An uncertainty aware fuzzy fractional model for thermo viscoelastic vibration of a size dependent microbeam

Authors

https://doi.org/10.48313/uda.v3i2.103

Abstract

This study develops a fractional phase framework for transient thermo viscoelastic vibrations of size-dependent uniaxial microbeams. Inherited damping, uncertain thermal loading, and micro scale stiffness effects are incorporated into a unified reduced-order model. The first mode reduction yields a fractional Kelvin Voight oscillator with temperature and size dependent effective stiffness. The fractional damping term is discretized using the  scheme, while the inertial dynamics is solved by central finite differences. Numerical results show that fractional order, thermal uncertainty, and size effects significantly affect the displacement and resonance behavior. The proposed model provides a transparent computational framework for coupling fractional dynamics, phase uncertainty, and vibration analysis at the micro/nano scale.

Keywords:

Fractional viscoelasticity, Thermo mechanical vibration, Size dependent microbeam, Fuzzy uncertainty, Caputo derivative

References

  1. [1] Bagley, R. L., & Torvik, P. J. (1983). A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of rheology, 27(3), 201–210. https://doi.org/10.1122/1.549724
  2. [2] Shimizu, N., & Zhang, W. (1999). Fractional calculus approach to dynamic problems of viscoelastic materials. JSME international journal series c mechanical systems, machine elements and manufacturing, 42(4), 825–837. https://doi.org/10.1299/jsmec.42.825
  3. [3] Mainardi, F. (2022). Fractional calculus and waves in linear viscoelasticity. World Scientific Publishing Company. https://doi.org/10.1142/p614
  4. [4] Di Paola, M., Heuer, R., & Pirrotta, A. (2013). Fractional visco-elastic Euler–Bernoulli beam. International journal of solids and structures, 50(22), 3505–3510. https://doi.org/10.1016/j.ijsolstr.2013.06.010
  5. [5] Freundlich, J. (2019). Transient vibrations of a fractional Kelvin-Voigt viscoelastic cantilever beam with a tip mass and subjected to a base excitation. Journal of sound and vibration, 438, 99–115. https://doi.org/10.1016/j.jsv.2018.09.006
  6. [6] Praharaj, R. K., & Datta, N. (2020). Dynamic response of Euler-Bernoulli beam resting on fractionally damped viscoelastic foundation subjected to a moving point load. Proceedings of the institution of mechanical engineers, part c: journal of mechanical engineering science, 234(24), 4801–4812. https://doi.org/10.1177/0954406220932597
  7. [7] Shitikova, M. V. (2022). Fractional operator viscoelastic models in dynamic problems of mechanics of solids: A review. Mechanics of solids, 57(1), 1–33. https://doi.org/10.3103/S0025654422010022
  8. [8] Bonfanti, A., Kaplan, J. L., Charras, G., & Kabla, A. (2020). Fractional viscoelastic models for power-law materials. Soft matter, 16(26), 6002–6020. https://doi.org/10.1039/d0sm00354a
  9. [9] Łabędzki, P. (2025). Fractional Kelvin–Voigt model for beam vibrations: Numerical simulations and approximation using a classical model. Electronics, 14(10), 1–18. https://doi.org/10.3390/electronics14101918
  10. [10] Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
  11. [11] Kaleva, O. (1987). Fuzzy differential equations. Fuzzy sets and systems, 24(3), 301–317. https://doi.org/10.1016/0165-0114(87)90029-7
  12. [12] Kaleva, O. (1985). On the convergence of fuzzy sets. Fuzzy sets and systems, 17(1), 53–65. https://doi.org/10.1016/0165-0114(85)90006-5
  13. [13] Buckley, J. J., & Feuring, T. (2000). Fuzzy differential equations. Fuzzy sets and systems, 110(1), 43–54. https://doi.org/10.1016/S0165-0114(98)00141-9
  14. [14] Allahviranloo, T., Ahmady, N., & Ahmady, E. (2007). Numerical solution of fuzzy differential equations by predictor–corrector method. Information sciences, 177(7), 1633–1647. https://doi.org/10.1016/j.ins.2006.09.015
  15. [15] Fu, G., Zhou, S., & Qi, L. (2019). A size-dependent Bernoulli-Euler beam model based on strain gradient elasticity theory incorporating surface effects. ZAMM - journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik, 99(6), e201800048. https://doi.org/10.1002/zamm.201800048
  16. [16] Yin, S., Xiao, Z., Deng, Y., Zhang, G., Liu, J., & Gu, S. (2021). Isogeometric analysis of size-dependent Bernoulli–Euler beam based on a reformulated strain gradient elasticity theory. Computers & structures, 253, 106577. https://doi.org/10.1016/j.compstruc.2021.106577
  17. [17] Park, S. K., & Gao, X. L. (2006). Bernoulli–Euler beam model based on a modified couple stress theory. Journal of micromechanics and microengineering, 16(11), 2355. https://doi.org/10.1088/0960-1317/16/11/015
  18. [18] Yang, F., Chong, A. C. M., Lam, D. C. C., & Tong, P. (2002). Couple stress based strain gradient theory for elasticity. International journal of solids and structures, 39(10), 2731–2743. https://doi.org/10.1016/S0020-7683(02)00152-X
  19. [19] Eringen AC, A., & Wegner JL, R. (2003). Nonlocal Continuum Field Theories. Springer. https://doi.org/10.1007/b97697
  20. [20] Awad, E., & El-Dhaba, A. R. (2026). The role of fractional calculus in modeling vibrations of a Timoshenko beam influenced by low and high thermal conduction. European journal of mechanics - a/solids, 118, 106100. https://doi.org/10.1016/j.euromechsol.2026.106100
  21. [21] W. Weaver, J., Timoshenko, S. P., & Young, D. H. (1991). Vibration problems in engineering. Wiley. https://www.wiley.com/en-us/shop/general-introductory-mechanical-engineering/vibration-problems-in-engineering-5th-edition-p-9780471632283
  22. [22] Meirovitch, L. (1975). Elements of vibration analysis. McGraw-Hill. https://books.google.com/books/about/Elements_of_Vibration_Analysis.html?id=PVm6QgAACAAJ
  23. [23] Nayfeh, A. H., & Mook, D. T. (2024). Nonlinear oscillations. John Wiley & Sons. https://www.wiley.com/en-us/shop/general-physics/nonlinear-oscillations-p-9783527617593
  24. [24] Warburton, G. B. (1993). Dynamics of structures11394‐7. McGraw-Hill. https://books.google.com/books/about/Dynamics_of_Structures.html?id=HxLakQEACAAJ
  25. [25] Caputo, M., Fabrizio, M., & others. (2015). A new definition of fractional derivative without singular kernel. Progress in fractional differentiation and applications, 1(2), 73–85. http://dx.doi.org/10.12785/pfda/010201

Downloads

Published

2026-06-25

Issue

Vol. 3 No. 2 (2026): Uncertainty Discourse and Applications

Section

Articles

How to Cite

Joorbonyan, Z., & Motallebi, S. (2026). An uncertainty aware fuzzy fractional model for thermo viscoelastic vibration of a size dependent microbeam. Uncertainty Discourse and Applications, 3(2), 178-187. https://doi.org/10.48313/uda.v3i2.103

Similar Articles

11-20 of 51

You may also start an advanced similarity search for this article.