Abstract
The paper deals with the stochastic dynamics of a vibroimpact single-degree-of-freedom system under a Gaussian white noise. The system is assumed to have a hard type impact against a one-sided motionless barrier, located at the system's equilibrium. The system is endowed with a fractional derivative element. An analytical expression for the system's mean squared response amplitude is presented and compared with the results of numerical simulations.
Issue Section:
Special Section Papers
References
1.
2.
Ibrahim
,
R. A.
,
Babitsky
, V.
I.
, and
Okuma
,
M.
, 2009
,
Vibro-Impact Dynamics of Ocean Systems and Related
Problems
, Springer
,
Berlin.3.
Dimentberg
,
M.
,
Yurchenko
,
D.
, and van
Ewijk
, O.
,
1998
, “Subharmonic Response of a
Quasi-Isochronous Vibroimpact System to a Randomly Disordered Periodic
Excitation
,” Nonlinear Dyn.
,
17
(2
), pp.
173
–186
.4.
Jacquelin
,
E.
,
Adhikari
,
S.
, and
Friswell
, M.
I.
, 2011
, “A
Piezoelectric Device for Impact Energy Harvesting
,”
Smart Mater. Struct.
,
20
(10
), p. 105008.5.
Zhang
,
Y.
,
Cai
, C.
S.
, and Zhang
,
W.
, 2014
,
“Experimental Study of a Multi-Impact Energy Harvester Under
Low Frequency Excitations
,” Smart Mater.
Struct.
, 23
(5
), p. 055002.6.
Masri
,
S. F.
, 1967
,
“Effectiveness of Two Particle Impact
Dampers
,” J. Acoust. Soc. Am.
,
41
(6
), pp.
1553
–1554
.7.
Lu
,
Z.
,
Masri
, S.
F.
, and Lu
,
X.
, 2011
,
“Studies of the Performance of Particle Dampers Attached to a
Two-Degrees-of-Freedom System Under Random Excitation
,”
J. Vib. Control
, 17
(10
),
pp. 1454
–1471
.8.
Pavlovskaia
,
E.
,
Hendry
, D.
C.
, and
Wiercigroch
,
M.
, 2015
,
“Modelling of High Frequency Vibro-Impact
Drilling
,” Int. J. Mech. Sci.
,
91
, pp. 110
–119
.9.
Dimentberg
,
M.
, and
Iourtchenko
, D.
V.
, 2004
,
“Random Vibrations With Impacts: A Review
,”
Nonlinear Dyn.
, 36
(2), pp.
229
–254
.10.
Dimentberg
,
M.
, and
Iourtchenko
, D.
V.
, 1999
,
“Towards Incorporating Impact Losses Into Random Vibration
Analyses: A Model Problem
,” Probab. Eng.
Mech.
, 14
(4
), pp.
323
–328
.11.
Iourtchenko
, D.
V.
, and
Song
, L.
L.
, 2006
, “Numerical
Investigation of a Response Probability Density Function of Stochastic
Vibroimpact Systems With Inelastic Impacts
,” Int. J.
Non-Linear Mech.
, 41
(3
), pp.
447
–455
.12.
Gemant
,
A.
, 1936, “A
Method of Analyzing Experimental Results Obtained From Elasto-Viscous
Bodies
,” J. Appl. Phys.
,
7
, pp. 311
–317
.13.
Bagley
,
R. L.
, and
Torvik
, P.
J.
, 1979
, “A
Generalized Derivative Model for an Elastomer Damper
,”
Shock Vib. Bull.
, 49
, pp.
135
–143
.https://www.researchgate.net/profile/Peter_Torvik/publication/4709831_A_generalized_derivative_model_for_an_elastomer_damper/links/55bf8d4708aed621de139986/A-generalized-derivative-model-for-an-elastomer-damper.pdf?origin=publication_detail14.
Bagley
,
R. L.
, and
Torvik
, P.
J.
, 1983
, “A
Theoretical Basis for the Application of Fractional
Calculus
,” J. Rheol.
,
27
(3
), pp.
201
–210
.15.
Bagley
,
R. L.
, and
Torvik
, P.
J.
, 1983
,
“Fractional Calculus—A Different Approach to the Analysis of
Viscoelastically Damped Structures
,” AIAA
J.
, 21
(5), pp.
741
–774
.16.
Bagley
,
R. L.
, and
Torvik
, P.
J.
, 1986
, “On the
Fractional Calculus Model of Viscoelastic Behavior
,”
J. Rheol.
, 30
(1
), pp.
133
–155
.17.
Nutting
,
P. G.
, 1921
,
“A New General Law Deformation
,” J.
Franklin Inst.
, 191
(5), pp.
678
–685
.http://ac.els-cdn.com/S0016003221901716/1-s2.0-S0016003221901716-main.pdf?_tid=9480453e-39cc-11e7-9c36-00000aacb35e&acdnat=1494893828_e9f85c51c0e04e386fad8188da13e1ff18.
Schmidt
,
A.
, and
Gaul
,
L.
, 2002
,
“Finite Element Formulation of Viscoelastic Constitutive
Equations Using Fractional Time Derivatives
,”
Nonlinear Dyn.
, 29
(1
), pp.
37
–55
.19.
Gonsovskii
, V.
L.
, and
Rossikhin
, Y.
A.
, 1973
,
“Stress Waves in a Viscoelastic Medium With a Singular
Hereditary Kernel
,” J. Appl. Mech. Tech.
Phys.
, 14
(4
), pp.
595
–597
.20.
Schiessel
,
H.
, and
Blumen
,
A.
, 1993
,
“Hierarchical Analogues to Fractional Relaxation
Equations
,” J. Phys. A
,
26
(19
), pp.
5057
–5069
.21.
Stiassnie
,
M.
, 1979
,
“On the Application of Fractional Calculus for the
Formulation of Viscoelastic Models
,” Appl. Math.
Modell.
, 3
(4
), pp.
300
–302
.22.
Mainardi
,
F.
, and
Gorenflo
,
R.
, 2007
,
“Time-Fractional Derivatives in Relaxation Processes: A
Tutorial Survey
,” Fractional Calculus Appl.
Anal.
, 10
(3
), pp.
269
–308
.http://www.diogenes.bg/fcaa/volume10/fcaa103/Mainardi_Gorenflo_survey103.pdf23.
Samko
,
G. S.
,
Kilbas
, A.
A.
, and
Marichev
, O.
I.
, 1993
, Fractional
Integrals and Derivatives
, Gordon and Breach
Science Publishers
, Amsterdam, The
Netherlands
.24.
Podlubny
,
I.
, 1999
,
On Solving Fractional Differential Equations by Mathematics, Science
and Engineering
, Academic Press
,
San Diego, CA.25.
Hilfer
,
R.
, 2000
,
Applications of Fractional Calculus in Physics
,
World Scientific
,
Singapore
.26.
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
.27.
Pirrotta
,
A.
,
Cutrona
,
S.
, Di
Lorenzo
, S.
, and
Di Matteo
,
A.
, 2015
,
“Fractional Visco-Elastic Timoshenko Beam Deflection Via
Single Equation
,” Int. J. Numer. Methods
Eng.
, 104
(9
), pp.
869
–886
.28.
Di
Lorenzo
, S.
,
Di Paola
,
M.
,
Pinnola
, F.
P.
, and
Pirrotta
,
A.
, 2014
,
“Stochastic Response of Fractionally Damped
Beams
,” Probab. Eng. Mech.
,
35
, pp. 37
–43
.29.
Di Paola
,
M.
,
Heuer
,
R.
, and
Pirrotta
,
A.
, 2013,
“Fractional Visco-Elastic Euler-Bernoulli
Beam
,” Int. J. Solids Struct.
,
50
(22–23
), pp.
3505
–3510
.30.
Pirrotta
,
A.
,
Cutrona
,
S.
, and Di
Lorenzo
, S.
,
2015
, “Fractional Visco-Elastic Timoshenko Beam
From Elastic Euler-Bernoulli Beam
,” Acta
Mech.
, 226
(1
), pp.
179
–189
.31.
Bucher
,
C.
, and
Pirrotta
,
A.
, 2015
,
“Dynamic Finite Element Analysis of Fractionally Damped
Structural Systems in the Time Domain
,” Acta
Mech.
, 226
(12
), pp.
3977
–3990
.32.
Alotta
,
G.
, Di
Paola
, M.
, and
Pirrotta
,
A.
, 2014
,
“Fractional Tajimi-Kanai Model for Simulating Earthquake
Ground Motion
,” Bull. Earthquake Eng.
(BEEE)
, 12
(6
), pp.
2495
–2506
.33.
Di
Matteo
, A.
,
Lo Iacono
,
F.
,
Navarra
,
G.
, and
Pirrotta
,
A.
, 2015
,
“Innovative Modeling of Tuned Liquid Column Damper
Motion
,” Commun. Nonlinear Sci. Numer.
Simul.
, 23
(1–3
), pp.
229
–244
.34.
Di Paola
,
M.
,
Failla
,
G.
, and
Pirrotta
,
A.
, 2012
,
“Stationary and Non-Stationary Stochastic Response of Linear
Fractional Viscoelastic Systems
,” Probab. Eng.
Mech.
, 28
, pp.
85
–90
.35.
Failla
,
G.
, and
Pirrotta
,
A.
, 2012
,
“On the Stochastic Response of a Fractionally-Damped Duffing
Oscillator
,” Commun. Nonlinear Sci. Numer.
Simul.
, 17
(12
), pp.
5131
–5142
.36.
Evangelatos
, G.
I.
, and
Spanos
, P.
D.
, 2011
, “An
Accelerated Newmark Scheme for Integrating the Equation of Motion of
Nonlinear Systems Comprising Restoring Elements Governed by Fractional
Derivatives
,” Recent Advances in Mechanics
,
Springer, Dordrecht, The Netherlands, pp.
159
–177
.37.
Huang
,
Z. L.
, and
Jin
, X.
L.
, 2009
, “Response
and Stability of a SDOF Strongly Nonlinear Stochastic System With Light
Damping Modeled by a Fraction Derivative
,” J. Sound
Vib.
, 319
(3–5
), pp.
1121
–1135
.38.
Spanos
,
P. D.
, and
Evangelatos
, G.
I.
, 2010
,
“Response of a Non-Linear System With Restoring Forces
Governed by Fractional Derivatives-Time Domain Simulation and Statistical
Linearization Solution
,” Soil Dyn. Earthquake
Eng.
, 30
(9
), pp.
811
–821
.Copyright © 2017 by ASME
You do not currently have access to this content.