## Abstract

To study the dynamic behavior of cable-stayed bridges, a linear multi-cable-stayed beam model is developed to investigate its in-plane and out-of-plane transverse vibrations. From the full-bridge perspective, an in-plane nonlinear single-mode discrete model is established. First, the in-plane and out-of-plane motion equations of the system and their boundary conditions are derived. Second, by employing the separation of variables method, the linear eigenvalue problems are solved. The influences of the mass ratio, stiffness ratio, and cable sag on the occurrence of global, local, and coupled vibration modes are studied. Third, frequency response, amplitude response, phase diagram, time history, and power spectrum are extracted to investigate the system’s nonlinear dynamic behaviors. The obtained results demonstrate that for the in-plane motion, the occurrence of global and local modes of the system depends on the mass and stiffness ratios between cable and beam significantly; for the out-of-plane motion, without the elastic support of the cable, the global modes occur, which can be suppressed by adjusting the mass and stiffness ratios between cable and beam but may in turn induce the cable’s local vibration modes; for the nonlinear analysis, the single-degree-of-freedom system behaves like a hardening spring. Its lower branch behaves more complicated than the higher one and has a double-periodic steady-state solution. The system with large damping ratio behaves shows weak hardening spring property.

### Similar content being viewed by others

## References

Fujino, Y., Warnitchai, P., Pacheco, B.M.: An experimental and analytical study of autoparametric resonance in a 3DOF model of cable-stayed-beam. Nonlinear Dynam.

**4**, 111–138 (1993)Warnitchai, P., Fujino, Y., Susumpow, T.: A nonlinear dynamic model for cables and its application to a cable-structure system. J. Sound Vib.

**187**(4), 695–712 (1995)Xia, Y., Fujino, Y.: Auto-parametric vibration of a cable-stayed beam structure under random excitation. J. Eng. Mech.

**132**(3), 279–286 (2006)Gattulli, V.A.P.: Planar motion of a cable-supported beam with feedback controlled actions. J. Intel. Mat. Syst. Str.

**8**, 767–774 (1997)Gattulli, V., Paolone, A.: A parametric analytical model for non-linear dynamics in cable-stayed beam. Earthq. Eng. Struct. D

**31**, 1281–1300 (2002)Fung, R.F., Lu, L.Y., Huang, S.C.: Dynamic modelling and vibration analysis of a flexible cable-stayed beam structure. J. Sound Vib.

**254**(4), 717–726 (2002)Lenci, S., Ruzziconi, L.: Nonlinear phnomena in the single-mode dynamics of a cable-supported beam. Int. J. Bifurcat. Chaos

**19**(3), 923–945 (2009)Zhu, J., Ye, G.R., Xiang, Y.Q., Chen, W.Q.: Dynamic behavior of cable-stayed beam with localized damage. J. Vib. Control

**17**(7), 1080–1089 (2010)Wei, M.H., Xiao, Y.Q., Liu, H.T.: Bifurcation and chaos of a cable-beam coupled system under simultaneous internal and external resonances. Nonlinear Dynam.

**67**(3), 1969–1984 (2011)Wei, M.H., Xiao, Y.Q., Liu, H.T., Lin, K.: Nonlinear responses of a cable-beam coupled system under parametric and external excitations. Arch. Appl. Mech.

**84**(2), 173–185 (2013)Wei, M.H., Lin, K., Jin, L., Zou, D.J.: Nonlinear dynamics of a cable-stayed beam driven by sub-harmonic and principal parametric resonance. Int. J. Mech. Sci.

**110**, 78–93 (2016)Wang, Z., Sun, C., Zhao, Y., Yi, Z.: Modeling and nonlinear modal characteristics of the cable-stayed beam. Eur. J. Mech. A-Solid.

**47**, 58–69 (2014)Wang, Z., Yi, Z., Luo, Y.: Energy-based formulation for nonlinear normal modes in cable-stayed beam. Appl. Math. Comput.

**265**, 176–186 (2015)Peng, J., Xiang, M., Wang, L., Xie, X., Sun, H., Yu, J.: Nonlinear primary resonance in vibration control of cable-stayed beam with time delay feedback. Mech. Syst. Signal Pr.

**137**(106488), 1–16 (2019)Wang, Z.: Modelling with Lagrange’s method and experimental analysis in cable-stayed beam. Int. J. Mech. Sci.

**176**(105518), 1–13 (2020)Liu, M., Zheng, L., Zhou, P., Xiao, H.: Stability and dynamics analysis of in-plane parametric vibration of stay cables in a cable-stayed bridge with superlong spans subjected to axial excitation. J. Aerospace Eng.

**33**(1), 04019106 (2020)Haibo, L., Jianjun, X.: Analysis on sensitivity of global mode of floating cablestayed bridge to CFRP cable. J. Dynam. Control

**16**(3), 250–257 (2018). (In Chinese)Zhang, Y., Fang, Z., Jiang, R., Xiang, Y., Long, H., Lu, J.: Static performance of a long-span concrete cable-stayed bridge subjected to multiple-cable loss during construction. J. Bridge Eng.

**25**(3), 04020002 (2020)Cao, D.Q., Song, M.T., Zhu, W.D., Tucker, R.W., Wang, C.H.T.: Modeling and analysis of the in-plane vibration of a complex cable-stayed bridge. J. Sound Vib.

**331**(26), 5685–5714 (2012)Song, M.T., Cao, D.Q., Zhu, W.D., Bi, Q.S.: Dynamic response of a cable-stayed bridge subjected to a moving vehicle load. Acta Mech.

**227**(10), 2925–2945 (2016)Kang, H.J., Guo, T.D., Zhao, Y.Y., Fu, W.B., Wang, L.H.: Dynamic modeling and in-plane 1:1:1 internal resonance analysis of cable-stayed bridge. Eur. J. Mech. A-Solid.

**62**, 94–109 (2017)Cong, Y., Kang, H., Guo, T.: Planar multimodal 1:2:2 internal resonance analysis of cable-stayed bridge. Mech. Syst. Signal Pr.

**120**, 505–523 (2019)Cong, Y., Kang, H., Guo, T.: Analysis of in-plane 1:1:1 internal resonance of a double cable-stayed shallow arch model with cables’ external excitations. Appl. Math. Mech. Engl.

**40**(7), 977–1000 (2019)Luongo, A., Rega, G., Vestroni, F.: Planar non-linear free vibrations of an elastic cable. Int. J. Nonlin. Mech.

**19**(1), 39–52 (1984)Irvine, H.M.: Cable Structures. MIT Press, Cambridge (1981)

Nayfeh, A.H.: Nonlinear Oscillations. Wiley, New York (1979)

Benedettini, F., Rega, G.: Non-linear dynamics of an elastic cable under planar excitation. Int. J. Nonlin. Mech.

**22**(6), 497–509 (1987)Kamel, M.M., Hamed, Y.S.: Nonlinear analysis of an elastic cable under harmonic excitation. Acta Mech.

**214**(3), 315–325 (2010)Nayfeh, A.H.: The response of non-linear single-degree-of-freedom systems to multifrequency excitations. J. Sound Vib.

**102**(3), 403–414 (1985)Zhao, Y., Guo, Z., Huang, C., Chen, L., Li, S.: Analytical solutions for planar simultaneous resonances of suspended cables involving two external periodic excitations. Acta Mech.

**229**(11), 4393–4411 (2018)Irvine, H.M., Caughey, T.K.: The linear theory of free vibrations of a suspended cable. P. Roy. Soc. A Math. Phy.

**341**(1626), 299–315 (1974)Wang, L., Zhao, Y.: Nonlinear interactions and chaotic dynamics of suspended cables with three-to-one internal resonances. Int. J. Solids Struct.

**43**(25–26), 7800–7819 (2006)

## Acknowledgements

This study is financially supported by National Science Foundation of China under Grant Nos. 11572117, 11502076, 11872176 and 11972151, and China Scholarship Council.

## Author information

### Authors and Affiliations

### Corresponding author

## Ethics declarations

### Conflict of interest

The authors declare that they have no conflict of interest.

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Appendix

### Appendix

For compact expression, the following integrals are introduced to define the coefficients \(c_{i}\) and the excitation amplitude \(F_{i}\) in Eq. (28).

where \(f_{i}\) are the amplitudes of external harmonic excitation applied to the components of the beam. Therefore, the coefficients \(c_{i}\) and \(F_{i}\) are expressed as

The coefficients in Eq. (43) are given as

## Rights and permissions

## About this article

### Cite this article

Cong, Y., Kang, H., Yan, G. *et al.* Modeling, dynamics, and parametric studies of a multi-cable-stayed beam model.
*Acta Mech* **231**, 4947–4970 (2020). https://doi.org/10.1007/s00707-020-02802-8

Received:

Revised:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s00707-020-02802-8