Bifurcation Analysis of a Vibro-Impact Viscoelastic Oscillator with Fractional Derivative Element

Article (PDF Available)inInternational Journal of Bifurcation and Chaos 28(14):1850170 · December 2018with 103 Reads
DOI: 10.1142/S0218127418501705
Cite this publication
Abstract
To the best of authors' knowledge, little work has been focused on the noisy vibro-impact systems with fractional derivative element. In this paper, stochastic bifurcation of a vibro-impact oscillator with fractional derivative element and a viscoelastic term under Gaussian white noise excitation is investigated. First, the viscoelastic force is approximately replaced by damping force and stiffness force. Thus the original oscillator is converted to an equivalent oscillator without a viscoelastic term. Second, the nonsmooth transformation is introduced to remove the discontinuity of the vibro-impact oscillator. Third, the stochastic averaging method is utilized to obtain analytical solutions of which the effectiveness can be verified by numerical solutions. We also find that the viscoelastic parameters, fractional coefficient and fractional derivative order can induce stochastic bifurcation.
December 24, 2018 14:49 WSPC/S0218-1274 1850170
International Journal of Bifurcation and Chaos, Vol. 28, No. 14 (2018) 1850170 (10 pages)
c
World Scientific Publishing Company
DOI: 10.1142/S0218127418501705
Bifurcation Analysis of a Vibro-Impact Viscoelastic
Oscillator with Fractional Derivative Element
Yong-Ge Yangand Wei Xu
Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi’an 710129, P. R. China
yonggeyang@163.com
weixu@nwpu.edu.cn
YangQuan Chen
Mechatronics, Embedded Systems and Automation Lab,
School of Engineering, University of California,
Merced, CA 95343, USA
ychen53@ucmerced.edu
Bingchang Zhou
Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi’an 710129, P. R. China
bczhou98@126.com
Received November 2, 2017; Revised June 30, 2018
To the best of authors’ knowledge, little work has been focused on the noisy vibro-impact
systems with fractional derivative element. In this paper, stochastic bifurcation of a vibro-impact
oscillator with fractional derivative element and a viscoelastic term under Gaussian white noise
excitation is investigated. First, the viscoelastic force is approximately replaced by damping
force and stiffness force. Thus the original oscillator is converted to an equivalent oscillator
without a viscoelastic term. Second, the nonsmooth transformation is introduced to remove the
discontinuity of the vibro-impact oscillator. Third, the stochastic averaging method is utilized to
obtain analytical solutions of which the effectiveness can be verified by numerical solutions. We
also find that the viscoelastic parameters, fractional coefficient and fractional derivative order
can induce stochastic bifurcation.
Keywords: Stochastic bifurcation; vibro-impact; viscoelastic oscillator; fractional derivative
element.
1. Introduction
Fractional calculus developing in parallel with the
classical calculus has received considerable atten-
tion in recent decades. Many authors have focused
on the dynamical response of fractional systems.
Huang extended the stochastic averaging method
to discuss the stochastic response and stability of
fractional nonlinear systems subject to Gaussian
white noise excitations [Huang & Jin,2009]. Chen
put forward an innovative bifurcation control
method based upon the fractional-order feedback
controller to control the stochastic jump bifurcation
Author for correspondence
1850170-1
December 24, 2018 14:49 WSPC/S0218-1274 1850170
Y.-G. Yang et al.
of noisy Duffing oscillator. Deng and Zhu devel-
oped a stochastic averaging method for quasi-
Hamiltonian systems under fractional Gaussian
noise [Deng & Zhu,2016]. Malara and Spanos
addressed the problem of determining the nonlin-
ear response of a fractional plate driven by ran-
dom loads [Malara & Spanos,2018]. Colinas-Armijo
and Di Paola used two methods to evaluate the
response of a linear viscoelastic material modeled
by the fractional Maxwell model and subject to a
Gaussian stochastic temperature process [Colinas-
Armijo et al.,2018]. Kougioumtzoglou proposed
a new multiple-input/single-output system identi-
fication approach for parameter identification of
fractional oscillators subject to incomplete nonsta-
tionary data [Kougioumtzoglou et al.,2017]. Ma
and Li established a local fractional center man-
ifold for a finite-dimensional fractional ordinary
differential system [Ma & Li,2016]. In addition,
comprehensive review papers have been completed
by Rossikhin [Rossikhin & Shitikova,1997,2010],
Machado [Machado et al.,2011] and Chen [Chen
et al.,2009; Li et al.,2017], respectively.
Vibro-impact systems, as a class of discontin-
uous and strongly nonlinear systems, can exhibit
complicated dynamical behaviors [Luo et al.,2006;
Di Bernardo et al.,2003]. Zhu discussed the stochas-
tic response of a vibro-impact Duffing system under
external Poisson impulses [Rossikhin,2015]. Xu
explored the stochastic response of an inelastic
vibro-impact system under Gaussian white noise
with the help of equivalent nonlinearization tech-
nique [Xu et al.,2014]. Iourtchenko and Song con-
sidered the stochastic vibro-impact systems with
one or two rigid barrier(s) by numerical simula-
tion [Iourtchenko & Song,2006]. Kumar investi-
gated the stochastic bifurcations of a Duffing–van
der Pol oscillator under random excitations [Kumar
et al.,2016a, 2016b, 2017]. Nguyen developed a
mathematical model of vibro-impact mobile sys-
tem to predict the progression rate of the system
[Nguyen et al.,2017]. Feng studied the chaotic sad-
dles of a nonlinear vibro-impact system using the
bisection procedure and an improved stagger-and-
step method [Feng et al.,2009]. Two nice overviews
of vibro-impact dynamics have been presented by
Namachchivaya [Namachchivaya & Park,2005] and
Dimentberg [Dimentberg & Iourtchenko,2004].
To the authors’ knowledge, little work was
focused on the dynamical systems with fractional
derivative elements. In this paper, we carry out a
bifurcation analysis of a vibro-impact viscoelastic
oscillator with fractional derivative element under
Gaussian white noise excitation.
2. Model and Its Simplification
We consider the vibro-impact viscoelastic system
with fractional derivative element under Gaussian
white noise excitation in the following form
¨x+Z+εβ1Dαx+εβ2f(x, ˙xx+ω2
0x
=ε1/2ξ(t),x>0,
˙x+=r˙x,x=0.
(1)
The variables ε,β1,β2and ω0are system
parameters; 0 <r1 is the coefficient of restitu-
tion factor; ˙xand ˙x+are the velocities just before
and after the impact, respectively. ξ(t) is Gaussian
white noise with zero mean and auto-correlation
E[ξ(t)ξ(t+τ)] = 2(τ). There are many defini-
tions for fractional derivatives [Li & Zeng,2015;
Ma & Li,2017, 2018], in this paper, Dαxis defined
as follows
Dαx=1
Γ(1 α)
d
dt t
0
x(tu)
uλ1du, (0 1).
(2)
The following viscoelastic model of Zcan be
expressed as:
Z=t
0
h(ts)x(s)ds,
where h(t) is the relaxation function which has the
following form
h(t)=
n
i=1
βiexpt
λi
=βexpt
λ.
Without loss of generality, let i=1.βiand λiare
the general elastic modulus and the relaxation time,
respectively. Then
Z=t
0
βexpts
λx(s)ds.
According to [Zhu & Cai,2011; Ling et al.,
2011], the viscoelastic force Zcan be replaced by
1850170-2
December 24, 2018 14:49 WSPC/S0218-1274 1850170
Vibro-Impact Viscoelastic Oscil lator with Fractional Derivative Element
the conservative force and the damping force:
Z=t
0
βexpts
λx(s)ds
=λβ
1+λ2ω2(xλ˙x)
=κ1xκ2˙x. (3)
Substituting Eq. (3) into the original system (1),
yields
¨x+εβ1Dαx+[εβ2f(x, ˙x)κ2x+ω2
1x
=ε1/2ξ(t),x>0,
˙x+=r˙x,x=0,
(4)
where
ω2
1=ω2
0+κ1.
The following transformation [Dimentberg &
Iourtchenko,2004; Zhuravlev,1976; Feng et al.,
2008] is introduced to remove the discontinuity in
Eq. (4)
x=x1=|y|,
˙x=x2ysgn(y),
¨xysgn(y).
(5)
Substituting Eq. (5) into Eq. (4) leads to the fol-
lowing equations:
¨ysgn(y)+εβ1Dα(|y|)
+[εβ2f(|y|,˙ysgn(y)) κ2ysgn(y)+ω2
1|y|
=ε1/2ξ(t),t=t,(6a)
˙y+=r˙y,t=t,(6b)
in which y(t)=0.
After multiplying Eq. (6a) by sgn(y), we get the
following formulas:
¨y+εβ1sgn(y)Dα(|y|)
+[εβ2f(|y|,˙ysgn(y)) κ2y+ω2
1y
=ε1/2ξ(t)sgn(y),t=t,(7a)
˙y+=r˙y,t=t.(7b)
Then, according to [Feng et al.,2009] we can
obtain the equivalent equation without impact term
of the original vibro-impact oscillator (1):
¨y+εβ1sgn(y)Dα(|y|)
+[εβ2f(|y|,˙ysgn(y)) κ2y
+(1ry|˙y|δ(y)+ω2
1y
=ε1/2ξ(t)sgn(y).(8)
3. Stochastic Averaging Procedure
Introduce the following transformation [Huang &
Jin,2009]
y(t)=A(t)co(t),
˙y(t)=A(t)ω1sin Ψ(t),
Ψ(t)=ω1t,
(9)
where A, Ψ, Φ are random processes. The equations
for the variables Aand Φ are
dA
dt =εF11(A, Φ) + εF12 (A, Φ)
+ε1/2G11(A, Φ)ξ(t),(10)
dΦ
dt =εF21(A, Φ) + εF22 (A, Φ)
+ε1/2G21(A, Φ)ξ(t),(11)
where
εF11 =εβ1sin Ψ
ω1
sgn(Acos Ψ)Dα(|Acos Ψ|),
εF12 =Asin2Ψ[εβ2f(|Acos Ψ|,1sin Ψ)
κ2+(1r)|−1sin Ψ|δ(Acos Ψ)],
εF21 =εβ1cos Ψ
1
Dα(Acos Ψ),
εF22 =sin Ψ cos Ψ[εβ2f(|Acos Ψ|,1sin Ψ)
κ2+(1r)|−1sin Ψ|δ(Acos Ψ)],
G1=sin Ψ
ω1
sgn(Acos Ψ),
G2=cos Ψ
1
sgn(Acos Ψ).
The averaged Itˆo equation for A(t)isoftheform
dA =m(A)dt +σ(A)dB(t),(12)
1850170-3
December 24, 2018 14:49 WSPC/S0218-1274 1850170
Y.-G. Yang et al.
where
m(A)=εF11 +F12 +D∂G11
∂A G11 +D∂G11
ΦG21
Ψ
,
(13)
σ2(A)=ε2DG2
11Ψ.(14)
To obtain the explicit expression, the key step
is to simplify the first term of Eq. (13). According
to the definition Eq. (2),
εF11Ψ=εβ1A
ω1Γ(1 α)sgn(Acos Ψ) sin Ψ
×d
dt t
0
|cos(Ψ ω1u)|
uαduΨ
.(15)
Then the cosine function in Eq. (15) can be
replaced by the following Fourier series
|cos θ|=2
π+
n=1
Bncos(2),(16)
where
Bn=4
π
(1)n
14n2.
According to Eqs. (15) and (16) and [Yurchenko
et al.,2017; Yang et al.,2018]
εF11Ψ=32εβ1A
π2ω1
sin απ
2
n=1
n2(21)α1
(1 4n2)2
≈−
32εβ1A
π2ω1
sin απ
2
15
n=1
n2(21)α1
(1 4n2)2.
The Fokker–Planck–Kolmogorov (FPK) equa-
tion corresponds to Eq. (12) as given by
∂p
∂t =
∂A[m(A)p]+ 1
2
2
∂A2[σ2(A)p].(17)
The boundary conditions for Eq. (17) are
p=c, c (−∞,+)asA=0,
p0,∂p
∂A 0,A→∞.
With the help of the aforementioned boundary
conditions, the stationary solution of Eq. (17) is
expected to be
p(A)= C
σ2(A)expA
0
2m(s)
σ2(s)ds,(18)
in which Cis the normalization constant. The joint
stationary PDF of the original displacement and
velo city p(x1,x
2) and corresponding marginal sta-
tionary PDFs p(x1)andp(x2) can be obtained
according to Eq. (18) and [Huang & Jin,2009; Yang
et al.,2015; Yang et al.,2017].
4. Example
To assess the accuracy of the proposed method, the
following oscillator is considered.
¨x+a1Dαx+Z+(b1+b2x2+b3˙x2x+ω2
0x
=ξ(t),x>0,
˙x+=r˙x,x=0,
(19)
where a1,b1,b2,b3and ω0are constants, ξ(t)is
Gaussian white noise with intensity 2D. The cor-
responding equivalent stochastic oscillator without
impact term is of the following form based on what
has been discussed in Sec. 2:
¨y+a1sgn(y)Dα(|y|)+[b4+b2y2+b3˙y2
+(1r)|˙y|δ(y)] ˙y+ω2
1y
=sgn(y)ξ(t),(20)
where
b4=b1+κ2,
ω2
1=λ2ω2
01+(1 + λ2ω2
0)2+4βλ3
2λ2.
The averaged drift and diffusion coefficients
are
m(A)=1
8b2A33
8b3ω2
1A3+b5A+D
22
1
,
(21)
σ2(A)= D
ω2
1
,(22)
in which
b5=b4
2(1 r)ω1
π
32a1
π2ω1
sin απ
2
15
n=1
n2(21)α1
(1 4n2)2.
1850170-4
December 24, 2018 14:49 WSPC/S0218-1274 1850170
Vibro-Impact Viscoelastic Oscil lator with Fractional Derivative Element
So, the stationary solution of oscillator (20) is
of the form
p(A)=2
1A
Dexpb5ω2
1
DA2
b2ω2
1
16D+3b3ω4
1
16DA4,
p(x1,x
2)=1
πD expb5ω2
1
Dx2
1+x2
2
ω2
1
b2ω2
1
16D+3b3ω4
1
16Dx2
1+x2
2
ω2
12.
4.1. Effectiveness of the proposed
approach
In this section, different levels of control parameters
b2,b3and noise intensity Dare considered to verify
the reliability and accuracy of the proposed tech-
nique. System parameters are fixed to be b1=0.09,
a1=0.01, α=0.5, λ=1.0, β=0.01, ω0=1.0,
r=0.95.The blue solid lines are theoretical pre-
dictions while discrete dots are numerical results in
the following figures, respectively.
First, noise intensity D=0.16 is fixed. Dif-
ferent levels of control parameters b2and b3are
considered then. Figure 1 gives the theoretical and
numerical results of the probability density func-
tions of amplitude, displacement and velocity for
different control parameter b2,Fig.2givesthe
results for different parameter b3. It can be seen
that the theoretical results agree well with those
from numerical results. It is also shown that the
effects of these two parameters b2and b3on the
system response are the same.
Then, to scrutinize the effect of the noise inten-
sity D, here, control parameters b2=0.09 and
b3=0.09 are fixed. Figure 3 gives the theoreti-
cal and numerical results of the probability density
functions of amplitude, displacement and velocity
for different noise intensity D. It can also be found
that the theoretical results and numerical results
are coincident. So, the reliability and accuracy of
the proposed technique are verified by Figs. 1–3.
4.2. Stochastic bifurcation analysis
In this paper, stochastic bifurcation refers to
stochastic P-bifurcation which occurs when the
shape of the stationary probability density function
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
A
p(A)
b2=0.36
b2=0.09
(a)
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
x1
p(x1)
b2=0.36
b2=0.09
(b)
−5 0 5
0
0.1
0.2
0.3
0.4
0.5
x2
p(x2)
b2=0.36
b2=0.09
(c)
Fig. 1. Probability density functions of amplitude, displace-
ment and velocity for different control parameter b2.
changes from unimodal to bimodal. This section
focuses on the analysis of stochastic bifurcation phe-
nomenon induced by viscoelastic parameters, frac-
tional coefficient and fractional derivative order.
First, we discuss the stochastic bifurcation
induced by viscoelastic parameter λ. The system
1850170-5
December 24, 2018 14:49 WSPC/S0218-1274 1850170
Y.-G. Yang et al.
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
A
p(A)
b3=0.07
b3=0.30
(a)
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
x1
p(x1)
b3=0.07
b3=0.30
(b)
−5 0 5
0
0.1
0.2
0.3
0.4
0.5
x2
p(x2)
b3=0.30
b3=0.07
(c)
Fig. 2. Probability density functions of amplitude, displace-
ment and velocity for different control parameter b3.
parameters are taken to be b1=0.032,b
2=0.004,
b3=0.004, a1=0.01, α=0.5, β=0.03,
ω0=1.0, D=0.01, r=0.955.Figure 4 gives
the joint stationary probability density p(x1,x
2)of
displacement and velocity for different viscoelastic
parameter λ. In order to better understand the
stochastic bifurcation, the corresponding section
graphs are presented in Fig. 5. An inspection of
these two figures clearly indicates that at λ=
0.9, the joint stationary probability density has
one peak and the corresponding section graph is
unimodal. At λ=0.1, the shape of the joint
stationary probability density changes to crater-like
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
A
p(A)
D=0.16
D=0.45
(a)
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
x1
p(x1)
D=0.45
D=0.16
(b)
−5 0 5
0
0.1
0.2
0.3
0.4
x2
p(x2)
D=0.16
D=0.45
(c)
Fig. 3. Probability density functions of amplitude, displace-
ment and velocity for different control parameter D.
1850170-6
December 24, 2018 14:49 WSPC/S0218-1274 1850170
Vibro-Impact Viscoelastic Oscil lator with Fractional Derivative Element
(a)
(b)
Fig. 4. The joint stationary probability density p(x1,x
2)of
displacement and velocity for different λ.(a)λ=0.9and
(b) λ=0.1.
structure and the corresponding section graph is
bimodal. This phenomenon indicates that stochas-
tic P-bifurcation takes place as λdecreases from
0.90 to 0.10.
Second, we explore the stochastic bifurcation
induced by viscoelastic parameter β. The system
parameters are taken to be b1=0.032, b2=0.004,
b3=0.004, a1=0.01, α=0.5, λ=1.0, ω0=1.0,
−6 −4 −2 0 2 4 6
0
0.05
0.1
0.15
0.2
0.25
0.3
x2
p(x2)
λ=0.90
λ=0.50
λ=0.10
λ=0.66
Fig. 5. Section graphs of stationary probability density
p(x1,x
2)whenx1=0.3fordierentλ.
(a)
(b)
Fig. 6. The joint stationary probability density of displace-
ment and velocity for different β.(a)β=0.03 and (b)
β=0.001.
D=0.01, r=0.955. Figure 6 gives the joint sta-
tionary probability density p(x1,x
2)fordierent
viscoelastic parameter β. Figure 7 shows the corre-
sponding section graphs. Based on the same analy-
sis, stochastic bifurcation takes place as viscoelastic
parameter βincreases from 0.03 to 0.001.
Third, we explore the stochastic bifurcation
induced by fractional coefficient a1. The system
−6 −4 −2 0 2 4 6
0
0.05
0.1
0.15
0.2
0.25
0.3
x2
p(x2)
β=−0.01
β=−0.02
β=−0.03
β=−0.001
Fig. 7. Section graphs of stationary probability density
p(x1,x
2)whenx1=0.3fordierentβ.
1850170-7
December 24, 2018 14:49 WSPC/S0218-1274 1850170
Y.-G. Yang et al.
(a)
(b)
Fig. 8. The joint stationary probability density of displace-
ment and velocity for different fractional coefficient a1.(a)
a1=0.001 and (b) a1=0.03.
parameters are taken to be b1=0.032, b2=0.004,
b3=0.004, α=0.3, λ=1.0, β=0.01, ω0=1.0,
D=0.01, r=0.952. Figure 8 gives the joint sta-
tionary probability density p(x1,x
2)fordierent
fractional coefficient a1. Figure 9 shows the corre-
sponding section graphs. Based on the same analy-
sis, stochastic bifurcation takes place as fractional
coefficient a1decreases from 0.001 to 0.03.
−6 −4 −2 0 2 4 6
0
0.05
0.1
0.15
0.2
0.25
0.3
x2
p(x2)
a1=−0.001
a1=−0.03
a1=−0.01
a1=−0.02
Fig. 9. Section graphs of stationary probability density
p(x1,x
2)whenx1=0.3fordierenta1.
(a)
(b)
Fig. 10. The joint stationary probability density of displace-
ment and velocity for different fractional order α.(a)α=0.1
and (b) α=0.9.
In the end, we explore the stochastic bifurca-
tion induced by fractional derivative order α.The
system parameters are taken to be b1=0.032, b2=
0.004, b3=0.004, a1=0.01, λ=1.0, β=0.01,
ω0=1.0, D=0.01, r=0.952.Figure 10 gives
the joint stationary probability density p(x1,x
2)for
different fractional derivative order α.Figure11
shows the corresponding section graphs. Based on
−6 −4 −2 0 2 4 6
0
0.05
0.1
0.15
0.2
0.25
0.3
x2
p(x2)
α=0.9
α=0.1
α=0.3
α=0.5
Fig. 11. Section graphs of stationary probability density
p(x1,x
2)whenx1=0.3 for different fractional order α.
1850170-8
December 24, 2018 14:49 WSPC/S0218-1274 1850170
Vibro-Impact Viscoelastic Oscil lator with Fractional Derivative Element
the same analysis, stochastic bifurcation takes place
as fractional derivative order αincreases from 0.1
to 0.9.
5. Conclusions
We discussed the stochastic bifurcation of a vibro-
impact oscillator with fractional derivative element
and a viscoelastic term under Gaussian white noise
excitation. The original oscillator is converted to
an equivalent oscillator without a viscoelastic term.
The stochastic averaging method and the nons-
mooth transformation are utilized to obtain the
analytical solutions the effectiveness of which can
be verified by numerical solutions. We also discussed
the stochastic bifurcation phenomenon induced by
the viscoelastic parameters, fractional coefficient
and fractional derivative order.
Acknowledgments
This work is supported by the National Natural Sci-
ence Foundation of China (NSFC) under the Grant
Nos. 11472212, 11532011 and by Shaanxi Natural
Science Foundation of China under the Grant No.
2017JM1038. The first author (Y.-G. Yang) would
like to thank the China Scholarship Council (No.
201606290189) and the hospitality of University of
California, Merced, during his visit from August
2016 to August 2018.
References
Chen, Y., Petras, I. & Xue, D. [2009] “Fractional order
control — A tutorial,” American Control Conf. 2009.
ACC 09, pp. 1397–1411.
Colinas-Armijo, N., Di Paola, M. & Di Matteo, A. [2018]
“Fractional viscoelastic behaviour under stochastic
temperature process,” Probab. Engin. Mech. 54, 37–
43.
Deng, M. & Zhu, W. [2016] “Response of MDOF strongly
nonlinear systems to fractional Gaussian noises,”
Chaos 26, 084313.
Di Bernardo, M., Kowalczyk, P. & Ordmark, A. [2003]
“Sliding bifurcations: A novel mechanism for the sud-
den onset of chaos in dry friction oscillators,” Int. J.
Bifurcation and Chaos 13, 2935–2948.
Dimentberg, M. & Iourtchenko, D. [2004] “Random
vibrations with impacts: A review,” Nonlin. Dyn. 36,
229–254.
Feng, J., Xu, W. & Wang, R. [2008] “Stochastic
responses of vibro-impact Duffing oscillator excited
by additive Gaussian noise,” J. Sound Vibr. 309, 730–
738.
Feng, J., Xu, W., Rong, H. & Wang, R. [2009] “Stochas-
tic responses of Duffing–van der Pol vibro-impact
system under additive and multiplicative random
excitations,” Int. J. Non-Lin. Mech. 44, 51–57.
Huang, Z. & Jin, X. [2009] “Response and stability of
a SDOF strongly nonlinear stochastic system with
light damping modeled by a fractional derivative,” J.
Sound Vibr. 319, 1121–1135.
Iourtchenko, D. V. & Song, L. L. [2006] “Numerical
investigation of a response probability density func-
tion of stochastic vibroimpact systems with inelastic
impacts,” Int. J. Non-Lin. Mech. 41, 447–455.
Kougioumtzoglou, I. A., dos Santos, K. R. & Comerford,
L. [2017] “Incomplete data based parameter identifi-
cation of nonlinear and time-variant oscillators with
fractional derivative elements,” Mech. Syst. Sign. Pro-
cess. 94, 279–296.
Kumar, P., Narayanan, S. & Gupta, S. [2016a] “Investi-
gations on the bifurcation of a noisy Duffing–van der
Pol oscillator,” Probab. Engin. Mech. 45, 70–86.
Kumar, P., Narayanan, S. & Gupta, S. [2016b] “Stochas-
tic bifurcations in a vibro-impact Duffing–van der Pol
oscillator,” Nonlin. Dyn. 85, 439–452.
Kumar, P., Narayanan, S. & Gupta, S. [2017] “Bifurca-
tion analysis of a stochastically excited vibro-impact
Duffing–van der Pol oscillator with bilateral rigid bar-
riers,” Int. J. Mech. Sci. 127, 103–117.
Li, C. & Zeng, F. [2015] Numerical Methods for Frac-
tional Calculus (CRC Press).
Li, Z., Liu, L., Dehghan, S., Chen, Y. & Xue, D. [2017]
“A review and evaluation of numerical tools for frac-
tional calculus and fractional order controls,” Int. J.
Contr. 90, 1165–1181.
Ling, Q., Jin, X. & Huang, Z. [2011] “Response and
stability of SDOF viscoelastic system under wide-
band noise excitations,” J. Franklin Instit. 348, 2026–
2043.
Luo, G., Chu, Y., Zhang, Y. & Zhang, J. [2006] “Double
Neimark–Sacker bifurcation and torus bifurcation of
a class of vibratory systems with symmetrical rigid
stops,” J. Sound Vibr. 298, 154–179.
Ma, L. & Li, C. [2016] “Center manifold of fractional
dynamical system,” J. Comput. Nonlin. Dyn. 11,
021010.
Ma, L. & Li, C. [2017] “On Hadamard fractional calcu-
lus,” Fract a ls 25, 1750033.
Ma, L. & Li, C. [2018] “On finite part integrals and
Hadamard-type fractional derivatives,” J. Comput.
Nonlin. Dyn. 13, 090905.
Machado, J. T., Kiryakova, V. & Mainardi, F. [2011]
“Recent history of fractional calculus,” Commun.
Nonlin.Sci.Numer.Simul.16, 1140–1153.
Malara, G. & Spanos, P. D. [2018] “Nonlinear random
vibrations of plates endowed with fractional derivative
elements,” Probab. Engin. Mech. 54, 2–8.
1850170-9
December 24, 2018 14:49 WSPC/S0218-1274 1850170
Y.-G. Yang et al.
Namachchivaya, N. S. & Park, J. H. [2005] “Stochastic
dynamics of impact oscillators,” J. Appl. Mech. 72,
862–870.
Nguyen, V.-D., Duong, T.-H., Chu, N.-H. & Ngo, Q.-
H. [2017] “The effect of inertial mass and excitation
frequency on a Duffing vibro-impact drifting system,”
Int. J. Mech. Sci. 124, 9–21.
Rossikhin, Y. A. & Shitikova, M. V. [1997] “Applications
of fractional calculus to dynamic problems of linear
and nonlinear hereditary mechanics of solids,” Appl.
Mech. Rev. 50, 15–67.
Rossikhin, Y. A. & Shitikova, M. V. [2010] “Application
of fractional calculus for dynamic problems of solid
mechanics: Novel trends and recent results,” Appl.
Mech. Rev. 63, 010801.
Rossikhin, Z. H. [2015] “Stochastic response of a
vibro-impact Duffing system under external Poisson
impulses,” Nonlin. Dyn. 82, 1001–1013.
Xu, M., Wang, Y., Jin, X. & Huang, Z. [2014] “Random
vibration with inelastic impact: Equivalent nonlin-
earization technique,” J. Sound Vibr. 333, 189–199.
Yang, Y., Xu, W., Jia, W. & Han, Q. [2015] “Station-
ary response of nonlinear system with Caputo-type
fractional derivative damping under Gaussian white
noise excitation,” Nonlin. Dyn. 79, 139–146.
Yang, Y., Xu, W., Sun, Y. & Xiao, Y. [2017] “Stochas-
tic bifurcations in the nonlinear vibro-impact system
with fractional derivative under random excitation,”
Commun. Nonlin. Sci. Numer. Simul. 42, 62–72.
Yang, Y., Xu, W. & Yang, G. [2018] “Bifurcation
analysis of a noisy vibro-impact oscillator with two
kinds of fractional derivative elements,” Chaos 28,
043106.
Yurchenko, D., Burlon, A., Di Paola, M., Failla, G. &
Pirrotta, A. [2017] “Approximate analytical mean-
square response of an impacting stochastic system
oscillator with fractional damping,” ASCE-ASME J.
Risk Uncertainty Engin. Syst.,Part B:Mech. Engin.
3, 030903.
Zhu, W. & Cai, G. [2011] “Random vibration of vis-
coelastic system under broad-band excitations,” Int.
J. Non-Lin. Mech. 46, 720–726.
Zhuravlev, V. [1976] “A method for analyzing vibration-
impact systems by means of special functions,” Mech.
Solids 11, 23–27.
1850170-10
This research hasn't been cited in any other publications.
  • Book
    Numerical Methods for Fractional Calculus presents numerical methods for fractional integrals and fractional derivatives, finite difference methods for fractional ordinary differential equations (FODEs) and fractional partial differential equations (FPDEs), and finite element methods for FPDEs. The book introduces the basic definitions and properties of fractional integrals and derivatives before covering numerical methods for fractional integrals and derivatives. It then discusses finite difference methods for both FODEs and FPDEs, including the Euler and linear multistep methods. The final chapter shows how to solve FPDEs by using the finite element method. This book provides efficient and reliable numerical methods for solving fractional calculus problems. It offers a primer for readers to further develop cutting-edge research in numerical fractional calculus. MATLAB® functions are available on the book’s CRC Press web page.
  • Article
    To the best of authors' knowledge, little work was referred to the study of a noisy vibro-impact oscillator with a fractional derivative. Stochastic bifurcations of a vibro-impact oscillator with two kinds of fractional derivative elements driven by Gaussian white noise excitation are explored in this paper. We can obtain the analytical approximate solutions with the help of non-smooth transformation and stochastic averaging method. The numerical results from Monte Carlo simulation of the original system are regarded as the benchmark to verify the accuracy of the developed method. The results demonstrate that the proposed method has a satisfactory level of accuracy. We also discuss the stochastic bifurcation phenomena induced by the fractional coefficients and fractional derivative orders. The important and interesting result we can conclude in this paper is that the effect of the first fractional derivative order on the system is totally contrary to that of the second fractional derivative order.
  • Article
    This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamardtype fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.
  • Article
    This paper deals with mechanical behaviour of a linear viscoelastic material modelled by the fractional Maxwell model and subject to a Gaussian stochastic temperature process. Two methods to evaluate the response in terms of strain of the mechanical behaviour under a deterministic stress of a material subjected to a varying temperature are presented. The first method consist in calculate the response making the material parameters change at each time step due to the temperature variation. The second, takes advantage of the Time Temperature Superposition Principle to lighten the calculations. Then a stochastic characterisation for the Time Temperature Superposition Principle method has been proposed for a Gaussian stochastic process.
  • Article
    This paper deals with the problem of determining the nonlinear response of a plate endowed with fractional derivative elements and exposed to random loads. It shows that an approximate solution of the nonlinear fractional partial differential equation governing the plate vibrations can be obtained via a statistical linearization based approach. The approach is implemented by considering a time-dependent representation of the response involving the eigen-functions of the linear problem. This representation allows deriving a nonlinear fractional ordinary differential equation governing the variation of the time-dependent part of the response, which is linearized in a mean square sense. Then, an iterative procedure provides the response statistics and power spectral density functions. Next, a Boundary Element Method is proposed for conducting relevant Monte Carlo data. The method is developed in conjunction with a Newmark integration scheme for estimating the response in the time domain given spectrum compatible realizations of the excitation. Monte Carlo data and statistical linearization solutions are calculated for square plates with simply supported stress-free edges, but problems involving other boundary conditions can be solved by the proposed approach.
  • Article
    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 results of numerical simulations.
  • Article
    This paper is devoted to the investigation of the Hadamard fractional calculus in three aspects. First, we study the semigroup and reciprocal properties of the Hadamard-type fractional operators. Then, the definite conditions of certain class of Hadamard-type fractional differential equations (HTFDEs) are proposed through the Banach contraction mapping principle. Finally, we prove a novel Gronwall inequality with weak singularity and analyze the dependence of solutions of HTFDEs on the derivative order and the perturbation terms along with the proposed initial value conditions. The illustrative examples are presented as well.
  • Article
    Various system identification techniques exist in the literature that can handle non-stationary measured time-histories, or cases of incomplete data, or address systems following a fractional calculus modeling. However, there are not many (if any) techniques that can address all three aforementioned challenges simultaneously in a consistent manner. In this paper, a novel multiple-input/single-output (MISO) system identification technique is developed for parameter identification of nonlinear and time-variant oscillators with fractional derivative terms subject to incomplete non-stationary data. The technique utilizes a representation of the nonlinear restoring forces as a set of parallel linear subsystems. In this regard, the oscillator is transformed into an equivalent MISO system in the wavelet domain. Next, a recently developed L 1-norm minimization procedure based on compressive sensing theory is applied for determining the wavelet coefficients of the available incomplete non-stationary input-output (excitation-response) data. Finally, these wavelet coefficients are utilized to determine appropriately defined time-and frequency-dependent wavelet based frequency response functions and related oscillator parameters. Several linear and nonlinear time-variant systems with fractional derivative elements are used as numerical examples to demonstrate the reliability of the technique even in cases of noise corrupted and incomplete data.
  • Article
    This paper presents the mathematical modeling and dynamical analysis to examine the coupled effect of inertial mass and excitation frequency on a Duffing vibro-impact device used for autogenous mobile systems. The Duffing oscillation with a nonlinear spring is utilized to generate impacts and thus to drive the whole system. Forward motion is observed in a wide range of the excited frequency and the mass ratio. The nonlinear interaction between the system body and the inertial mass can provide a backward motion of the system for a narrow range of the mass ratio and the frequency of excitation. The proposed model has validated by an experimental implementation, and thus can be used as a useful tool to optimize the system dynamics. Bifurcation study is carried out to provide a better understanding for design and selection of the system parameters. The mathematical model of the system with dimensionless parameters allows extending the results to both large- and micro-scale applications.