1 Introduction

The analysis of vibrations in a cable with discrete elements, e.g., a concentrated mass, a translational spring, a viscous damper, and a harmonic oscillator, remains the focus of many theoretical studies and experimental research projects. The results of these investigations are frequently published and discussed. Literature review has indicated a number of methods for the modeling and analysis of the vibrations in a cable with discrete elements. These methods cover both precise methods, which are based on the analytical solutions and obtained from the classical sources such as the Green function, the Laplace transformation, and the Fourier transform, and approximate methods, which are obtained from the finite element method, the finite difference method, the Ritz method, and the Galerkin method. Sergev and Iwan^[1] analyzed, both theoretically and experimentally, the eigenproblem of a cable with several concentrated masses. The equation of motion was formulated for a cable consisting of a number of segments, at the ends of which several concentrated masses were attached. However, they did not focus on the effects caused by the cable sag. Cheng and Perkins^[2] used the Hamilton principle of least action to formulate the equations of motion describing the linear dynamics of an elastic cable with a small sag and a single concentrated mass. They also considered the eigenproblems of two approximate models for the cable. Lin and Perkins^[3] presented the results of theoretical analyses and experimental research into the eigenproblem of a cable with several concentrated masses. The analyses were based on a hybrid analytical-numerical method for a cable with discrete masses. Al-Qassab and Nair^[4] used the Wavelet-Galerkin method to analyze the free vibration in a cable with an attached concentrated mass, and compared the results with those obtained from the classical Fourier method. Wang and Rega^[5] investigated a three-dimensional (3D) equation of motion for a cable with a moving mass, and examined the displacement of a cable with a moving mass with the Newmark method. Their analyses included the influence of such factors as the inertial force, the mass, the cable sag, and the velocity of the moving mass. Main and Jones^[6], Cu et al.^[7], and Sun and Chen^[8] demonstrated theoretical analyses on the effects of the positioning of a viscous damper along the cable axis on the eigenfrequencies, the eigenforms, and the damping characteristics. The analyses were based on the analytical formulae for an assumed model of taut string, with regard neither to the cable sag nor to the effects of the bending stiffness. Zhou et al.^[9] showed the results of experimental research on how to affect the damping parameters of a cable by changing the position of the viscous damper along the cable axis. Their tests were performed on a full scale cable, and the results were compared with the analytical solutions with and without the consideration of the internal stiffness of the damper and the linear and non-linear damping characterization. Zhou et al.^[10] analyzed the eigenfrequencies and damping characteristics of a taut cable with discrete elements, i.e., a damper and a translational spring. Yu and Xu^[11] formulated an algorithm to analyze the cable vibrations both in and out of the sag plane for a cable with oil dampers. The algorithm was based on an original hybrid method, and could be used to analyze an inclined cable with the consideration of its sag, the direction in which the damper was attached, the damper stiffness, etc. Wu and Cai^[12] presented the analytical formulae of a composite equation of motion, and analyzed the free vibration in a horizontal taut cable with a tuned mass damper (TMD), which might be attached in any location along the cable. Chen et al.^[13] investigated the cable vibrations considering the cable resistance to bending, in configurations with lateral and rotational dampers, which might be attached at any position along the axis of the cable. Mekki and Auricchio^[14] examined a linear model of a cable with a small sag and a shape-memory alloy (SMA) damper, evaluated numerical analyses, and demonstrated that an SMA damper ensured more efficient reduction in both the free vibration and the excited vibration as compared with a classical TMD damper. The issues related to the modeling of several cables cross-tied with discrete elements have been discussed in Refs. [15]-[19]. Cross-tied cables are frequently used in cable-stayed bridges in order to reduce the cable vibrations. In such cases, the cables are cross-tied with either flexible ropes or rigid elements. A translational spring model and/or a viscous damper model frequently function as a mathematical model for the cross-tie of a cable. The dynamic parameters of a network of cables interconnected with the additional elements largely depend on the installation location, the stiffness, and the damping characteristics of the cross-ties.

In this paper, the vibrations in the cable provided with discrete elements are modeled and analyzed with the commonly known Lagrange multiplier formalism. This method has been frequently used to model and analyze the vibrations in various complex mechanical systems, and has been extensively discussed in the literature^[20-31]. The issue of formulating and solving the problems of free vibrations in a system composed of a cable with discrete elements has been approached from a perspective similar to the methods used in the case of a rod with discrete elements^[20-31]. The modeling method used here is known in the literature as continuous-discrete modeling, with which some selected fragments of the object are modeled with an actual object where the parameters of the elements are continuously distributed, while other fragments of the object are modeled with discrete elements^[20-36]. In this case, a linear model of a cable with a small sag serves as a continuous model, while such elements as a translational spring, a concentrated mass, and a harmonic oscillator serve as the discrete elements. The method is based on the analytical solutions related to constituent elements, which, when once derived, can be used to formulate the equations describing various complex systems compatible with the structure of an actual object. The advantage of this modeling method used in complex systems lies in the fact that formulating and calculating an eigenfrequency equation do not require formulating and calculating the complex equations in an exact analytical form. Based on this model, it is possible to solve the problem of free vibrations in such a way that the results will be closer to those obtained from an exact method rather than an approximate method^{[2, 4-5]}. An additional advantage of the modeling method offered in this paper is that the calculations are based on the linear model of a cable with a small sag. Interestingly, the cable is still modeled as a taut string^{[1, 6-8]}. The string is a particular type of cables, and has a negligible sag and a negligible bending stiffness. A string model can be used to simplify the calculations greatly. However, such a model fails to correctly describe the behavior of a number of actual structural members, e.g., cables in cable-stayed bridges, and should not be used in their modeling. The above review of literature allows an observation that the analyses are limited with regard to both the type of the additional elements and their number. In Refs. [2] and [4]-[5], only the analyses of cable vibrations with one concentrated mass were attached, while in Refs. [6]-[9], only one attached damper was focused. In Refs. [1] and [3], the cable vibrations with several attached concentrated masses were analyzed. In Ref. [10], the analysis of cable vibrations with two types of discrete elements, i.e., a damper and a translational spring, was presented. In Ref. [13], the effects of lateral and rotational dampers were analyzed. The systems presented in this paper are more complex. They involve discrete elements of various types, e.g., a concentrated mass, a translational spring, and a harmonic oscillator, which may be used at any location along the cable axis regardless the number of the elements. After appropriate modifications, the formulated equations can be also used to analyze the effects of other types of discrete elements, e.g., rotational spring and additional support.

2 Formulated problem

The considered system is schematically represented in Fig. 1(a). It consists of a simply supported cable with a small sag (0 ≤ d/L ≤ 1/8) and the span L. The system also consists of the following discrete elements: a translational spring with a constant elasticity K, a concentrated mass M, and a harmonic oscillator characterized by the parameters K_o and M_o, which are attached to the cable. The additional elements are connected to the cable at x_i (i=1, 2, 3). Figure 1(b) shows the discrete elements prior to their connection to the basic system, i.e., the cable.

As a part of the assumed linear theory for the lateral vibration in a cable in the plane with a small sag and without discrete elements, the kinetic energy and the potential energy may be expressed as follows^[37-38]:

(1)

(2)

where w(x, t) is the vertical component of the in-plane motion from the static profile y(x). H is the normal tension in the cable due to static load. H_d is the additional horizontal component of the cable tension due to cable movement. λ² is the dimensionless parameter combining geometrical and deformational characteristics, as proposed by Irvine and Caughey^[37] and Irvine^[38]. g is the gravitational acceleration. m is the cable mass per unit of length. ρ is the density of the material of the cable. A is the cross-sectional area of the cable. E is Young's modulus. The calculations are based on an assumption that m, ρ, E, and A are constant along the length of the cable. The dynamic tension in the cable due to the cable movement can be represented with appropriate assumptions if

(3)

where d/L≪1, and d is the value of y(x) at x=L/2. It is under the assumption that the static small cable sag has a parabolic profile, and may be expressed as follows^[37-38]:

(4)

At the same time, the dimensionless parameter proposed by Irvine and Caughey^[37] and Irvine^[38] may be represented as follows:

(5)

Based on the solution to the eigen problem in a cable without discrete elements, the lateral displacement in the sag plane w(x, t) may be formulated as follows:

(6)

where are the ith eigenform of a cable in the sag plane, and are time functions. When Eq. (6) is inserted into Eqs. (1) and (2), respectively, if a finite number n of eigenforms is considered in the solution, the cable energy (1) and (2) may be represented as follows:

(7)

(8)

where

(9)

(10)

The introduced designations M_yi and K_yi do not contain any discrete element attached to the cable. By using Eqs. (7) and (8), the total kinetic energy T_y as well as the total potential energy V_y of the system, which includes all of the elements shown in Fig. 1(b), may be represented as follows^[20-32]:

(11)

(12)

Additional coordinates w_i (i=1, 2, 3, 4) describing the movement of the ends of the additional discrete elements are introduced, in accordance with the diagram shown in Fig. 1(b), in order to form the constraint equations defining the coupling of the system elements. The constraint equations f_r describing a system composed of a cable and the discrete elements attached to the cable at x_r (r=1, 2, 3) (see Fig. 1(a)) may be described as

(13)

The Lagrangian kinetic potential for such a system composed of a cable in the sag plane and the additional elements will take the following form:

(14)

where λ_yr (r=1, 2, 3, …, R) are the Lagrangian multipliers, and R is the number of the discrete elements (in this case, R=3). The equations of motion for the investigated system may be derived by using the Lagrangian equations, and may be presented as follows:

(15)

(16)

(17)

(18)

(19)

If the harmonic solution to the systems of Eqs. (15)−(19) is assumed as follows:

(20)

(21)

(22)

and Eqs. (20)-(22) are inserted into Eqs. (15)-(19), respectively, the values of amplitudes ξ_i and w_k may be calculated by functions of the Lagrangian multiplier amplitudes λ_yr. After necessary transformations, we have

(23)

(24)

(25)

(26)

(27)

If Eqs. (23)-(27) are subsequently inserted into Eq. (14), a system of R equations is obtained, i.e.,

(28)

which may be formulated in a matrix form as follows:

(29)

where δ_kr is the Kronecker delta, Λ_y=(λ_y1, λ_y2, λ_y3)^T, and C_y is the square matrix of a degree dependent on the number of discrete elements. In this case, C_y takes the following form:

(30)

where

(31)

(32)

Determining the eigenfrequencies ω of a system composed of a cable with discrete elements is possible on the condition that there is a non-zero solution to the system of Eq. (29), i.e.,

(33)

For the determined frequencies ω in the system, it is possible to determine the corresponding eigenforms. The eigenform in a complex system is based on the prior equation, and can be obtained after transformations^[30], i.e.,

(34)

Y(x) will be completely determined after the respective systems of Eq. (28) are solved with regard to λ_yr. It is worth mentioning that, in the discussed system, the coefficients C_kr^y describe a flexible cable in the sag plane, while the coefficients ε_k^z describe the type of the discrete elements attached to the cable. The dimensions of C_y depend on the number of the discrete elements. Therefore, if the considered system is composed of a number discrete elements which are different from that of the system presented in Fig. 1, C_y should be modified to a dimension equal to the number of the discrete elements, being supplemented with the coefficients ε_k^y as required. The newly defined matrix C_y serves as a basis to determine the eigenfrequencies ω_r in a new continuous discrete system.

It should be stressed that, this analysis is based on the commonly known linear equations of motion for a cable with a small sag, describing uncoupled in-plane and out-plane lateral vibrations^[37-38]. In other words, it is assumed that the cable vibrations in the sag plane do not affect the cable vibrations out of the sag plane, and thus the two types may be analyzed independently.

Functions Y_i(x), which are required in further analyses and describe the eigenforms in the sag plane in the case without discrete elements, may be represented using the following commonly known relationships^[37-38]:

(ⅰ) Antisymmetric eigenform (even eigenfrequencies)

The eigenform corresponding to the ith eigenfrequency is described as follows:

(35)

where A_2i is the vibration amplitude, and The eigenfrequency is given by

(36)

which signifies the eigenfrequency of the ith antisymmetric eigenform.

(ⅱ) Symmetric eigenform (odd eigenfrequencies)

The eigenform, which corresponds to the ith eigenfrequency, is described as follows:

(37)

where A_2i-1 is the vibration amplitude. The additional nondimensional frequency ω_i is calculated from

(38)

The additional nondimensional frequency corresponds to the ith eigenfrequency of the cable with symmetric in-plane vibrations through

(39)

which signifies the eigenfrequency of the ith symmetric eigenform.

Thus, when λ² is small, no difference is observed for the in-plane and out-plane eigenfrequencies and eigenforms^[37-39]. The linear equations for the vibrations in cables, as presented in this section, are a fundamental and obligatory calculation model in most bridge structures^[39]. However, it should be again stressed that, due to practical aspects and ease of calculation, the cable is still frequently modeled as a taut string, i.e., without the consideration of its sag and bending stiffness^[39].

3 Sample results of numerical calculations

The numerical calculations are based on a system presented in Fig. 1, which is composed of a cable with a small sag or several discrete elements. In the first three examples, the investigations covered a system composed of a cable with one discrete element, i.e., a concentrated mass, a translational spring, or a harmonic oscillator. In the second example, the analyzed system consisted of a cable with three discrete elements attached simultaneously. The plotted results of the numerical calculations represent the eigenfrequencies of the analyzed complex systems, with respect to the point at which a particular discrete element is attached. The plots also show the eigenforms of the complex systems corresponding to some of the eigenfrequencies. The figures show the symmetric and antisymmetric eigenforms in the sag plane. The eigenforms presented in the figures have been normalized to the value of 1 by dividing each of the presented eigenforms by its maximum value. The presented results of the numerical calculations are obtained from the relationships shown in Section 2, transforming them appropriately for the analyzed individual complex systems. Figure 2 shows the first three eigenfrequencies corresponding to the symmetric and antisymmetric eigenforms in the sag plane, depending on the Irvine parameter λ². In this case, the cable is analyzed without the attached discrete elements. The eigenfrequencies obtained directly from Eqs. (36) and (37) are marked in Fig. 2 with continuous and dashed lines, whereas the frequencies obtained from the algorithm presented in Section 2 are marked with red dots, which agree well with each other.

In the first example, a system composed of a cable with an attached concentrated mass is analyzed. Figures 3 and 4 show the first three eigenfrequencies corresponding to the symmetric and antisymmetric eigenforms in the sag plane, depending on the location described by the non-dimensional coordinate x₂/L and attached by the concentrated mass. The values of the calculated eigenfrequencies ω_i are presented in relation to the number π. In this and the following examples, the first three frequencies are marked in the figures, respectively, in solid, dashed, and dash-dotted curves.

Figures 5, 6 and 7, 8 show the first two standardized symmetric eigenforms of the system Y_2i(x)/A_2i with the mass located at x₂/L=0.5 and x₂/L=0.25, respectively. The calculations are performed with three different Irvine parameters. Figures 9 and 10 show the first two standardized antisymmetric free vibration forms of the system Y_2i-1(x)/A_2i-1 with the mass located at x₂/L=0.25. It can be seen that the mass located at x₂/L=0.5 does not affect either the eigenfrequency or the corresponding antisymmetric eigenform, because it is located at the eigenform node. It should be additionally emphasized that the Irvine parameter does not affect the antisymmetric eigenforms.

In the second example, the analysis covers the behavior of a system composed of a cable with a translational spring. Figures 11 and 12 show the first three eigenfrequencies corresponding to the symmetric and antisymmetric eigenforms in the sag plane, depending on the position described by the non-dimensional coordinate x₁/L and attached by the translational spring. The values of the calculated eigenfrequencies ω_2i corresponding to the symmetric eigenforms and the eigenfrequencies ω_2i corresponding to the antisymmetric eigenforms are also presented in relation to the number π.

Figures 13, 14 and 15, 16 show the first two standardized symmetric eigenforms of the system Y_2i(x)/A_2i with the translational spring located at x₁/L=0.5 and x₁/L=0.25, respectively. The calculations are also performed with three different Irvine parameters.

Figures 17 and 18 show the first two standardized antisymmetric eigenforms of the system Y_2i-1(x)/A_2i-1 with the translational spring located at x₁/L=0.25.

In the third example, a system composed of a cable with a harmonic oscillator is analyzed. Figures 19 and 20 show the relationship between the first three eigenfrequencies corresponding to the symmetric and antisymmetric eigenforms in the sag plane, depending on the position described by the non-dimensional coordinate x₃/L and attached by the harmonic oscillator. The values of the calculated eigenfrequencies ω_2i corresponding to the symmetric eigenforms and the calculated eigenfrequencies ω_2i corresponding to the antisymmetric eigenforms are also presented in relation to the number π. As the case in the previous examples, the calculations are performed for three Irvine parameters, having respective values of λ²=1, λ²=10, and λ²=100.

In the next example, the analysis covers the behavior of a system composed of a cable with three discrete elements, i.e., a mass, a spring, and a harmonic oscillator. Figure 21 shows the first three eigenfrequencies corresponding to the symmetric eigenforms of the complex system. These eigenfrequencies are represented as a function of x₁/L, indicating the attachment positions of the cable, where K=50 and M=1 at x₂/L=0.25 while K_o=10 and M_o=5 at x₃/L=0.5. Figure 22 shows the first three eigenfrequencies, corresponding to the symmetric eigenforms of the complex system. These eigenfrequencies are represented as a function of x₂/L, indicating the attachment positions of the cable, where M=1 and K=50 at x₁/L=0.25 while K_o=10 and M_o=5 at x₃/L=0.5. As the case in the previous examples, the calculations are performed for three various Irvine parameters.

The analysis of the obtained results allows an observation that the values of the parameters describing the additional discrete elements, i.e., M, K, M_o, and K_o, and the change in the coordinates designating the attachment points of the elements to the cable have a significant effect on both the eigenfrequencies and the eigenforms corresponding to these eigenfrequencies. The above observation is particularly evident in the cases of symmetric in-plane eigenforms (see Eqs.(5)-(8) and Figs. 13-16). The obtained results are consistent with an already known fact in the literature that increasing the mass causes a decrease in the eigenfrequency (see Figs. 3 and 4) while increasing the stiffness of the translational spring causes an increase in the eigenfrequency (see Figs. 11 and 12). In the third example with a harmonic oscillator, a significant effect of the mass parameter on the eigenfrequency change is noticeable. As can be seen, a significant increase in the stiffness parameter results in a slight increase in the eigenfrequency. It is necessary to emphasize the significant effects on the eigenfrequencies for the parameters describing the individual additional elements and the coordinates of the connection points between these elements, where the three discrete elements are attached (see Figs. 21 and 22). Also, Figs. 3, 11, and 19 clearly indicate the generally known effect of the Irvine parameter on the change in the eigenfrequencies corresponding to the symmetric eigenforms. Allowing for the Irvine parameter of λ²=10 in the calculations only slightly modifies the first eigenfrequency, while does not have any effect on the change in the second and third eigenfrequencies. However, allowing for the Irvine parameter of λ²=100 causes a significant effect on the first eigenfrequency while a small effect on the change in the second and third frequencies. The numerical analysis confirms that, when λ²>4π², the first eigenfrequency of the first symmetric in-plane mode is larger than that of the first antisymmetric in-plane mode, while the first symmetric in-plane mode has two internal nodes along the span (see Figs. 5 and 15). Furthermore, both the first and second symmetric eigenforms have two internal nodes along the span (see Figs. 5 and 15). Additionally, it has been demonstrated that the eigenfrequencies (see Figs. 4, 12, and 20) and the corresponding antisymmetric eigenforms (see Figs. 9, 10, 17, and 18) do not depend on the geometrical and material characteristics included in the Irvine parameter.

4 Summary and conclusions

The primary purpose of this work is to formulate and test a method for the continuous-discrete modeling of a cable with additional elements as well as to investigate what influence the positions of various additional elements along the cable axis have on the eigenfrequencies and eigenforms of a complex system. The proposed complex system modeling method is advantageous. One advantage is to allow the eigenfrequency equation to be formulated and solved in an uncomplicated manner. The analytical solutions need to be developed only once for the constituent elements of the system, and serve to formulate equations for modeling various systems composed of any type and any number of discrete elements by specifying the coordinates designating the attached points between these elements and the cable. Another advantage lies in the fact that the method is based on a linear model of a cable with a small sag subject to the dynamic strain under exactly described movement, in accordance with the theory of elastic deformation. The analytical solutions used for the constituent elements of the system do not introduce additional sources of inaccuracies to the models of complex systems as the case with methods discretizing the whole model, and thus allow results representing the behavior of actual structures with great accuracy.

The present work confirms that the presented algorithm may be an effective tool for the analysis of free vibrations in a cable with such discrete elements as a concentrated mass, a translational spring, and a harmonic oscillator. The discrete elements have been chosen so as to represent the elements actually incorporated in a number of actual structures, e.g., devices limiting cable vibrations, devices used for the installation, tension, and replacement of cables, and technical inspection baskets moving on the cables and cable cross-ties. This work also shows that additional elements may have a significant effect on the dynamic characteristics of the investigated complex system. In the case of symmetric in-plane eigenfrequencies and eigenforms, the geometrical and material characteristics included in the Irvine parameter, which allow for the dynamic tension observed in the cable due to the movement of the cable, have a significant effect on both the quantitative and the qualitative changes in the analyzed eigenfrequencies and eigenforms. This fact is particularly evident when a large Irvine parameter is incorporated. A proper selection of the type, location, and characteristics of discrete elements allows a purposeful modification for the frequency and form of free vibrations in the cable, e.g., to detune the frequency resonance of the structure.

The algorithm shown in this paper can be easily modified so as to allow analyses of systems composed of a cable with any type and any number of discrete elements, which may include other types than shown here, e.g., a rotational spring and an additional support. In such cases, the dimension of the matrix C_y should be modified to match the number of the additional elements and supplemented as required with the coefficients ε_k describing the type of the discrete elements attached to the cable. Based on Eq. (33) and the newly defined matrix C_y, it is possible to determine the eigenfrequency ω of a new complex system. Equation (34) can be used to obtain the eigenforms for each of the determined frequencies. Due to length limitations, this paper does not include the analyses and results of the calculations for the out-plane eigenfrequencies and eigenforms, which are less complicated than those for the in-plane symmetric and antisymmetric eigenfrequencies and eigenforms. Nevertheless, the equations related to the out-plane eigenfrequencies and eigenforms can be easily derived by introducing appropriate formulas into the kinetic and potential energy. Uncomplicated modifications of the presented algorithm also enable the analysis of free vibrations in a system of several cables connecting each other with other elements. The above fact is relevant in practice, as the cross-ties between cables are frequently used in bridge engineering in order to reduce the excessive cable vibrations.