A discrete model of a rope with spiral springs in joints is considered, the aim being to include transverse elasticity of the rope. Elastic characteristic of the springs is derived on the basis of simple geometrical formulas and the classical curvature-bending moment relationship for beams. Lagrange’s equations of motion are presented and their complexity is discussed from the computational point of view. Numerical experiments are performed for a system with both scleronomic and rheonomic constraints. The influence of the elasticity on behaviour of the model is analyzed. Results validity is examined in terms of basic energy principles.

A mechanical system composed of two weakly coupled oscillators under harmonic excitation is considered. Its main part is a vibro-impact unit composed of a linear oscillator with an internally colliding small block. This block is coupled with the secondary part being a damped linear oscillator. The mathematical model of the system has been presented in a non-dimensional form. The analytical studies are restricted to the case of a periodic steady-state motion with two symmetric impacts per cycle near 1:1 resonance. The multiple scales method combined with the sawtooth-function-based modelling of the non-smooth dynamics is employed. A conception of the stability analysis of the periodic motions suited for this theoretical approach is presented. The frequency–response curves and force–response curves with stable and unstable branches are determined, and the interplay between various model parameters is investigated. The theoretical predictions related to the motion amplitude and the range of stability of the periodic steady-state response are verified via a series of numerical experiments and computation of Lyapunov exponents. Finally, the limitations and extensibility of the approach are discussed.

Two-dimensional motion of a rope fixed at one end is considered. The Rigid Finite Element Method (RFEM) is reviewed and applied to obtain a model of the rope, including its elastic and dissipative properties. Equations of motion are derived without the small displacement assumption, using the Lagrange equations. The resulting model is compared to another one, being derived within the framework of standard analytical mechanics methods and the Lagrange formalism. Advantages of the RFE approach are discussed from a computational point of view. The presented, alternative model can be a basis for e cient numerical simulations, which seem to be useful in further, comparative studies of the rope dynamics.