Nonlinear modal interaction between longitudinal and bending vibrations of a beam resonator under periodic thermal loading
DOI:
https://doi.org/10.21638/spbu01.2022.212Abstract
The paper investigates the nonlinear modal interaction of longitudinal and bending vibrations of a beam resonator under periodic thermal loading. The mode of parametric oscillations is investigated under conditions of internal multiple resonance between some flexural and longitudinal forms of free oscillations of the resonator. The possibility of generation in the system of the longitudinal-bending mode was found, the frequency of the slow envelope of which essentially depends on the parameter of the internal frequency detuning, which is directly related to the magnitude of external disturbances subject to high-precision measurement.Keywords:
object-oriened modeling, dynamics of a vehicle, rolling disc on the horizontal plane dynamics, dynamics of the wheelset, joint constraint
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References
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 Y. Acoustic waves generated by a laser line pulse in a transversely isotropic cylinder. Appl. Phys. Lett. 82, 4379 (2003). https://doi.org/10.1063/1.1583135
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 (6), 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 J. Phys. 20, 013026 (2018).
7. Li Ch., Guan G., Zhang F., Nabi Gh., Wang R.K., Huang Zh. Laser induced surface acoustic wave combined with phase sensitive optical coherence tomography for superficial tissue characterization: a solution for practical application. Biomed. Opt. Express. 5 (5), 1403–1419 (2014). https://doi.org/10.1364/BOE.5.001403
8. Phinney L.M., Klody K.A., Sackos J. T., Walraven J.A. Damage of MEMS thermal actuators heated by laser irradiation. Proceedings of Reliability, Packaging, Testing, and Characterization of MEMS/MOEMS IV 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 Ch., H¨ubner R., Pahner D., Dauderstadt U.A. In situ bow change of Alalloy MEMS micromirrors during 248-nm laser irradiation. J. 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 (1–3), 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., Zant H. S. J. van der, Steeneken P.G. Opto-thermally excited multimode parametric resonance in graphene membranes. Scientific Reports 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. 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
16. Carvalho E.Ch., Gon¸calves 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
17. Ribeiro E. A. R., Lenci S., Mazzilli C. E. N. Modal localization 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
18. Lenci S. Isochronous Beams by an Inclined Roller Support. J. Appl. Mech. 85 (9), 091008 (2018). https://doi.org/10.1115/1.4040453
19. Lacarbonara W., Rega G., Nayfeh A.H. Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems. International Journal of Non-Linear Mechanics 38 (6), 851–872 (2003). https://doi.org/10.1016/S0020-7462(02)00033-1
20. 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
21. 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
22. Leamy M. J., Gottlieb O. Internal resonances in whirling strings involving longitudinal dynamics and material non-linearities. Journal of Sound and Vibration 236 (4), 683–703 (2000). https://doi.org/10.1006/jsvi.2000.3039
23. Srinil N., Rega G. Nonlinear longitudinal/transversal modal interactions in highly extensible suspended cables. Journal of Sound and Vibration 310 (1–2), 230–242 (2008). https://doi.org/10.1016/j.jsv.2007.07.056
24. Yang X.D., Zhang W. Nonlinear dynamics of axially moving beam with coupled longitudinaltransversal vibrations. Nonlinear Dyn. 78, 2547–2556 (2014). https://doi.org/10.1007/s11071-014-1609-5
25. 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
26. Indeitsev D.A., Osipova E.V. A two-temperature model of optical excitation of acoustic waves in conductors. Dokl. Phys. 62, 136–140 (2017). https://doi.org/10.1134/S1028335817030065
27. Sun Yu., Liu Sh., Rao Zh., Li Yu., 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
28. Wen Ch., 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
29. 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
30. 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
31. 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
32. 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 optothermal excitation of N/MEMS. Journal of Sound and Vibration 509, 116247 (2021). https://doi.org/ 10.1016/j.jsv.2021.116247
33. Nayfeh A.H., Moo D.T. Nonlinear Oscillations. Wiley (1995).
References
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 Y. Acoustic waves generated by a laser line pulse in a transversely isotropic cylinder. Appl. Phys. Lett. 82, 4379 (2003). https://doi.org/10.1063/1.1583135
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 (6), 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 J. Phys. 20, 013026 (2018).
7. Li Ch., Guan G., Zhang F., Nabi Gh., Wang R.K., Huang Zh. Laser induced surface acoustic wave combined with phase sensitive optical coherence tomography for superficial tissue characterization: a solution for practical application. Biomed. Opt. Express. 5 (5), 1403–1419 (2014). https://doi.org/10.1364/BOE.5.001403
8. Phinney L.M., Klody K.A., Sackos J. T., Walraven J.A. Damage of MEMS thermal actuators heated by laser irradiation. Proceedings of Reliability, Packaging, Testing, and Characterization of MEMS/MOEMS IV 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 Ch., H¨ubner R., Pahner D., Dauderstadt U.A. In situ bow change of Alalloy MEMS micromirrors during 248-nm laser irradiation. J. 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 (1–3), 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., Zant H. S. J. van der, Steeneken P.G. Opto-thermally excited multimode parametric resonance in graphene membranes. Scientific Reports 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. 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
16. Carvalho E.Ch., Gon¸calves 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
17. Ribeiro E. A. R., Lenci S., Mazzilli C. E. N. Modal localization 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
18. Lenci S. Isochronous Beams by an Inclined Roller Support. J. Appl. Mech. 85 (9), 091008 (2018). https://doi.org/10.1115/1.4040453
19. Lacarbonara W., Rega G., Nayfeh A.H. Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems. International Journal of Non-Linear Mechanics 38 (6), 851–872 (2003). https://doi.org/10.1016/S0020-7462(02)00033-1
20. 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
21. 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
22. Leamy M. J., Gottlieb O. Internal resonances in whirling strings involving longitudinal dynamics and material non-linearities. Journal of Sound and Vibration 236 (4), 683–703 (2000). https://doi.org/10.1006/jsvi.2000.3039
23. Srinil N., Rega G. Nonlinear longitudinal/transversal modal interactions in highly extensible suspended cables. Journal of Sound and Vibration 310 (1–2), 230–242 (2008). https://doi.org/10.1016/j.jsv.2007.07.056
24. Yang X.D., Zhang W. Nonlinear dynamics of axially moving beam with coupled longitudinaltransversal vibrations. Nonlinear Dyn. 78, 2547–2556 (2014). https://doi.org/10.1007/s11071-014-1609-5
25. 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
26. Indeitsev D.A., Osipova E.V. A two-temperature model of optical excitation of acoustic waves in conductors. Dokl. Phys. 62, 136–140 (2017). https://doi.org/10.1134/S1028335817030065
27. Sun Yu., Liu Sh., Rao Zh., Li Yu., 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
28. Wen Ch., 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
29. 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
30. 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
31. 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
32. 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 optothermal excitation of N/MEMS. Journal of Sound and Vibration 509, 116247 (2021). https://doi.org/ 10.1016/j.jsv.2021.116247
33. Nayfeh A.H., Moo D.T. Nonlinear Oscillations. Wiley (1995).
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Published
2022-07-06
How to Cite
Morozov, N. F., Indeitsev, D. A., Lukin, A. V., Popov, I. A., & Shtukin, L. V. (2022). 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(2), 317–337. https://doi.org/10.21638/spbu01.2022.212
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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.
