A discretised planar tether model is proposed for the Motorised Momentum Exchange Tether (MMET) in which axial elasticity is accommodated. The model uses a generalised co-ordinate deﬁning angular motion of the tether

about its centre of mass, as it travels at constant velocity on a circular orbit in the Earth’s equatorial plane and a generalised coordinate depicting the

elastic part of the tether length. The system comprises a symmetrical double payload conﬁguration, with outrigger counter inertia, and it is shown that including axial elasticity permits an enhanced level of modelling accuracy for the tether both in librating and spinning modes.  A simulation has been devised in MATLAB and SIMULINK for diﬀerent data cases. This work will be used later within a spin-up control system and will act as a precursor for

an in-depth study into the multi-scale dynamics of MMET tethers and space webs, on more complicated orbits.  This, in turn, will be assimilated within new mission architectures

Cartmell, M.

Chen, Y.

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Dynamical modelling of the motorised momentum exchange tether incorporating axial elastic effects

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       Chen, Y. and Cartmell, M. (2007) Dynamical modelling of the motorised momentum exchange tether incorporating axial elastic effects. In: Advanced Problems in Mechanics: Russian Academy of Sciences, 20-28 June 2007, St. Petersburg, Russia.  http://eprints.gla.ac.uk/30502/  Deposited on: 15 June 2010   Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk Dynamical Modelling of the Motorised MomentumExchange Tether Incorporating Axial ElasticEffectsYi Chen* Matthew P. Cartmell**yichen@mech.gla.ac.uk matthewc@mech.gla.ac.ukAbstractA discretised planar tether model is proposed for the Motorised Momen-tum Exchange Tether (MMET) in which axial elasticity is accommodated.The model uses a generalised co-ordinate defining angular motion of the tetherabout its centre of mass, as it travels at constant velocity on a circular or-bit in the Earth’s equatorial plane and a generalised coordinate depicting theelastic part of the tether length. The system comprises a symmetrical doublepayload configuration, with outrigger counter inertia, and it is shown thatincluding axial elasticity permits an enhanced level of modelling accuracy forthe tether both in librating and spinning modes. A simulation has been de-vised in MATLAB and SIMULINK for different data cases. This work willbe used later within a spin-up control system and will act as a precursor foran in-depth study into the multi-scale dynamics of MMET tethers and spacewebs, on more complicated orbits. This, in turn, will be assimilated withinnew mission architectures.1 IntroductionThe motorised momentum exchange tether (MMET) was first proposed by Cartmell[1] and based broadly on the conceptual schematic of the system shown in Figure 1.The model and concept were developed further, Ziegler and Cartmell [2], which ledto proposals by Ziegler [3], who took physical angular co-ordinates for the generalisedco-ordinates representing spin and tilt, and a further angular co-ordinate definingback-spin of the propulsion motor’s stator components, known henceforth as theoutrigger sub-system.In Figure 2, there are two generalised co-ordinate systems, one is an Earth centredX − Y − Z global co-ordinate system, the other is the x0 − y0 − z0 relative rotatingco-ordinate system. Two payload masses at each end, M1 and M2, are connectedby the tether sub-span, where the distance from the tether’s centre of mass M0 toeach end mass is denoted by L. R, R1 , and R2 are the distances from E (0 , 0 , 0 ) toM0 (x0 , y0 , z0 ),M1 (x1 , y1 , z1 ), andM2 (x2 , y2 , z2 ), respectively, withM1 = M2 = MP ,and M0 = MM . The centre of the Earth is denoted by E (Ex ,Ey ,Ez ), which isFigure 1: Conceptual Schematic of the Motorised Momentum Exchange Tetherdefined as the origin of the X − Y − Z system. The origin of the x0 − y0 − z0rotating system is located at the centre of the facility mass. The X − Y planeand the x0 − y0 plane lie coplanar to the orbit plane, where the Z and z0 axes areperpendicular to the orbit plane. The X axis is aligned along the direction of theperigee of the orbit, and the x0 axis is an extension of R. ψ is the in-plane pitchangle, and this denotes the angle from the x0 axis to the projection of the tetheronto the orbit plane. α is the out-of-plane angle, from the projection of the tetheronto the orbit plane to the tether, and is always within a plane normal to the orbitplane. θ is the circular orbit angular position, effectively the true anomaly. In thecase of a planar circular orbit, the R , α and θ are constants.2 Modelling MMET ElasticityA discretised model of the motorised momentum exchange system is shown in Fig-ure 3. The configuration shown could potentially offer high performance for somedesign data cases and orbital settings. The model contains both the dumb-bell char-acteristics, as discussed in previous publications [1] [2] [3], along with new dynamicswhich are representative of axial elasticity effects. The discretised MMET systemcomprises a symmetrical double payload configuration, with an outrigger counterinertia, as shown in Figures 1 and 3. The axial elasticity is defined by a string ofmass-spring-damper systems, where the masses are connected together serially withsprings and dampers, in which, k1 = . . .= kN+2 , c1 = . . .= cN+2 and m1 = . . .=mN . The discretised MMET model in Figure 3 based on a discretisation scheme inwhich the locations of the point masses along the length of the tether sub-span canbe calculated by means of equation (1). The location table of 1 to 20 point massesis given in Figure 4. The location of each point mass within the tether is given by2Figure 2: Planar Circular Orbit MMET System Schematic and Coordinatesequation (1):li (i) =(2i− 1)LT2N(i = 1, 2, ..., N) (1)Figures 3 and 4 show the locations defined by equation (1) for the point massesassumed for modelling the tether. The full length of the tether is LT in equation (2).L0 is the original length of each sub-span of the tether when balanced symmetrically,as seen in Figures 3 and 4. The start point is at i = 0 on the left, and the end is theLT point at the right. Figure 4 shows the mass point locations within the full-lengthof LT , the values of N in the list are the scaled locations of the point masses alongthe full length LT , where N is the number of discrete point masses considered. Anexample for N = 2 is given in Figure 4, where N = 2 means there are 2 point massesin the tether, which are located at 0.25LT and 0.75LT , respectively. According toprevious research [2][3] on the tether’s potential energy, there should be at least 15point masses to approximate adequately the full tether, and so Figure 4 gives a 1 to20 point mass location scheme. The elastic length of the discretised MMET modelis given by equation (3), ∆xi is the relative deformation between two mass pointsconnected directly as defined in equation (7), and the time variant length L(t) ofthe tether is the sum of the static length, L0 , and the elastic length, Lx , of thediscretised tether as given by equation (4).3Figure 3: MMET Discretisation StrategyLT = 2L (t) (2)Lx (t) =N+1∑i=1∆xi (t) (3)L (t) = L0 + Lx (4)On this basis equation (1) can be re-written as equation (5),li (t, i) =(2i− 1)LT2N=(2i− 1) (L0 + Lx)N(5)The elastic motion of the symmetrical tether system is considered to be dis-tributed symmetrically along each tether sub-span, and so equation (6) transfersthe location of each point mass from the end-point-tether coordinate system, withits generalised coordinate of i from 0 to LT in equation (5) to the motor-point-tethercoordinate system, with its generalised coordinate of x from 0 to L in equation (6).The relative deformation between mass-points mi and mi+1 is given by equation (7),and is also shown in Figure 3 and Figure 5.lx (t, i) = |li (t, i)− L| =∣∣∣∣(2i− 1−N) (L0 + Lx)N∣∣∣∣ (6)∆xi = lx(t, i+ 1)− lx(t, i) (7)In the discretised planar tether model, environmental effects such as solar ra-diation, aerodynamic drag and electrodynamic forces, that may also influence themodelling, are assumed to be negligible, and the motor consists of a rotor, which is4attached to the propulsion tethers and a stator which can more relative to the rotorby means of a suitable bearing. The power supplies, control systems, and communi-cation equipment are likely to be fitted within the stator assembly in any practicalinstallation. The stator provides the necessary reaction that is required for the rotorto spin-up in a friction free environment. The motor torque is assumed here to re-main coplanar with the propulsive tethers and payloads. The tether and the motorare connected by a spring-damper of as in Figure 3, when the tether moves out ofthe orbital plane, the motor drive axis remains orthogonal to the tether plane, andmotor will similarly rotate about its centre of mass. In the planar discretised model,the Cartesian components of the end masses of M0 , M1 and M2 in Figure 2 aregiven by equation (8),(9) and (10):x0y0z0 =R (t) cos θR (t) sin θ0 (8)x1y1z1 =x0 + (L0 + Lx) cosαcos (θ + ψ)y0 + (L0 + Lx) cosαcos (θ + ψ)z0 + (L0 + Lx) sinα (9)x2y2z2 =x0 − (L0 + Lx) cosαcos (θ + ψ)y0 − (L0 + Lx) cosαcos (θ + ψ)z0 − (L0 + Lx) sinα (10)where R,θ, α,ψ, Lx could be defined as independent generalised coordinates, andR, R1 ,R2 are the distances from the Earth’s centre to M0 , M1 and M2 . R1 and R2are given by equation (11). In the current system, a planar circular orbit model hasbeen chosen and so R ,α and θ are henceforth considered to be constants.R1 =√(L0 + Lx)2 +R2 + 2 (L0 + Lx)R cosα cosψR2 =√(L0 + Lx)2 +R2 − 2 (L0 + Lx)R cosα cosψ(11)T =12MP(x˙12 + y˙12)+12MP(x˙22 + y˙22)+12MM(x˙02 + y˙02)+12ρA (L0 + Lx)(x˙2T1 + y˙2T1)+12ρA (L0 + Lx)(x˙2T2 + y˙2T2)+(IP + IT +12IM)(ψ˙ + θ˙)2(12)The discretised planar model is based on aspects of the spin-up of the point-mass planar model, suggested by Ziegler and Cartmell [2] and stated in the resultsof work by Ziegler [3]. The kinetic energy of a motorised tether, taking into accountthe translation and rotation of each component, is given by equation (12), and thepotential energy is given by equations (13) or (14), incorporating gravitational andelastic strain energy components.5U = −µMPR1− µMPR2− µMMR− µρA(L0+Lx)∫−(L0+Lx)(R2 + l2 + 2lR cosψ)− 12 dl + SE= −µMPR1− µMPR2− µMMR+µρA lnR cosψ − (L0 + Lx) +√R2 + (L0 + Lx)2 − 2 (L0 + Lx)R cosψR cosψ + (L0 + Lx) +√R2 + (L0 + Lx)2 + 2 (L0 + Lx)R cosψ+ SE(13)Note that ln() is explicit in equation (13), and that this term has a numericalsingularity at ψ = npi, which means for the case of a spinning motorised tetherproblems will be encountered when numerically integrating. To avoid the singu-larity situation, an alternative description of the potential energy is derived, whichdiscretises the tether mass, and so the potential energy is re-stated by equation (14).U = −µMPR1− µMPR2− µMMR−N∑i=1µρA (L0 + Lx)N√R2 +((2i− 1) (L0 + Lx)2N)2+2 (2i− 1)R (L0 + Lx)2N cosψ−N∑i=1µρA (L0 + Lx)N√R2 +((2i− 1) (L0 + Lx)2N)2− 2 (2i− 1)R (L0 + Lx)2N cosψ+ SE(14)The SE term is the strain energy of the discretised tether and is defined inequation (16), CE is an assumed dissipation function based on Rayleigh dampingand defined in equation (17). The mass moments of inertia can be derived from firstprinciples, and are given by equations (15),IP =12MP r2P , IM =12MMr2M , IT =112ρA (L0 + Lx)(3r2T + (L0 + Lx)2) (15)where, IP is the mass moment of inertia of the payload,rp is the radius of payload,IT is the mass moment of inertia of the tether,rT is the radius of the tether, IM is themass moment of inertia of the motor, rM is the radius of the motor.In the discretisedmodel, energy is stored as potential energy in the assumed spring elements. Thestrain energy for tether elasticity is defined in equation (16).SE =N+1∑i=112ki∆x2i =12keqL2x (16)6Figure 4: Point Mass Location SchemeDamping in each discretised elastic element is assumed to be classical linearviscous in form and is given by,CE =N+1∑i=112ci∆x˙i2 =12ceqL˙x2(17)Equation (18) defines the equivalent spring stiffness keq, and the damping coeffi-cient is ceq for each tether sub-span. A is the tether tube cross-sectional area, statedin equation (19). Also f0 is the tensile force at the end of each tether sub-span, asdefined by equation (20).keq =kiN + 1, ceq =ciN + 1(18)A = pir2T (19)fi(t) = ψ˙2(t)MP + ρA∣∣∣∣(2i− 1−N)(L0 + Lx)N∣∣∣∣2∣∣∣∣(2i− 1−N)(L0 + Lx)N∣∣∣∣ (20)In Figure 6 the tether has a payload at each end and rotates due to the motor,thereby experiencing centripetal acceleration. The tether’s tensile force related tothe tether stress is given by equation (20). Calculating the stress, Ziegler and7Figure 5: The Relative Deformation Between Mass-pointsCartmell [4], provides a basis for Figure 6 in order to show the stress distributionwithin the tether sub-spans. This shows the relationship between tether stress, pointmass location, and the angular velocity.It can be note that ψ is defined as the planar pitch angle, and τ is the motortorque, which is assumed to be a constant here. Equations (12), (14) are substitutedinto Lagrange’s equation, leading to the dynamical equations of motion in ψ andLx .Since assumptions have been made that there are two non-conservative forces areacting on the system, namely, τ , the motor torque, and QLx , the generalised forceassociated with the generalised coordinate Lx , then a classical linear viscous dampingterm as a result of assuming a Rayleigh type dissipation function in equation (17)is also assumed in equation (22).N∑i=1µρpiRr2T (2i− 1)(L0 + Lx)2sinψ2N2(R2 − (2i− 1)Rcosψ(L0 + Lx)N+(2i− 1)2(L0 + Lx)24N2) 32−N∑i=1µρpiRr2T (2i− 1)(L0 + Lx)2sinψ2N2(R2 +(2i− 1)Rcosψ(L0 + Lx)N+(2i− 1)2 (L0 + Lx)24N2) 32+16(θ¨ + ψ¨)(3MMr2M + 6MP r2P + 4piρr2T (L0 + Lx)3 + 12MP (L0 + Lx)2 + 3piρr4T (L0 + Lx))+16(θ˙ + ψ˙)L˙x(12ρpir2T (L0 + Lx)2 + 24MP (L0 + Lx) + 3piρr4T)− µRMP (L0 + Lx)sinψ(1(R2 + 2R(L0 + Lx)cosψ + (L0 + Lx)2)32− 1(R2 − 2R(L0 + Lx)cosψ + (L0 + Lx)2)32)= τ(21)8Figure 6: Stress Distribution within Tether Sub-span(2MP +12ρpir2T (L0 + Lx))L¨x + 3ρpir2T L˙x −(2MP (L0 + Lx)− 14ρpir4T − ρpir2T (L0 + Lx)2)(θ˙ + ψ˙)2− ρpir2T(R˙2 +R2θ˙2)+k0LxN + 1+ µMP(L0 + Lx)−Rcosψ(R2 − 2(L0 + Lx)Rcosψ + (L0 + Lx)2) 32+(L0 + Lx) +Rcosψ(R2 + 2(L0 + Lx)Rcosψ + (L0 + Lx)2)32+ µρpir2TN∑i=1(L0 + Lx)((2i− 1)2(L0 + Lx)2N2− (2i− 1)RcosψN)2N(R2 − (2i− 1)Rcosψ(L0 + Lx)N+(2i− 1)2(L0 + Lx)24N2) 32−1N(R2 − (2i− 1)Rcosψ(L0 + Lx)N+(2i− 1)2(L0 + Lx)24N2) 12+N∑i=1(L0 + Lx)((2i− 1)2(L0 + Lx)2N2− (2i− 1)RcosψN)2N(R2 +(2i− 1)Rcosψ(L0 + Lx)N+(2i− 1)2(L0 + Lx)24N2) 32−1N(R2 +(2i− 1)Rcosψ(L0 + Lx)N+(2i− 1)2(L0 + Lx)24N2) 12+ ceqL˙x = 0(22)9Figure 7: Ψ and Lx Simulation Graphs,τ = 2.5× 108Nm3 Case StudyUnless stated otherwise all of the results were generated with the following sys-tem parameters: the number of point masses N = 20, gravitational constant µ =3.9877848×1014 m3s−2, propulsion tether mass of payload MP = 1000 kg, mass ofmotor facility MM = 5000 kg, static length of propulsion tether L0 = 10000 m, cir-cular orbit with eccentricity e = 0, periapsis distance rper = 6890 ×103 m, apoapsisdistance rapo = rper , radius of tether tube rT = 0.1 m, radius of facility of motorrM = 0.5 m, radius of payload rP = 0.5 m, the original tether tube cross-sectionalarea A = 62.83×10−6 m2 , the tether density ρ = 970 kg/m3 , initial angular ψ0 =-0.9 rad, initial angular velocity of ψ˙0 = 0.001 rad/s, motor torque τ = 2.5 × 108Nm in Figure 7 and τ = 2.5× 1012 Nm in Figure 8, damping coefficient ci = 2000Ns/m, stiffness ki= 20000 N/m, acceleration of gravity g = 9.81 m/s2, The periodof elliptical orbit T = 2pi( rapo+rper2)32/µ12 , Numerical results were obtained by inte-grating equations (21) and (22), in MATLAB/SIMULINK with a fourth-fifth-orderFehlberg Runge-Kutta method using a relative error of 10−6 .Figure 7 shows the time responses for the pitch angle ψ, and the axial displace-ment of the tether, Lx, along with their corresponding phase spaces. It is clear thatover the integration time shown both quantities exhibit what looks to be steady10Figure 8: Ψ and Lx Simulation Graphs,τ = 2.5× 1012Nmstate oscillation. Physically this equates to libration of the tether, and an axial os-cillation about a constant tensile offset from the nominal length of around 5 metres,with an amplitude of approximately +/- 5 metres about that offset value. For anunstretched 10 km sub-span the constant offset represents a stretch of 0.05%, sowith the oscillation superimposed on this the tether oscillates between the nominalunstretched length and a maximum extension of 0.1%. This is agreeably small inpercentage terms but could still have practical implications for payload delivery andretrieval should the tether be required to work in the librating mode. The phaseplots show limit cycle behaviour for each coordinate, which tends to corroborate theinterpretations of steady-state. Figure 8 illustrates the system responses once thetether pitch angle has bifurcated out of the periodic libration and into monotonicspin. This response is highly nonlinear but shows a good build-up of rotation ina short time. The axial response shows a gradual growth in tensile stretch overa fraction of a metre and then oscillation once again, about a small tensile offsetvalue of just over 1 m. Although the integration time shown is limited it looks asthough the axial coordinate could, in time, reach a steady-state oscillation. How-11ever, the phase plot for the axial coordinate is harder to decipher and a longer timeis needed to get accurate clarification of the qualitative aspects of this response. Itis interesting to note that the frequency of axial oscillation in Figure 8 is very muchhigher than that of Figure 7, which is one reason why the plots in Figure 8 are for ashorter integration time, noting that all detail would otherwise be lost. The torqueis increased 10000 times to provide the bifurcation from the case of Figures 7 to thatof Figure 8.4 ConclusionsThe work in this paper has shown that including axial elasticity within an MMETmodel has a significant bearing on overall performance and that this effect shouldnot be ignored in future, particularly for control. All subsequent analyses for controlapplications will henceforth include axial compliance within the modelling.AcknowledgementsThe authors would like to acknowledge the support provided to the first author bythe Overseas Research Students Awards Scheme and the Scholarship awarded by theUniversity of Glasgow’s Faculty of Engineering.References[1] M. P. Cartmell, (1998), ”Generating Velocity Increments by means of a SpinningMotorised Tether”, 34th AIAA/ASME/SAE/ASEE Joint propulsion Conferenceand Exhibit, Cleveland Conference Centre, Cleveland, Ohio, USA, AIAA-98-3739.[2] Ziegler, S.W., and Cartmell, M.P., (2001), ”Using Motorised Tethers for Pay-load Orbital Transfer”, Journal of Spacecraft and Rockets, 38 (6), November /December, pp 904-913.[3] Spencer Wilson Ziegler, (2003), ”The Rigid-body Dynamics of Tethers in Space”,PhD Dissertation, Department of Mechanical Engineering, University of Glas-gow.* Postgraduate student ** ProfessorDepartment of Mechanical Engineering, University of Glasgow, Glasgow, G12 8QQ,Scotland, UK.12

Generating Velocity Increments by means of a Spinning Motorised Tether&quot;,

The Rigid-body Dynamics of Tethers in Space&quot;,

Using Motorised Tethers for Payload Orbital Transfer&quot;,

http://eprints.gla.ac.uk/30502/1/id30502.pdf

Dynamical modelling of the motorised momentum exchange tether incorporating axial elastic effects

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