Abstract
In this chapter, the vibration behaviour of cables is considered. The starting point is to consider horizontal cables, which are initially assumed to be inextensible. Of particular importance is cable sag, the static displacement of a cable due to gravity. Sag results in cables having complex dynamic behaviour. This is seen when the nonlinear equations of motion for an inclined cable are developed. Inclined cables are important for applications such as cable-stayed bridges. Galerkin’s method is used to convert the nonlinear partial differential equations into a set of modal equations in which the linear terms are decoupled. However, modal coupling remains in the nonlinear terms. These nonlinear coupled terms lead to internal resonance, such as autoparametric resonance. This type of resonance can be observed for cable-stay bridges when certain combinations of external excitation frequency, deck frequency and cable mode frequency occur. In the final part of the chapter, two case studies of cable vibration are considered. Firstly the techniques of averaging, multiple scales and normal forms are compared when applied to the analysis of a single mode of vibration of an inclined cable. Then the modal interaction between the second in- and out-of-plane modes is considered when the cable is subjected to a vertical support motion close to the second natural frequency. This second case study uses the normal forms technique to find the backbone curves for the system and then to identify a region in which there is response from both the directly excited second in-plane mode and the auto-parametrically excited out-of-plane mode.
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Notes
- 1.
To validate this equation for odd \(n\), note that \(\phi _{n}^{\prime \prime }=(B_{n}/\ell )^{2}[(1-\sec (B_{n}/2))^{-1}-\phi _{n}]\) and that \(\int _{0}^{\ell }\phi _{n}\mathrm{d}x=[1-\sec (B_{n}/2)]^{-1}(\ell B_{n}^{2}/\lambda ^{2})\). Substituting these into Eq. (7.55) gives \(\omega _{n}^{2}= [T_{sx}B_{n}^{2}/(\rho A\ell ^{2})](\int _{0}^{\ell }\phi _{n}^{2}\mathrm{d}x)/(\int _{0}^{\ell }\phi _{n}^{2}\mathrm{d}x)\). So \(\omega _{n}^{2}=T_{sx}B_{n}^{2}/(\rho A\ell ^{2})\) which gives the same relationship for \(B_{n}\) as its definition given in Eq. (7.46).
- 2.
The definition of secular terms can be found in Sect. 4.4.2.
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Wagg, D., Neild, S. (2015). Cables. In: Nonlinear Vibration with Control. Solid Mechanics and Its Applications, vol 218. Springer, Cham. https://doi.org/10.1007/978-3-319-10644-1_7
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