Nonlinear modal interaction between longitudinal and bending vibrations of a beam resonator under periodic thermal loading

Authors

  • Nikita F. Morozov St Petersburg State University, 7–9, Universitetskaya nab., St Petersburg, 199034, Russian Federation; Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, 61, V.O., Bolshoi pr., St Petersburg, 199178, Russian Federation
  • Dmitriy A. Indeitsev Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, 61, V.O., Bolshoi pr., St Petersburg, 199178, Russian Federation; Peter the Great St Petersburg Polytechnic University, 29, ul. Polytechnicheskaya, StPetersburg, 195251, Russian Federation
  • Alexei V. Lukin Peter the Great St Petersburg Polytechnic University, 29, ul. Polytechnicheskaya, StPetersburg, 195251, Russian Federation
  • Ivan A. Popov Peter the Great St Petersburg Polytechnic University, 29, ul. Polytechnicheskaya, StPetersburg, 195251, Russian Federation
  • Lev V. Shtukin St Petersburg State University, 7–9, Universitetskaya nab., St Petersburg, 199034, Russian Federation; Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, 61, V.O., Bolshoi pr., St Petersburg, 199178, Russian Federation

DOI:

https://doi.org/10.21638/spbu01.2022.212

Abstract

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

Downloads

Download data is not yet available.
 

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).

Downloads

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

Issue

Vol. 9 No. 2 (2022)

Section

Mechanics

License

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.