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⛄ 内容介绍
In the context of finite element model updating using output-only vibration test data, natural frequencies and mode shapes are used as validation criteria. Consequently, the correct pairing of experimentally obtained and numerically derived natural frequencies and mode shapes is important. In many cases, only limited spatial information is available and noise is present in the measurements. Therefore, the automatic selection of the most likely numerical mode shape corresponding to a particular experimentally identified mode shape can be a difficult task. The most common criterion for indicating corresponding mode shapes is the modal assurance criterion. Unfortunately, this criterion fails in certain cases and is not reliable for automatic approaches. In this paper, the purely mathematical modal assurance criterion will be enhanced by additional physical information from the numerical model in terms of modal strain energies. A numerical example and a benchmark study with experimental data are presented to show the advantages of the proposed energy-based criterion in comparison to the traditional modal assurance criterion.
⛄ 完整代码
%% Clustered Energy Based Modal Assurance Criterion (MAC)
%% Reference
% Brehm, M., Zabel, V., & Bucher, C. (2010). An automatic mode pairing
% strategy using an enhanced modal assurance criterion based on modal
% strain energies. Journal of Sound and Vibration, 329(25), 5375-5392.
%
% <https://doi.org/10.1016/j.jsv.2010.07.006>
%% Description
% The above reference proposes a mode pairing strategy using an enhanced
% modal assurance criterion based on modal strain energies. The procedure
% analyzed in section 3 (Mode assignment using energy-based modal assurance
% criterion) of this reference is programmed in this submission. This
% verification script verifies a specific result of section 4 (Benchmark
% study: cantilever truss), i.e. the fraction of the total modal strain
% energy of numerical eigenmode 1 that corresponds to vertical degrees of
% freedom, shown in Figure 7.
%% Structural eigenvalue analysis
% Load stiffness matrix
load('K.mat','K')
%%
% Load stiffness matrix
load('M.mat','M')
%%
% Eigenvalue analysis
[V,D]=eig(K,M);
%%
% Eigenfrequencies (cycles/time)
f=sqrt(diag(D))/(2*pi);
f(1)
%%
% Check equation (6)
M6=V'*M*V;
figure()
imagesc(M6)
%%
% Check equation (7)
M7=V'*K*V;
figure()
imagesc(M7)
%% Modal assurance criterion
% Apply equation (1)
mac=zeros(size(V,2));
for i=1:size(V,2)
for j=1:size(V,2)
mac(i,j)=(abs(V(:,i)'*V(:,j)))^2/((V(:,i)'*V(:,i))*(V(:,j)'*V(:,j)));
end
end
%%
% Define clustering (horizontal and vertical DOFs)
cl={1:2:20,2:2:20};
%%
% Define eigenmode ID
j=1;
%%
% Define cluster ID
k=2;
%%
% Apply equation (10)
MSEjk=0;
for l=1:numel(cl)
MSEjk=MSEjk+1/2*V(cl{k},j)'*K(cl{k},cl{l})*V(cl{l},j);
end
%%
% Apply equation (11)
MSEj=0;
for k=1:numel(cl)
for l=1:numel(cl)
MSEj=MSEj+1/2*V(cl{k},j)'*K(cl{k},cl{l})*V(cl{l},j);
end
end
%%
% Apply equation (12). Compare the result with the relative modal strain
% energy for vertical degrees of freedom of the 1st numerical mode of
% Figure 7 of the above reference.
PI_jk=MSEjk/MSEj
⛄ 运行结果
⛄ 参考文献
Brehm, M., Zabel, V., & Bucher, C. (2010). An automatic mode pairing strategy using an enhanced modal assurance criterion based on modal strain energies. Journal of Sound and Vibration, 329(25), 5375-5392.
