О термооптическом возбуждении параметрических колебаний микробалочных резонаторов. I
DOI:
https://doi.org/10.21638/spbu01.2023.212Аннотация
Настоящая статья является первой частью работы, посвященной исследованию нелинейной динамики параметрически возбуждаемых изгибных колебаний защемленной с двух концов микробалки - базового чувствительного элемента перспективного класса микродатчиков различных физических величин - при лазерном термооптическом воздействии в форме периодически генерируемых импульсов, действующих на некоторую часть поверхности балочного элемента. Найдено аналитическое решение задачи теплопроводности для установившегося гармонического распределения температуры в объеме резонатора. Определены статические и динамические компоненты силовых факторов температурной природы - температурных осевой силы и изгибающего момента. С помощью метода Галёркина выполнена дискретизация нелинейных связанных уравнений в частных производных, описывающих продольно-изгибные колебания резонатора. С применением асимптотического метода многих масштабов получено приближенное аналитическое решение для задачи нелинейной динамики системы в условиях главного параметрического резонанса.Ключевые слова:
нелинейная динамика, параметрические колебания, балка Бернулли - Эйлера, модальное взаимодействие, лазерное термооптическое воздействие
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Литература
1. Vorobyev R. I., Sergeichev I. V., Karabutov A. A., Mironova E. A., Savateeva E. V., Akhatov I. Sh. Application of the Optoacoustic Method to Assess the Effect of Voids on the Crack Resistance of Structural Carbon Plastics. Acoust. Phys. 66, 132-136 (2020). https://doi.org/10.1134/S1063771020020153
2. Yan G., Raetz S., Chigarev N., Blondeau Ja., Gusev V. E., Tournat V. Cumulative fatigue damage in thin aluminum films evaluated non-destructively with lasers via zero-group-velocity Lamb modes. NDT & E International 116, 102323 (2020). https://doi.org/10.1016/j.ndteint.2020.102323
3. Pan Yu., Rossignol C., Audoin B. Acoustic waves generated by a laser line pulse in cylinders; Application to the elastic constants measurement. J. Acoust. Soc. Am. 115 (4), 1537-1545 (2004). https://doi.org/10.1121/1.1651191
4. Chow G., Uchaker E., Cao G., Wang Ju. Laser-induced surface acoustic waves: An alternative method to nanoindentation for the mechanical characterization of porous nanostructured thin film electrode media. Mechanics of Materials 91, 333-342 (2015). https://doi.org/10.1016/J.MECHMAT.2015.10.005
5. Champion A., Bellouard Y. Direct volume variation measurements in fused silica specimens exposed to femtosecond laser. Optical Materials Express 2, 789-798 (2012). https://doi.org/10.1364/OME.2.000789
6. Otsuka P. H., Mezil S., Matsuda O., Tomoda M., Maznev A. A., Gan T., Fang N., Boechler N., Gusev V. E., Wright O. B. Time-domain imaging of gigahertz surface waves on an acoustic metamaterial. New Journal of Physics 20, 013026 (2018). https://doi.org/10.1088/1367-2630/AA9298
7. Li C., Guan G., Zhang F., Nabi G., Wang R. K., Huang Z. Laser induced surface acoustic wave combined with phase sensitive optical coherence tomography for superficial tissue characterization: a solution for practical application. Biomedical Optics Express 5 (5), 1403-1418 (2014). https://doi.org/10.1364/BOE.5.001403
8. Phinney L. M., Klody K. A., Sackos Jo. T., Walraven Je. A. Damage of MEMS thermal actuators heated by laser irradiation. Reliability, Packaging, Testing and Characterization of MEMS/MOEMS IV. Proceedings of MOEMS-MEMS Micro and Nanofabrication, 2005, San Jose, California, United States, 5716, 81-88 (2005). https://doi.org/10.1117/12.594408
9. Serrano J. R., Phinney L. M. Displacement and Thermal Performance of Laser-Heated Asymmetric MEMS Actuators. Journal of Microelectromechanical Systems 17 (1), 166-174 (2008). https://doi.org/10.1109/JMEMS.2007.911945
10. Mai A., Bunce C., H¨ubner R., Pahner D., Dauderst¨adt U. A. In situ bow change of Al-alloy MEMS micromirrors during 248-nm laser irradiation. Journal of Micro/Nanolithography, MEMS and MOEMS 15 (3), 035502 (2016). https://doi.org/10.1117/1.JMM.15.3.035502
11. Zook J. D., Burns D.W., Herb W. R., Guckel H., Kang J.W., Ahn Y. Optically excited self-resonant microbeams. Sensors and Actuators A: Physical 52 (1), 92-98 (1996). https://doi.org/10.1016/0924-4247(96)80131-2
12. Yang T., Bellouard Y. Laser-induced transition between nonlinear and linear resonant behaviors of a micromechanical oscillator. Phys. Rev. Applied 7, 064002 (2017). https://doi.org/10.1103/PhysRevApplied.7.064002
13. Dolleman R. J., Houri S., Chandrashekar A., Alijani F., van der Zant H. S. J., Steeneken P. G. Opto-thermally excited multimode parametric resonance in graphene membranes. Sci. Rep. 8, 9366 (2018). https://doi.org/10.1038/s41598-018-27561-4
14. Zehnder A. T., Rand R. H., Krylov S. Locking of electrostatically coupled thermo-optically driven MEMS limit cycle oscillators. International Journal of Non-linear Mechanics 102, 92-100 (2018). https://doi.org/10.1016/J.IJNONLINMEC.2018.03.009
15. Bhaskar A., Shayak B., Rand R. H., Zehnder A. T. Synchronization characteristics of an array of coupled MEMS limit cycle oscillators. International Journal of Non-Linear Mechanics 128, 103634 (2021). https://doi.org/10.1016/j.ijnonlinmec.2020.103634
16. Morozov N. F., Tovstik P. E. Dynamic loss of stability of a rod under longitudinal load lower than the Eulerian load. Dokl. Phys. 58, 510-513 (2013). https://doi.org/10.1134/S102833581311013X
17. Carvalho E. C., Goncalves P. B., Rega G. Multiple internal resonances and nonplanar dynamics of a cruciform beam with low torsional stiffness. International Journal of Solids and Structures 121, 117-134 (2017). https://doi.org/10.1016/j.ijsolstr.2017.05.020
18. Ribeiro E. A. R., Lenci S., Mazzilli C. E. N. Modal localisation in a beam modelled as a continuous system: A discussion on the use of auxiliary oscillators. Journal of Sound and Vibration 485, 115595 (2020). https://doi.org/10.1016/j.jsv.2020.115595
19. Lenci S. Isochronous beams by an inclined Roller Support. Journal of Applied Mechanics 85 (9), 091008 (2018). https://doi.org/10.1115/1.4040453
20. Lacarbonara W., Rega G., Nayfeh A. H. Resonant nonlinear normal modes. Part I: analytical treatment for structural one-dimensional systems. Int. J. Non-Linear Mech. 38 (6), 851-872 (2003). https://doi.org/10.1016/S0020-7462(02)00033-1
21. Manevitch L. New approach to beating phenomenon in coupled nonlinear oscillatory chains. Arch. Appl. Mech. 77, 301-312 (2007). https://doi.org/10.1007/s00419-006-0081-1
22. Kambali P. N., Pandey A. K. Nonlinear coupling of transverse modes of a fixed-fixed microbeam under direct and parametric excitation. Nonlinear Dynamics 87, 1271-1294 (2017). https://doi.org/10.1007/s11071-016-3114-5
23. Clementi F., Lenci S., Rega G. 1:1 internal resonance in a two d. o. f. complete system: a comprehensive analysis and its possible exploitation for design. Meccanica 55, 1309-1332 (2020). https://doi.org/10.1007/s11012-020-01171-9
24. Ruzziconi L., Jaber N., Kosuru L., Bellaredj M. L., Younis M. I. Experimental and theoretical investigation of the 2:1 internal resonance in the higher-order modes of a MEMS microbeam at elevated excitations. Journal of Sound and Vibration 499, 115983 (2021). https://doi.org/10.1016/j.jsv.2021.115983
25. Leamy M. J., Gottlieb O. Internal resonances in whirling strings involving longitudinal dynamics and material non-linearities. Journal of Sound and Vibration 236, 683-703 (2000). https://doi.org/10.1006/jsvi.2000.3039
26. Kovriguine D. A., Maugin G. A., Potapov A. I. Multiwave nonlinear couplings in elastic structures. Part I. One-dimensional examples. International Journal of Solids and Structures 39, 5571-5583 (2002). https://doi.org/10.1016/S0020-7683(02)00365-7
27. Kovriguine D. A., Maugin G. A., Potapov A. I. Multiwave non-linear couplings in elastic structures. Part II: Two-dimensional example. Journal of Sound and Vibration 263 (5), 1055-1069 (2003). https://doi.org/10.1016/S0022-460X(03)00274-8
28. Srinil N., Rega G. Nonlinear longitudinal/transversal modal interactions in highly extensible suspended cables. Journal of Sound and Vibration 310, 230-242 (2008). https://doi.org/10.1016/j.jsv.2007.07.056
29. Yang X.-D., Zhang W. Nonlinear dynamics of axially moving beam with coupled longitudinaltransversal vibrations. Nonlinear Dynamics 78, 2547-2556 (2014). https://doi.org/10.1007/s11071-014-1609-5
30. Беляев А. К., Морозов Н. Ф., Товстик П. Е., Товстик Т. П. Устойчивость гибкого вертикального стержня на вибрирующем основании. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 5 (63), вып. 3, 477-488 (2018). https://doi.org/10.21638/11701/spbu01.2018.311
31. Беляев А. К., Ма Ч.-Ч., Морозов Н. Ф., Товстик П. Е., Товстик Т. П., Шурпатов А. О. Динамика стержня при продольном ударе телом. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 4 (62), вып. 3, 506-515 (2017).
32. Saetta E., Settimi V., Rega G. Minimal thermal modeling of two-way thermomechanically coupled plates for nonlinear dynamics investigation. Journal of Thermal Stresses 43 (3), 345-371 (2020). https://doi.org/10.1080/01495739.2019.1704669
33. Indeitsev D. A., Osipova E. V. A Two-temperature model of optical excitation of acoustic waves in conductors. Dokl. Phys. 62 (3), 136-140 (2017). https://doi.org/10.1134/S1028335817030065
34. Sun Y., Liu S., Rao Z., Li Y., Yang J. Thermodynamic response of beams on Winkler foundation irradiated by moving laser pulses. Symmetry 10 (8), 328 (2018). https://doi.org/10.3390/sym10080328
35. Wen C., Tang L., Yang G. Buckling and post-buckling of pinned Euler beams on weakened Winkler foundation under thermal loading. Journal of Thermal Stresses 43 (5), 529-542 (2020). https://doi.org/10.1080/01495739.2020.1734128
36. Gu B., He T. Investigation of thermoelastic wave propagation in Euler - Bernoulli Beam via nonlocal strain gradient elasticity and G-N Theory. Journal of Vibration Engineering & Technologies 9 (5), 715-724 (2021). https://doi.org/10.1007/s42417-020-00277-4
37. Morozov N. F., Indeitsev D. A., Lukin A. V., Popov I. A., Privalova O. V., Shtukin L. V. Stability of the Bernoulli - Euler Beam in coupled electric and thermal fields. Dokl. Phys. 63, 342-347 (2018). https://doi.org/10.1134/S1028335818080086
38. Morozov N. F., Indeitsev D. A., Lukin A. V., Popov I. A., Privalova O. V., Semenov B. N., Shtukin L. V. Bernoulli - Euler beam under action of a moving thermal source: characteristics of the dynamic behavior. Dokl. Phys. 64, 185-188 (2019). https://doi.org/10.1134/S1028335819040050
39. Morozov N. F., Indeitsev D. A., Lukin A. V., Popov I. A., Privalova O. V., Shtukin L. V. Stability of the Bernoulli - Euler Beam under the action of a moving thermal source. Dokl. Phys. 65, 67-71 (2020). https://doi.org/10.1134/S102833582002007X
40. Morozov N. F., Indeitsev D. A., Lukin A. V., Popov I. A., Shtukin L. V. Nonlinear interaction of longitudinal and transverse vibrations of a rod at an internal combinational resonance in view of opto-thermal excitation of N/MEMS. Journal of Sound and Vibration 509, 116247 (2021). https://doi.org/10.1016/j.jsv.2021.116247
41. Морозов Н. Ф., Индейцев Д. А., Лукин А. В., Попов И. А., Штукин Л. В. Нелинейное модальное взаимодействие продольных и изгибных колебаний балочного резонатора при периодическом тепловом нагружении. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 9 (67), вып. 2, 317-337 (2022). https://doi.org21638/spbu01.2022.212
42. Tang D.W., Araki N. Wavy, wavelike, diffusive thermal responses of finite rigid slabs to highspeed heating of laser-pulses. International Journal of Heat and Mass Transfer 42, 855-860 (1999). https://doi.org/10.1016/S0017-9310(98)00244-0
43. Cole K., Beck J., Haji-Sheikh A., Litkouhi B. Heat Conduction Using Green’s Functions. Taylor & Francis (2011).
References
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2. Yan G., Raetz S., Chigarev N., Blondeau Ja., Gusev V. E., Tournat V. Cumulative fatigue damage in thin aluminum films evaluated non-destructively with lasers via zero-group-velocity Lamb modes. NDT & E International 116, 102323 (2020). https://doi.org/10.1016/j.ndteint.2020.102323
3. Pan Yu., Rossignol C., Audoin B. Acoustic waves generated by a laser line pulse in cylinders; Application to the elastic constants measurement. J. Acoust. Soc. Am. 115 (4), 1537-1545 (2004). https://doi.org/10.1121/1.1651191
4. Chow G., Uchaker E., Cao G., Wang Ju. Laser-induced surface acoustic waves: An alternative method to nanoindentation for the mechanical characterization of porous nanostructured thin film electrode media. Mechanics of Materials 91, 333-342 (2015). https://doi.org/10.1016/J.MECHMAT.2015.10.005
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6. Otsuka P. H., Mezil S., Matsuda O., Tomoda M., Maznev A. A., Gan T., Fang N., Boechler N., Gusev V. E., Wright O. B. Time-domain imaging of gigahertz surface waves on an acoustic metamaterial. New Journal of Physics 20, 013026 (2018). https://doi.org/10.1088/1367-2630/AA9298
7. Li C., Guan G., Zhang F., Nabi G., Wang R. K., Huang Z. Laser induced surface acoustic wave combined with phase sensitive optical coherence tomography for superficial tissue characterization: a solution for practical application. Biomedical Optics Express 5 (5), 1403-1418 (2014). https://doi.org/10.1364/BOE.5.001403
8. Phinney L. M., Klody K. A., Sackos Jo. T., Walraven Je. A. Damage of MEMS thermal actuators heated by laser irradiation. Reliability, Packaging, Testing and Characterization of MEMS/MOEMS IV. Proceedings of MOEMS-MEMS Micro and Nanofabrication, 2005, San Jose, California, United States, 5716, 81-88 (2005). https://doi.org/10.1117/12.594408
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10. Mai A., Bunce C., H¨ubner R., Pahner D., Dauderst¨adt U. A. In situ bow change of Al-alloy MEMS micromirrors during 248-nm laser irradiation. Journal of Micro/Nanolithography, MEMS and MOEMS 15 (3), 035502 (2016). https://doi.org/10.1117/1.JMM.15.3.035502
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12. Yang T., Bellouard Y. Laser-induced transition between nonlinear and linear resonant behaviors of a micromechanical oscillator. Phys. Rev. Applied 7, 064002 (2017). https://doi.org/10.1103/PhysRevApplied.7.064002
13. Dolleman R. J., Houri S., Chandrashekar A., Alijani F., van der Zant H. S. J., Steeneken P. G. Opto-thermally excited multimode parametric resonance in graphene membranes. Sci. Rep. 8, 9366 (2018). https://doi.org/10.1038/s41598-018-27561-4
14. Zehnder A. T., Rand R. H., Krylov S. Locking of electrostatically coupled thermo-optically driven MEMS limit cycle oscillators. International Journal of Non-linear Mechanics 102, 92-100 (2018). https://doi.org/10.1016/J.IJNONLINMEC.2018.03.009
15. Bhaskar A., Shayak B., Rand R. H., Zehnder A. T. Synchronization characteristics of an array of coupled MEMS limit cycle oscillators. International Journal of Non-Linear Mechanics 128, 103634 (2021). https://doi.org/10.1016/j.ijnonlinmec.2020.103634
16. Morozov N. F., Tovstik P. E. Dynamic loss of stability of a rod under longitudinal load lower than the Eulerian load. Dokl. Phys. 58, 510-513 (2013). https://doi.org/10.1134/S102833581311013X
17. Carvalho E. C., Goncalves P. B., Rega G. Multiple internal resonances and nonplanar dynamics of a cruciform beam with low torsional stiffness. International Journal of Solids and Structures 121, 117-134 (2017). https://doi.org/10.1016/j.ijsolstr.2017.05.020
18. Ribeiro E. A. R., Lenci S., Mazzilli C. E. N. Modal localisation in a beam modelled as a continuous system: A discussion on the use of auxiliary oscillators. Journal of Sound and Vibration 485, 115595 (2020). https://doi.org/10.1016/j.jsv.2020.115595
19. Lenci S. Isochronous beams by an inclined Roller Support. Journal of Applied Mechanics 85 (9), 091008 (2018). https://doi.org/10.1115/1.4040453
20. Lacarbonara W., Rega G., Nayfeh A. H. Resonant nonlinear normal modes. Part I: analytical treatment for structural one-dimensional systems. Int. J. Non-Linear Mech. 38 (6), 851-872 (2003). https://doi.org/10.1016/S0020-7462(02)00033-1
21. Manevitch L. New approach to beating phenomenon in coupled nonlinear oscillatory chains. Arch. Appl. Mech. 77, 301-312 (2007). https://doi.org/10.1007/s00419-006-0081-1
22. Kambali P. N., Pandey A. K. Nonlinear coupling of transverse modes of a fixed-fixed microbeam under direct and parametric excitation. Nonlinear Dynamics 87, 1271-1294 (2017). https://doi.org/10.1007/s11071-016-3114-5
23. Clementi F., Lenci S., Rega G. 1:1 internal resonance in a two d. o. f. complete system: a comprehensive analysis and its possible exploitation for design. Meccanica 55, 1309-1332 (2020). https://doi.org/10.1007/s11012-020-01171-9
24. Ruzziconi L., Jaber N., Kosuru L., Bellaredj M. L., Younis M. I. Experimental and theoretical investigation of the 2:1 internal resonance in the higher-order modes of a MEMS microbeam at elevated excitations. Journal of Sound and Vibration 499, 115983 (2021). https://doi.org/10.1016/j.jsv.2021.115983
25. Leamy M. J., Gottlieb O. Internal resonances in whirling strings involving longitudinal dynamics and material non-linearities. Journal of Sound and Vibration 236, 683-703 (2000). https://doi.org/10.1006/jsvi.2000.3039
26. Kovriguine D. A., Maugin G. A., Potapov A. I. Multiwave nonlinear couplings in elastic structures. Part I. One-dimensional examples. International Journal of Solids and Structures 39, 5571-5583 (2002). https://doi.org/10.1016/S0020-7683(02)00365-7
27. Kovriguine D. A., Maugin G. A., Potapov A. I. Multiwave non-linear couplings in elastic structures. Part II: Two-dimensional example. Journal of Sound and Vibration 263 (5), 1055-1069 (2003). https://doi.org/10.1016/S0022-460X(03)00274-8
28. Srinil N., Rega G. Nonlinear longitudinal/transversal modal interactions in highly extensible suspended cables. Journal of Sound and Vibration 310, 230-242 (2008). https://doi.org/10.1016/j.jsv.2007.07.056
29. Yang X.-D., Zhang W. Nonlinear dynamics of axially moving beam with coupled longitudinaltransversal vibrations. Nonlinear Dynamics 78, 2547-2556 (2014). https://doi.org/10.1007/s11071-014-1609-5
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33. Indeitsev D. A., Osipova E. V. A Two-temperature model of optical excitation of acoustic waves in conductors. Dokl. Phys. 62 (3), 136-140 (2017). https://doi.org/10.1134/S1028335817030065
34. Sun Y., Liu S., Rao Z., Li Y., Yang J. Thermodynamic response of beams on Winkler foundation irradiated by moving laser pulses. Symmetry 10 (8), 328 (2018). https://doi.org/10.3390/sym10080328
35. Wen C., Tang L., Yang G. Buckling and post-buckling of pinned Euler beams on weakened Winkler foundation under thermal loading. Journal of Thermal Stresses 43 (5), 529-542 (2020). https://doi.org/10.1080/01495739.2020.1734128
36. Gu B., He T. Investigation of thermoelastic wave propagation in Euler - Bernoulli Beam via nonlocal strain gradient elasticity and G-N Theory. Journal of Vibration Engineering & Technologies 9 (5), 715-724 (2021). https://doi.org/10.1007/s42417-020-00277-4
37. Morozov N. F., Indeitsev D. A., Lukin A. V., Popov I. A., Privalova O. V., Shtukin L. V. Stability of the Bernoulli - Euler Beam in coupled electric and thermal fields. Dokl. Phys. 63, 342-347 (2018). https://doi.org/10.1134/S1028335818080086
38. Morozov N. F., Indeitsev D. A., Lukin A. V., Popov I. A., Privalova O. V., Semenov B. N., Shtukin L. V. Bernoulli - Euler beam under action of a moving thermal source: characteristics of the dynamic behavior. Dokl. Phys. 64, 185-188 (2019). https://doi.org/10.1134/S1028335819040050
39. Morozov N. F., Indeitsev D. A., Lukin A. V., Popov I. A., Privalova O. V., Shtukin L. V. Stability of the Bernoulli - Euler Beam under the action of a moving thermal source. Dokl. Phys. 65, 67-71 (2020). https://doi.org/10.1134/S102833582002007X
40. Morozov N. F., Indeitsev D. A., Lukin A. V., Popov I. A., Shtukin L. V. Nonlinear interaction of longitudinal and transverse vibrations of a rod at an internal combinational resonance in view of opto-thermal excitation of N/MEMS. Journal of Sound and Vibration 509, 116247 (2021). https://doi.org/10.1016/j.jsv.2021.116247
41. Morozov N. I., Indeitsev D. A., Lukin A. V., Popov I. A., Shtukin L. V. Nonlinear modal interaction between longitudinal and bending vibrations of a beam resonator under periodic thermal loading. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 9 (67), iss. 2, 317-337 (2022). https://doi.org21638/spbu01.2022.212 (In Russian) [Eng. transl.: Vestnik St Petersburg University. Mathematics 55, iss. 2, 212-228 (2022). https://doi.org/10.1134/S106345412202008X].
42. Tang D.W., Araki N. Wavy, wavelike, diffusive thermal responses of finite rigid slabs to highspeed heating of laser-pulses. International Journal of Heat and Mass Transfer 42, 855-860 (1999). https://doi.org/10.1016/S0017-9310(98)00244-0
43. Cole K., Beck J., Haji-Sheikh A., Litkouhi B. Heat Conduction Using Green’s Functions. Taylor & Francis (2011).
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Опубликован
10.05.2023
Как цитировать
Морозов, Н. Ф., Индейцев, Д. А., Лукин, А. В., Попов, И. А., & Штукин, Л. В. (2023). О термооптическом возбуждении параметрических колебаний микробалочных резонаторов. I. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия, 10(2), 315–333. https://doi.org/10.21638/spbu01.2023.212
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Механика
Лицензия
Статьи журнала «Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия» находятся в открытом доступе и распространяются в соответствии с условиями Лицензионного Договора с Санкт-Петербургским государственным университетом, который бесплатно предоставляет авторам неограниченное распространение и самостоятельное архивирование.
