Abstract
In this paper, an effective numerical Bernstein polynomials operational matrix is exploited to study the nonlinear forced vibration of fractional viscoelastic curved beam with viscoelastic nonlinear boundary conditions, for the first time. The Caputo fractional derivative is employed to incorporate viscoelastic material having nonlinear behavior. Based on Euler–Bernoulli beam theory and von Kármán geometric nonlinearity, the governing equation of viscoelastic curved beam which is a nonlinear integro-partial fractional differential equation is introduced. Based on the collocation method of Bernstein polynomials approximation, the generalized integer-order operational matrices of differentiation and integration as well as fractional-order operational matrix of differentiation are derived. Using Bernstein polynomials operational matrices (BPOM), the nonlinear static problem is discretized; then, it is solved via Newton's method. To compute the linear vibration mode, the linear vibration problem is discretized via BPOM and it is solved as a linear eigenvalue problem. Discretized by the Galerkin approximation, the fractional-order nonlinear integro-partial differential equation is transformed into a nonlinear fractional-order duffing-type equation. The fractional-order duffing equation is discretized using BPOM resulting nonlinear eigenvalue problem, which can be easily solved by pseudo-arc length continuation algorithm. The effects of the Caputo fractional derivative order, foundation parameters, amplitude of initial curvature, viscoelastic parameters and amplitude of excitation force on the nonlinear forced vibration of the viscoelastic beam are investigated through a detailed parametric study. The proposed procedure is supportive in the analysis and design of curved viscoelastic structure with non-classical viscoelastic boundary conditions under the dynamic mechanical loads.
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References
Abo-Bakr, R.M., Mohamed, N.A., Mohamed, S.A.: Meta-heuristic algorithms for solving nonlinear differential equations based on multivariate Bernstein polynomials. Soft. Comput. 26(2), 605–619 (2022). https://doi.org/10.1007/s00500-021-06535-1
Abo-bakr, R.M., Mohamed, N., Eltaher, M.A., Emam, S.: Multi-objective optimization for snap-through response of spherical shell panels. Appl. Math. Model. (2023). https://doi.org/10.1016/j.apm.2023.12.014
Alfadil, H., Abouelregal, A.E., Marin, M., Carrera, E.: Goufo-Caputo fractional viscoelastic photothermal model of an unbounded semiconductor material with a cylindrical cavity. Mech. Adv. Mater. Struct. 21, 1–14 (2023). https://doi.org/10.1080/15376494.2023.2278181
Ansari, R., Faraji Oskouie, M., Rouhi, H.: Studying linear and nonlinear vibrations of fractional viscoelastic Timoshenko micro-/nano-beams using the strain gradient theory. Nonlinear Dyn. 87, 695–711 (2017). https://doi.org/10.1007/s11071-016-3069-6
Bagley, R.L., Torvik, P.J.: Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J. 23, 918–925 (1985). https://doi.org/10.2514/3.9007
Calaf-Chica, J., Cea-González, V., García-Tárrago, M.J., Gómez-Gil, F.J.: Fractional viscoelastic models for the estimation of the frequency response of rubber bushings based on relaxation tests. Results Eng. 20, 101465 (2023). https://doi.org/10.1016/j.rineng.2023.101465
Cui, Y., Qu, J., Han, C., Cheng, G., Zhang, W., Chen, Y.: Shifted Bernstein-Legendre polynomial collocation algorithm for numerical analysis of viscoelastic Euler–Bernoulli beam with variable order fractional model. Math. Comput. Simul 200, 361–376 (2022). https://doi.org/10.1016/j.matcom.2022.04.035
Dang, R., Chen, Y.: Fractional modelling and numerical simulations of variable-section viscoelastic arches. Appl. Math. Comput. 409, 126376 (2021). https://doi.org/10.1016/j.amc.2021.126376
Di Paola, M., Heuer, R., Pirrotta, A.: Fractional visco-elastic Euler–Bernoulli beam. Int. J. Solids Struct. 50(22–23), 3505–3510 (2013). https://doi.org/10.1016/j.ijsolstr.2013.06.010
Doha, E.H., Bhrawy, A.H., Saker, M.A.: Integrals of Bernstein polynomials: an application for the solution of high even-order differential equations. Appl. Math. Lett.Lett. 24(4), 559–565 (2011). https://doi.org/10.1016/j.aml.2010.11.013
Galucio, A.C., Deü, J.F., Ohayon, R.: Finite element formulation of viscoelastic sandwich beams using fractional derivative operators. Comput. Mech. 33, 282–291 (2004). https://doi.org/10.1007/s00466-003-0529-x
Han, C., Chen, Y., Liu, D.Y., Boutat, D.: Numerical analysis of viscoelastic rotating beam with variable fractional order model using shifted Bernstein–Legendre polynomial collocation algorithm. Fractal Fract. 5(1), 8 (2021). https://doi.org/10.3390/fractalfract5010008
Han, C., Chen, Y., Cheng, G., Serra, R., Wang, L., Feng, J.: Numerical analysis of axially non-linear viscoelastic string with the variable fractional order model by using Bernstein polynomials algorithm. Int. J. Comput. Math. 99(3), 537–552 (2022). https://doi.org/10.1080/00207160.2021.1924367
Hao, Y., Zhang, M., Cui, Y., Cheng, G., Xie, J., Chen, Y.: Dynamic analysis of variable fractional order cantilever beam based on shifted Legendre polynomials algorithm. J. Comput. Appl. Math. 423, 114952 (2023). https://doi.org/10.1016/j.cam.2022.114952
Javadi, M., Rahmanian, M.: Nonlinear vibration of fractional Kelvin-Voigt viscoelastic beam on nonlinear elastic foundation. Commun. Nonlinear Sci. Numer. Simul. 98, 105784 (2021). https://doi.org/10.1016/j.cnsns.2021.105784
Jin, S., Xie, J., Qu, J., Chen, Y.: A numerical method for simulating viscoelastic plates based on fractional order model. Fractal Fract. 6(3), 150 (2022). https://doi.org/10.3390/fractalfract6030150
Kanda, K., Maruyama, T.: Theoretical analysis of forced Lamb waves using the method of multiple scales and Green’s function method. Acta Mech. (2023). https://doi.org/10.1007/s00707-023-03573-8
Li, X., Sha, A., Jiao, W., Song, R., Cao, Y., Li, C., Liu, Z.: Fractional derivative Burgers models describing dynamic viscoelastic properties of asphalt binders. Constr. Build. Mater. 408, 133552 (2023). https://doi.org/10.1016/j.conbuildmat.2023.133552
Li, Y., Wang, H., Zheng, X.: Analysis of a fractional viscoelastic Euler–Bernoulli beam and identification of its piecewise continuous polynomial order. Fract. Cal. Appl. Anal. 26(5), 2337–2360 (2023). https://doi.org/10.1007/s13540-023-00193-w
Loghman, E., Bakhtiari-Nejad, F., Kamali, A., Abbaszadeh, M., Amabili, M.: Nonlinear vibration of fractional viscoelastic micro-beams. Int. J. Non-Linear Mech. 137, 103811 (2021). https://doi.org/10.1016/j.ijnonlinmec.2021.103811
Loghman, E., Kamali, A., Bakhtiari-Nejad, F., Abbaszadeh, M., Amabili, M.: On the combined Shooting–Pseudo–Arclength method for finding frequency response of nonlinear fractional-order differential equations. J. Sound Vib. 516, 116521 (2022). https://doi.org/10.1016/j.jsv.2021.116521
Mohamed, N.A., Shanab, R.A., Eltaher, M.A., Abdelrahman, A.A.: Vibration response of viscoelastic perforated higher-order nanobeams rested on an elastic substrate under moving load. Acta Mech. (2023). https://doi.org/10.1007/s00707-023-03776-z
Mohamed, S.A., Mohamed, N., Abo-bakr, R.M., Eltaher, M.A.: Multi-objective optimization of snap-through instability of helicoidal composite imperfect beams using Bernstein polynomials method. Appl. Math. Model. 120, 301–329 (2023). https://doi.org/10.1016/j.apm.2023.03.034
Paunović, S., Cajić, M., Karličić, D., Mijalković, M.: A novel approach for vibration analysis of fractional viscoelastic beams with attached masses and base excitation. J. Sound Vib. 463, 114955 (2019). https://doi.org/10.1016/j.jsv.2019.114955
Permoon, M.R., Haddadpour, H., Shakouri, M.: Nonlinear vibration analysis of fractional viscoelastic cylindrical shells. Acta Mech. 231, 4683–4700 (2020). https://doi.org/10.1007/s00707-020-02785-6
Qing, J., Zhou, S., Wu, J., Shao, M., Tang, J.: Parametric resonance of an axially accelerating viscoelastic membrane with a fractional model. Commun. Nonlinear Sci. Numer. Simul. 130, 107691 (2024). https://doi.org/10.1016/j.cnsns.2023.107691
Song, J.P., She, G.L.: Nonlinear resonance and chaotic dynamic of rotating graphene platelets reinforced metal foams plates in thermal environment. Arch. Civ. Mech. Eng. 24(1), 1–31 (2024). https://doi.org/10.1007/s43452-023-00846-w
Song, J.P., She, G.L., He, Y.J.: Nonlinear forced vibration of axially moving functionally graded cylindrical shells under hygro-thermal loads. Geomech. Eng. 36(2), 99 (2024). https://doi.org/10.12989/gae.2024.36.2.099
Sun, L., Chen, Y.: Numerical analysis of variable fractional viscoelastic column based on two-dimensional Legendre wavelets algorithm. Chaos Solitons Fractals 152, 111372 (2021). https://doi.org/10.1016/j.chaos.2021.111372
Suzuki, J.L., Kharazmi, E., Varghaei, P., Naghibolhosseini, M., Zayernouri, M.: Anomalous nonlinear dynamics behavior of fractional viscoelastic beams. J. Comput. Nonlinear Dyn. 16(11), 111005 (2021). https://doi.org/10.1115/1.4052286
Vazirzadeh, M., Rouzegar, J., Heydari, M.H.: A refined fractional viscoelastic model for vibration analysis of moderately-thick plates. Mech. Res. Commun. (2024). https://doi.org/10.1016/j.mechrescom.2023.104224
Wang, L., Chen, Y.M.: Shifted-Chebyshev-polynomial-based numerical algorithm for fractional order polymer visco-elastic rotating beam. Chaos Solitons Fractals 132, 109585 (2020). https://doi.org/10.1016/j.chaos.2019.109585
Wang, Y., Chen, Y.: Shifted Legendre polynomials algorithm used for the dynamic analysis of viscoelastic pipes conveying fluid with variable fractional order model. Appl. Math. Model. 81, 159–176 (2020). https://doi.org/10.1016/j.apm.2019.12.011
Xu, Y., Wei, P., Zhao, L.: Flexural waves in nonlocal strain gradient high-order shear beam mounted on fractional-order viscoelastic Pasternak foundation. Acta Mech. 233(10), 4101–4118 (2022). https://doi.org/10.1007/s00707-022-03334-z
Yang, A., Zhang, Q., Qu, J., Cui, Y., Chen, Y.: Solving and numerical simulations of fractional-order governing equation for micro-beams. Fractal Fract. 7(2), 204 (2023). https://doi.org/10.3390/fractalfract7020204
Ye, S.Q., Mao, X.Y., Ding, H., Ji, J.C., Chen, L.Q.: Nonlinear vibrations of a slightly curved beam with nonlinear boundary conditions. Int. J. Mech. Sci. 168, 105294 (2020). https://doi.org/10.1016/j.ijmecsci.2019.105294
Zhang, Y.W., She, G.L.: Combined resonance of graphene platelets reinforced metal foams cylindrical shells with spinning motion under nonlinear forced vibration. Eng. Struct. 300, 117177 (2024). https://doi.org/10.1016/j.engstruct.2023.117177
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Mohamed, N., Eltaher, M.A., Mohamed, S.A. et al. Bernstein polynomials in analyzing nonlinear forced vibration of curved fractional viscoelastic beam with viscoelastic boundaries. Acta Mech (2024). https://doi.org/10.1007/s00707-024-03954-7
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DOI: https://doi.org/10.1007/s00707-024-03954-7