J. Cent. South Univ. Technol. (2008) 15(s1): 192-196

DOI: 10.1007/s11771-008-344-9

Non-linear dynamics analysis of in-plane motion for suspended cable under concentrated load

CHEN Sheng-ming(陈胜铭), CHEN Zi-li(陈自力), LUO Ying-she(罗迎社)

(Institute of Rheological Mechanics and Material Engineering, Central South University of Forestry and Technology, Changsha 410004, China)

Abstract: The non-linear equations of strings under a concentrated load were derived. The formulae of the linear frequency and the governing equation of the primary resonance were obtained by Galerkin and Multiple-dimensioned method. The reason of the loss of load in practical engineering was addressed. The bifurcation graphics and the relationship graphics of bifurcate point with concentrated load and the span length of the cable were obtained by calculating example. The results show that formula of the linear frequency of the suspended cable is different from that of the string.

Key words: Galerkin method; multiple-dimensioned method; nonlinear vibration; natural frequency; vibration; bifurcation condition

1 Introduction

Cable, as a fabric of flexibility, lightness, and high strength, has been widely used in long-span structures, such as cable-stayed bridges, guyed towers, cableways, and cable-cranes. Due to the non-linear feature of the material, large deformation and sagging, the dynamics study of the cable is very complicated and is still remaining as a key research in mathematics, mechanics, and engineering sector. YAMAGUCHI et al^[1] studied the time response of a cable under harmonic excitation  and concluded that the dynamics nature of cable was greatly affected by the geometric and physical parameters. The effects of non-linearity on planar/non-planar dynamic motion of a sagged cable were investigated by HAGEDORN et al^[2]. RAO et al^[3] studied the internal resonance and nonlinear response under periodic excitation. PERKINS^[4] studied nonlinear response of the multi-model interactions under parametric and forced excitation. BENEDETHNI et al^[5] conducted a research in the nonlinear oscillation of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions. The dynamic stability problem of a flag sagged cable subjected to an axial periodic load was investigated by TAKAHASHI^[6]. Many researches^[7-10] on transverse motions of parametrically excited moving strings were concentrated on equilibrium or periodic vibrations and their stabilities.

In Ref.[11], the non-linear equations of the cable under a moving concentrated load and self-weight were derived based on the nonlinear equations of the string. It was indicated that the natural vibration frequency of the cable is different from that of the string because of the concentrated load, and the natural vibration frequency has a close connection with the position of the concentrated load. The reason of cabin in practical engineering was explained. By using Galerkin and Multiscale method, the formula of the linear frequency and the governing equations of the primary resonance were obtained. The results show that the formula of the linear frequency of the suspended cable is different from that of the string. By using the numerical calculation, the bifurcation graphics and the relationship graphics of bifurcate point with concentrated load and the span length of the cable were obtained.

2 Fundamental equation

The ends of an infinitesimal length of string were labeled by F and G in the un-deformed position and F′ and G′ in the deformed position as shown in Fig.1.

The displacements of F is given by

                (1)

and the displacement of G is given by

     (2)

where  u, v and w are the displacements of the point in

Fig.1 Element of string in deformed and undeformed position

the directions of x, y and z, respectively; i, j and k are the unit vectors in the directions of x, y and z, respectively.

The length of the deformed segment is given by

           (3)

and the unit vector parallel to the deformed segment is given by

         (4)

The string has a tension N₀ initially. During the motion, the length of the string changes and hence the tension is given by

                       (5)

The momentum equation for the string is given by^[10]

                 (6)

where E is the elastic modulus; ρ is the density; A is the cross-sectional area of the string in stationary state.

Using Eqns.(4) and (5), expanding for small u, v, and w, and keeping up to cubic terms, the following equations can be obtained.

     (7)

(8)

(9)

where c₁ and c₂ stand for the velocities of the longitudinal and transverse elastic waves, respectively, which are defined by

                            (10)

Considering parametric excitations, the boundary conditions of u are expressed in the following forms:

                          (11)

Using these boundary conditions, and considering only the transverse vibration and neglecting the non-planar vibration, the longitudinal inertia u_ff can be neglected and also c₁＞＞ c₂ can be under assumption, the following equation is obtained.

            (12)

Adding linear damping forces and external excitations, Eqn.(12) can be rewritten as

(13)

where  μ is the damping coefficient and g(x, t) is the function of external excitation.

If there is a motivation F on the concentrated load point, the function of external excitation has the form:

    (14)

where  M is the mass of the concentrated load; Q_y is unit length mass of string in x direction; δ is the Derrick function; F is the excited function; x_d and v_dtt are the coordinate and acceleration of the action point of the concentrated load respectively.

Substituting Eqn.(14) into Eqn.(13), there is

    (15)

Eqn.(15) is the governing equation of the string under a moving concentrated load.

3 Solution of Galerkin method

The solution of Eqn.(15) is solved by Galerkin method as follows. Let

                      (16)

where  ε^1/2 is a small dimensionless quantity in the order of the amplitudes of the motion. Substituting Eqn.(16) into Eqn.(15) and using the orthogonality of the linear modes, there is

    　      (17)

where

,      (18)

        (19)

Substituting Eqn.(19) into (17), there is

      (20)

From Eqn.(20), the linear frequency of the system can be obtained:

                 (21)

From Eqn.(21), it is noted that the coordinate of the action point of the concentrated load is relevant to the linear frequency. The least of every mode of the linear frequency is

　　    　　　　　　　　 (22)

and the action point relevant to the concentrated load is

              (23)

From Eqn.(21), the least of the first mode of linear frequency presents if x_d = 1/2, and the least of the second mode of the linear frequency presents at the points of x_d = 1/4 and x_d = 3l/4. The appearance point of the least of the every mode of the linear frequency can be obtained through Eqn.(23).

4 Primary resonance solution with ulti-scale method

If there is an excitation at the action point of the concentrated load and the excitation frequency of the load approaches to that of the primary resonance, the forced vibration of the system appears. If the string solids are at the ends, p=0. By using Multi-scale method, there is

　  (24)

where  Ω= w_0s+εσ                           (25)

For fixed s, Eqn.(20) can be expressed as

          (26)

where

，

(27)

Writing the solution of Eqn.(25) as

             (28)

where  T₀ = t, T₁ = εt.

Substituting Eqn.(28) into (26), so

                   (29)

where　D₀=?/?T₀, D₁=?/?T₁.

Equating coefficients of like powers ε,

Order ε⁰

                         (30)

Order ε¹

           (31)

The solution of Eqn.(30) can be written in the form

                  (32)

where  c is the conjugate function of all of the items of the left. Substituting Eqn.(32) into Eqn.(31). Eqn.(31) can be rewritten as

(33)

From Eqn.(33), it is found the secular terms are eliminated from the ζ_n1 if

       (34)

Letting

                          (35)

Substituting Eqn.(35) into Eqn.(34), and separating the results into real and imaginary parts, so

           (36)

    (37)

where

,(38)

For steady-state motion a_n and v are constant, Letting , rewriting Eqns.(36), (37) as

               (39)

                  (40)

where  a_n and C are the functions of ε.

From Eqns.(39) and (40), if n≠s a_n=0. Therefore, for all of a_n≠0, n=s.

Using Eqn.(39), v_n can be obtained. For undamped mode, μ_n=0, so

                                  (41)

Letting b_s=ε^1/2a_s, ?_sw0 = εσ = Ω-w_0s          (42)

Eqn.(40) is changed as

                    (43)

where

              (44)

b_s and ?_sw0 are parameters independent to ε.

The graphic of b_s0—?_sw0 can be obtained by solving Eqn.(43) numerically.

5 Bifurcation analysis

Rewriting Eqn.(43) as

                  (45)

Deriving both side of Eqn.(45) with ?_sw0, there is

            (46)

For the sth mode bifurcation point,,

             (47)

If (,), b_s=0. This

illustrates that the excitation at the action point of the concentrated load has no influence on these points. b_s0 expresses numeric value of b_s of mode bifurcation point.

Substituting Eqns.(44) and (47) into Eqn.(45), there is

              (48)

where {b_s0, ?_sw0} is the bifurcation point. Its place is different for different s and ?_sw0.

6 Numerical example

A concentrated load worked on a fixed end cable with a length of 100 m, a diameter of 10 mm, and E = 210 GPa. Deriving from Eqn.(22), the following conclusions can be obtained, and show in Table 1 and Figs.2-4.

Table 1 Frequency range of different mode and coordinate of bifurcation point

Fig.2 First mode bifurcation graphics

Fig.3 Relationship between coordinate of bifurcation point ?_1w0 and F₀

Fig.4 Relationship between initial tension T₀ and coordinate of bifurcation point ?_1w0

7 Conclusions

1) The non-linear equations of cables under a moving concentrated load are derived.

2) By using Galerkin and Multiple-dimensioned method, the formulas of the linear frequency and the governing equations of the primary resonance are obtained, the results show that the formula of the linear frequency of the suspended cable is different from that of the string.

3) By using the numerical calculation, the bifurcation graphics and the relationship graphics of bifurcation point with concentrated load and the span length of the cable are obtained.

References

[1] HAGEDORN P, SCHAFER B. On nonlinear free vibrations of an elastic cable [J]. International Journal of Non-linear Mechanicals, 1980, 15(3): 333-340.

[2] RAO G V, IYENGAR R. Internal resonance and non-linear response of a cable under periodic excitation [J]. Journal of Sound and Vibration, 1991, 149(3): 25-41.

[3] PERKINS N C. Modal internactions in the nonlinear response of elastic cable under parametric/external excitation [J]. International Journal on Non-linear Mechanics, 1992, 27(2): 233-250.

[4] BENEDETHNI F, REGA G, ALAGGIO R. Non-linear oscillation of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions [J]. Journal of Sound and Vibration, 1995, 182(4): 775-798.

[5] TAKAHASHI K. Dynamic stability of cables subjected to an axial periodic load [J]. Journal of Sound and Vibration, 1991, 144(2): 323-330.

[6] PAKDEMIRLI M, UISOY A G. Stability analysis of an axially accelerating string [J]. Journal of Sound and Vibration, 1997, 203(2): 815-832.

[7] ZHANG L, ZU J W. Nonlinear vibration of parametrically excited moving belts, part 1: Dynamic response [J]. Journal of Applied Mechanics, 1999, 66(2): 396-402.

[8] ZHANG L, ZU J W. Nonlinear vibration of parametrically excited moving belts, part 2: Stability analysis [J]. Journal of Applied Mechanics, 1999, 66(4): 403-409.

[9] ZHAO W J, CHEN L Q. A numerical algorithm for non-linear parametric vibration analysis of a visco-elastic moving belt [J]. International Journal of Science and Numerical Simulations, 2002, 3(2): 129-134.

[10] NAYFEH A H, MOOK D T. Nonlinear oscillations [M]. New York: Wiley Inter-Science Publication, 1979.

(Edited by YANG Hua)

Foundation item: Project(10672191) supported by the National Natural Science Foundation of China; Project(06y028) supported by Central South University of Forestry and Technology; Project(2008050B) supported by the Scientific Research Fund of Central South University of Forestry and Technology

Received date: 2008-06-25; Accepted date: 2008-08-05

Corresponding author: CHEN Zi-li, PhD, Professor; Tel: +86-731-5623376; E-mail: czl0843@163.com