Learning Forum

[biao.zhou]

Hi, everybody.

I want to construct the mass-normalized complex-valued cyclic modes based on the real-valued cyclic modes extracted from cyclic sysmetric modal analysis. But I have a problem about the mass-normalization of the complex cyclic modes.

 

 Below is the equation I used for expanding the reference sector mode (in complex
 cyclic coordinates)
% 
%   Ux(n) =  e^((n-1)k*theta) + (Uxa + i*Uxb)
%   Uy(n) =  e^((n-1)k*theta) + (Uya + i*Uyb)
%   Uz(n) =  e^((n-1)k*theta) + (Uza + i*Uzb)
% 
% Where:
%       Uxa: the value from base sector in Ansys
%       Uxb: the value from duplicate sector in Ansys
%       k: is the Harmonic Index
%       theta: is the sector angle
%       n = sector number(1 to N total number of sectors)
% 
 Afterwards, in order to normalize the modes with repect to the mass matrix of the full structure,  Ux(n),Uy(n),Uz(n) for are divided by: 
% 
%       sqrt(N) if k = 0 or k = N/2,
%       sqrt(N/2) : for all other Harmonic Indices
% 
At this point, these Ux,Uy,Uz are in the complex form.

It has been verified that either the real part or the imaginary part of U = [Ux;Uy;Uz] is normalized  with repect to the mass matrix of the full structure. 

 

However, these complex-valued cyclic modes gives the following product:

U^H*M*U = [2,  2i; -2i, 2 ] (only for harmonic index with double modes. For harmonic index 0 with a single mode,  the results are correct. See the image below).

 

So my problem is how to get the complex-valued cyclic modes normalized to the mass matrix of the full structure?

Any comments and help will be highly appreciated.

[biao.zhou]

Problem solved.