Skip to main content
SpringerLink
Log in

Magneto-electro-elastic modelling and nonlinear vibration analysis of bi-directional functionally graded beams

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In the paper, a novel magneto-electro-elastic model of bi-directional (2D) functionally graded materials (FGMs) beams is developed for investigating the nonlinear dynamics. It is shown that the asymmetric modes induced by the 2D FGMs may significantly affect the nonlinear dynamic responses, which is tremendously different from previous studies. Taking into account the geometric nonlinearity, the nonlinear equation of motion and associated boundary conditions for the beams are derived according to the Hamilton’s principle. The natural frequencies and numerical modes of the beams are calculated by the generalized differential quadrature method. The frequency responses of the nonlinear forced vibration are constructed based on the Galerkin technique incorporating with the incremental harmonic balance approach. The influences of the material distributions, length–thickness ratio, electric voltage, magnetic potential as well as boundary condition on the nonlinear resonant frequency and response amplitude are discussed in details. It is notable that increasing in the axial and thickness FG indexes, negative electric potential and positive magnetic potential can lead to decline the nonlinear resonance frequency and amplitude peak, which is usually applied to accurately design the multi-ferroic composite structures. Furthermore, the nonlinear characteristics of motion can be regulated by tuning/tailoring the 2D FG materials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27

Similar content being viewed by others

Data availability

The datasets supporting the conclusion of this article are included within the article.

Abbreviations

1D:

One-directional

2D:

Bi-directional

C–C:

Clamped–clamped

C–H:

Clamped–hinged

FG:

Functionally graded

FGM:

Functionally graded material

FGMs:

Functionally graded materials

FGMEE:

Functional graded magneto-electro-elastic

GDQM:

Generalized differential quadrature method

H–H:

Hinged–hinged

IHB:

Incremental harmonic balance

MEE:

Magneto-electro-elastic

References

  1. Ding, H., Chen, L.Q.: Natural frequencies of nonlinear vibration of axially moving beams. Nonlinear Dyn. 63(1), 125–134 (2011)

    Article  MathSciNet  Google Scholar 

  2. Tan, X., Ding, H., Chen, L.Q.: Nonlinear frequencies and forced responses of pipes conveying fluid via a coupled Timoshenko model. J. Sound Vib. 455, 241–255 (2019)

    Article  Google Scholar 

  3. Tang, Y.Q., Chen, L.Q., Yang, X.D.: Parametric resonance of axially moving Timoshenko beams with time-dependent speed. Nonlinear Dyn. 58(4), 715–724 (2009)

    Article  Google Scholar 

  4. Mao, X.Y., Ding, H., Chen, L.Q.: Steady-state response of a fluid-conveying pipe with 3:1 internal resonance in supercritical regime. Nonlinear Dyn. 86(2), 795–809 (2016)

    Article  Google Scholar 

  5. Ding, H., Lu, Z.Q., Chen, L.Q.: Nonlinear isolation of transverse vibration of pre-pressure beams. J. Sound Vib. 442, 738–751 (2019)

    Article  Google Scholar 

  6. Chen, L.Q., Zu, J.W.: Solvability condition in multi-scale analysis of gyroscopic continua. J. Sound Vib. 309(1–2), 338–342 (2008)

    Article  Google Scholar 

  7. Zang, J., Cao, R.Q., Fang, B., Zhang, Y.W.: A vibratory energy harvesting absorber using integration of a lever-enhanced nonlinear energy sink and a levitation magnetoelectric energy harvester. J. Sound Vib. 484, 115534 (2020)

    Article  Google Scholar 

  8. Zhang, W., Wang, D.M., Yao, M.H.: Using Fourier differential quadrature method to analyze transverse nonlinear vibrations of an axially accelerating viscoelastic beam. Nonlinear Dyn. 78(2), 839–856 (2014)

    Article  MathSciNet  Google Scholar 

  9. Ding, H., Chen, L.Q.: On two transverse nonlinear models of axially moving beams. Sci. China Ser. E 52(3), 743–751 (2009)

    Article  Google Scholar 

  10. Yang, X.D., Zhang, W.: Nonlinear dynamics of axially moving beam with coupled longitudinal–transversal vibrations. Nonlinear Dyn. 78(4), 2547–2556 (2014)

    Article  Google Scholar 

  11. Chen, L.Q., Zhang, Y.L., Zhang, G.C., Ding, H.: Evolution of the double-jumping in pipes conveying fluid flowing at the supercritical speed. Int. J. Nonlinear Mech. 58(1), 11–21 (2014)

    Article  Google Scholar 

  12. Spaldin, N.A., Fiebig, M.: The renaissance of magnetoelectric multiferroics. Science 309, 391–392 (2005)

    Article  Google Scholar 

  13. Chen, J.Y., Guo, J., Pan, E.: Wave propagation in magneto-electro-elastic multilayered plates with nonlocal effect. J. Sound Vib. 400, 550–563 (2017)

    Article  Google Scholar 

  14. Yuan, J., Huang, Y., Chen, W., Pang, E., Kang, G.: Theory of dislocation loops in multilayered anisotropic solids with magneto-electro-elastic couplings. J. Mech. Phys. Solids 125, 440–471 (2019)

    Article  MathSciNet  Google Scholar 

  15. Ma, J., Ke, L.L., Wang, Y.S.: Frictionless contact of a functionally graded magneto-electro-elastro layered half-plane under a conducting punch. Int. J. Solids Struct. 51, 2791–2806 (2014)

    Article  Google Scholar 

  16. Wu, T.H., Li, X.Y., Chen, X.H.: Three-dimensional closed-form solution to elliptical crack problem in magneto-electro-elasticity: electrically and magnetically induced Maxwell stress boundary condition. Int. J. Solids Struct. 202, 729–744 (2020)

    Article  Google Scholar 

  17. Ansari, R., Gholami, R.: Nonlocal free vibration in the pre- and post-buckled states of magneto-electro-thermo elastic rectangular nanoplates with various edge conditions. Smart Mater. Struct. 25(9), 095033 (2016)

    Article  Google Scholar 

  18. Vinyas, M.: Vibration control of skew magneto-electro-elastic plates using active constrained layer damping. Compos. Struct. 208, 600–617 (2019)

    Article  Google Scholar 

  19. Birman, V., Byrd, L.W.: Modeling and analysis of functionally graded materials and structures. Appl. Mech. Rev. 60, 195–216 (2007)

    Article  Google Scholar 

  20. Huang, D.J., Ding, H.J., Chen, W.Q.: Analytical solution for functionally graded magneto-electro-elastic plane beams. Int. J. Eng. Sci. 45(2), 467–485 (2007)

    Article  Google Scholar 

  21. Wang, Y., Xu, R.Q., Ding, H.J.: Axisymmetric bending of functionally graded circular magneto-electro-elastic plates. Eur. J. Mech. A Solid 30, 999–1011 (2011)

    Article  MathSciNet  Google Scholar 

  22. Wu, C.P., Tsai, Y.H.: Dynamic responses of functionally graded magneto-electro-elastic shells with closed-circuit surface conditions using the method of multiple scales. Eur. J. Mech. A Solid 29(2), 166–181 (2010)

    Article  MathSciNet  Google Scholar 

  23. Arefi, M., Zenkour, A.M.: A simplified shear and normal deformations nonlocal theory for bending of functionally graded piezomagnetic sandwich nanobeams in magneto-thermo-electric environment. J. Sandw. Struct. Mater. 18, 624–651 (2016)

    Article  Google Scholar 

  24. Żur, K.K., Arefi, M., Kim, J., Reddy, J.N.: Free vibration and buckling analyses of magneto-electro-elastic FGM nanoplates based on nonlocal modified higher-order sinusoidal shear deformation theory. Compos. Part B Eng. 182, 107601 (2020)

    Article  Google Scholar 

  25. Li, Z., Wang, Q.S., Qin, B., Zhong, R., Yu, H.L.: Vibration and acoustic radiation of magneto-electro-thermal-elastic functionally graded porous plates in the multi-physics fields. Int. J. Mech. Sci. 185, 105850 (2020)

    Article  Google Scholar 

  26. Liu, H., Lyu, Z.: Modeling of novel nanoscale mass sensor made of smart FG magneto-electro-elastic nanofilm integrated with graphene layers. Thin Walled Struct. 151, 106749 (2020)

    Article  Google Scholar 

  27. Nejad, M.Z., Hadi, A.: Non-local analysis of free vibration of bi-directional functionally graded Euler–Bernoulli nano-beams. Int. J. Eng. Sci. 105, 1–11 (2016)

    Article  MathSciNet  Google Scholar 

  28. Tang, Y., Yang, T.Z.: Post-buckling behavior and nonlinear vibration analysis of a fluid-conveying pipe composed of functionally graded material. Compos. Struct. 185, 393–400 (2018)

    Article  Google Scholar 

  29. Shafiei, N., Mirjavadi, S.S., MohaselAfshari, B., Rabby, S., Kazemi, M.: Vibration of two dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams. Comput. Methods Appl. Mech. Eng. 322, 615–632 (2017)

    Article  MathSciNet  Google Scholar 

  30. Tang, Y., Lv, X.F., Yang, T.Z.: Bi-directional functionally graded beams: asymmetric modes and nonlinear free vibration. Compos. Part B Eng. 156, 319–331 (2019)

    Article  Google Scholar 

  31. Tang, Y., Ding, Q.: Nonlinear vibration analysis of a bi-directional functionally graded beam under hygro-thermal loads. Compos. Struct. 225, 111076 (2019)

    Article  Google Scholar 

  32. Chen, X.C., Zhang, X.L., Lu, Y.X., Li, Y.H.: Static and dynamic analysis of the postbuckling of bi-directional functionally graded material microbeams. Int. J. Mech. Sci. 151, 424–443 (2019)

    Article  Google Scholar 

  33. Esmaeilzadeh, M., Kadkhodayan, M.: Dynamic analysis of stiffened bi-directional functionally graded plates with porosities under a moving load by dynamic relaxation method with kinetic damping. Aerosp. Sci. Technol. 93, 105333 (2019)

    Article  Google Scholar 

  34. Shu, C., Chew, Y.T.: On the equivalence of generalized differential quadrature and highest order finite difference scheme. Comput. Methods Appl. Mech. Eng. 155, 249–260 (1998)

    Article  MathSciNet  Google Scholar 

  35. Ebrahimi, F., Barati, M.R.: Porosity-dependent vibration analysis of piezo-magnetically actuated heterogeneous nanobeams. Mech. Syst. Signal Process. 93(9), 445–459 (2017)

    Article  Google Scholar 

  36. Ibrahim, S.M., Patel, B.P., Nath, Y.: Modified shooting approach to the non-linear periodic forced response of isotropic/composite curved beams. Int. J. Non-Linear Mech. 44, 1073–1084 (2009)

    Article  Google Scholar 

  37. Wu, Z.H., Zhang, Y.M., Yao, G., Yang, Z.: Nonlinear primary and super-harmonic responses of functionally graded carbon nanotube reinforced composite beams. Int. J. Mech. Sci. 153–154, 321–340 (2019)

    Article  Google Scholar 

  38. Cheung, Y.K.: Application of the incremental harmonic balance method to cubic non-linearity systems. J. Sound Vib. 140, 273–286 (1990)

    Article  Google Scholar 

  39. Bolotin, V.V.: The dynamic stability of elastic systems. Am. J. Phys. 33, 752–753 (1965)

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 11902001, 12072221, 11672187, 11802201), China Postdoctoral Science Foundation (No. 2018M641643), the Anhui Provincial Natural Science Foundation (Nos. 1908085QA13, 1808085ME128), the Middle-aged Top-notch Talent and Innovative Team Support Programs of Anhui Polytechnic University.

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Anhui Polytechnic University, Wuhu, 241000, China

    Ye Tang

  2. Department of Mechanics, Tianjin University, Tianjin, 300350, China

    Ye Tang, Tao Wang & Zhi-Sai Ma

  3. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, China

    Tianzhi Yang

Authors
  1. Ye Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhi-Sai Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tianzhi Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tianzhi Yang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, Y., Wang, T., Ma, ZS. et al. Magneto-electro-elastic modelling and nonlinear vibration analysis of bi-directional functionally graded beams. Nonlinear Dyn 105, 2195–2227 (2021). https://doi.org/10.1007/s11071-021-06656-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06656-0

Keywords

Search

Navigation