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2D harmonic axisymmetric eigenfrequency study in solid mechanics - why complex mode shapes?
Posted 14 avr. 2021, 13:39 UTC−4 Structural & Acoustics, Acoustics & Vibrations Version 5.6 5 Replies
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Dear Comsol Community,
I am using Comsol to calculate eigenfrequencies and mode shapes of a wheel in the Solid Mechanics toolbox and take advantage of the 2D axisymmetry. I am runnig harmonic studies with different circumferential mode numbers m.
From the study I get complex-valued mode shapes and natural frequencies for m>0. The natural frequencies have a negligible imaginary part and the mode shapes (i.e. the displacement field components u,v,w) are either purely real or imaginary or some complex number, see Screenshot1. I am only aware of complex modes for non-proportionally damped or rotating mechanical systems. Maybe I miss something here, I would have not expected them in this type of study. From what I know about the theory, using a Fourier series of cos(m.theta) and sin(m.theta) for the variation around the circumference (or theta direction) yields real-valued mass/stiffness matrices and modes (apart from rigid body motion with perhaps imaginary natural frequencies).
What I would like to know is:
1) What is the physical meaning of the complex mode shapes and frequencies for this type of study is and how do I have to treat them here? Or is it just something mathematical?
2) Is there a possibility to directly or indirectly obtain the corresponding real-valued mode shapes from the complex ones in COMSOL (or convert them somehow)? Or do I have to do this outside of COMSOL?
3) If the complex modes are not supposed to be there, could you maybe tell me, what I am doing wrong? Because it might be a fundamental error then :-)
I am very happy for any help you can provide regarding my questions and I want to thank you for your support. Stay safe!
With best wishes,
Christopher Knuth
PS: I added a simplified version of my model (with exchanged geometry and courser mesh which both did not influence the results in being complex or not). In the "Results" tab the complex displacement field components u,v,w can be seen in "Tables", as well as the system matrices (where I found the stiffness matrix to be having purely real and imaginary components which is, if not an error in my model, due to the formulation COMSOL uses, I suppose).
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The quantities u,v,w are phasor quantities, that is, they have both magnitude and phase.
u = 1+j0 and v = 0+j1 just means v is 90 degrees out of phase with u.
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As Dave says, the complex values indicate phasors. For a damped problem, this means a temporal lag between different quantities.
For the case of circumferential mode extension in axial symmetry, the phasor implies a shift around the circumference. If, for example u is real, while v is purely imaginary, then
where 
is the circumferential coordinate.
By using a Revolution 2D dataset you can see the 'physical' mode in 3D. Such a dataset and corresponding plot comes as a default when you solve an Eigenfrequency study with circumferential mode extension selected.
-------------------Henrik Sönnerlind
COMSOL
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Dear Dave and Dear Henrik,
many thanks for your quick response. Your answers make sense, thanks for explaining. When looking at the sequence of real and imaginary numbers of the three phasors u,v,w (displacements in r-,-,z-direction respectively), see Screenshot1 in the original post, it is in good alignment with the theory/equations I was aware of. If written as Fourier series, the equations I mean (taken from M. Petyt, Introduction to Finite Element Vibration Analysis 2nd edition Chapter 5) are of the format:
where the quantities with tilde denote antisymmetric distribution with respect to position 
and quantities without tilde are symmetric. (The sign could vary depending on how the overall scheme is formulated in case someone is wondering about the minus). At some modes u and w are purely real and v is imaginary, leading to sin and cos as above.
When solving a full 3D model, each mode for m>0 has multiplicty 2; one with the symmetric and antisymmetric variation in , see Screenshot2. I assume, I just need to swap the relationship you posted Henrik (based on what can also be seen in the second part of the sum within the equations I posted). Is that correct? Just to be sure, I bring everything into the right context.
Sometimes a tiny real or imaginary part is present (many orders of magnitude smaller). I assume, it is just a spurious numerical effect and can be ignored. But for some modes, the phasors have a real and imaginary part of about same order of magnitude (see mode 2,4,6,8,9 in Screenshot1 above). This is no longer in alignment with the equations, when having sin or cos distribution in theta with an additional initial phase (if I see this right). Do I have to treat them differently to return a purely real-valued number? Can I even return it in that case without deleting the overall phase relationship between the DOFs for such a mode? I hope this makes sense.
Another thing I was wondering, does this effect the mass normalisation I applied in the solver (at non purely real or imaginary mode shapes)? In the end, I am interested in exporting the mass-normalised mode shapes at discrete locations for further studies.
I know this is a couple more follow-up questions. I am hoping I am not mixing something up and/or missing your point. I'm sorry for referring to the equations which might be different to how COMSOL solves the problem and can't be used as reference, but I found them useful to explain this to myself. Any more help I would very much appreciate. Many thanks again.
Best wishes,
Christopher
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Modes can be computed with an arbitrary offset in the circumferential direction. Thus, you may see complex numbers, rather than purely real and imaginary. If you look at the phase angle, using for example arg(u) and arg(v), you will see that the difference is still 90 degrees. If you look at the same mode in the 3D representation, you will find that it is not aligned with the x- and y-axes.
As you mention, some modes are duplicated in a true 3D setting. You do, however, only get one eigenfrequency / mode pair in the axisymmetric model.
The modes are properly mass normalized. You can check that by computing integrals like
solid.rho*(withsol('sol3',u,setind(lambda,m1))*withsol('sol3',conj(u),setind(lambda,m2))+withsol('sol3',v,setind(lambda,m1))*withsol('sol3',conj(v),setind(lambda,m2))+withsol('sol3',w,setind(lambda,m1))*withsol('sol3',conj(w),setind(lambda,m2)))
Here, m1 and m2 are parameters that represent the mode numbers. When m1 = m2, such an integral will evaluate to 1.000; when they differ, the integral gives numerical noise. Thus, the modes are both orthogonal and normalized with respect to the mass.
-------------------Henrik Sönnerlind
COMSOL
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Dear Henrik,
thank you for your additional response, I haven't thought about the arbitrary offset. It makes sense to me now. Thanks Henrik, and thanks Dave, for both of your help.
Best wishes,
Christopher