This paper presents approximate series solutions for nonlinear free vibration of suspended cables via the LindstedtPoincare method and homotopy analysis method, respectively. Firstly, taking into account the geometric nonlinearity of the suspended cable as well as the quasistatic assumption, a mathematical model is presented. Secondly, two analytical methods are introduced to obtain the approximate series solutions in the case of nonlinear free vibration. Moreover, small and large sagtospan ratios and initial conditions are chosen to study the nonlinear dynamic responses by these two analytical methods. The numerical results indicate that frequency amplitude relationships obtained with different analytical approaches exhibit some quantitative and qualitative differences in the cases of motions, mode shapes, and particular sagtospan ratios. Finally, a detailed comparison of the differences in the displacement fields and cable axial total tensions is made.
As a basic and significant structural element, the suspended cable has been widely applied in many mechanical systems and engineering fields [
The suspended cable is a typical weakly nonlinear continuous system which contains the quadratic and cubic nonlinearity terms, and the nonlinear free vibration of the system has been studied through many analytical methods in the previous researches. Recently, Hagedorn and Schäfer [
Hence, in order to overcome the limitations in both the perturbation method and the nonperturbation method, Liao [
On the other hand, there are some comparisons between the homotopy analysis method and perturbation method. Specifically, Yuan and Li [
To the best knowledge of the authors, no specific study has addressed a comparison of homotopy analysis method and other perturbation method in the case of nonlinear free vibrations of the suspended cable. However, the homotopy analysis method is different from those presented in the papers published before on the same problem, so this research will focus on the comparison of these methods. The paper consists of four sections: firstly, the nonlinear free vibration equations of motion are derived by applying the Homilton’s principle and quasistatic assumption, and then the multimode expansion of the displacement is introduced to obtain a discrete cable model. Secondly, the approximate series solutions obtained with LindstedtPoincare method and homotopy analysis method are constructed, respectively. Moreover, in the section of the numerical analysis, time histories, frequency amplitude curves, displacement fields, and axial tension forces of suspended cables are compared and illustrated. Finally, some conclusions are made at the end of the paper.
Figure
Two different configurations of the suspended cable.
By applying the Hamilton’s principle and the quasistatic stretching assumption, we could express the nonlinear partial differential equation of motion without considering the bending, torsional, and shear rigidities as [
The corresponding boundary conditions are written as
In this study, because the sagtospan ratio is sufficiently small (
In order to make the subsequent section more general, the following nondimensional quantities are adopted [
As a result, (
Eliminating the nonlinear terms, the mode shapes, and frequencies can be ascertained by solving linearized equation of motion. Therefore, the inplane
On the other hand, the
Assuming that the suspended cable is a multidegreeoffreedom (MDOF) dynamic system, which is composed of symmetric and antisymmetric modes with respect to the midspan, the Galerkin method is employed to simplify the nonlinear oscillation equation of motion. Considering the boundary conditions, the solutions of (
Therefore, a set of nonlinear ordinary differential equations are yielded by substituting (
This section begins with the approximate series solutions for the nonlinear free vibrations obtained with the LindstedtPoincare method, followed by the homotopy analysis method. For the sake of simplicity, only singlemode model (symmetric mode or antisymmetric mode) is considered.
Firstly, a new independent variable is introduced, which is
By assuming an expansion for
Following the method of LindstedtPoincare [
Then, we could obtain the relation between the nonlinear frequency
Substituting (
In the following, the nonlinear free response of the suspended cable is explored by homotopy analysis method which transforms a nonlinear problem into an infinite number of linear problems with an embedding parameter
Introducing a new time scale
In (
Under the new time scale transformation, the new form of (
Therefore, the corresponding initial conditions are
Given the fact that the free oscillations of a conservative system could be expressed by a series of periodic functions which satisfy the initial conditions:
Considering the rule of solution expression and initial conditions in (
To construct the homotopy function, one may define the linear auxiliary operator as
According to (
For the sake of simplicity, we choose
Therefore, with the increase of the embedding parameter
By using the Taylor series expansion and considering the deformation derivatives, we will obtain
The
For the sake of brevity and simplicity, the following vectors are defined:
Differentiating the zeroth order deformation equation
Moreover, the right hand side of the
Therefore, the general periodic solution
Here, we take
In order to satisfy the rule of solution expression, the coefficients
The solutions of (
Eliminating the secular term and considering the expression of linear operator, the first order deformation equation becomes
It is easy to solve the linear ordinary differential equation with the initial conditions (
Following the same procedure, the
Finally, it should be pointed out that, on the one hand, only the zeroth order algebraic equations are nonlinear and all the higher order equations are linear. On the other hand, compared with the results obtained with LindstedtPoincare method ((
The dimensional parameters and material properties of the suspended cable are chosen as follows [
As mentioned in the previous section, the auxiliary parameter
Effect of the auxiliary parameter
Nevertheless, it should be mentioned that the convergence tests or proofs are significant and important for homotopy analysis method. Yet, only several sagtospan ratios and initial conditions are involved, and the convergent regions could not be checked one by one. Moreover, just as mentioned by Liao [
Once the auxiliary parameter
Comparison of the series solution
Generally speaking, both the frequency amplitude relationship and the effect of the nonlinearities on the law of motion are two important aspects that need to be examined and analyzed [
At the beginning, Table
Three nondimensional parameters of suspended cables.
(a)  (b)  (c)  (d)  


0.002  0.02  0.04  0.08 

9.418  94.181  188.361  376.723 

0.0024 (0.0156)  2.4110 (0.4945)  19.2882 (1.3987)  154.3060 (3.9560) 
Figure
Comparisons of frequency amplitude relationships obtained with the 1st order homotopy analysis method and the 4th order LindstedtPoincare method for the first four modes: (a)
In Figure
As the sagtospan ratio increases (
Figure
Furthermore, Figure
Inplane displacement fields obtained with homotopy analysis method and LindstedtPoincare method in the case of the first symmetric mode: (a)
In the following, a comparison of displacement fields obtained with these two analytical approaches is made. Substituting (
It should be mentioned that, as to the first three cases, the Irvine parameter
In the fields of engineering, the tension force of the suspended cable plays a very important role. Therefore, the cable total tension obtained with different analytic methods is studied and analyzed in this section. According to Srinil et al. [
Figure
Time histories of the nondimensional cable tension force for the case of the first symmetric mode: (a)
In this research, the nonlinear free vibrations of the singlemode model of the suspended cable are studied via the LindstedtPoincare method, homotopy analysis method, and numerical integrations and only the first two symmetric and antisymmetric modes are considered. Moreover, the numerical results and discussions are extended from a taut string (
The homotopy analysis method does not depend on any small parameter assumption and provides us with a convenient way to ensure the convergence for the series solutions. It is found that above a certain value of response amplitude, it still continues to agree well with the results of the numerical integrations of the ODEs, whereas the LindstedtPoincare solutions partly fail. On the one hand, in the case of the taut string (small sagtospan ratio), these two analytical methods make no difference. On the other hand, as to the high order vibration modes, large response amplitudes, and sagtospan ratios, these two approaches may lead to some quantitative and qualitative differences in the frequency amplitude relationships. However, the results obtained with the homotopy analysis method are in good agreement with the ones obtained by using numerical integrations in the whole range of response amplitude. Furthermore, the homotopy analysis method has an advantage over the original one in the accuracy of the estimation of the displacement fields and second order harmonic to the time history of the cable axial tension.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work was supported by the National Natural Science Foundation of China (nos. 11032004 and 11102063). The authors would like to thank the anonymous reviewers for their constructive comments and suggestions on the earlier version of this paper.