This paper studies a position feedback control strategy for controlling a higher order drifting oscillator which could be used in modelling vibro-impact drilling. Special attention is given to two control issues, eliminating bistability and suppressing chaos, which may cause inefficient and unstable drilling. Numerical continuation methods implemented via the continuation platform COCO are adopted to investigate the dynamical response of the system. Our analyses show that the proposed controller is capable of eliminating coexisting attractors and mitigating chaotic behaviour of the system, providing that its feedback control gain is chosen properly. Our investigations also reveal that, when the slider’s property modelling the drilled formation changes, the rate of penetration for the controlled drilling can be significantly improved.

Vibrations of cables with a small bending stiffness concerns many engineering applications such as the fatigue assessment of stay cables. With the finite element (FE) method, the analysis can be performed with nonlinear truss elements, but bending effects are not taken into account. Otherwise, beam elements can be used, but the smallness of the bending stiffness may lead to numerical instability and mediocre results in boundary regions. In this context, the paper presents an alternative method to calculate the time evolution of the profile of bending moments in boundary layers of cables, avoiding heavy FE analysis. The developments combine the theory of vibrations of extensible rods with asymptotic methods. The equations are decoupled between the slow dynamics of the boundary regions and the fast dynamics of the span. Then, a composite solution is constructed by means of a matched asymptotic procedure.

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