Structural Engineering and Mechanics

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Non-stationary vibration and super-harmonic resonances of nonlinear viscoelastic nano-resonators

  • Ajri, Masoud (School of Mechanical Engineering, College of Engineering, University of Tehran) ;
  • Rastgoo, Abbas (School of Mechanical Engineering, College of Engineering, University of Tehran) ;
  • Fakhrabadi, Mir Masoud Seyyed (School of Mechanical Engineering, College of Engineering, University of Tehran)
  • Received : 2018.04.09
  • Accepted : 2019.03.12
  • Published : 2019.06.10
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Abstract

This paper analyzes the non-stationary vibration and super-harmonic resonances in nonlinear dynamic motion of viscoelastic nano-resonators. For this purpose, a new coupled size-dependent model is developed for a plate-shape nano-resonator made of nonlinear viscoelastic material based on modified coupled stress theory. The virtual work induced by viscous forces obtained in the framework of the Leaderman integral for the size-independent and size-dependent stress tensors. With incorporating the size-dependent potential energy, kinetic energy, and an external excitation force work based on Hamilton's principle, the viscous work equation is balanced. The resulting size-dependent viscoelastically coupled equations are solved using the expansion theory, Galerkin method and the fourth-order Runge-Kutta technique. The Hilbert-Huang transform is performed to examine the effects of the viscoelastic parameter and initial excitation values on the nanosystem free vibration. Furthermore, the secondary resonance due to the super-harmonic motions are examined in the form of frequency response, force response, Poincare map, phase portrait and fast Fourier transforms. The results show that the vibration of viscoelastic nanosystem is non-stationary at higher excitation values unlike the elastic ones. In addition, ignoring the small-size effects shifts the secondary resonance, significantly.

Keywords

References

  1. Aifantis, E.C. (1999), Strain Gradient Interpretation of Size Effects, Springer, Germany.
  2. Ajri, M., Fakhrabadi, M.M.S. and A. Rastgoo (2018a), "Analytical solution for nonlinear dynamic behavior of viscoelastic nanoplates modeled by consistent couple stress theory", Latin American J. Solids Struct., 15(9),1-23. http://doi.org/10.1590/1679-78254918.
  3. Ajri, M., Fakhrabadi, M.M.S. and A. Rastgoo (2018b), "Primary and secondary resonance analyses of viscoelastic nanoplates based on strain gradient theory", J. Appl. Mech., 10(10), 1850109. https://doi.org/10.1142/S1758825118501090
  4. Akbas, S.D. (2016), "Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium", Smart Struct. Syst., 18(6), 1125-1143. http://doi.org/10.12989/sss.2016.18.6.1125.
  5. Amabili, M. (2004), "Nonlinear vibrations of rectangular plates with different boundary conditions: theory and experiments", Comput. Struct., 82(31-32), 2587-2605. https://doi.org/10.1016/j.compstruc.2004.03.077.
  6. Asghari, M. (2012), "Geometrically nonlinear micro-plate formulation based on the modified couple stress theory", J. Eng. Sci., 51, 292-309. https://doi.org/10.1016/j.ijengsci.2011.08.013.
  7. Babaei, A., Noorani, M.R.S. and Ghanbari, A. (2017), "Temperature-dependent free vibration analysis of functionally graded micro-beams based on the modified couple stress theory", Microsyst. Technol., 23(10), 4599-4610. https://doi.org/10.1007/s00542-017-3285-0.
  8. Baghelani, M. (2016), "Design of a multi-frequency resonator for UHF multiband communication applications", Microsyst. Technol., 22(10), 2543-2548. https://doi.org/10.1007/s00542-015-2639-8.
  9. Braun, T., Barwich, V., Ghatkesar, M.K., Bredekamp, A.H., Gerber, C., Hegner, M. and Lang, H.P. (2005), "Micromechanical mass sensors for biomolecular detection in a physiological environment", Physical Review E, 72(3), 031907. https://doi.org/10.1103/PhysRevE.72.031907.
  10. Budakian, R., Mamin, H. and Rugar, D. (2006), "Spin manipulation using fast cantilever phase reversals", Appl. Phys. Lett., 89(11), 113113. https://doi.org/10.1063/1.2349311.
  11. Christensen, R.M. and Freund, L. (1971), "Theory of viscoelasticity", J. Appl. Mech., 38, 720. https://doi.org/10.1115/1.3408900
  12. Domaneschi, M., Limongelli, M.P. and Martinelli, L. (2013), "Vibration based damage localization using MEMS on a suspension bridge model", Smart Struct. Syst., 12(6), 679-694. https://doi.org/10.12989/sss.2013.12.6.679.
  13. Ehyaei, J. and Akbarizadeh, M.R. (2017), "Vibration analysis of micro composite thin beam based on modified couple stress", Struct. Eng. Mech., 64(4), 403-411. https://doi.org/10.12989/sem.2017.64.4.403.
  14. Ekinci, K., Huang, X. and Roukes, M. (2004), "Ultrasensitive nanoelectromechanical mass detection", Appl. Phys. Lett., 84(22), 4469-4471. https://doi.org/10.1063/1.1755417.
  15. Elwenspoek, M. and Jansen, H.V. (2004), Silicon Micromachining, Cambridge University Press, United Kingdom.
  16. Fu, H. and Qian, Y. (2018), "Study on a Multi-Frequency Homotopy Analysis Method for Period-Doubling Solutions of Nonlinear Systems", J. Bifurcation Chaos, 28(04), 1850049. https://doi.org/10.1142/S0218127418500499.
  17. Fu, Y. and Zhang, J. (2009), "Nonlinear static and dynamic responses of an electrically actuated viscoelastic microbeam", Acta Mechanica Sinica, 25(2), 211-218. https://doi.org/10.1007/s10409-008-0216-4.
  18. Ghadiri, M., Mahinzare, M., Shafiei, N. and Ghorbani, K. (2017), "On size-dependent thermal buckling and free vibration of circular FG Microplates in thermal environments", Microsyst. Technol., 23(10), 4989-5001. https://doi.org/10.1007/s00542-017-3308-x.
  19. Ghayesh, M.H. and Amabili, M. (2012), "Nonlinear dynamics of axially moving viscoelastic beams over the buckled state", Comput. Struct., 112, 406-421. https://doi.org/10.1016/j.compstruc.2012.09.005.
  20. Ghayesh, M.H., Farokhi, H. and Amabili, M. (2013), "Nonlinear behaviour of electrically actuated MEMS resonators", J. Eng. Sci., 71, 137-155. https://doi.org/10.1016/j.ijengsci.2013.05.006.
  21. Ghayesh, M.H., Farokhi, H., Hussain, S., Gholipour, A. and Arjomandi, M. (2017), "A size-dependent nonlinear third-order shear-deformable dynamic model for a microplate on an elastic medium", Microsyst. Technol., 23(8), 3281-3299. https://doi.org/10.1007/s00542-016-3096-8.
  22. Gurtin, M.E. and Murdoch, A.I. (1975), "A continuum theory of elastic material surfaces", Arch Rat. Mech. Anal., 57(4), 291-323. https://doi.org/10.1007/BF00261375.
  23. Hashemi, S.H., Mehrabani, H. and Ahmadi-Savadkoohi, A. (2015), "Exact solution for free vibration of coupled double viscoelastic graphene sheets by viscoPasternak medium", Compos. Part B, 78, 377-383. https://doi.org/10.1016/j.compositesb.2015.04.008.
  24. Huang, N.E., Shen, Z. and Long, S.R. (1999), "A new view of nonlinear water waves: the Hilbert spectrum", Annual Rev. Fluid Mech., 31(1), 417-457. https://doi.org/10.1146/annurev.fluid.31.1.417
  25. Huang, N.E., Shen, Z., Long, S.R., Wu, M.C., Shih, H.H., Zheng, Q., Yen, N.C., Tung, C.C. and Liu, H.H. (1998). "The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis", Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. https://doi.org/10.1098/rspa.1998.0193.
  26. Huang, X., Feng, X., Zorman, C., Mehregany, M. and Roukes, M. (2005), "VHF, UHF and microwave frequency nanomechanical resonators", New J. Phys., 7(1), 247. https://doi.org/10.1088/1367-2630/7/1/247
  27. Huber, T.M., Abell, B.C., Mellema, D.C., Spletzer, M. and Raman, A. (2010), "Mode-selective noncontact excitation of microcantilevers and microcantilever arrays in air using the ultrasound radiation force", Appl. Phys. Lett., 97(21), 214101. https://doi.org/10.1063/1.3521256.
  28. Hwang, D.G., Chae, Y.M., Choi, N., Cho, I.J., Kang, J.Y. and Lee, S.H. (2017), "Label-free detection of prostate specific antigen (PSA) using a bridge-shaped PZT resonator", Microsyst. Technol., 23(5), 1207-1214. https://doi.org/10.1007/s00542-015-2804-0.
  29. Jamalpoor, A., Bahreman, M. and Hosseini, M. (2017), "Free transverse vibration analysis of orthotropic multi-viscoelastic microplate system embedded in visco-Pasternak medium via modified strain gradient theory", J. Sandwich Struct. Mater., https://doi.org/10.1177/1099636216689384.
  30. JE., L. (1989), Boundary Stabilization of Thin Plates, SIAM, Philadelphia, USA.
  31. Jiang, J.W., Wang, J.S. and Li, B. (2009), "Young's modulus of graphene: a molecular dynamics study", Phys. Rev. B, 80(11), 113405. https://doi.org/10.1103/PhysRevB.80.113405.
  32. Jomehzadeh, E., Noori, H. and Saidi, A. (2011), "The sizedependent vibration analysis of micro-plates based on a modified couple stress theory", Physica E: Low-dimensional Systems and Nanostructures, 43(4), 877-883. https://doi.org/10.1016/j.physe.2010.11.005.
  33. Kalyanaraman, R., Rinaldi, G., Packirisamy, M. and Bhat, R. (2013), "Equivalent area nonlinear static and dynamic analysis of electrostatically actuated microstructures", Microsyst. Technol., 19(1), 61-70. https://doi.org/10.1007/s00542-012-1621-y.
  34. Karlicic, D., Kozic, P. and Pavlovic, R. (2014), "Free transverse vibration of nonlocal viscoelastic orthotropic multi-nanoplate system (MNPS) embedded in a viscoelastic medium", Compos. Struct., 115, 89-99. https://doi.org/10.1016/j.compstruct.2014.04.002.
  35. Ke, L.-L., Wang, Y.-S., Yang, J. and Kitipornchai, S. (2012), "Free vibration of size-dependent Mindlin microplates based on the modified couple stress theory", J. Sound Vib., 331(1), 94-106. https://doi.org/10.1016/j.jsv.2011.08.020.
  36. Khaniki, H.B. and Hosseini-Hashemi, S. (2017), "Dynamic response of biaxially loaded double-layer viscoelastic orthotropic nanoplate system under a moving nanoparticle", J. Eng. Sci., 115, 51-72. https://doi.org/10.1016/j.ijengsci.2017.02.005.
  37. Lam, D.C., Yang, F., Chong, A., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids, 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X.
  38. Leaderman, H. (1962), "Large longitudinal retarded elastic deformation of rubberlike network polymers", Transac. Soc. Rheology, 6(1), 361-382. https://doi.org/10.1122/1.548932.
  39. Lee, H.J., Zhang, P. and Bravman, J.C. (2005), "Stress relaxation in free-standing aluminum beams", Thin Solid Films, 476(1), 118-124. https://doi.org/10.1016/j.tsf.2004.10.001.
  40. Leng, H. and Lin, Y. (2011), "A MEMS/NEMS sensor for human skin temperature measurement", Smart Struct. Syst., 8(1), 53-67. https://doi.org/10.12989/sss.2011.8.1.053.
  41. Li, M., Tang, H.X. and Roukes, M.L. (2007), "Ultra-sensitive NEMS-based cantilevers for sensing, scanned probe and very high-frequency applications", Nature Nanotechnol., 2(2), 114. https://doi.org/10.1038/nnano.2006.208
  42. Liu, J., Zhang, Y. and Fan, L. (2017), "Nonlocal vibration and biaxial buckling of double-viscoelastic-FGM-nanoplate system with viscoelastic Pasternak medium in between", Phys. Lett. A, 381(14), 1228-1235. https://doi.org/10.1016/j.physleta.2017.01.056.
  43. Lopez, G. (2013), "Diamond as a solid state quantum computer with a linear chain of nuclear spins system", arXiv preprint arXiv:1310.0750. https://doi.org/10.4236/jmp.2014.51009.
  44. Lou, J. and He, L. (2015), "Closed-form solutions for nonlinear bending and free vibration of functionally graded microplates based on the modified couple stress theory", Compos. Struct., 131, 810-820. https://doi.org/10.1016/j.compstruct.2015.06.031.
  45. Ma, H., Gao, X.L. and Reddy, J. (2011), "A non-classical Mindlin plate model based on a modified couple stress theory", Acta Mechanica, 220(1-4), 217-235. https://doi.org/10.1007/s00707-011-0480-4
  46. Mindlin, R. and Tiersten, H. (1962), "Effects of couple-stresses in linear elasticity", Arch Rat. Mech. Anal., 11(1), 415-448. https://doi.org/10.1007/BF00253946.
  47. Mockensturm, E.M. and Guo, J. (2005), "Nonlinear vibration of parametrically excited, viscoelastic, axially moving strings", J. Appl. Mech., 72(3), 374-380. https://doi.org/10.1115/1.1827248
  48. Mohammadimehr, M., Navi, B.R. and Arani, A.G. (2015), "Free vibration of viscoelastic double-bonded polymeric nanocomposite plates reinforced by FG-SWCNTs using MSGT, sinusoidal shear deformation theory and meshless method", Compos. Struct., 131, 654-671. https://doi.org/10.1016/j.compstruct.2015.05.077.
  49. Naik, A., Hanay, M., Hiebert, W., Feng, X. and Roukes, M. (2009), "Towards single-molecule nanomechanical mass spectrometry", Nature Nanotechnol., 4(7), 445. https://doi.org/10.1038/nnano.2009.152
  50. Niyogi, A. (1973), "Nonlinear bending of rectangular orthotropic plates", J. Solid. Struct., 9(9), 1133-1139. https://doi.org/10.1016/0020-7683(73)90020-6.
  51. Pan, Z. and Chen, J. (2017), "Measurements of pedestrian", Struct. Eng. Mech., 63(6).
  52. Paolino, P. and Bellon, L. (2009), "Frequency dependence of viscous and viscoelastic dissipation in coated micro-cantilevers from noise measurement", Nanotechnol., 20(40), 405705. https://doi.org/10.1088/0957-4484/20/40/405705
  53. Park, W.T., Han, S.C., Jung, W.Y. and Lee, W.H. (2016), "Dynamic instability analysis for S-FGM plates embedded in Pasternak elastic medium using the modified couple stress theory", Steel Compos. Struct., 22(6), 1239-1259. https://doi.org/10.12989/scs.2016.22.6.1239.
  54. Qian, Y. and Fu, H. (2017), "Research for coupled van der Pol systems with parametric excitation and its application", Zeitschrift fur Naturforschung A, 72(11), 1009-1020. https://doi.org/10.1515/zna-2017-0249.
  55. Qian, Y., Fu, H. and Guo, J. (2018), "Weakly resonant double Hopf bifurcation in coupled nonlinear systems with delayed freedback and application of homotopy analysis method", J. Low Frequency Noise, Vib. Active Control, ttps://doi.org/10.1177/1461348418765975.
  56. Qian, Y. and Zhang, Y. (2017), "Optimal extended homotopy analysis method for Multi-Degree-of-Freedom nonlinear dynamical systems and its application", Struct. Eng. Mech., 61(1), 105-116. https://doi.org/10.12989/sem.2017.61.1.105.
  57. Reddy, J. and Kim, J. (2012), "A nonlinear modified couple stressbased third-order theory of functionally graded plates", Compos. Struct., 94(3), 1128-1143. https://doi.org/10.1016/j.compstruct.2011.10.006.
  58. Sato, M., Hubbard, B., Sievers, A., Ilic, B., Czaplewski, D. and Craighead, H. (2003), "Observation of locked intrinsic localized vibrational modes in a micromechanical oscillator array", Physical Rev. Lett., 90(4), https://doi.org/10.1103/PhysRevLett.90.044102.
  59. Saulson, P.R. (1990), "Thermal noise in mechanical experiments", Physical Review D, 42(8), 2437. https://doi.org/10.1103/PhysRevD.42.2437.
  60. Setoodeh, A. and Rezaei, M. (2017), "Large amplitude free vibration analysis of functionally graded nano/micro beams on nonlinear elastic foundation", Struct. Eng. Mech., 61(2), 209-220. https://doi.org/10.12989/sem.2017.61.2.209.
  61. Shim, S.-B., Imboden, M. and Mohanty, P. (2007), "Synchronized oscillation in coupled nanomechanical oscillators", Science, 316(5821), 95-99. https://doi.org/10.1126/science.1137307.
  62. Shinozuka, M., Chou, P.H., Kim, S., Kim, H., Karmakar, D. and Lu, F. (2010), "Non-invasive acceleration-based methodology for damage detection and assessment of water distribution system", Smart Struct. Syst., 6(6).
  63. Smart, J. and Williams, J. (1972), "A comparison of single-integral non-linear viscoelasticity theories", J. Mech. Phys. Solids, 20(5), 313-324. https://doi.org/10.1016/0022-5096(72)90027-0.
  64. Su, Y., Wei, H., Gao, R., Yang, Z., Zhang, J., Zhong, Z. and Zhang, Y. (2012), "Exceptional negative thermal expansion and viscoelastic properties of graphene oxide paper", Carbon, 50(8), 2804-2809. https://doi.org/10.1016/j.carbon.2012.02.045.
  65. Tajaddodianfar, F., Yazdi, M.R.H. and Pishkenari, H.N. (2017), "Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method", Microsyst. Technol., 23(6), 1913-1926. https://doi.org/10.1007/s00542-016-2947-7.
  66. Tang, Y.Q. and Chen, L.Q. (2012), "Parametric and internal resonances of in-plane accelerating viscoelastic plates", Acta Mechanica, 223(2), 415-431. https://doi.org/10.1007/s00707-011-0567-y.
  67. Teh, K.S. and Lin, L. (1999), "Time-dependent buckling phenomena of polysilicon micro beams", Microelectronic. J., 30(11), 1169-1172. https://doi.org/10.1016/S0026-2692(99)00081-6.
  68. Toupin, R.A. (1962), "Elastic materials with couple-stresses", Arch Rat. Mech. Anal., 11(1), 385-414. https://doi.org/10.1007/BF00253945.
  69. Tuck, K., Jungen, A., Geisberger, A., Ellis, M. and Skidmore, G. (2005), "A study of creep in polysilicon MEMS devices", J. Eng. Mater. Technol., 127(1), 90-96. https://doi.org/10.1016/j.physe.2014.11.007.
  70. Wang, Y., Li, F.M. and Wang, Y.Z. (2015), "Nonlinear vibration of double layered viscoelastic nanoplates based on nonlocal theory", Physica E: Low-dimensional Systems and Nanostructures, 67, 65-76. https://doi.org/10.1016/j.physe.2014.11.007.
  71. Yan, X., Brown, W., Li, Y., Papapolymerou, J., Palego, C., Hwang, J. and Vinci, R. (2009), "Anelastic stress relaxation in gold films and its impact on restoring forces in MEMS devices", J. Microelectromech. Syst., 18(3), 570-576. https://doi.org/10.1109/JMEMS.2009.2016280
  72. Yang, F., Chong, A., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", J. Solid. Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X.
  73. Yang, Y., Callegari, C., Feng, X., Ekinci, K. and Roukes, M. (2006), "Zeptogram-scale nanomechanical mass sensing", Nano Lett., 6(4), 583-586. https://doi.org/10.1021/nl052134m.