Dave Greve
Certified Consultant
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Posted:
3 years ago
14 avr. 2021, 22:00 UTC−4
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.
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.
Henrik Sönnerlind
COMSOL Employee
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Posted:
3 years ago
15 avr. 2021, 02:19 UTC−4
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
![](data:image/png;base64,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)
![](data:image/png;base64,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)
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
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
u = u_0 \mathrm{cos} (\phi)
v = v_0 \mathrm{sin} (\phi)
where \phi 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.
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Posted:
3 years ago
15 avr. 2021, 10:55 UTC−4
Updated:
3 years ago
16 avr. 2021, 05:14 UTC−4
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:
![](data:image/png;base64,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)
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
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-,\phi-,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:
[u,w]^T = \sum_{m=1}^\infty {[u_m,w_m]^T cos(m\phi)} - \sum_{m=1}^\infty {[\tilde{u}_m,\tilde{w}_m]^T sin(m\phi)}
v = \sum_{m=1}^\infty {v_m sin(m\phi)} - \sum_{m=1}^\infty {\tilde{v}_m cos(m\phi)}
where the quantities with tilde denote antisymmetric distribution with respect to position \phi=0 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 \phi, 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
Henrik Sönnerlind
COMSOL Employee
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Posted:
3 years ago
19 avr. 2021, 02:07 UTC−4
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
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.
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Posted:
3 years ago
19 avr. 2021, 09:36 UTC−4
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
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