Uncertain dynamic responses of fuzzy fractionally damped beams have been studied using the newly developed double parametric form of fuzzy numbers subject to unit step and impulse loads. Uncertainties are assumed to be present in the initial conditions, and these are modeled through triangular convex normalized fuzzy sets. Using the alpha-cut form, the corresponding beam equation is first converted to an interval-based fuzzy equation. Next, it has been transformed to crisp form by applying a double-parametric form of fuzzy numbers. Finally, homotopy perturbation method (HPM) has been used to solve the same for obtaining the general fuzzy responses. Various numerical examples are taken into consideration. The results are plotted.

References

References
1.
Samko
,
S. G.
,
Kilbas
,
A. A.
, and
Marichev
,
O. I.
,
1993
,
Fractional Integrals and Derivatives: Theory and Application
,
Gordon and Breach Science Publishers
,
Langhorne, PA
.
2.
Miller
,
K. S.
, and
Ross
,
B.
,
1993
,
An Introduction to the Fractional Calculus and Fractional Differential Equations
,
John Wiley and Sons
,
New York
.
3.
Oldham
,
K. B.
, and
Spanier
,
J.
,
1974
,
The Fractional Calculus
,
Academic Press
,
New York
.
4.
Kiryakova
,
V. S.
,
1993
,
Generalized Fractional Calculus and Applications
,
Longman Scientific and Technical
,
London
.
5.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
New York
.
6.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier
,
New York
.
7.
Suarez
,
L. E.
, and
Shokooh
,
A.
,
1997
, “
An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives
,”
ASME J. Appl. Mech.
,
64
(
3
), pp.
629
635
.
8.
Yuan
,
L.
, and
Agrawal
,
O. P.
,
2002
, “
A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives
,”
ASME J. Vib. Acoust.
,
124
(
2
), pp.
321
324
.10.1115/1.1448322
9.
Behera
,
D.
, and
Chakraverty
,
S.
,
2013
, “
Numerical Solution of Fractionally Damped Beam by Homotopy Perturbation Method
,”
Cent. Eur. J. Phys.
,
11
(
6
), pp.
792
798
.10.2478/s11534-013-0201-9
10.
Chakraverty
,
S.
, and
Behera
,
D.
,
2013
, “
Dynamic Responses of Fractionally Damped Mechanical System Using Homotopy Perturbation Method
,”
Alexandria Eng. J.
,
52
(
3
), pp.
557
562
.10.1016/j.aej.2013.04.007
11.
Odibat
,
Z. M.
, and
Momani
,
S.
,
2008
, “
An Algorithm for the Numerical Solution of Differential Equations of Fractional Order
,”
J. Appl. Math. Inf.
,
26
(
1–2
), pp.
15
27
.
12.
Jumarie
,
G.
,
2009
, “
Table of Some Basic Fractional Calculus Formulae Derived From a Modified Riemann-Liouville Derivative for Non-Differentiable Functions
,”
Appl. Math. Lett.
,
22
(
3
), pp.
378
385
.10.1016/j.aml.2008.06.003
13.
Wei
,
Z.
,
Li
,
Q.
, and
Che
,
J.
,
2010
, “
Initial Value Problems for Fractional Differential Equations Involving Riemann-Liouville Sequential Fractional Derivative
,”
J. Math. Anal. Appl.
,
367
(
1
), pp.
260
272
.10.1016/j.jmaa.2010.01.023
14.
Qian
,
D.
,
Li
,
C.
,
Agarwal
,
R. P.
, and
Wong
,
P. J. Y.
,
2010
, “
Stability Analysis of Fractional Differential System With Riemann-Liouville Derivative
,”
Math. Comput. Modell.
,
52
(
5–6
), pp.
862
874
.10.1016/j.mcm.2010.05.016
15.
Rahimy
,
M.
,
2010
, “
Applications of Fractional Differential Equations
,”
Appl. Math. Sci.
,
4
(
50
), pp.
2453
2461
.
16.
Zadeh
,
L.
,
1965
, “
Fuzzy sets
,”
Inf. Control
,
8
(
3
), pp.
338
353
.10.1016/S0019-9958(65)90241-X
17.
Hanss
,
M.
, and
Turrin
,
S.
,
2010
, “
A Fuzzy-Based Approach to Comprehensive Modelling and Analysis of Systems With Epistemic Uncertainties
,”
Struct. Saf.
,
32
(
6
), pp.
433
441
.10.1016/j.strusafe.2010.06.003
18.
Rao
,
M. V. R.
,
Pownuk
,
A.
,
Vandewalle
,
S.
, and
Moens
,
D.
,
2010
, “
Transient Response of Structures With Uncertain Structural Parameters
,”
Struct. Saf.
,
32
(
6
), pp.
449
460
.10.1016/j.strusafe.2010.05.001
19.
Farkas
,
L.
,
Moens
,
D.
,
Donders
,
S.
, and
Vandepitte
,
D.
,
2012
, “
Optimisation Study of a Vehicle Bumper Subsystem With Fuzzy Parameters
,”
Mech. Syst. Signal Process.
,
32
, pp.
59
68
.10.1016/j.ymssp.2011.11.014
20.
Behera
,
D.
, and
Chakraverty
,
S.
,
2013
, “
Fuzzy Analysis of Structures With Imprecisely Defined Properties
,”
Comput. Model. Eng. Sci.
,
96
(
5
), pp.
317
337
.
21.
Behera
,
D.
, and
Chakraverty
,
S.
,
2013
, “
Fuzzy Finite Element Analysis of Imprecisely Defined Structures With Fuzzy Nodal Force
,”
Eng. Appl. Artif. Intell.
,
26
(
10
), pp.
2458
2466
.10.1016/j.engappai.2013.07.021
22.
Behera
,
D.
, and
Chakraverty
,
S.
,
2014
, “
Solving Fuzzy Complex System of Linear Equations
,”
Inf. Sci.
,
277
, pp.
154
162
.10.1016/j.ins.2014.02.014
23.
Tapaswini
,
S.
, and
Chakraverty
,
S.
,
2014
, “
Non-Probabilistic Solution of Uncertain Vibration Equation of Large Membranes Using Adomian Decomposition Method
,”
Sci. World J.
,
2014
, pp.
1
11
.10.1155/2014/308205
24.
Agrawal
,
R. P.
,
Lakshmikantham
,
V.
, and
Nieto
,
J. J.
,
2010
, “
On the Concept of Solution for Fractional Differential Equations With Uncertainty
,”
Nonlinear Anal.
,
72
(
6
), pp.
2859
2862
.10.1016/j.na.2009.11.029
25.
Arshad
,
S.
, and
Lupulescu
,
V.
,
2011
, “
On the Fractional Differential Equations With Uncertainty
,”
Nonlinear Anal.
,
74
(
11
), pp.
3685
3693
.10.1016/j.na.2011.02.048
26.
Mohammed
,
O. H.
,
Fadhel
,
S. F.
, and
Fajer
,
A. A. K.
,
2011
, “
Differential Transform Method for Solving Fuzzy Fractional Initial Vale Problems
,”
J. Basrah Res.
,
37
(
4
), pp.
158
170
.
27.
Wang
,
H.
, and
Liu
,
Y.
,
2011
, “
Existence Results for Fractional Fuzzy Differential Equations With Finite Delay
,”
Int. Math. Forum
,
6
(
51
), pp.
2535
2538
.
28.
Salahshour
,
S.
,
Allahviranloo
,
T.
, and
Abbasbandy
,
S.
,
2012
, “
Solving Fuzzy Fractional Differential Equations by Fuzzy Laplace Transforms
,”
Commun. Nonlinear Sci. Numer. Simul.
,
17
(
3
), pp.
1372
1381
.10.1016/j.cnsns.2011.07.005
29.
Karthikeyan
,
K.
, and
Chandran
,
C.
,
2011
, “
Existence Results for Functional Fractional Fuzzy Impulsive Differential Equations
,”
Int. J. Contemp. Math. Sci.
,
6
(
39
), pp.
1941
1954
.
30.
Jeong
,
J. U.
,
2010
, “
Existence Results for Fractional Order Fuzzy Differential Equations With Infinite Delay
,”
Int. Math. Forum
,
5
(
65
), pp.
3221
3230
.
31.
Ahmad
,
M. Z.
,
Hasan
,
M. K.
, and
Abbasbandy
,
S.
,
2013
, “
Solving Fuzzy Fractional Differential Equations Using Zadeh’s Extension Principle
,”
Sci. World J.
,
2013
, pp.
1
11
.10.1155/2013/454969.
32.
He
,
J. H.
,
1999
, “
Homotopy Perturbation Technique
,”
Comput. Methods Appl. Mech. Eng.
,
178
(
3–4
), pp.
257
262
.10.1016/S0045-7825(99)00018-3
33.
He
,
J. H.
,
2000
, “
A Coupling Method of a Homotopy Technique and a Perturbation Technique for Non-Linear Problems
,”
Int. J. Nonlinear Mech.
,
35
(
1
), pp.
37
43
.10.1016/S0020-7462(98)00085-7
34.
Ross
,
T. J.
,
2004
,
Fuzzy Logic With Engineering Applications
,
John Wiley & Sons
,
New York
.
35.
Zimmermann
,
H. J.
,
2001
,
Fuzzy Set Theory and its Application
,
Kluwer Academic Publishers
,
London
.
36.
Zu-feng
,
L.
, and
Xiao-yan
,
T.
,
2007
, “
Analytical Solution of Fractionally Damped Beam by Adomian Decomposition Method
,”
Appl. Math. Mech.
,
28
(
2
), pp.
219
228
.10.1007/s10483-007-0210-zs
You do not currently have access to this content.