Is there a way to find the Eigenvectors and Eigenvalues when there is unknown values in a complex damping matrix , using theoretical methods ?
Is it also possible to be done in MATLAB ?
Hi Farzad
I assume you are referring to FE of a bounded region?
There are two kinds of modes that can be computed, Real modes and Complex modes. The former is standing waves that conserve energy, i.e. with node position (=minimum response) that stand still. The latter uses damping information and describes energy transport across the system and, hence, has node positions that move.
As indicated by their names, real modes have real valued response while complex modes have complex values response.
Real modes can be computed without any damping information.
As indicated by Giuseppe Pennisi damping can be added afterwards with the tacit assumption that it is evenly distributed and light. Other common damping models are viscous modal damping and hysteretic material damping.
The former (viscous) is a weak approximation of sound radiation, the latter (hysteretic) a weak appoximation of internal material friction. The Raleigh damping model mentioned above is used for mathematical convenience and does not originate from any physical damping mechanism.
In real life, damping can be many things. Some ramblings of mine on this topic can be found here https://qringtech.com/2014/06/22/designed-damping-types-mechanisms-application-limitation/
Sincerely
Claes
Hello Farzad,
If I correctly understand your question, you do not need the damping matrix in order to calculate the eigenvalues and the eigenvectors of the system, i.e.structural natural frequencies and structural natural modes are slightly affected by the damping (you can look into the single DOF oscillator case). Usually, when the damping matrix is present, methods are used to split it in two parts proportional to the mass and stiffness matrices respectively (C=alpha*M+beta*K).
Hope this helps.
Best,
GP
Hi Farzad
I assume you are referring to FE of a bounded region?
There are two kinds of modes that can be computed, Real modes and Complex modes. The former is standing waves that conserve energy, i.e. with node position (=minimum response) that stand still. The latter uses damping information and describes energy transport across the system and, hence, has node positions that move.
As indicated by their names, real modes have real valued response while complex modes have complex values response.
Real modes can be computed without any damping information.
As indicated by Giuseppe Pennisi damping can be added afterwards with the tacit assumption that it is evenly distributed and light. Other common damping models are viscous modal damping and hysteretic material damping.
The former (viscous) is a weak approximation of sound radiation, the latter (hysteretic) a weak appoximation of internal material friction. The Raleigh damping model mentioned above is used for mathematical convenience and does not originate from any physical damping mechanism.
In real life, damping can be many things. Some ramblings of mine on this topic can be found here https://qringtech.com/2014/06/22/designed-damping-types-mechanisms-application-limitation/
Sincerely
Claes
Hi Farzad Torabi
as already suggested by Dr Claes Richard Fredö , if you are dealing with complex modes you can review the methods adopted for example when studying visco-elastic material or in general look for nonclassical damped system. There are numerical methods you can certainly implement in MatLab to handle this class of problem. Look for example to "Classical Damping, Non-Classical Damping and Complex Modes" by Henri P. Gavin, or refer to " Eigenvalue and eigenvector determination of non-classically damped dynamic systems " by Cronin1990, "Complex-damped dynamic systems in the time and frequency domains" Bonisoli and Mottershead 2004. You may want to read also "A tutorial on complex eigenvalues" by Lallement and Inman just to mention few works.
As far as the unkown coefficients of the damping matrix, I think you can proceed in two ways: 1) guess suitable damping values based on previous experiences or equivalently look at other data published for similar material/ problem; 2) if you have experimental data then you can fit the latter with your numerical prediction and tune the damping matrix coefficients to match the experimental response.
Regards,
Francesco
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