Dynamics of a space vehicle with elastic deploying tether

The mechanics of motorised momentum exchange tethers when applied to active debris removal from LEO

“…The three equations of motion are derived using a Lagrangian derivation similar to that of section 1, and emerge in the following forms, noting that damping of the form and is subsequently added into the equations in l 1 and l 2 , respectively, in order to get the numerical results presented below in Figures 12 and 13. (17) (18) (19) Numerical integration of equations (17)- (19) is Smooth and well controlled deployment is evident for both sub-spans in Figures 12(a) and 12(b), with a brake actuated during the interval of 1870 to 1910 s. Many other deployment characteristics can be identified for this system but it is considered that the profiles shown here are indicative of a desirable deployment function. Figure 13 shows the response of the system in where a small amplitude libration is evident about a 2 radian offset during deployment until braking starts, and then a large swing and subsequent return to small angle libration about the offset of 2 radians, at around 6000 s. Clearly once the two sub-spans are fully deployed then the deployer can be locked, effectively returning the system to a single degree of freedom model with freedom to revert to any chosen form of motion after around 2000s.…”

“…In all cases it is possible to deploy from the spool by means of active propulsion or ejection of the free end out from the supply module, but in the case of motorised tethers, which by definition librate or spin, then there is the additional option of some form of centripetal deployment. Centripetal deployment is examined in the last section of the paper and although this is not a new concept per se, and the general principles have been discussed in some detail by Danilin et al [19] and Levin [20], the modelling of a variable length motorised tether undergoing centripetal deployment is discussed for the first time by Kirrane [21]. In this paper we concentrate on developing a simple but useful precursor model for the application of a motorised momentum exchange tether to ADR and place that in the context of orbital modelling, and deployment from an as-delivered package on-orbit.…”

Three dimensional dynamics of a flexible Motorised Momentum Exchange Tether

“…The next category is represented by a sequence of elements which allows some form of flexibility in the model where [15,16,17] studied a lumped mass model connected by massless springs. A bead model was used by Avanzini and Fedi [18] for massive a tether in modelling a multi-tethered satellite formation.…”

“…The tether is modelled as hinged rigid bodies which are connected with massless springs and dampers in order to be able to model precisely the aerodynamics and gravitational forces, and the moment, with a limited number of elements which, in turn, give a reduction in the computational cost. Two examples of motion, the swinging of a cable and the plane motion of a space vehicle with a deploying tether system on orbit, have been studied to verify the mathematical model and computer code, and also to estimate the accuracy of calculation [17]. Cartmell and McKenzie [21] remarked on the important point made by Danilin et al [17] that tether element forces cannot be compressive, so the numerical solution algorithm has to accommodate this.…”

Dynamics of variable-length tethers with application to tethered satellite deployment

“…To overcome the difficulty in obtaining the accurate numerical solution of the dynamics of tethered satellite systems due to complexity of the problem, several simplified models have been developed to approximate the dynamics of deploying tethers with end satellites. For instance, Danilin et al [5] and Nakaya et al [6] simulated a deploying tether using a lumped mass model. Steiner et al [7] and Williams [8,9] used a spring model to study the deployment and retrieval processes of satellites.…”