General Mechanical

General Mechanical

Normalization of Complex-valued Cyclic Modes

    • 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.

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