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Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds
Journal of Sound and Vibration 331 (2012) 1612–1623
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Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds Guo-Ce Zhang a, Hu Ding a,n, Li-Qun Chen a,b, Shao-Pu Yang c a Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China b Department of Mechanics, Shanghai University, Shanghai 200444, China c Shijiazhuang Tiedao University, Shijiazhuang 050043, China
a r t i c l e in f o
abstract
Article history: Received 9 April 2011 Received in revised form 20 October 2011 Accepted 3 December 2011 Handling Editor: L.G. Tham Available online 31 December 2011
The present paper investigates the steady-state periodic response of an axially moving viscoelastic beam in the supercritical speed range. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. It is assumed that the excitation of the forced vibration is spatially uniform and temporally harmonic. Under the quasi-static stretch assumption, a nonlinear integro-partialdifferential equation governs the transverse motion of the axially moving beam. The equation is cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. For a beam constituted by the Kelvin model, the primary resonance is analyzed via the Galerkin method under the simply supported boundary conditions. Based on the Galerkin truncation, the finite difference schemes are developed to verify the results via the method of multiple scales. Numerical simulations demonstrate that the steady-state periodic responses exist in the transverse vibration and a resonance with a softening-type behavior occurs if the external load frequency approaches the linear natural frequency in the supercritical regime. The effects of the viscoelastic damping, external excitation amplitude, and nonlinearity on the steady-state response amplitude for the first mode are illustrated. & 2011 Elsevier Ltd. All rights reserved.
1. Introduction The wide diffusion of axially moving systems in industrial processes, civil, mechanical and automotive applications has motivated intense research activities. Elevator cables, power transmission band, aerial cableways and belt saws are some examples of one-dimensional axially moving continua [1–5]. These systems can be modeled as axially moving beams if the bending stiffness is considered [6,7]. Therefore, the related investigations are important to design these devices. The axial speed greatly affects the dynamic behaviors of the moving system. Wickert noticed that the unstable straight equilibrium configuration bifurcates into multiple equilibrium positions for a translating beam above the critical velocity [8]. Hwang and Perkins investigated the effects of an initial curvature due to supporting wheels and pulleys on the bifurcation and stability of equilibrium in the supercritical speed regime and underlined the system sensitivity to initial imperfections [9,10]. Considering simply supported boundary conditions, Pellicano and Vestroni analyzed the dynamic behavior of an axially moving beam subjected to an axial transport of mass by using a high dimension discrete model
n
Corresponding author. Tel.: þ 86 21 56337273, fax: þ 86 21 36033287. E-mail address: [email protected] (H. Ding).
0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.12.004
G.-C. Zhang et al. / Journal of Sound and Vibration 331 (2012) 1612–1623
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obtained by a Galerkin procedure [11]. They investigated the forced response of an axially moving beam under harmonic excitation in the range of velocities greater than the first critical velocity [12]. The work focused on exploring the dynamics of a traveling beam, a simple viscous damping mechanism subjected to a transverse load with simple supports via the Galerkin method when its main parameters vary in the supercritical velocity range. Parker discussed supercritical phenomena in moving strings and found that the elastically supported strings always exhibit divergence instability above ¨ z calculated the natural frequencies of axially moving beams under clamped– the first critical speed [13]. Pakdemirli and O clamped boundary conditions in the supercritical regime, and found the beam unstable at those velocities [14]. In the supercritical regime, the unstable straight equilibrium configuration bifurcates for a translating beam is analogous to a buckled beam problem [15]. For each non-trivial equilibrium solutions, the governing equation is cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform, and the first natural frequency of the axially moving beam is calculated by a perturbation method [8]. Under simply supported boundary conditions, the non-trivial equilibriums and natural frequencies of the transverse model for the free vibration were calculated in the supercritical regime [16,17]. As we know, there have been no investigations about the steady-state periodic response of the forced vibration of axially moving viscoelastic beams in the supercritical regime. This paper focuses on the forced vibration and stability for an axially moving beam with a supercritical transport speed excited by the external vibration which is assumed to be spatially uniform and temporally harmonic. The Kelvin model had been used for describing the viscoelastic materials to investigate the forced vibration of axially moving systems [18–22]. As exact solutions are usually unavailable, approximate analytical methods are widely applied to investigate nonlinear vibration of axially moving beams. These methods include the asymptotic method of Krylov, Bogoliubov, and Mitropolsky [8], the Lindstedt–Poincare´ method [23], the method of normal forms [24], complex modes [25], the method of multiple scales [26–28]. Under the critical traveling speed, Chen and Ding showed the steady-state transverse response in coupled planar vibration of axially moving viscoelastic beams [29]. In the present paper, Galerkin method will be used to analyze the governing equation, a nonlinear integro-partial-differential equation for an axially translating beam in the supercritical transport speed ranges. As we know, based on the first-order Galerkin discretization, Ravindra and Zhu analyzed the chaotic behavior of axially accelerating beams [30]. Based on 4-term Galerkin method, Chen et al. analyzed the regular and chaotic vibrations of an axially moving viscoelastic string [21], and later on, Chen and Yang investigated nonlinear dynamics of axially accelerating viscoelastic beams [31]. Zhang studied the transverse motion of an axially moving viscoelastic string via the Galerkin method [22]. Riedel and Tan studied the forced response of a nonlinear axially moving strip with coupled transverse and longitudinal motion and applied the method of multiple scales to examine the internal resonance based on the Galerkin truncation [32]. In this paper, based on the Galerkin truncation, the frequency–response curves for the periodic forced vibration are obtained via using the multiple-scale method and the finite difference method (FDM), respectively. The present paper is organized as follows. Section 2 establishes the governing equation for the transverse motion of an axially moving beam in the supercritical regime. Section 3 carries out a three time-scales perturbation analysis of the equation obtained in Section 2 via 1-term Galerkin method. Section 4 combines the 4-term Galerkin truncation and the finite difference method to solve the stable steady-state response from the governing equations presented in Section 2. Section 5 compares the results of the multiple-scale methods in Section 3 and the finite difference method in Section 4. Section 6 ends the paper with the concluding remarks. 2. The mathematical model Consider a uniform axially traveling beam with simply supported boundary conditions at the constant transport speed
G between both ends separated by distance L. V is the transverse displacement field, T is the time and X is the spatial coordinate along the axis of the beam, r is the density, I is the moment of inertia, P0 is the initial tension, A is the crosssectional area. Under certain conditions, the transverse motion can be decoupled from the longitudinal motion so that a nonlinear partial-differential model is obtained to govern the transverse motion. It is assumed that the external transverse load is a spatially uniformly distributed periodic force. B and O are the amplitude and the frequency of the external excitation subjected to the beam, respectively. The equation of motion can be read as [33,34] " # 2 @2 V @2 V @ @V @2 M 2@ V þG ðP þ rA 2 þ 2G þ A s Þ ¼ B cosðOTÞ (1) 0 2 @X@T @X @X @T @X @X 2 where sðTÞ and MðX,TÞ are the axially disturbed stress and bending moment, respectively. The material particle of the moving beam experiences local ð@2 V=@T 2 Þ, Coriolis 2Gð@2 V=@X@TÞ, and centripetal G2 ð@2 V=@X 2 Þ acceleration components. The Kelvin viscoelastic model is used to describe the viscoelastic properties of the materials. So, the constitution relation is
sðT Þ ¼ EeK þ L
@eK @T
(2)
where E is Young’s modulus of the beam material, L is the dynamic viscosity and eK ðTÞ is the axial strain satisfying Z 1 L @V 2 eK ðTÞ ¼ dX (3) 2L 0 @X
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Such a condition is caused by the quasi-static stretch assumption. Considering the Kelvin model of the viscoelastic material, the moment–curvature relationship can be expressed as MðX,TÞ ¼ EI
@2 V @X
2
þ IL
@3 V
(4)
@X 2 @T
Substituting Eqs. (2)–(4) into Eq. (1) yields an integro-partial-differential equation for the transverse motion of the axially moving viscoelastic beam " # 2 @2 V @2 V @2 V @4 V @5 V 2@ V þG rA þ2G P 0 2 þ EI 4 þ IL 4 2 2 @X@T @T @X @X @X @X @T ¼
Z
EA @2 V 2L @X 2
L 0
2 Z @V AL @2 V L @V @2 V dX þ B cosðOTÞ dX þ 2 @X L @X 0 @X @X@T
Introducing the dimensionless variables and parameters as follows: sffiffiffiffiffiffiffi V X BL T P0 , v¼ , x¼ , b¼ , t¼ L L P0 L rA sffiffiffiffiffiffiffi rA , P0
o ¼ OL
a¼
L3
IL pffiffiffiffiffiffiffiffiffiffiffi , rAP0
2
k1 ¼
(5)
sffiffiffiffiffiffiffi rA , P0
g¼G
EA , P0
2
kf ¼
EI P0 L2
(6)
where x is the dimensionless spatial coordinate and t is the dimensionless time. Eq. (5) can be transformed into the dimensionless equation 2
v, tt þ 2gv, xt þðg2 1Þv, xx þ kf v, xxxx þ av, xxxxt 2
k1 v, xx 2
¼
Z 0
1
ak21
v, 2x dx þ
2 kf
v, xx
Z
1
v, x v, xt dx þb cosðotÞ
(7)
0
where a comma preceding x or t denotes the partial differentiation with respect to x or t. Consider a steel beam with modulus of elasticity E¼2.1 1011 Pa and density r ¼7850 kg/m3. Let the beam length L¼1 m, the dynamic viscosity L ¼634,402 N s/m2, the initial tension P0 ¼7850 N, the axial speed ! ¼206.88 m/s, and the rectangular cross-section of the beam with the width W¼ 13.49 mm and the height H¼27.71 mm. Then Eq. (6) yields the dimensionless axial speed g ¼4.0, the flexural stiffness kf ¼0.8, the nonlinear coefficient k1 ¼ 100, and the viscoelastic coefficient a ¼0.0001 [29]. Consider the dimensionless simply supported boundary conditions vð0,tÞ ¼ vð1,tÞ ¼ 0,
v, xx ð0,tÞ ¼ v, xx ð1,tÞ ¼ 0
(8)
In the supercritical transport speed regime qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g 4 1 þ p2 k2f
(9)
^ The trivial equilibrium position vðxÞ ¼ 0 becomes unstable and bifurcates into multiple equilibrium positions. Analytical formulae for the non-trivial equilibrium solutions and the critical speeds of Eq. (7) were shown in the Ref. [8]. The maximum positive equilibrium is ^ vðxÞ ¼ AS sinðpxÞ
(10)
where AS ¼
2 pk1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2 1p2 k2f
(11)
^ In fact, vðxÞ is also a non-trivial equilibrium solution. This paper only shows the results of the replacement ^ ^ vðx,tÞ-vðx,tÞ þ vðxÞ, as those via vðx,tÞ-vðx,tÞvðxÞ are the completely same under small vibrations. The substitution ^ vðx,tÞ-vðx,tÞ þ vðxÞ in Eq. (7) yields the equation of motion 2
2
v, tt þ2gv, xt þ p2 kf v, xx þ kf v, xxxx þ av, xxxxt ¼
ak21 2 kf
v, xx AS p2 sinðpxÞ
2 k1 v, xx AS p2 sinðpxÞ 2
Z
Z
1 0
½v, 2x þ2pAS v, x cosðpxÞdx
1
½v þ AS sinðpxÞv, xxt dx þ b cosðotÞ 0
(12)
G.-C. Zhang et al. / Journal of Sound and Vibration 331 (2012) 1612–1623
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In the present investigation, both the trial and weight functions are chosen as eigenfunctions of a stationary beam under the boundary condition (8). Suppose the solution of Eq. (12) have the following form [21,22,30–32]: N X
vðx,tÞ ¼
qk ðtÞ sinðkpxÞ
(13)
k¼1
where qk(t) are generalized displacements of the beam. Substituting Eq. (13) into Eq. (12) yields N X
q€ k sinðkpxÞ þ 2pg
k¼1
N X
kq_ k cosðkpxÞ þ ap4
k¼1
N X
4 2 k q_ k sinðkpxÞ þ p4 kf
k¼1
¼ b cosðotÞ
N X
4
2
ðk k Þqk sinðkpxÞ
k¼1
" # ! 2 N X k1 p4 2a 2 AS sinðpxÞ þ k qk sinðkpxÞ I1 þ 2 I2 4 k k¼1
(14)
f
where the dot denotes differentiation with respect to time and N X
I1 ðtÞ ¼ 2AS q1 þ
2
(15)
k qk q_ k
(16)
k q2k
k¼1
I2 ðtÞ ¼ AS q_ 1 þ
N X
2
k¼1
Using the Galerkin method, both sides of the Eq. (14) are multiplied by sin(mpx), m¼1,2,y,N, and the results are then integrated over the domain [0,1]. Then the following set of second-order ordinary differential equations is obtained h i ! N kq _ k 1ð1Þ1 þ k X 4 p4 k21 2a 4_ €q 1 þ 4g ðAS þ q1 Þ I1 þ 2 I2 þ ap q 1 ¼ b cosðotÞ (17) 2 p 4 1k kf k¼2
q€ m þ 4g
N X
h i mkq_ k 1ð1Þm þ k 2
k ¼ 1,kam
¼
m2 k
2
þ ap4 m4 q_ m þ p4 kf ðm4 m2 Þqm
2½1ð1Þm 1 2a 2 b cosðotÞ m2 p4 k1 qm I1 þ 2 I2 mp 4 k
! (18)
f
where m¼ 2,3,y,N. Setting qm ¼ 0 in Eq. (18) leads to the simplification based on the first-order Galerkin truncation which will be solved by the multiple-scale method. While N ¼4 and q4a0, Eqs. (17) and (18) lead to the fourth-order Galerkin truncation. 3. The multiple-scale method Substituting Eqs. (15) and (16) into Eq. (17) yields ! " # 2 2 2 k A2 k A2 p4 4bcosðotÞ k1 p4 2a q€ 1 þ ap4 1 þ 1 2S q_ 1 þ 1 S q1 ¼ 3AS q21 þ q31 þ 2 ð2AS q1 þq21 Þq_ 1 2 p 4 2kf kf
(19)
A detuning parameter s with the small parameter e is introduced to quantify the deviation of o from the fundamental frequency o1, and o is described by
o ¼ o1 þ e2 s
(20)
e2a and e3b are introduced to instead of a and b, respectively [35]. To determine a second-order uniform expansion of the solution to Eq. (19) by using the method of multiple scales, introduce the time scales as T 0 ¼ t,
T 1 ¼ et,
T 2 ¼ e2 t
(21)
As the solution does not depend on the scale T1 [35], the solution can be expanded in powers of the small parametere q1 ðtÞ ¼ eq10 ðT 0 ,T 2 Þ þ e2 q11 ðT 0 ,T 2 Þ þ e3 q12 ðT 0 ,T 2 Þ þ
(22)
Substituting Eqs. (20)–(22) into Eq. (19) and equating coefficients of like powers of e yields the order e @2 q10 @T 20
þ
p4 k21 A2S 2
q10 ¼ 0
(23)
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the order e2 @2 q11 @T 20
þ
p4 k21 A2S 2
2
q11 ¼
3p4 k1 AS 2 q10 4
(24)
the order e3 @2 q12 @T 20
þ
p4 k21 A2S
ap4 1 þ
2 2 k1 A2S 2 2kf
q12 ¼
!
4bcosðotÞ
p 4 2 k1 AS
@q10 3p @T 0 2
2
@2 q10 @T 0 @T 2
q10 q11
p4 k21 4
q310
(25)
The solution to Eq. (23) can be written as q10 ðT 0 ,T 2 Þ ¼ a1 cosðo1 T 0 þ b1 Þ ¼ YðT 2 Þ eio1 T 0 þ cc
(26)
where cc represents the complex conjugate of the preceding term and
o1 ¼
p2 k1 AS pffiffiffi 2
YðT 2 Þ ¼ 12a1 ðT 2 Þ eib1 ðT 2 Þ
(27) (28)
Substituting Eq. (26) into Eq. (24) yields @2 q11 @T 0
2
þ
p4 k21 A2S 2
2
q11 ¼
3k1 AS p4 2 2io1 T 0 ðY e þYYÞ þ cc 4
(29)
whose solution can be expressed as q11 ¼
1 2 2io1 T 0 3 Y e YY þ cc 2AS 2AS
Substituting Eqs. (26) and (30) into Eq. (25) yields " # ! 2 2 @2 q12 p4 k1 A2S 2b isT 2 k1 A2S 2 4 2 4 _ q12 ¼ þ e 2ion Y ion ap 1þ Y þ 3k1 p Y Y eion T 0 þcc þ NST 2 2 p @T 0 2 2kf
(30)
(31)
where the dot indicates the derivative with respect to the scale T2 and NST represents those terms that does not produce secular terms. Eliminating the secular terms from Eq. (31) yields the complex-valued modulation equation ! 2 2b isT 2 k1 A2S 2 4 _ e 2io1 Y io1 ap 1 þ (32) Y þ 3k1 p4 Y 2 Y ¼ 0 2 p 2kf Substituting Eq. (28), the polar form of Y, into Eq. (32), and separating the real and imaginary parts in Eq. (32), the realvalued modulation equations can be obtained as follows: 8 2 > k1 A2S 2b 1 4 > sin y ap 2 þ a1 < a_ 1 ¼ po 1 2 4 1 kf (33) 2 4 > > 2 1p : y_ 1 ¼ po2ba cos y1 þ s þ 3k8o a 1 1 1 1 where
y1 ðT 2 Þ ¼ sT 2 b1 ðT 2 Þ
(34)
Based on the 1-term Galerkin, Substituting Eq. (26) into Eq. (22), and then substituting Eq. (22) into Eq. (13), a1 represents the amplitude A1 of the steady-state response at the center of the axially moving beam. For the steady-state response in the resonance near the fundamental frequency, the amplitude a1 and the new phase angle y1 in Eq. (33) are constant. Hence the amplitude a1 and the phase y1 of the steady-state response satisfy 8 2 > k1 A2S 2b 1 4 > sin y ap 2 þ a1 ¼ 0 < po 1 2 4 1 kf (35) 4 2 > > : po2ba cos y1 þ s þ 38pok1 a21 ¼ 0 1 1 1 Elimination of the new phase angle y1 from Eq. (35) leads to the frequency–response relation between the detuning parameter and the amplitude of the steady-state response for the first mode vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 a2 p8 ð2k2f þ k21 A2S Þ2 3k1 p4 2 u b s¼ a1 7 t4 (36) 4 8o1 po1 a1 16kf
G.-C. Zhang et al. / Journal of Sound and Vibration 331 (2012) 1612–1623
0.003
1617
1-term Galerkin
A1
0.002
0.001
0.000 -0.3
-0.2
-0.1
0.0
0.1
0.2
Fig. 1. Frequency–response curve with a softening-type behavior.
0.003
0.005
1-term Galerkin
kf=0.8
0.004
0.002
0.003
=4.0
A1
A1
kf=0.7
0.002
0.001 kf=0.6 0.000 -0.3
1-term Galerkin
=3.5
-0.2
-0.1
0.0
=4.5
0.001 0.1
0.2
0.000 -1.0
-0.5
0.0
0.5
Fig. 2. Effects of system parameters on the frequency–response curve: (a) effects of the flexural stiffness and (b) effects of the axial speed.
The first term in the right-hand side of Eq. (36) demonstrates that the frequency–response curves exhibit a softeningtype behavior. Fig. 1 shows the curve with the axial speed g ¼4.0, the flexural stiffness kf ¼ 0.8, the nonlinear coefficient k1 ¼100, the viscosity coefficient a ¼0.0001 and the excitation amplitude b¼ 0.001. The solid lines exhibit the frequency– response relation and the dotted lines exhibit the softening-type behavior in Fig. 1. In contrast, a hardening-type behavior exists for subcritical speeds in the Refs. [29,33]. It is distinct that the change of transport speed will dramatically affect on the dynamic behavior. The second term in the right-hand side of Eq. (36) demonstrates that there is a least upper bound on a1 whose supremum is not affected by the nonlinear coefficient k1 without ignoring an inverse ratio relationship between k1 and the maximum static equilibrium AS. Further, there is a simple linear relationship between the supremum of the approximate amplitude a1 of the steady-state response and the viscosity coefficient (or the external excitation amplitude). As both have different impacts, the supremum increases with the increasing excitation amplitude, but the viscosity coefficient is on the contrary. In addition, the effects of the flexural stiffness and the axial speed on the frequency–response curve are investigated. The former is shown in Fig. 2(a) with the parameters g ¼4.0, b¼0.001, k1 ¼100, a ¼0.0001 and the changing flexural stiffness kf ¼0.6, 0.7, and 0.8, respectively. The other is shown in Fig. 2(b) with the parameters g ¼4.0, b¼0.001, k1 ¼100, a ¼0.0001 and the changing axial speed g ¼3.5, 4.0, and 4.5, respectively.
4. The finite difference method The stable steady-state response of the forced vibration for an axially moving viscoelastic beam in the supercritical speed range can be determined by solving the governing equations numerically. The central finite difference method is employed to find an approximate solution of the initial-value problems (17) and (18). Finite-difference methods provide a powerful approach to solving differential equations and are widely used in many fields of applied sciences. Introducing the equispaced mesh grids with time step t t j ¼ jt
ðj ¼ 0,1,2,. . .,TÞ
(37)
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G.-C. Zhang et al. / Journal of Sound and Vibration 331 (2012) 1612–1623 j
and denoting the function values qk(t) at tj as qk. Eqs. (17) and (18) can be discretized into, respectively [16]
h i jþ1 N k q qkj1 1ð1Þ1 þ k X qj þ 1 q1j1 k qj1þ 1 2qj1 þ qj1 1 þ ap4 1 þ 2g t2 2 2t tð1k Þ k¼2 ! 2 4 4 k p 2a ¼ bcosðojtÞ 1 ðAS þ qj1 Þ Ij1 þ 2 Ij2 p 4 k
(38)
f
j1 qjmþ 1 2qjm þqm
t
2
mkðqjkþ 1 qkj1 Þ½1ð1Þm þ k
N X
þ 2g
2
tðm2 k Þ
k ¼ 1,kam
2
2½1ð1Þ m2 p4 k1 qjm b cosðojtÞ mp 4 m
2
þ p4 kf ðm4 m2 Þqjm ¼
qjmþ 1 qj1 m 2t ! 2a Ij1 þ 2 Ij2 kf
þ am4 p4
(39)
where the corresponding changes are Ij1 ¼ 2AS qj1 þ
N X
ðkqjk Þ2
(40)
k¼1
Ij2 ¼
" # N X 1 2 j jþ1 j1 AS ðqj1þ 1 qj1 Þ þ k q ðq q Þ 1 k k k 2t k¼1
Introducing linear equations to compute the long history of time
(41)
h
jþ1 N k q 1ð1Þ1 þ k qj1 X ap4 j þ 1 2qj1 q1j1 ap4 j1 k k þ ¼ þ q 2 g q 2t 1 t2 2t 1 t2 tð1k2 Þ k¼2
þ
i
1
4
p
2
b cosðojtÞ
k1 p4 2a ðAS þ qj1 Þ Ij1 þ 2 Ij2 4 k
!
(42)
f
h
N mkðqjkþ 1 qj1 Þ 1ð1Þm þ k X am4 p4 j þ 1 2qjm qj1 am4 p4 j1 k m þ ¼ þ q 2 g q m m 2 2 2t 2t t t tðm2 k2 Þ k ¼ 1,kam
i
1
2
p4 kf ðm4 m2 Þqjm þ
2 2 1ð1Þm m2 k1 p4 qjm b cosðojtÞ mp 4
Ij1 þ
2a 2
kf
!
Ij2
(43)
In all numerical examples here, the initial conditions are vðx,0Þ ¼ 0:0001 sinðpxÞ,v, t ðx,0Þ ¼ 0
(44)
Substituting Eq. (13) into Eq. (44) yields q1 ð0Þ ¼ 0:0001,
qm ð0Þ ¼ q_ 1 ð0Þ ¼ q_ m ð0Þ ¼ 0,
m ¼ 2,3,. . .,N
(45)
For given N and temporal step t, the function values qk(t) with respect to t from Eqs. (42) and (43) under the initial conditions (45) can be calculated. Numerical results demonstrate that results caused by N ¼ 4 are similar with those when N ¼8. The fourth-order Galerkin truncation would be showed in the whole paper with the temporal step t ¼0.00001. In order to investigate the steady-state response for the forced vibration, the time history of the dynamic response at the center point of the moving beam with an external periodic transverse load is exhibited in Fig. 3 with the system parameters g ¼4.0, kf ¼0.8, k1 ¼100, a ¼0.0001, b¼0.001, the excitation frequency o ¼10 and T¼108. Numerical results show that the beam moving periodically after a short transition time from Fig. 3. Such a periodic motion is referred to as the steady-state response. As a matter of fact, the periodic steady-state response is not sensitive to the initial conditions (44) or (45). In the following, the amplitude A1 of the stable steady-state response is defined by the amplitude of the beam center displacement when the dimensionless time is between 900 and 1000. In the numerical calculation, A1 is half of the maximum displacement subtracted by the minimum displacement for the steady-state motion. It can also be seen as the average value of both. The steady-state amplitude responses are numerically calculated based on 4-trem Galerkin truncation with N ¼4. Concentrated on the first resonance, the relationship between the stable amplitude of the steady-state response and the frequency of the external excitation is illustrated in Fig. 4. Based on the numerical solutions to Eq. (12) by assuming that the axial speed g ¼4.0, the flexural stiffness kf ¼0.8, the nonlinear coefficient k1 ¼100, the viscosity coefficient a ¼0.0001 and the amplitude of the external excitation b¼0.001, Fig. 4(a) illustrates the amplitudes of the stable steady-state responses with the changing frequency of the external excitation. Obviously, the amplitude of the steady-state periodic response depends on the load frequency. The numerical results disclose that a resonance may occur if the load frequency o approaches the natural frequency of the linear elastic beam axially moving at the supercritical speed. The jumping phenomenon, a typical nonlinear phenomenon, occurs in the
G.-C. Zhang et al. / Journal of Sound and Vibration 331 (2012) 1612–1623
0.000
-0.001
0
100
200 t
0.001
300
-0.001 100
110
120 t
130
140
0
5
10 t
0.001
4-term Galerkin
0.000
4-term Galerkin
0.000
-0.001
400
v(0.5, t)
v(0.5, t)
0.001
4-term Galerkin
v(0.5, t)
v(0.5, t)
0.001
1619
15
20
4-term Galerkin
0.000
-0.001 995
996
997
998
999
1000
t
Fig. 3. The time history of the dynamic response with an external periodic excitation: (a) the full time history of the response; (b) local magnification of (a) at initial phase; (c) local magnification of (a) at transient phase; (d) local magnification of (a) at steady-state phase.
first resonance. A softening-type behavior is illustrated over the supercritical range. In contrast, a hardening-type behavior exists for subcritical speeds while using the same parameters except speeds [29]. The effects of system parameters on the stable frequency–response curve are illustrated in Fig. 4 as well. Fig. 4(b) illustrates the effects of the nonlinear coefficient on the stable steady-state response amplitude for Eq. (12) with k1 ¼75, 100, and 150. Fig. 4(c) shows the effects of the dynamic viscosity on the first resonance with a ¼0.0001, 0.0002, and 0.0003. Fig. 4(c) demonstrates that the amplitude for the first mode decreases with the increasing dynamic viscosity. The effects of the amplitude of the external excitation on the first resonance are shown in Fig. 4(d) with b¼ 0.001, 0.002, and 0.003. The amplitude of the stable steady-state response increases with the variable amplitude of the external excitation. Fig. 4(e) illustrates the effects of the flexural stiffness kf on the first resonance with kf ¼0.6, 0.7, and 0.8. Fig. 4(f) illustrates the effects of the axial speed on the first resonance with g ¼3.5, 4.0, and 4.5. Fig. 4 demonstrates that the different natural frequencies for the linear systems with the changing axial speeds or stiffness coefficients incur quantitatively differences among the stable steady-state responses. In order to prove the convergence and precision of the finite difference method, classical Runge–Kutta method is attached to solve Eqs. (17) and (18). They can be converted into first-order ordinary differential equations after introducing a substitution pk ¼ q_ k (k¼1,2,y,N). Then the dynamic response can be calculated by the fourth stage Runge–Kutta method. Considered the axially moving beam with g ¼4.0, kf ¼0.8, k1 ¼100 and a ¼0.0001, Fig. 5 shows the comparisons between the finite difference method and the fourth stage Runge–Kutta method based on 4-term Galerkin. The value of the external excitation amplitude in Fig. 5(a) and (b) is b¼0.001 and 0.002, respectively. The numerical results demonstrate that the amplitudes of the steady-state responses from both methods almost coincide in the resonance.
5. Comparisons 5.1. Between multi-scale method and FDM Lyapunov linearized stability theory can be employed to obtain the stability conditions of the steady-state responses. Using the same analytical method with the Ref. [35], the stability of the steady-state response can be determined by Eq. (33) on the condition that a1a0. The Jacobian matrix of the right-hand functions of Eq. (33) should be calculated at the fixed point defined by Eq. (35) in the first place. According to the Routh–Hurwitz criterion, the system defined by the foregoing Jacobian matrix is stable as long as all eigenvalues of the matrix are with negative real parts. Lyapunov stability theory guarantees that the stability of a nonlinear system coincides with that of its linearized system. Thus, the instability
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0.002
0.002
4-term Galerkin
k1=75
4-term Galerkin
k1=100
0.000
A1
A1
k1=150 0.001
9.4
9.6
9.8
10.0
10.2
0.001
0.000
10.4
9.4
9.6
9.8
0.002
0.003
4-term Galerkin
=0.0001
10.4
b=0.002
=0.0002
b=0.001
A1
A1
10.2
4-term Galerkin
b=0.003 0.002
0.001
10.0
0.001 =0.0003
0.000
9.6
9.8
0.002
10.0
0.000
10.2
9.0
9.5
10.0
0.002
4-term Galerkin kf=0.8
=4.0
9.0
A1
A1
0.000
kf=0.6
9.5
10.0
11.0
4-term Galerkin
=3.5
kf=0.7 0.001
10.5
0.000
10.5
=4.5
0.001
7
8
9
10
11
12
Fig. 4. Effects of system parameters on the stable frequency–response curve: (a) the stable frequency–response curve; (b) effects of the nonlinear coefficient; (c) effects of the dynamic viscosity; (d) effects of the excitation amplitude; (e) effects of the flexural stiffness; and (f) effects of the axial speed.
b=0.001
0.001
0.0024
FDM Runge-Kutta
4-term Galerkin
A1
A1
0.002
FDM Runge-Kutta
4-term Galerkin
0.0012 b=0.002
0.000
0.0000 9.0
9.5
10.0
10.5
11.0
9.0
9.5
10.0
10.5
Fig. 5. Comparisons between FDM and the fourth stage Runge–Kutta method: (a) b ¼ 0.001 and (b) b ¼ 0.002.
11.0
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boundary is directly given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 2 2 2 2 2 2 3k1 p4 2 p4 u t9k1 a4 a ð2kf þ k1 AS Þ s¼ a1 7 4 4o1 4 o21 1 kf
(46)
From Eq. (20), the detuning parameter s quantifies the deviation of o from the fundamental frequency o1, which is the first frequency of the free vibration described by Eq. (12) without the nonlinear terms, the dynamic viscosity term and the external excitation term. By considering the different orders Galerkin truncation, the frequencies o1 of the linear system can be numerically computed for an axially moving beam with g ¼4.0, kf ¼0.8, and k1 ¼100 [17]. By taking N ¼4 as an example, the fundamental frequency o1 ¼9.92. Based on the first-order Galerkin method, Fig. 6 shows the comparisons of the supercritical detuning-response curves via the finite difference method (FDM) and the multi-scale method. The dots represent the stable steady-state responses to Eqs. (42) and (43) via the finite difference method, the solid lines represent the theoretical steady-state responses to Eq. (36) via the multi-scale method, and the dotted lines represent the analytical instability boundaries of the supercritical steady-state response curves from Eq. (46). In Fig. 6(a), kf ¼0.8, a ¼0.0001, k1 ¼100, b¼0.001 and g ¼4.0. In Fig. 6(b), kf ¼0.8, a ¼0.0001, k1 ¼150, b¼ 0.001 and g ¼4.0. In Fig. 6(c), kf ¼0.8, a ¼0.0002, k1 ¼100, b¼0.001 and g ¼4.0. In Fig. 6(d), kf ¼0.8, a ¼0.0001, k1 ¼ 100, b¼0.002 and g ¼4.0. In Fig. 6(e), kf ¼0.7, a ¼0.0001, k1 ¼100, b¼0.001 and g ¼4.0. In Fig. 6(f),
0.004 0.003 0.002 0.001 0.000
-0.8
-0.6
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-0.2
0.2
0.000 -0.8
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Multi-Scale FDM
0.003
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0.002 0.001
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Multi-Scale FDM
0.003 0.002
0.0
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Multi-Scale FDM
0.003 A1
A1
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A1
0.0
Multi-Scale FDM
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0.002 0.001
0.001 0.000
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Multi-Scale FDM
0.003 A1
A1
0.004
Multi-Scale FDM
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0.0
0.2
0.4
0.000 -0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
Fig. 6. Comparisons of the analytical and numerical result based on 1-term Galerkin: (a) kf ¼0.8,a ¼ 0.0001, k1 ¼ 100, b¼ 0.001 and g ¼4.0; (b) kf ¼ 0.8, a ¼ 0.0001, k1 ¼150, b ¼0.001 and g ¼ 4.0; (c) kf ¼ 0.8, a ¼0.0002, k1 ¼ 100, b ¼0.001 and g ¼ 4.0; (d) kf ¼ 0.8, a ¼ 0.0001, k1 ¼ 100, b¼ 0.002 and g ¼4.0; (e) kf ¼ 0.7, a ¼ 0.0001, k1 ¼100, b ¼ 0.001 and g ¼4.0; and (f) kf ¼ 0.8, a ¼ 0.0001, k1 ¼ 100, b ¼0.001 and g ¼3.5.
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0.0024 b=0.001
0.0012
1-term Galerkin 2-term Galerkin 4-term Galerkin
b=0.002 0.002 A1
A1
0.003
1-term Galerkin 2-term Galerkin 4-term Galerkin
0.001
0.0000 -0.3
-0.2
-0.1
0.0
0.1
0.2
0.000 -0.3
-0.2
-0.1
0.0
0.1
0.2
Fig. 7. Comparisons among the 1-term, 2-term and 4-term Galerkin based on FDM: (a) b ¼0.001; and (b) b ¼0.002.
kf ¼0.8, a ¼0.0001, k1 ¼100, b ¼0.001 and g ¼3.5. It can be remarked that the stable detuning-response curves and the instability boundaries do not intersect each other in Fig. 6. Fig. 6 also indicates that the theoretical outcomes conform to those calculated by the finite difference method based on the 1-term Galerkin method. Observed from Eq. (46), the instability boundaries have no relation to the amplitude of the external excitation. Fig. 6(a) and (d) help to explain this point exactly. Hence, the instability of the responses may not exist if the excitation amplitude is small enough. In addition, Eq. (46) easily leads to the result that the detuning-response curves will be stable if s 40. It means that jumping phenomena may only happen at the less frequency than the fundamental one under the very small vibration. This information can also be gained from Fig. 6. In fact, there will be no jumping phenomenon if the viscosity coefficient is large enough, which may be deduced from Eqs. (36) and (46). 5.2. Between different truncation terms Considered the axially moving beam with g ¼4.0, kf ¼0.8, k1 ¼100 and a ¼0.0001, Fig. 7 shows the comparisons of the supercritical detuning-response curves among the Galerkin procedures with different orders. The value of the external excitation amplitude in Fig. 7(a) and (b) is b ¼0.001 and 0.002, respectively. The amplitudes of the stable steady-state responses for the axially moving viscoelastic beam with the changing frequency of the external excitation are showed in Fig. 7, where the dashed lines, the dots and the solid lines represent the supercritical steady-state responses from the 1-term Galerkin method, the 2-term Galerkin method and the 4-term Galerkin method, respectively. It is obvious that 1-term Galerkin truncation can yield the convincing primary resonance. However, the results of the 2-term truncation are closer to those of the 4-term truncation than the 1-term Galerkin method. On the whole, the difference of the natural frequencies among the different orders Galerkin method incurs quantitatively differences on the amplitudes of the stable steady-state responses in the supercritical region. 6. Conclusions This paper is devoted to the steady-state periodic response of the forced vibration for an axially moving viscoelastic beam in the supercritical speed range. For this motion, the model is cast in the standard form of continuous gyroscopic systems. The relations between the load frequency and the amplitude of the steady-state response are calculated from an integro-partial-differential equation via the Galerkin method under the simply supported boundary conditions. Based on the 1-term Galerkin truncation, the method of multiple scales is applied to the nonlinear equation to determine steadystate responses. The finite difference schemes are developed to verify the approximate analytical results. The numerical results demonstrate that the steady-state periodic responses exist in the transverse vibration under the periodic transverse loads in the supercritical regime. Both the analytical and the numerical calculations demonstrate that, the forced resonances occur when the frequency of the external excitation is close to the linear natural frequency. A softening-type behavior is illustrated over the supercritical range. The amplitude of the steady-state response increases with the variable amplitude of the external excitation, and decreases with the dynamic viscosity.
Acknowledgments This work was supported by the National Outstanding Young Scientists Fund of China (Project no. 10725209), the National Science Foundation of China (Project nos. 10932006 and 10902064), Shanghai Rising-Star Program (no. 11QA1402300), Innovation Program of Shanghai Municipal Education Commission (no. 12YZ028), and Shanghai Leading Academic Discipline Project (no. S30106).
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Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds Guo-Ce Zhang a, Hu Ding a,n, Li-Qun Chen a,b, Shao-Pu Yang c a Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China b Department of Mechanics, Shanghai University, Shanghai 200444, China c Shijiazhuang Tiedao University, Shijiazhuang 050043, China
a r t i c l e in f o
abstract
Article history: Received 9 April 2011 Received in revised form 20 October 2011 Accepted 3 December 2011 Handling Editor: L.G. Tham Available online 31 December 2011
The present paper investigates the steady-state periodic response of an axially moving viscoelastic beam in the supercritical speed range. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. It is assumed that the excitation of the forced vibration is spatially uniform and temporally harmonic. Under the quasi-static stretch assumption, a nonlinear integro-partialdifferential equation governs the transverse motion of the axially moving beam. The equation is cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. For a beam constituted by the Kelvin model, the primary resonance is analyzed via the Galerkin method under the simply supported boundary conditions. Based on the Galerkin truncation, the finite difference schemes are developed to verify the results via the method of multiple scales. Numerical simulations demonstrate that the steady-state periodic responses exist in the transverse vibration and a resonance with a softening-type behavior occurs if the external load frequency approaches the linear natural frequency in the supercritical regime. The effects of the viscoelastic damping, external excitation amplitude, and nonlinearity on the steady-state response amplitude for the first mode are illustrated. & 2011 Elsevier Ltd. All rights reserved.
1. Introduction The wide diffusion of axially moving systems in industrial processes, civil, mechanical and automotive applications has motivated intense research activities. Elevator cables, power transmission band, aerial cableways and belt saws are some examples of one-dimensional axially moving continua [1–5]. These systems can be modeled as axially moving beams if the bending stiffness is considered [6,7]. Therefore, the related investigations are important to design these devices. The axial speed greatly affects the dynamic behaviors of the moving system. Wickert noticed that the unstable straight equilibrium configuration bifurcates into multiple equilibrium positions for a translating beam above the critical velocity [8]. Hwang and Perkins investigated the effects of an initial curvature due to supporting wheels and pulleys on the bifurcation and stability of equilibrium in the supercritical speed regime and underlined the system sensitivity to initial imperfections [9,10]. Considering simply supported boundary conditions, Pellicano and Vestroni analyzed the dynamic behavior of an axially moving beam subjected to an axial transport of mass by using a high dimension discrete model
n
Corresponding author. Tel.: þ 86 21 56337273, fax: þ 86 21 36033287. E-mail address: [email protected] (H. Ding).
0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.12.004
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obtained by a Galerkin procedure [11]. They investigated the forced response of an axially moving beam under harmonic excitation in the range of velocities greater than the first critical velocity [12]. The work focused on exploring the dynamics of a traveling beam, a simple viscous damping mechanism subjected to a transverse load with simple supports via the Galerkin method when its main parameters vary in the supercritical velocity range. Parker discussed supercritical phenomena in moving strings and found that the elastically supported strings always exhibit divergence instability above ¨ z calculated the natural frequencies of axially moving beams under clamped– the first critical speed [13]. Pakdemirli and O clamped boundary conditions in the supercritical regime, and found the beam unstable at those velocities [14]. In the supercritical regime, the unstable straight equilibrium configuration bifurcates for a translating beam is analogous to a buckled beam problem [15]. For each non-trivial equilibrium solutions, the governing equation is cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform, and the first natural frequency of the axially moving beam is calculated by a perturbation method [8]. Under simply supported boundary conditions, the non-trivial equilibriums and natural frequencies of the transverse model for the free vibration were calculated in the supercritical regime [16,17]. As we know, there have been no investigations about the steady-state periodic response of the forced vibration of axially moving viscoelastic beams in the supercritical regime. This paper focuses on the forced vibration and stability for an axially moving beam with a supercritical transport speed excited by the external vibration which is assumed to be spatially uniform and temporally harmonic. The Kelvin model had been used for describing the viscoelastic materials to investigate the forced vibration of axially moving systems [18–22]. As exact solutions are usually unavailable, approximate analytical methods are widely applied to investigate nonlinear vibration of axially moving beams. These methods include the asymptotic method of Krylov, Bogoliubov, and Mitropolsky [8], the Lindstedt–Poincare´ method [23], the method of normal forms [24], complex modes [25], the method of multiple scales [26–28]. Under the critical traveling speed, Chen and Ding showed the steady-state transverse response in coupled planar vibration of axially moving viscoelastic beams [29]. In the present paper, Galerkin method will be used to analyze the governing equation, a nonlinear integro-partial-differential equation for an axially translating beam in the supercritical transport speed ranges. As we know, based on the first-order Galerkin discretization, Ravindra and Zhu analyzed the chaotic behavior of axially accelerating beams [30]. Based on 4-term Galerkin method, Chen et al. analyzed the regular and chaotic vibrations of an axially moving viscoelastic string [21], and later on, Chen and Yang investigated nonlinear dynamics of axially accelerating viscoelastic beams [31]. Zhang studied the transverse motion of an axially moving viscoelastic string via the Galerkin method [22]. Riedel and Tan studied the forced response of a nonlinear axially moving strip with coupled transverse and longitudinal motion and applied the method of multiple scales to examine the internal resonance based on the Galerkin truncation [32]. In this paper, based on the Galerkin truncation, the frequency–response curves for the periodic forced vibration are obtained via using the multiple-scale method and the finite difference method (FDM), respectively. The present paper is organized as follows. Section 2 establishes the governing equation for the transverse motion of an axially moving beam in the supercritical regime. Section 3 carries out a three time-scales perturbation analysis of the equation obtained in Section 2 via 1-term Galerkin method. Section 4 combines the 4-term Galerkin truncation and the finite difference method to solve the stable steady-state response from the governing equations presented in Section 2. Section 5 compares the results of the multiple-scale methods in Section 3 and the finite difference method in Section 4. Section 6 ends the paper with the concluding remarks. 2. The mathematical model Consider a uniform axially traveling beam with simply supported boundary conditions at the constant transport speed
G between both ends separated by distance L. V is the transverse displacement field, T is the time and X is the spatial coordinate along the axis of the beam, r is the density, I is the moment of inertia, P0 is the initial tension, A is the crosssectional area. Under certain conditions, the transverse motion can be decoupled from the longitudinal motion so that a nonlinear partial-differential model is obtained to govern the transverse motion. It is assumed that the external transverse load is a spatially uniformly distributed periodic force. B and O are the amplitude and the frequency of the external excitation subjected to the beam, respectively. The equation of motion can be read as [33,34] " # 2 @2 V @2 V @ @V @2 M 2@ V þG ðP þ rA 2 þ 2G þ A s Þ ¼ B cosðOTÞ (1) 0 2 @X@T @X @X @T @X @X 2 where sðTÞ and MðX,TÞ are the axially disturbed stress and bending moment, respectively. The material particle of the moving beam experiences local ð@2 V=@T 2 Þ, Coriolis 2Gð@2 V=@X@TÞ, and centripetal G2 ð@2 V=@X 2 Þ acceleration components. The Kelvin viscoelastic model is used to describe the viscoelastic properties of the materials. So, the constitution relation is
sðT Þ ¼ EeK þ L
@eK @T
(2)
where E is Young’s modulus of the beam material, L is the dynamic viscosity and eK ðTÞ is the axial strain satisfying Z 1 L @V 2 eK ðTÞ ¼ dX (3) 2L 0 @X
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Such a condition is caused by the quasi-static stretch assumption. Considering the Kelvin model of the viscoelastic material, the moment–curvature relationship can be expressed as MðX,TÞ ¼ EI
@2 V @X
2
þ IL
@3 V
(4)
@X 2 @T
Substituting Eqs. (2)–(4) into Eq. (1) yields an integro-partial-differential equation for the transverse motion of the axially moving viscoelastic beam " # 2 @2 V @2 V @2 V @4 V @5 V 2@ V þG rA þ2G P 0 2 þ EI 4 þ IL 4 2 2 @X@T @T @X @X @X @X @T ¼
Z
EA @2 V 2L @X 2
L 0
2 Z @V AL @2 V L @V @2 V dX þ B cosðOTÞ dX þ 2 @X L @X 0 @X @X@T
Introducing the dimensionless variables and parameters as follows: sffiffiffiffiffiffiffi V X BL T P0 , v¼ , x¼ , b¼ , t¼ L L P0 L rA sffiffiffiffiffiffiffi rA , P0
o ¼ OL
a¼
L3
IL pffiffiffiffiffiffiffiffiffiffiffi , rAP0
2
k1 ¼
(5)
sffiffiffiffiffiffiffi rA , P0
g¼G
EA , P0
2
kf ¼
EI P0 L2
(6)
where x is the dimensionless spatial coordinate and t is the dimensionless time. Eq. (5) can be transformed into the dimensionless equation 2
v, tt þ 2gv, xt þðg2 1Þv, xx þ kf v, xxxx þ av, xxxxt 2
k1 v, xx 2
¼
Z 0
1
ak21
v, 2x dx þ
2 kf
v, xx
Z
1
v, x v, xt dx þb cosðotÞ
(7)
0
where a comma preceding x or t denotes the partial differentiation with respect to x or t. Consider a steel beam with modulus of elasticity E¼2.1 1011 Pa and density r ¼7850 kg/m3. Let the beam length L¼1 m, the dynamic viscosity L ¼634,402 N s/m2, the initial tension P0 ¼7850 N, the axial speed ! ¼206.88 m/s, and the rectangular cross-section of the beam with the width W¼ 13.49 mm and the height H¼27.71 mm. Then Eq. (6) yields the dimensionless axial speed g ¼4.0, the flexural stiffness kf ¼0.8, the nonlinear coefficient k1 ¼ 100, and the viscoelastic coefficient a ¼0.0001 [29]. Consider the dimensionless simply supported boundary conditions vð0,tÞ ¼ vð1,tÞ ¼ 0,
v, xx ð0,tÞ ¼ v, xx ð1,tÞ ¼ 0
(8)
In the supercritical transport speed regime qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g 4 1 þ p2 k2f
(9)
^ The trivial equilibrium position vðxÞ ¼ 0 becomes unstable and bifurcates into multiple equilibrium positions. Analytical formulae for the non-trivial equilibrium solutions and the critical speeds of Eq. (7) were shown in the Ref. [8]. The maximum positive equilibrium is ^ vðxÞ ¼ AS sinðpxÞ
(10)
where AS ¼
2 pk1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2 1p2 k2f
(11)
^ In fact, vðxÞ is also a non-trivial equilibrium solution. This paper only shows the results of the replacement ^ ^ vðx,tÞ-vðx,tÞ þ vðxÞ, as those via vðx,tÞ-vðx,tÞvðxÞ are the completely same under small vibrations. The substitution ^ vðx,tÞ-vðx,tÞ þ vðxÞ in Eq. (7) yields the equation of motion 2
2
v, tt þ2gv, xt þ p2 kf v, xx þ kf v, xxxx þ av, xxxxt ¼
ak21 2 kf
v, xx AS p2 sinðpxÞ
2 k1 v, xx AS p2 sinðpxÞ 2
Z
Z
1 0
½v, 2x þ2pAS v, x cosðpxÞdx
1
½v þ AS sinðpxÞv, xxt dx þ b cosðotÞ 0
(12)
G.-C. Zhang et al. / Journal of Sound and Vibration 331 (2012) 1612–1623
1615
In the present investigation, both the trial and weight functions are chosen as eigenfunctions of a stationary beam under the boundary condition (8). Suppose the solution of Eq. (12) have the following form [21,22,30–32]: N X
vðx,tÞ ¼
qk ðtÞ sinðkpxÞ
(13)
k¼1
where qk(t) are generalized displacements of the beam. Substituting Eq. (13) into Eq. (12) yields N X
q€ k sinðkpxÞ þ 2pg
k¼1
N X
kq_ k cosðkpxÞ þ ap4
k¼1
N X
4 2 k q_ k sinðkpxÞ þ p4 kf
k¼1
¼ b cosðotÞ
N X
4
2
ðk k Þqk sinðkpxÞ
k¼1
" # ! 2 N X k1 p4 2a 2 AS sinðpxÞ þ k qk sinðkpxÞ I1 þ 2 I2 4 k k¼1
(14)
f
where the dot denotes differentiation with respect to time and N X
I1 ðtÞ ¼ 2AS q1 þ
2
(15)
k qk q_ k
(16)
k q2k
k¼1
I2 ðtÞ ¼ AS q_ 1 þ
N X
2
k¼1
Using the Galerkin method, both sides of the Eq. (14) are multiplied by sin(mpx), m¼1,2,y,N, and the results are then integrated over the domain [0,1]. Then the following set of second-order ordinary differential equations is obtained h i ! N kq _ k 1ð1Þ1 þ k X 4 p4 k21 2a 4_ €q 1 þ 4g ðAS þ q1 Þ I1 þ 2 I2 þ ap q 1 ¼ b cosðotÞ (17) 2 p 4 1k kf k¼2
q€ m þ 4g
N X
h i mkq_ k 1ð1Þm þ k 2
k ¼ 1,kam
¼
m2 k
2
þ ap4 m4 q_ m þ p4 kf ðm4 m2 Þqm
2½1ð1Þm 1 2a 2 b cosðotÞ m2 p4 k1 qm I1 þ 2 I2 mp 4 k
! (18)
f
where m¼ 2,3,y,N. Setting qm ¼ 0 in Eq. (18) leads to the simplification based on the first-order Galerkin truncation which will be solved by the multiple-scale method. While N ¼4 and q4a0, Eqs. (17) and (18) lead to the fourth-order Galerkin truncation. 3. The multiple-scale method Substituting Eqs. (15) and (16) into Eq. (17) yields ! " # 2 2 2 k A2 k A2 p4 4bcosðotÞ k1 p4 2a q€ 1 þ ap4 1 þ 1 2S q_ 1 þ 1 S q1 ¼ 3AS q21 þ q31 þ 2 ð2AS q1 þq21 Þq_ 1 2 p 4 2kf kf
(19)
A detuning parameter s with the small parameter e is introduced to quantify the deviation of o from the fundamental frequency o1, and o is described by
o ¼ o1 þ e2 s
(20)
e2a and e3b are introduced to instead of a and b, respectively [35]. To determine a second-order uniform expansion of the solution to Eq. (19) by using the method of multiple scales, introduce the time scales as T 0 ¼ t,
T 1 ¼ et,
T 2 ¼ e2 t
(21)
As the solution does not depend on the scale T1 [35], the solution can be expanded in powers of the small parametere q1 ðtÞ ¼ eq10 ðT 0 ,T 2 Þ þ e2 q11 ðT 0 ,T 2 Þ þ e3 q12 ðT 0 ,T 2 Þ þ
(22)
Substituting Eqs. (20)–(22) into Eq. (19) and equating coefficients of like powers of e yields the order e @2 q10 @T 20
þ
p4 k21 A2S 2
q10 ¼ 0
(23)
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G.-C. Zhang et al. / Journal of Sound and Vibration 331 (2012) 1612–1623
the order e2 @2 q11 @T 20
þ
p4 k21 A2S 2
2
q11 ¼
3p4 k1 AS 2 q10 4
(24)
the order e3 @2 q12 @T 20
þ
p4 k21 A2S
ap4 1 þ
2 2 k1 A2S 2 2kf
q12 ¼
!
4bcosðotÞ
p 4 2 k1 AS
@q10 3p @T 0 2
2
@2 q10 @T 0 @T 2
q10 q11
p4 k21 4
q310
(25)
The solution to Eq. (23) can be written as q10 ðT 0 ,T 2 Þ ¼ a1 cosðo1 T 0 þ b1 Þ ¼ YðT 2 Þ eio1 T 0 þ cc
(26)
where cc represents the complex conjugate of the preceding term and
o1 ¼
p2 k1 AS pffiffiffi 2
YðT 2 Þ ¼ 12a1 ðT 2 Þ eib1 ðT 2 Þ
(27) (28)
Substituting Eq. (26) into Eq. (24) yields @2 q11 @T 0
2
þ
p4 k21 A2S 2
2
q11 ¼
3k1 AS p4 2 2io1 T 0 ðY e þYYÞ þ cc 4
(29)
whose solution can be expressed as q11 ¼
1 2 2io1 T 0 3 Y e YY þ cc 2AS 2AS
Substituting Eqs. (26) and (30) into Eq. (25) yields " # ! 2 2 @2 q12 p4 k1 A2S 2b isT 2 k1 A2S 2 4 2 4 _ q12 ¼ þ e 2ion Y ion ap 1þ Y þ 3k1 p Y Y eion T 0 þcc þ NST 2 2 p @T 0 2 2kf
(30)
(31)
where the dot indicates the derivative with respect to the scale T2 and NST represents those terms that does not produce secular terms. Eliminating the secular terms from Eq. (31) yields the complex-valued modulation equation ! 2 2b isT 2 k1 A2S 2 4 _ e 2io1 Y io1 ap 1 þ (32) Y þ 3k1 p4 Y 2 Y ¼ 0 2 p 2kf Substituting Eq. (28), the polar form of Y, into Eq. (32), and separating the real and imaginary parts in Eq. (32), the realvalued modulation equations can be obtained as follows: 8 2 > k1 A2S 2b 1 4 > sin y ap 2 þ a1 < a_ 1 ¼ po 1 2 4 1 kf (33) 2 4 > > 2 1p : y_ 1 ¼ po2ba cos y1 þ s þ 3k8o a 1 1 1 1 where
y1 ðT 2 Þ ¼ sT 2 b1 ðT 2 Þ
(34)
Based on the 1-term Galerkin, Substituting Eq. (26) into Eq. (22), and then substituting Eq. (22) into Eq. (13), a1 represents the amplitude A1 of the steady-state response at the center of the axially moving beam. For the steady-state response in the resonance near the fundamental frequency, the amplitude a1 and the new phase angle y1 in Eq. (33) are constant. Hence the amplitude a1 and the phase y1 of the steady-state response satisfy 8 2 > k1 A2S 2b 1 4 > sin y ap 2 þ a1 ¼ 0 < po 1 2 4 1 kf (35) 4 2 > > : po2ba cos y1 þ s þ 38pok1 a21 ¼ 0 1 1 1 Elimination of the new phase angle y1 from Eq. (35) leads to the frequency–response relation between the detuning parameter and the amplitude of the steady-state response for the first mode vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 a2 p8 ð2k2f þ k21 A2S Þ2 3k1 p4 2 u b s¼ a1 7 t4 (36) 4 8o1 po1 a1 16kf
G.-C. Zhang et al. / Journal of Sound and Vibration 331 (2012) 1612–1623
0.003
1617
1-term Galerkin
A1
0.002
0.001
0.000 -0.3
-0.2
-0.1
0.0
0.1
0.2
Fig. 1. Frequency–response curve with a softening-type behavior.
0.003
0.005
1-term Galerkin
kf=0.8
0.004
0.002
0.003
=4.0
A1
A1
kf=0.7
0.002
0.001 kf=0.6 0.000 -0.3
1-term Galerkin
=3.5
-0.2
-0.1
0.0
=4.5
0.001 0.1
0.2
0.000 -1.0
-0.5
0.0
0.5
Fig. 2. Effects of system parameters on the frequency–response curve: (a) effects of the flexural stiffness and (b) effects of the axial speed.
The first term in the right-hand side of Eq. (36) demonstrates that the frequency–response curves exhibit a softeningtype behavior. Fig. 1 shows the curve with the axial speed g ¼4.0, the flexural stiffness kf ¼ 0.8, the nonlinear coefficient k1 ¼100, the viscosity coefficient a ¼0.0001 and the excitation amplitude b¼ 0.001. The solid lines exhibit the frequency– response relation and the dotted lines exhibit the softening-type behavior in Fig. 1. In contrast, a hardening-type behavior exists for subcritical speeds in the Refs. [29,33]. It is distinct that the change of transport speed will dramatically affect on the dynamic behavior. The second term in the right-hand side of Eq. (36) demonstrates that there is a least upper bound on a1 whose supremum is not affected by the nonlinear coefficient k1 without ignoring an inverse ratio relationship between k1 and the maximum static equilibrium AS. Further, there is a simple linear relationship between the supremum of the approximate amplitude a1 of the steady-state response and the viscosity coefficient (or the external excitation amplitude). As both have different impacts, the supremum increases with the increasing excitation amplitude, but the viscosity coefficient is on the contrary. In addition, the effects of the flexural stiffness and the axial speed on the frequency–response curve are investigated. The former is shown in Fig. 2(a) with the parameters g ¼4.0, b¼0.001, k1 ¼100, a ¼0.0001 and the changing flexural stiffness kf ¼0.6, 0.7, and 0.8, respectively. The other is shown in Fig. 2(b) with the parameters g ¼4.0, b¼0.001, k1 ¼100, a ¼0.0001 and the changing axial speed g ¼3.5, 4.0, and 4.5, respectively.
4. The finite difference method The stable steady-state response of the forced vibration for an axially moving viscoelastic beam in the supercritical speed range can be determined by solving the governing equations numerically. The central finite difference method is employed to find an approximate solution of the initial-value problems (17) and (18). Finite-difference methods provide a powerful approach to solving differential equations and are widely used in many fields of applied sciences. Introducing the equispaced mesh grids with time step t t j ¼ jt
ðj ¼ 0,1,2,. . .,TÞ
(37)
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G.-C. Zhang et al. / Journal of Sound and Vibration 331 (2012) 1612–1623 j
and denoting the function values qk(t) at tj as qk. Eqs. (17) and (18) can be discretized into, respectively [16]
h i jþ1 N k q qkj1 1ð1Þ1 þ k X qj þ 1 q1j1 k qj1þ 1 2qj1 þ qj1 1 þ ap4 1 þ 2g t2 2 2t tð1k Þ k¼2 ! 2 4 4 k p 2a ¼ bcosðojtÞ 1 ðAS þ qj1 Þ Ij1 þ 2 Ij2 p 4 k
(38)
f
j1 qjmþ 1 2qjm þqm
t
2
mkðqjkþ 1 qkj1 Þ½1ð1Þm þ k
N X
þ 2g
2
tðm2 k Þ
k ¼ 1,kam
2
2½1ð1Þ m2 p4 k1 qjm b cosðojtÞ mp 4 m
2
þ p4 kf ðm4 m2 Þqjm ¼
qjmþ 1 qj1 m 2t ! 2a Ij1 þ 2 Ij2 kf
þ am4 p4
(39)
where the corresponding changes are Ij1 ¼ 2AS qj1 þ
N X
ðkqjk Þ2
(40)
k¼1
Ij2 ¼
" # N X 1 2 j jþ1 j1 AS ðqj1þ 1 qj1 Þ þ k q ðq q Þ 1 k k k 2t k¼1
Introducing linear equations to compute the long history of time
(41)
h
jþ1 N k q 1ð1Þ1 þ k qj1 X ap4 j þ 1 2qj1 q1j1 ap4 j1 k k þ ¼ þ q 2 g q 2t 1 t2 2t 1 t2 tð1k2 Þ k¼2
þ
i
1
4
p
2
b cosðojtÞ
k1 p4 2a ðAS þ qj1 Þ Ij1 þ 2 Ij2 4 k
!
(42)
f
h
N mkðqjkþ 1 qj1 Þ 1ð1Þm þ k X am4 p4 j þ 1 2qjm qj1 am4 p4 j1 k m þ ¼ þ q 2 g q m m 2 2 2t 2t t t tðm2 k2 Þ k ¼ 1,kam
i
1
2
p4 kf ðm4 m2 Þqjm þ
2 2 1ð1Þm m2 k1 p4 qjm b cosðojtÞ mp 4
Ij1 þ
2a 2
kf
!
Ij2
(43)
In all numerical examples here, the initial conditions are vðx,0Þ ¼ 0:0001 sinðpxÞ,v, t ðx,0Þ ¼ 0
(44)
Substituting Eq. (13) into Eq. (44) yields q1 ð0Þ ¼ 0:0001,
qm ð0Þ ¼ q_ 1 ð0Þ ¼ q_ m ð0Þ ¼ 0,
m ¼ 2,3,. . .,N
(45)
For given N and temporal step t, the function values qk(t) with respect to t from Eqs. (42) and (43) under the initial conditions (45) can be calculated. Numerical results demonstrate that results caused by N ¼ 4 are similar with those when N ¼8. The fourth-order Galerkin truncation would be showed in the whole paper with the temporal step t ¼0.00001. In order to investigate the steady-state response for the forced vibration, the time history of the dynamic response at the center point of the moving beam with an external periodic transverse load is exhibited in Fig. 3 with the system parameters g ¼4.0, kf ¼0.8, k1 ¼100, a ¼0.0001, b¼0.001, the excitation frequency o ¼10 and T¼108. Numerical results show that the beam moving periodically after a short transition time from Fig. 3. Such a periodic motion is referred to as the steady-state response. As a matter of fact, the periodic steady-state response is not sensitive to the initial conditions (44) or (45). In the following, the amplitude A1 of the stable steady-state response is defined by the amplitude of the beam center displacement when the dimensionless time is between 900 and 1000. In the numerical calculation, A1 is half of the maximum displacement subtracted by the minimum displacement for the steady-state motion. It can also be seen as the average value of both. The steady-state amplitude responses are numerically calculated based on 4-trem Galerkin truncation with N ¼4. Concentrated on the first resonance, the relationship between the stable amplitude of the steady-state response and the frequency of the external excitation is illustrated in Fig. 4. Based on the numerical solutions to Eq. (12) by assuming that the axial speed g ¼4.0, the flexural stiffness kf ¼0.8, the nonlinear coefficient k1 ¼100, the viscosity coefficient a ¼0.0001 and the amplitude of the external excitation b¼0.001, Fig. 4(a) illustrates the amplitudes of the stable steady-state responses with the changing frequency of the external excitation. Obviously, the amplitude of the steady-state periodic response depends on the load frequency. The numerical results disclose that a resonance may occur if the load frequency o approaches the natural frequency of the linear elastic beam axially moving at the supercritical speed. The jumping phenomenon, a typical nonlinear phenomenon, occurs in the
G.-C. Zhang et al. / Journal of Sound and Vibration 331 (2012) 1612–1623
0.000
-0.001
0
100
200 t
0.001
300
-0.001 100
110
120 t
130
140
0
5
10 t
0.001
4-term Galerkin
0.000
4-term Galerkin
0.000
-0.001
400
v(0.5, t)
v(0.5, t)
0.001
4-term Galerkin
v(0.5, t)
v(0.5, t)
0.001
1619
15
20
4-term Galerkin
0.000
-0.001 995
996
997
998
999
1000
t
Fig. 3. The time history of the dynamic response with an external periodic excitation: (a) the full time history of the response; (b) local magnification of (a) at initial phase; (c) local magnification of (a) at transient phase; (d) local magnification of (a) at steady-state phase.
first resonance. A softening-type behavior is illustrated over the supercritical range. In contrast, a hardening-type behavior exists for subcritical speeds while using the same parameters except speeds [29]. The effects of system parameters on the stable frequency–response curve are illustrated in Fig. 4 as well. Fig. 4(b) illustrates the effects of the nonlinear coefficient on the stable steady-state response amplitude for Eq. (12) with k1 ¼75, 100, and 150. Fig. 4(c) shows the effects of the dynamic viscosity on the first resonance with a ¼0.0001, 0.0002, and 0.0003. Fig. 4(c) demonstrates that the amplitude for the first mode decreases with the increasing dynamic viscosity. The effects of the amplitude of the external excitation on the first resonance are shown in Fig. 4(d) with b¼ 0.001, 0.002, and 0.003. The amplitude of the stable steady-state response increases with the variable amplitude of the external excitation. Fig. 4(e) illustrates the effects of the flexural stiffness kf on the first resonance with kf ¼0.6, 0.7, and 0.8. Fig. 4(f) illustrates the effects of the axial speed on the first resonance with g ¼3.5, 4.0, and 4.5. Fig. 4 demonstrates that the different natural frequencies for the linear systems with the changing axial speeds or stiffness coefficients incur quantitatively differences among the stable steady-state responses. In order to prove the convergence and precision of the finite difference method, classical Runge–Kutta method is attached to solve Eqs. (17) and (18). They can be converted into first-order ordinary differential equations after introducing a substitution pk ¼ q_ k (k¼1,2,y,N). Then the dynamic response can be calculated by the fourth stage Runge–Kutta method. Considered the axially moving beam with g ¼4.0, kf ¼0.8, k1 ¼100 and a ¼0.0001, Fig. 5 shows the comparisons between the finite difference method and the fourth stage Runge–Kutta method based on 4-term Galerkin. The value of the external excitation amplitude in Fig. 5(a) and (b) is b¼0.001 and 0.002, respectively. The numerical results demonstrate that the amplitudes of the steady-state responses from both methods almost coincide in the resonance.
5. Comparisons 5.1. Between multi-scale method and FDM Lyapunov linearized stability theory can be employed to obtain the stability conditions of the steady-state responses. Using the same analytical method with the Ref. [35], the stability of the steady-state response can be determined by Eq. (33) on the condition that a1a0. The Jacobian matrix of the right-hand functions of Eq. (33) should be calculated at the fixed point defined by Eq. (35) in the first place. According to the Routh–Hurwitz criterion, the system defined by the foregoing Jacobian matrix is stable as long as all eigenvalues of the matrix are with negative real parts. Lyapunov stability theory guarantees that the stability of a nonlinear system coincides with that of its linearized system. Thus, the instability
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G.-C. Zhang et al. / Journal of Sound and Vibration 331 (2012) 1612–1623
0.002
0.002
4-term Galerkin
k1=75
4-term Galerkin
k1=100
0.000
A1
A1
k1=150 0.001
9.4
9.6
9.8
10.0
10.2
0.001
0.000
10.4
9.4
9.6
9.8
0.002
0.003
4-term Galerkin
=0.0001
10.4
b=0.002
=0.0002
b=0.001
A1
A1
10.2
4-term Galerkin
b=0.003 0.002
0.001
10.0
0.001 =0.0003
0.000
9.6
9.8
0.002
10.0
0.000
10.2
9.0
9.5
10.0
0.002
4-term Galerkin kf=0.8
=4.0
9.0
A1
A1
0.000
kf=0.6
9.5
10.0
11.0
4-term Galerkin
=3.5
kf=0.7 0.001
10.5
0.000
10.5
=4.5
0.001
7
8
9
10
11
12
Fig. 4. Effects of system parameters on the stable frequency–response curve: (a) the stable frequency–response curve; (b) effects of the nonlinear coefficient; (c) effects of the dynamic viscosity; (d) effects of the excitation amplitude; (e) effects of the flexural stiffness; and (f) effects of the axial speed.
b=0.001
0.001
0.0024
FDM Runge-Kutta
4-term Galerkin
A1
A1
0.002
FDM Runge-Kutta
4-term Galerkin
0.0012 b=0.002
0.000
0.0000 9.0
9.5
10.0
10.5
11.0
9.0
9.5
10.0
10.5
Fig. 5. Comparisons between FDM and the fourth stage Runge–Kutta method: (a) b ¼ 0.001 and (b) b ¼ 0.002.
11.0
G.-C. Zhang et al. / Journal of Sound and Vibration 331 (2012) 1612–1623
1621
boundary is directly given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 2 2 2 2 2 2 3k1 p4 2 p4 u t9k1 a4 a ð2kf þ k1 AS Þ s¼ a1 7 4 4o1 4 o21 1 kf
(46)
From Eq. (20), the detuning parameter s quantifies the deviation of o from the fundamental frequency o1, which is the first frequency of the free vibration described by Eq. (12) without the nonlinear terms, the dynamic viscosity term and the external excitation term. By considering the different orders Galerkin truncation, the frequencies o1 of the linear system can be numerically computed for an axially moving beam with g ¼4.0, kf ¼0.8, and k1 ¼100 [17]. By taking N ¼4 as an example, the fundamental frequency o1 ¼9.92. Based on the first-order Galerkin method, Fig. 6 shows the comparisons of the supercritical detuning-response curves via the finite difference method (FDM) and the multi-scale method. The dots represent the stable steady-state responses to Eqs. (42) and (43) via the finite difference method, the solid lines represent the theoretical steady-state responses to Eq. (36) via the multi-scale method, and the dotted lines represent the analytical instability boundaries of the supercritical steady-state response curves from Eq. (46). In Fig. 6(a), kf ¼0.8, a ¼0.0001, k1 ¼100, b¼0.001 and g ¼4.0. In Fig. 6(b), kf ¼0.8, a ¼0.0001, k1 ¼150, b¼ 0.001 and g ¼4.0. In Fig. 6(c), kf ¼0.8, a ¼0.0002, k1 ¼100, b¼0.001 and g ¼4.0. In Fig. 6(d), kf ¼0.8, a ¼0.0001, k1 ¼ 100, b¼0.002 and g ¼4.0. In Fig. 6(e), kf ¼0.7, a ¼0.0001, k1 ¼100, b¼0.001 and g ¼4.0. In Fig. 6(f),
0.004 0.003 0.002 0.001 0.000
-0.8
-0.6
-0.4
-0.2
0.2
0.000 -0.8
0.4
-0.4
-0.2
0.002
0.0
0.2
0.4
Multi-Scale FDM
0.003
0.001
0.002 0.001
-0.8
-0.6
-0.4
-0.2
0.004
0.0
0.2
0.000 -0.8
0.4
-0.6
-0.4
-0.2
0.004
Multi-Scale FDM
0.003 0.002
0.0
0.2
0.4
Multi-Scale FDM
0.003 A1
A1
-0.6
0.004
A1
A1
0.0
Multi-Scale FDM
0.003
0.002 0.001
0.001 0.000
0.002 0.001
0.004
0.000
Multi-Scale FDM
0.003 A1
A1
0.004
Multi-Scale FDM
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.000 -0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
Fig. 6. Comparisons of the analytical and numerical result based on 1-term Galerkin: (a) kf ¼0.8,a ¼ 0.0001, k1 ¼ 100, b¼ 0.001 and g ¼4.0; (b) kf ¼ 0.8, a ¼ 0.0001, k1 ¼150, b ¼0.001 and g ¼ 4.0; (c) kf ¼ 0.8, a ¼0.0002, k1 ¼ 100, b ¼0.001 and g ¼ 4.0; (d) kf ¼ 0.8, a ¼ 0.0001, k1 ¼ 100, b¼ 0.002 and g ¼4.0; (e) kf ¼ 0.7, a ¼ 0.0001, k1 ¼100, b ¼ 0.001 and g ¼4.0; and (f) kf ¼ 0.8, a ¼ 0.0001, k1 ¼ 100, b ¼0.001 and g ¼3.5.
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0.0024 b=0.001
0.0012
1-term Galerkin 2-term Galerkin 4-term Galerkin
b=0.002 0.002 A1
A1
0.003
1-term Galerkin 2-term Galerkin 4-term Galerkin
0.001
0.0000 -0.3
-0.2
-0.1
0.0
0.1
0.2
0.000 -0.3
-0.2
-0.1
0.0
0.1
0.2
Fig. 7. Comparisons among the 1-term, 2-term and 4-term Galerkin based on FDM: (a) b ¼0.001; and (b) b ¼0.002.
kf ¼0.8, a ¼0.0001, k1 ¼100, b ¼0.001 and g ¼3.5. It can be remarked that the stable detuning-response curves and the instability boundaries do not intersect each other in Fig. 6. Fig. 6 also indicates that the theoretical outcomes conform to those calculated by the finite difference method based on the 1-term Galerkin method. Observed from Eq. (46), the instability boundaries have no relation to the amplitude of the external excitation. Fig. 6(a) and (d) help to explain this point exactly. Hence, the instability of the responses may not exist if the excitation amplitude is small enough. In addition, Eq. (46) easily leads to the result that the detuning-response curves will be stable if s 40. It means that jumping phenomena may only happen at the less frequency than the fundamental one under the very small vibration. This information can also be gained from Fig. 6. In fact, there will be no jumping phenomenon if the viscosity coefficient is large enough, which may be deduced from Eqs. (36) and (46). 5.2. Between different truncation terms Considered the axially moving beam with g ¼4.0, kf ¼0.8, k1 ¼100 and a ¼0.0001, Fig. 7 shows the comparisons of the supercritical detuning-response curves among the Galerkin procedures with different orders. The value of the external excitation amplitude in Fig. 7(a) and (b) is b ¼0.001 and 0.002, respectively. The amplitudes of the stable steady-state responses for the axially moving viscoelastic beam with the changing frequency of the external excitation are showed in Fig. 7, where the dashed lines, the dots and the solid lines represent the supercritical steady-state responses from the 1-term Galerkin method, the 2-term Galerkin method and the 4-term Galerkin method, respectively. It is obvious that 1-term Galerkin truncation can yield the convincing primary resonance. However, the results of the 2-term truncation are closer to those of the 4-term truncation than the 1-term Galerkin method. On the whole, the difference of the natural frequencies among the different orders Galerkin method incurs quantitatively differences on the amplitudes of the stable steady-state responses in the supercritical region. 6. Conclusions This paper is devoted to the steady-state periodic response of the forced vibration for an axially moving viscoelastic beam in the supercritical speed range. For this motion, the model is cast in the standard form of continuous gyroscopic systems. The relations between the load frequency and the amplitude of the steady-state response are calculated from an integro-partial-differential equation via the Galerkin method under the simply supported boundary conditions. Based on the 1-term Galerkin truncation, the method of multiple scales is applied to the nonlinear equation to determine steadystate responses. The finite difference schemes are developed to verify the approximate analytical results. The numerical results demonstrate that the steady-state periodic responses exist in the transverse vibration under the periodic transverse loads in the supercritical regime. Both the analytical and the numerical calculations demonstrate that, the forced resonances occur when the frequency of the external excitation is close to the linear natural frequency. A softening-type behavior is illustrated over the supercritical range. The amplitude of the steady-state response increases with the variable amplitude of the external excitation, and decreases with the dynamic viscosity.
Acknowledgments This work was supported by the National Outstanding Young Scientists Fund of China (Project no. 10725209), the National Science Foundation of China (Project nos. 10932006 and 10902064), Shanghai Rising-Star Program (no. 11QA1402300), Innovation Program of Shanghai Municipal Education Commission (no. 12YZ028), and Shanghai Leading Academic Discipline Project (no. S30106).
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