Study On The Causes And Propagating Characteristics Of Stress Waves In A Deep-sea Mining Deployment Cable

With the development of world economy,the demand for resources is also rising.More attention has been laid on deep-sea high-grade mineral resources due to the decrease of land resources.Thus deep-sea mining system became the focus of the research of all countries.Compared to common shallow-water deployment systems,the vertical deployment distance for deep-sea mining system is longer.The deployment system for deep-sea mining system also need to bear a greater load compared with the deployment system for manned or unmanned vehicles.So more challenges might be identified for the deployment process of deep-sea mining system.The security of the entire deployment system thus became an important topic.With certain displacement ratio,damping ratio,stress wave damping ratio and frequency ratio,or when the deployment system encounters sudden force generated by wind,wave or current,a large move-ment of the deployment system may occur and the deployment cable may become slack,which would lead to a sudden load of high amplitude propagating in the form of stress wave,which may be dangerous to the deployment system.So it is of great importance for the safety of sys-tem that the cause and propagation characteristics of the high amplitude stress wave be deeply studied.Theoretical analytical calculation and numerical calculation are combined in the present paper.The cause and production of high amplitude sudden load,with the propagation of stress wave taken into consideration,the spectral characteristics of the propagation of stress wave,the temporal characteristics of the propagation of stress wave,the impact of the propagation of stress wave on the maximum stress in the cable and the impact of different cable properties are studied in the present paper.The mathematical model was built relatively considering and ignoring the effect of stress wave propagation.The cable was modeled as a one-dimension continuum and its single de-gree of freedom(SDOF)dynamic equations were established.The motion of the system with different dimensionless displacement ratios,damping ratios,stress wave damping ratios and frequency ratios were calculated numerically.Slack-taut critical transition curves were plotted.When the slack-taut occurs,a high loading rate stress load will be generated.It will propagate as a stress wave in the deployment system.The research indicates that when the effect of stress wave propagation is neglected,all critical transition curves pass trough the point where the fre-quency ratio is(?)/2 and displacement ratio is 1.When the frequency ratio is less than(?)/2,the critical displacement ratio increases with the increase of damping ratio,otherwise the criti-cal displacement ratio decreases with the increase of damping ratio.When the frequency ratio is less than 1,the critical displacement ratio increases with the increase of frequency ratio and it reaches a peak at unity frequency ratio.Afterwards,when the frequency ratio is greater than 1,the critical displacement ratio becomes lower at higher frequency ratio.The stability of the deployment system might be augmented through increasing its displacement ratio at the same frequency ratio.When the critical displacement is greater than 1,the stableness of the system can be obtained by increasing or decreasing frequency ratio,however when it' s less than 1,only by decreasing frequency ratio might the system be stable.When the stress wave propagation ef?fect is considered,not all of the critical curves pass through the point where the frequency ratio is(?)/2 and displacement ratio is 1.When the stress wave damping ratio is significant,the local maximum near unit frequency ratio might disappear,the critical displacement become thus a monotonically increasing curve,and the system is always taut and stable when its displacement ratio is greater than 1.The deployment cable has been modeled as a one-dimensional elastic uniform continuum moving in three-dimensional space.Its equations of motion are resolved nonlinearly by adopt-ing Hamilton' s Principle.The spectral characteristics of the propagation of stress wave were calculated numerically and analyzed in detail and the frequency limit curves,frequency relation curves,dispersion curves and group velocity curves are obtained.Four distinct frequency bands are identified for longitudinal waves,depending on cable' s property and the vertical and hori-zontal forces acting on the cable.Partly pass-bands and stop-band exist at low frequencies.The larger frequency limit decreases with the increase of longitudinal tension and increases with the rise of cl,which is only related to the cable' s property.The relationship between the larger fre-quency limit and the transversal tension is relatively complicated.Generally,the limit increases with the increase of transversal tension at high longitudinal tension.However,the frequency limit decreases with the increase of transversal tension when the longitudinal tension is small.The smaller frequency limit does not decrease monotonically with the increase of longitudinal tension.A local maximum can be identified.The longitudinal tension where the local maximum occurs is the critical tension,start from which the frequency limit becomes complex.If both of the frequency limits of the system are purely real,the frequency band between these two fre-quency limits is stop band.Otherwise if they are complex,that band stays partly passing-band.At short wavelength-high frequency limit,the phase velocity of longitudinal and transversal propagating stress waves tends to the phase velocity calculated by adopting the classical model-the phase velocity of the stress wave propagating in a one-dimension semi-infinite bar,at the limit,the system becomes non-dispersive.At pass-band,the longitudinal stress wave may prop-agate along the cable and its amplitude decreases at a certain attenuation ratio.The system is dispersive.At partly pass-bands,two one-directional local evanescent waves excited by the source with different wavenumbers take place instead of two traveling longitudinal stress waves propagating along the cable.At stop-band,no longitudinal waves propagate either,only two evanescent waves towards two different directions can be found.Transversal stress waves always propagate along the cable at any frequencies,no stop-band or partly pass-band can be observed.The system is also dispersive for transversal stress waves taking the nonlinear relation between the curvature of deployment cable and the cable arc coordinate.The group velocity of both transversal and longitudinal stress waves tend to their phase velocity at high frequencies.How-ever,dispersion relation of transversal stress waves is anomalous.Thus a strong-discontinuous wave front will form during the propagating,which may cause a huge dangerous stress-strain change at the wave front and therefore may be undesirable for the deployment system.The motion of the deployment system in three-dimensional space was modeled.The impact of hydrodynamic force,damping and material properties on system was taken into considera-tion.The equations of motion were deducted by adopting Hamilton' s Principle.The response of system in time domain was obtained by numerical calculation.The production of sudden load in deployment cable,the propagation characteristics of stress wave,the influence of stress wave propagation on the maximum tension of the cable and the characteristics of stress loading rate during the propagation were analyzed.The impact of different cable properties on propagation characteristics and maximum tension was also analyzed in detail.The results indicates that,when the system falls in the unstable area,a sudden load will occur due to the slack-taut effect,a stress wave of high loading rate and large amplitude will be produced and propagate along the cable.The order of magnitude of an entire stress-wave propagating time in a short-distance swallow water deployment system is less than order of magnitude of the characteristic time of the system.The propagation of the stress wave is thus difficult to be identified and it has little in-fluence on the force curve and the maximum tension on the system.It acts like a high-frequency addition on the force curve.However,the order of magnitude of an entire stress-wave propagat-ing time in a long-distance deep water deployment system is the same as the order of magnitude of the characteristic time of the system.The effect of its propagation can be clearly identified and it has a significant impact on the time plot and the maximum tension on the cable.When the deployment system is in the stable area where it is always taut,no slack-taut and its rele-vant stress wave propagation of sudden impact will occur.The sudden distribution due to other causes will still propagate in form of stress wave in the system and will reflect at two ends of the cable.Its amplitude will attenuate exponentially during the propagation.When the deployment is in unstable area,the slack-taut effect will generate a high-loading-rate and large-amplitude sudden tension.The tension will propagate in form of stress wave and reflect at two ends.The properties of the stress wave,such as phase velocity,etc.corresponds the properties calculated in the spectral analysis.Meanwhile,the propagation of stress wave changes the distribution of maximum tension in the cable.The maximum tension at the middle of deployment system keeps still,however,due to the effect of stress wave reflection,the maximum tension near two ends of the cable is larger.For the part near the suspension point near sea-level,its maximum tension increases with the decrease of the distance between the point and the suspension point.At the suspension point,it reaches a maximum of about two times of the tension at middle of cable.For the part near the suspension point near the load,its maximum tension also increases with the decrease of the distance between the point and the suspension point.It reaches a local maximum before the suspension point and then drops.The viscoelasticity reduces the tension-increasing effect due to the stress wave.With the increase of viscoelasticity,the ratio of the maximum tension at the suspension point and the average tension at middle of cable decreases exponentially and tends to one.The points where the tension starts to start near two end points apart from the suspension point with the increase of viscoelasticity.The viscoelasticity of cable reduces the loading rate of the front of stress wave during the propagation,which makes the sys-tem safer.However,the viscoelasticity of deployment cable increases the initial loading tension at the debut of loading,which endangers the system.