Research Article  Open Access
Jiangen Lv, Zhicheng Yang, Xuebin Chen, Quanke Wu, Xiaoxia Zeng, "Modeling and 1 : 1 Internal Resonance Analysis of CableStayed Shallow Arches", Shock and Vibration, vol. 2020, Article ID 7080927, 16 pages, 2020. https://doi.org/10.1155/2020/7080927
Modeling and 1 : 1 Internal Resonance Analysis of CableStayed Shallow Arches
Abstract
In this paper, an analytical model of a cablestayed shallow arch is developed in order to investigate the 1 : 1 internal resonance between modes of a cable and a shallow arch. Integrodifferential equations with quadratic and cubic nonlinearities are used to model the inplane motion of a simple cablestayed shallow arch. Nonlinear dynamic responses of a cablestayed shallow arch subjected to external excitations with simultaneous 1 : 1 internal resonances are investigated. Firstly, the Galerkin method is used to discretize the governing nonlinear integralpartialdifferential equations. Secondly, the multiple scales method (MSM) is used to derive the modulation equations of the system under external excitation of the shallow arch. Thirdly, the equilibrium, the periodic, and the chaotic solutions of the modulation equations are also analyzed in detail. The frequency and forceresponse curves are obtained by using the Newton–Raphson method in conjunction with the pseudoarclength pathfollowing algorithm. The cascades of perioddoubling bifurcations leading to chaos are obtained by applying numerical simulations. Finally, the effects of key parameters on the responses are examined, such as initial tension, inclined angle of the cable, and rise and inclined angle of shallow arch. The comprehensive numerical results and research findings will provide essential information for the safety evaluation of cablesupported structures that have widely been used in civil engineering.
1. Introduction
Cablesupported structures have been widely used in civil engineering for its light weight, flexibleness, and low fundamental damping [1, 2]. Because of their large amplitude vibrations under environmental loads, these structures are susceptible to loss of serviceability or even failure. It is quite crucial to understand the dynamic characteristics of cablesupported structures under environmental loads [3, 4].
According to previous studies, the support motion effect on cablesupported systems is complicated and thus has been simplified as a boundary excitation in the analysis process. Benedettini et al. [5] established a fourdegreeoffreedom discrete cable model for modal couplings under vertical/outofplane support motions. Guo et al. [6] investigated cable’s mode interactions under vertical support motions by boundary resonant modulation. Lee et al. [7] studied free vibration of a rotating curved beam with elastically restrained roots. The twotoone internal resonance between nonlinear normal modes of an elastically constrained shallow arch is examined [8]. In these cases, the assumption would be reasonable if the mass of the support is much larger than that of cable. Moreover, several studies have been carried out on the dynamic characteristic of cablestayed beam. AbdelGhaffar et al. [3, 9, 10] presented the refined finite element models for cablestayed bridge dynamics to show the importance of nonlinear dynamical analysis and of cable beam interaction. Fujino et al. [11–13] proposed to study cable beam interactions using a Ritztype analytical model based on the test function representative of global and local modes and through an experimental setup of a cablestayed beam. Gattulli et al. [14] presented a parametric investigation of both linear and nonlinear characteristics of a simple cablestayed beam and explored the interaction mechanisms of cable beams by means of numerical simulations. In particular, in order to completely explore the interaction mechanisms, they presented a precise comparison between the findings obtained by an analytical model for a simple geometrically nonlinear cablestayed beam [15] and those obtained by both finite element analysis and experiments on a physical model [16], and they also investigated the localization and veering in the dynamics of cablestayed bridges by a cablestayed beam model [17]. Wei et al. [18] studied bifurcation and chaos of a cable beamcoupled system under simultaneous internal and external resonances. Kang et al. [19] numerically investigated the nonlinear dynamic response of the stay cable in cablestayed beam subjected to parametrical and forced excitations. Generally speaking, the nonlinear dynamic problem has been solved by approximate method in much literature, and some researchers validated the approximate results by numerical simulations and/or experimental method. The static configuration of a beam was considered, and the nonlinear modal properties of cablestayed beam with the direct approach and discretization approach were investigated on the basis of the exact mode shapes of cablestayed beam [20]. Srinil and Rega [21] presented spacetime numerical simulation and validation of analytical predictions for the finiteamplitude forced dynamics of suspended cables. Luongo and Zulli [22] validated the strongly modulated response of the string and the NES by the Galerkin model.
Many of the studies mentioned above are focused on the dynamic behavior of cablestayed beams, and little work has been done on dynamic studies of cablestayed shallow arches. With the development of longspan structure, cablestayed shallow arches are gradually widely used in civil engineering, especially in the arch bridge construction process and other longspan building structure. The main advantages of the cablestayed shallow arch system are that this system has a good use of the compressive strength of the arch and the tensile strength of the cable. However, the structures of the system thus become complex. In spite of this, the cablestayed shallow arch system has also attracted the attention of many researchers. Ju and Guo [23] studied the inplane elastic buckling behavior of a cablestayed shallow arch structure. Ai et al. [24] proposed a practical calculation method of the critical lateral flexure load of cablestayed shallow arch systems, and the effect of the stay cable on the lateral stability of a cablearch system was discussed. Zhao and Kang [25, 26] investigated the inplane and outofplane free vibration of a cablesupported arch structure and its inherent properties. Most researches on dynamic characteristics of cablestayed arches did not take into account the nonlinearity. There are only a few studies that considered geometrical nonlinearity: Lv et al. [27] established the governing equations of cablestayed arch structures, and the possible internal resonance of cablestayed arch structures was also investigated. Kang et al. [28] investigated the inplane 1 : 1 : 1 internal resonance between three first modes of the shallow arch and two cables under both external primary and subharmonic resonance.
In cablestayed beam structures, two different modal shapes can be basically identified: some modal shapes mainly involve the dynamic beam deflection, while the cable seems to be quasistatically dragged by the beam tip; in other modes, the deflection is extremely localized in the cable domain, with negligible beam participation in the modal shape [17]. Similarly, the cabledominant vibration can be classified as local modes, while the archdominant vibration may be named global modes (since the arch ideally represents the main structure of a cablestayed arch structure). For a specific cablesupported arch structure, there may be three main resonance mechanisms: 1 : 1 internal resonance, twotoone internal resonance, and onetotwo internal resonance. Hence, this paper is devoted to the investigation of the 1 : 1 internal resonance of cablestayed shallow arches. The governing equation for the cablestayed shallow arch is derived by using Hamilton’s principle and solved by the Galerkin method and multiple scales method. The effects of initial tension, inclined angle of the cable and arch, and arch geometry on the 1 : 1 internal resonance of the cablestayed shallow arch are studied comprehensively.
2. Modeling and Equations of Motion
A cablestayed shallow arch is used as the supporting member in the arch bridge system (Figure 1), and its first construction stage model containing a shallow arch and a single cable is shown in Figure 2. The static and dynamic models used in following analyses for the cablestayed shallow arch are shown in Figure 3. The cable and shallow arch are homogeneous and rigidly connected at the end. Shear inertia and rotary deformation of the shallow arch are neglected. The rise to length of arch is small (i.e., , namely, shallow arch); similarly, the sag d_{c} to length l_{c} of cable is small (i.e., ). The shallow arch is subjected to a harmonic load, which can be expressed in the form , is the amplitude of the transverse excitation applied to the shallow arch, and is the excitation frequency. and are the longitudinal and transverse displacements of the cable, respectively, and l_{c} its span length, d_{c} its sag, and θ_{c} its inclined angle; and are the longitudinal and transverse displacements of shallow arch, respectively, and its span lengths, its rise, and its inclined angle.
(a)
(b)
To ensure continuity at connection point between the cable and the shallow arch and projecting along the and directions, the following equations are obtained:where m_{c} denotes the mass per unit length of the cable, c_{c} its damping coefficient, its static displacement, E_{c}A_{c} its axial stiffness, and H its initial tension. denotes mass per unit length of the shallow arch, its damping coefficients, its axial stiffness, its static displacement, its flexural stiffness, and N its initial axial force. Equations of motion, which govern the transverse vibrations, are obtained by the classical extended Hamilton’s principle:where
As can be seen from equations (2) and (3), compared with a cablestayed beam system, equations of the cablestayed shallow arch system are more complex. The cable’s equation of motion of the cablestayed beam system is in agreement with that of the cablestayed shallow arch system, while the shallow arch’s equation motion of the cablestayed shallow arch is different from that of beam in the cablestayed beam because of the geometric curvature of shallow arch, and it is consistent with that equation in [29]. As a result of these different from the cablestayed beam, it is very necessary to study the nonlinear dynamic characteristics of the cablestayed shallow arch structure.
The geometric and mechanical boundary conditions at the junction can be written as follows:
As shown in Figure 4, the cable is anchored at the junction S; is the deformed position of S; ; ; ; and are the shear forces acting on the left and right of the junction, respectively; M and are the bending moment acting on the left and right of the junction, separately; and and are the components of the tension of cable in the longitudinal and transverse directions, respectively.
(a)
(b)
2.1. Galerkin Discretization Procedure
Using the discretization approach, Lacarbonara et al. [30] have obtained the general modulation equations of the displacement in distributedparameter systems. Similarly, following the methodologies in [30], the discretization approach is used to obtain the modulation equations governing the nonlinear dynamic behavior of cablestayed shallow arches with 1 : 1 internal resonances. Then, the inplane transverse displacements and are approximately expressed in the following form:where and are the mode shape functions of the cable and shallow arch, respectively, with i = 1, 2, 3, … referring to the natural modes of the linearized problem (e.g., Figure 5). and are the generalized coordinates of the cable and shallow arch.
(a)
(b)
Equations (6) and (7) are substituted into equations (2) and (3) and integrated where required, and the Galerkin method is used to obtain a nonlinear model with the following system of differential equations for the cable and shallow arch motions:where the coefficients are defined in the Appendix, is the amplitude of the transverse excitation applied to the shallow arch, and is the excitation frequency.
2.2. Perturbation Analysis
The multiple scale method [31] is used to seek a firstorder uniform expansion of the solution of equations (8) and (9) of the following form:where (i = 0, 1). T_{0} is the fast time scale associated with changes occurring at , , and , and T_{1} is the slow time scale associated with modulations in the amplitudes and phases caused by nonlinearity, damping, and resonance. In terms of T_{0} and T_{1}, the time derivatives becomewhere (i = 0, 1). Substituting them into equations (8) and (9) and then equating coefficients of the same power of ε, we obtain the following differential equations:
The general solution of equations (12) and (13) can be expressed aswhere and with a_{n}, b_{m} and θ_{n}, θ_{m} denoting the steadystate amplitudes and the phases of motion, respectively. cc denotes a complex conjugate of the preceding term. Substituting equations (16) and (17) into equations (14) and (15), we getwhere the prime indicates the derivative with respect to T_{1}. The functions A_{n} and B_{m} can be determined so that the solutions to equations (18) and (19) do not contain secular terms or smalldivisor terms caused by resonance.
As shown in equations (18) and (19), there are many types of resonance in cablestayed shallow arches, such as primary resonance, 1 : 1 internal resonance, 2 : 1 internal resonances, and 1 : 2 internal resonance.
2.3. 1 : 1 Internal Resonance
As the 1 : 1 internal resonance is the most common form of internal resonance in modal interaction, next, the case is analyzed when 1 : 1 internal resonance exists between modes of the cable and shallow arch () in the presence of primary resonance of shallow arch (). These resonant relationships, which have been confirmed numerically, can be described as follows:where and are two detuning parameters. Substituting equation (20) into equations (18) and (19) and eliminating secular terms yield the following equations:
Substituting A_{n} and B_{m} into equations (21) and (22) and then separating the real and imaginary components give the governing equations for the amplitudes a_{n} and b_{m} and the phases and :where and . The steadystate solutions correspond to constant solutions, i.e., . Hence, the following set of nonlinear algebraic equations is obtained that is solved numerically to obtain the fixed points of the system:
From equations (24)–(27), we obtain possible solutions besides the trivial solution. Two mode solutions are considered: and ; equations (24) and (25) are squared, the squared results are then added together, and the process is repeated for equations (24)–(27), giving the following frequencyresponse equations:where
In order to assess stability of the steadystate solution, it is necessary to obtain the modulation equations in the Cartesian form. Therefore, A_{n} and B_{m} are written in Cartesian form:
Substituting A_{n} and B_{m} from equation (30) into equations (21) and (22), the following modulation equations in the form of Cartesian form can be obtained by separating the results into real and imaginary parts:
The periodic motion of cablestayed shallow arch is mathematically expressed as the steadystate solutions of equations (31)–(33). In order to obtain the steadystate solution, we can assume that in equations (31)–(33), and the Newton–Raphson method can be applied to solve these four nonlinear equations. After the steadystate Γ_{0} is determined (Γ_{0} = {}^{T}, T is the transpose), the stability of the steadystate solution can then be assessed by applying a classical linearization method [32]. If the steadystate solution is not zero, a new solution Γ = Γ_{0} + ΔΓ is substituted into these equations. The result is then expanded in a Taylor series about Γ_{0}, and only the linear terms in the disturbance are retained to yield:where [J] is called the Jacobian matrix. The stability of the steadystate solution is determined by the eigenvalues of the Jacobian matrix. If all the eigenvalues of Jacobian matrix have a negative real part, the steadystate solution is stable. If the real part of one of the eigenvalues of Jacobian matrix is positive, the steadystate solution is unstable.
3. Numerical Discussion
The cablestayed shallow arch case with following physical parameters is taken from a particular working bridge in China, as shown in Figure 1. The dimensional parameters and material properties of the cable and shallow arch in the first construction stage are listed in Tables 1 and 2, respectively. Moreover, the static equilibrium equation of the shallow arch is approximated by the parabolic function as and that of the cable is .


By using the finite element method (FEM), the first ten local and global frequencies of the cablestayed shallow arch are obtained, as shown in Figure 6. From the figure, the 1 : 1 internal resonance may occur between the local and global modes when and . Therefore, in this paper, the case of 1 : 1 internal resonance between the 10th local mode and the 4th global mode is studied. The solid and dashed lines of the frequency and forceresponse curves in next sections indicate the stable and unstable solutions, respectively.
3.1. Equilibrium Solutions, Stability, and Validation
Figure 7 shows variation in the response amplitudes of the cable and shallow arch with the detuning parameter σ_{2} when = 2.0 ( = 118.6 kN/m) and , where SNB and HB represent the saddlenode and Hopf bifurcation, respectively. For the right branch, the composite structure loses its stability due to a Hopf bifurcation at HB1 (). The system then regains its stability through another Hopf bifurcation at HB2 (). Due to the effect of internal resonance, the cable that is not directly stimulated has a larger amplitude. As described in Figure 7, in this case, the cable vibration of cable dominates the nonlinear response. On the contrary, in order to verify the approximate analytical solutions obtained with multiple scale method (MSM), Runge–Kutta method (RM) is applied to the original ODEs in equations (8) and (9), and the corresponding numerical solutions are also presented in Figure 7. As shown in Figure 7, the solid lines are stable solutions obtained from MSM, dashed lines are unstable solutions, and circles are the solution obtained from RM. According to the error analysis, the present results have a good agreement with the numerical results obtained from RM (Table 3).
(a)
(b)

Figure 8 shows the forceresponse curves with various excitation amplitudes and detuning parameters. When the detuning parameter σ_{2} < 0, there is a singlevalued relationship between the response and excitation amplitudes. When σ_{2} 0, there is a relatively small range of multiple solutions of the system because of the saddlenode bifurcations at SNB1 and SNB2. There are two stable solutions and an unstable solution between SNB1 and SNB2. When increases from SNB1 to SNB2, the two stable equilibrium solutions coexist.
(a)
(b)
Figure 9 shows the time histories of the cable and shallow arch, obtained by integration of two degreesoffreedom ordinary differential equations in equations (8) and (9) under the initial conditions where the initial speed and displacement are zeros.
(a)
(b)
3.2. Dynamic Solutions, Bifurcation, and Chaos
The Cartesian form of the equations has the following standard form:
The periodic solutions and chaotic solutions can be obtained by the shooting method. After the periodic solution x_{0} is constructed, the branch of periodic solution is tracked by pseudoarclength pathfollowing algorithm, and its stability is determined using the Floquet theory.
Figure 10 shows the phase portraits and Poincare maps of the dynamic solutions onto the plane as the detuning parameter σ_{2} slowly varies. As σ_{2} increases from 0.05, a oneperiod limit cycle develops, as shown in Figure 10(a) (σ_{2} = 0.05587). The stable 1P solution then loses its stability via a perioddoubling bifurcation, and a twoperiod solution is shown in Figure 10(b) (σ_{2} = 0.05602). As σ_{2} increases further, the 2P solution undergoes a cascade of perioddoubling bifurcations at σ_{2} = 0.05635 (4P). Finally, this cascade of perioddoubling bifurcations leads to chaos. A representative chaotic attractor is shown in Figure 10(d).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 11 shows the phase portraits and Poincare maps of the dynamic solutions onto the plane as the detuning parameter σ_{2} slowly varies. When σ_{2} increases past Hopf bifurcation point (HB1), a small limit cycle develops and grows in size, deforms, and then undergoes a sequence of perioddoubling bifurcations leading to chaos, as shown in Figures 11(a)–11(d).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 12 shows the bifurcation diagrams as the detuning parameter σ_{2} is varied in the range . The values of other parameters are all fixed when there is 1 : 1 internal resonance and simultaneous primary resonance. As seen in Figure 12, as the detuning parameter σ_{2} slowly increases in the narrow region , and motion of the system undergoes the following process: chaotic motion periodic1 motion periodic2 motion chaotic motion. In addition, it should be mentioned that the chaos region on the left of Figure 12 belonged to another bifurcation interval.
(a)
(b)
3.3. Effects of Key Parameters
In this section, the effects of the key parameters of the cable and shallow arch on the frequencyresponse curves of the cablestayed shallow arch are investigated. The frequencyresponse curves shown in Figures 13–16 are obtained by numerical calculation with different values of parameters. These key parameters include the initial tension, the inclined angle of the cable, and the rise and inclined angle of shallow arch.
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
In order to investigate the effect of the initial tension H of the cable on the dynamic behavior of the system, Figure 13 illustrates the frequencyresponse curves under different values of detuning parameter σ_{2} when and . As shown in Figure 13, the initial tension governs the multivalue region of the frequencyresponse curves. The resonance interval of the system is increased when the initial tension of the cable increases. With the increase in initial tension, the response amplitudes of the system also increase. Overall, the initial tension has a significant effect on the frequencyresponse curves of the cable, as shown in Figure 13(a), whereas the influence of parameter on the frequencyresponse curves of the shallow arch is not susceptible as seen in Figure 13(b).
Generally, the inclined angle θ_{c} of the cable plays an important role in the dynamic behavior of the cablestayed shallow arch. Figure 14 shows the effect of the inclined angle of the cable on the frequencyresponse curves of the cablestayed system when and . As expected, increasing or decreasing the inclined angle causes significant changes in the frequencyresponse curves. As can be seen in Figure 14, as the inclined angle is increased, the response amplitudes of the system decrease and the hardening spring behavior of the resonance increases. Additionally, increasing the inclined angle delays the jump phenomenon of the frequencyresponse curves. It should be noted that the threevalued region of the cable rapidly narrows and the fivevalued region gets larger. That is, the resonance interval of the system decreases with the increase in the inclined angle. However, the inclined angle parameter has no obvious impact on the multivalue range of the shallow arch.
Figure 15 shows the effect of the shallow arch rise on the dynamic behavior of the cablestayed shallow arch with and . As shown in Figure 15, a backbone curve exists for the three different frequencyresponse curves with each different arch rise. The fivevalued region of the cable slow decreases as the arch rise value is decreased, which is in a similar trend to that of the initial tension on the dynamic response. However, in contrast with the results of the initial tension on the dynamic behavior, the arch rise has a slight effect on the response amplitude of the resonance.
In the cablestayed shallow arch structure, the shallow arch inclined angle is another key parameter of the structure. Figure 16 shows the effect of shallow arch inclined angle on the dynamic behavior of the system when and . As shown in Figure 16, the arch elevation has a significant influence on the multivalue region of the cable, and the jump phenomenon of the shallow arch can be delayed by decreasing the shallow arch inclined angle. In particular, the multivalue range of the cable rapidly expands for shallow arch inclined angle above 45°. The shallow arch inclined angle has little effect on the multivalue region of the shallow arch.
4. Conclusions
The 1 : 1 internal resonance characteristics of cablestayed shallow have been investigated in this paper. Analytical solution for 1 : 1 internal resonance and primary resonance has been derived for the cablestayed shallow arch. The effects of initial tension, inclined angle, arch rise, and elevation on the internal resonance responses have been discussed in detail. It is found that the 1 : 1 internal resonance of cablestayed shallow arch may occur when the natural frequency of local mode for the cable is close to that of global mode for the shallow arch. Numerical results also show that periodic motion of the system may lose its stability due to the periodicdoubling bifurcation. It is also found that the effects of the inclined angle of both cable and shallow arch on the dynamic characteristics of the cablestayed shallow arch system are quite remarkable, while the effects of the rise of shallow arch and the initial tension of the cable are not. The comprehensive numerical results and research findings will provide essential information for the safety evaluation of cablesupported structures that are widely used in civil engineering.
Appendix
where
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
The study was supported by the National Natural Science Foundation of China (Grant no. 51678247).
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