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Piezoelectric Eigenfrequency analysis - Decay factor - Understanding results
Posted Dec 11, 2012, 3:18 a.m. EST MEMS & Nanotechnology, MEMS & Piezoelectric Devices, Modeling Tools & Definitions, Parameters, Variables, & Functions, Results & Visualization Version 4.3 4 Replies
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Hi,
I carried out successfully an eigenfrequency analysis of a piezoelectric structure and now I need some help in understanding the meaning of the results.
These are the six eigenfrequencies that I found:
2968.3458428909757-1.8095493631772506i
13608.900380896357+915.4349497985088i
15335.000591497948+424.2799489321054i
40886.81667544398+7309.84485583443i
44541.79244857697+162.59187825549208i
80139.68435108368-3760.1058813209215i
and these are the six eigenvalues (= lambda):
-11.369733971331478-18650.666986680182i
5751.847426252672-85507.24292011866i
2665.82954126111-96352.65040209018i
45929.10979594117-256899.4457924949i
1021.5949005216398-279864.33586834144i
-23625.442026955163-503532.4872367387i
According to the COMSOL documentantion, I found the following relations to post-process these data (please, correct me, if I am wrong):
1. eigenfrequency = -1*lambda / (2*pi*i)
2. mode_frequency = abs( imag(-1*lambda) / (2*pi) )
3. Quality_factor = imag(lambda) / (2*real(lambda) )
4. decay_factor = real(lambda) (also named elesewhere "damping in time")
By applying these formulas: I am finding, correctly, positive values for "mode_frequencies" but I am getting some "quality_factors" and "decay_factors" that are positive and others that are negative.
What does it physically mean? Have I found stable (positive eigenvalues) and unstable (negative eigenvalues) modes for my structure? Or simply is there anything wrong with my simulations/results?
Thank you very much.
PS: I applied no loads to structure (only the constraints for the fixed end of the beam and the ground and the terminal conditions for the piezo).
I carried out successfully an eigenfrequency analysis of a piezoelectric structure and now I need some help in understanding the meaning of the results.
These are the six eigenfrequencies that I found:
2968.3458428909757-1.8095493631772506i
13608.900380896357+915.4349497985088i
15335.000591497948+424.2799489321054i
40886.81667544398+7309.84485583443i
44541.79244857697+162.59187825549208i
80139.68435108368-3760.1058813209215i
and these are the six eigenvalues (= lambda):
-11.369733971331478-18650.666986680182i
5751.847426252672-85507.24292011866i
2665.82954126111-96352.65040209018i
45929.10979594117-256899.4457924949i
1021.5949005216398-279864.33586834144i
-23625.442026955163-503532.4872367387i
According to the COMSOL documentantion, I found the following relations to post-process these data (please, correct me, if I am wrong):
1. eigenfrequency = -1*lambda / (2*pi*i)
2. mode_frequency = abs( imag(-1*lambda) / (2*pi) )
3. Quality_factor = imag(lambda) / (2*real(lambda) )
4. decay_factor = real(lambda) (also named elesewhere "damping in time")
By applying these formulas: I am finding, correctly, positive values for "mode_frequencies" but I am getting some "quality_factors" and "decay_factors" that are positive and others that are negative.
What does it physically mean? Have I found stable (positive eigenvalues) and unstable (negative eigenvalues) modes for my structure? Or simply is there anything wrong with my simulations/results?
Thank you very much.
PS: I applied no loads to structure (only the constraints for the fixed end of the beam and the ground and the terminal conditions for the piezo).
4 Replies Last Post Dec 12, 2012, 8:50 a.m. EST
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Posted:
1 decade ago
Hi
my first comment is: pls note that you should not consider more than 5-6 digits for numerical FEM (due to the way numbers are presented in the binary system) and most sover settings do not give you more than 3-4 digits anyhow, by default.
Then I agree so far with your formulas, but if you look up the "equations" (see options preferences equation view on) you will see how COMSOL converts eigenvalues to eigenfrequencies
the damping normally gives you phase lag (or advance for electric contribution) and a slight frequency shift for the absolute value. But pls note that you get only some electrical phase lag by adding PZT electric connections (sort or impedance load). The PZT material has itself some damping, perhaps even 1% or more (all depends on how its made, its size ...)
--
Good luck
Ivar
my first comment is: pls note that you should not consider more than 5-6 digits for numerical FEM (due to the way numbers are presented in the binary system) and most sover settings do not give you more than 3-4 digits anyhow, by default.
Then I agree so far with your formulas, but if you look up the "equations" (see options preferences equation view on) you will see how COMSOL converts eigenvalues to eigenfrequencies
the damping normally gives you phase lag (or advance for electric contribution) and a slight frequency shift for the absolute value. But pls note that you get only some electrical phase lag by adding PZT electric connections (sort or impedance load). The PZT material has itself some damping, perhaps even 1% or more (all depends on how its made, its size ...)
--
Good luck
Ivar
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Posted:
1 decade ago
Hi Ivar,
thank you for your kind reply and your advice.
If I correctly understand, the positive (negative) sign in the quality/decay factor physically indicates only a phase lag (advance) in the contribution of the mode to the overall system response.
However, as you said, I normally expect only phase lags in the electrical response of my piezoelectric system, so, how should I physically treat the modes indicating a phase advance?
I'd consider them as unstable modes (considering the Control Theory).
Or could this kind of results (i.e. phase advance in modes) descend from any numerical errors (mesh dimension, tolerances and so on)?
thank you for your kind reply and your advice.
If I correctly understand, the positive (negative) sign in the quality/decay factor physically indicates only a phase lag (advance) in the contribution of the mode to the overall system response.
However, as you said, I normally expect only phase lags in the electrical response of my piezoelectric system, so, how should I physically treat the modes indicating a phase advance?
I'd consider them as unstable modes (considering the Control Theory).
Or could this kind of results (i.e. phase advance in modes) descend from any numerical errors (mesh dimension, tolerances and so on)?
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Posted:
1 decade ago
Hi
I would also have expected phase lag first of all, but you have electrical feedback there, (if you have a short or high impedance or somewhere in between). But I agree, I remain slightly suspicious.
One way try with a pure structural damping and check, then change the PZT circuit impedance, can you make any sense out of it ?
--
Good luck
Ivar
I would also have expected phase lag first of all, but you have electrical feedback there, (if you have a short or high impedance or somewhere in between). But I agree, I remain slightly suspicious.
One way try with a pure structural damping and check, then change the PZT circuit impedance, can you make any sense out of it ?
--
Good luck
Ivar
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Posted:
1 decade ago
Dear Ivar,
I followed your advice of running several eigenfrequency analyses of the same device with different electrical loads (Node: "Terminal", Terminal type: "Terminated", Terminal power: 1 W and Characteristic impedance is my parameter).
I find that, for several values of the Characteristic impedance (ranging from 50 ohm to 1e+6 ohm), the frequency modes slightly change (in the order of few hertz), while the quality factor and the decay factor exhibit large changes (remaining approximately in the same order of magnitude, but sometimes switching their sign). I continue to get modes with phase lag and modes with phase advance.
However, the application of a pure mechanical load (without "terminal" and " ground" as electrical conditions) changes slightly the frequency of the modes (always in the order of few hertz), but the quality factors and the decay factors are affected in the same way as changing the output impedance and I still have modes with phase lag and modes with phase advance.
Changes in the mesh and in the tolerances of the solver don't affect the final result.
I followed your advice of running several eigenfrequency analyses of the same device with different electrical loads (Node: "Terminal", Terminal type: "Terminated", Terminal power: 1 W and Characteristic impedance is my parameter).
I find that, for several values of the Characteristic impedance (ranging from 50 ohm to 1e+6 ohm), the frequency modes slightly change (in the order of few hertz), while the quality factor and the decay factor exhibit large changes (remaining approximately in the same order of magnitude, but sometimes switching their sign). I continue to get modes with phase lag and modes with phase advance.
However, the application of a pure mechanical load (without "terminal" and " ground" as electrical conditions) changes slightly the frequency of the modes (always in the order of few hertz), but the quality factors and the decay factors are affected in the same way as changing the output impedance and I still have modes with phase lag and modes with phase advance.
Changes in the mesh and in the tolerances of the solver don't affect the final result.