Nonlinear Dynamics of Cable Structures

Cable structures — ranging from long‐span bridges and aerial tramways to tensioned membrane roofs and offshore mooring lines — exhibit rich dynamic behaviour driven by inherent geometric nonlinearity and time‐varying tension. When subjected to transverse loading, end‐motion excitation or aerodynamic forcing, cables can undergo large‐amplitude oscillations that depart substantially from the predictions of linear theory. Key features include amplitude-dependent natural frequencies, internal resonances between modes, parametric instabilities arising from axial motion of end supports and aeroelastic galloping in exposed spans. Analysis techniques such as the method of multiple scales, harmonic balance, perturbation expansions and direct numerical integration have been used to derive reduced‐order models capturing nonlinear modal coupling, bifurcation phenomena and limit-cycle responses. Understanding these effects is vital for ensuring the safety and serviceability of cable‐supported infrastructure, for reducing fatigue damage in dynamic environments and for guiding the design of active or passive control measures to mitigate excessive vibration.

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Recent studies of taut inclined cables under combined direct and parametric excitation have demonstrated that geometric nonlinearity leads to stiffening effects and strong modal interactions. Analyses based on scaling and averaging have produced non-dimensional algebraic equations governing steady-state amplitudes in multiple modes, revealing conditions under which in-plane and out-of-plane modes exchange energy and how cable sag, even when small, significantly alters both response amplitudes and stability boundaries.

Revisited modelling of sagged cables subject to support motion has explored multimodal nonlinear oscillations near internal resonances. By employing a reduced-degree‐of-freedom description and multi-time‐scale methods, researchers have shown that symmetric and antisymmetric support displacements can trigger complex bifurcation scenarios, leading to transition between periodic, quasi-periodic and chaotic regimes. Numerical simulations confirm analytical predictions and highlight the influence of sag profile and excitation amplitude on the emergence of large, coupled amplitude responses.

Investigations into in-plane aeroelastic galloping of shallow flexible cables under steady flow have applied continuum modelling with Rayleigh‐type damping and quasi-steady aerodynamic forcing. Perturbation procedures reveal how internal damping and quadratic nonlinearities affect both the critical wind speed for galloping onset and ensuing limit-cycle amplitudes. Results emphasise the importance of cable inclination, ice accretion and damping distribution in suppressing or exacerbating wind-induced oscillations.

Nonlinear Dynamics of Cable Structures publication trend

The graph below shows the total number of articles in nonlinear dynamics of cable structures across all publications each year (not limited to Nature Index journals).

Technical terms

Parametric excitation: Time-varying system parameters (for example, tension due to axial support motion) that can induce dynamic instabilities and subharmonic or combination resonances.

Modal coupling: Interaction between vibrational modes arising from nonlinear terms in the equations of motion, allowing energy transfer and mixed‐mode oscillations.

Geometric nonlinearity: Nonlinear relationship between deformation and restoring force when displacements are large, producing amplitude-dependent stiffness and frequency shifts.

Sag: Static curvature of a cable under self-weight, which modifies its tension distribution and fundamental mode shapes.

Bifurcation: Qualitative change in system response (for example, transition from stable equilibrium to periodic oscillation) when a control parameter crosses a critical threshold.

References

1. Multi-modal vibration amplitudes of taut inclined cables due to direct and/or parametric excitation. Journal of Sound and Vibration (2016).
2. Revisited modelling and multimodal nonlinear oscillations of a sagged cable under support motion. Meccanica (2016).
3. A Continuum Approach to the Nonlinear In‐Plane Galloping of Shallow Flexible Cables. Advances in Mathematical Physics (2019).