A Fourier series-based approach for the size-dependent free vibration of viscoelastic microbeams
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Abstract
Abstract In this study, the free vibration behavior of viscoelastic microbeams was investigated by combining the Kelvin–Voigt viscoelastic model with the Modified Couple Stress theory. A semi-analytical method based on the Fourier series and Stokes transformations was developed for the analysis. The proposed approach provides highly accurate results under various boundary conditions due to its structure, which requires solving only a single eigenvalue problem. The convergence analysis of the method was performed to determine the optimal number of terms, and calculations were performed with the number of terms that would yield the most reliable results. Furthermore, the results were compared with studies in literature and confirmed to exhibit high accuracy at the micro scale. Furthermore, the relationship between frequency and damping according to the Kelvin–Voigt model was examined in detail; it was found that an increment in the viscous damping coefficient causes damping to rise linearly and the natural frequency to decrease. It was observed that the damping effect also became more pronounced due to the increase in micro-level stiffness in the system caused by the boost in the material length scale parameter. In conclusion, the developed method and the findings obtained provide important engineering data for the dynamic analysis of viscoelastic microbeams and the design of micro-scale structural elements, while also contributing to a deeper understanding of the frequency–damping interaction in viscoelastic systems.
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OpenAlex topics: Nonlocal and gradient elasticity in micro/nano structures · Composite Structure Analysis and Optimization · Thermoelastic and Magnetoelastic Phenomena
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References (65)
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1. Abbasion, S., Rafsanjani, A., Avazmohammadi, R., Farshidianfar, A.: Free vibration of microscaled Timoshenko beams. Appl. Phys. Lett. 95, 143122 (2009). https://doi.org/10.1063/1.324614310.1063/1.3246143 ↗
2. Abdelrahman, A.A., Esen, I., Eltaher, M.A.: Vibration response of Timoshenko perforated microbeams under accelerating load and thermal environment. Appl. Math. Comput. 407, 126307 (2021). https://doi.org/10.1016/j.amc.2021.12630710.1016/j.amc.2021.126307 ↗
3. Akbarzadeh Khorshidi, M.: Postbuckling of viscoelastic micro/nanobeams embedded in visco-Pasternak foundations based on the modified couple stress theory. Mech. Time-Depend. Mater. 25, 265–278 (2021). https://doi.org/10.1007/s11043-019-09439-810.1007/s11043-019-09439-8 ↗
4. Allahkarami, F., Nikkhah-bahrami, M., Saryazdi, M.G.: Magneto-thermo-mechanical dynamic buckling analysis of a FG-CNTs-reinforced curved microbeam with different boundary conditions using strain gradient theory. Int. J. Mech. Mater. des. 14, 243–261 (2018). https://doi.org/10.1007/s10999-017-9374-510.1007/s10999-017-9374-5 ↗
5. Ansari, R., Faraji Oskouie, M., Sadeghi, F., Bazdid-Vahdati, M.: Free vibration of fractional viscoelastic Timoshenko nanobeams using the nonlocal elasticity theory. Physica E 74, 318–327 (2015). https://doi.org/10.1016/j.physe.2015.07.01310.1016/j.physe.2015.07.013 ↗
6. Arda, M., Aydogdu, M.: Dynamic analysis of a viscoelastic nanobeam. Key Eng. Mater. 799, 223–229 (2019). https://doi.org/10.4028/www.scientific.net/KEM.799.22310.4028/www.scientific.net/KEM.799.223 ↗
7. Attia, M.A., Abdel Rahman, A.A.: On vibrations of functionally graded viscoelastic nanobeams with surface effects. Int. J. Eng. Sci. 127, 1–32 (2018). https://doi.org/10.1016/j.ijengsci.2018.02.00510.1016/j.ijengsci.2018.02.005 ↗
8. Attia, M.A., Emam, S.A.: Electrostatic nonlinear bending, buckling and free vibrations of viscoelastic microbeams based on the modified couple stress theory. Acta Mech. 229, 3235–3255 (2018). https://doi.org/10.1007/s00707-018-2162-y10.1007/s00707-018-2162-y ↗
9. Attia, M.A., Mahmoud, F.F.: Analysis of viscoelastic Bernoulli–Euler nanobeams incorporating nonlocal and microstructure effects. Int. J. Mech. Mater. des. 13, 385–406 (2017). https://doi.org/10.1007/s10999-016-9343-410.1007/s10999-016-9343-4 ↗
10. Babaei, H., Eslami, M.R.: Thermally induced nonlinear stability and imperfection sensitivity of temperature- and size-dependent FG porous micro-tubes. Int. J. Mech. Mater. des. 17, 381–401 (2021). https://doi.org/10.1007/s10999-021-09531-310.1007/s10999-021-09531-3 ↗
11. Babaei, A., Noorani, M.-R., Ghanbari, A.: Temperature-dependent free vibration analysis of functionally graded micro-beams based on the modified couple stress theory. Microsyst. Technol. 23, 4599–4610 (2017). https://doi.org/10.1007/s00542-017-3285-010.1007/s00542-017-3285-0 ↗
12. Bahrami, A.: A wave-based computational method for free vibration, wave power transmission and reflection in multi-cracked nanobeams. Compos. Part B Eng. 120, 168–181 (2017). https://doi.org/10.1016/j.compositesb.2017.03.05310.1016/j.compositesb.2017.03.053 ↗
13. Balci, M.N.: Axial forced vibration problem for a viscoelastic nanorod involving radially graded porosity. J. Braz. Soc. Mech. Sci. Eng. 47, 203 (2025). https://doi.org/10.1007/s40430-025-05497-x10.1007/s40430-025-05497-x ↗
14. Banerjee, J.R.: Free vibration of micro-beams and frameworks using the dynamic stiffness method and modified couple stress theory, In: Modern trends in structural and solid mechanics 2, pp. 79–107. Wiley, (2021) https://doi.org/10.1002/9781119831860.ch410.1002/9781119831860.ch4 ↗
15. Civalek, Ö., Dastjerdi, S., Akbaş, ŞD., Akgöz, B.: Vibration analysis of carbon nanotube-reinforced composite microbeams. Math. Methods Appl. Sci. (2021). https://doi.org/10.1002/mma.706910.1002/mma.7069 ↗
16. Dai, H.L., Wang, Y.K., Wang, L.: Nonlinear dynamics of cantilevered microbeams based on modified couple stress theory. Int. J. Eng. Sci. 94, 103–112 (2015). https://doi.org/10.1016/j.ijengsci.2015.05.00710.1016/j.ijengsci.2015.05.007 ↗
17. Dehrouyeh-Semnani, A.M., Dehrouyeh, M., Torabi-Kafshgari, M., Nikkhah-Bahrami, M.: An investigation into size-dependent vibration damping characteristics of functionally graded viscoelastically damped sandwich microbeams. Int. J. Eng. Sci. 96, 68–85 (2015). https://doi.org/10.1016/j.ijengsci.2015.07.00810.1016/j.ijengsci.2015.07.008 ↗
18. Ebrahimi, F., Barati, M.R.: Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory. Compos. Struct. 159, 433–444 (2017). https://doi.org/10.1016/j.compstruct.2016.09.09210.1016/j.compstruct.2016.09.092 ↗
19. Ebrahimi, F., Barati, M.R.: Damping vibration behavior of viscoelastic porous nanocrystalline nanobeams incorporating nonlocal–couple stress and surface energy effects. Iran. J. Sci. Technol. Trans. Mech. Eng. 43, 187–203 (2019). https://doi.org/10.1007/s40997-017-0127-810.1007/s40997-017-0127-8 ↗
20. El-Borgi, S., Rajendran, P., Friswell, M.I., Trabelssi, M., Reddy, J.N.: Torsional vibration of size-dependent viscoelastic rods using nonlocal strain and velocity gradient theory. Compos. Struct. 186, 274–292 (2018). https://doi.org/10.1016/j.compstruct.2017.12.00210.1016/j.compstruct.2017.12.002 ↗
21. Eltaher, M.A., Shanab, R.A., Mohamed, N.A.: Analytical solution of free vibration of viscoelastic perforated nanobeam. Arch. Appl. Mech. 93, 221–243 (2023). https://doi.org/10.1007/s00419-022-02184-410.1007/s00419-022-02184-4 ↗
22. Fang, J., Gu, J., Wang, H., Zhang, X.: Thermal effect on vibrational behaviors of rotating functionally graded microbeams. Eur. J. Mech. - A/Solids 75, 497–515 (2019). https://doi.org/10.1016/j.euromechsol.2019.03.00510.1016/j.euromechsol.2019.03.005 ↗
23. Ghayesh, M.H.: Dynamics of functionally graded viscoelastic microbeams. Int. J. Eng. Sci. 124, 115–131 (2018). https://doi.org/10.1016/j.ijengsci.2017.11.00410.1016/j.ijengsci.2017.11.004 ↗
24. Ghayesh, M.H., Farajpour, A.: Vibrations of shear deformable FG viscoelastic microbeams. Microsyst. Technol. 25, 1387–1400 (2019). https://doi.org/10.1007/s00542-018-4184-810.1007/s00542-018-4184-8 ↗
25. Hajikarimi, P., Nejad, F.M.: Applications of viscoelasticity: bituminous materials characterization and modeling, Elsevier, (2021). https://www.sciencedirect.com/book/monograph/9780128212103/applications-of-viscoelasticity (accessed March 4, 2026).10.1016/B978-0-12-821210-3.00008-5 ↗
26. Hosseini, S.M.H., Beni, Y.T.: Free vibration analysis of rotating piezoelectric/flexoelectric microbeams. Appl. Phys. A 129, 330 (2023). https://doi.org/10.1007/s00339-023-06615-z10.1007/s00339-023-06615-z ↗
27. Hosseini-Hashemi, S., Behdad, S., Fakher, M.: Vibration analysis of two-phase local/nonlocal viscoelastic nanobeams with surface effects. Eur. Phys. J. plus 135, 190 (2020). https://doi.org/10.1140/epjp/s13360-020-00148-710.1140/epjp/s13360-020-00148-7 ↗
28. Imani, M., Hamzehei-Javaran, S., Shojaee, S., Daraei, B.: Vibration analysis of functionally graded curved microbeams using modified strain gradient finite elements. J. Sound Vib. 618, 119298 (2025). https://doi.org/10.1016/j.jsv.2025.11929810.1016/j.jsv.2025.119298 ↗
29. Jin, Q., Ren, Y.: Coupled resonance of FGM nanotubes transporting super-critical high-speed pulsatile flow under forced vibration: size-dependence and bifurcation topology. Comput. Methods in Appl. Mech. Eng. 404, 115834 (2023). https://doi.org/10.1016/j.cma.2022.11583410.1016/j.cma.2022.115834 ↗
30. Jin, Q., Yuan, F.-G., Yu, D., Wen, J., Ren, Y.: Size-dependent longitudinal–transverse mode interaction of fluid-conveying nanotubes under base excitation. Nonlinear Dyn. 112, 6181–6204 (2024). https://doi.org/10.1007/s11071-024-09345-w10.1007/s11071-024-09345-w ↗
31. Kadioglu, H.G., Uzun, B., Yayli, M.O.: Dynamic analysis of a porous timoshenko microbeam considering the rotational restraints. Thin-Walled Struct. (2025a). https://doi.org/10.1016/j.tws.2025.11440810.1016/j.tws.2025.114408 ↗
32. Kadioglu, H.G., Uzun, B., Yayli, M.O.: On the torsional dynamics of viscoelastic nanotubes on time-dependent boundary conditions using nonlocal elasticity theory. Adv. in Nano Res. 18, 537–547 (2025b). https://doi.org/10.12989/anr.2025.18.6.53710.12989/anr.2025.18.6.537 ↗
33. Kadıoğlu, H.G., Yaylı, M.Ö.: Analysis of torsional vibration in viscoelastic functionally graded nanotubes with viscoelastic constraints using doublet mechanics theory. Int. J. Mech. Mater. des. (2025). https://doi.org/10.1007/s10999-025-09802-310.1007/s10999-025-09802-3 ↗
34. Kadıoğlu, H.G., Uzun, B., Civalek, Ö., Yaylı, M.Ö.: Dynamic behavior of micromorphic viscoelastic Euler–Bernoulli beams under viscoelastic boundary constraints. Acta Mech. 236, 4475–4490 (2025a). https://doi.org/10.1007/s00707-025-04364-z10.1007/s00707-025-04364-z ↗
35. Kadıoğlu, H.G., Yaylı, M.Ö., Uzun, B.: A new iterative method for free vibration analysis of a FG Timoshenko beam. Int. J. Eng. Appl. Sci. 17, 17–43 (2025b). https://doi.org/10.24107/ijeas.162591210.24107/ijeas.1625912 ↗
36. Kar, U.K., Srinivas, J.: Dynamic analysis and identification of bi-directional functionally graded elastically supported cracked microbeam subjected to thermal shock loads. Eur. J. Mech. - A/Solids 99, 104930 (2023). https://doi.org/10.1016/j.euromechsol.2023.10493010.1016/j.euromechsol.2023.104930 ↗
37. Karamanli, A., Vo, T.P.: Bending, vibration, buckling analysis of bi-directional FG porous microbeams with a variable material length scale parameter. Appl. Math. Model. 91, 723–748 (2021). https://doi.org/10.1016/j.apm.2020.09.05810.1016/j.apm.2020.09.058 ↗
38. Kong, S.: A review on the size-dependent models of micro-beam and micro-plate based on the Modified Couple Stress Theory. Arch. Comput. Methods Eng. 29, 1–31 (2022). https://doi.org/10.1007/s11831-021-09567-w10.1007/s11831-021-09567-w ↗
39. Kong, S., Zhou, S., Nie, Z., Wang, K.: The size-dependent natural frequency of Bernoulli-Euler micro-beams. Int. J. Eng. Sci. 46, 427–437 (2008). https://doi.org/10.1016/j.ijengsci.2007.10.00210.1016/j.ijengsci.2007.10.002 ↗
40. Lei, Y., Adhikari, S., Friswell, M.I.: Vibration of nonlocal Kelvin–Voigt viscoelastic damped Timoshenko beams. Int. J. Eng. Sci. 66, 1–13 (2013a). https://doi.org/10.1016/j.ijengsci.2013.02.00410.1016/j.ijengsci.2013.02.004 ↗
41. Lei, Y., Murmu, T., Adhikari, S., Friswell, M.I.: Dynamic characteristics of damped viscoelastic nonlocal Euler-Bernoulli beams. Eur. J. Mech. - A/Solids 42, 125–136 (2013b). https://doi.org/10.1016/j.euromechsol.2013.04.00610.1016/j.euromechsol.2013.04.006 ↗
42. Loghman, E., Bakhtiari-Nejad, F., Kamali E., A., Abbaszadeh, M., Amabili, M.: Nonlinear vibration of fractional viscoelastic micro-beams. Int. J. of Non-Linear Mech. 137, 103811 (2021). https://doi.org/10.1016/j.ijnonlinmec.2021.10381110.1016/j.ijnonlinmec.2021.103811 ↗
43. Ma, H.M., Gao, X.-L., Reddy, J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56, 3379–3391 (2008). https://doi.org/10.1016/j.jmps.2008.09.00710.1016/j.jmps.2008.09.007 ↗
44. Ma, H.M., Gao, X.-L., Reddy, J.N.: A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech. 220, 217–235 (2011). https://doi.org/10.1007/s00707-011-0480-410.1007/s00707-011-0480-4 ↗
45. Nguyen, N.-D., Nguyen, T.-N., Nguyen, T.-K., Vo, T.P.: A Chebyshev-Ritz solution for size-dependent analysis of the porous microbeams with various boundary conditions. Int. J. Mech. Mater. des. 19, 861–881 (2023). https://doi.org/10.1007/s10999-023-09666-510.1007/s10999-023-09666-5 ↗
46. Nguyen, N.-D., Bui, V.-T., Trinh, L.C., Le, Q.-C.: Elastic foundation effect on the small-scale analysis of functionally graded porous microbeams using a modified strain gradient theory. Int. J. Mech. Mater. des. 21, 231–259 (2025). https://doi.org/10.1007/s10999-024-09735-310.1007/s10999-024-09735-3 ↗
47. Park, S.K., Gao, X.-L.: Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16, 2355 (2006). https://doi.org/10.1088/0960-1317/16/11/01510.1088/0960-1317/16/11/015 ↗
48. Reddy, J.N.: Microstructure-dependent couple stress theories of functionally graded beams. J. Mech. Phys. Solids 59, 2382–2399 (2011). https://doi.org/10.1016/j.jmps.2011.06.00810.1016/j.jmps.2011.06.008 ↗
49. Sabahi, M.A., Saidi, A.R.: On nonlinear vibrations of Timoshenko FG porous micropipes in thermal environment: analysis and optimization. Int. J. Mech. Mater. des. 21, 799–821 (2025). https://doi.org/10.1007/s10999-025-09764-610.1007/s10999-025-09764-6 ↗
50. Sheng, G.G., Wang, X.: Nonlinear forced vibration of functionally graded Timoshenko microbeams with thermal effect and parametric excitation. Int. J. Mech. Sci. 155, 405–416 (2019). https://doi.org/10.1016/j.ijmecsci.2019.03.01510.1016/j.ijmecsci.2019.03.015 ↗
51. Togun, N., Bağdatli, S.M.: Application of modified couple-stress theory to nonlinear vibration analysis of nanobeam with different boundary conditions. J. Vib. Eng. Technol. 12, 6979–7008 (2024). https://doi.org/10.1007/s42417-024-01294-310.1007/s42417-024-01294-3 ↗
52. Trinh, L.C., Vo, T.P., Thai, H.-T., Nguyen, T.-K.: Size-dependent vibration of bi-directional functionally graded microbeams with arbitrary boundary conditions. Compos. Part B Eng. 134, 225–245 (2018). https://doi.org/10.1016/j.compositesb.2017.09.05410.1016/j.compositesb.2017.09.054 ↗
53. Vo, D., Suttakul, P., Rungamornrat, J., Nanakorn, P.: Static analysis of planar arbitrarily curved microbeams with the modified couple stress theory and Euler-Bernoulli beam model. Appl. Math. Model. 112, 358–390 (2022). https://doi.org/10.1016/j.apm.2022.08.00510.1016/j.apm.2022.08.005 ↗
54. Wang, L.: Size-dependent vibration characteristics of fluid-conveying microtubes. J. Fluids Struct. 26, 675–684 (2010). https://doi.org/10.1016/j.jfluidstructs.2010.02.00510.1016/j.jfluidstructs.2010.02.005 ↗
55. Wang, C.M., Zhang, Y.Y., He, X.Q.: Vibration of nonlocal Timoshenko beams. Nanotechnology 18, 105401 (2007). https://doi.org/10.1088/0957-4484/18/10/10540110.1088/0957-4484/18/10/105401 ↗
56. Xu, Y., Xu, Z.-D., Guo, Y.-Q., Jia, H., Huang, X., Wen, Y.: Mathematical modeling and test verification of viscoelastic materials considering microstructures and ambient temperature influence. Mech. Adv. Mater. Struct. 29, 7063–7074 (2022). https://doi.org/10.1080/15376494.2021.199268910.1080/15376494.2021.1992689 ↗
57. Xu, Y., Xu, Z., Guo, Y., Sarwar, W., She, W., Geng, Z.: Study on viscoelastic materials at micro scale pondering supramolecular interaction impacts with DMA tests and fractional derivative modeling. J of Applied Polymer Sci 140, e53660 (2023). https://doi.org/10.1002/app.5366010.1002/app.53660 ↗
58. Xu, Y., Xu, Z.-D., Hu, H., Guo, Y.-Q., Huang, X.-H., Zhang, Z.-W., Zhang, T., Xu, C.: Experiment, simulation, and theoretical investigation of a new type of interlayer connections enhanced viscoelastic damper. Int. J. Str. Stab. Dyn. 25, 2550045 (2025a). https://doi.org/10.1142/S021945542550045210.1142/S0219455425500452 ↗
59. Xu, Y., Mao, M., Huang, X.-H., Guo, Y.-Q., Yang, Y., Dong, Y.-R., Shah, S.A.A., Gao, X., Usman, M., Jeong, S.-H.: Comparative investigation of dynamic damping and fracture in viscoelastic dampers with two different interfacial bonding types: molecular dynamics simulations and device tests. J. Build. Eng. 116, 114582 (2025b). https://doi.org/10.1016/j.jobe.2025.11458210.1016/j.jobe.2025.114582 ↗
60. Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002). https://doi.org/10.1016/S0020-7683(02)00152-X10.1016/S0020-7683(02)00152-X ↗
61. Yao, L.Q., Ji, C.J., Shen, J.P., Li, C.: Free vibration and wave propagation of axially moving functionally graded Timoshenko microbeams. J. Braz. Soc. Mech. Sci. Eng. 42, 137 (2020). https://doi.org/10.1007/s40430-020-2206-910.1007/s40430-020-2206-9 ↗
62. Yin, L., Qian, Q., Wang, L., Xia, W.: Vibration analysis of microscale plates based on modified couple stress theory. Acta Mech. Solida Sinica 23, 386–393 (2010). https://doi.org/10.1016/S0894-9166(10)60040-710.1016/S0894-9166(10)60040-7 ↗
63. Zhang, Y., Pang, M., Fan, L.: Analyses of transverse vibrations of axially pretensioned viscoelastic nanobeams with small size and surface effects. Phys. Lett. A 380, 2294–2299 (2016). https://doi.org/10.1016/j.physleta.2016.05.01610.1016/j.physleta.2016.05.016 ↗
64. Zhen, Y.-X., Wen, S.-L., Tang, Y.: Free vibration analysis of viscoelastic nanotubes under longitudinal magnetic field based on nonlocal strain gradient Timoshenko beam model. Physica E 105, 116–124 (2019). https://doi.org/10.1016/j.physe.2018.09.00510.1016/j.physe.2018.09.005 ↗
65. Zhou, Y., Shi, X., Zhang, Y.: Multiscale bending model of graphene based on asymmetric elasticity: The role of microscopic bond rotation in macroscopic deformation. Int. J. Eng. Sci. 220, 104434 (2026). https://doi.org/10.1016/j.ijengsci.2025.10443410.1016/j.ijengsci.2025.104434 ↗