Steady-state periodic responses for a vibration system with distributed order derivatives are investigated, where the fractional derivative operator is utilized. The response to complex harmonic excitation is derived and the amplitude–frequency and phase–frequency relations are obtained. For a periodic excitation, we decompose it into the Fourier series, and then make use of the principle of superposition and the results of harmonic excitations to obtain the response. Finally, we examine three numerical examples by using the proposed method.
AtanackovicTM (2002) A generalized model for the uniaxial isothermal deformation of a viscoelastic body. Acta Mechanica159: 77–86.
2.
AtanackovicTM (2003) On a distributed derivative model of a viscoelastic body. Comptes Rendus Mecanique331: 687–692.
3.
AtanackovićTMPilipovićSStankovićBet al. (2014) Fractional Calculus with Applications in Mechanics, New York: Wiley.
4.
BăleanuDMustafaOG (2010) On the global existence of solutions to a class of fractional differential equations. Computers and Mathematics with Applications59: 1835–1841.
5.
BaleanuDMachadoJATLuoACJ (2012) Fractional Dynamics and Control, New York: Springer.
6.
BăleanuDMustafaOGAgarwalRP (2010) An existence result for a superlinear fractional differential equation. Applied Mathematics Letters23: 1129–1132.
7.
BagleyRLTorvikPJ (1979) A generalized derivative model for an elastomer damper. Shock and Vibration Bulletin49: 135–143.
8.
BagleyRLTorvikPJ (2000a) On the existence of the order domain and the solution of distributed order equations, Part I. International Journal of Applied Mathematics2: 865–882.
9.
BagleyRLTorvikPJ (2000b) On the existence of the order domain and the solution of distributed order equations, part II. International Journal of Applied Mathematics2: 965–987.
10.
BeyerHKempfleS (1995) Definition of physically consistent damping laws with fractional derivatives. ZAMM–Journal of Applied Mathematics and Mechanics75: 623–635.
11.
CaputoM (2001) Distributed order differential equations modelling dielectric induction and diffusion. Fractional Calculus and Applied Analysis4: 421–442.
12.
DuanJS (2005) Time- and space-fractional partial differential equations. Journal of Mathematical Physics46: 13504–13511.
13.
DuanJSQiuX (2014) The periodic solution of Stokes’ second problem for viscoelastic fluids as characterized by a fractional constitutive equation. Journal of Non-Newtonian Fluid Mechanics205: 11–15.
14.
DuanJSWangZFuSZ (2014) The zeros of the solutions of the fractional oscillation equation. Fractional Calculus and Applied Analysis17: 10–22.
15.
DuanJSWangZLiuYLet al. (2003) Eigenvalue problems for fractional ordinary differential equations. Chaos, Solitons and Fractals46: 46–53.
16.
FriedrichCBraunH (1994) Linear viscoelastic behaviour of complex polymeric materials: A fractional mode representation. Colloid and Polymer Science272: 1536–1546.
17.
GaoXYuJ (2005) Chaos in the fractional order periodically forced complex Duffing’s oscillators. Chaos, Solitons and Fractals24: 1097–1104.
18.
HilferR (2000) Applications of Fractional Calculus in Physics, Singapore: World Scientific.
19.
HuangCDuanJS (2016) Steady-state response to periodic excitation in fractional vibration system. Journal of Mechanics32: 25–33.
20.
KaslikESivasundaramS (2012) Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions. Nonlinear Analysis: Real World Applications13: 1489–1497.
21.
KoellerRC (1984) Applications of fractional calculus to the theory of viscoelasticity. Journal of Applied Mechanics51: 299–307.
22.
LiCPengG (2004) Chaos in Chen’s system with a fractional order. Chaos, Solitons and Fractals22: 443–450.
23.
LiuLLDuanJS (2015) A detailed analysis for the fundamental solution of fractional vibration equation. Open Mathematics13: 826–838.
24.
MainardiF (2010) Fractional Calculus and Waves in Linear Viscoelasticity, London: Imperial College.
25.
MoghaddamBPMachadoJAT (2016) A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations. Computers & Mathematics with Applications. Epub ahead of print 3 August 2016. DOI:10.1016/j.camwa.2016.07.010.
26.
MonjeCAChenYQVinagreBMet al. (2010) Fractional-order Systems and Controls, Fundamentals and Applications, London: Springer.
27.
PodlubnyI (1999) Fractional Differential Equations, San Diego: Academic.
28.
SchiesselHMetzlerRBlumenAet al. (1995) Generalized viscoelastic models: Their fractional equations with solutions. Journal of Physics A: Mathematical and General28: 6567–6584.
29.
Scott-BlairGW (1949) A Survey of General and Applied Rheology, London: Pitman.
30.
ShengHChenYQiuT (2012) Fractional Processes and Fractional-Order Signal Processing Techniques and Applications, London: Springer.
31.
SoonCMCoimbraCFMKobayashiMH (2005) The variable viscoelasticity oscillator. Annalen der Physik14: 378–389.
32.
TanWXuM (2002) Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model. Acta Mechanica Sinica18: 342–349.
33.
WuGCBaleanuDXieHPet al. (2016) Chaos synchronization of fractional chaotic maps based on the stability condition. Physica A460: 374–383.
34.
YaghoobiSMoghaddamBPIvazK (2016) An efficient cubic spline approximation for variable-order fractional differential equations with time delay. Nonlinear Dynamics. Epub ahead of print 22 September 2016. DOI:10.1007/s11071-016-3079-4.
