In this paper, we present a generalized model of a viscoelastic beam, which takes into account the influence of axial forces and incorporates a fractional constitutive relationship. In addition, we propose a novel numerical calculation method for analyzing fractional-order viscoelastic beams. This method takes the transverse displacement and bending moment of the beam as state variables, transforms the beam model into a discrete state space equation through the application of the central difference method, and utilizes an improved precise integration algorithm to solve the equation. To evaluate the performance of the method, the responses of the beam under two types of excitations, namely uniformly distributed transverse load and the motion of the support at both ends, are calculated under fixed hinge conditions. The results demonstrate that the present method has excellent accuracy and convergence, and also reveal some nonlinear phenomena of the system.
AbhyankarNSHallEKHanagudSV (1993) Chaotic vibrations of beams: Numerical solution of partial differential equations. Journal of Applied Mechanics60(1): 167–174.
2.
AnsariRFaraji OskouieMGholamiR (2016) Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory. Physica E Low-dimensional Systems and Nanostructures75: 266–271.
3.
ArgyrisJBelubekianVOvakimyanN, et al. (1996) Chaotic vibrations of a nonlinear viscoelastic beam. Chaos, Solitons, and Fractals7(2): 151–163.
4.
BagleyRTorvikP (1979) A generalized derivative model for an elastomer damper. Shock and Vibration49(2): 135–143.
5.
BoonrodYMohsenR (2019) Legendre wavelets approach for numerical solutions of distributed order fractional differential equations. Applied Mathematical Modelling70: 350–364.
6.
BourneMWinklerJSuY (2017) The computation of the degree of an approximate greatest common divisor of two Bernstein polynomials. Applied Numerical Mathematics111: 17–35.
7.
ChenLChengC (2001) Vibration analysis of viscoelastic beams with fractional derivative constitute relation. Chinese Quarterly of Mechanics22(4): 5.
8.
ChenLQChengCJ (2000) Dynamical behavior of nonlinear viscoelastic beams. Applied Mathematics and Mechanics21(9): 995–1001.
9.
ChenN (2002a) Dynamical behaviors of nonlinear viscoelastic buckled beams. Journal of Theoretical and Applied Mechanics22(4): 5.
10.
ChenN (2002b) Numerical investigation on dynamical behaviors of a buckled beam under axial harmonic excitation. Journal of Nanjing Forestry University26(2): 5.
11.
ChenYKeXWeiY (2015) Numerical algorithm to solve system of nonlinear fractional differential equations based on wavelets method and the error analysis. Applied Mathematics and Computation251: 475–488.
12.
CortésFElejabarrietaMJ (2007) Homogenised finite element for transient dynamic analysis of unconstrained layer damping beams involving fractional derivative models. Computational Mechanics40(2): 313–324.
13.
FoucartSLasserreJB (2019) Computation of Chebyshev polynomials for union of intervals. Computational Methods and Function Theory19(4): 625–641.
14.
JavadiMRahmanianM (2021) Nonlinear vibration of fractional kelvin–Voigt viscoelastic beam on nonlinear elastic foundation. Communications in Nonlinear Science and Numerical Simulation Epub ahead of print July 2021; DOI: 10.1016/j.cnsns.2021.105784.
15.
KoellerRC (1984) Applications of fractional calculus to the theory of viscoelasticity. Journal of Applied Mechanics51(2): 299–307.
16.
LeiYAdhikariSFriswellMI (2013) Vibration of nonlocal kelvin–Voigt viscoelastic damped Timoshenko beams. International Journal of Science and Engineering66-67(1-13): 1–13.
17.
LeungAYTYangHXZhuP, et al. (2013) Steady state response of fractionally damped nonlinear viscoelastic arches by residue harmonic homotopy. Computers & Structures121: 10–21.
18.
LewandowskiRBaumM (2015) Dynamic characteristics of multilayered beams with viscoelastic layers described by the fractional Zener model. Archive of Applied Mechanics85(12): 1793–1814.
19.
LiangZFTangXY (2007) Analytical solution of fractionally damped beam by Adomian decomposition method. Applied Mathematics and Mechanics28(2): 219–228.
20.
LoghmanEBakhtiari-NejadFKamaliE A, et al. (2021b) Nonlinear vibration of fractional viscoelastic micro-beams. International Journal of Non-Linear Mechanics Epub ahead of print 1 September 2021. DOI: 10.1016/j.ijnonlinmec.2021.103811.
21.
LoghmanEKamaliABakhtiari-NejadF, et al. (2021a) Nonlinear free and forced vibrations of fractional modeled viscoelastic FGM micro-beam. Applied Mathematical Modelling92: 297–314.
22.
MasoudSEghtesadMFaridM, et al. (2013) Large deflection of viscoelastic beams using fractional derivative model. Journal of Mechanical Science and Technology27(4): 1063–1070.
23.
OlgaM (2019) Stability approach to the fractional variational iteration method used for the dynamic analysis of viscoelastic beams. Journal of Computational and Applied Mathematics346: 261–276.
24.
PritzT (2015) Analysis of four-parameter fractional derivative model of real solid materials. Journal of Sound and Vibration195(1): 103–115.
25.
PuYFSiarryPZhouJL, et al. (2014) A fractional partial differential equation based multiscale denoising model for texture image. Mathematical Methods in the Applied Sciences37(12): 1784–1806.
26.
SakranMRA (2019) Numerical solutions of integral and integro-differential equations using Chebyshev polynomials of the third kind. Applied Mathematics and Computation351: 66–82.
27.
ShenMChenSLiuF (2018) Unsteady MHD flow and heat transfer of fractional Maxwell viscoelastic nanofluid with Cattaneo heat flux and different particle shapes. Chinese Journal of Physics56(3): 1199–1211.
28.
SuireGCederbaumG (1995) Periodic and chaotic behavior of viscoelastic nonlinear (elastica) bars under harmonic excitations. International Journal of Mechanical Sciences37(7): 753–772.
29.
SweilamNHNagyAMEl-SayedAA (2015) Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation. Chaos, Solitons, and Fractals73: 141–147.
30.
WangJXuTZWangGW (2018) Numerical algorithm for time-fractional Sawada-Kotera equation and Ito equation with Bernstein polynomials. Applied Mathematics and Computation338: 1–11.
31.
XuHJiangX (2017) Creep constitutive models for viscoelastic materials based on fractional derivatives. Computers & Mathematics With Applications73(6): 1377–1384.
32.
XuJChenYTaiY, et al. (2020a) Vibration analysis of sandwich beams with viscoelastic coating described by fractional constitutive equation. Mechanics of Advanced Materials and Structures29(3): 429–439.
33.
XuJChenYTaiY, et al. (2020b) Vibration analysis of complex fractional viscoelastic beam structures by the wave method. International Journal of Mechanical Sciences Epub ahead of print 1 February 2020. DOI: 10.1016/j.ijmecsci.2019.105204.
34.
XuJChenYTaiY, et al. (2022) New control strategy for suppressing the local vibration of sandwich beams based on the wave propagation method. Journal of Intelligent Material Systems and Structures33(1): 231–247.
35.
YinHChenNZhaoC, et al. (2010) A numerical algorithm for the dynamic equation of the fractional derivative viscoelasticity structure. Journal of Nanjing Forestry University34(2): 115–118.
36.
ZhangGWuZLiY (2019) Nonlinear Dynamic Analysis of fractional damped viscoelastic beams. International Journal of Structural Stability and Dynamics Epub ahead of print November 2019. DOI: 10.1142/S0219455419501293.
37.
ZhangZHuangWLiH, et al. (2017) Research progress on the application of viscoelastic damper materials and damping structure. Environmental Protection and Technology23(6): 5.
38.
Zheng-youZGen-guoLChang-junC (2002) Quasi-static and dynamical analysis for viscoelastic Timoshenko beam with fractional derivative constitutive relation. Applied Mathematics and Mechanics23(1): 1–12.
