On opto-thermally excited parametric oscillations of microbeam resonators. I
DOI:
https://doi.org/10.21638/spbu01.2023.212Abstract
The present article is the first part of the work devoted to investigation of the nonlinear dynamics of parametrically excited flexural vibrations of a clamped-clamped microbeam - the basic sensitive element of a promising class of microsensors of various physical quantities - under laser thermooptical action in the form of periodically generated pulses acting on a certain part of the surface of the beam element. An analytical solution of the heat transfer problem is found for the steady harmonic distribution of temperature in the volume of the resonator. The static and dynamic components of temperature-induced axial force and bending moment are determined. Using the Galerkin method, the discretization of nonlinear coupled partial differential equations describing the longitudinal-flexural oscillations of the resonator is performed. Using the asymptotic method of multiple time-scales, an approximate analytical solution is obtained for the nonlinear dynamics problem under the conditions of primary parametric resonance.Keywords:
nonlinear dynamics, parametric oscillations, Bernoulli - Euler beam, modal interaction, laser-induced opto-thermal excitation
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Литература
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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
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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
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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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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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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
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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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2023-05-10
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Morozov, N. F., Indeitsev, D. A., Lukin, A. V., Popov, I. A., & Shtukin, L. V. (2023). On opto-thermally excited parametric oscillations of microbeam resonators. I. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 10(2), 315–333. https://doi.org/10.21638/spbu01.2023.212
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
