بررسی اثر میرایی ترموالاستیک در مدل غیرخطی تشدیدکننده های میکروالکترومکانیکی با روش تربیع دیفرانسیلی
نویسندگان
1 انشگاه شاهرود
2 دانشگاه علوم و فنون دریایی خرمشهر
چکیده
در این مقاله ، یک مدل غیرخطی میکروتیر دو سر گیردار تحت اثر بار الکترواستا ت یک، اثر ات کشش درون صفح ه ای و میرایی ترموالاستیک در نظر گرفته شده است. فرکانس ارتعاش آزاد با استفاده از گسسته سازی براساس روش تربیع دیفرانسیلی به دست آمده، که به دلیل اتلاف انرژی ناشی از میرایی ترموالاستیک، این فرکانس کمیتی مختلط است. با تفکیک مقادیر حقیقی و موهومی فرکانس می توان ضریب کیفیت میرایی ترموالاستیک را محاسبه کرد. اثرات کشش درون صفحه ای و میرایی ترموالاستیک در مقایسه با نتایج مقالات مرجع صحه گذاری شدند. تغییرات میرایی ترموالاستیک برحسب مدول الاستیسیته، ضریب انبساط گرمایی و متغیرهای هندسی شامل ضخامت، طول و فاصله خازنی بررسی شده و نتایج مدلهای خطی و غیرخطی در ولتاژهای بالا مقایسه شده اند. همچنین نشان داده شده است که در ولتاژهای بالا مدل خطی در محاسبه میرایی ترموالاستیک خطای زیادی دارد و در این ولتاژها باید از مدل غیرخطی استفاده شود .
کلیدواژه‌ها
عنوان مقاله [English]
On the Effect of Thermoelastic Damping in Nonlinear Micro Electro Mechanical Resonators using Differential Quadrature Method
نویسندگان [English]
- A. Karami Mohammadi 1
- N. AleAli 2
1
2
چکیده [English]
: In this paper, a nonlinear model of clamped-clamped microbeam actuated by electrostatic load with stretching and thermoelastic effects is presented. Free vibration frequency is calculated by discretization based on DQ method. Frequency is a complex value due to the thermoelastic effect that dissipates the energy. By separating the real and imaginary parts of frequency, quality factor of thermoelastic damping is calculated. Both stretching and thermoelastic effects are validated against the results of the reference papers. The variations of thermoelastic damping versus elasticity modulus, coefficient of thermal expansion and geometrical parameters such as thickness, gap distance, and length are investigated and these results are compared in the linear and nonlinear models for high values of voltage. Also, this paper shows that since for high values of electrostatic voltage the linear model reveals a large error for calculating the thermoelastic damping, the nonlinear model should be used for this purpose.
کلیدواژه‌ها [English]
- Microbeam vibration
- nonlinear vibration
- thermoelastic damping
- differential quadrature method
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144305, 2006.
2. Zener, C., “Internal Friction in Solids I. Theory of
Internal Friction in Reeds”, Physical Review, Vol. 52,
pp. 230-235, 1937.
3. Zener, C., “Internal Friction in Solids II. General
Theory of Thermoelastic Internal Friction”, Physical
Review, Vol. 53, pp. 90-99, 1938.
4. Alblas, J. B., “A Note on the Theory of
Thermoelastic Damping”, Journal of Thermal
Stresses, Vol. 4, Issue 3-4, pp. 333-355 , 1981.
5. Lifshitz, R., and Roukes, M. L., “Thermoelastic
Damping in Micro- and Nanomechanical Systems”,
Physical Review B, Vol. 61, No. 8, pp. 5600-5609,
1999.
6. Thomas, H., Metcalf Bradford B., Douglas M., and
Brian, H., “Thermoelastic Damping in
Micromechanical Resonators”, Applied Physics
Letters, Vol. 95, 061903, 2009.
7. Prabhakar, S., and Vengallatore, S., “Theory of
Thermoelastic Damping in Micromechanical
Resonators with Two-Dimensional Heat
Conduction”, Journal of Microelectromechanical
Systems, Vol. 17, No. 2, pp. 494-502, 2008.
8. Yun-Bo, Y., and Matin, M. A., “Eigenvalue Solution
of Thermoelastic Damping in Beam Resonators
Using a Finite Element Analysis”, Journal of
Vibration and Acoustics, Vol. 129, pp. 478-483,
2007.
9. Yun-Bo Y., “Geometric Effects on Thermoelastic
Damping in MEMS Resonators”, Journal of Sound
and Vibration, Vol. 309, pp. 588-599, 2008.
10.Yun-Bo, Yi., “Finite Element Analysis of Thermoelastic
Damping in Contour-Mode Vibrations of Micro- and
Nanoscale Ring, Disk, and Elliptical Plate
Resonators”, Journal of Vibration and Acoustics,
Vol. 132, Issue 4, 041015, 2010.
11. Raffaele, A., Claudia, C., Corigliano, A., and Frangi,
A., “Solid Damping in Micro Electro Mechanical
Systems”, Meccanica, Vol. 43, pp. 419-428, 2008.
12. Serra, E., and Bonaldi, E., “A Finite Element
Formulation for Thermoelastic Damping Analysis”,
International Journal for Numerical Methods in
Engineering, Vol. 78, Issue 6, pp. 671-691, 2009.
13. Jinbok, C., Maenghyo, C., and Jaewook, R.,
“Efficient Prediction of the Quality Factors of
Micromechanical Resonators”, Journal of Sound and
Vibration, Vol. 329, Issue 1, pp. 84-95, 2009.
14. Nayfeh, A. H., and Younis, M. I., “Modeling and
Simulations of Thermoelastic Damping in
Microplates”, Journal of Micromechanics and
Microengineering, Vol. 14, pp. 1711–1717, 2004.
15. Feng, Y., and Bert, C. W., “Application of the
Quadrature Method to Flexural Vibration Analysis of
a Geometrically Nonlinear Beam”, Nonlinear
Dynamics, Vol. 156, pp. 3-18, 1993.
16. Guo, Q., and Zhong, H., “Nonlinear Vibration
Analysis of Beams by a Spline-based Differential
Quadrature Method”, Journal of Shock and
Vibration, Vol. 269, pp. 413-420, 2004.
17. Zhong, H., and Guo, Q. “Nonlinear Vibration
Analysis of Timoshenko Beams using the
Differential Quadrature Method”, Nonlinear
Dynamics, Vol. 32, pp. 223-234, 2003 .
18. Han, K. M., Xiao, J. B., and Du, Z. M., “Differential
Quadrature Method for Mindlin Plates on Winkler
Foundations”, International Journal of Mechanical
Sciences, Vol. 38, pp. 405-421, 1996.
19. Liew, K. M., Han, J. B., and Xiao, Z. M.,
10 روش های عددی در مهندسی، سال ۳۴ ، شماره ١، تابستان ١٣٩٤
“Differential Quadrature Method for Thick
Symmetric Cross-Ply Laminates with First-Order
Shear Flexibility”, International Journal of Solids
and Structures, Vol. 33, pp. 2647-2658, 1996.
20. Liew, K. M., and Han, J. B., “A Four-Node
Differential Quadrature Method for Straight-Sided
Quadrilateral Reissner/Mindlin Plates”, Communications
in Numerical Methods in Engineering, Vol. 13, pp.
73-81, 1997.
21. Han, J. B., and Liew, K. M., “An Eight-Node
Curvilinear Differential Quadrature Formulation for
Reissner/Mindlin Plates”, Computer Methods in
Applied Mechanics and Engineering, Vol. 141, pp.
265-280, 1997.
22. Li, J. J., and Cheng, C. J., “Differential Quadrature
Method for Nonlinear Free Vibration of Orthotropic
Plate with Finite Deformation and Transverse Shear
Effect”, Journal of Sound and Vibration, Vol. 281,
pp. 295-309, 2005.
23. Malekzadeh, P., “Differential Quadrature Large
Amplitude Free Vibration Analysis of Laminated
Skew Plates Based on FSDT”, Composite Structures,
Vol. 83, Issue 2, pp. 189-200, 2008.
24. Malekzadeh, P., and Karami, G., “Large Amplitude
Flexural Vibration Analysis of Tapered Plates with
Edges Elastically Restrained Against Rotation Using
DQM”, Engineering Structures, Vol. 30, Issue 10,
pp. 2850-2858, 2008.
25. Malekzadeh, P., “Three-Dimensional Free Vibration
Analysis of Thick Functionally Graded Plates on
Elastic Foundations”, Composite Structures, Vol. 89,
Issue 3, pp. 367-373, 2009.
26.Malekzadeh, P., and Vosoughi, A. R., “DQM Large
Amplitude Vibration of Composite Beamson
Nonlinear Elastic Foundations with Restrained
Edges”, Communications in Nonlinear Science and
Numerical Simulation, Vol. 14, pp. 906–915, 2009.
27. Nayfeh, A. H., and Frank, P. P., Linear and Nonlinear
Structural Mechanics, New Jersey, John Wiley &
Sons, pp. 382-388, 2004.
28. Sarma, B. S., and Varadan, T. K., “Lagrange-Type
Formulation for Finite Element Analysis of Non-
Linear Beam Vibrations”, Journal of Sound and
Vibration, Vol. 86, pp. 61-70, 1938.
29. Sun, Y., and Saka, M., “Thermoelastic Damping in
Micro-Scale Circular Plate Resonators”, Journal of
Sound and Vibration, 329, pp. 328–337, 2010.
144305, 2006.
2. Zener, C., “Internal Friction in Solids I. Theory of
Internal Friction in Reeds”, Physical Review, Vol. 52,
pp. 230-235, 1937.
3. Zener, C., “Internal Friction in Solids II. General
Theory of Thermoelastic Internal Friction”, Physical
Review, Vol. 53, pp. 90-99, 1938.
4. Alblas, J. B., “A Note on the Theory of
Thermoelastic Damping”, Journal of Thermal
Stresses, Vol. 4, Issue 3-4, pp. 333-355 , 1981.
5. Lifshitz, R., and Roukes, M. L., “Thermoelastic
Damping in Micro- and Nanomechanical Systems”,
Physical Review B, Vol. 61, No. 8, pp. 5600-5609,
1999.
6. Thomas, H., Metcalf Bradford B., Douglas M., and
Brian, H., “Thermoelastic Damping in
Micromechanical Resonators”, Applied Physics
Letters, Vol. 95, 061903, 2009.
7. Prabhakar, S., and Vengallatore, S., “Theory of
Thermoelastic Damping in Micromechanical
Resonators with Two-Dimensional Heat
Conduction”, Journal of Microelectromechanical
Systems, Vol. 17, No. 2, pp. 494-502, 2008.
8. Yun-Bo, Y., and Matin, M. A., “Eigenvalue Solution
of Thermoelastic Damping in Beam Resonators
Using a Finite Element Analysis”, Journal of
Vibration and Acoustics, Vol. 129, pp. 478-483,
2007.
9. Yun-Bo Y., “Geometric Effects on Thermoelastic
Damping in MEMS Resonators”, Journal of Sound
and Vibration, Vol. 309, pp. 588-599, 2008.
10.Yun-Bo, Yi., “Finite Element Analysis of Thermoelastic
Damping in Contour-Mode Vibrations of Micro- and
Nanoscale Ring, Disk, and Elliptical Plate
Resonators”, Journal of Vibration and Acoustics,
Vol. 132, Issue 4, 041015, 2010.
11. Raffaele, A., Claudia, C., Corigliano, A., and Frangi,
A., “Solid Damping in Micro Electro Mechanical
Systems”, Meccanica, Vol. 43, pp. 419-428, 2008.
12. Serra, E., and Bonaldi, E., “A Finite Element
Formulation for Thermoelastic Damping Analysis”,
International Journal for Numerical Methods in
Engineering, Vol. 78, Issue 6, pp. 671-691, 2009.
13. Jinbok, C., Maenghyo, C., and Jaewook, R.,
“Efficient Prediction of the Quality Factors of
Micromechanical Resonators”, Journal of Sound and
Vibration, Vol. 329, Issue 1, pp. 84-95, 2009.
14. Nayfeh, A. H., and Younis, M. I., “Modeling and
Simulations of Thermoelastic Damping in
Microplates”, Journal of Micromechanics and
Microengineering, Vol. 14, pp. 1711–1717, 2004.
15. Feng, Y., and Bert, C. W., “Application of the
Quadrature Method to Flexural Vibration Analysis of
a Geometrically Nonlinear Beam”, Nonlinear
Dynamics, Vol. 156, pp. 3-18, 1993.
16. Guo, Q., and Zhong, H., “Nonlinear Vibration
Analysis of Beams by a Spline-based Differential
Quadrature Method”, Journal of Shock and
Vibration, Vol. 269, pp. 413-420, 2004.
17. Zhong, H., and Guo, Q. “Nonlinear Vibration
Analysis of Timoshenko Beams using the
Differential Quadrature Method”, Nonlinear
Dynamics, Vol. 32, pp. 223-234, 2003 .
18. Han, K. M., Xiao, J. B., and Du, Z. M., “Differential
Quadrature Method for Mindlin Plates on Winkler
Foundations”, International Journal of Mechanical
Sciences, Vol. 38, pp. 405-421, 1996.
19. Liew, K. M., Han, J. B., and Xiao, Z. M.,
10 روش های عددی در مهندسی، سال ۳۴ ، شماره ١، تابستان ١٣٩٤
“Differential Quadrature Method for Thick
Symmetric Cross-Ply Laminates with First-Order
Shear Flexibility”, International Journal of Solids
and Structures, Vol. 33, pp. 2647-2658, 1996.
20. Liew, K. M., and Han, J. B., “A Four-Node
Differential Quadrature Method for Straight-Sided
Quadrilateral Reissner/Mindlin Plates”, Communications
in Numerical Methods in Engineering, Vol. 13, pp.
73-81, 1997.
21. Han, J. B., and Liew, K. M., “An Eight-Node
Curvilinear Differential Quadrature Formulation for
Reissner/Mindlin Plates”, Computer Methods in
Applied Mechanics and Engineering, Vol. 141, pp.
265-280, 1997.
22. Li, J. J., and Cheng, C. J., “Differential Quadrature
Method for Nonlinear Free Vibration of Orthotropic
Plate with Finite Deformation and Transverse Shear
Effect”, Journal of Sound and Vibration, Vol. 281,
pp. 295-309, 2005.
23. Malekzadeh, P., “Differential Quadrature Large
Amplitude Free Vibration Analysis of Laminated
Skew Plates Based on FSDT”, Composite Structures,
Vol. 83, Issue 2, pp. 189-200, 2008.
24. Malekzadeh, P., and Karami, G., “Large Amplitude
Flexural Vibration Analysis of Tapered Plates with
Edges Elastically Restrained Against Rotation Using
DQM”, Engineering Structures, Vol. 30, Issue 10,
pp. 2850-2858, 2008.
25. Malekzadeh, P., “Three-Dimensional Free Vibration
Analysis of Thick Functionally Graded Plates on
Elastic Foundations”, Composite Structures, Vol. 89,
Issue 3, pp. 367-373, 2009.
26.Malekzadeh, P., and Vosoughi, A. R., “DQM Large
Amplitude Vibration of Composite Beamson
Nonlinear Elastic Foundations with Restrained
Edges”, Communications in Nonlinear Science and
Numerical Simulation, Vol. 14, pp. 906–915, 2009.
27. Nayfeh, A. H., and Frank, P. P., Linear and Nonlinear
Structural Mechanics, New Jersey, John Wiley &
Sons, pp. 382-388, 2004.
28. Sarma, B. S., and Varadan, T. K., “Lagrange-Type
Formulation for Finite Element Analysis of Non-
Linear Beam Vibrations”, Journal of Sound and
Vibration, Vol. 86, pp. 61-70, 1938.
29. Sun, Y., and Saka, M., “Thermoelastic Damping in
Micro-Scale Circular Plate Resonators”, Journal of
Sound and Vibration, 329, pp. 328–337, 2010.
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