Dynamic Responses Of Multi-Cable Parallel Suspension System In Deep Shaft Construction

The multi-cable parallel suspension system is the most important auxiliary system when constructing a vertical shaft.It consists of a suspended platform,several suspension cables and their winder drums.By controlling all the sinking winches,the platform can be lifted up or lowerd down in the shaft.However,on the one hand,due to the flexibility of the suspension cables,the coupled relations between all the suspension cables and the excitations act on the system,the dynamic responses of the platform might be quite complicated.On the other hand,due to the driving differences between all the sinking winches,there might be a large length difference of the cables,thus,one or more cable may become slack,and when the slack cable become tensioned again,a shock will be applied on the platform,which can cause severe accidents of the sinking system.So,it is necessary to investigate the dynamic responses of the multicable parallel suspension system,explore the cable dynamics influence on the suspension platform,find the non-smooth behaviuor of the platform when the system involves slack cables,thereby give an accurate prediction of the dynamic responses of the system,as well as provide a theoretical foundation for the design of levelling control strategy for the suspension platform.Firstly,in order to accurately describe the dynamic responses and cable tension characters of the multi-cable suspension system,an accurate dynamical model of the system is built.In this model,cables are treated as continuum,their longitudinal,lateral,torsional and longitudinal-torsional coupled characters are included,as well as the 6 degrees of freedom of the platform.By using the Lagrange’s equation with constraints and with the help of the assumed mode method,the equations of motion of the system are obtained and expressed in the form of differential algebraic equations.Then,a second order accurate numerical solving method,the generalized-α method,is employed to numerically solve the obtained equations.By a mutual comparison with an ADAMS simulation,the accuracy and validity of the model and the numerical results are preliminarily validated.Based on the mathematical model,a thorough analysis on the dynamic responses of the system and the cable tension characters are presented,through which one can tell that the mechanical model with uniformly distributed mass can generally describe the dynamic characters of a multi-cable suspension system.A more proper cable arrangement pattern is also proposed.Then,in order to simulate the dynamic responses of the multi-cable parallel suspension system when involving slack cables,a non-smooth dynamic model of the system is built,in which both the unilateral character of the cable in longitudinal direction and the bilateral characters in lateral and torsional directions are considered.Based on the complementary relationship between the cable tension and the geometric matching condition of the cable and the platform,the slack cable is described mathematically,and the corresponding non-smooth dynamic equations are also derived.In order to solve the derived non-smooth dynamic equations which consists both the unilateral constraints and bilateral constraints,the equations are rewritten at velocity level,and a non-smooth generalized-α method is employed here to numerically solve the derived equations by modifying the constraint equations expressions at velocity level.Again,an ADAMS simulation is carried out to verify the proposed mechanical model and numerical method.Afterwards,based on the mechanical model and the numerical method,a thorough analysis is presented to investigate the non-smooth characters of the multi-cable suspension system.Through the analysis,one can find that the cable will not become slack easily when at a long cable length.A length differences control range is also given corresponding to different cable lengths.Finally,in order to analyze the dynamic responses of the multi-cable suspension system when the platform moves with guiding devices,a non-smooth mechanical model for such scenario is built by neglecting the lateral and torsional characters of the system.Following the method presented previously,the dynamic responses of the system is analyzed,and the pressure properties of the platform applied on the shaft are derived.The lateral natural frequency of the system with and without guiding devices are derived numerically,and the longitudinal natural frequencies of the system with different cable length are derived by both numerical method and analytical method.The simulation results have shown that the platform mass has little effect on the second or higher order longitudinal natural frequencies,and a probable longitudinal resonance due to the excitation of the sinking machines is analyzed.