1 Introduction

Nowadays, cable-stayed bridges are generally the first choice to cross roads, rivers and valleys because of their good ratio of economy to span and the mature construction techniques. They often undergo various complex environmental loads during their service period. Hence, understanding their mechanical properties under different working conditions has been attracting many researchers.

Many scholars have been putting their interests on dynamics of the shallow arch structures, which are used to model the bridge deck. Generally, a pre-arch in the planar configuration of long-span bridges is designed to match the demand of drainage. Hence, the shallow arch rather than the straight beam is used to model the bridge deck, where its geometric nonlinearity caused by the initial configuration is considered. However, the shallow arch may lose its stability by “snap buckling”, in which the structure suddenly jumps from one stable equilibrium configuration to another one^[1]. As early as in 1979, Plaut^[2] investigated the influence of load position on stability of the shallow arches. Levitas et al.^[3] studied the global dynamic stability of a shallow elastic arch subject to a distributed constant load by Poincaré-like simple cell mapping. Then, Breslavsky et al.^[4] analyzed the nonlinear modes of snap-through motions of a shallow arch. Cai et al.^[5] studied the in-plane elastic buckling of fixed shallow parabolic arches subject to a vertical uniform load based on the virtual work formulation to delineate the relation between arch slenderness and buckling modes. They further provided accurate solutions for the symmetric and antisymmetric buckling of fixed shallow parabolic arches and approximate solutions for symmetric buckling loads of the shallow parabolic arches. Pi and Bradford^[6] investigated the nonlinear dynamic buckling of shallow circular arches under a sudden uniform radial load. Chen and Lin^[7] studied the dynamic snap-through of a shallow arch under a moving point load in order to explore the effect of the load speed on the snap-through. Then, they investigated the effects of the elastic foundation on the snap-through buckling of the shallow arch under a moving point load^[8] and the stability of the shallow arch with one end moving at the constant speed^[9]. Abdelgawad et al.^[10] investigated the static and dynamic snap-through of the shallow arch resting on a two-parameter elastic foundation under a point load moving at a constant speed. Moreover, Ha et al.^[11] investigated the stability of a shallow arch under the constant load. Plaut^[12] analyzed snap-through of arches and buckled beams under unilateral displacement control. In addition, many other scholars had studied the nonlinear dynamic behaviors of the shallow arch. Blair et al.^[13] investigated the dynamic response of a shallow arch to the harmonic force with the method of harmonic balance coupled with a continuation scheme and Floquet analysis. Lakrad and Schiehlen^[14] investigated the effects of slowly varying periodic parametric excitation on a hinged shallow arch. Furthermore, due to the initial static configuration of the shallow arch, the internal resonance may be activated. Under this circumstance, the energy transfer between the involved resonant modes may occur, which results in the rich and complex dynamic behaviors^[15]. Tien et al.^[16-17] investigated the weakly nonlinear one-to-one and one-to-two internal resonances of a two-degree-of-freedom shallow arch subject to the simple harmonic excitation. Malhotra and Namachchivaya^[18-19] studied the global bifurcations of the shallow arch structure subject to a spatially and temporally varying load under the conditions of principal sub-harmonic resonance and one-to-one and one-to-two internal resonance near single-mode periodic motions. Deng et al.^[20] studied dynamic impact factors for simply-supported bridges due to vehicle braking, where a three-dimensional vehicle-bridge coupled model was established. Then, Deng et al.^[21] developed a new multi-point tire model for studying the bridge-vehicle coupled vibration compared with the existing single-point tire model and disk model using the numerical simulation. Bi and Dai^[22] analyzed the nonlinear dynamics and bifurcations of the shallow arch subject to the periodic excitation with the internal resonance. Chen and Liao^[23] investigated the motion of the shallow arch with one end pinned in space while the other end was attached to a mass and simultaneously supported by a spring at the attached mass through experimental and theoretical analyses. Using Galerkin's method and the numerical integration, Yi et al.^[24] studied the nonlinear dynamic behaviors of a viscoelastic shallow arch. Moreover, Yi et al.^[25] investigated the modal interaction activations and the two-to-one internal resonance of the shallow arch with both ends vertically elastically constrained. In addition, Ding et al.^[26] studied the steady-state periodic response of the forced vibration of a traveling viscoelastic beam under the 3:1 internal resonance. The direct multi-scale method was used to determine the steady-state response amplitudes, and Galerkin's method was used to verify the approximate solutions. They held that the effect of the internal resonance cannot be ignored.

Meanwhile, cable dynamics has also attracted much attention in the past decades. A detailed review of cables' dynamic study history can be found in the related paper by Rega^[27]. Here, some studies on cable's resonant dynamics are reviewed. Pakdemirli et al.^[28] analyzed the one-to-one internal resonance between the in-plane and out-of-plane modes and the primary resonance of the in-plane mode of cables. Zhao et al.^[29] examined cable's one-to-one resonant interaction dynamics subject to a harmonic excitation using Galerkin's method and the multi-scale method. Srinil et al.^[30] and Srinil and Rega^[31] investigated the two-to-one internal resonance, multi-modal and free dynamics of horizontal/inclined cables and presented the approximate closed form solutions for small sagged cables. Using the exterior matrix method, Paulsen and Manning^[32] established a theory to compute the eigenfrequencies of the structure composed of a series of inclined cables. Zhao and Wang^[33] conducted further studies on the suspended cable's three-to-one internal resonance by using the multi-scale method to directly attack the nonlinear partial differential equation and boundary conditions. Based upon the boundary resonant modulation concept, cable's triad and two-to-one mode interactions under support motions are solved in a unified way, through attacking the continuous dynamic equations directly with the multi-scale method by Guo et al.^[34]. Next, Guo et al.^[35] investigated cable's nonlinear coupled dynamics excited by out-of-plane support motions, established a boundary modulation formulation using multi-scale expansions, and analyzed the cable's two-to-one nonlinear resonant responses based on the boundary modulation formulation technique. Moreover, Guo et al.^[36] studied cable dynamics under non-ideal longitudinal support motions/excitations. By introducing a large support/cable mass ratio, the cable-support dynamic interaction was properly modeled, and a cable-support coupled model was established using multi-scale expansions and asymptotic approximations. It is observed that only the single member, such as cables or a shallow arch, is concerned in the above works.

In addition, some composite structures have also been investigated, such as the cable-stayed beam systems. Fujino et al.^[37] and Fujino and Xia^[38] studied the dynamics of a three-degree-of-freedom model of a cable-stayed beam. Gattulli et al.^[39] investigated the parametric influence on linear and nonlinear behaviors of a cable-stayed beam model. Lenci and Ruzziconi^[40] investigated the nonlinear vibrations in the single-mode dynamics of a cable-supported beam. The cable-stayed beam structure, consisting of a single cable and a beam, was concerned in these works, but it could not reveal the interaction among those different cables and the beam.

Actually, the stayed cables are susceptible to the external load, especially the wind and traffic loads, due to their large flexibility, relatively small mass and very low inherent damping. Hence, the weak external interference may cause large vibrations of cables or the overall system. It is necessary to build a more complex model to explore the dynamic interactions among different cables and the deck. In our previous work^[41], a double cable-stayed shallow arch model of the cable-stayed bridge was proposed, and its nonlinear dynamic behaviors were investigated when the boundary excitation is applied to the right end of the shallow arch. In this paper, the nonlinear dynamic behaviors of the double cable-stayed shallow arch model when cables are subject to external harmonic excitations will be investigated, with the condition of simultaneously one-to-one-to-one internal resonance among the lowest modes of cables and the shallow arch and external primary resonance of cables. This work is organized as follows. Firstly, the geometric configuration of the double cable-stayed shallow arch model is given, and the planar dynamic equations governing the motion of cables and the shallow arch are presented. Then, Galerkin's method is used to deduce the ordinary differential equations (ODEs) of the system, and the multi-scale method is used to deduce its modulation equations. Thirdly, the numerical analysis of the double cable-stayed shallow arch system is conducted. Frequency- and force-response curves under different conditions are presented to explore the rich dynamic behaviors. Finally, some interesting phenomena are obtained, and conclusions are given.

2 Mathematical model

The double cable-stayed shallow arch model is described in Fig. 1. Here, both ends of the shallow arch are hinged. The upper ends of two cables are also hinged, and the lower ends are jointed with the shallow arch at the points s₁ and s₂, respectively. Some assumptions are given as follows: (ⅰ) the lateral vibration of the shallow arch is not considered; (ⅱ) the effect of motion of towers on the oscillation of the system is neglected, due to the fact that experiment measurements and the finite element analysis of a real cable-stayed bridge demonstrate that motion of towers is minimal vibration^[42-43]; (ⅲ) the flexural, torsional, shear strain, and the longitudinal inertia force of cables are ignorable; (ⅳ) the axial extensions of cables are described by Lagrangian strain of the centerline^[39].

For the two Cartesian coordinates: Oys and as shown in Fig. 1, U_j and V_j denote the axial and transverse displacements of the jth cable, respectively. m_j is its mass per unit length. is its axial stiffness. H_j is its static cable tension. l_j is its length of span. is its damping ratio. V_a denotes the transverse displacement of the shallow arch with respect to the static equilibrium configuration. m_a is its mass per unit length. is its bending stiffness. l_a is its length of span. is its damping ratio. denotes the angle between the overall vertical direction and the axial coordinate of cables. Y_a and Y_j denote the initial equilibrium configurations of cables and the shallow arch, respectively. In order to obtain a non-dimensional form, the following non-dimensional quantities are defined:

(1)

where is a given frequency, such as 1 or the first structural modal frequency. According to the classical extended Hamilton's principle, the planar transverse vibration equations of the shallow arch^[19] and cables^[39] are obtained as follows:

(2)

(3)

where , and is the dynamic tension of the cable acting on the shallow arch, which is caused by the interconnection between cables and the shallow arch at the joint points. P_j and are the amplitudes and frequency of the external harmonic excitation, respectively. The prime indicates differentiation with respect to the coordinates s and x.

For the reduced double cable-stayed shallow arch model, it should satisfy the following boundary and continuation conditions:

(4)

In addition, the relevant mechanical equilibrium condition at the joint points should also be satisfied. The excitation mechanism induced by the angle-variation of the cable tension during vibration is considered. In the vibration process, the shear difference of the shallow arch at the joint points s_j will be balanced with the dynamic forces of cables, which can be assumed as the external excitation to the shallow arch, namely, the term in Eq.(2), which has the following expression:

(5)

where is the Dirac delta function. is the angle-variation of cables between the actual vibration location and their reference configuration as shown in Fig. 1(b). Under the assumption of the small value of , the approximation that can be adopted. is the uniform dynamic elongation of cables given as follows:

(6)

Based on these previous assumptions, the displacement expressions of cables and the shallow arch can be written in the following form:

(7)

(8)

where and are the generalized coordinates, and and are the first-order eigen-functions or trial functions. The linearized governing equations and the influence of key parameters on the eigenvalue problem were investigated by Cong et al.^[44]. In that work, the piecewise modal function formulas of the shallow arch that satisfy both the displacement and mechanical conditions were derived. Inserting and into Eqs.(2) and (3) and imposing Galerkin's integral, the following nonlinear ODEs of the double cable-stayed shallow arch model can be obtained as follows:

(9)

(10)

where and . and are the relevant coefficients of corresponding polynomials of the above ODEs, given in Appendix A. Compared with our previous work^[41], the new cubic terms, namely, the last six terms in the governing equation of the shallow arch shown in Eq.(9), appear but the forced and parametric excitation terms vanish, and in cables' equations, the forced excitation is applied as shown in Eq.(10). Moreover, it should be noted that, compared with the cable-beam system, the initial geometrical configuration of the shallow arch generates the quadratic nonlinearity of the system, which can be seen in the expression of b₁₃ in Eq.(9).

3 Perturbation analysis

In order to obtain a uniformly valid approximate solution of this problem, the excitation term should be regarded as the same order with the damping and the nonlinearity. According to the multi-scale method^[45], a small bookkeeping parameter is artificially introduced into Eqs.(9) and (10). Then, the following equations are achieved:

(11)

(12)

where , , and .

The uniform expansions of g and q_j can be expressed in the following forms:

(13)

(14)

where is the fast time scale, and is the slow time scale. Substituting Eqs.(13) and (14) into Eqs.(11) and (12) and equating the coefficients of the same order of , the following differential equations can be obtained:

(15)

(16)

(17)

(18)

(19)

(20)

where , and .

As known, the general solutions of Eqs.(15) and (16) at the order have the following forms:

(21)

(22)

where c.c. denotes the complex conjugate of the preceding terms. are the unknown complex functions of the slow time scale, which are assumed to be the polar form as follows:

(23)

where a_n and are the amplitude and the phase angle of A_n, respectively. By substituting the solutions at the orders and into Eqs.(19) and (20), the solutions including the secular terms at the order can be obtained, which are given in Appendix B. Furthermore, the detuning parameters , , and are introduced to describe the nearness of the external primary and internal resonances, defined as

(24)

By eliminating the secular terms and separating the real and imaginary parts, the coupled modulation equations in the polar form of the system can be obtained as follows:

(25)

(26)

(27)

(28)

(29)

(30)

where , , , and

(31)

The stable equilibrium solutions of the modulation equations (25)–(30) corresponding to the steady periodic motion of cables and the shallow arch can be determined by setting α_z,T₂ = 0. Then, the Newton-Raphson method is used to solve the modulation equations. Stability of motion can be determined by eigenvalues of Jacobian matrix of the linearized equilibrium equations in Eqs.(25)–(30).

4 Discussion

In this section, the following parameters are used for the reduced double cable-stayed shallow arch model to explore its nonlinear dynamic behaviors under the one-to-one-to-one internal and external primary resonances. For cables: Young's modulus Pa, cross-section area m², mass per unit length kg/m, their initial cable tension N, their incline degree , the length of cable m; for the shallow arch: Young’s modulus E_a = 2 × 10¹¹ Pa, second moment of area I_a = 1.2 m⁴, cross-section area A_a = 2.158 m², mass per unit length ma = 1.694 × 10⁴ kg/m, length of span l_a = 300 m. The initial equilibrium configuration of the shallow arch can be given in the form of function y_a = (h_a/2)(1 − cos(2πs))^[15], where h_a is the non-dimensional rise of arch, which can be used to adjust the frequency of the shallow arch in order to obtain different internal resonance combinations between cables and the shallow arch. The relational figure of frequencies of cables and the shallow arch is given in Fig. 2.

Additionally, some typical non-dimensional parameters of cables and the shallow arch are calculated and presented in Table 1.

4.1 One of cables subject to harmonic excitation

For the sake of presentation, the cable subject to the external harmonic load is named as Cable 1, and the other is named as Cable 2. Figures 3–5 show the frequency-response curves of the shallow arch (a₁), Cable 1 (a₂), and Cable 2 (a₃) with the external excitation amplitude P = 0.1, respectively. It is firstly seen that the double peaks are observed in the frequency-response curves of the shallow arch. The double peaks bend to the lower direction of the horizontal axis of the detuning parameter σ, which implies the softening property as shown in Fig. 3. The behaviors of two identical cables are obviously different when one (here is Cable 1) of them is subject to the external excitation. Cable 1 shows a similar softening property since the only one peak of the frequency-response curves also bends to the lower direction of the horizontal axis, as shown in Fig. 4. However, Cable 2 exhibits a more complex behavior since there are almost triple peaks of the frequency-response curves, as shown in Fig. 5, where two of them bend to the lower direction of the horizontal axis, and the other one bends to the higher direction. Generally, bending of the frequency-response curves means multiple solutions of the system, which in turn causes the jump phenomenon as shown in these figures. Although multiple solutions of the present system are obtained, and the double and triple peaks are observed, the double jump phenomena are observed only in the frequency-response curves of Cable 2, as shown in Fig. 5. The main cause for the phenomenon is a multi-solution overlap of Cable 1 and the shallow arch. It can be seen that three branches of the solutions as σ > 0.15 are almost overlapped as shown in Figs. 3 and 4, where two of them are stable and the other is unstable.

Different types of bifurcations are also identified in the frequency-response curves of the system as shown in Figs. 3–5. Their dynamic behaviors can be analyzed from two paths. If the excitation frequency increases from a relatively small value, the lower branch of solutions of the system loses their stability through a Hopf bifurcation (HB1) and then regains stability through another Hopf bifurcation (HB2). Then, a jump phenomenon caused by a saddle node bifurcation (SN1) is observed, and the solutions lose their stability at another Hopf bifurcation (HB4) with the further increasing of the detuning parameter σ. Then, the solutions become stable till another Hopf bifurcation (HB3) and larger with the increasing of the detuning parameter σ till to 0.5, except that of the shallow arch. On the contrary, if the excitation frequency decreases from a relatively large value, the frequency-responses of the shallow arch are very small and almost zero, but the response of Cable 1, is relatively large and decreases. Extraordinarily, the solution of Cable 2 increases with the decreasing of the detuning parameter σ in a small range from 0.5 to 0.17. This can be explained that the energy coming from the external excitation is transferring from Cable 1 to Cable 2 through the shallow arch, and its exchange rate is increasing because of the increasing slope. The another jump phenomenon only for Cable 2 is observed by a new saddle node bifurcation (SN2), but no jumps are observed in frequency-response curves of Cable 1 and the shallow arch due to the multi-solution overlap as discussed above. The Hopf bifurcation (HB3) makes the solutions unstable as the detuning parameter σ decreases to a certain value corresponding to the HB3 point, where the double periodic bifurcation and torus bifurcation (TB) will be observed in a small frequency range with the further decreasing of the detuning parameter σ. The solutions regain stability through a new Hopf bifurcation (HB4), and they all increase with the decreasing of the detuning parameter σ. Form the above discussion, it can be seen that because of the internal resonance among cables and the shallow arch, the vibration of one cable subject to excitation can induce vibrations and jump phenomena of other cables and the shallow arch (deck), which can be used to interpret the vibration of multi-cable of cable-stayed bridge in practice.

Furthermore, the comparison of choosing the common sine function and the exact piecewise function (given again in Eq.(32)) as the trial functions of the shallow arch in Galerkin's integration is firstly investigated for the purpose to explore validity and consistency of the two results. The modal shape of the exact piecewise function of the shallow arch was given in Ref.[46], where it had been proved that the exact piecewise function can satisfy boundary and continuous conditions of the system. Figures 3(b)–5(b) show the frequency-response curves of response amplitudes of two cables and the shallow arch by choosing sine functions as the trial functions in Galerkin's integration. It is seen that there is just a slight difference between these two results. The frequencies of excitation inducing bifurcation only drift to the right side a little. Besides, frequency-responses of the system obtained by these two methods have almost identical behaviors. Based on the comparison, the following conclusion can be made. The two different trial functions have a good consistency at the displacement response level of the system, and all of them can be used to deduce the ODEs of the double cable-stayed shallow arch model with Galerkin's method. Choosing the piecewise function as the trial function of the shallow arch might have a better accuracy because both the displacement and mechanical continuous conditions, especially at the joint points s₁ and s₂, are satisfied simultaneously. However, due to the simple expression and the highly computational efficiency, the common sine function is regarded as the trial function in the following analysis:

(32)

where is the term that depends on the initial static configuration of the shallow arch, and the other coefficients can be seen in our previous work^[44].

Figures 6–8 show the force-response curves of two cables and the shallow arch by fixing the external detuning parameter σ to be a constant, respectively. From the following figures, it is apparently found that the dynamic behaviors of the present system when σ ≥ 0 are more complex than that as σ < 0. Only one stable branch of the solution is obtained when σ < 0 as shown in Fig. 8 while the multi-solutions and bifurcations are observed when σ ≥ 0 as shown in Figs. 6 and 7. The major reason may be that the initial deflection given in the system is small when σ < 0, which confirms the importance of initial deflection in the nonlinear system. Actually, the other solution may be found when the other initial conditions are given. Furthermore, the force-response curves of Cable 2 and the shallow arch are similar but different from those of Cable 1. From a view of energy transfer, it might be that two cables are identical but the energy sources sustaining the vibration of Cable 1 and the other two members, namely Cable 2 and the shallow arch, are different. The energy inducing vibration of Cable 1 is coming from the external resonance, which leads to a relatively large response. However, the energy sustaining the motion of Cable 2 is coming from the excitation of the shallow arch, and the energy sustaining the vibration of the shallow arch is coming from Cable 1, indirectly. The response amplitude of the shallow arch is very small (almost close to zero) because the total energy coming from Cable 1 is not enough to support its large vibration, and some of them are transferred to Cable 2. It is interesting that, why can the small energy coming from the shallow arch support the large vibration of Cable 2 as shown Figs. 6(b)–8(b)? The main reason may be that Cable 2 is excited by the parametric and forced load at its lower end. The parametric and forced excitation can induce the large vibration even the excitation force is very small^[47]. This implies that it is almost invalid to control the vibration of the deck in order to control the vibration of the cable.

The bifurcation and jump in the force-response curves of the system are also observed in Figs. 6 and 7. There are two changing routes of solutions when the excitation force increases from zero as described in Fig. 6. One way is that, with the increasing of the excitation force, a jump induced by a saddle node bifurcation (SN1) appears, and then the response amplitude of Cable 1 decreases, but the response amplitudes of the other two members increase in a small force range; then all response amplitudes increase with the increasing of the excitation force till another saddle node bifurcation (SN2) occurs, which triggers another jump phenomenon; all the response amplitudes increase with the further increasing of the excitation force. The other way is that the saddle bifurcation (SN3) makes the solutions jump to the upper stable branch and then increase with the further increasing of the excitation force. On the contrary, there are also two changing routes of solutions when the excitation force decreases from 0.5. Additionally, the saddle node bifurcations (SN1 and SN2) and Hopf bifurcations (HB1, HB2, HB3, and HB4) are also identified in the force-response curves in Fig. 7, which should be given more attention. It is seen that, from the amplitude-response curves when σ < 0 as shown in Fig. 8, increasing the frequencies of members of cable-stayed bridge can effectively reduce the generation of complex dynamic behaviors, which can be used to compress the vibration of cables in design.

4.2 Both cables subject to harmonic excitation

In this section, nonlinear dynamic behaviors of the double cable-stayed shallow arch system with both two cables subject to the external harmonic excitations are investigated. Similarly, the frequency-response and force-response curves are used to explore dynamic behaviors of the system. Figure 9 presents frequency-response curves of cables and the shallow arch with the same parameters mentioned above and the identical external excitation force as . It should be noted that a new triple-peak is firstly observed in frequency-response curves of the shallow arch, and the triple-peak bends to the lower direction of the horizontal axis of detuning the parameter σ for the softening property in Fig. 9(a). In Fig. 9(b), it can be seen that there are four stable branches of solutions since there are almost four peaks of the frequency-response curves, where two of them arise from the TB. Hence, although only three stable branches of solutions of the shallow arch are observed, there are actually four stable branches as σ < 0 in Fig. 9(a), where two branches arising from the TB are overlapped. Additionally, it is also seen that, for σ > 0, the jump phenomena of cables disappear since there is just one stable branch of the solution. By comparing curves in Fig. 9 with those in Figs. 3–5, it is seen that the Hopf bifurcation in the lower stable branch of solutions also disappears since another excitation is applied on Cable 2. Therefore, the dynamic behaviors of the system depend on not only the dynamic model but also its applied excitation. The excitation applied on Cable 2 not only increases the stable branches of solutions but also controls some of the bifurcations.

Accordingly, the force-response curves of the system in this case are presented. Figure 10 shows the force-response curves of two cables and the shallow arch as σ=0.3. The Hopf bifurcation and jump phenomenon in the force-response curves of the system are observed again. With the increasing of the excitation force, the response amplitudes of cables and the shallow arch increase sharply and slightly in a small range, respectively, and then lose stability through the Hopf bifurcation (HB), after a jump phenomenon can be observed with the further increasing of the excitation force. On the other hand, with the decreasing of the excitation force from 0.5, the response amplitudes of cables and the shallow arch decrease till a small value where they jump to zero.

Figures 11 and 12 present force-response curves of cables and the shallow arch as σ = −0.5, respectively. It should be noted again that there are four stable branches for cables, but only three stable branches for the shallow arch are observed as shown in these figures. The force-response curves are relatively simple. With the increasing of the excitation force, the response amplitude of the shallow arch increases, but those of two cables might increase or decrease, which depends on the initial deflection of cables.

Generally, the applied load on each cable might be different since the environmental load is very complex in engineering. In the following, the case of two external excitations with different amplitudes by fixing is considered, and their force-response curves are given. Figures 13 and 14 show the force-response curves of cables and the shallow arch on the conditions of and , respectively. These curves are relatively simple. The jump phenomena of cables and the shallow arch disappear in Fig. 13 compared with the former case in Fig. 10. The Hopf bifurcations of them appear in Fig. 14 compared with the former cases in Figs. 11 and 12. Moreover, it can be seen that the responses of the shallow arch and Cable 2 always increase monotonically but that of Cable 1 might increase or decrease as shown in Figs. 14(e) and 14(h) when the excitation frequency is less than the natural frequency of cables. The asymmetric excitation and the complex intercoupling might cause different energy transfer between the shallow arch and different cables. The downtrends of force response curve should be taken seriously, which tell us that the dynamic response and the excitation amplitude may not be necessarily proportional.

5 Conclusions

The partial differential equations governing in-plane motion of a double cable-stayed shallow arch model are derived when harmonic excitations are applied to cables, and the nonlinear dynamic behaviors of the system with one-to-one-to-one internal resonance and the external primary resonance of cables are investigated. Galerkin's integral method is used to obtain ODEs, and the multi-scale method is used to derive modulation equations governing the amplitude and phase of the system. Furthermore, the frequency- and force-response curves are given to investigate in-plane nonlinear dynamic behaviors of the system.

Moreover, in order to consider the effect of a trial function on a nonlinear system, the common sine function and the exact piecewise function are chosen as the trial functions in Galerkin's integration, respectively. The validity and accuracy are further verified by analyzing nonlinear dynamic responses of the system. Additionally, the influence of asymmetry caused by the external excitation on the dynamic behaviors of the system is considered. Some interesting and meaningful conclusions can be drawn as follows:

(ⅰ) There only exists a slight difference between the results obtained by two different trial functions, i.e., sine and piecewise functions. The exact piecewise function as the trial function of the shallow arch has a better accuracy because both the geometrical and mechanical conditions are considered.

(ⅱ) The multi-peaks of the frequency-responses of both the shallow arch and two cables are firstly and simultaneously observed in the double cable-stayed shallow arch system.

(ⅲ) The behaviors of two identical cables are different under the condition of the external excitation applied to one of cables, because one is excited by a forced excitation, and the other one is excited by the parametric and forced excitation coming from the motion of the shallow arch. Due to the internal resonance and the complex intercoupling between cables and the shallow arch, the cable that does not bear the load directly can behave larger responses, which only has the double jump phenomenon in the frequency-response curves.

(ⅳ) Under different loading conditions, the frequency-responses of cables may behave a softening or hardening property while those of the shallow arch always behave a softening property. When cables are subject to the symmetry excitation, dynamic behaviors of the system are relatively simple, and some Hopf bifurcations vanish.

(ⅴ) Different initial deflections of two cables can lead to different variation trends (the downtrend or uptrend) of their response amplitudes and generate different jump phenomena. In addition, the overlapping solution can make the jump phenomenon disappear.

From the above discussion and conclusions, it is seen that the one-to-one-to-one internal resonance among cables and the shallow arch induces energy transfer among them and simultaneously causes complex dynamic behaviors, such as jump phenomena and bifurcations. Moreover, different initial disturbance and the symmetry of external excitations can also affect the dynamic behaviors of the system. Hence, the nonlinear dynamic analysis should be paid more attention in engineering design of the cable-stayed bridge.