In-Plane Free Vibrations of an Inclined Taut Cable

In-Plane Free Vibrations of an Inclined Taut Cable Xiaodong Zhou e-mail: [email protected]

Shaoze Yan e-mail: [email protected]

Fulei Chu1 e-mail: [email protected] Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, China

1

The investigation of the free vibrations of inclined taut cables has been a significant subject due to their wide applications in various engineering fields. For this subject, accurate analytical expression for the natural modes and the natural frequencies is of great importance. In this paper, the free vibration of an inclined taut cable is further investigated by accounting for the factor of the weight component parallel to the cable chord. Two coupled linear differential equations describing two-dimensional in-plane motion of the cable are derived based on Newton’s law. By variable substitution, the equation of the transverse motion becomes a Bessel equation of zero order when the equation of longitudinal motion is ignored. Solving the Bessel equation with the given boundary conditions, a set of explicit formulae is presented, which is more accurate for determining the natural frequencies and the modal shapes of an inclined taut cable. The accuracy of the proposed formulae is validated by numerical results obtained by the Galerkin method. The influences of two characteristic parameters ␭ and ␧ on the natural frequencies and modal shapes of an inclined taut cable are studied. The results are discussed and compared with those of other literatures. It appears that the present theory has an advantage over others in the aspect of accuracy, and may be used as a base for the correct analysis of linear and nonlinear dynamics of cable structures. 关DOI: 10.1115/1.4003397兴

Introduction

Cables are promising structural elements in engineering applications because of their high flexibility, light weight, and high packaging efficiency. They are widely used in many mechanical systems and civil structures. A thorough and exact analysis of cable dynamics is of great importance for the design of cable structures, among which the derivation of accurate analytical expressions for the natural modes and the natural frequencies of taut cables is particularly important. The main advantage of having these expressions is that the influence of the cable principal parameters can be studied clearly. Other advantages include that the correct calculation of natural frequencies and modal shapes can be obtained, and the expressions can be used as the bases for nonlinear dynamics analysis. The linear theory on free vibrations of suspended taut cables supported at both ends has been developed for a long time. Through different methods, such as Newton’s laws, Hamilton’s principle, or the finite element method, various kinds of cable motion equations had been built and then linearized based on small amplitude assumption. These works could well demonstrate the vibration characteristics of a suspended small-sag cable with two supports located at the same level. However, for the case of an inclined cable, namely, one support is some distance below the other, most of the research tended to ignore the weight component parallel to the cable chord to simplify the analysis, which often led to inaccuracy in the definition of static profiles of suspended inclined taut cables and in the derivation process of governing differential equations. Early linear cable theories with the assumption of inextensibility failed to reconcile with the taut string theory. Later, Irvine and Caughey 关1兴 presented their work on transverse vibrations of a suspended extensible cable with supports located at the same level. It was shown that the symmetric in-plane modes were heavily dependent on an elastogeometry parameter. The frequencies of the symmetric in-plane modes varied with this parameter, 1 Corresponding author. Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received February 26, 2010; final manuscript received September 27, 2010; published online March 24, 2011. Assoc. Editor: Weidong Zhu.

Journal of Vibration and Acoustics

which led to the existence of the cross-over points when the frequencies of the symmetric in-plane modes were equal to those of the neighbor antisymmetric modes. Irvine 关2兴 extended the theory of the horizontal cable to cover the case of an inclined cable simply by a coordinate transformation and therefore neglected the weight component parallel to the cable chord. His new theory was a modified version of the horizontal cable results and gave the same natural vibration properties as those of a horizontal cable. Henghold et al. 关3兴 made an extensive analysis of the threedimensional free vibration of a single span cable by employing a nonlinear finite element technique. They presented an empirical formula that gave an approximation to the lowest natural frequency only while other results had to be analyzed from plots and data tables. Perkins and Mote 关4兴 derived three nonlinear equations of the cable dynamics based on Hamilton’s principle for traveling cables with arbitrary initial sag, and the governing equations were linearized and discretized using the Galerkin method. The natural frequencies and modal shapes were obtained but the numerical calculation of infinite series had to be implemented. The frequencies and modes of the suspended cable were also calculated by Srinil et al. 关5兴 using the numerical method. Considering the coupling between tangential and normal displacements, Triantafyllou 关6兴 and Triantafyllou and Grinfogel 关7兴 derived a set of more general linearized equations of inclined taut cables, and the final asymptotic closed-form solutions were obtained as the sum of the slow and fast solutions by recognizing two distinct physical mechanisms of the cable vibration. It was shown that the natural frequencies did not cross-over due to the inclination, and the modes could become hybrid modes, a mixture of symmetric and antisymmetric shapes, with a significant effect on the dynamic tension. They stated further that this phenomenon was accompanied by the curvature variation along the catenary static profile of inclined taut cable, which would not happen when the parabola assumption is adopted. Shih and Tadjbakhsh 关8兴 used the Galerkin procedure with Fourier series to approximate the dynamic displacements of incompressible cable under free vibration. AlQassab and Nair 关9兴 used the wavelet-Galerkin method to study the free vibration of a shallow and slack catenary cable suspended horizontally. Wang et al. 关10兴 investigated the multimode dynamics of a shallow suspended cable using the kinematically con-

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densed model. Rega and Srinil 关11兴 investigated the forced oscillations of sagged inclined cables at avoidance regions in the associated frequency spectra. Presenting a modification of the expressions for the in-plane natural frequencies of an inclined cable developed by Irvine, Wu et al. 关12兴 derived an explicit formula to demonstrate the “avoid crossover” phenomenon that Triantafyllou mentioned. In some special applications, such as towed cable systems, oceanographic cable structures, and elevators, the displacements of cables are large and the boundary conditions are various, and some more complex modeling methods were established to meet these practical needs. Introducing flexural stiffness, Sun and Leonard 关13兴 derived a general set of three-dimensional dynamic field equations for a cable segment based on the classical Euler– Kirchhoff theory, which could remove the potential singularity when the cable tension vanished. Gobat et al. 关14,15兴 proposed a new time integration procedure based on the generalized-␣ method, which exhibited superior stability properties compared with other algorithms. They used this method to study the dynamics of a hanging chain and the ocean cable structures numerically. Goyal et al. 关16,17兴 provided a general dynamical formulation and numerical solution procedure to investigate the three-dimensional nonlinear dynamics of Kirchhoff rods, and extended this theory to the analysis of dynamic formation of deoxyribose nucleic acid 共DNA兲 loops and the loops in marine cables. Considering nonhomogeneous and discontinuous changes in stiffness, Goyal and Perkins 关18兴 developed a computational rod model to analyze the DNA looping. Stump et al. 关19兴 also used the large deflection theory of elastic rods with circular cross section to analyze the writhing of undersea cables and closed loops. Zhu and Chen 关20兴 analyzed the vibration and control of elevator cables. Using the method of characteristic transformation, Zhu and Zheng 关21兴 obtained the exact response of a translating string with arbitrarily varying length under general excitation. This paper can be considered as an extension of the work of Irvine 关2兴 and Wu et al. 关12兴 to the general case of inclined taut cables. In this study, an improved cable theory and the in-plane free vibration analysis for a freely hanging inclined taut cable are presented. This paper is organized as follows. In Sec. 2, a set of two coupled linear differential equations describing twodimensional in-plane motion of a supported inclined cable are presented, in which the weight component parallel to the cable chord is taken into account. The terms are composed of chordwise cable self-weight, which can be denoted by a small parameter bringing some new additional vibration properties of taut cables. In Sec. 3, by variable substitution, the equation of vertical motion is transformed to a Bessel equation of zero order and then the explicit analytical solution of nondimensional natural frequencies and in-plane modes are obtained. In Secs. 4 and 5, the two coupled differential equations obtained in Sec. 2 are solved using the numerical procedure of the Galerkin method. The cable natural frequencies and modal shapes for in-plane motions, which are determined from the eigensolutions of the discretized equations, are compared with the analytical results. In Sec. 6, vibration analysis of an inclined taut cable is carried out and some new characteristics are discussed. The results and some concluding remarks are summarized in Sec. 7.

2 Equilibrium Equations and Dynamic Motions of a Supported Inclined Cable Figure 1 shows an elastic cable suspended between point A and point B that is some distance below point A. The axis y is perpendicular to the chord line AB, so that x measures the distance along the chord from A and y measures the distance to the profile from the chord and perpendicular to it. The dynamic behavior of an extensible inclined taut cable can be simplified when the solution is assumed to consist of a static part and small oscillations around 031001-2 / Vol. 133, JUNE 2011

Fig. 1 Definition diagram for cable static profile and vibrations: „a… coordinate definition of an inclined cable and „b… equilibrium of a differential cable element

this mean position. In the coordinate system Axy, considering the chordwise component of the self-weight, the static profile of the inclined cable can be written in the form as 关2兴 y=

冉冊冉冊

mgl cos ␣ 1 mgl x 1− x− 12 H 2H l

冉冊冉冊

2

sin 2␣ 1 −

2x l

1−

x x l 共1兲

+¯

where m is the cable mass per unit length in static equilibrium, g is the gravity constant, l is the span of the cable between two supports, ␣ is the inclination angle, and H is the chordwise component of static cable tension T. The symbols “…” stands for small value terms containing the product of high power of mgl / H. By making Eq. 共1兲 nondimensional, the following equation is obtained as:

再

冎

1 ␧ ¯y = ¯x共1 − ¯x兲 1 − 共1 − 2x ¯ 兲 + O共␧2兲 2 3

共2兲

where ¯y = y / 共mgl2 cos ␣ / H兲, ¯x = x / l, and ␧ = mgl sin ␣ / H is a small parameter since the static profile of the taut cable is close to the chord, and the term mgl must be a small fraction of H. In order to investigate the dynamic behaviors of a cable, a small arbitrary disturbance from its static equilibrium position is assumed for the cable, as shown in Fig. 1. It is known from literature 关1兴 that the transverse horizontal motion is uncoupled from the in-plane motion so that the dynamic governing equations are derived for a two-dimensional cable, in which only the two inplane displacements, the longitudinal displacement u in the x direction and the normal displacement v in the y direction, are taken into account. From Newton’s law, the governing equations of the in-plane motion can be obtained from Fig. 1共b兲 as

冦

冋冉冋冉

⳵ 共T + ␶兲 ⳵s ⳵ 共T + ␶兲 ⳵s

dx ⳵ u + ds ⳵ s dy ⳵ v + ds ⳵ s

冊册冊册

⳵ 2u =0 ⳵ t2 ⳵ 2v + mg cos ␣ − m 2 = 0 ⳵t + mg sin ␣ − m

冧

共3兲

where ␶共s , t兲 is the additional cable tension and s is the Lagrangian coordinate along the arc length in static equilibrium. By accounting for the weight component parallel to the cable chord, the chordwise component of static cable tension H will not be constant. It can be expressed as Transactions of the ASME


H共x兲 = HA − xmg sin ␣

共4兲

where HA is the chordwise component of the static cable tension at the support A. In the following part, the symbol HA in Eq. 共4兲 will be substituted by H for the simplicity of expression and the effect on the accuracy of Eqs. 共1兲 and 共2兲 caused by this substitution is very small. Substituting Eq. 共4兲 into Eq. 共3兲, using H = Tdx / ds and h = ␶dx / ds, neglecting terms of second and higher orders, and removing the static equilibrium terms, Eq. 共3兲 could be written as

⳵u ⳵h ⳵u ⳵u 共H − xmg sin ␣兲 2 − mg sin ␣ + −m 2 =0 ⳵x ⳵x ⳵t ⳵x 2

2

共H − xmg sin ␣兲

共5兲

冉冊

共7兲

where E is the modulus of elasticity of the cable and A is the section area of the cable. By regarding h as a function of time alone, Eq. 共7兲 may be integrated with the boundary conditions to give

冋

冦

¯h 共␳ + ␧兲sin ¯v共x ¯兲 = 2 ¯ ␻

册

= ˜h

冉冊

冋冉冊

冉冊册冉冊

¯␤ ¯␤ 2␻ 15 ␻ ¯ + ␧2共␤ + 2兲tan + 2␧2␻ 8 ␧ ␧ 共15兲

Equations 共14兲 and 共15兲 are the formulae for the in-plane modal shapes and natural circular frequencies of an inclined taut cable. As mentioned above, the mechanism through which the chordwise Journal of Vibration and Acoustics

共9兲

h共t兲 = ˜hei␻t

共10兲

d˜v d2˜v + m␻2˜v 2 − mg sin ␣ dx dx

冉

mg cos ␣ mgl sin ␣ 2mg sin ␣ 1− + x H H H

冊

共11兲

The dimensionless forms of Eqs. 共8兲 and 共11兲 could be obtained as ¯兲共1 − ␧x

d¯v d2¯v ¯ 2¯v = ¯h共1 − ␧ + 2␧x ¯兲 −␧ +␻ 2 ¯ ¯ dx dx ¯h = − ␭2

冕

1

0

d2¯y ¯vdx ¯ ¯2 dx

共12兲共13兲

¯ = ␻l / 共H / m兲1/2, ¯h = ˜h / H, and ␭2 where ¯v = ˜v / 共mgl2 cos ␣ / H兲, ␻ 2 = 共mgl cos ␣ / H兲 l / 共HLe / EA兲. The two independent parameters ␭ and ␧ are of fundamental importance in the dynamic response of inclined taut cables. It is seen that ␭ involves cable geometry and elasticity while only the geometric effect is included in ␧. The parameter ␭ is the same elastogeometry parameter that was used as the single parameter to calculate in-plane natural frequencies and modal shapes by Irvine 关2兴, and the parameter ␧ has the same meaning with the identical parameter used by Wu et al. 关12兴. Equation 共12兲, accompanied by Eq. 共13兲, is the linear equation through which the steady-state system response can be investigated. The solution of Eq. 共12兲, with the boundary conditions ¯v共0兲 = ¯v共1兲 = 0, can be obtained as 共The process is provided in Appendix A.兲

冉冊

¯5 ¯␤ ¯␤ ␧2 3 ␻ ␻ ␻ ¯ +␻ ¯ 2共␤ + 2兲 tan = 1 + ␻ − ␧2 cot 2 3 ␭ ␧ ␧ ¯ 2 cot − 2␧␤␻

v共x,t兲 = ˜v共s兲ei␻t

冋

册

¯ ¯ 2␻ 2␻ ¯ 兲 − 共␳ − ␧兲sin ¯兲共1 − 冑1 − ␧x 共冑1 − ␧ − 冑1 − ␧x ␧ ␧ ¯兲 + 共␳ − ␧ + 2␧x ¯␤ 2␻ sin ␧

where ␳ and ␤ are defined in Appendix A. Equations 共13兲 and 共14兲 are used together to eliminate ¯h and to obtain the following equation:

冉冊

共8兲

where i = −1 and ␻ is the natural circular frequency of vibration. Substituting Eqs. 共1兲 and 共8兲–共10兲 into Eq. 共6兲 leads to the ordinary differential equations as

Analytical Solution for the Steady-State Response

h共dx/ds兲3 ⳵ u dy ⳵ v = + EA ⳵ x dx ⳵ x

0

d2y vdx dx2

2

共H − xmg sin ␣兲

For a suspended taut cable, the ratio of sag to span is small and the amplitude of the longitudinal motion is substantially smaller than that of the normal motion, hence, Eq. 共5兲 can be neglected in the derivation of analytical solution for simplicity. To solve the solution of Eq. 共6兲, h has to be evaluated. A linearized cable equation, which provides for the compatibility between the changes in cable tension and the changes in cable geometry when the cable is displaced from its original equilibrium profile, can be written as 关1兴

l

and

共6兲

3

冕

where Le ⯝ l关1 + 共mgl cos ␪ / H兲2 / 8兴. It is assumed that the solutions have the forms as

⳵v ⳵ ⳵ 2v ⳵ 2v dy + h −m 2 =0 2 − mg sin ␣ ⳵ x ⳵ x dx ⳵t ⳵x

Equations 共5兲 and 共6兲 are the governing equations of the inplane motion for a supported inclined taut cable. When ␣ is equal to zero, they will be reduced to Irvine’s equation of motion for the horizontal cable 关1兴, and if the weight component parallel to the cable chord is neglected, Wu’s equations 关12兴 can be recovered from Eqs. 共5兲 and 共6兲.

EA Le

h=−

冧

共14兲

weight component influences the free vibration frequencies and modal shapes of the cable can be divided into two aspects: the changing of the static profile and the changing of the chordwise component of static cable tension H. In order to further investigate these two aspects of the mechanism, on one hand, the static profile defined by dimensionless Eq. 共2兲 is approximated to a parabola by neglecting high order terms, and Eqs. 共14兲 and 共15兲 can then be simplified as

再

冋

册冋

¯ ¯ ¯2 2␻ 2␻ ␻ ¯v共x ¯ 兲 = 共1 − ␧/4兲sin ¯ 兲 − sin 共1 − 冑1 − ␧x 共冑1 − ␧ ¯h ␧ ␧ ¯兲 − 冑1 − ␧x

册冎冉冊 /sin

¯␤ 2␻ +1 ␧

共16兲

JUNE 2011, Vol. 133 / 031001-3


Table 1 Comparisons of the first five natural circular frequencies of an inclined taut cable between present analytical solution and Galerkin method „␣ = 10 deg… Freq. No. 共rad/s兲

H = 81,370N

H = 73,200N

H = 68,350N

H = 45,000N

Analytical solution

Galerkin method

Analytical solution

Galerkin method

Analytical solution

Galerkin method

Analytical solution

Galerkin method

2.302 2.792 3.759 4.605 5.794

2.296 2.793 3.755 4.603 5.795

2.181 2.836 3.800 4.363 5.511

2.174 2.838 3.795 4.362 5.513

2.106 2.819 3.890 4.213 5.344

2.098 2.822 3.885 4.212 5.346

1.698 2.403 3.397 4.099 5.095

1.685 2.408 3.397 4.114 5.106

␻1 ␻2 ␻3 ␻4 ␻5

冉冊

冉冊

¯␤ ¯3 ¯␤ 2␻ ␧␤ ␧2 ␻ ␻ ¯ + − cot 2 = ␻ + 共␤ + 2兲tan ␭ ␧ 4 ␧ ¯ 8␻

兵P其 = 关P11, P12, . . . , PN1 , P21, P22, . . . , PN2 兴T

共17兲

关M兴 =

on the other hand, if the terms containing parameter ␧ at left side of Eq. 共12兲 are ignored, Wu’s results 关12兴 can be obtained as

冋

¯2 ␻ ¯v共x ¯ 兲 = 共1 − ␧ + 2␧x ¯ 兲 − 共1 − ␧兲cos ␻ ¯ ¯x ¯h

冉冉冊

− tan

tan

4

冊册

关K兴 =

¯ ¯ ␻ ␻ ¯ ¯x + ␧ cot sin ␻ 2 2

共18兲

冉冊

¯ ¯ ¯ 4 ␻ ␻ ␧2 ␧2 ␻ ␧2 + − 2 = 1+ − 3 2 ␻ 2 ¯ /2 tan共␻ ¯ /2兲 ␭ 2

3

共19兲

The coupled Eqs. 共5兲 and 共6兲 can also be solved more accurately by using the Galerkin method to obtain the solutions of both longitudinal motion and normal motion. In the procedure of Galerkin method, basis functions satisfying the boundary conditions are used. To this end, the in-plane displacements u and v are assumed in the N-term separable series of the form N

u共x,t兲 =

兺P

1

n共t兲Qn共x兲

共20兲

2

n共t兲Qn共x兲

共21兲

n=1 N

v共x,t兲 =

兺P n=1

where the P1n共t兲 and P2n共t兲 are generalized coordinates, and Qn共x兲 = sin共n␲x / L兲 guarantees the satisfaction of the boundary conditions. Substitution of Eqs. 共20兲 and 共21兲 into Eqs. 共5兲 and 共6兲 and the application of the Galerkin method provide a set of 2N coupled ordinary differential equations as 关M兴兵P¨ 其 + 关K兴兵P其 = 0

共22兲

where

册册

M11

0

0

M22

K11 K12 K21 K22

共24兲共25兲

The overdot indicates the derivative with respect to time t. The coefficients of the submatrices in 关M兴 and 关K兴 are provided in Appendix B. The in-plane natural frequencies can be obtained numerically by solving the eigenvalue problem of Eq. 共22兲, and the modal shapes are obtained from the eigenvectors through Eqs. 共20兲 and 共21兲.

5

Solution Using Galerkin Procedure

冋冋

共23兲

Validation of the Present Analytical Expressions

In order to verify the applicability and performance of the analytical expressions, comparisons of the analytical solution with the solution obtained by using the Galerkin method are made. Unless stated otherwise, the cable under investigation has the following material and physical parameters: m = 5.46 kg/ m, E = 200 GPa, l = 330 m, A = 6.427⫻ 10−4 m2. The values of the two parameters ␭ and ␧ will change with the inclination angle ␣ and the chordwise static cable tension H. In the actual construction process, the cable is placed in supports, prestressed to a given tension and then anchored off, therefore, in the following analysis, the inclination angle is fixed at first and the chordwise static cable tension is changed gradually. The lowest five modal frequencies of the cable with the inclination angle ␣ = 10 deg and ␣ = 60 deg are calculated, respectively, by Eq. 共15兲 and the Galerkin method with the number of retained terms N = 10. The results are listed in Tables 1 and 2. It is clear that the numerical and analytical results are in good agreement.

6

Results Analysis

After testing the proposed frequency formula, vibration analysis of an inclined taut cable is carried out and discussed as follows. It can be seen from Eq. 共15兲 that the frequencies of the in-plane free

Table 2 Comparisons of the first five natural circular frequencies of an inclined taut cable between present analytical solution and Galerkin method „␣ = 60 deg… Freq. No. 共rad/s兲

␻1 ␻2 ␻3 ␻4 ␻5

H = 81,370N

H = 73,200N

H = 68,350N

H = 45,000N

Analytical solution

Galerkin method

Analytical solution

Galerkin method

Analytical solution

Galerkin method

Analytical solution

Galerkin method

1.792 2.216 3.357 4.419 5.532

1.792 2.214 3.356 4.419 5.533

1.863 2.099 3.185 4.166 5.218

1.863 2.098 3.185 4.166 5.219

1.899 2.047 3.085 4.008 5.022

1.898 2.046 3.084 4.008 5.024

1.550 2.137 2.963 3.168 3.984

1.547 2.140 2.962 3.168 3.988

031001-4 / Vol. 133, JUNE 2011

Transactions of the ASME


Fig. 2 The first four nondimensional frequency ratios versus ␭ / ␲ for different values of ε

vibration are dependent on two parameters ␭ and ␧, both of which allow for the material and physical parameters of a cable, as well as for the inclination angle ␣ and the chordwise static cable tension H. Figure 2 shows the first four nondimensional frequency ratios as a function of parameter ␭ for different ␧. It can be found from Fig. 2 that for different figures, when ␭ / ␲ is not in a certain interval, the nondimensional frequencies of each mode are obviously affected by the parameter ␧.When the values of ␭ / ␲ fall into these interval, the increasing rate of nondimensional frequencies gradually decrease. This phenomenon can be explained qualitatively by accounting for the variation of dynamic tension with the parameter ␭. The dynamic tension, which has been discussed in Ref. 关7兴, is greatly amplified at values of ␭ / ␲ = 2n, so that in the interval near this point, the changing rate of the chordwise static cable tension becomes unimportant compared with the changing rate of dynamic tension and therefore its influence on the nondimensional frequency can be ignored. The first four nondimensional frequency ratios as a function of parameter ␧ for different values of ␭ / ␲ are shown in Fig. 3. As indicated in the figure, the first four nondimensional frequency ratios vary with the changing of ␧. For a certain value of ␭ / ␲, the first four nondimensional frequency ratios would monotonically decrease with the increasing of ␧ except for the second mode. An interesting fact can be seen from Fig. 3共b兲, where in the nondimensional frequencies of the second mode keep unchanged with parameter ␧ at the value of ␭ / ␲ = 1.98. As mentioned before, it is the chordwise static cable tension H that can be easily changed during the actual construction process. With the changing of H, the value of parameters ␭ and ␧ will be changed simultaneously. Figure 4 shows the changing of the first four nondimensional frequency ratios versus parameter ␭ when H is changed gradually. The result of Wu’s 关12兴 is also shown in the same plot for comparison. It can be seen that the result of the present theory is considerably different from that of Wu’s especially for the higher order frequencies. The biggest difference is that the nondimensional frequencies of odd modes will decrease with the increasing of ␭ / ␲ when the value of ␭ / ␲ is relatively Journal of Vibration and Acoustics

small while the nondimensional frequencies of even modes will still increase, as demonstrated by Wu, but with different values. When the two frequencies are close to each other, a veering also appears, as predicted by Triantafyllou 关6兴 and Wu et al. 关12兴. The general trend is that the nondimensional frequencies of present result are somewhat lower than that of Wu’s in varying degrees at different values of ␭ / ␲. The comparisons of the first three natural circular frequencies of an inclined taut cable between the present analysis and Wu’s result are shown in Table 3. It can be seen that the natural circular frequencies calculated by Wu’s equation is higher than that of the present for every case. The errors between the two results increase with the increasing of the inclined angle, and the maximum relative error 12.1% is observed, corresponding to the first mode when ␣ = 60 deg and ␭ / ␲ = 3.5 deg. By simplifying the static profile to a parabola, the relation between the nondimensional frequency and the parameter ␭ can be determined by Eq. 共17兲 and the first four nondimensional frequencies are plotted in Fig. 5. The biggest change lies in the veering zone with each two nondimensional frequencies becoming very close. The enlarged views of the first two veering zones with the comparisons to the results of Eq. 共15兲 are depicted in Fig. 6. Figure 7 shows the transition of the first two longitudinal modal 共LM兲 and the first two transverse modal 共TM兲 shapes of an inclined taut cable with an inclination angle of 60 deg, which are obtained by the Galerkin method. The chordwise static cable tension H is changed, as in Table 2, and the values of ␭ / ␲ are also provided, with reference to Fig. 6. The transverse modal shapes given by Eq. 共14兲 are also shown in these figures for comparison with the result obtained by the Galerkin method. It is shown that the transverse modal shapes given by Eq. 共14兲 coincide well with the results of the Galerkin method. These figures also clearly demonstrate the transitions of modal shapes between symmetric mode and antisymmetric mode with the changes of the two parameters ␭ and ␧. This result is in accordance with JUNE 2011, Vol. 133 / 031001-5


Fig. 3 The first four nondimensional frequency ratios versus ε for different values of ␭ / ␲

that in literature 关6兴, namely, in the veering zone of Fig. 4, the modal shapes are neither symmetric nor antisymmetric but hybrid modes.

7

Fig. 4 The changing of the first four nondimensional frequency ratios versus parameter ␭ by Eq. „15…

Conclusions

Taking the factor of the weight component parallel to the cable chord into account, an improved cable theory on the free vibration of an inclined taut cable is developed. By variable substitution, two coupled linear differential equations describing twodimensional in-plane motion of the cable become one Bessel equation of zero order when the equation of longitudinal motion is ignored. The explicit formulae for determining the modal shapes and natural frequencies of an inclined taut cable are presented by solving the Bessel equation. It can be seen from these explicit formulae that the factors, which affect the free vibration frequencies and modal shapes, can be condensed as two parameters ␭ and ␧. The accuracy of the proposed explicit formulae is validated by numerical results obtained by the Galerkin method and compared with those of other literatures. It appears that the present theory has an advantage over others in the aspect of accuracy.

Table 3 Comparisons of the first three natural circular frequencies of an inclined taut cable between present analysis and Wu’s result ␭ / ␲ = 1.6 Method

␣ = 10 deg Present result Wu’s result Error 共%兲 ␣ = 30 deg Present result Wu’s result Error 共%兲 ␣ = 60 deg Present result Wu’s result Error 共%兲

␭ / ␲ = 2.2

␭ / ␲ = 3.5

Mode 1

Mode 2

Mode 3

Mode 1

Mode 2

Mode 3

Mode 1

Mode 2

Mode 3

2.389 2.395 0.25

2.745 2.764 0.69

4.188 4.215 0.64

2.463 2.483 0.81

3.857 3.885 0.73

4.929 4.97 0.83

2.105 2.128 1.09

2.819 2.844 0.89

3.891 3.908 0.44

2.271 2.288 0.75

2.599 2.656 2.19

3.963 4.046 2.09

2.311 2.369 2.51

2.528 2.555 1.07

3.639 3.725 2.36

1.962 2.034 3.67

2.652 2.729 2.9

3.693 3.745 1.41

1.863 1.898 1.88

2.099 2.222 5.86

3.185 3.363 5.59

1.821 1.932 6.1

2.103 2.174 3.38

2.919 3.103 6.3

1.517 1.70 12.1

2.113 2.291 8.42

2.976 3.119 4.81

031001-6 / Vol. 133, JUNE 2011



¯vh共z兲 = C1J0共z兲 + C2Y 0共z兲

共A3兲

where J0共z兲 is the first kind Bessel function of zero order, Y 0共z兲 is the second kind Bessel function of zero order, and C1 and C2 are constants. When z is large, the two Bessel functions of Eq. 共A3兲 are approximated to 关22兴 J0共z兲 ⯝

Y 0共z兲 ⯝

冑冉冊冑冉冊 2 ␲ cos z − ␲z 4

共A4兲

2 ␲ sin z − ␲z 4

共A5兲

A particular solution of Eq. 共A2兲 is Fig. 5 The changing of the first four nondimensional frequency ratios versus parameter ␭ by Eq. „17…

With these explicit formulae, it can be found that the natural frequencies of each mode are obviously affected by the two parameters ␭ and ␧. However, the influence of the parameter ␧ on the modal frequency can be ignored in the interval, where the dynamic tension is greatly amplified. A further analysis indicates that the form of the cable static profile mainly influences the veering zone with little effect on other zones. Accurate analytical expressions for the natural modes and the natural frequencies are very important in engineering applications, and the proposed analysis is an accurate and useful tool to the design of inclined cable structures and may be used as a base to the correct analysis of linear and nonlinear dynamics of cable structures.

Acknowledgment

¯v p共z兲 =

¯v共z关x ¯ 兴兲 =

1

冑z

关C1 cos共z兲 + C2 sin共z兲兴 +

¯ 2␻ 冑1 − ␧x¯ ␧

共A1兲

Eq. 共12兲 becomes a nonhomogeneous Bessel equation of zero order as follows:

冉

¯h ␧2 2 d ¯v 1 d¯v + ¯v = 2 3 − ␧ − z 2 + ¯ ¯2 dz z dz ␻ 2␻ 2

冊

共A2兲

The homogeneous solution of Eq. 共A2兲 is a linear combination of two Bessel functions that can be written as

冉

¯h 2␧2 ␧2 2 3 − ␧ + − z ¯2 ¯ 2 2␻ ¯2 ␻ ␻

冊

With the boundary conditions ¯v共z关0兴兲 = ¯v共z关1兴兲 = 0, the unknown constants C1 and C2 of Eq. 共A7兲 can be obtained as C1 =

¯h 关共␳ + ␧兲冑z共1兲sin关z共0兲兴 − 共␳ − ␧兲冑z共0兲sin关z共1兲兴兴 ¯2 ¯ ␤/␧兲 ␻ sin共2␻ 共A8兲 ¯h 关共␳ − ␧兲冑z共0兲cos关z共1兲兴 − 共␳ + ␧兲冑z共1兲cos关z共0兲兴兴 ¯2 ¯ ␤/␧兲 ␻ sin共2␻ 共A9兲

where

Appendix A

¯兴 = z关x

共A6兲

共A7兲

␳=1+

By variable substitution, such as

冊

From Eqs. 共A3兲–共A6兲, the whole solution of Eq. 共A2兲 is given by

C2 =

This research is supported by the Natural Science Foundation of China 共Grant Nos. 50425516 and 50875149兲 and China High Technology Research and Development Program by the Ministry of Science and Technology 共Grant No. 2009AA04Z401兲.

冉

¯h 2␧2 ␧2 2 3−␧+ 2 − z 2 ¯ ¯ ¯2 ␻ ␻ 2␻

2␧2 , ¯2 ␻

␤ = 冑1 − ␧ − 1

By substituting Eqs. 共A8兲 and 共A9兲 into Eq. 共A7兲 and then using the substitution Eq. 共14兲, the solution of Eq. 共12兲, with the boundary conditions ¯v共0兲 = ¯v共1兲 = 0, can be obtained.

Appendix B The coefficients of the submatrices in 关M兴 and 关K兴 are M 11共i, j兲 = M 22共i, j兲 =

冕

l

mQiQ jdx

共B1兲

0

Fig. 6 The enlarged views of the two veering zones


JUNE 2011, Vol. 133 / 031001-7


Fig. 7 The first two longitudinal and transverse modal shapes of an inclined taut cable with an inclination angle of 60 deg

冕再

冋冉冊

l

K11共i, j兲 =

mg sin ␣Q⬘j − EA

册冎

0

dx ds

冉冊冉冊册冕再冋冋冉冊冉冊册冎 l

3

K22共i, j兲 =

mg sin ␣ − 2EA

0

+ H − xmg sin ␣ Q⬙j Qidx

共B2兲

− EA

dx ds

3

dy dx

2

dx ds

3

d2y dy Q⬘ dx2 dx j

+ H − xmg sin ␣ Q⬙j Qidx 共B4兲

where i , j = 1 , 2 , . . . , N, and the expressions for dy / dx, d y / dx2, and dx / ds can be obtained from Eq. 共1兲, the prime indicates the derivative with respect to the coordinate x. 2

冕冉冊冉 l

K12共i, j兲 = K21共i, j兲 =

− EA

0

dx ds

3

冊

d2y dy Q⬘ + Q⬙ Qidx dx2 j dx i 共B3兲

031001-8 / Vol. 133, JUNE 2011

References 关1兴 Irvine, H. M., and Caughey, T. K., 1974, “The Linear Theory of Free Vibra-



tions of a Suspended Cable,” Proc. R. Soc. London, Ser. A, 341, pp. 299–315. 关2兴 Irvine, H. M., 1978, “Free Vibration of Inclined Cables,” J. Struct. Div., 104共2兲, pp. 343–347. 关3兴 Henghold, W. M., Russell, J. J., and Morgan, J. D., 1977, “Free Vibrations of Cable in Three Dimensions,” J. Struct. Div., 103共5兲, pp. 1127–1136. 关4兴 Perkins, N. C., and Mote, C. D., 1987, “Three-Dimensional Vibration of Travelling Elastic Cables,” J. Sound Vib., 114共2兲, pp. 325–340. 关5兴 Srinil, N., Rega, G., and Chuchepsakul, S., 2007, “Two-to-One Resonant Multi-Modal Dynamics of Horizontal/Inclined Cables. Part I: Theoretical Formulation and Model Validation,” Nonlinear Dyn., 48共3兲, pp. 231–252. 关6兴 Triantafyllou, M. S., 1984, “The Dynamic of Taut Inclined Cables,” Q. J. Mech. Appl. Math., 37共3兲, pp. 421–440. 关7兴 Triantafyllou, M. S., and Grinfogel, L., 1986, “Natural Frequencies and Modes of Inclined Cables,” J. Struct. Eng., 112共1兲, pp. 139–148. 关8兴 Shih, B., and Tadjbakhsh, I. G., 1984, “Small-Amplitude Vibrations of Extensible Cables,” J. Eng. Mech., 110共4兲, pp. 569–576. 关9兴 Al-Qassab, M., and Nair, S., 2004, “Wavelet-Galerkin Method for the Free Vibrations of an Elastic Cable Carrying an Attached Mass,” J. Sound Vib., 270共1–2兲, pp. 191–206. 关10兴 Wang, L., Zhao, Y., and Rega, G., 2009, “Multimode Dynamics and Out-ofPlane Drift in Suspended Cable Using the Kinematically Condensed Model,” ASME J. Vibr. Acoust., 131, p. 061008. 关11兴 Rega, G., and Srinil, N., 2007, “Nonlinear Hybrid-Mode Resonant Forced Oscillations of Sagged Inclined Cables at Avoidances,” ASME J. Comput. Nonlinear Dyn., 2, pp. 324–336. 关12兴 Wu, Q., Takahashi, K., and Nakamura, S., 2005, “Formulae for Frequencies and Modes of In-Plane Vibrations of Small-Sag Inclined Cables,” J. Sound Vib., 279共3–5兲, pp. 1155–1169.


关13兴 Sun, Y., and Leonard, J. W., 1998, “Dynamics of Ocean Cables With Local Low-Tension Regions,” Ocean Eng., 25共6兲, pp. 443–463. 关14兴 Gobat, J. I., Grosenbaugh, M. A., and Triantafyllou, M. S., 2002, “Generalized-␣ Time Integration Solutions for Hanging Chain Dynamics,” J. Eng. Mech., 128共6兲, pp. 677–687. 关15兴 Gobat, J. I., and Grosenbaugh, M. A., 2006, “Time-Domain Numerical Simulation of Ocean Cable Structures,” Ocean Eng., 33共10兲, pp. 1373–1400. 关16兴 Goyal, S., Perkins, N. C., and Lee, C. L., 2005, “Nonlinear Dynamics and Loop Formation in Kirchhoff Rods With Implications to the Mechanics of DNA and Cables,” J. Comput. Phys., 209共1兲, pp. 371–389. 关17兴 Goyal, S., Perkins, N. C., and Lee, C. L., 2008, “Non-Linear Dynamic Intertwining of Rods With Self-Contact,” Int. J. Non-Linear Mech., 43共1兲, pp. 65–73. 关18兴 Goyal, S., and Perkins, N. C., 2008, “Looping Mechanics of Rods and DNA With Non-Homogeneous and Discontinuous Stiffness,” Int. J. Non-Linear Mech., 43共10兲, pp. 1121–1129. 关19兴 Stump, D. M., Fraser, W. B., and Gates, K. E., 1998, “The Writhing of Circular Cross-Section Rods: Undersea Cables to DNA Supercoils,” Proc. R. Soc. London, Ser. A, 454, pp. 2123–2156. 关20兴 Zhu, W. D., and Chen, Y., 2006, “Theoretical and Experimental Investigation of Elevator Cable Dynamics and Control,” ASME J. Vibr. Acoust., 128, pp. 66–78. 关21兴 Zhu, W. D., and Zheng, N. A., 2008, “Exact Response of a Translating String With Arbitrarily Varying Length Under General Excitation,” ASME J. Appl. Mech., 75共3兲, p. 031003. 关22兴 Abramowitz, M., and Stegun, I. A., 1972, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, Dover, New York.

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