Abstract
The dynamic response of fractionally damped viscoelastic plates subjected to a moving point load is investigated. In order to capture the viscoelastic dynamic behavior more accurately, the material is modeled using the fractionally damped Kelvin–Voigt model (rather than the integer-type viscoelastic model). The Riemann–Liouville fractional derivative of order 0 < α ≤ 1 is applied. Galerkin's method and Newton–Raphson technique are used to evaluate the natural frequencies and corresponding damping coefficients. The structure is subject to a moving point load, traveling at different speeds. The modal summation technique is applied to generate the dynamic response of the plate. The influence of the order of the fractional derivative on the free and transient vibrations is studied for different velocities of the moving load. The results are compared with those using the classical integer-type Kelvin–Voigt viscoelastic model. The results show that an increase in the order of the fractional derivative causes a significant decrease in the maximum dynamic amplification factor, especially in the “dynamic zone” of the normalized sweep time. The dynamic behavior of the plate is verified with ansys.
Issue Section:
Research Papers
References
1.
Newman
, M. K.
, 1959
, “Viscous Damping in Flexural Vibrations of Bars
,” ASME J. Appl. Mech.
, 26
, pp. 367
–376
. 10.1115/1.36439722.
Gürgöze
, M.
, 1987
, “Parametric Vibrations of a Viscoelastic Beam (Maxwell Model) Under Steady Axial Load and Transverse Displacement Excitation at One End
,” J. Sound Vib.
, 115
(2
), pp. 329
–338
. 10.1016/0022-460X(87)90476-73.
Mahmoodi
, S. N.
, Khadem
, S. E.
, and Kokabi
, M.
, 2007
, “Non-Linear Free Vibrations of Kelvin–Voigt Visco-Elastic Beams
,” Int. J. Mech. Sci.
, 49
(6
), pp. 722
–732
. 10.1016/j.ijmecsci.2006.10.0054.
Mainardi
, F.
, and Spada
, G.
, 2011
, “Creep, Relaxation and Viscosity Properties for Basic Fractional Models in Rheology
,” Eur. Phys. J. Spec. Top.
, 193
(1
), pp. 133
–160
. 10.1140/epjst/e2011-01387-15.
Bagley
, R. L.
, and Torvik
, P. J.
, 1983
, “A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity
,” J. Rheol.
, 27
(3
), pp. 201
–210
. 10.1122/1.5497246.
Bagley
, R. L.
, and Torvik
, P. J.
, 1986
, “On the Fractional Calculus Model of Viscoelastic Behavior
,” J. Rheol.
, 30
(1
), pp. 133
–155
. 10.1122/1.5498877.
Meral
, F. C.
, Royston
, T. J.
, and Magin
, R.
, 2010
, “Fractional Calculus in Viscoelasticity: An Experimental Study
,” Commun. Nonlinear Sci.
, 15
(4
), pp. 939
–945
. 10.1016/j.cnsns.2009.05.0048.
Adolfsson
, K.
, Enelund
, M.
, and Olsson
, P.
, 2005
, “On the Fractional Order Model of Viscoelasticity
,” Mech. Time-Depend. Mater.
, 9
(1
), pp. 15
–34
. 10.1007/s11043-005-3442-19.
Di Paola
, M.
, Pirrotta
, A.
, and Valenza
, A.
, 2011
, “Visco-Elastic Behavior Through Fractional Calculus: An Easier Method for Best Fitting Experimental Results
,” Mech. Mater.
, 43
(12
), pp. 799
–806
. 10.1016/j.mechmat.2011.08.01610.
Ramirez
, L. E. S.
, and Coimbra
, C. F. M.
, 2007
, “A Variable Order Constitutive Relation for Viscoelasticity
,” Ann. Phys.
, 16
(7–8
), pp. 543
–552
. 10.1002/andp.20071024611.
Eldred
, L. B.
, Baker
, W. P.
, and Palazotto
, A. N.
, 1995
, “Kelvin-Voigt Versus Fractional Derivative Model as Constitutive Relations for Viscoelastic Materials
,” AIAA J.
, 33
(3
), pp. 547
–550
. 10.2514/3.1247112.
Sunny
, M. R.
, Kapania
, R. K.
, Moffitt
, R. D.
, Mishra
, A.
, and Goulbourne
, N.
, 2010
, “A Modified Fractional Calculus Approach to Model Hysteresis
,” ASME J. Appl. Mech.
, 77
(3
), p. 031004
. 10.1115/1.400041313.
Caputo
, M.
, 1967
, “Linear Models of Dissipation Whose Q Is Almost Frequency Independent
,” Geophys. J. Int.
, 13
(5
), pp. 529
–539
. 10.1111/j.1365-246X.1967.tb02303.x14.
Enelund
, M.
, and Olsson
, P.
, 1999
, “Damping Described by Fading Memory—Analysis and Application to Fractional Derivative Models
,” Int. J. Solids Struct.
, 36
(7
), pp. 939
–970
. 10.1016/S0020-7683(97)00339-915.
Caputo
, M.
, and Mainardi
, F.
, 1971
, “A New Dissipation Model Based on Memory Mechanism
,” Pure Appl. Geophys.
, 91
(1
), pp. 134
–147
. 10.1007/BF0087956216.
Bagley
, R.
, 2007
, “On the Equivalence of the Riemann-Liouville and the Caputo Fractional Order Derivatives in Modeling of Linear Viscoelastic Materials
,” Fract. Calc. Appl. Anal.
, 10
(2
), pp. 123
–126
.17.
Rossikhin
, Y.
, Shitikova
, M.
, and Shcheglova
, T.
, 2010
, “Forced Vibrations of a Nonlinear Oscillator With Weak Fractional Damping
,” J. Mech. Mater. Struct.
, 4
(9
), pp. 1619
–1636
. 10.2140/jomms.2009.4.161918.
Rossikhin
, Y. A.
, 2010
, “Reflections on Two Parallel Ways in the Progress of Fractional Calculus in Mechanics of Solids
,” ASME Appl. Mech. Rev.
, 63
(1
), p. 010701
. 10.1115/1.400024619.
Rossikhin
, Y. A.
, and Shitikova
, M. V
, 2019
, “Fractional Calculus in Structural Mechanics,” Handbook of Fractional Calculus With Applications
, Vol. 7
, D.
Baleanu
, and A.
Mendes Lopes
, eds., De Gruyter
, Berlin
, pp. 159
–192
.20.
21.
Markert
, R.
, and Seidler
, M.
, 2001
, “Analytically Based Estimation of the Maximum Amplitude During Passage Through Resonance
,” Int. J. Solids Struct.
, 38
(10–13
), pp. 1975
–1992
. 10.1016/S0020-7683(00)00147-522.
Weaver
, W.
, Jr., Timoshenko
, S. P.
, and Young
, D. H.
, 1990
, Vibration Problems in Engineering
, John Wiley & Sons
, New York
.23.
Frýba
, L.
, 2013
, Vibration of Solids and Structures Under Moving Loads
, Springer Science & Business Media
, Berlin
.24.
Abu-Mallouh
, R.
, Abu-Alshaikh
, I.
, Zibdeh
, H. S.
, and Ramadan
, K.
, 2012
, “Response of Fractionally Damped Beams With General Boundary Conditions Subjected to Moving Loads
,” Shock Vib.
, 19
(3
), pp. 333
–347
. 10.1155/2012/32142125.
Freundlich
, J. K.
, 2016
, “Dynamic Response of a Simply Supported Viscoelastic Beam of a Fractional Derivative Type to a Moving Force Load
,” J. Theor. Appl. Mech.
, 54
(4
), pp. 1433
–1445
. 10.15632/jtam-pl.54.4.143326.
Hedrih
, K.
, 2005
, “Partial Fractional Differential Equations of Creeping and Vibrations of Plate and Their Solutions (First Part)
,” J. Mech. Behav. Mater.
, 16
(4/5
), p. 305
.27.
Rossikhin
, Y. A.
, and Shitikova
, M. V.
, 2006
, “Analysis of Damped Vibrations of Linear Viscoelastic Plates With Damping Modeled With Fractional Derivatives
,” Signal Process.
, 86
(10
), pp. 2703
–2711
. 10.1016/j.sigpro.2006.02.01628.
Rossikhin
, Y. A.
, and Shitikova
, M. V.
, 2011
, “The Analysis of the Impact Response of a Thin Plate via Fractional Derivative Standard Linear Solid Model
,” J. Sound Vib.
, 330
(9
), pp. 1985
–2003
. 10.1016/j.jsv.2010.11.01029.
Ingman
, D.
, and Suzdalnitsky
, J.
, 2008
, “Response of Viscoelastic Plate to Impact
,” ASME J. Vib. Acoust.
, 130
(1
), p. 011010
. 10.1115/1.273141630.
Katsikadelis
, J. T.
, and Babouskos
, N. G.
, 2010
, “Post-Buckling Analysis of Viscoelastic Plates With Fractional Derivative Models
,” Eng. Anal. Bound. Elem.
, 34
(12
), pp. 1038
–1048
. 10.1016/j.enganabound.2010.07.00331.
Rossikhin
, Y. A.
, Shitikova
, M. V.
, and Trung
, P. T.
, 2016
, “Application of the Fractional Derivative Kelvin-Voigt Model for the Analysis of Impact Response of a Kirchhoff-Love Plate
,” WSEAS Trans. Math.
, 15
, pp. 498
–501
.32.
Katsikadelis
, J. T.
, 2015
, “Generalized Fractional Derivatives and Their Applications to Mechanical Systems
,” Arch. Appl. Mech.
, 85
(9–10
), pp. 1307
–1320
. 10.1007/s00419-014-0969-033.
Ari
, M.
, Faal
, R. T.
, and Zayernouri
, M.
, 2019
, “Vibrations Suppression of Fractionally Damped Plates Using Multiple Optimal Dynamic Vibration Absorbers
,” Int. J. Comput. Math.
, 97
(4
), pp. 1
–24
. 10.1080/00207160.2019.159479234.
Permoon
, M. R.
, Haddadpour
, H.
, and Javadi
, M.
, 2018
, “Nonlinear Vibration of Fractional Viscoelastic Plate: Primary, Subharmonic, and Superharmonic Response
,” Int. J. Nonlinear Mech.
, 99
, pp. 154
–164
. 10.1016/j.ijnonlinmec.2017.11.01035.
Shermergor
, T. D.
, 1966
, “On the Use of Fractional Differentiation Operators for the Description of Elastic-After Effect Properties of Materials
,” J. Appl. Mech. Tech. Phys.
, 7
(6
), pp. 85
–87
. 10.1007/BF0091434736.
Samko
, S. G.
, Kilbas
, A. A.
, and Marichev
, O. I.
, 1993
, Fractional Integrals and Derivatives
, Gordon and Breach Science Publishers
, Yverdon Yverdon-les-Bains, Switzerland
.37.
Kiasat
, M. S.
, Zamani
, H. A.
, and Aghdam
, M. M.
, 2014
, “On the Transient Response of Viscoelastic Beams and Plates on Viscoelastic Medium
,” Int. J. Mech. Sci.
, 83
, pp. 133
–145
. 10.1016/j.ijmecsci.2014.03.00738.
Podlubny
, I.
, 2000
, “Matrix Approach to Discrete Fractional Calculus
,” Fract. Calc. Appl. Anal.
, 3
(4
), pp. 359
–386
.39.
Podlubny
, I.
, Chechkin
, A.
, Skovranek
, T.
, Chen
, Y.
, and Jara
, B. M. V.
, 2009
, “Matrix Approach to Discrete Fractional Calculus II: Partial Fractional Differential Equations
,” J. Comput. Phys.
, 228
(8
), pp. 3137
–3153
. 10.1016/j.jcp.2009.01.01440.
Timoshenko
, S. P.
, and Woinowsky-Krieger
, S.
, 1959
, Theory of Plates and Shells
, McGraw-Hill
, New York
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