 | hbm.harm = setupHarm(hbm.harm); 8 | 9 | %% Options 10 | if ~isfield(hbm,'options') 11 | hbm.options = struct(); 12 | end 13 | hbm.options = setupOptions(hbm.options); 14 | if hbm.options.bUseStandardHBM && prod(hbm.harm.NHarm) > 0 15 | error('Cannot use standard HBM code in case of more than one fundemental') 16 | end 17 | 18 | %% Dependence 19 | if ~isfield(hbm,'dependence') 20 | hbm.dependence = struct(); 21 | end 22 | hbm.dependence = setupDependence(hbm.dependence); 23 | 24 | %% Scaling 25 | if ~isfield(hbm,'scaling') 26 | hbm.scaling = struct(); 27 | end 28 | hbm.scaling = setupScaling(hbm.scaling); 29 | 30 | %% Continuation 31 | if ~isfield(hbm,'cont') 32 | hbm.cont = struct(); 33 | end 34 | hbm.cont = setupCont(hbm.cont); 35 | 36 | %% Problem definition 37 | problem = setupProblem(problem,hbm); 38 | 39 | if ~isfield(problem,'sparsity') 40 | hbm.sparsity = ones(hbm.harm.NComp*problem.NDof); 41 | else 42 | hbm.sparsity = repmat(problem.sparsity(1:problem.NDof,1:problem.NDof),hbm.harm.NComp); 43 | end 44 | 45 | %% Precompute matrices 46 | hbm.lin = setupLin(hbm.harm,problem); 47 | hbm.nonlin = setupNonlin(hbm.harm,problem); 48 | 49 | hbm.harm = default_missing(hbm.harm,{'iHarmPlot'},{1:hbm.harm.NFreq}); 50 | 51 | if ~isfield(problem,'RDofPlot') 52 | if ~isfield(problem,'iDofPlot') 53 | problem.iDofPlot = 1:problem.NDof; 54 | end 55 | 56 | R = zeros(length(problem.iDofPlot),problem.NDof); 57 | for j = 1:length(problem.iDofPlot) 58 | R(j,problem.iDofPlot(j)) = 1; 59 | end 60 | 61 | problem.RDofPlot = R; 62 | elseif size(problem.RDofPlot,2) ~= problem.NDof 63 | error('Wrong size for RDofPlot') 64 | end 65 | 66 | function options = setupOptions(options) 67 | if ~isfield(options,'bUseStandardHBM'), options.bUseStandardHBM = 0; end 68 | if ~isfield(options,'bVerbose'), options.bVerbose = 1; end 69 | if ~isfield(options,'bPlot'), options.bPlot = 1; end 70 | 71 | if ~isfield(options,'solver'), options.solver = 'ipopt'; end 72 | 73 | function dependence = setupDependence(dependence) 74 | dependence = default_missing(dependence,{'x','xdot','xddot','w','u','udot','uddot'},{true,false,false,false,false,false,false}); 75 | 76 | function scaling = setupScaling(scaling) 77 | scaling = default_missing(scaling,{'method','tol'},{'max',1E-6}); 78 | 79 | function cont = setupCont(cont) 80 | cont = default_missing(cont,{'method','bUpdate','step0','min_step','max_step','ftol','xtol','c', 'C','maxfail','num_iter_increase','num_iter_reduce'},{'predcorr',true,1E-3, 1E-6, 5E-3, 1E-6,1E-6,0.5, 1.05,4,10,3}); 81 | 82 | if ~isfield(cont,'predcorr'), cont.predcorr = struct(); end 83 | cont.predcorr = default_missing(cont.predcorr,{'predictor','corrector','bMoorePenrose','solver','maxit'},{'linear','pseudo',true,'ipopt',30}); 84 | 85 | function problem = setupProblem(problem,hbm) 86 | if ~isfield(problem,'name') 87 | problem.name = ''; 88 | end 89 | problem.NDof = size(problem.K,2); 90 | problem.NInput = size(problem.Ku,2); 91 | 92 | f = {'K','M','C','G'}; 93 | for i = 1:length(f) 94 | if ~isfield(problem,f{i}) 95 | problem.(f{i}) = zeros(problem.NDof); 96 | elseif size(problem.(f{i}),1) ~= problem.NDof || size(problem.(f{i}),2) ~= problem.NDof 97 | error('Wrong size for %s matrix',f{i}) 98 | end 99 | end 100 | 101 | if ~isfield(problem,'F0') 102 | problem.('F0') = zeros(problem.NDof,1); 103 | elseif length(problem.('F0')) ~= problem.NDof 104 | error('Wrong size for %s matrix','F0') 105 | end 106 | 107 | 108 | f = {'Ku','Cu','Mu'}; 109 | for i = 1:length(f) 110 | if ~isfield(problem,f{i}) 111 | problem.(f{i}) = zeros(problem.NDof,problem.NInput); 112 | elseif size(problem.(f{i}),1) ~= problem.NDof || size(problem.(f{i}),2) ~= problem.NInput 113 | error('Wrong size for %s matrix',f{i}) 114 | end 115 | end 116 | 117 | try 118 | States = empty_states(problem); 119 | out = feval(problem.model,'output',States,hbm,problem); 120 | problem.NOutput = length(out); 121 | catch 122 | error('Error detected in non-linear function') 123 | end 124 | 125 | if ~isfield(problem,'update') 126 | problem.update = []; 127 | end 128 | 129 | if isfield(problem,'res') 130 | f = {'input','output','iHarm'}; 131 | 132 | if ~isfield(problem.res,f{i}) 133 | error('Missing field %s from resonance condition',f{i}) 134 | end 135 | 136 | switch problem.res.input 137 | case 'unity' 138 | problem.res.NInput = 1; 139 | case 'fe' 140 | problem.res.NInput = problem.NDof; 141 | case {'u','udot','uddot'} 142 | problem.res.NInput = problem.NInput; 143 | otherwise 144 | error('invalid input') 145 | end 146 | 147 | if ~isfield(problem.res,'RInput') 148 | if ~isfield(problem.res,'iInput') 149 | problem.res.iInput = 1:problem.res.NInput; 150 | end 151 | 152 | R = zeros(length(problem.res.iInput),problem.res.NInput); 153 | for j = 1:length(problem.res.iInput) 154 | R(j,problem.res.iInput(j)) = 1; 155 | end 156 | 157 | problem.res.RInput = R; 158 | elseif size(problem.res.RInput,2) ~= problem.res.NInput 159 | error('Wrong size for RInput') 160 | end 161 | 162 | switch problem.res.output 163 | case 'none' 164 | problem.res.NOutput = 1; 165 | case {'x','xdot','xddot','fnl'} 166 | problem.res.NOutput = problem.NDof; 167 | otherwise 168 | error('invalid output') 169 | end 170 | 171 | if ~isfield(problem.res,'ROutput') 172 | if ~isfield(problem.res,'iOutput') 173 | problem.res.iOutput = 1:problem.res.NOutput; 174 | end 175 | 176 | R = zeros(length(problem.res.iOutput),problem.res.NOutput); 177 | for j = 1:length(problem.res.iOutput) 178 | R(j,problem.res.iOutput(j)) = 1; 179 | end 180 | 181 | problem.res.ROutput = R; 182 | elseif size(problem.res.ROutput,2) ~= problem.res.NOutput 183 | error('Wrong size for ROutput') 184 | end 185 | 186 | if ~isfield(problem.res,'wMin') 187 | problem.res.wMin = -Inf; 188 | end 189 | if ~isfield(problem.res,'wMax') 190 | problem.res.wMax = Inf; 191 | end 192 | end 193 | 194 | function States = empty_states(problem) 195 | States.w0 = NaN; 196 | States.wBase = NaN; 197 | States.t = 0; 198 | 199 | States.x = zeros(problem.NDof,1); 200 | States.xdot = zeros(problem.NDof,1); 201 | States.xddot = zeros(problem.NDof,1); 202 | 203 | States.u = zeros(problem.NInput,1); 204 | States.udot = zeros(problem.NInput,1); 205 | States.uddot = zeros(problem.NInput,1); 206 | 207 | function s = default_missing(s,f,d) 208 | for i = 1:length(f) 209 | if ~isfield(s,f{i}) 210 | s.(f{i}) = d{i}; 211 | end 212 | end -------------------------------------------------------------------------------- /Generalised/hbm_nonlinear3d.m: -------------------------------------------------------------------------------- 1 | function varargout = hbm_nonlinear3d(command,hbm,problem,w0,Xp,Up) 2 | if hbm.options.bUseStandardHBM 3 | [varargout{1:nargout}] = hbm_nonlinear(command,hbm,problem,w0(1),Xp,Up); 4 | return; 5 | end 6 | 7 | NInput = problem.NInput; 8 | NDof = problem.NDof; 9 | 10 | NFreq = hbm.harm.NFreq; 11 | NComp = hbm.harm.NComp; 12 | NDofTot = NComp*NDof; 13 | NInputTot = NComp*NInput; 14 | 15 | %work out the time domain 16 | X = unpackdof(Xp,NFreq-1,NDof); 17 | U = unpackdof(Up,NFreq-1,NInput); 18 | 19 | %get the time series 20 | States = hbm_states3d(w0,X,U,hbm); 21 | 22 | %push through the nl system 23 | States.f = feval(problem.model,'nl',States,hbm,problem); 24 | 25 | %finally convert into the frequency domain 26 | F = hbm.nonlin.FFT*States.f.'; 27 | Fp = packdof(F); 28 | 29 | if ~iscell(command) 30 | command = {command}; 31 | end 32 | 33 | varargout = cell(1,length(command)); 34 | 35 | ijacobx = hbm.nonlin.hbm.ijacobx; 36 | ijacobu = hbm.nonlin.hbm.ijacobu; 37 | 38 | for o = 1:length(command) 39 | switch command{o} 40 | case 'func' 41 | %all done 42 | varargout{o} = Fp; 43 | case 'derivW' %df_dw 44 | if ~hbm.dependence.w 45 | Dw = repmat({zeros(NDofTot,1)},1,2); 46 | else 47 | dfhbm_dw = hbm_derivatives('nl','w',States,hbm,problem); 48 | if isfield(problem,'derivW') 49 | Dw = feval(problem.derivW,dfhbm_dw,hbm,problem); 50 | else 51 | for n = 1:2 52 | Dw{n} = packdof(hbm.nonlin.FFT*dfhbm_dw{n}); 53 | end 54 | end 55 | end 56 | varargout{o} = Dw; 57 | case 'jacobX' %df_dX = Jx*X 58 | if ~hbm.dependence.x 59 | Jx = zeros(NDofTot,NDofTot); 60 | else 61 | States.df_dx = hbm_derivatives('nl','x',States,hbm,problem); 62 | 63 | if isfield(problem,'jacobX') 64 | Jx = feval(problem.jacobX,States,hbm,problem); 65 | else 66 | Jx = sum(hbm.nonlin.hbm.Jx.*States.df_dx(ijacobx,ijacobx,:),3); 67 | end 68 | end 69 | varargout{o} = Jx; 70 | case 'jacobU' %df_dU = Ju*U 71 | if ~hbm.dependence.u 72 | Ju = zeros(NDofTot,NInputTot); 73 | else 74 | States.df_du = hbm_derivatives('nl','u',States,hbm,problem); 75 | 76 | if isfield(problem,'jacobU') 77 | Ju = feval(problem.jacobU,States.df_du,hbm,problem); 78 | else 79 | Ju = sum(hbm.nonlin.hbm.Ju.*States.df_du(ijacobx,ijacobu,:),3); 80 | end 81 | end 82 | varargout{o} = Ju; 83 | case 'jacobXdot' %df_dX = w*Jxdot*X 84 | if ~hbm.dependence.xdot 85 | Jxdot = repmat({zeros(NDofTot,NDofTot)},1,2); 86 | else 87 | States.df_dxdot = hbm_derivatives('nl','xdot',States,hbm,problem); 88 | 89 | if isfield(problem,'jacobXdot') 90 | Jxdot = feval(problem.jacobXdot,States,hbm,problem); 91 | else 92 | for n = 1:2 93 | Jxdot{n} = sum(hbm.nonlin.hbm.Jxdot{n}.*States.df_dxdot(ijacobx,ijacobx,:),3); 94 | end 95 | end 96 | end 97 | varargout{o} = Jxdot; 98 | case 'jacobUdot' %df_dU = w*Judot*U 99 | if ~hbm.dependence.udot 100 | Judot = repmat({zeros(NDofTot,NInputTot)},1,2); 101 | else 102 | States.df_dudot = hbm_derivatives('nl','udot',States,hbm,problem); 103 | 104 | if isfield(problem,'jacobUdot') 105 | Judot = feval(problem.jacobUdot,States,hbm,problem); 106 | else 107 | for n = 1:2 108 | Judot{n} = sum(hbm.nonlin.hbm.Judot{n}.*States.df_dudot(ijacobx,ijacobu,:),3); 109 | end 110 | end 111 | end 112 | varargout{o} = Judot; 113 | case 'jacobXddot' %df_dX = w^2*Jxddot*X 114 | if ~hbm.dependence.xddot 115 | Jxddot = repmat({zeros(NDofTot,NDofTot)},1,3); 116 | else 117 | States.df_dxddot = hbm_derivatives('nl','xddot',States,hbm,problem); 118 | 119 | if isfield(problem,'jacobXdot') 120 | Jxddot = feval(problem.jacobXdot,States,hbm,problem); 121 | else 122 | for n = 1:3 123 | Jxddot{n} = sum(hbm.nonlin.hbm.Jxddot{n}.*States.df_dxddot(ijacobx,ijacobx,:),3); 124 | end 125 | end 126 | end 127 | varargout{o} = Jxddot; 128 | case 'jacobUddot' %df_dU = w^2*Juddot*U 129 | if ~hbm.dependence.uddot 130 | Juddot = repmat({zeros(NDofTot,NInputTot)},1,3); 131 | else 132 | States.df_duddot = hbm_derivatives('nl','uddot',States,hbm,problem); 133 | 134 | if isfield(problem,'jacobUddot') 135 | Juddot = feval(problem.jacobUddot,States,hbm,problem); 136 | else 137 | for n = 1:3 138 | Juddot{n} = sum(hbm.nonlin.hbm.Juddot{n}.*States.df_duddot(ijacobx,ijacobu,:),3); 139 | end 140 | end 141 | end 142 | varargout{o} = Juddot; 143 | case 'floquet1xdot' 144 | if ~hbm.dependence.xdot 145 | D1 = zeros(NDofTot,NDofTot); 146 | else 147 | States.df_dxdot = hbm_derivatives('nl','xdot',States,hbm,problem); 148 | 149 | if isfield(problem,'floquet1xdot') 150 | D1 = feval(problem.floquet1xdot,States,hbm,problem); 151 | else 152 | D1 = sum(hbm.nonlin.hbm.Jx.*States.df_dxdot(ijacobx,ijacobx,:),3); 153 | end 154 | end 155 | varargout{o} = D1; 156 | case 'floquet1xddot' 157 | if ~hbm.dependence.xddot 158 | D1dd = repmat({zeros(NDofTot,NDofTot)},1,2); 159 | else 160 | States.df_dxddot = hbm_derivatives('nl','xddot',States,hbm,problem); 161 | 162 | if isfield(problem,'floquet1xddot') 163 | D1dd = feval(problem.floquet1xddot,States,hbm,problem); 164 | else 165 | for n = 1:2 166 | D1dd{n} = sum(hbm.nonlin.hbm.Jxdot{n}.*States.df_dxddot(ijacobx,ijacobx,:),3); 167 | end 168 | end 169 | end 170 | varargout{o} = D1dd; 171 | case 'floquet2' 172 | if ~hbm.dependence.xddot 173 | D2 = zeros(NDofTot); 174 | else 175 | States.df_dxddot = hbm_derivatives('nl','xddot',States,hbm,problem); 176 | 177 | if isfield(problem,'floquet2') 178 | D2 = feval(problem.floquet2,States,hbm,problem); 179 | else 180 | D2 = sum(hbm.nonlin.hbm.Jx.*States.df_dxddot(ijacobx,ijacobx,:),3); 181 | end 182 | end 183 | varargout{o} = D2; 184 | end 185 | end -------------------------------------------------------------------------------- /Functions/hbm_amp_plot.m: -------------------------------------------------------------------------------- 1 | function hbm_amp_plot(command,hbm,problem,results) 2 | persistent fig hSuccess hWarn hErr X A 3 | w0 = problem.w0; 4 | 5 | if hbm.cont.bUpdate 6 | switch command 7 | case 'init' 8 | if ~isempty(fig) && ishandle(fig) 9 | close(fig) 10 | end 11 | 12 | if isempty(results) 13 | X = zeros(hbm.harm.NFreq,problem.NDof); 14 | A = NaN; 15 | else 16 | X = results.X; 17 | A = results.A; 18 | end 19 | [xlin, Alin] = getLinearReponse(hbm,problem,X,w0); 20 | 21 | xlin = mtimesx(xlin,problem.RDofPlot'); 22 | Xplot = mtimesx(X,problem.RDofPlot'); 23 | 24 | [fig,hSuccess,hWarn,hErr] = createFRF(hbm,problem,Xplot,A,xlin,Alin); 25 | 26 | case {'data','err','warn'} 27 | if ~ishandle(fig(1)) 28 | [xlin, Alin] = getLinearReponse(hbm,problem,X(:,:,1),w0); 29 | xlin = mtimesx(xlin,problem.RDofPlot'); 30 | Xplot = mtimesx(X,problem.RDofPlot'); 31 | [fig,hSuccess,hWarn,hErr] = createFRF(hbm,problem,Xplot,A,xlin,Alin); 32 | end 33 | 34 | X(:,:,end+1) = results.X; 35 | A(:,end+1) = results.A; 36 | 37 | Xplot = mtimesx(X(:,:,end-1:end),problem.RDofPlot'); 38 | Xabs = abs(Xplot(:,:,end)); 39 | Xph = unwrap(angle(Xplot)); Xph = Xph(:,:,end); 40 | Aplot = A(end); 41 | 42 | if any(strcmpi(command,{'data','warn'})) 43 | update_handles(hSuccess,Xabs,Xph,Aplot,hbm,problem) 44 | %update our progress 45 | 46 | if strcmpi(command,'warn') 47 | %warning, overlay in blue 48 | update_handles(hWarn,Xabs,Xph,Aplot,hbm,problem) 49 | end 50 | 51 | %reset the error points 52 | for i = 1:length(hbm.harm.iHarmPlot) 53 | for j = 1:size(problem.RDofPlot,1) 54 | for k = 1:2 55 | set(hWarn{k}(i,j),'xdata',NaN,'ydata',NaN); 56 | set(hErr{k}(i,j) ,'xdata',NaN,'ydata',NaN); 57 | end 58 | end 59 | end 60 | else 61 | %error 62 | X(:,:,end) = []; 63 | A(:,end) = []; 64 | update_handles(hErr,Xabs,Xph,Aplot,hbm,problem) 65 | end 66 | 67 | drawnow 68 | case 'close' 69 | close(fig) 70 | hSuccess = []; 71 | hWarn = []; 72 | hErr = []; 73 | end 74 | end 75 | 76 | function update_handles(han,Xabs,Xph,A,hbm,problem) 77 | for i = 1:length(hbm.harm.iHarmPlot) 78 | for j = 1:size(problem.RDofPlot,1) 79 | a = [get(han{1}(i,j),'xdata'),A]; 80 | mag = [get(han{1}(i,j),'ydata'),permute(Xabs(hbm.harm.iHarmPlot(i),j,:),[1 3 2])]; 81 | ph = [get(han{2}(i,j) ,'ydata'),permute(Xph(hbm.harm.iHarmPlot(i),j,:),[1 3 2])]; 82 | set(han{1}(i,j),'xdata',a,'ydata',mag); 83 | set(han{2}(i,j) ,'xdata',a,'ydata',ph); 84 | end 85 | end 86 | 87 | function [fMag,hSuccess,hWarn,hErr] = createFRF(hbm,problem,x,A,xlin,Alin) 88 | fMag = figure('Name',['Amp: ' problem.name]); 89 | 90 | for i = 1:length(hbm.harm.iHarmPlot) 91 | for j = 1:size(problem.RDofPlot,1) 92 | tmp = subplot(size(problem.RDofPlot,1),length(hbm.harm.iHarmPlot),(j-1)*length(hbm.harm.iHarmPlot) + i,'Parent',fMag); 93 | Xij = squeeze(xlin(hbm.harm.iHarmPlot(i),j,:)); 94 | [tmp2,hLin{1}(i,j),hLin{2}(i,j)] = plotyy(tmp,Alin,abs(Xij),Alin,unwrap(angle(Xij))); 95 | ax{1}(i,j) = tmp2(1); ax{2}(i,j) = tmp2(2); 96 | hold(tmp2(1), 'on'); 97 | hold(tmp2(2), 'on'); 98 | end 99 | end 100 | 101 | for i = 1:length(hbm.harm.iHarmPlot) 102 | for j = 1:size(problem.RDofPlot,1) 103 | hSuccess{1}(i,j) = plot(ax{1}(i,j),A,abs(squeeze(x(hbm.harm.iHarmPlot(i),j,:))),'g.-'); 104 | hSuccess{2}(i,j) = plot(ax{2}(i,j),A,unwrap(angle(squeeze(x(hbm.harm.iHarmPlot(i),j,:)))),'g.-'); 105 | 106 | for k = 1:2 107 | hWarn{k}(i,j) = plot(ax{k}(i,j),NaN,NaN,'b.'); 108 | hErr{k}(i,j) = plot(ax{k}(i,j),NaN,NaN,'r.'); 109 | 110 | %xlim(ax{k}(i,j),wlim(hbm.harm.iHarmPlot(i),:)); 111 | set(ax{k}(i,j),'XLimMode','auto') 112 | set(ax{k}(i,j),'YLimMode','auto') 113 | end 114 | 115 | if j==size(problem.RDofPlot,1) 116 | xlabel(ax{1}(i,j),'A (-)') 117 | end 118 | if j==1 119 | title(ax{1}(i,j),harmonicName(hbm,i)) 120 | end 121 | 122 | if i == 1 123 | ylabel(ax{1}(i,j),sprintf('|Dof #%d|',j)) 124 | end 125 | 126 | if i == length(hbm.harm.iHarmPlot) 127 | ylabel(ax{2}(i,j),sprintf('\\angle Dof #%d',j)) 128 | end 129 | end 130 | end 131 | 132 | function s = harmonicName(hbm,i) 133 | k1 = hbm.harm.kHarm(hbm.harm.iHarmPlot(i),1)*hbm.harm.rFreqBase(1); 134 | k2 = hbm.harm.kHarm(hbm.harm.iHarmPlot(i),2)*hbm.harm.rFreqBase(2); 135 | if k1 == 0 136 | s1 = ''; 137 | elseif k1 == 1 138 | s1 = '\omega_1'; 139 | else 140 | s1 = sprintf('%s\\omega_1',num2frac(k1)); 141 | end 142 | if k2 == 0 143 | s2 = ''; 144 | elseif k2 == 1 145 | if ~isempty(s1) 146 | s2 = '+\omega_2'; 147 | else 148 | s2 = '\omega_2'; 149 | end 150 | elseif k2 == -1 151 | s2 = '-\omega_2'; 152 | else 153 | if ~isempty(s1) 154 | s2 = sprintf('%+s\\omega_2',num2frac(k2)); 155 | else 156 | s2 = sprintf('%s\\omega_2',num2frac(k2)); 157 | end 158 | end 159 | 160 | if isempty(s2) 161 | s = s1; 162 | elseif isempty(s1) 163 | s = s2; 164 | else 165 | s = [s1 ' ' s2]; 166 | end 167 | 168 | if isempty(s) 169 | s = '0'; 170 | end 171 | 172 | function [xlin, Alin] = getLinearReponse(hbm,problem,X,w) 173 | %find the linearised contribution to the stiffness/damping due from the non-linearity 174 | w0 = w*hbm.harm.rFreqRatio + hbm.harm.wFreq0; 175 | wB = w0.*hbm.harm.rFreqBase; 176 | w = hbm.harm.kHarm*wB'; 177 | 178 | x0 = X(1,:).'; 179 | U = feval(problem.excite,hbm,problem,w0); 180 | u0 = U(1,:).'; 181 | 182 | States = hbm_states3d(w0,X,U,hbm); 183 | 184 | Alin = linspace(problem.A0,problem.AEnd,1000); 185 | xlin = zeros(hbm.harm.NFreq,problem.NDof,length(Alin)); 186 | 187 | %now loop over all the amplitudes 188 | 189 | for i = 1:length(Alin) 190 | States_i = scale_inputs(States,Alin(i)); 191 | States_i.f = feval(problem.model,'nl',States_i,hbm,problem); 192 | 193 | [K_nl, C_nl, M_nl] = hbm_derivatives('nl',{'x','xdot','xddot'},States_i,hbm,problem); 194 | [Ku_nl,Cu_nl,Mu_nl] = hbm_derivatives('nl',{'u','udot','uddot'},States_i,hbm,problem); 195 | 196 | K_nl = mean(K_nl,3); C_nl = mean(C_nl,3); M_nl = mean(M_nl,3); 197 | Ku_nl = mean(Ku_nl,3); Cu_nl = mean(Cu_nl,3); Mu_nl = mean(Mu_nl,3); 198 | 199 | M = problem.M + M_nl; 200 | G = problem.G; 201 | C = problem.C + C_nl; 202 | K = problem.K + K_nl; 203 | 204 | Mu = problem.Mu + Mu_nl; 205 | Cu = problem.Cu + Cu_nl; 206 | Ku = problem.Ku + Ku_nl; 207 | 208 | NFreq = hbm.harm.NFreq; 209 | 210 | for k = 1:NFreq 211 | Fe = (Ku + 1i*w(k)*Cu - w(k)^2*Mu)*Alin(i)*U(k,:).'; 212 | H = K + 1i*w(k)*(C + w0(1)*G) - w(k)^2 * M; 213 | xlin(k,:,i) = (H\Fe).'; 214 | end 215 | end 216 | 217 | xlin(1,:,:) = xlin(1,:,:) + x0.'; 218 | 219 | function States = scale_inputs(States,A) 220 | f = {'u','udot','uddot'}; 221 | for i = 1:length(f) 222 | States.(f{i}) = States.(f{i}) * A; 223 | end -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # HarmLAB 2 | This toolbox offers an implementation of the Generalised Harmonic Balance method in MATLAB, supporting up to 2 base frequencies. This library can find periodic and quasi-periodic solutions of non-linear ODEs expressed in the following general form: 3 | 4 | $$M_x \ddot{x} + C_x \dot{x} + K_x x + f_0 + f_{nl}\left({x},\dot{{x}},\ddot{{x}},{u},\dot{{u}},\ddot{{u}},\omega\right)={f}_e (t) = {M}_u\ddot{{u}} + {C}_u\dot{{u}}+{K}_u{u}$$ 5 | 6 | with an output given by: 7 | 8 | $$y = g(x, \dot{x}, \ddot{x}, u, \dot{u}, \ddot{u} )$$ 9 | 10 | This toolbox has been developed to solve problems in structural dynamics, so much of the notation and terminology used stems from this field. However, it could equally be applied to problems in a variety of other disciplines. 11 | 12 | The input is assumed to be of the following multi-harmonic form: 13 | 14 | $${u} = A \sum_{k} \Re\left({U}_{k} e^{k\omega t}\right)$$ 15 | 16 | where the variable $A$ controls the excitation level. 17 | 18 | The output is assumed to contain content at the same frequencies, so can be expressed as: 19 | 20 | $${x} = \sum_{k} \Re\left({X}_{k} e^{k\omega t}\right)$$ 21 | 22 | This toolbox supports both Standard HBM with a single base frequency $\omega$ or Generalised HBM with 2 base frequencies which are interpendent: 23 | 24 | $$ \begin{bmatrix}\omega_1 & \omega_2\end{bmatrix} = \begin{bmatrix}\lambda_1 & \lambda_2\end{bmatrix} \Omega $$ 25 | 26 | Analytical jacbobians have been implemented throughout to speed up the execution. 27 | 28 | # Problem definition 29 | ## `problem` structure 30 | The `problem` structure defines the system being simulated. This must have the following fields for the linear parts of the system: 31 | - The mass, stiffness and damping matrices `K`, `C`, `M` 32 | - The constant term `F0` (which defaults to `zeros`) 33 | - The matrices for the excitation `Ku`, `Cu`, `Mu` 34 | 35 | The number of DOF and inputs is inferred from the dimensions of the matrices. 36 | 37 | The non-linear parts must be specified via the `model` callback which must have the form: 38 | 39 | ``` MATLAB 40 | function varargout = test_model(part,States,hbm,problem) 41 | switch part 42 | case 'nl' 43 | %f_nl 44 | case 'output' 45 | %y 46 | ... 47 | end 48 | end 49 | ``` 50 | 51 | where `part` can be one of: 52 | - `nl` which means this function should return $f_{nl}$ 53 | - `nl_x`, `nl_xdot`, `nl_xddot` which means this function should return $\frac{\partial f_{nl}}{\partial x}$ etc 54 | - `nl_u`, `nl_udot`, `nl_uddot` which means this function should return $\frac{\partial f_{nl}}{\partial u}$ etc 55 | - `output` which means this function should return the output $y$ 56 | 57 | The excitation must be specified via the `excite` callback which must has the form: 58 | 59 | ``` MATLAB 60 | function U = test_excite(hbm, problem, w0) 61 | ... 62 | end 63 | ``` 64 | and returns the $U_k$. 65 | 66 | You can also store other useful information in the `problem` structure which is needed by your model (eg model parameters). 67 | 68 | ### `problem.res` structure 69 | For resonance problems (using `hbm_res` or `hbm_bb`), it is necessary to configure the term to be maximised. This can be of the following form: 70 | 71 | $$ \frac{\|X_k|}{|F_{e,k}|} $$ 72 | 73 | The `res` structure should have the following fields to set the numerator and denominator: 74 | - `iHarm` which sets harmonic to use the terms from 75 | - `output` which can be either `x` (or its derivatives),`fnl`, or `none` (to have no output). 76 | - `input` which can be either `u` (or its derivatives), `fe` or `unity` (to have no denominator). 77 | - `NInput` which selects which DOF from `x` to use 78 | - `NOutput` which selects which index from `fe` to use 79 | 80 | ## HBM structure 81 | The other key structure needed to solve problems is the `hbm` structure, which stores information about the harmonics and other options. This has the following fields: 82 | - `harm` contains information about harmonics 83 | - `dependence` contains information about the form of $f_{nl}$ 84 | - `cont` contains settings for the continuation algorithm 85 | - `options` contains settings for the continuation algorithm 86 | 87 | ### `harm` structure 88 | This must have the following fields: 89 | 90 | - `rFreqRatio` is the ratio of the base harmonics to the base frequency. ie $\lambda$. For HBM standard problems `rFreqRatio` should be set to 1.0. For GHBM problems this should be vector. 91 | - `NHarm` is the number of harmonics to include for each base frequency. This should have the same length as `rFreqRatio`. 92 | - `Nfft` is the number of FFT points to use for the AFT transformation. This should be set to a power of 2 and have the same length as `rFreqRatio`. 93 | - `iHarmPlot` sets which harmonic to plot on the FRF when using `hbm_frf` or `hbm_bb`. Typically you want to see the first harmonic so this should be set to 1. 94 | 95 | ### `dependence` structure 96 | This should have the following fields: 97 | - `x`, `xdot`, `xddot` if the non-linearity has a dependence on the state and its derivatives 98 | - `u`, `udot`, `uddot` if the non-linearity has an explicit dependence on the state and its derivatives 99 | - `w` if the non-linearity has an explicit frequency dependence 100 | 101 | The value should be set to `1.0` or `True` where there is a dependence, and `0.0` or `False` otherwise 102 | ### `cont` structure 103 | This is used to configure settings of the continuation algorithm. This should have the following fields: 104 | - `method`: this should be one of the following values: 105 | - `predcorr` for a predictor-corrector algorithm (default) 106 | - `none` to simply step in frequency for `hbm_frf` or amplitude for `hbm_amp` and `hbm_bb` 107 | If method is `predcorr`, you can set the following additional settings in the `cont.predcorr` structure: 108 | - `solver` to choose which non-linear solver to use (`fipopt` to use IPOPT or `fsolve` to use the inbuilt solver) 109 | - `step0` which sets the initial step size 110 | - `min_step`/`max_step` which sets the minimum and maximum step size 111 | - `num_iter_increase`/ `num_iter_reduce` which sets the threshold for increasing/decreasing the step size depending on the number of iterations 112 | - `C` and `c` which sets the ratio to increase/decrease stepsize after a successful/unsuccessful step 113 | 114 | ### `options` structure 115 | This sets other more general options and can have the following fields: 116 | - `bUseStandardHBM`: force the solver to use the standard HBM when there is a single base frequency. 117 | - `bVerbose`: toggle whether to suppress output to console 118 | - `bPlot`: toggle whether to suppress plots 119 | 120 | ## Usage 121 | Once `problem` and `hbm` have been defined, you must call the setup code: 122 | 123 | ``` MATLAB 124 | [hbm,problem] = setuphbm(hbm,problem); 125 | ``` 126 | You can then call one of the following functions to solve a problem using HBM. 127 | 128 | ### One-off problems 129 | The most simple case is to find a periodic solution for a given excitation. Two different functions are provided to solve such problems, depending on the asumptions made about the base frequency: 130 | 131 | * `hbm_solve` : this assumes the base frequency $\Omega$ is fixed to a known value. This is the simplest and most common case. This can be called as follows: 132 | ``` MATLAB 133 | sol = hbm_solve(hbm,problem,w0,A); 134 | ``` 135 | where `sol` contains information about the solution including the components of $x$, $f_{nl}$ and $u$ at each harmonic 136 | * `hbm_res` : this assumes that the base frequency $\Omega$ can vary in order to find the maximum response at resonance. This is configured by the `problem.res` field: 137 | ``` MATLAB 138 | sol = hbm_res(hbm,problem,w0,A,X0); 139 | ``` 140 | If you do not have initial guess for `X0`, then this can be omitted or set to `[]`. 141 | 142 | ### Continuation problems 143 | Each of the different types of one-off problem can then be solved over a range of frequencies. If the base frequency is assumed fixed, there are two potential continuation parameters: 144 | 145 | * ```hbm_frf```: The base frequency $\Omega$ is used as the the continuation parameter, keeping the amplitude $A$ fixed. This yields a non-linear frequency response. 146 | ``` MATLAB 147 | sol = hbm_frf(hbm,problem,A,w0,X0,wEnd,XEnd); 148 | ``` 149 | * ```hbm_amp```: The base frequency $\Omega$ is kept fixed, and the amplitude $A$ is used as the the continuation parameter. 150 | ``` MATLAB 151 | sol = hbm_frf(hbm,problem,A,w0,X0,wEnd,XEnd); 152 | ``` 153 | If the base frequency is allowed to vary, and chosen to satisfy an objective, then there is only one potential continuation parameter: 154 | 155 | * ```hbm_bb```: The excitation amplitude $A$ is the continuation parameter, and the base frequency $\Omega$ is chosen to satisfy the objective. This yields the "backbone" curve. 156 | ``` MATLAB 157 | sol = hbm_bb(bb,problem,A0,w0,X0,Aend,wEnd,XEnd); 158 | ``` 159 | If you not have an intial guess for `X0` or `XEnd`, then this can be set to `[]` for any of these functions. 160 | -------------------------------------------------------------------------------- /Functions/hbm_frf_plot.m: -------------------------------------------------------------------------------- 1 | function hbm_frf_plot(command,hbm,problem,results) 2 | persistent fig hSuccess hWarn hErr X W 3 | if hbm.cont.bUpdate 4 | switch command 5 | case 'init' 6 | if ~isempty(fig) && ishandle(fig) 7 | close(fig) 8 | end 9 | 10 | if isempty(results) 11 | Xi = zeros(hbm.harm.NFreq,problem.NDof); 12 | wi = NaN; 13 | else 14 | Xi = results.X; 15 | wi = results.w; 16 | end 17 | W = getfrequencies(wi,hbm); 18 | X = Xi; 19 | 20 | [xlin, wlin] = getLinearReponse(hbm,problem,Xi,wi,results.A); 21 | 22 | xlin = mtimesx(xlin,problem.RDofPlot'); 23 | Xplot = mtimesx(X,problem.RDofPlot'); 24 | 25 | [fig,hSuccess,hWarn,hErr] = createFRF(hbm,problem,Xplot,W,xlin,wlin); 26 | 27 | case {'data','err','warn'} 28 | if ~ishandle(fig(1)) 29 | [xlin, wlin] = getLinearReponse(hbm,problem,X(:,:,1),W(1),results.A); 30 | xlin = mtimesx(xlin,problem.RDofPlot'); 31 | Xplot = mtimesx(X,problem.RDofPlot'); 32 | [fig,hSuccess,hWarn,hErr] = createFRF(hbm,problem,Xplot,W,xlin,wlin); 33 | end 34 | 35 | X(:,:,end+1) = results.X; 36 | W(:,end+1) = getfrequencies(results.w,hbm); 37 | 38 | Xplot = mtimesx(X(:,:,end-1:end),problem.RDofPlot'); 39 | Xabs = abs(Xplot(:,:,end)); 40 | Xph = unwrap(angle(Xplot)); Xph = Xph(:,:,end); 41 | Wfreq = W(:,end); 42 | 43 | if any(strcmpi(command,{'data','warn'})) 44 | update_handles(hSuccess,Xabs,Xph,Wfreq,hbm,problem) 45 | %update our progress 46 | 47 | if strcmpi(command,'warn') 48 | %warning, overlay in blue 49 | update_handles(hWarn,Xabs,Xph,Wfreq,hbm,problem) 50 | end 51 | 52 | %reset the error points 53 | for i = 1:length(hbm.harm.iHarmPlot) 54 | for j = 1:size(problem.RDofPlot,1) 55 | for k = 1:2 56 | set(hWarn{k}(i,j),'xdata',NaN,'ydata',NaN); 57 | set(hErr{k}(i,j) ,'xdata',NaN,'ydata',NaN); 58 | end 59 | end 60 | end 61 | else 62 | %error 63 | X(:,:,end) = []; 64 | W(:,end) = []; 65 | update_handles(hErr,Xabs,Xph,Wfreq,hbm,problem) 66 | end 67 | drawnow 68 | case 'close' 69 | close(fig) 70 | hSuccess = []; 71 | hWarn = []; 72 | hErr = []; 73 | end 74 | end 75 | 76 | function update_handles(han,Xabs,Xph,W,hbm,problem) 77 | for i = 1:length(hbm.harm.iHarmPlot) 78 | for j = 1:size(problem.RDofPlot,1) 79 | w = [get(han{1}(i,j),'xdata'),W(hbm.harm.iHarmPlot(i),:)]; 80 | mag = [get(han{1}(i,j),'ydata'),permute(Xabs(hbm.harm.iHarmPlot(i),j,:),[1 3 2])]; 81 | ph = [get(han{2}(i,j) ,'ydata'),permute(Xph(hbm.harm.iHarmPlot(i),j,:),[1 3 2])]; 82 | ph = unwrap(ph); 83 | set(han{1}(i,j),'xdata',w,'ydata',mag); 84 | set(han{2}(i,j) ,'xdata',w,'ydata',ph); 85 | end 86 | end 87 | 88 | function ws = getfrequencies(w0,hbm) 89 | w = hbm.harm.rFreqBase'.*(hbm.harm.rFreqRatio'*w0 + hbm.harm.wFreq0'); 90 | ws = abs(hbm.harm.kHarm(:,1)*w(1,:) + hbm.harm.kHarm(:,2)*w(2,:)); 91 | ws(ws == 0) = w0; 92 | 93 | function [fMag,hSuccess,hWarn,hErr] = createFRF(hbm,problem,x,w,xlin,wlin0) 94 | fMag = figure('Name',['FRF: ' problem.name]); 95 | 96 | wlin = getfrequencies(wlin0,hbm); 97 | wlim = getfrequencies([problem.wMin problem.wMax],hbm); 98 | 99 | for i = 1:length(hbm.harm.iHarmPlot) 100 | for j = 1:size(problem.RDofPlot,1) 101 | tmp = subplot(size(problem.RDofPlot,1),length(hbm.harm.iHarmPlot),(j-1)*length(hbm.harm.iHarmPlot) + i,'Parent',fMag); 102 | Xij = squeeze(xlin(hbm.harm.iHarmPlot(i),j,:)); 103 | wij = wlin(hbm.harm.iHarmPlot(i),:); 104 | [tmp2,hLin{1}(i,j),hLin{2}(i,j)] = plotyy(tmp,wij,abs(Xij),wij,unwrap(angle(Xij))); 105 | ax{1}(i,j) = tmp2(1); ax{2}(i,j) = tmp2(2); 106 | hold(tmp2(1), 'on'); 107 | hold(tmp2(2), 'on'); 108 | end 109 | end 110 | 111 | for i = 1:length(hbm.harm.iHarmPlot) 112 | for j = 1:size(problem.RDofPlot,1) 113 | hSuccess{1}(i,j) = plot(ax{1}(i,j),w(hbm.harm.iHarmPlot(i),:),abs(squeeze(x(hbm.harm.iHarmPlot(i),j,:))),'g.-'); 114 | hSuccess{2}(i,j) = plot(ax{2}(i,j),w(hbm.harm.iHarmPlot(i),:),unwrap(angle(squeeze(x(hbm.harm.iHarmPlot(i),j,:)))),'m.-'); 115 | 116 | for k = 1:2 117 | hWarn{k}(i,j) = plot(ax{k}(i,j),NaN,NaN,'b.'); 118 | hErr{k}(i,j) = plot(ax{k}(i,j),NaN,NaN,'r.'); 119 | 120 | %xlim(ax{k}(i,j),wlim(hbm.harm.iHarmPlot(i),:)); 121 | set(ax{k}(i,j),'XLimMode','auto') 122 | set(ax{k}(i,j),'YLimMode','auto') 123 | end 124 | 125 | if j==size(problem.RDofPlot,1) 126 | xlabel(ax{1}(i,j),'\omega (rads)') 127 | end 128 | if j==1 129 | title(ax{1}(i,j),harmonicName(hbm,i)) 130 | end 131 | 132 | if i == 1 133 | ylabel(ax{1}(i,j),sprintf('|Dof #%d|',j)) 134 | end 135 | 136 | if i == length(hbm.harm.iHarmPlot) 137 | ylabel(ax{2}(i,j),sprintf('\\angle Dof #%d',j)) 138 | end 139 | end 140 | end 141 | 142 | function s = harmonicName(hbm,i) 143 | k1 = hbm.harm.kHarm(hbm.harm.iHarmPlot(i),1)*hbm.harm.rFreqBase(1); 144 | k2 = hbm.harm.kHarm(hbm.harm.iHarmPlot(i),2)*hbm.harm.rFreqBase(2); 145 | if k1 == 0 146 | s1 = ''; 147 | elseif k1 == 1 148 | s1 = '\omega_1'; 149 | else 150 | s1 = sprintf('%s\\omega_1',num2frac(k1)); 151 | end 152 | if k2 == 0 153 | s2 = ''; 154 | elseif k2 == 1 155 | if ~isempty(s1) 156 | s2 = '+\omega_2'; 157 | else 158 | s2 = '\omega_2'; 159 | end 160 | elseif k2 == -1 161 | s2 = '-\omega_2'; 162 | else 163 | if ~isempty(s1) 164 | s2 = sprintf('%+s\\omega_2',num2frac(k2)); 165 | else 166 | s2 = sprintf('%s\\omega_2',num2frac(k2)); 167 | end 168 | end 169 | 170 | if isempty(s2) 171 | s = s1; 172 | elseif isempty(s1) 173 | s = s2; 174 | else 175 | s = [s1 ' ' s2]; 176 | end 177 | 178 | if isempty(s) 179 | s = '0'; 180 | end 181 | 182 | function [xlin, wlin] = getLinearReponse(hbm,problem,X,w,A) 183 | %find the linearised contribution to the stiffness/damping due from the non-linearity 184 | w0 = w*hbm.harm.rFreqRatio + hbm.harm.wFreq0; 185 | 186 | x0 = X(1,:).'; 187 | U = A*feval(problem.excite,hbm,problem,w0); 188 | u0 = U(1,:).'; 189 | 190 | %compute a LU table of frequency dependent stiffness etc 191 | wLU = linspace(problem.wMin,problem.wMax,10); 192 | for i = 1:length(wLU) 193 | w0 = wLU(i) * hbm.harm.rFreqRatio + hbm.harm.wFreq0; 194 | States = hbm_states3d(w0,X,U,hbm); 195 | States.f = feval(problem.model,'nl',States,hbm,problem); 196 | 197 | [K_nl, C_nl, M_nl] = hbm_derivatives('nl',{'x','xdot','xddot'},States,hbm,problem); 198 | [Ku_nl,Cu_nl,Mu_nl] = hbm_derivatives('nl',{'u','udot','uddot'},States,hbm,problem); 199 | 200 | %find average stiffness etc over time 201 | K_lu(:,:,i) = mean(K_nl,3); C_lu(:,:,i) = mean(C_nl,3); M_lu(:,:,i) = mean(M_nl,3); 202 | Ku_lu(:,:,i) = mean(Ku_nl,3); Cu_lu(:,:,i) = mean(Cu_nl,3); Mu_lu(:,:,i) = mean(Mu_nl,3); 203 | end 204 | 205 | %now loop over all the frequencies 206 | wlin = linspace(problem.wMin,problem.wMax,1000); 207 | xlin = zeros(hbm.harm.NFreq,problem.NDof,length(wlin)); 208 | 209 | NFreq = hbm.harm.NFreq; 210 | 211 | %work out frequencies 212 | w = getfrequencies(wlin,hbm); 213 | for i = 1:length(wlin) 214 | w0 = wlin(i) * hbm.harm.rFreqRatio + hbm.harm.wFreq0; 215 | 216 | %interpolate into LU table 217 | M_nl = interpx(wLU,M_lu,wlin(i)); 218 | C_nl = interpx(wLU,C_lu,wlin(i)); 219 | K_nl = interpx(wLU,K_lu,wlin(i)); 220 | 221 | Mu_nl = interpx(wLU,Mu_lu,wlin(i)); 222 | Cu_nl = interpx(wLU,Cu_lu,wlin(i)); 223 | Ku_nl = interpx(wLU,Ku_lu,wlin(i)); 224 | 225 | M = problem.M + M_nl; 226 | G = problem.G; 227 | C = problem.C + C_nl; 228 | K = problem.K + K_nl; 229 | 230 | Mu = problem.Mu - Mu_nl; 231 | Cu = problem.Cu - Cu_nl; 232 | Ku = problem.Ku - Ku_nl; 233 | 234 | U = A*feval(problem.excite,hbm,problem,w0); 235 | 236 | for k = 1:NFreq 237 | Fe = (Ku + 1i*w(k,i)*Cu - w(k,i)^2*Mu)*U(k,:).'; 238 | H = K + 1i*w(k,i)*(C+w0(1)*G) - w(k,i)^2 * M; 239 | xlin(k,:,i) = (H\Fe).'; 240 | end 241 | end 242 | xlin(1,:,:) = xlin(1,:,:) + x0.'; -------------------------------------------------------------------------------- /Functions/hbm_amp.m: -------------------------------------------------------------------------------- 1 | function [results,curr] = hbm_amp(hbm,problem,w0,A0,X0,AEnd,XEnd) 2 | problem.type = 'amp'; 3 | problem.w0 = w0; 4 | 5 | %first solve @ A0 6 | sol = hbm_solve(hbm,problem,w0,A0,X0); 7 | x0 = packdof(sol.X); 8 | if any(isnan(abs(x0(:)))) 9 | error('Failed to solve initial problem') 10 | end 11 | z0 = [x0;A0]; 12 | 13 | init.X = sol.X; 14 | init.w = w0; 15 | init.A = A0; 16 | 17 | sol = hbm_solve(hbm,problem,w0,AEnd,XEnd); 18 | xEnd = packdof(sol.X); 19 | if any(isnan(abs(xEnd(:)))) 20 | error('Failed to solve final problem') 21 | end 22 | zEnd = [xEnd;AEnd]; 23 | 24 | hbm.bIncludeNL = 1; 25 | 26 | if isfield(problem,'xscale') 27 | xscale = [problem.xscale'; repmat(problem.xscale',hbm.harm.NFreq-1,1)*(1+1i)]; 28 | problem.Xscale = packdof(xscale)*sqrt(length(xscale)); 29 | problem.Ascale = mean([A0 AEnd]); 30 | problem.Fscale = problem.Xscale*0+1; 31 | problem.Zscale = [problem.Xscale; problem.Ascale]; 32 | problem.Jscale = (1./problem.Fscale(:))*problem.Zscale(:)'; 33 | bUpdateScaling = 0; 34 | else 35 | problem = hbm_scaling(problem,hbm,init); 36 | bUpdateScaling = 1; 37 | end 38 | 39 | AMax = max(AEnd,A0); 40 | AMin = min(AEnd,A0); 41 | 42 | problem.AMax = AMax; 43 | problem.AMin = AMin; 44 | problem.A0 = A0; 45 | problem.AEnd = AEnd; 46 | 47 | 48 | prog.Status = 'success'; 49 | 50 | switch hbm.cont.method 51 | case 'none' 52 | hbm_amp_plot('init',hbm,problem,init); 53 | Ascale = mean([A0 AEnd]); 54 | 55 | pred.step = hbm.cont.step0; 56 | direction = sign(AEnd-A0)*Ascale; 57 | 58 | problem.Xscale = 0*problem.Xscale + 1; 59 | problem.Ascale = 1; 60 | problem.Zscale = 0*problem.Zscale + 1; 61 | 62 | z = z0; 63 | J = hbm_amp_jacobian(z,hbm,problem); 64 | t = get_tangent(J); 65 | 66 | corr.step = 0; 67 | corr.it = 0; 68 | 69 | curr = hbm_amp_results(z,t,pred,corr,hbm,problem); 70 | results = curr; 71 | zprev = z0; 72 | 73 | Asol = A0; 74 | zsol = z0; 75 | 76 | while Asol(end) <= AMax && Asol(end) >= AMin 77 | Apred = Asol(end) + pred.step*direction; 78 | if Apred > AMax || Apred < AMin 79 | break; 80 | end 81 | 82 | iPredict = max(length(Asol)-6,1):length(Asol); 83 | if length(iPredict) > 1 84 | zpred = interp1(Asol(iPredict),zsol(:,iPredict)',Apred,'pchip','extrap')'; 85 | else 86 | zpred = zsol(:,end); 87 | zpred(end) = Apred; 88 | end 89 | 90 | %now try to solve 91 | xpred = zpred(1:end-1); 92 | Xpred = unpackdof(xpred,hbm.harm.NFreq-1,problem.NDof); 93 | sol = hbm_solve(hbm,problem,w0,Apred,Xpred); 94 | sol.x = packdof(sol.X); 95 | 96 | z = [sol.x; sol.A]; 97 | t = z - zprev; 98 | corr.step = norm2(z - zprev); 99 | 100 | curr(end+1) = hbm_amp_results(z,t,pred,corr,hbm,problem); 101 | 102 | if ~any(isnan(curr(end).z)) 103 | curr(end).flag = 'Success'; 104 | pred.step = min(max(pred.step * hbm.cont.C,hbm.cont.min_step),hbm.cont.max_step); 105 | results(end+1) = curr(end); 106 | Asol(end+1) = results(end).A; 107 | zsol(:,end+1) = results(end).z; 108 | hbm_amp_plot('data',hbm,problem,results(end)); 109 | prog.NFail = 0; 110 | if Asol(end) >= AMax || Asol(end) <= AMin 111 | break; 112 | end 113 | zprev = curr(end).z; 114 | else 115 | curr(end).flag = 'Fail'; 116 | pred.step = pred.step * hbm.cont.c; 117 | prog.NFail = prog.NFail + 1; 118 | hbm_amp_plot('err',hbm,problem,curr(end)); 119 | if prog.NFail > hbm.cont.maxfail 120 | prog.Status = 'Too many failed iterations'; 121 | break; 122 | end 123 | end 124 | end 125 | 126 | %add on final point 127 | t = zEnd - zprev; 128 | 129 | pred.step = norm(zEnd - zprev); 130 | corr.step = norm(zEnd - zprev); 131 | corr.it = 0; 132 | curr(end+1) = hbm_amp_results(zEnd,t,pred,corr,hbm,problem); 133 | results(end+1) = curr(end); 134 | 135 | hbm_amp_plot('close',hbm,problem,[]); 136 | 137 | case 'predcorr' 138 | prog.NStep = 1; 139 | prog.NFail = 0; 140 | prog.NIter = 0; 141 | 142 | switch hbm.cont.predcorr.corrector 143 | case 'pseudo' 144 | case 'arclength' 145 | Jstr = [hbm.sparsity 0*x0+1; 146 | 0*x0'+1 1]; 147 | switch hbm.cont.predcorr.solver 148 | case 'ipopt' 149 | ipopt_opt.print_level = 0; 150 | ipopt_opt.maxit = hbm.cont.predcorr.maxit; 151 | ipopt_opt.ftol = hbm.cont.ftol; 152 | ipopt_opt.xtol = hbm.cont.xtol; 153 | ipopt_opt.jacob = @hbm_arclength_jacobian; 154 | ipopt_opt.jacobstructure = Jstr; 155 | case 'fsolve' 156 | fsolve_opt = optimoptions('fsolve',... 157 | 'Display','off',... 158 | 'FunctionTolerance',hbm.cont.ftol,... 159 | 'StepTolerance',hbm.cont.xtol,... 160 | 'SpecifyObjectiveGradient',true,... 161 | 'MaxIterations',hbm.cont.predcorr.maxit); 162 | end 163 | end 164 | 165 | Zprev = z0./problem.Zscale; 166 | F = hbm_amp_constraints(Zprev,hbm,problem); 167 | J = hbm_amp_jacobian(Zprev,hbm,problem); 168 | Tprev = get_tangent(J); 169 | Tprev = Tprev * sign(Tprev(end)) * sign(AEnd - A0); 170 | 171 | Zend = zEnd./problem.Zscale; 172 | J = hbm_amp_jacobian(Zend,hbm,problem); 173 | Tend = get_tangent(J); 174 | Tend = Tend * sign(Tend(end)) * sign(AEnd - A0); 175 | 176 | pred.step = hbm.cont.step0; 177 | corr.step = pred.step; 178 | corr.it = 0; 179 | 180 | curr = hbm_amp_results(Zprev,Tprev,pred,corr,hbm,problem); 181 | results = curr; 182 | 183 | hbm_amp_plot('init',hbm,problem,init); 184 | fprintf('STEP PRED CORR STATUS INFO ITER TOT AMP ') 185 | fprintf('Z(%d) ',1:length(z0)) 186 | fprintf('\n') 187 | 188 | fprintf('%3d %6.4f %6.4f %s %3s %3d %3d %6.2f ',prog.NStep,pred.step,corr.step,'S','Ini',corr.it,prog.NIter,results(end).A) 189 | fprintf('%+5.2e ',results(end).z) 190 | fprintf('\n') 191 | 192 | zsol = results.z; 193 | tsol = results.t; 194 | 195 | while norm(Zprev - Zend) > hbm.cont.max_step 196 | 197 | %predictor 198 | switch hbm.cont.predcorr.predictor 199 | case 'linear' 200 | Zpred = Zprev + pred.step*Tprev; 201 | case 'quadratic' 202 | if length(results) < 5 203 | Zpred = Zprev + pred.step*Tprev; 204 | else 205 | Zpred = polynomial_predictor(Zsol(:,end-2:end),Tsol(:,end-2:end),pred.step); 206 | end 207 | case 'cubic' 208 | if length(results) < 5 209 | Zpred = Zprev + pred.step*Tprev; 210 | else 211 | Zpred = polynomial_predictor(Zsol(:,end-3:end),Tsol(:,end-3:end),pred.step); 212 | end 213 | end 214 | corr.it = 0; 215 | Zlast = Zpred + Inf; 216 | 217 | %corrector 218 | switch hbm.cont.predcorr.corrector 219 | case 'pseudo' 220 | bConverged = 0; 221 | Z = Zpred; 222 | T = Tprev; 223 | 224 | while corr.it <= hbm.cont.predcorr.maxit 225 | Zlast = Z; 226 | J = hbm_amp_jacobian(Z,hbm,problem); 227 | F = hbm_amp_constraints(Z,hbm,problem); 228 | if hbm.cont.predcorr.bMoorePenrose 229 | Z = Zlast - J\F; 230 | else 231 | B = [J; T']; 232 | R = [J*T; 0]; 233 | Q = [F; 0]; 234 | W = T - B\R; 235 | T = normalise(W); 236 | Z = Zlast - B\Q; 237 | end 238 | corr.it = corr.it + 1; 239 | if ~(any(abs(Z - Zlast) > hbm.cont.xtol) || any(abs(F) > hbm.cont.ftol)) 240 | bConverged = 1; 241 | break; 242 | end 243 | end 244 | if hbm.cont.predcorr.bMoorePenrose 245 | T = get_tangent(J); 246 | T = sign(T'*Tprev)*T; 247 | end 248 | case 'arclength' 249 | corr.Zprev = Zprev; 250 | corr.Tprev = Tprev; 251 | corr.step = pred.step; 252 | switch hbm.cont.predcorr.solver 253 | case 'ipopt' 254 | [Z,info] = fipopt('',Zpred,@hbm_arclength_constraints,ipopt_opt,hbm,problem,corr); 255 | corr.it = info.iter; 256 | bConverged = info.status == 0; 257 | case 'fsolve' 258 | [Z,F,status,out] = fsolve(@hbm_arclength_constraints,Zpred,fsolve_opt,hbm,problem,corr); 259 | corr.it = out.iterations + 1; 260 | bConverged = status == 1; 261 | end 262 | J = hbm_amp_jacobian(Z,hbm,problem); 263 | T = get_tangent(J); 264 | T = sign(T'*Tprev)*T; 265 | end 266 | corr.step = norm(Z - Zprev)*sign((Z-Zprev)'*Tprev); 267 | 268 | prog.NStep = prog.NStep + 1; 269 | prog.NIter = prog.NIter + corr.it; 270 | 271 | %prepare for plots 272 | curr(end+1) = hbm_amp_results(Z,T,pred,corr,hbm,problem); 273 | 274 | if bConverged && curr(end).sCorr >= hbm.cont.min_step && curr(end).sCorr <= hbm.cont.max_step && curr(end).A > 0 275 | %success 276 | status = 'S'; 277 | hbm_amp_plot('data',hbm,problem,curr(end)); 278 | 279 | if corr.it <= hbm.cont.num_iter_increase 280 | pred.step = min(curr(end).sPred * hbm.cont.C,hbm.cont.max_step/curr(end).sCorr*curr(end).sPred); 281 | curr(end).flag = 'Success: Increasing step size'; 282 | info = 'Inc'; 283 | elseif corr.it >= hbm.cont.num_iter_reduce 284 | pred.step = max(curr(end).sPred / hbm.cont.C,hbm.cont.min_step/curr(end).sCorr*curr(end).sPred); 285 | curr(end).flag = 'Success: Reducing step size'; 286 | info = 'Red'; 287 | else 288 | curr(end).flag = 'Success'; 289 | info = ''; 290 | end 291 | 292 | %store the data 293 | results(end+1) = curr(end); 294 | 295 | if bUpdateScaling 296 | problem = hbm_scaling(problem,hbm,results(end)); 297 | end 298 | 299 | zsol(:,end+1) = results(end).z; 300 | tsol(:,end+1) = results(end).t; 301 | Zsol = zsol./(repmat(problem.Zscale,1,size(zsol,2))); 302 | Tsol = normalise(tsol./(repmat(problem.Zscale,1,size(tsol,2)))); 303 | 304 | Zend = zEnd./problem.Zscale; 305 | 306 | Zprev = Zsol(:,end); 307 | Tprev = Tsol(:,end); 308 | 309 | prog.NFail = 0; 310 | else 311 | %failed 312 | hbm_amp_plot('err',hbm,problem,curr(end)); 313 | status = 'F'; 314 | 315 | if curr(end).A < 0 316 | %gone to negative frequencies (somehow) 317 | curr(end).flag = 'Failed: Negative frequency'; 318 | info = 'Neg'; 319 | elseif corr.it <= hbm.cont.predcorr.maxit 320 | %converge but step size is unacceptable 321 | if curr(end).sCorr < 0 322 | %backwards 323 | pred.step = max(pred.step * hbm.cont.c,hbm.cont.min_step); 324 | curr(end).flag = 'Failed: Wrong direction'; 325 | info = 'Bwd'; 326 | elseif curr(end).sCorr > hbm.cont.max_step 327 | %too large 328 | pred.step = max(0.9 * pred.step * hbm.cont.max_step / curr(end).sCorr,hbm.cont.min_step); 329 | curr(end).flag = 'Failed: Step too large'; 330 | info = 'Lrg'; 331 | elseif curr(end).sCorr < hbm.cont.min_step 332 | %too small 333 | pred.step = min(1.1 * pred.step * hbm.cont.min_step / curr(end).sCorr,hbm.cont.max_step); 334 | curr(end).flag = 'Failed: Step too small'; 335 | info = 'Sml'; 336 | else 337 | pred.step = max(pred.step * hbm.cont.c,hbm.cont.min_step); 338 | curr(end).flag = 'Failed: Other error'; 339 | info = 'Oth'; 340 | end 341 | else 342 | %failed to converge 343 | if any(abs(Z - Zlast) > hbm.cont.xtol) 344 | curr(end).flag = 'Failed: No convergence'; 345 | info = 'xtl'; 346 | elseif any(abs(F) > hbm.cont.ftol) 347 | curr(end).flag = 'Failed: Constraints violated'; 348 | info = 'ftl'; 349 | else 350 | curr(end).flag = 'Failed'; 351 | info = ''; 352 | end 353 | pred.step = max(pred.step * hbm.cont.c,hbm.cont.min_step); 354 | end 355 | 356 | prog.NFail = prog.NFail + 1; 357 | if prog.NFail > hbm.cont.maxfail 358 | prog.Status = 'Too many failed iterations'; 359 | break 360 | elseif curr(end).A < 0 361 | prog.Status = 'Negative frequency'; 362 | break 363 | end 364 | end 365 | fprintf('%3d %6.4f %6.4f %s %3s %3d %3d %6.2f ',prog.NStep,curr(end).sPred,curr(end).sCorr,status,info,corr.it,prog.NIter,curr(end).A) 366 | fprintf('%+5.2e ',curr(end).z) 367 | fprintf('\n') 368 | end 369 | 370 | %add on final point 371 | pred.step = norm(Zend - Zprev); 372 | corr.step = norm(Zend - Zprev); 373 | corr.it = 0; 374 | curr(end+1) = hbm_amp_results(Zend,Tend,pred,corr,hbm,problem); 375 | results(end+1) = curr(end); 376 | 377 | hbm_amp_plot('close',hbm,problem,[]); 378 | end 379 | 380 | NPts = length(results); 381 | 382 | if ~isfield(results,'W') 383 | for i = 1:NPts 384 | w0 = results(i).w*hbm.harm.rFreqRatio + hbm.harm.wFreq0; 385 | results(i).W = hbm.harm.kHarm*(hbm.harm.rFreqBase.*w0)'; 386 | end 387 | end 388 | 389 | if ~isfield(results,'L') 390 | results = hbm_floquet(hbm,problem,results); 391 | end 392 | 393 | results = hbm_excitation_forces(problem,results); 394 | 395 | %% Predictor 396 | function X_extrap = polynomial_predictor(Z,dZ,s_extrap) 397 | s = norm2(diff(Z,[],2)); 398 | s = cumsum([0 s]); 399 | N = length(s); 400 | if ~isempty(dZ) 401 | [A,Ad] = poly_mat(s,N); 402 | p = [A;Ad]\[Z dZ]'; 403 | else 404 | A = poly_mat(s,N); 405 | p = A\Z'; 406 | end 407 | B = poly_mat(s(end) + s_extrap,N); 408 | X_extrap = (B*p)'; 409 | 410 | function [A,Ad] = poly_mat(s,N) 411 | A = zeros(length(s),N); 412 | Ad = zeros(length(s),N); 413 | for i = 1:N 414 | A(:,i) = s.^(i-1); 415 | if nargout > 1 416 | if i > 1 417 | Ad(:,i) = (i-1)*s.^(i-2); 418 | else 419 | Ad(:,i) = 0*s; 420 | end 421 | end 422 | end 423 | 424 | %% Constraints and Jacobian 425 | function c = hbm_amp_constraints(Z,hbm,problem) 426 | %unpack the inputs 427 | x = Z(1:end-1).*problem.Xscale; 428 | A = Z(end).*problem.Ascale; 429 | w = problem.w0; 430 | 431 | w0 = w * hbm.harm.rFreqRatio + hbm.harm.wFreq0; 432 | 433 | U = A*feval(problem.excite,hbm,problem,w0); 434 | u = packdof(U); 435 | 436 | c = hbm_balance3d('func',hbm,problem,w,u,x); 437 | c = c ./ problem.Fscale; 438 | 439 | function J = hbm_amp_jacobian(Z,hbm,problem) 440 | %unpack the inputs 441 | x = Z(1:end-1).*problem.Xscale; 442 | A = Z(end).*problem.Ascale; 443 | w = problem.w0; 444 | 445 | w0 = w * hbm.harm.rFreqRatio + hbm.harm.wFreq0; 446 | 447 | U = A*feval(problem.excite,hbm,problem,w0); 448 | u = packdof(U); 449 | 450 | Jx = hbm_balance3d('jacob' ,hbm,problem,w,u,x); 451 | Da = hbm_balance3d('derivA',hbm,problem,w,u,x); 452 | 453 | J = [Jx Da]; 454 | J = J .* problem.Jscale; 455 | 456 | %% Arclength files 457 | function [c,J] = hbm_arclength_constraints(Z,hbm,problem,corr) 458 | c = hbm_amp_constraints(Z,hbm,problem); 459 | sgn = sign((Z - corr.Zprev)' * corr.Tprev); 460 | s = norm(Z - corr.Zprev) * sgn; 461 | c(end+1) = s - corr.step; 462 | if nargout > 1 463 | J = hbm_arclength_jacobian(Z,hbm,problem,corr); 464 | end 465 | 466 | function J = hbm_arclength_jacobian(Z,hbm,problem,corr) 467 | J = hbm_amp_jacobian(Z,hbm,problem); 468 | sgn = sign((Z - corr.Zprev)' * corr.Tprev); 469 | J(end+1,:) = sgn*((Z - corr.Zprev)'+eps)/(1*(norm(Z - corr.Zprev)+eps)); 470 | 471 | %% Utilities 472 | function y = norm2(x) 473 | y = sqrt(sum(x.^2,1)); 474 | 475 | function y = normalise(x) 476 | scale = repmat(norm2(x),size(x,1),1); 477 | y = x ./scale; 478 | 479 | function t = get_tangent(J) 480 | [U,S,V] = svd(J,0); 481 | t = V(:,end); 482 | t = t./norm(t); 483 | 484 | function curr = hbm_amp_results(Z,tangent,pred,corr,hbm,problem) 485 | A = Z(end).*problem.Ascale; 486 | x = Z(1:end-1).*problem.Xscale; 487 | w = problem.w0; 488 | t = normalise(tangent.*problem.Zscale); 489 | 490 | curr.z = [x; A]; 491 | curr.t = t; 492 | 493 | curr.sCorr = corr.step; 494 | curr.sPred = pred.step; 495 | curr.it = corr.it; 496 | curr.flag = ''; 497 | 498 | w0 = w*hbm.harm.rFreqRatio + hbm.harm.wFreq0; 499 | 500 | curr.w = w; 501 | curr.X = unpackdof(x,hbm.harm.NHarm,problem.NDof); 502 | curr.U = A*feval(problem.excite,hbm,problem,w0); 503 | curr.F = hbm_output3d(hbm,problem,curr.w,curr.U,curr.X); 504 | curr.A = A; -------------------------------------------------------------------------------- /Functions/hbm_bb.m: -------------------------------------------------------------------------------- 1 | function [results,curr] = hbm_bb(hbm,problem,A0,w0,X0,AEnd,wEnd,XEnd) 2 | problem.type = 'bb'; 3 | 4 | %first solve @ A0,w0 5 | sol = hbm_res(hbm,problem,w0,A0,X0); 6 | x0 = packdof(sol.X); 7 | if any(isnan(abs(x0(:)))) 8 | error('Failed to solve initial problem') 9 | end 10 | z0 = [x0;w0;A0]; 11 | 12 | init.X = sol.X; 13 | init.H = sol.H; 14 | init.w = sol.w; 15 | init.A = A0; 16 | 17 | sol = hbm_res(hbm,problem,w0,AEnd,XEnd); 18 | xEnd = packdof(sol.X); 19 | if any(isnan(abs(xEnd(:)))) 20 | error('Failed to solve final problem') 21 | end 22 | zEnd = [xEnd;sol.w;AEnd]; 23 | 24 | hbm.bIncludeNL = 1; 25 | 26 | if isfield(problem,'xscale') 27 | xscale = [problem.xscale'; repmat(problem.xscale',hbm.harm.NFreq-1,1)*(1+1i)]; 28 | problem.Xscale = packdof(xscale)*sqrt(length(xscale)); 29 | problem.wscale = mean([w0 wEnd]); 30 | problem.Ascale = mean([A0 AEnd]); 31 | problem.Fscale = [problem.Xscale*0+1;1]; 32 | problem.Zscale = [problem.Xscale; problem.wscale; problem.Ascale]; 33 | problem.Jscale = (1./problem.Fscale(:))*problem.Zscale(:)'; 34 | bUpdateScaling = 0; 35 | else 36 | problem = hbm_scaling(problem,hbm,init); 37 | bUpdateScaling = 1; 38 | end 39 | 40 | AMax = max(AEnd,A0); 41 | AMin = min(AEnd,A0); 42 | 43 | problem.AMax = AMax; 44 | problem.AMin = AMin; 45 | problem.A0 = A0; 46 | problem.AEnd = AEnd; 47 | 48 | 49 | prog.Status = 'success'; 50 | 51 | switch hbm.cont.method 52 | case 'none' 53 | hbm_bb_plot('init',hbm,problem,init); 54 | Ascale = mean([A0 AEnd]); 55 | 56 | pred.step = hbm.cont.step0; 57 | direction = sign(AEnd-A0)*Ascale; 58 | 59 | problem.Xscale = 0*problem.Xscale + 1; 60 | problem.wscale = 1; 61 | problem.Ascale = 1; 62 | problem.Zscale = 0*problem.Zscale + 1; 63 | 64 | z = z0; 65 | J = hbm_bb_jacobian(z,hbm,problem); 66 | t = get_tangent(J); 67 | 68 | corr.step = 0; 69 | corr.it = 0; 70 | 71 | curr = hbm_bb_results(z,t,pred,corr,hbm,problem); 72 | results = curr; 73 | zprev = z0; 74 | 75 | Asol = A0; 76 | zsol = z0; 77 | 78 | while Asol(end) <= AMax && Asol(end) >= AMin 79 | Apred = Asol(end) + pred.step*direction; 80 | if Apred > AMax || Apred < AMin 81 | break; 82 | end 83 | 84 | iPredict = max(length(Asol)-6,1):length(Asol); 85 | if length(iPredict) > 1 86 | zpred = interp1(Asol(iPredict),zsol(:,iPredict)',Apred,'pchip','extrap')'; 87 | else 88 | zpred = zsol(:,end); 89 | zpred(end) = Apred; 90 | end 91 | 92 | %now try to solve 93 | xpred = zpred(1:end-2); 94 | wpred = zpred(end-1); 95 | Xpred = unpackdof(xpred,hbm.harm.NFreq-1,problem.NDof); 96 | sol = hbm_res(hbm,problem,wpred,Apred,Xpred); 97 | sol.x = packdof(sol.X); 98 | 99 | z = [sol.x; sol.w; sol.A]; 100 | t = z - zprev; 101 | corr.step = norm2(z - zprev); 102 | 103 | curr(end+1) = hbm_bb_results(z,t,pred,corr,hbm,problem); 104 | 105 | if ~any(isnan(curr(end).z)) 106 | curr(end).flag = 'Success'; 107 | pred.step = min(max(pred.step * hbm.cont.C,hbm.cont.min_step),hbm.cont.max_step); 108 | results(end+1) = curr(end); 109 | Asol(end+1) = results(end).A; 110 | zsol(:,end+1) = results(end).z; 111 | hbm_bb_plot('data',hbm,problem,results(end)); 112 | prog.NFail = 0; 113 | if Asol(end) >= AMax || Asol(end) <= AMin 114 | break; 115 | end 116 | zprev = curr(end).z; 117 | else 118 | curr(end).flag = 'Fail'; 119 | pred.step = pred.step * hbm.cont.c; 120 | prog.NFail = prog.NFail + 1; 121 | hbm_bb_plot('err',hbm,problem,curr(end)); 122 | if prog.NFail > hbm.cont.maxfail 123 | prog.Status = 'Too many failed iterations'; 124 | break; 125 | end 126 | end 127 | end 128 | 129 | %add on final point 130 | t = zEnd - zprev; 131 | 132 | pred.step = norm(zEnd - zprev); 133 | corr.step = norm(zEnd - zprev); 134 | corr.it = 0; 135 | curr(end+1) = hbm_bb_results(zEnd,t,pred,corr,hbm,problem); 136 | results(end+1) = curr(end); 137 | 138 | hbm_bb_plot('close',hbm,problem,[]); 139 | 140 | case 'predcorr' 141 | prog.NStep = 1; 142 | prog.NFail = 0; 143 | prog.NIter = 0; 144 | 145 | switch hbm.cont.predcorr.corrector 146 | case 'pseudo' 147 | case 'arclength' 148 | Jstr = [hbm.sparsity 0*x0+1; 149 | 0*x0'+1 1]; 150 | switch hbm.cont.predcorr.solver 151 | case 'ipopt' 152 | ipopt_opt.print_level = 0; 153 | ipopt_opt.maxit = hbm.cont.predcorr.maxit; 154 | ipopt_opt.ftol = hbm.cont.ftol; 155 | ipopt_opt.xtol = hbm.cont.xtol; 156 | ipopt_opt.jacob = @hbm_arclength_jacobian; 157 | ipopt_opt.jacobstructure = Jstr; 158 | case 'fsolve' 159 | fsolve_opt = optimoptions('fsolve',... 160 | 'Display','off',... 161 | 'FunctionTolerance',hbm.cont.ftol,... 162 | 'StepTolerance',hbm.cont.xtol,... 163 | 'SpecifyObjectiveGradient',true,... 164 | 'MaxIterations',hbm.cont.predcorr.maxit); 165 | end 166 | end 167 | 168 | Zprev = z0./problem.Zscale; 169 | F = hbm_bb_constraints(Zprev,hbm,problem); 170 | J = hbm_bb_jacobian(Zprev,hbm,problem); 171 | Tprev = get_tangent(J); 172 | Tprev = Tprev * sign(Tprev(end)) * sign(AEnd - A0); 173 | 174 | Zend = zEnd./problem.Zscale; 175 | J = hbm_bb_jacobian(Zend,hbm,problem); 176 | Tend = get_tangent(J); 177 | Tend = Tend * sign(Tend(end)) * sign(AEnd - A0); 178 | 179 | pred.step = hbm.cont.step0; 180 | corr.step = pred.step; 181 | corr.it = 0; 182 | 183 | curr = hbm_bb_results(Zprev,Tprev,pred,corr,hbm,problem); 184 | results = curr; 185 | 186 | hbm_bb_plot('init',hbm,problem,init); 187 | fprintf('STEP PRED CORR STATUS INFO ITER TOT FREQ AMP ') 188 | fprintf('Z(%d) ',1:length(z0)) 189 | fprintf('\n') 190 | 191 | fprintf('%3d %6.4f %6.4f %s %3s %3d %3d %6.2f %6.2f ',prog.NStep,pred.step,corr.step,'S','Ini',corr.it,prog.NIter,results(end).w,results(end).A) 192 | fprintf('%+5.2e ',results(end).z) 193 | fprintf('\n') 194 | 195 | zsol = results.z; 196 | tsol = results.t; 197 | 198 | while norm(Zprev - Zend) > hbm.cont.max_step 199 | 200 | %predictor 201 | switch hbm.cont.predcorr.predictor 202 | case 'linear' 203 | Zpred = Zprev + pred.step*Tprev; 204 | case 'quadratic' 205 | if length(results) < 5 206 | Zpred = Zprev + pred.step*Tprev; 207 | else 208 | Zpred = polynomial_predictor(Zsol(:,end-2:end),Tsol(:,end-2:end),pred.step); 209 | end 210 | case 'cubic' 211 | if length(results) < 5 212 | Zpred = Zprev + pred.step*Tprev; 213 | else 214 | Zpred = polynomial_predictor(Zsol(:,end-3:end),Tsol(:,end-3:end),pred.step); 215 | end 216 | end 217 | corr.it = 0; 218 | Zlast = Zpred + Inf; 219 | 220 | %corrector 221 | switch hbm.cont.predcorr.corrector 222 | case 'pseudo' 223 | bConverged = 0; 224 | Z = Zpred; 225 | T = Tprev; 226 | 227 | while corr.it <= hbm.cont.predcorr.maxit 228 | Zlast = Z; 229 | J = hbm_bb_jacobian(Z,hbm,problem); 230 | F = hbm_bb_constraints(Z,hbm,problem); 231 | if hbm.cont.predcorr.bMoorePenrose 232 | Z = Zlast - J\F; 233 | else 234 | B = [J; T']; 235 | R = [J*T; 0]; 236 | Q = [F; 0]; 237 | W = T - B\R; 238 | T = normalise(W); 239 | Z = Zlast - B\Q; 240 | end 241 | corr.it = corr.it + 1; 242 | if ~(any(abs(Z - Zlast) > hbm.cont.xtol) || any(abs(F) > hbm.cont.ftol)) 243 | bConverged = 1; 244 | break; 245 | end 246 | end 247 | if hbm.cont.predcorr.bMoorePenrose 248 | T = get_tangent(J); 249 | T = sign(T'*Tprev)*T; 250 | end 251 | case 'arclength' 252 | corr.Zprev = Zprev; 253 | corr.Tprev = Tprev; 254 | corr.step = pred.step; 255 | switch hbm.cont.predcorr.solver 256 | case 'ipopt' 257 | [Z,info] = fipopt('',Zpred,@hbm_arclength_constraints,ipopt_opt,hbm,problem,corr); 258 | corr.it = info.iter; 259 | bConverged = info.status == 0; 260 | case 'fsolve' 261 | [Z,F,status,out] = fsolve(@hbm_arclength_constraints,Zpred,fsolve_opt,hbm,problem,corr); 262 | corr.it = out.iterations + 1; 263 | bConverged = status == 1; 264 | end 265 | J = hbm_bb_jacobian(Z,hbm,problem); 266 | T = get_tangent(J); 267 | T = sign(T'*Tprev)*T; 268 | end 269 | corr.step = norm(Z - Zprev)*sign((Z-Zprev)'*Tprev); 270 | 271 | prog.NStep = prog.NStep + 1; 272 | prog.NIter = prog.NIter + corr.it; 273 | 274 | %prepare for plots 275 | curr(end+1) = hbm_bb_results(Z,T,pred,corr,hbm,problem); 276 | 277 | if bConverged && curr(end).sCorr >= hbm.cont.min_step && curr(end).sCorr <= hbm.cont.max_step && curr(end).A > 0 278 | %success 279 | status = 'S'; 280 | hbm_bb_plot('data',hbm,problem,curr(end)); 281 | 282 | if corr.it <= hbm.cont.num_iter_increase 283 | pred.step = min(curr(end).sPred * hbm.cont.C,hbm.cont.max_step/curr(end).sCorr*curr(end).sPred); 284 | curr(end).flag = 'Success: Increasing step size'; 285 | info = 'Inc'; 286 | elseif corr.it >= hbm.cont.num_iter_reduce 287 | pred.step = max(curr(end).sPred / hbm.cont.C,hbm.cont.min_step/curr(end).sCorr*curr(end).sPred); 288 | curr(end).flag = 'Success: Reducing step size'; 289 | info = 'Red'; 290 | else 291 | curr(end).flag = 'Success'; 292 | info = ''; 293 | end 294 | 295 | %store the data 296 | results(end+1) = curr(end); 297 | 298 | if bUpdateScaling 299 | problem = hbm_scaling(problem,hbm,results(end)); 300 | end 301 | 302 | zsol(:,end+1) = results(end).z; 303 | tsol(:,end+1) = results(end).t; 304 | Zsol = zsol./(repmat(problem.Zscale,1,size(zsol,2))); 305 | Tsol = normalise(tsol./(repmat(problem.Zscale,1,size(tsol,2)))); 306 | 307 | Zend = zEnd./problem.Zscale; 308 | 309 | Zprev = Zsol(:,end); 310 | Tprev = Tsol(:,end); 311 | 312 | prog.NFail = 0; 313 | else 314 | %failed 315 | hbm_bb_plot('err',hbm,problem,curr(end)); 316 | status = 'F'; 317 | 318 | if curr(end).A < 0 319 | %gone to negative frequencies (somehow) 320 | curr(end).flag = 'Failed: Negative frequency'; 321 | info = 'Neg'; 322 | elseif corr.it <= hbm.cont.predcorr.maxit 323 | %converge but step size is unacceptable 324 | if curr(end).sCorr < 0 325 | %backwards 326 | pred.step = max(pred.step * hbm.cont.c,hbm.cont.min_step); 327 | curr(end).flag = 'Failed: Wrong direction'; 328 | info = 'Bwd'; 329 | elseif curr(end).sCorr > hbm.cont.max_step 330 | %too large 331 | pred.step = max(0.9 * pred.step * hbm.cont.max_step / curr(end).sCorr,hbm.cont.min_step); 332 | curr(end).flag = 'Failed: Step too large'; 333 | info = 'Lrg'; 334 | elseif curr(end).sCorr < hbm.cont.min_step 335 | %too small 336 | pred.step = min(1.1 * pred.step * hbm.cont.min_step / curr(end).sCorr,hbm.cont.max_step); 337 | curr(end).flag = 'Failed: Step too small'; 338 | info = 'Sml'; 339 | else 340 | pred.step = max(pred.step * hbm.cont.c,hbm.cont.min_step); 341 | curr(end).flag = 'Failed: Other error'; 342 | info = 'Oth'; 343 | end 344 | else 345 | %failed to converge 346 | if any(abs(Z - Zlast) > hbm.cont.xtol) 347 | curr(end).flag = 'Failed: No convergence'; 348 | info = 'xtl'; 349 | elseif any(abs(F) > hbm.cont.ftol) 350 | curr(end).flag = 'Failed: Constraints violated'; 351 | info = 'ftl'; 352 | else 353 | curr(end).flag = 'Failed'; 354 | info = ''; 355 | end 356 | pred.step = max(pred.step * hbm.cont.c,hbm.cont.min_step); 357 | end 358 | 359 | prog.NFail = prog.NFail + 1; 360 | if prog.NFail > hbm.cont.maxfail 361 | prog.Status = 'Too many failed iterations'; 362 | break 363 | elseif curr(end).A < 0 364 | prog.Status = 'Negative frequency'; 365 | break 366 | end 367 | end 368 | fprintf('%3d %6.4f %6.4f %s %3s %3d %3d %6.2f %6.2f ',prog.NStep,curr(end).sPred,curr(end).sCorr,status,info,corr.it,prog.NIter,curr(end).w,curr(end).A) 369 | fprintf('%+5.2e ',curr(end).z) 370 | fprintf('\n') 371 | end 372 | 373 | %add on final point 374 | pred.step = norm(Zend - Zprev); 375 | corr.step = norm(Zend - Zprev); 376 | corr.it = 0; 377 | curr(end+1) = hbm_bb_results(Zend,Tend,pred,corr,hbm,problem); 378 | results(end+1) = curr(end); 379 | 380 | hbm_bb_plot('close',hbm,problem,[]); 381 | end 382 | 383 | NPts = length(results); 384 | 385 | if ~isfield(results,'W') 386 | for i = 1:NPts 387 | w0 = results(i).w*hbm.harm.rFreqRatio + hbm.harm.wFreq0; 388 | results(i).W = hbm.harm.kHarm*(hbm.harm.rFreqBase.*w0)'; 389 | end 390 | end 391 | 392 | if ~isfield(results,'L') 393 | results = hbm_floquet(hbm,problem,results); 394 | end 395 | 396 | results = hbm_excitation_forces(problem,results); 397 | 398 | %% Predictor 399 | function X_extrap = polynomial_predictor(Z,dZ,s_extrap) 400 | s = norm2(diff(Z,[],2)); 401 | s = cumsum([0 s]); 402 | N = length(s); 403 | if ~isempty(dZ) 404 | [A,Ad] = poly_mat(s,N); 405 | p = [A;Ad]\[Z dZ]'; 406 | else 407 | A = poly_mat(s,N); 408 | p = A\Z'; 409 | end 410 | B = poly_mat(s(end) + s_extrap,N); 411 | X_extrap = (B*p)'; 412 | 413 | function [A,Ad] = poly_mat(s,N) 414 | A = zeros(length(s),N); 415 | Ad = zeros(length(s),N); 416 | for i = 1:N 417 | A(:,i) = s.^(i-1); 418 | if nargout > 1 419 | if i > 1 420 | Ad(:,i) = (i-1)*s.^(i-2); 421 | else 422 | Ad(:,i) = 0*s; 423 | end 424 | end 425 | end 426 | 427 | %% Constraints and Jacobian 428 | function c = hbm_bb_constraints(Z,hbm,problem) 429 | %unpack the inputs 430 | x = Z(1:end-2).*problem.Xscale; 431 | w = Z(end-1).*problem.wscale; 432 | A = Z(end).*problem.Ascale; 433 | 434 | w0 = w * hbm.harm.rFreqRatio + hbm.harm.wFreq0; 435 | 436 | U = A*feval(problem.excite,hbm,problem,w0); 437 | u = packdof(U); 438 | 439 | c = hbm_balance3d('func',hbm,problem,w,u,x); 440 | c(end+1) = resonance_condition(hbm,problem,w,x,A); 441 | c = c ./ problem.Fscale; 442 | 443 | function J = hbm_bb_jacobian(Z,hbm,problem) 444 | 445 | %unpack the inputs 446 | x = Z(1:end-2).*problem.Xscale; 447 | w = Z(end-1).*problem.wscale; 448 | A = Z(end).*problem.Ascale; 449 | 450 | w0 = w * hbm.harm.rFreqRatio + hbm.harm.wFreq0; 451 | 452 | U = A*feval(problem.excite,hbm,problem,w0); 453 | u = packdof(U); 454 | 455 | Jx = hbm_balance3d('jacob' ,hbm,problem,w,u,x); 456 | Dw = hbm_balance3d('derivW',hbm,problem,w,u,x); 457 | Da = hbm_balance3d('derivA',hbm,problem,w,u,x); 458 | 459 | [~,drdx,drdw,drdA] = resonance_condition(hbm,problem,w,x,A); 460 | 461 | J = [Jx Dw Da; 462 | drdx drdw drdA]; 463 | J = J .* problem.Jscale; 464 | 465 | %% Arclength files 466 | function [c,J] = hbm_arclength_constraints(Z,hbm,problem,corr) 467 | c = hbm_bb_constraints(Z,hbm,problem); 468 | 469 | sgn = sign((Z - corr.Zprev)' * corr.Tprev); 470 | s = norm(Z - corr.Zprev) * sgn; 471 | c(end+1) = s - corr.step; 472 | 473 | if nargout > 1 474 | J = hbm_arclength_jacobian(Z,hbm,problem,corr); 475 | end 476 | 477 | function J = hbm_arclength_jacobian(Z,hbm,problem,corr) 478 | J = hbm_bb_jacobian(Z,hbm,problem); 479 | sgn = sign((Z - corr.Zprev)' * corr.Tprev); 480 | J(end+1,:) = sgn*((Z - corr.Zprev)'+eps)/(1*(norm(Z - corr.Zprev)+eps)); 481 | 482 | %% Utilities 483 | function y = norm2(x) 484 | y = sqrt(sum(x.^2,1)); 485 | 486 | function y = normalise(x) 487 | scale = repmat(norm2(x),size(x,1),1); 488 | y = x ./scale; 489 | 490 | function t = get_tangent(J) 491 | [U,S,V] = svd(J,0); 492 | t = V(:,end); 493 | t = t./norm(t); 494 | 495 | function curr = hbm_bb_results(Z,tangent,pred,corr,hbm,problem) 496 | A = Z(end).*problem.Ascale; 497 | w = Z(end-1).*problem.wscale; 498 | x = Z(1:end-2).*problem.Xscale; 499 | t = normalise(tangent.*problem.Zscale); 500 | 501 | curr.z = [x;w;A]; 502 | curr.t = t; 503 | 504 | curr.sCorr = corr.step; 505 | curr.sPred = pred.step; 506 | curr.it = corr.it; 507 | curr.flag = ''; 508 | 509 | w0 = w*hbm.harm.rFreqRatio + hbm.harm.wFreq0; 510 | 511 | curr.w = w; 512 | curr.X = unpackdof(x,hbm.harm.NHarm,problem.NDof); 513 | curr.U = A*feval(problem.excite,hbm,problem,w0); 514 | curr.F = hbm_output3d(hbm,problem,curr.w,curr.U,curr.X); 515 | curr.A = A; 516 | 517 | u = packdof(curr.U); 518 | curr.H = hbm_objective('complex',hbm,problem,w,x,u); 519 | 520 | 521 | function [r,drdx,drdw,drdA] = resonance_condition(hbm,problem,w0,x,A) 522 | U = A*feval(problem.excite,hbm,problem,w0); 523 | u = packdof(U); 524 | 525 | r = hbm_objective({'derivW'},hbm,problem,w0,x,u); 526 | r = -problem.res.sign*r; 527 | 528 | if nargout > 1 529 | h = 1E-10; 530 | x0 = x; 531 | for i = 1:length(x) 532 | x = x0; 533 | x(i) = x(i) + h; 534 | drdx(i) = (resonance_condition(hbm,problem,w0,x,A) - r)/h; 535 | end 536 | 537 | drdw = (resonance_condition(hbm,problem,w0+h,x0,A) - r)/h; 538 | drdA = (resonance_condition(hbm,problem,w0,x0,A+h) - r)/h; 539 | end -------------------------------------------------------------------------------- /Functions/hbm_frf.m: -------------------------------------------------------------------------------- 1 | function [results,curr] = hbm_frf(hbm,problem,A,w0,X0,wEnd,XEnd) 2 | problem.type = 'frf'; 3 | problem.A = A; 4 | 5 | %first solve @ w0 6 | sol = hbm_solve(hbm,problem,w0,A,X0); 7 | x0 = packdof(sol.X); 8 | u0 = packdof(sol.U); 9 | f0 = packdof(sol.F); 10 | if any(isnan(abs(x0(:)))) 11 | results = struct('X',x0,... 12 | 'A',A,... 13 | 'U',u0,... 14 | 'F',f0,... 15 | 'w',w0,... 16 | 'err','Failed to solve initial problem'); 17 | return; 18 | end 19 | z0 = [x0;w0]; 20 | 21 | init.X = sol.X; 22 | init.w = w0; 23 | init.A = A; 24 | 25 | sol = hbm_solve(hbm,problem,wEnd,A,XEnd); 26 | xEnd = packdof(sol.X); 27 | uEnd = packdof(sol.U); 28 | fEnd = packdof(sol.F); 29 | if any(isnan(abs(xEnd(:)))) 30 | results = struct('X',xEnd,... 31 | 'A',A,... 32 | 'U',uEnd,... 33 | 'F',fEnd,... 34 | 'w',wEnd,... 35 | 'err','Failed to solve final problem'); 36 | return; 37 | end 38 | zEnd = [xEnd;wEnd]; 39 | 40 | hbm.bIncludeNL = 1; 41 | 42 | if isfield(problem,'xscale') 43 | xscale = [problem.xscale'; repmat(problem.xscale',hbm.harm.NFreq-1,1)*(1+1i)]; 44 | problem.Xscale = packdof(xscale)*sqrt(length(xscale)); 45 | problem.wscale = mean([w0 wEnd]); 46 | problem.Fscale = problem.Xscale*0+1; 47 | 48 | problem.Zscale = [problem.Xscale; problem.wscale]; 49 | problem.Jscale = (1./problem.Fscale(:))*problem.Zscale(:)'; 50 | bUpdateScaling = 0; 51 | else 52 | problem = hbm_scaling(problem,hbm,init); 53 | bUpdateScaling = 1; 54 | end 55 | 56 | wMax = max(wEnd,w0); 57 | wMin = min(wEnd,w0); 58 | 59 | problem.wMax = wMax; 60 | problem.wMin = wMin; 61 | problem.w0 = w0; 62 | problem.wEnd = wEnd; 63 | 64 | prog.Status = 'success'; 65 | 66 | switch hbm.cont.method 67 | case 'none' 68 | if hbm.options.bPlot 69 | hbm_frf_plot('init',hbm,problem,init); 70 | end 71 | if ~isempty(problem.update) 72 | feval(problem.update, 0); 73 | end 74 | 75 | wscale = mean([w0 wEnd]); 76 | 77 | pred.step = hbm.cont.step0; 78 | direction = sign(wEnd-w0)*wscale; 79 | 80 | problem.Xscale = 0*problem.Xscale + 1; 81 | problem.wscale = 1; 82 | problem.Zscale = 0*problem.Zscale + 1; 83 | 84 | z = z0; 85 | J = hbm_frf_jacobian(z,hbm,problem); 86 | t = get_tangent(J); 87 | 88 | corr.step = 0; 89 | corr.it = 0; 90 | 91 | curr = hbm_frf_results(z,t,pred,corr,hbm,problem); 92 | results = curr; 93 | zprev = z0; 94 | 95 | wsol = w0; 96 | zsol = z0; 97 | while wsol(end) >= wMin && wsol(end) <= wMax 98 | wpred = wsol(end) + pred.step*direction; 99 | if wpred > wMax || wpred < wMin 100 | break; 101 | end 102 | 103 | iPredict = max(length(wsol)-6,1):length(wsol); 104 | if length(iPredict) > 1 105 | zpred = interp1(wsol(iPredict),zsol(:,iPredict)',wpred,'pchip','extrap')'; 106 | else 107 | zpred = zsol(:,end); 108 | zpred(end) = wpred; 109 | end 110 | 111 | %now try to solve 112 | xpred = zpred(1:end-1); 113 | Xpred = unpackdof(xpred,hbm.harm.NFreq-1,problem.NDof); 114 | sol = hbm_solve(hbm,problem,wpred,A,Xpred); 115 | sol.x = packdof(sol.X); 116 | 117 | z = [sol.x; sol.w]; 118 | t = z - zprev; 119 | corr.step = norm2(z - zprev); 120 | 121 | curr(end+1) = hbm_frf_results(z,t,pred,corr,hbm,problem); 122 | 123 | %unpack outputs 124 | if ~any(isnan(curr(end).z)) 125 | curr(end).flag = 'Success'; 126 | pred.step = min(max(pred.step * hbm.cont.C,hbm.cont.min_step),hbm.cont.max_step); 127 | results(end+1) = curr(end); 128 | wsol(end+1) = results(end).w; 129 | zsol(:,end+1) = results(end).z; 130 | if hbm.options.bPlot 131 | hbm_frf_plot('data',hbm,problem,results(end)); 132 | end 133 | 134 | prog.NFail = 0; 135 | if wsol(end) >= wMax || wsol(end) <= wMin 136 | break; 137 | end 138 | zprev = curr(end).z; 139 | 140 | if ~isempty(problem.update) 141 | feval(problem.update, (results(end).w - w0)/(wEnd-w0)); 142 | end 143 | else 144 | curr(end).flag = 'Failed: No convergence'; 145 | pred.step = pred.step * hbm.cont.c; 146 | prog.NFail = prog.NFail + 1; 147 | if hbm.options.bPlot 148 | hbm_frf_plot('err',hbm,problem,curr(end)); 149 | end 150 | if prog.NFail > hbm.cont.maxfail 151 | prog.Status = 'Too many failed iterations'; 152 | break; 153 | end 154 | end 155 | end 156 | 157 | %add on final point 158 | t = zEnd - zprev; 159 | 160 | pred.step = norm(zEnd - zprev); 161 | corr.step = norm(zEnd - zprev); 162 | corr.it = 0; 163 | curr(end+1) = hbm_frf_results(zEnd,t,pred,corr,hbm,problem); 164 | results(end+1) = curr(end); 165 | 166 | if hbm.options.bPlot 167 | hbm_frf_plot('close',hbm,problem,[]); 168 | end 169 | if ~isempty(problem.update) 170 | feval(problem.update, 1); 171 | end 172 | 173 | case 'predcorr' 174 | prog.NStep = 1; 175 | prog.NFail = 0; 176 | prog.NIter = 0; 177 | 178 | switch hbm.cont.predcorr.corrector 179 | case 'pseudo' 180 | case 'arclength' 181 | Jstr = [hbm.sparsity 0*x0+1; 182 | 0*x0'+1 1]; 183 | switch hbm.cont.predcorr.solver 184 | case 'ipopt' 185 | ipopt_opt.print_level = 0; 186 | ipopt_opt.maxit = hbm.cont.predcorr.maxit; 187 | ipopt_opt.ftol = hbm.cont.ftol; 188 | ipopt_opt.xtol = hbm.cont.xtol; 189 | ipopt_opt.jacob = @hbm_arclength_jacobian; 190 | ipopt_opt.jacobstructure = Jstr; 191 | case 'fsolve' 192 | fsolve_opt = optimoptions('fsolve',... 193 | 'Display','off',... 194 | 'FunctionTolerance',hbm.cont.ftol,... 195 | 'StepTolerance',hbm.cont.xtol,... 196 | 'SpecifyObjectiveGradient',true,... 197 | 'MaxIterations',hbm.cont.predcorr.maxit); 198 | end 199 | end 200 | 201 | Zprev = z0./problem.Zscale; 202 | F = hbm_frf_constraints(Zprev,hbm,problem); 203 | J = hbm_frf_jacobian(Zprev,hbm,problem); 204 | Tprev = get_tangent(J); 205 | Tprev = Tprev * sign(Tprev(end)) * sign(wEnd - w0); 206 | 207 | Zend = zEnd./problem.Zscale; 208 | J = hbm_frf_jacobian(Zend,hbm,problem); 209 | Tend = get_tangent(J); 210 | Tend = Tend * sign(Tend(end)) * sign(wEnd - w0); 211 | 212 | pred.step = hbm.cont.step0; 213 | corr.step = pred.step; 214 | corr.it = 0; 215 | 216 | curr = hbm_frf_results(Zprev,Tprev,pred,corr,hbm,problem); 217 | results = curr; 218 | 219 | if hbm.options.bPlot 220 | hbm_frf_plot('init',hbm,problem,init); 221 | end 222 | if ~isempty(problem.update) 223 | feval(problem.update, 0); 224 | end 225 | 226 | if hbm.options.bVerbose 227 | fprintf('STEP PRED CORR STATUS INFO ITER TOT FREQ ') 228 | fprintf('Z(%d) ',1:length(z0)) 229 | fprintf('\n') 230 | fprintf('%3d %6.4f %6.4f %s %3s %3d %3d %6.2f ',prog.NStep,pred.step,corr.step,'S','Ini',corr.it,prog.NIter,results(end).w) 231 | fprintf('%+5.2e ',results(end).z) 232 | fprintf('\n') 233 | end 234 | 235 | zsol = results.z; 236 | tsol = results.t; 237 | 238 | while norm(Zprev - Zend) > hbm.cont.max_step 239 | 240 | %predictor 241 | switch hbm.cont.predcorr.predictor 242 | case 'linear' 243 | Zpred = Zprev + pred.step*Tprev; 244 | case 'quadratic' 245 | if length(results) < 5 246 | Zpred = Zprev + pred.step*Tprev; 247 | else 248 | Zpred = polynomial_predictor(Zsol(:,end-2:end),Tsol(:,end-2:end),pred.step); 249 | end 250 | case 'cubic' 251 | if length(results) < 5 252 | Zpred = Zprev + pred.step*Tprev; 253 | else 254 | Zpred = polynomial_predictor(Zsol(:,end-3:end),Tsol(:,end-3:end),pred.step); 255 | end 256 | end 257 | corr.it = 0; 258 | Zlast = Zpred + Inf; 259 | 260 | %corrector 261 | switch hbm.cont.predcorr.corrector 262 | case 'pseudo' 263 | bConverged = 0; 264 | Z = Zpred; 265 | T = Tprev; 266 | 267 | while corr.it <= hbm.cont.predcorr.maxit 268 | Zlast = Z; 269 | J = hbm_frf_jacobian(Z,hbm,problem); 270 | F = hbm_frf_constraints(Z,hbm,problem); 271 | if hbm.cont.predcorr.bMoorePenrose 272 | Z = Zlast - J\F; 273 | else 274 | B = [J; T']; 275 | R = [J*T; 0]; 276 | Q = [F; 0]; 277 | W = T - B\R; 278 | T = normalise(W); 279 | Z = Zlast - B\Q; 280 | end 281 | corr.it = corr.it + 1; 282 | if ~(any(abs(Z - Zlast) > hbm.cont.xtol) || any(abs(F) > hbm.cont.ftol)) 283 | bConverged = 1; 284 | break; 285 | end 286 | end 287 | if hbm.cont.predcorr.bMoorePenrose 288 | T = get_tangent(J); 289 | T = sign(T'*Tprev)*T; 290 | end 291 | case 'arclength' 292 | corr.Zprev = Zprev; 293 | corr.Tprev = Tprev; 294 | corr.step = pred.step; 295 | switch hbm.cont.predcorr.solver 296 | case 'ipopt' 297 | [Z,info] = fipopt('',Zpred,@hbm_arclength_constraints,ipopt_opt,hbm,problem,corr); 298 | corr.it = info.iter; 299 | bConverged = info.status == 0; 300 | case 'fsolve' 301 | [Z,F,status,out] = fsolve(@hbm_arclength_constraints,Zpred,fsolve_opt,hbm,problem,corr); 302 | corr.it = out.iterations + 1; 303 | bConverged = status == 1; 304 | end 305 | J = hbm_frf_jacobian(Z,hbm,problem); 306 | T = get_tangent(J); 307 | T = sign(T'*Tprev)*T; 308 | end 309 | corr.step = norm(Z - Zprev)*sign((Z-Zprev)'*Tprev); 310 | 311 | prog.NStep = prog.NStep + 1; 312 | prog.NIter = prog.NIter + corr.it; 313 | 314 | %prepare for plots 315 | curr(end+1) = hbm_frf_results(Z,T,pred,corr,hbm,problem); 316 | 317 | if bConverged && curr(end).sCorr >= hbm.cont.min_step && curr(end).sCorr <= hbm.cont.max_step && curr(end).w > 0 318 | %success 319 | status = 'S'; 320 | if hbm.options.bPlot 321 | hbm_frf_plot('data',hbm,problem,curr(end)); 322 | end 323 | 324 | if corr.it <= hbm.cont.num_iter_increase 325 | pred.step = min(curr(end).sPred * hbm.cont.C,hbm.cont.max_step/curr(end).sCorr*curr(end).sPred); 326 | curr(end).flag = 'Success: Increasing step size'; 327 | info = 'Inc'; 328 | elseif corr.it >= hbm.cont.num_iter_reduce 329 | pred.step = max(curr(end).sPred / hbm.cont.C,hbm.cont.min_step/curr(end).sCorr*curr(end).sPred); 330 | curr(end).flag = 'Success: Reducing step size'; 331 | info = 'Red'; 332 | else 333 | curr(end).flag = 'Success'; 334 | info = ''; 335 | end 336 | 337 | %store the data 338 | results(end+1) = curr(end); 339 | 340 | if bUpdateScaling 341 | problem = hbm_scaling(problem,hbm,results(end)); 342 | end 343 | 344 | zsol(:,end+1) = results(end).z; 345 | tsol(:,end+1) = results(end).t; 346 | Zsol = zsol./(repmat(problem.Zscale,1,size(zsol,2))); 347 | Tsol = normalise(tsol./(repmat(problem.Zscale,1,size(tsol,2)))); 348 | 349 | Zend = zEnd./problem.Zscale; 350 | 351 | Zprev = Zsol(:,end); 352 | Tprev = Tsol(:,end); 353 | 354 | prog.NFail = 0; 355 | 356 | if ~isempty(problem.update) 357 | feval(problem.update, (results(end).w - w0)/(wEnd-w0)); 358 | end 359 | else 360 | %failed 361 | if hbm.options.bPlot 362 | hbm_frf_plot('err',hbm,problem,curr(end)); 363 | end 364 | status = 'F'; 365 | 366 | if curr(end).w < 0 367 | %gone to negative frequencies (somehow) 368 | curr(end).flag = 'Failed: Negative frequency'; 369 | info = 'Neg'; 370 | elseif corr.it <= hbm.cont.predcorr.maxit 371 | %converge but step size is unacceptable 372 | if curr(end).sCorr < 0 373 | %backwards 374 | pred.step = max(pred.step * hbm.cont.c,hbm.cont.min_step); 375 | curr(end).flag = 'Failed: Wrong direction'; 376 | info = 'Bwd'; 377 | elseif curr(end).sCorr > hbm.cont.max_step 378 | %too large 379 | pred.step = max(0.9 * pred.step * hbm.cont.max_step / curr(end).sCorr,hbm.cont.min_step); 380 | curr(end).flag = 'Failed: Step too large'; 381 | info = 'Lrg'; 382 | elseif curr(end).sCorr < hbm.cont.min_step 383 | %too small 384 | pred.step = min(1.1 * pred.step * hbm.cont.min_step / curr(end).sCorr,hbm.cont.max_step); 385 | curr(end).flag = 'Failed: Step too small'; 386 | info = 'Sml'; 387 | else 388 | pred.step = max(pred.step * hbm.cont.c,hbm.cont.min_step); 389 | curr(end).flag = 'Failed: Other error'; 390 | info = 'Oth'; 391 | end 392 | else 393 | %failed to converge 394 | if any(abs(Z - Zlast) > hbm.cont.xtol) 395 | curr(end).flag = 'Failed: No convergence'; 396 | info = 'xtl'; 397 | elseif any(abs(F) > hbm.cont.ftol) 398 | curr(end).flag = 'Failed: Constraints violated'; 399 | info = 'ftl'; 400 | else 401 | curr(end).flag = 'Failed'; 402 | info = ''; 403 | end 404 | pred.step = max(pred.step * hbm.cont.c,hbm.cont.min_step); 405 | end 406 | 407 | prog.NFail = prog.NFail + 1; 408 | if prog.NFail > hbm.cont.maxfail 409 | prog.Status = 'Too many failed iterations'; 410 | break 411 | elseif curr(end).w < 0 412 | prog.Status = 'Negative frequency'; 413 | break 414 | end 415 | end 416 | 417 | if hbm.options.bVerbose 418 | fprintf('%3d %6.4f %6.4f %s %3s %3d %3d %6.2f ',prog.NStep,curr(end).sPred,curr(end).sCorr,status,info,corr.it,prog.NIter,curr(end).w) 419 | fprintf('%+5.2e ',curr(end).z) 420 | fprintf('\n') 421 | end 422 | end 423 | 424 | %add on final point 425 | pred.step = norm(Zend - Zprev); 426 | corr.step = norm(Zend - Zprev); 427 | corr.it = 0; 428 | curr(end+1) = hbm_frf_results(Zend,Tend,pred,corr,hbm,problem); 429 | results(end+1) = curr(end); 430 | 431 | if hbm.options.bPlot 432 | hbm_frf_plot('close',hbm,problem,[]); 433 | end 434 | if ~isempty(problem.update) 435 | feval(problem.update, 1); 436 | end 437 | end 438 | 439 | NPts = length(results); 440 | 441 | if ~isfield(results,'W') 442 | for i = 1:NPts 443 | w0 = results(i).w*hbm.harm.rFreqRatio + hbm.harm.wFreq0; 444 | results(i).W = hbm.harm.kHarm*(hbm.harm.rFreqBase.*w0)'; 445 | end 446 | end 447 | 448 | if ~isfield(results,'L') 449 | results = hbm_floquet(hbm,problem,results); 450 | end 451 | 452 | results = hbm_excitation_forces(problem,results); 453 | 454 | %% Predictor 455 | function X_extrap = polynomial_predictor(Z,dZ,s_extrap) 456 | s = norm2(diff(Z,[],2)); 457 | s = cumsum([0 s]); 458 | N = length(s); 459 | if ~isempty(dZ) 460 | [A,Ad] = poly_mat(s,N); 461 | p = [A;Ad]\[Z dZ]'; 462 | else 463 | A = poly_mat(s,N); 464 | p = A\Z'; 465 | end 466 | B = poly_mat(s(end) + s_extrap,N); 467 | X_extrap = (B*p)'; 468 | 469 | function [A,Ad] = poly_mat(s,N) 470 | A = zeros(length(s),N); 471 | Ad = zeros(length(s),N); 472 | for i = 1:N 473 | A(:,i) = s.^(i-1); 474 | if nargout > 1 475 | if i > 1 476 | Ad(:,i) = (i-1)*s.^(i-2); 477 | else 478 | Ad(:,i) = 0*s; 479 | end 480 | end 481 | end 482 | 483 | %% Constraints and Jacobian 484 | function c = hbm_frf_constraints(Z,hbm,problem) 485 | %unpack the inputs 486 | x = Z(1:end-1).*problem.Xscale; 487 | w = Z(end).*problem.wscale; 488 | A = problem.A; 489 | 490 | w0 = w * hbm.harm.rFreqRatio + hbm.harm.wFreq0; 491 | 492 | U = A*feval(problem.excite,hbm,problem,w0); 493 | u = packdof(U); 494 | 495 | c = hbm_balance3d('func',hbm,problem,w,u,x); 496 | c = c ./ problem.Fscale; 497 | 498 | function J = hbm_frf_jacobian(Z,hbm,problem) 499 | %unpack the inputs 500 | x = Z(1:end-1).*problem.Xscale; 501 | w = Z(end).*problem.wscale; 502 | A = problem.A; 503 | 504 | w0 = w * hbm.harm.rFreqRatio + hbm.harm.wFreq0; 505 | 506 | U = A*feval(problem.excite,hbm,problem,w0); 507 | u = packdof(U); 508 | 509 | Jx = hbm_balance3d('jacob' ,hbm,problem,w,u,x); 510 | Dw = hbm_balance3d('derivW',hbm,problem,w,u,x); 511 | 512 | J = [Jx Dw]; 513 | J = J .* problem.Jscale; 514 | 515 | %% Arclength files 516 | function [c,J] = hbm_arclength_constraints(Z,hbm,problem,corr) 517 | c = hbm_frf_constraints(Z,hbm,problem); 518 | sgn = sign((Z - corr.Zprev)' * corr.Tprev); 519 | s = norm(Z - corr.Zprev) * sgn; 520 | c(end+1) = s - corr.step; 521 | if nargout > 1 522 | J = hbm_arclength_jacobian(Z,hbm,problem,corr); 523 | end 524 | 525 | function J = hbm_arclength_jacobian(Z,hbm,problem,corr) 526 | J = hbm_frf_jacobian(Z,hbm,problem); 527 | sgn = sign((Z - corr.Zprev)' * corr.Tprev); 528 | J(end+1,:) = sgn*((Z - corr.Zprev)'+eps)/(1*(norm(Z - corr.Zprev)+eps)); 529 | 530 | %% Utilities 531 | function y = norm2(x) 532 | y = sqrt(sum(x.^2,1)); 533 | 534 | function y = normalise(x) 535 | scale = repmat(norm2(x),size(x,1),1); 536 | y = x ./scale; 537 | 538 | function t = get_tangent(J) 539 | [U,S,V] = svd(J,0); 540 | t = V(:,end); 541 | t = t./norm(t); 542 | 543 | function curr = hbm_frf_results(Z,tangent,pred,corr,hbm,problem) 544 | w = Z(end).*problem.wscale; 545 | x = Z(1:end-1).*problem.Xscale; 546 | X = unpackdof(x,hbm.harm.NHarm,problem.NDof); 547 | A = problem.A; 548 | t = normalise(tangent.*problem.Zscale); 549 | 550 | curr.z = [x; w]; 551 | curr.t = t; 552 | 553 | curr.sCorr = corr.step; 554 | curr.sPred = pred.step; 555 | curr.it = corr.it; 556 | curr.flag = ''; 557 | 558 | w0 = w*hbm.harm.rFreqRatio + hbm.harm.wFreq0; 559 | 560 | curr.w = w; 561 | curr.U = A*feval(problem.excite,hbm,problem,w0); 562 | curr.X = X; 563 | curr.F = hbm_output3d(hbm,problem,curr.w,curr.U,curr.X); 564 | curr.A = A; -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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"Knowingly relying" means you have 508 | actual knowledge that, but for the patent license, your conveying the 509 | covered work in a country, or your recipient's use of the covered work 510 | in a country, would infringe one or more identifiable patents in that 511 | country that you have reason to believe are valid. 512 | 513 | If, pursuant to or in connection with a single transaction or 514 | arrangement, you convey, or propagate by procuring conveyance of, a 515 | covered work, and grant a patent license to some of the parties 516 | receiving the covered work authorizing them to use, propagate, modify 517 | or convey a specific copy of the covered work, then the patent license 518 | you grant is automatically extended to all recipients of the covered 519 | work and works based on it. 520 | 521 | A patent license is "discriminatory" if it does not include within 522 | the scope of its coverage, prohibits the exercise of, or is 523 | conditioned on the non-exercise of one or more of the rights that are 524 | specifically granted under this License. You may not convey a covered 525 | work if you are a party to an arrangement with a third party that is 526 | in the business of distributing software, under which you make payment 527 | to the third party based on the extent of your activity of conveying 528 | the work, and under which the third party grants, to any of the 529 | parties who would receive the covered work from you, a discriminatory 530 | patent license (a) in connection with copies of the covered work 531 | conveyed by you (or copies made from those copies), or (b) primarily 532 | for and in connection with specific products or compilations that 533 | contain the covered work, unless you entered into that arrangement, 534 | or that patent license was granted, prior to 28 March 2007. 535 | 536 | Nothing in this License shall be construed as excluding or limiting 537 | any implied license or other defenses to infringement that may 538 | otherwise be available to you under applicable patent law. 539 | 540 | 12. No Surrender of Others' Freedom. 541 | 542 | If conditions are imposed on you (whether by court order, agreement or 543 | otherwise) that contradict the conditions of this License, they do not 544 | excuse you from the conditions of this License. If you cannot convey a 545 | covered work so as to satisfy simultaneously your obligations under this 546 | License and any other pertinent obligations, then as a consequence you may 547 | not convey it at all. For example, if you agree to terms that obligate you 548 | to collect a royalty for further conveying from those to whom you convey 549 | the Program, the only way you could satisfy both those terms and this 550 | License would be to refrain entirely from conveying the Program. 551 | 552 | 13. Use with the GNU Affero General Public License. 553 | 554 | Notwithstanding any other provision of this License, you have 555 | permission to link or combine any covered work with a work licensed 556 | under version 3 of the GNU Affero General Public License into a single 557 | combined work, and to convey the resulting work. The terms of this 558 | License will continue to apply to the part which is the covered work, 559 | but the special requirements of the GNU Affero General Public License, 560 | section 13, concerning interaction through a network will apply to the 561 | combination as such. 562 | 563 | 14. Revised Versions of this License. 564 | 565 | The Free Software Foundation may publish revised and/or new versions of 566 | the GNU General Public License from time to time. Such new versions will 567 | be similar in spirit to the present version, but may differ in detail to 568 | address new problems or concerns. 569 | 570 | Each version is given a distinguishing version number. If the 571 | Program specifies that a certain numbered version of the GNU General 572 | Public License "or any later version" applies to it, you have the 573 | option of following the terms and conditions either of that numbered 574 | version or of any later version published by the Free Software 575 | Foundation. If the Program does not specify a version number of the 576 | GNU General Public License, you may choose any version ever published 577 | by the Free Software Foundation. 578 | 579 | If the Program specifies that a proxy can decide which future 580 | versions of the GNU General Public License can be used, that proxy's 581 | public statement of acceptance of a version permanently authorizes you 582 | to choose that version for the Program. 583 | 584 | Later license versions may give you additional or different 585 | permissions. However, no additional obligations are imposed on any 586 | author or copyright holder as a result of your choosing to follow a 587 | later version. 588 | 589 | 15. Disclaimer of Warranty. 590 | 591 | THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY 592 | APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT 593 | HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY 594 | OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, 595 | THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 596 | PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM 597 | IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF 598 | ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 599 | 600 | 16. Limitation of Liability. 601 | 602 | IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING 603 | WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS 604 | THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY 605 | GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE 606 | USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF 607 | DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD 608 | PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), 609 | EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF 610 | SUCH DAMAGES. 611 | 612 | 17. Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | --------------------------------------------------------------------------------