A position equation of saddle-node for end-excited suspended cables under primary resonance

Systems exhibiting Saddle-Node (SN) bifurcations are often characterized by drastic amplitude and phase jumps, representing a crucial state in engineering scenarios. The accurate and efficient prediction of SN points is fundamental for the comprehensive understanding and control of dynamical systems. This paper derives an equation for locating SN points based on the dimensionless governing equation for the in-plane primary resonance of a suspended cable. It reveals that the SN points for the cable are influenced by three key parameters: the cable’s effective nonlinearity Γem, the excitation parameter F2, and damping ratio. Importantly, when the cable is subjected solely to horizontal end excitation, the product of Γem and F2 emerges as a new parameter, Λem. The effects of parameter Λem (Γem), damping, axial, and vertical excitation amplitudes on SN points are investigated. Findings indicate that these key parameters more significantly affect the SN1 (the one near peck point) than SN2, and slight variations in Λem (Γem) or vertical excitation amplitude can lead to substantial alterations in SN1. The effect of Λem (Γem) on SN1 is asymmetric, with the values of σ and a being significantly higher when Λem (Γem) is positive than when negative. The computational results of the SN position equation closely align with experimental observations and the literature, demonstrating good computational efficiency.