Signal separation of displacement multimodal superposition in structural dynamics

clc; clear;

% 参数定义

L = 10;

v = 1;

A1 = 1;

A2 = 0.1;

x0 = L / 3;

% 时间轴

t = linspace(0, 10, 1000);

U_t = A1 * sin(pi * v * t / L) * sin(pi * x0 / L) + ...

A2 * sin(2 * pi * v * t / L) * sin(2 * pi * x0 / L);

figure;

plot(t, U_t, 'r-', 'LineWidth', 2);

xlabel('时间 t (s)');

ylabel('位移 U(t)');

grid on;

For such signals, is there any method to separate these two signals from the perspective of data?

Answer 1

[Mathieu NOE]

hello

you could do that :

clc; clear;
L = 10;         
v = 1;              
A1 = 1;             
A2 = 0.1;           
x0 = L / 3;     
t = linspace(0, 10, 1000);  
u1 = A1 * sin(pi * v * t / L) * sin(pi * x0 / L);
u2 = A2 * sin(2 * pi * v * t / L) * sin(2 * pi * x0 / L);

U_t = u1 + u2;

% extraction based on cos / sin projection 
%  u2 = 1 period signal 
omega = 2 * pi * v / L;
u2c = 2*sum(u2.*cos(omega * t ))/numel(t);
u2s = 2*sum(u2.*sin(omega * t ))/numel(t);
u2_estimated = u2c*cos(omega * t ) + u2s*sin(omega * t );

u1_estimated = U_t - u2_estimated;

% plot the estimated signals with a downsampling factor of r
r = 25;
ind = 1:r:numel(t);


figure;
plot(t, U_t, 'r-', 'LineWidth', 2);
hold on 

plot(t, u1, 'b-', 'LineWidth', 1);
plot(t(ind), u1_estimated(ind), 'b*', 'LineWidth', 2);

plot(t, u2, 'c-', 'LineWidth', 1);
plot(t(ind), u2_estimated(ind), 'c*', 'LineWidth', 2);

xlabel('t (s)');
ylabel('U(t)');
grid on;

legend ('U_t','u1','u1 estimated','u2','u2 estimated');

●

[华]

Thank you for your reply. Suppose I only have a one-dimensional superimposed signal with an unknown expression and an unknown function form. Is there any method to separate such a signal?

[Mathieu NOE]

you could use a fft approach but you need to have a longer signal - here the lowest frequency has only half a period

in practice you would need probably a record long enough so that the lowest frequency has a couple of periods

[华]

You're absolutely right, that's indeed the case. However, if it's impossible to obtain signals over multiple periods, is there any method for separation? I've thought about it too, but precisely because of the particularity of the signal, I haven't found a good method.

[Mathieu NOE]

maybe we have a solution if you can ensure that the signal has exactly a multiple of half periods (like here we have half + one period)

if you have a time duration which is random, except the fft I don't see how you could do this work

BTW, have you looked at EMD :

emd - Empirical mode decomposition - MATLAB

Empirical Mode Decomposition - File Exchange - MATLAB Central

GitHub - macolominas/CEEMDAN: A MATLAB package for CEEMDAN (Complete Ensemble Empirical Mode Decomposition with Adaptive Noise)

(haven't tried yet )

there is also a 2D version :

Bidimensional Multivariate Empirical Mode Decomposition - File Exchange - MATLAB Central

[华]

This work is actually about modal separation in structural dynamics. I have tried many signal separation methods, including EMD, but failed to achieve separation. EEMD seems to have some effect, but it is not very ideal and is affected by parameters. Therefore, I want to find a perfect solution.

[Mathieu NOE]

maybe this could help ?

Least action criteria for blind separation of structural modes | Mechanics & Industry

[华]

Thank you for providing the paper. I have carefully reviewed the literature, but one difference is that my data is only in the form of a one-dimensional vector rather than a matrix, so it cannot be decomposed in the same way as a matrix.

[Mathieu NOE]

hello again

can we assume that start and end points have always y = 0 value ? (fixed or pinned end points ?)

[华]

Yes, both ends are fixed. This refers to the mode superposition of a simply supported beam in dynamics. How to decouple it?