Asymptotic analysis on steady-state response of axially accelerating beam constituted by the standard linear solid model
The transverse bending vibrations of an axially accelerating viscoelastic beam are studied. A material of beam is constituted by the standard viscoelastic model. The method of multiple scales is applied to determine the steady-state response of beam. The numerical examples illustrate the asymptotic...
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Wang, B. Qiu, J. 2019-09-12T18:36:04Z 2019-09-12T18:36:04Z 2017 Asymptotic analysis on steady-state response of axially accelerating beam constituted by the standard linear solid model / B. Wang, J. Qiu // Прикладная механика. — 2017. — Т. 53, № 4. — С. 138-144. — Бібліогр.: 18 назв. — англ. 0032-8243 https://nasplib.isofts.kiev.ua/handle/123456789/158784 The transverse bending vibrations of an axially accelerating viscoelastic beam are studied. A material of beam is constituted by the standard viscoelastic model. The method of multiple scales is applied to determine the steady-state response of beam. The numerical examples illustrate the asymptotic solution. Вивчено поперечні згинні коливання в'язкопружної балки, яка допускає рухи вздовж осі. Матеріал балки припускається деформівним за стандартною триконстантною моделлю теорії в'язкопружності. Для визначення стаціонарної реакції балки застосовано метод багатьох масштабів. Числові приклади ілюструють асимптотичний розв'язок. en Інститут механіки ім. С.П. Тимошенка НАН України Прикладная механика Asymptotic analysis on steady-state response of axially accelerating beam constituted by the standard linear solid model Асимптотический анализ стационарной реакции балки с ускорением вдоль оси и с использованием стандартной линейной модели деформируемого тела Article published earlier |
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Asymptotic analysis on steady-state response of axially accelerating beam constituted by the standard linear solid model |
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Asymptotic analysis on steady-state response of axially accelerating beam constituted by the standard linear solid model Wang, B. Qiu, J. |
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Asymptotic analysis on steady-state response of axially accelerating beam constituted by the standard linear solid model |
| title_full |
Asymptotic analysis on steady-state response of axially accelerating beam constituted by the standard linear solid model |
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Asymptotic analysis on steady-state response of axially accelerating beam constituted by the standard linear solid model |
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Asymptotic analysis on steady-state response of axially accelerating beam constituted by the standard linear solid model |
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asymptotic analysis on steady-state response of axially accelerating beam constituted by the standard linear solid model |
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Wang, B. Qiu, J. |
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Wang, B. Qiu, J. |
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2017 |
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English |
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Прикладная механика |
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Інститут механіки ім. С.П. Тимошенка НАН України |
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Article |
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Асимптотический анализ стационарной реакции балки с ускорением вдоль оси и с использованием стандартной линейной модели деформируемого тела |
| description |
The transverse bending vibrations of an axially accelerating viscoelastic beam are studied. A material of beam is constituted by the standard viscoelastic model. The method of multiple scales is applied to determine the steady-state response of beam. The numerical examples illustrate the asymptotic solution.
Вивчено поперечні згинні коливання в'язкопружної балки, яка допускає рухи вздовж осі. Матеріал балки припускається деформівним за стандартною триконстантною моделлю теорії в'язкопружності. Для визначення стаціонарної реакції балки застосовано метод багатьох масштабів. Числові приклади ілюструють асимптотичний розв'язок.
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0032-8243 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/158784 |
| citation_txt |
Asymptotic analysis on steady-state response of axially accelerating beam constituted by the standard linear solid model / B. Wang, J. Qiu // Прикладная механика. — 2017. — Т. 53, № 4. — С. 138-144. — Бібліогр.: 18 назв. — англ. |
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2025-11-25T20:40:52Z |
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2025-11-25T20:40:52Z |
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| fulltext |
2017 ПРИКЛАДНАЯ МЕХАНИКА Том 53, № 4
138 ISSN0032–8243. Прикл. механика, 2017, 53, № 4
B . W a n g 1 , J . Q i u
ASYMPTOTIC ANALYSIS ON STEADY-STATE RESPONSE
OF AXIALLY ACCELERATING BEAM CONSTITUTED
BY THE STANDARD LINEAR SOLID MODEL
1School of Mechanical Engineering, Shanghai Institute of Technology,
Shanghai 200235 ;wang@live.com
Abstract: The transverse bending vibrations of an axially accelerating viscoelastic
beam are studied. A material of beam is constituted by the standard viscoelastic model. The
method of multiple scales is applied to determine the steady-state response of beam. The
numerical examples illustrate the asymptotic solution.
Key words: viscoelastic beam, axial acceleration, standard viscoelastic model, steady-
state response of beam.
1, Introduction.
Due to their technological importance, the vibrations of axially moving beams have
been investigated by many researchers. Transverse parametric vibration of axially accelerat-
ing elastic beams has been extensively analyzed since first study by Pasin [1]. Wickert and
Mote reviewed the literature on axially moving materials [2]. Öz and Pakdemirli [3] em-
ployed the method of multiple scales to study dynamic stability of an axially accelerating
beam with small bending stiffness. In addition to elastic beams, axially accelerating
viscoelastic beams have recently been investigated. Chen and Yang [4] applied the method
of multiple scales to obtain analytically the stability boundaries. Yang and Chen [5] presents
vibration and stability of an axially moving beam constituted by the viscoelastic constitutive
law of an integral type. Chen and Yang [6] presents analytically vibration and stability of an
axially moving beam constrained by simple supports with rotational springs. Chen and
Wang [7] presented the comparison between the analytical and the numerical results
parametric resonances via the differential quadrature method. In Ref. [4 – 7], the Kelvin
model containing the partial time derivative was used to describe the viscoelastic behavior
of beam materials. As parametric vibration excited by the variation of the axial tension or
axial velocity, large transverse motion of axially moving beams may occur under certain
conditions. Geometrical nonlinearity produced due to axially stretching of beam, can’t be
neglected when large transverse displacement takes place. Chen and Yang investigated the
steady-state response and their stability of two nonlinear models of axially moving
viscoelastic beams [8]. Chen and Ding investigates steady-state periodical response for
planar vibration of axially moving viscoelastic beams with two nonlinear models subjected
external transverse loads [9]. Compared with Kelvin model, standard linear solid model is
more typical and representative, mean while this model can degenerate to the Kevin or
Maxwell model by varying alternative of the stiffness of beam. Wang and Chen investigated
the linear stability of axially accelerating beams via asymptotic analysis [10]. In present
investigation, the standard linear solid model viscoelastic material is developed for two
nonlinear models of axially accelerating viscoelastic beams. The method of multiple scales
139
is applied to solution of governing equation and steady-state response of nonlinear axially
moving beams.
2. Equation of motion.
A uniform axially moving viscoelastic beam, with density ρ, cross-sectional area A, and
initial tension P, travels at time-dependent axial velocity γ(t) between two transversely mo-
tionless ends separated by distance l. Consider only the bending vibration of beam in a ref-
erence frame described by the transverse displacement v(x, t), where t is the time and x is the
axial coordinate. The equation of motion in the transverse direction can be derived from
Newton’ second law as
2
2
, ,xx x
d v
A M P A v
xdt
, (1)
where (·),x denotes partial differentiation with respect to x. σ(x, t) is the disturbing stress, and
the bending moment M(x, t) is defined by
, , ,
A
M x t z x z t dA , (2)
where the z-x plane is regarded as the principal plane of bending, and σ (x, z, t) is the normal
stress. The standard linear solid model is adopted to describe the viscoelastic property of the
material of beam. The stress-strain relationship of the model is expressed in a differential
form as [11 – 14]
1 2 1 2 1, , , , , , , , ,
d d
E E x z t x z t E E x z t E x z t
dt dt
(3)
where both E1 and E2 are stiffness constants, ε(x, z, t) is the axial strain, and η is viscous
damping. The standard linear solid model can be employed to describe the behavior of linear
viscoelastic materials of solid type with limited creep deformation. It can reduce to Kelvin
model (E1→∞ and E2≠0) or Maxwell model (E1 ≠ 0 and E2 = 0). For small deflections, the
strain-displacement relation is
2
2
,
, ,
v x t
x z t z
x
. (4)
Lagrangian strain is employed as a finite measure to account for geometric nonlinearity
due to small but finite stretching of beam. For one-dimensional problems, the disturbing
stress σ (x, t) in equation (1) is still described by the standard linear solid model
1 2 1 2 1, , , ,L L
d d
E E x t x t E E x t E x t
dt dt
, (5)
where εL(x, t) is the Lagrangian strain,
21
,
2L xv . (6)
Introduce the material time derivative by defining differential operator d / dt as
d
dt t x
. (7)
Inserting Eq. (7) in (1), (3) and (5) yields
2, , 2 , , , , ,tt x xt xx xx x xA v v v v M P A v ; (8)
1 2 1 2, , , ,t x t xE E E E ; (9)
140
1 2 1 2 L L L, , , ,t x t xE E E E . (10)
Eqs. (8) – (10) are the governing equation of an axially moving viscoelastic beam. If the
spatial variation of the tension is rather small compared with the initial tension, the exact
form of the disturbing tension Aσ can be replaced by its spatially averaged value
0
1
d
l
A x
l
[15, 16]. Then Eq. (8) leads to
2
0
1
, , 2 , , , d ,
l
tt x xt xx xx xxA v v v v M P A x v
l
. (11)
Let the dimensionless variables be:
2
; ; ; ;
v x P A
v x t t c
l PAll
2A
1 2
1
, , , d ; , ; ;
A P
x t z x z t A x t
Pl P E E Al
(12)
1 2 1 1 2 1
2 2
1 21 2
; ; ; ,a b c d
IE E IE E E A E A
E E E E
P E E PPl E E Pl
where I is the moment of inertial, book keeping device is a small dimensionless parameter
accounting for the fact that the transverse displacement and the viscous damping is very
small. Eqs. (8) – (10) can be respectively cast into the dimensionless form of governing
equation
2, , 2 , 1 , , , ,tt x xt xx xx x xv cv cv c v v ; (13)
, , , , ,t x a xx b xxt xxxc E v E v cv ; (14)
2 2 21 1
, , , , , , ,
2 2t x c x d x t x xc E v E v c v . (15)
We suppose that the beam is constrained at both ends by simple supports with rotational
springs whose stiffness constants are k respectively, the boundary conditions are expressed
in dimensionless form as follows then:
0, 0; , 0, , 0, 0; 1, 0; , 1, , 1, 0.xx x xx xv t v t kv t v t v t kv t (16)
In the present investigation, the axial velocity is assumed to be a small simple harmonic
variation about the constant mean speed
0 1 sinc t c c t , (17)
where c0 is the constant mean speed, and c1 and are respectively the disturbed amplitude
and the excitation frequency, all in the dimensionless form. Here the bookkeeping device
is used to indicate the fact that disturbed amplitude is small, with the same order as the di-
mensionless viscosity.
Substitution of Eqs. (14), (15) and (17) in Eq. (13) and neglect of higher order terms in
the resulting equation yield the following nonlinear partial-differential equation
2
1 1 0
3
, , , , , cos 2 sin , ,
2tt t c x xx x xt xxMv Gv Kv E v v c v t c t v c v
141
2
0, , ,b a xxxxt xxxxxE E v c v O (18)
where the mass, gyroscopic, and linear stiffness operators are, respectively, defined as follows:
2 4
2
0 0 2 4
; 2 ; 1 .aM I G c K c E
x x x
(19)
Resting on Eq. (11), the nonlinear integro-partial-differential equation below can be cast
into by means of the previous similar procedures.
1
2
1 1 0
0
2
0
1
, , , , d , cos 2 sin , ,
2
, , .
tt t c xx x x xt xx
b a xxxxt xxxxx
Mv Gv Kv E v v x c v t c t v c v
E E v c v O
(20)
3. Multi-scale procedure with numerical examples.
Both equations (18) and (20) are regarded as gyroscopic continuous system with weakly
nonlinear and parametric disturbances. Using the method of multiple scales, defining a slow
time scale T = ε t, and looking for an asymptotic solution of the form
2
0 1, ; , , , ,v x t v x t T v x t T O . (21)
Substitution of Eq. (21) in Eq. (18), equating coefficients of each power of ε to zero, we obtain
0 0 0, , 0tt tMv Gv Kv ; (22)
1 1 1 1 0 0 0 0 1 0 0 0
2
0 0 0 0 0
, , cos , 2 , 2 , 2 sin , ,
3
, , , , .
2
tt t x tT xT xt xx
c x xx a b xxxxt xxxxx
Mv Gv Kv tv v v t v v
E v v E E v v
(23)
We assume that the system is in summation parametric resonance and in order to ex-
press the nearness of the excitation frequency to sum of arbitrary two natural frequencies,
and define a detuning parameter μ through the relation
n m , (24)
where ωm and ωn are respectively the mth and nth natural frequencies of the undisturbed
gyroscopic continuous system (22) under boundary condition (16).
Wickert and Mote [2] have obtained the solution to Eq. (22)
i i
0 , , e em nt t
m m n nv x t T x A T x A T cc , (25)
where cc stands for the complex conjugate of all preceding terms on the right hand of an
equation. The expressions of modal functions m(x) and n(x) had given in Ref. [3].
Substituting Eqs. (24) and (25) into Eq. (23), we obtain
1 1 1 0 1 0 1
1
, , 2
2
iT iT
tt t m m m m n n n m n nMv Gv Kv i A e A ie A
22 23
2 3
2 c m m m m m m m c m n n n m m n n m n nE A A E A A
4 5
0 0 0 12mi t iT
m a b m m m n n n n m mA E E i e i A i A e
(26)
22 23
2 3
2 c n n n n n n n c n m n m m m m n m n mE A A E A A
142
4 5
1 0
1
,
2
ni tiT
m m m n n a b n n ne A A E E i e
where the dot and the prime denote differentiation with respect to T and x respectively, and
NST stands for the terms that will not bring secular terms into the solution. Eq. (26) has a
bounded solution only if a solvability condition holds. The solvability condition demands
the following orthogonal relationships [17,18]
2 2
0 1 0
2 4 5
0
0 1 0
1 3
2 2 2
2 2
3 , 0;
1 3
2 2
2 2
iT
m m m m n n n m n c m m m m m m m
c m n n n m m n n m n n m a b m m m m
iT
n n n n m m m n m c
A i A e i E A A
E A A A E E i
A i A e i E
2 2
2 4 5
0
2
3 , 0.
n n n n n n n
c n m n m m m m n m n m n a b n n n n
A A
E A A A E E i
(2
7)
Substituting the polar form
;m ni i
m m n nA a e A a e (28)
into Eq. (27), where ak (T) and βk(T) (k = m, n) are respectively the amplitude and phase of
the steady-state response in the summation parametric resonance. By the means of separat-
ing real and imaginary parts of the resulting equations, we arrive at the relationship between
ak(T) and μ to determine steady-state response of nonlinear system.
Specify system parameters of an axially moving beam with Ea=0,64, c0=2,0 and k = 2
under boundary condition (16). The first two natural frequencies of undisturbed system (22)
are ω1=8,1570 and ω2=32,9441. Fig. 1 depicts the relationship between the amplitude and
the detuning parameter in the summation parametric resonance, in which the solid and
dashed lines stand for stable and unstable amplitudes, respectively. In the figures, c1=0,2,
Ec=400, α=0,0001 and E1 =E2=3. Eq. (12) gives Eb = Ea (E1 +E2) / E2 = 1,28.
(a) (b)
Fig. 1. The amplitude and the detuning parameter relationship in the summation resonance: (a)
the first order and (b) the second order.
In the summation resonance, only the trivial solution exists and is stable for μ<μ1. At μ=μ1
the trivial solution losses its stability and a stable nontrivial solution occurs. At μ=μ2 the
unstable trivial solution becomes stable again, and an unstable nontrivial solution bifurcates.
μ1 μ2 μ1 μ2
143
The instability interval in the first order is larger than in the second, which indicates that
effect of the low order is more significant.
(a) (b)
Fig. 2. The effect of the axial velocity variation amplitude in the summation resonance: (a) the
first order and (b) the second order.
Fig. 2 illustrates the effect of the axial velocity variation amplitude in the summation
resonance, and the increasing velocity variation amplitude leads to the larger instability in-
terval. Fig. 3 show the comparison of the two nonlinear models, and the nontrivial solution
amplitude of partial-differential equation model is smaller, but both of them possess the
same instability intervals and changing trend with related parameters.
(a) (b)
Fig. 3. Comparison of two nonlinear models in the summation resonance: (a) the first order and
(b) the second order.
4. Conclusions.
With conclusions: (1) There exists an instability interval of the detuning on which
straight equilibrium is unstable, and the instability interval of lower order resonance is more
than the higher order one. The first nontrivial solution is always stable but the second one.
(2) Increasing velocity variation amplitude lead to the larger instability interval. (3) By the
means of the comparison of the two nonlinear models, both of them have the same tenden-
cies and instability intervals with changing relevant parameters. Besides partial-differential
equation model has smaller amplitude of nontrivial steady-state response.
Р Е ЗЮМ Е . Вивчено поперечні згинні коливання в’язкопружної балки яка допускає рухи
вздовж осі. Матеріал балки припускається деформівним за стандартною триконстантною моделлю
теорії в’язкопружності. Для визначення стаціонарної реакції балки застосовано метод багатьох масш-
табів. Числові приклади ілюструють асимптотичний розв‘язок.
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___________________
From the Editorial Board: The article correspond to submitted manuscript.
Поступила 24.12.2015 Утверждена в печать 14.03.2017
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