Eigenfrequency and Temperature in a Cantilever

Francisco Munoz
Hello,

I'm using COMSOL for some months. I'm simulating the eigenfrequency of a clamped-free beam (cantilever MEMS).
The value of the eigenfrequency at room temperautre is correct (about 6.4MHz) but the increase in temperature produces an increase in the frequency.
Theorically and experimentally, if the temperature rises on the structure, the eigenfrequency decreases.

I tried to solve it including a HT module but there was no change. What is wrong?


Thanks. Kind Regards.
-Francisco

PS I simulated a clamped-clamped beam (bridge) and it worked correctly.


9 Replies Last Post 2017年5月22日 GMT-0400上午8:12
Henrik Sönnerlind COMSOL Employee
Posted: 5 years ago 2013年1月30日 GMT-0500上午3:53
Hi,

This is a rather interesting question, which turns up now and then.

The clamped and cantilever cases are controlled by completely different physical phenomena.

For a clamped beam, a large axial compressive stress is introduced because of the restrained expansion. At 100 degrees this stress is of the same order of magnitude as the yield stress for a metal. According to beam theory, a tensile axial stress will increase the stiffness and a compressive stress will decrease it. This will then affect the natural frequency just as when you tune a string in an instrument. This effect is large, and results according to the theory will be produced by COMSOL.

The cantilever case is much more difficult, and the influence of temperature on frequency is orders of magnitude smaller. Since the cantilever is free to expand, no stresses will be introduced by the thermal expansion (except possibly locally at the clamped end). So any changes in natural frequency comes from either

a) Changes in geometrical dimensions causes by thermal expansion

or

b) Changes in material properties caused by the change in temperature

If we first study case a), then a natural frequency can be written as



where EI is the bending stiffness and m is the mass per unit length. During thermal expansion, the length in each directions is changed by a factor



The new natural frequency is thus



Since the thermal expansion is small,



This gives a (very) small increase in natural frequency with temperature. The order of magnitude is 0.1% per 100 degrees. COMSOL cannot exactly capture this effect using a Linear Elastic material even if the frequency shift has the right order of magnitude. This is caused by some subtleties in the formulation of the thermal expansion together with a geometric nonlinear formulation. With an equivalent hyperelastic material (Saint-Venant material) COMSOL will return a result corresponding very well to the derivation above.

In reality, the variation of E with temperature (usually decreasing) is a much more important effect. Also, the internal damping of the material can change with temperature. So in real life the natural frequency will decrease.

Regards,
Henrik
Hi, This is a rather interesting question, which turns up now and then. The clamped and cantilever cases are controlled by completely different physical phenomena. For a clamped beam, a large axial compressive stress is introduced because of the restrained expansion. At 100 degrees this stress is of the same order of magnitude as the yield stress for a metal. According to beam theory, a tensile axial stress will increase the stiffness and a compressive stress will decrease it. This will then affect the natural frequency just as when you tune a string in an instrument. This effect is large, and results according to the theory will be produced by COMSOL. The cantilever case is much more difficult, and the influence of temperature on frequency is orders of magnitude smaller. Since the cantilever is free to expand, no stresses will be introduced by the thermal expansion (except possibly locally at the clamped end). So any changes in natural frequency comes from either a) Changes in geometrical dimensions causes by thermal expansion or b) Changes in material properties caused by the change in temperature If we first study case a), then a natural frequency can be written as [math]f_i = \frac{k_i}{L^2} \sqrt{ \frac{EI}{m} }[/math] where EI is the bending stiffness and m is the mass per unit length. During thermal expansion, the length in each directions is changed by a factor [math]q = 1 + \alpha \Delta T[/math] The new natural frequency is thus [math]f_i(\Delta T) = \frac{k_i}{L^2q^2} \sqrt{ \frac{EIq^4}{\frac{m}{q}} } = \sqrt{q} f_i(0)[/math] Since the thermal expansion is small, [math]\sqrt{q} \approx 1 +\frac{1}{2} \alpha \Delta T[/math] This gives a (very) small increase in natural frequency with temperature. The order of magnitude is 0.1% per 100 degrees. COMSOL cannot exactly capture this effect using a Linear Elastic material even if the frequency shift has the right order of magnitude. This is caused by some subtleties in the formulation of the thermal expansion together with a geometric nonlinear formulation. With an equivalent hyperelastic material (Saint-Venant material) COMSOL will return a result corresponding very well to the derivation above. In reality, the variation of E with temperature (usually decreasing) is a much more important effect. Also, the internal damping of the material can change with temperature. So in real life the natural frequency will decrease. Regards, Henrik

Nagi Elabbasi Certified Consultant
Posted: 5 years ago 2013年1月30日 GMT-0500上午11:43
That was very informative Henrik, thanks.

I want to add that when the temperature of a clamped or cantilevered beam increases the supports typically experience the same or a similar temperature increase, and they will expand as well. In that case, the local stresses around a clamped end (where all displacements are fixed) are not physical. They are much higher than what the beam will really experience, and also may affect the natural frequencies (especially when you’re looking for small changes in frequency). Interestingly, when you use beam elements instead of solid elements that issue goes away.

Nagi Elabbasi
Veryst Engineering
That was very informative Henrik, thanks. I want to add that when the temperature of a clamped or cantilevered beam increases the supports typically experience the same or a similar temperature increase, and they will expand as well. In that case, the local stresses around a clamped end (where all displacements are fixed) are not physical. They are much higher than what the beam will really experience, and also may affect the natural frequencies (especially when you’re looking for small changes in frequency). Interestingly, when you use beam elements instead of solid elements that issue goes away. Nagi Elabbasi Veryst Engineering

Posted: 5 years ago 2013年1月30日 GMT-0500下午1:02
Hi

or you replace the fixed BC by a prescribed displacement, taking into account the alpha of the support material and the interface temperature, but you might need to define a "centre of expansion" to do fine measurements

--
Good luck
Ivar
Hi or you replace the fixed BC by a prescribed displacement, taking into account the alpha of the support material and the interface temperature, but you might need to define a "centre of expansion" to do fine measurements -- Good luck Ivar

Francisco Munoz
Posted: 5 years ago 2013年1月30日 GMT-0500下午3:56
Hello Henrik,

really thanks for your answer.

I have a question about the Young modulus. In fact the Young's modulus value decreases when the temperature rises. Does COMSOL consider/contemplate Young's modulus variation with temperature or is it a fixed value? if it doesn't contemplate, is it possible/easy to include?


Thank you again. Kind regards.
-Francisco
Hello Henrik, really thanks for your answer. I have a question about the Young modulus. In fact the Young's modulus value decreases when the temperature rises. Does COMSOL consider/contemplate Young's modulus variation with temperature or is it a fixed value? if it doesn't contemplate, is it possible/easy to include? Thank you again. Kind regards. -Francisco

Posted: 5 years ago 2013年1月31日 GMT-0500上午12:58
Hi

that depends on your material chosen, several of the materials in the material library have a T dependence build in (you add an interpolation function under the material node) your model becomes non linear but generally solver over some more time, just enough to get another coffee ;)

--
Good luck
Ivar
Hi that depends on your material chosen, several of the materials in the material library have a T dependence build in (you add an interpolation function under the material node) your model becomes non linear but generally solver over some more time, just enough to get another coffee ;) -- Good luck Ivar

Henrik Sönnerlind COMSOL Employee
Posted: 5 years ago 2013年1月31日 GMT-0500上午2:51
For this case, where you just run a parametric sweep over temperatures, no extra overhead is caused by having a temperature dependent Young's modulus. The problem is still linear (in this sense) for each parameter value.

Regards,
Henrik


Hi

that depends on your material chosen, several of the materials in the material library have a T dependence build in (you add an interpolation function under the material node) your model becomes non linear but generally solver over some more time, just enough to get another coffee ;)

--
Good luck
Ivar


For this case, where you just run a parametric sweep over temperatures, no extra overhead is caused by having a temperature dependent Young's modulus. The problem is still linear (in this sense) for each parameter value. Regards, Henrik [QUOTE] Hi that depends on your material chosen, several of the materials in the material library have a T dependence build in (you add an interpolation function under the material node) your model becomes non linear but generally solver over some more time, just enough to get another coffee ;) -- Good luck Ivar [/QUOTE]

Francisco Munoz
Posted: 5 years ago 2013年1月31日 GMT-0500下午3:32
Thank you all for your answers, they were very helpful.


Kind regards,
-Francisco
Thank you all for your answers, they were very helpful. Kind regards, -Francisco

Henrik Sönnerlind COMSOL Employee
Posted: 6 months ago 2017年4月26日 GMT-0400上午4:45
Hi,

For readers of this topic:

From version 5.3, even the small frequency shift caused by the only the change in size due to thermal expansion is accurately captured if you do a prestressed eigenfrequency analysis.

Regards,
Henrik
Hi, For readers of this topic: From version 5.3, even the small frequency shift caused by the only the change in size due to thermal expansion is accurately captured if you do a prestressed eigenfrequency analysis. Regards, Henrik

Henrik Sönnerlind COMSOL Employee
Posted: 5 months ago 2017年5月22日 GMT-0400上午8:12
Hi,

This blog post contains more details:

www.comsol.com/blogs/how-to-analyze-eigenfrequencies-that-change-with-temperature/

Regards,
Henrik
Hi, This blog post contains more details: https://www.comsol.com/blogs/how-to-analyze-eigenfrequencies-that-change-with-temperature/ Regards, Henrik

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