## 2D harmonic axisymmetric eigenfrequency study in solid mechanics - why complex mode shapes?

Posted 14/04/2021, 13:39 GMT-4 Mechanical, 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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