Hi all,
I was simulating eigen frequencies of an arbitratry 3D object in COMSOL Multiphysics. I obtained many frequency values including pure imaginary, real and complex frequencies.
1) I don't understand the meaning of pure imaginary and complex frequencies. Imaginary part are meant to denote decaying component. But decay is in time domain right? Is it supposed to show up in eigen frequency analysis?
2) Some of the real frequency values are close to the theoretical estimations but not exact. I presume it maybe due to theoretical assumptions or imperfections in meshing. Additionally, it is not just one single value for an expected vibrational mode, rather multiple values. Would anybody help me to understand these obtained results? Or are these wrong results?
Dear all
This is great fun. Here is what I have been taught.
Real modes are standing waves, i.e. waves with zero net flux of energy across the system. Real modes are good approximations when damping is light and evenly distributed and when edge effects do not absorb energy. A real mode can be thought of as a storage of vibration energy with zero leakage (damping). A real mode can be visualized as a frozen pattern, i.e. be printed on paper.
For real modes, damping is added to the response summation as an afterthought and we get away with this only when damping is light.
Complex modes are modes in which there is a net flux of energy from one part of the system to another part of the system. You find complex modes, e.g. when coupling (acoustic) fluid/structure, when you locate a strong damper (e.g. a dashpot) somwehere in the system and whenever damping is significant (high) as you then have spatial decay. A complex mode cannot be shown as a fixed pattern and must be animated.
Splitting the complex mode into real and imaginary parts, the real part shows the standing wave pattern while the imaginary part shows the energy flux (or vice versa depending on how you choose to normalize).
Perhaps somewhat confusing. Substructuring using local real modes for parts A and B can form complex modes for the joined system A+B (as is the case in acoustic/structure models).
Complex modes can be inferred when vibration couples to transport processes such as flow or rotation. A complex which picks up energy from the transpoort process then has negative damping, e.g. as is the case for things like brake squeal, flutter, flute type of sound etc.
Modes appear in pairs as the standing wave pattern forms from at least two directions. This is easily understood in rotating systems, e.g. the acoustic mode inside a tyre where the air rotates with the tyre. One wave travels with the rotation and the other against rotation and the single acoustic mode (at standstill) splits into a pair of modes. Damping has a similar effect as the wave pattern can bounce and form a mode from different directions.
Last. Complex modes are referred to as an end product, i.e. they cannot as readily be summed as can real modes.
On a more practical note. A FE software vendor once referred solving for complex modes to be 'a hobby for gentlemen'. I have to agree as in some FE systems, running the same input data deck twice produces slightly different results and as tweaking the frequency range definitely modifies results. Complex indeed.
I agree with Nils & the rest of you guys. No damping & solving for complex modes can be done but the modes should have a very tiny real part when your model is correct.
Hope this helps
Claes
Did you consider some kind of damping, e.g. modal damping, viscous damping, material damping in your finite element model ?
If not, the eigenfrequencies should be real due to the characteristics of your system matrices.
The stiffness and mass matrix should be symmetric and positive (semi-)definite.
If you have used the Ansatz \exp{\lambda t} a pure imaginary frequency is an oscillating mode. A positive real part means an unstable mode; a negative real part is related to a stable mode.
No, I did not consider any damping. I just run the direct eigen frequency study of the object in a free surrounding. The only input I had to give was the elasticity matrix for the anisotropic solid model, which is symmetric, positive definite and real.
For every complex frequencies, I get the conjugate pair (not for the pure imaginary).
In fact, Nils is right. However, if you do not have damping in your model and you get complex pairs of eigenvalues, then the system is solving the eigenvalue in the state-space form. But, in this case (no damping included), the real part will be vanishingly small (technically zero). You may need to check the real part.
Something must be wrong with the model (I suppose). Without damping you must to have only pairs of imaginary parts.
What is the physical interpretation of imaginary parts appearing in pairs?
Dear all
This is great fun. Here is what I have been taught.
Real modes are standing waves, i.e. waves with zero net flux of energy across the system. Real modes are good approximations when damping is light and evenly distributed and when edge effects do not absorb energy. A real mode can be thought of as a storage of vibration energy with zero leakage (damping). A real mode can be visualized as a frozen pattern, i.e. be printed on paper.
For real modes, damping is added to the response summation as an afterthought and we get away with this only when damping is light.
Complex modes are modes in which there is a net flux of energy from one part of the system to another part of the system. You find complex modes, e.g. when coupling (acoustic) fluid/structure, when you locate a strong damper (e.g. a dashpot) somwehere in the system and whenever damping is significant (high) as you then have spatial decay. A complex mode cannot be shown as a fixed pattern and must be animated.
Splitting the complex mode into real and imaginary parts, the real part shows the standing wave pattern while the imaginary part shows the energy flux (or vice versa depending on how you choose to normalize).
Perhaps somewhat confusing. Substructuring using local real modes for parts A and B can form complex modes for the joined system A+B (as is the case in acoustic/structure models).
Complex modes can be inferred when vibration couples to transport processes such as flow or rotation. A complex which picks up energy from the transpoort process then has negative damping, e.g. as is the case for things like brake squeal, flutter, flute type of sound etc.
Modes appear in pairs as the standing wave pattern forms from at least two directions. This is easily understood in rotating systems, e.g. the acoustic mode inside a tyre where the air rotates with the tyre. One wave travels with the rotation and the other against rotation and the single acoustic mode (at standstill) splits into a pair of modes. Damping has a similar effect as the wave pattern can bounce and form a mode from different directions.
Last. Complex modes are referred to as an end product, i.e. they cannot as readily be summed as can real modes.
On a more practical note. A FE software vendor once referred solving for complex modes to be 'a hobby for gentlemen'. I have to agree as in some FE systems, running the same input data deck twice produces slightly different results and as tweaking the frequency range definitely modifies results. Complex indeed.
I agree with Nils & the rest of you guys. No damping & solving for complex modes can be done but the modes should have a very tiny real part when your model is correct.
Hope this helps
Claes
Thank you all,
It was of really great help and got many things clarified. I guess I can proceed with my study.
The following paper might be very useful.
http://www.maths.manchester.ac.uk/~higham/papers/hmtg08.pdf
Dear Thomas,
Please, check if your model exhibits the first 6 zero modes (gross motion of the body).
These modes should appear at 0 Hz, but due to numerical approximation, some of them can be complex.
If there is no damping in your model, this could be a reason for having complex modes.
You just need to neglect them or apply appropriate boundary conditions to remove body gross motion.
Do you have fluid flow in your problem? That sort of problem would be the main use for COMSOL multiphysics, ie, to combine a structural problem with a fluid problem, and in that case the eigenvalue is always going to be complex, with the real part being the damping and the imaginary part being the natural frequency, both a function of flow speed.
A positive definite matrix will yield real non zero frequencies (i.e. a real and stable structure)
Complex frequencies imply some damping (in the time domain – yes) of the associated real part (frequency)
Multiple values will be caused by system symmetries or numerical computation errors caused by the eigen solver (e.g. Lanczos method w/out vector reorthognalizations)
The real part of eigenvalues gives an index (or measure) of the rate of decay (or divergence) of the mode of interest.
The imaginary part can give an index (or measure) of the rate of oscillation of the dynamic analysis if it is nonzero. These complex eigenvalues have become an integral part of many engineering applications, including they can be used as an index for measuring the effect of damping on system dynamic analysis or determining the stability of the system when it contains sources of energy such as fluid, sliding contact and rotating components. In the case of nonclassically damped systems, the eigenvectors cannot be normalized to be real vectors and therefore these mode shapes contain complex numbers.
The relative magnitudes of complex mode shapes indicate which parts of the structural system are the most active in the corresponding The imaginary parts of complex mode shapes show the lag time or phase differences between the DOFs of structural systems.
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