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A computation method for nonlinear vibration of axially accelerating viscoelastic strings
Applied Mathematics and Computation 162 (2005) 305–310 www.elsevier.com/locate/amc
A computation method for nonlinear vibration of axially accelerating viscoelastic strings Li-Qun Chen
a,b,*
, Wei-Jia Zhao
b,c
a
b
Department of Mechanics, Shanghai University, Shanghai 200436, China Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China c Department of Mathematics, Qingdao University, Qingdao 266071, China
Abstract A numerical algorithm is proposed for computing nonlinear vibration of axially accelerating viscoelastic strings. Based on independent functions, the variational principle is used to discretize the governing equation into a set of differential/algebraic equations. Numerical examples are presented. Ó 2004 Elsevier Inc. All rights reserved.
1. Introduction Axially moving strings can represent many engineering devices such as power transmission belts, elevator cables, plastic films, magnetic tapes, paper sheets, textile fibers, band saws, aerial cable tramways, and crane hoist cables. Despite many advantages of these devices, vibrations associated with the devices have limited their applications. Thus understanding vibrations of axially moving strings is important for the design of the devices. There are several comprehensive survey papers reviewing the state-of-the-art in different time phases of investigations related to vibrations of axially moving strings [1,2].
* Corresponding author. Address: Department of Mechanics, Shanghai University, Shanghai 200436, China. E-mail address: [email protected] (L.-Q. Chen).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.12.100
306
L.-Q. Chen, W.-J. Zhao / Appl. Math. Comput. 162 (2005) 305–310
Numerical computation is an effective approach to understand the vibrations of axially moving strings. In recent years, some numerical methods were developed. The Galerkin method was applied [3,4], and the finite difference method was also modified to calculate the vibration of strings moving at a constant axial speed [5]. In present paper, a numerical algorithm is proposed to calculate nonlinear vibration of an axially accelerating string. The variational principle is used to discretize the governing equation into a set of differential/ algebraic equations.
2. The governing equation Taking into account the translation effect by the material derivative and the nonliearity by the Lagrangian strain, one can drive the governing equation, a nonlinear partial differential equation from NewtonÕs second law [6,7] o2 V o2 V oV T o2 V q 2 þ 2qmT þ q_mT þ qm2T ot otox ox A ox2 ! 2 ! o 1 oV oV E ¼ 0; ð1Þ ox 2 ox ox where x denotes spatial Cartesian coordinate, t represents time, mT is the transport speed of the string, q is the volume density and A is the cross-section area. V is the displacement of the belt in the transverse direction, and the differential operator E is defined as E ¼ E0 þ g
o : ot
ð2Þ
The boundary conditions of (1) are given as following homogeneous ones V ðt; 0Þ ¼ V ðt; LÞ ¼ 0: By introducing the following variable transformations 1=2 1=2 V x T A qA m¼ ; 1¼ ; s¼t ; s ¼ r ; c ¼ mT : L L qAL2 T T
ð3Þ
ð4Þ
Eqs. (1) and (2) can be transformed into the non-dimensional forms o2 m o2 m om o2 m 2 _ þ c þ ðc þ 2c 1Þ os2 oso1 o1 o12 2 ! o om o 1 om r ¼ B þ g1 ¼r o1 o1 os 2 o1
ð5Þ
L.-Q. Chen, W.-J. Zhao / Appl. Math. Comput. 162 (2005) 305–310
307
with boundary conditions mðs; 0Þ ¼ mðs; 1Þ ¼ 0:
ð6Þ
3. Discretization Take m and r as unknown functions. Denote H the linear space of continuous differentiable functions in [0,1] with inner product Z 1 ðf ; gÞ ¼ f ð1Þgð1Þ d1: ð7Þ 0
Let f/1 ðrÞ; . . . ; /n ðrÞg be n independent functions in H0 ¼ f/j/ 2 H
and
/ð0Þ ¼ /ð1Þ ¼ 0g
ð8Þ
and let n X
mh ðx; tÞ ¼
ai ðtÞui ðxÞ
ai ðtÞ 2 C 2 ;
i¼1
rh ðx; tÞ ¼
n X
bi ðtÞu0i ðxÞ
bi ðtÞ 2 C 0
ð9Þ
i¼1
be approximating functions of m and r in H0 ½0; 1. Substituting m and r in Eq. (5) by mh and rh , using the variational principle and integrating by parts, one obtains the following differential/algebraic equation system n X
ðui ; uk Þa00i þ 2c
i¼1
¼
n X n Z X i¼1
Z
i¼1
i¼1
i¼1
1
u0i u0j u0k d1aj bi ðk ¼ 1; 2; . . . ; nÞ
0
j¼1
n X n BX 2 i¼1 j¼1
n n n X X X ðu0i ; uk Þa0i þ c_ ðu0i ; uk Þai þ ð1 c2 Þ ðu0i ; u0k Þai
1
u0i u0j u0k d1aj ai 0
þ g1
n X n Z X i¼1
j¼1
1
u0i u0j u0k d1aj a0i ¼
0
n X
ðu0i ; u0k Þbi
i¼1
ð10Þ or in matrix forms M n a00 þ 2cG n a0 þ c_ G n a þ ðc2 1ÞK n a ¼ ðUaÞb B ðUaÞa þ g1 ðUaÞa0 ¼ M n b: 2
ð11Þ
308
L.-Q. Chen, W.-J. Zhao / Appl. Math. Comput. 162 (2005) 305–310
Initial values satisfy n X li ui ð1Þ ¼ gn ð1Þ; i¼1 n X i¼1 n X
gi ui ð1Þ ¼ g1n ð1Þ;
ð12Þ
mi u0i ð1Þ ¼ hn ð1Þ;
i¼0
where gn , g1n and hn are the initial values of m, m0 and r. The coefficient vectors l, g and m that are the initial values of a, a0 and b, can be obtained by the least squares method. It can easily be proved that M n and K n are symmetry and positive definite, so Eq. (11) or (8) is an index 1 differential/algebraic equation system. For a set of given consist initial values, the solution of the equation is exist and unique. Efficient algorithms can be found in the literature [8]. 4. Numerical examples Let 0 ¼ 10 < 11 < 1n ¼ 1 be spatial knots, h ¼ 1l 1l1 ¼ const. Choose Hermite trial functions as the base f/1 ð1Þ; . . . ; /n ð1Þg so that M, G, K, U and a are all 5 diagonal matrixes. Since the problem is non-linear and stiff, but is easy to perform algebraic computations for its banded structure, implicit 4-order Runge–Kutta method [8] is chosen to solve differential/algebraic equation system (11) and the Newton iterative algorithm is adopted in the computation process.
Fig. 1. Vibration response for x ¼ 2p.
L.-Q. Chen, W.-J. Zhao / Appl. Math. Comput. 162 (2005) 305–310
309
Fig. 2. Vibration response for x ¼ 0:2p.
In the following numerical example, the material and geometric data are chosen as E ¼ 3:0 109 N/m2 , g ¼ 1:5 106 sN/m2 , q ¼ 7:68 103 kg/m3 , T =A ¼ 7:5 106 N/m2 and L ¼ 1m. Let the axially velocity be mT ¼ 0:5 þ 0:05 sin xt
ð13Þ
and initial values be mð0; xÞ ¼ 0;
mt ð0; xÞ ¼ 0:02 sinðxxÞ:
ð14Þ
Vibration responses for different excitation frequencies are shown in Figs. 1 and 2.
5. Conclusions This paper present a computation method for nonlinear vibration of axially moving strings. Based on a set of independent functions, the governing equation, an nonlinear partial-differential equation, is cast into a set of differential/algebraic equation, which can be numerically solved by effective algorithms. Numerical examples demonstrate the effectiveness of the procedure proposed.
Acknowledgements The research is supported by the National Natural Science Foundation of China (Project No. 10172056).
310
L.-Q. Chen, W.-J. Zhao / Appl. Math. Comput. 162 (2005) 305–310
References [1] J.A. Wickert, C.D. Mote Jr., Current research on the vibration and stability of axially moving materials, Shock Vib. Dig. 20 (1988) 3–13. [2] L.Q. Chen, Analysis and control of transverse vibrations of axially moving strings. ASME Appl. Mech. Rev., in press. [3] R.F. Fung, J.S. Huang, Y.C. Chen, The transient amplitude of the viscoelastic traveling string: an integral constitutive law, J. Sound Vib. 201 (1997) 153–167. [4] L.Q. Chen, W.J. Zhao, J.W. Zu, Transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law. J. Sound Vib., in press. [5] L.Q. Chen, W.J. Zhao, A numerical method for simulating transverse vibrations of axially moving strings. Appl. Math. Comp., in press. [6] L.Q. Chen, J.W. Zu, J. Wu, X.D. Yang, Transverse vibrations of an axially accelerating viscoelastic string with geometric nonlinearity, J. Eng. Math. 48 (2004) 171–182. [7] L.Q. Chen, J. Wu, J.W. Zu, Asymptotic nonlinear behaviors in transverse vibration of an axially accelerating viscoelastic string. Nonlinear Dyn., in press. [8] K.E. Brenan, S.L. Campbell, L.R. Petzold, Numerical solution of initial-value problems in differential-algebraic equations, North Holland, SIAM, 1996.
A computation method for nonlinear vibration of axially accelerating viscoelastic strings Li-Qun Chen
a,b,*
, Wei-Jia Zhao
b,c
a
b
Department of Mechanics, Shanghai University, Shanghai 200436, China Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China c Department of Mathematics, Qingdao University, Qingdao 266071, China
Abstract A numerical algorithm is proposed for computing nonlinear vibration of axially accelerating viscoelastic strings. Based on independent functions, the variational principle is used to discretize the governing equation into a set of differential/algebraic equations. Numerical examples are presented. Ó 2004 Elsevier Inc. All rights reserved.
1. Introduction Axially moving strings can represent many engineering devices such as power transmission belts, elevator cables, plastic films, magnetic tapes, paper sheets, textile fibers, band saws, aerial cable tramways, and crane hoist cables. Despite many advantages of these devices, vibrations associated with the devices have limited their applications. Thus understanding vibrations of axially moving strings is important for the design of the devices. There are several comprehensive survey papers reviewing the state-of-the-art in different time phases of investigations related to vibrations of axially moving strings [1,2].
* Corresponding author. Address: Department of Mechanics, Shanghai University, Shanghai 200436, China. E-mail address: [email protected] (L.-Q. Chen).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.12.100
306
L.-Q. Chen, W.-J. Zhao / Appl. Math. Comput. 162 (2005) 305–310
Numerical computation is an effective approach to understand the vibrations of axially moving strings. In recent years, some numerical methods were developed. The Galerkin method was applied [3,4], and the finite difference method was also modified to calculate the vibration of strings moving at a constant axial speed [5]. In present paper, a numerical algorithm is proposed to calculate nonlinear vibration of an axially accelerating string. The variational principle is used to discretize the governing equation into a set of differential/ algebraic equations.
2. The governing equation Taking into account the translation effect by the material derivative and the nonliearity by the Lagrangian strain, one can drive the governing equation, a nonlinear partial differential equation from NewtonÕs second law [6,7] o2 V o2 V oV T o2 V q 2 þ 2qmT þ q_mT þ qm2T ot otox ox A ox2 ! 2 ! o 1 oV oV E ¼ 0; ð1Þ ox 2 ox ox where x denotes spatial Cartesian coordinate, t represents time, mT is the transport speed of the string, q is the volume density and A is the cross-section area. V is the displacement of the belt in the transverse direction, and the differential operator E is defined as E ¼ E0 þ g
o : ot
ð2Þ
The boundary conditions of (1) are given as following homogeneous ones V ðt; 0Þ ¼ V ðt; LÞ ¼ 0: By introducing the following variable transformations 1=2 1=2 V x T A qA m¼ ; 1¼ ; s¼t ; s ¼ r ; c ¼ mT : L L qAL2 T T
ð3Þ
ð4Þ
Eqs. (1) and (2) can be transformed into the non-dimensional forms o2 m o2 m om o2 m 2 _ þ c þ ðc þ 2c 1Þ os2 oso1 o1 o12 2 ! o om o 1 om r ¼ B þ g1 ¼r o1 o1 os 2 o1
ð5Þ
L.-Q. Chen, W.-J. Zhao / Appl. Math. Comput. 162 (2005) 305–310
307
with boundary conditions mðs; 0Þ ¼ mðs; 1Þ ¼ 0:
ð6Þ
3. Discretization Take m and r as unknown functions. Denote H the linear space of continuous differentiable functions in [0,1] with inner product Z 1 ðf ; gÞ ¼ f ð1Þgð1Þ d1: ð7Þ 0
Let f/1 ðrÞ; . . . ; /n ðrÞg be n independent functions in H0 ¼ f/j/ 2 H
and
/ð0Þ ¼ /ð1Þ ¼ 0g
ð8Þ
and let n X
mh ðx; tÞ ¼
ai ðtÞui ðxÞ
ai ðtÞ 2 C 2 ;
i¼1
rh ðx; tÞ ¼
n X
bi ðtÞu0i ðxÞ
bi ðtÞ 2 C 0
ð9Þ
i¼1
be approximating functions of m and r in H0 ½0; 1. Substituting m and r in Eq. (5) by mh and rh , using the variational principle and integrating by parts, one obtains the following differential/algebraic equation system n X
ðui ; uk Þa00i þ 2c
i¼1
¼
n X n Z X i¼1
Z
i¼1
i¼1
i¼1
1
u0i u0j u0k d1aj bi ðk ¼ 1; 2; . . . ; nÞ
0
j¼1
n X n BX 2 i¼1 j¼1
n n n X X X ðu0i ; uk Þa0i þ c_ ðu0i ; uk Þai þ ð1 c2 Þ ðu0i ; u0k Þai
1
u0i u0j u0k d1aj ai 0
þ g1
n X n Z X i¼1
j¼1
1
u0i u0j u0k d1aj a0i ¼
0
n X
ðu0i ; u0k Þbi
i¼1
ð10Þ or in matrix forms M n a00 þ 2cG n a0 þ c_ G n a þ ðc2 1ÞK n a ¼ ðUaÞb B ðUaÞa þ g1 ðUaÞa0 ¼ M n b: 2
ð11Þ
308
L.-Q. Chen, W.-J. Zhao / Appl. Math. Comput. 162 (2005) 305–310
Initial values satisfy n X li ui ð1Þ ¼ gn ð1Þ; i¼1 n X i¼1 n X
gi ui ð1Þ ¼ g1n ð1Þ;
ð12Þ
mi u0i ð1Þ ¼ hn ð1Þ;
i¼0
where gn , g1n and hn are the initial values of m, m0 and r. The coefficient vectors l, g and m that are the initial values of a, a0 and b, can be obtained by the least squares method. It can easily be proved that M n and K n are symmetry and positive definite, so Eq. (11) or (8) is an index 1 differential/algebraic equation system. For a set of given consist initial values, the solution of the equation is exist and unique. Efficient algorithms can be found in the literature [8]. 4. Numerical examples Let 0 ¼ 10 < 11 < 1n ¼ 1 be spatial knots, h ¼ 1l 1l1 ¼ const. Choose Hermite trial functions as the base f/1 ð1Þ; . . . ; /n ð1Þg so that M, G, K, U and a are all 5 diagonal matrixes. Since the problem is non-linear and stiff, but is easy to perform algebraic computations for its banded structure, implicit 4-order Runge–Kutta method [8] is chosen to solve differential/algebraic equation system (11) and the Newton iterative algorithm is adopted in the computation process.
Fig. 1. Vibration response for x ¼ 2p.
L.-Q. Chen, W.-J. Zhao / Appl. Math. Comput. 162 (2005) 305–310
309
Fig. 2. Vibration response for x ¼ 0:2p.
In the following numerical example, the material and geometric data are chosen as E ¼ 3:0 109 N/m2 , g ¼ 1:5 106 sN/m2 , q ¼ 7:68 103 kg/m3 , T =A ¼ 7:5 106 N/m2 and L ¼ 1m. Let the axially velocity be mT ¼ 0:5 þ 0:05 sin xt
ð13Þ
and initial values be mð0; xÞ ¼ 0;
mt ð0; xÞ ¼ 0:02 sinðxxÞ:
ð14Þ
Vibration responses for different excitation frequencies are shown in Figs. 1 and 2.
5. Conclusions This paper present a computation method for nonlinear vibration of axially moving strings. Based on a set of independent functions, the governing equation, an nonlinear partial-differential equation, is cast into a set of differential/algebraic equation, which can be numerically solved by effective algorithms. Numerical examples demonstrate the effectiveness of the procedure proposed.
Acknowledgements The research is supported by the National Natural Science Foundation of China (Project No. 10172056).
310
L.-Q. Chen, W.-J. Zhao / Appl. Math. Comput. 162 (2005) 305–310
References [1] J.A. Wickert, C.D. Mote Jr., Current research on the vibration and stability of axially moving materials, Shock Vib. Dig. 20 (1988) 3–13. [2] L.Q. Chen, Analysis and control of transverse vibrations of axially moving strings. ASME Appl. Mech. Rev., in press. [3] R.F. Fung, J.S. Huang, Y.C. Chen, The transient amplitude of the viscoelastic traveling string: an integral constitutive law, J. Sound Vib. 201 (1997) 153–167. [4] L.Q. Chen, W.J. Zhao, J.W. Zu, Transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law. J. Sound Vib., in press. [5] L.Q. Chen, W.J. Zhao, A numerical method for simulating transverse vibrations of axially moving strings. Appl. Math. Comp., in press. [6] L.Q. Chen, J.W. Zu, J. Wu, X.D. Yang, Transverse vibrations of an axially accelerating viscoelastic string with geometric nonlinearity, J. Eng. Math. 48 (2004) 171–182. [7] L.Q. Chen, J. Wu, J.W. Zu, Asymptotic nonlinear behaviors in transverse vibration of an axially accelerating viscoelastic string. Nonlinear Dyn., in press. [8] K.E. Brenan, S.L. Campbell, L.R. Petzold, Numerical solution of initial-value problems in differential-algebraic equations, North Holland, SIAM, 1996.