**Stabilization of the External Tank for Use as a Large Space Platform**

W. D. Kelly – Triton Systems, DBA

Houston, Texas

High Frontier Conference, Space Studies Institute, Princeton, N. J., 1997.

### Abstract

Use of magnetic passive dampers is recommended to stabilize the Space Shuttle External Tank in free flight for on-orbit applications. Longitudinal alignment along the radius vector allows for gravity gradient passive stabilization, but, without a damping mechan-ism, there is no inherent means of eliminating disturbance accelerations and buildup of pendular motions lead to spin or tumbling. Review of previous magnetic, passive dampers is provided drawing from the Long Duration Exposure Facility (LDEF) flight and Space Station program studies for early assembly flights. Basic physical models of the damper-spacecraft combination are discussed. and preliminary damper design analysis is demonstrated.

### Introduction

The alchemists of the Middle Ages who sought to transform lead into gold would feel quite at home with space designers seeking to transform expended rocket propellant tanks into useful space payloads. For decades designers have considered spent rocket stages for space applications, thus combining structural weight and payload, the Skylab space station a notable example. Since Space Shuttle formulation as a space plane with an expendable, External Tank (ET) jettisoned at suborbital velocity, there have been numerous proposals for carrying the ET (Fig.-1) into orbit for use as a space platform. Advocates have proposed its large fuel and oxidizer tanks as pressurized laboratories1,2 or micro-g processing plants; they also note the dynamic stability of a symmetric cylindrical form (gravity gradient stabili-zation), the prospect of scavenging residual propellants, using intertank position mounts for external payloads and even exploiting the ET as an on-orbit scrap metal source. Studies have included a gamma ray observatory (ET-GRO), industrial processing facilities, an orbital fluid storage tank, a “strong back” for mounting other devices, a counter weight for tethered boost (and consequent ET de-orbit) of other payloads and simple, quick manned space station concepts. Mission plans have ranged from concepts requiring only one Shuttle flight and release before operations (e.g., ET-GRO) to those involving revisits or a sequence of assembly missions.

Whether ET applications involve a single or multiple Shuttle flights for setting up space platforms, proposals still encounter thorny problems. Functional simplicity of the ET prevents tapping into readily available utilities for propulsion, attitude control, communications and power generation; ET structures and materials were chosen to store propellant for a 10 minute ascent – and little else. To counter these issues, designers have proposed a number of fixes, but to our knowledge, none has suggested use of magnetic dampers for augmenting passive stabilization. This is our principal point of discussion.

Cylindrical or ellipsoidal satellites with corresponding mass properties can be stabilized with respect to the local vertical – local horizontal (LV-LH) if their principal (inertia) axes are placed near stability points: the minimum inertial axis along the radius vector, the intermediate with the velocity vector and the maximum with the normal to the orbital plane (Ipitch > Iroll > Iyaw or IN>IT>IR). Gravity gradient forces tend to counter deflections in pitch and roll ( rotations about the orbital tangential velocity vector and orbital normal, if the body coordinates deflections from local coordinates are small), but neither low earth orbit (LEO) aerodynamics or rigid body mechanics provide any inherent damping to perturbations induced. There can be a host of reasons for a spacecraft to begin oscillations: variations in the minuscule aerodynamic moments, departure from circular orbit due to Keplerian eccentricity or gravity field oblateness effects, thermal buckling with passage in and out of the Earth’s shadow, fluid venting, initial spacecraft release conditions or subsequent mechanical history. Generally, to counter these effects, designers employ active means of control such as attitude thrusters, reaction wheels or control moment gyros (CMGs) which presuppose fuel systems or power supplies. Even magnetic torquer bars, used on the Hubble Space Telescope to desaturate the reaction wheel primary control system, require current from an external power source to regulate their response.

An early Space Shuttle payload employed a notable exception to these “active” control schemes; in April, 1984 was launched the Long Duration Exposure Facility (LDEF, Fig.-2 ), a passively stabilized 11-ton cylindrical satellite with a 12-sided longitudinal cross-section. This satellite orbited for several years in LEO without tumbling or accumulating large angular rates until January 1990 ( STS-32) retrieval. The presence of large bar magnets embedded in one of two concentric spheres accounted for its stability; the cluster positioned itself in alignment, (described in 3-6) with the terrestrial magnetic field, impeded by a viscous fluid layer between the two spherical shells (Fig.-3). From the perspective of the spacecraft, if the magnetic damper is sized appropriately, perturbing accelerations and motions from the LVLH are damped and eventually decay. A terrestrial analog to a spacecraft with a passive magnetic damper would be a ship with an anchor that simply impedes its tendency to drift by dragging on a muddy sea bottom; the anchor does not really “anchor” the ship in this case, but slows it down.

Beside application to the LDEF, this stabilization method was extensively studied for use on the first Space Station Freedom assembly flight; perhaps for several flights if power and control were unavailable and the application proved effective. During these studies, means were developed to determine the size of the magnetic dampers required and simulate their performance on orbit at various levels of fidelity. The author participated in the Space Station control studies as well as several earlier, unrelated ET applications studies7 in which dampers were not considered; he sees no reason that LDEF and Space Station experience can not be applied to the ET stabilization problem. For that matter, magnetic dampers would probably work as effectively on other spent upper stages (e.g., Titan, Ariane or Proton), if similar applications were proposed.

In the summer of 1986, this analyst and several colleagues8-10 examined tentative schemes for passive stabilization of early Space Station assembly flights, concluding that magnetic passive dampers were the best means available. Subsequent analyses drew on the earlier LDEF work of Old Dominion University researchers Heinbockel and Breedlove3,4 as well as Dr. Karen Castle’s preceding LDEF contractor work at the Johnson Space Center. We first assessed damper gain requirements with root locus stability analyses, then developed 6-degree of freedom simulations of spacecraft damper control. Our initial simulations employed simple cylinder and plate aerodynamics and a dipole model of terrestrial magnetism without offset from the earth’s center, but inclined to the axis of rotation11. Later simulations included a detailed geophysical field model and aerodynamics closely based on CAD modeling12. Pilot-in-the-loop flight simulator runs modeled thruster plume impingement effects of the Orbiter separation maneuver as well as residual rates from separation with the Shuttle Remote Manipulator System (RMS)13. Finally, studies were undertaken to determine the effect of modeling both the spacecraft and the field tracking magnet. Once again, a root locus concept was developed as well as detailed dynamic simulations 14, 15.

The dynamic simulations noted above involve various levels of fidelity, but analyses with preliminary design tools are just as important as detailed configuration examinations. There are hardware limitations in constructing stiff magnetic damper systems, but in theory a magnetic damper could become so tight that it would align the vehicle with the magnetic pole. At lower gains steady state errors could also be very high, precluding the use of mission oriented instrumen-tation, or else the action of seeking magnetic alignment could induce resonant oscillations of exactly the sort which mission planners sought to avoid. Root locus plots, for example, identify suitable control gains, elimininating need for global search with 6-DOF simulation. Formulation of the linearized system in the Laplace domain gives an indication of time dependent behavior based on initial conditions. Lyapunov stability analysis, based on Lagrangian and Hamiltonian dynamics, determines initial deflection and rate bounds for tumbling or spinning about a point of stability. If rate and position stability limits can be determined from this analysis, deployment conditions for a passively stabilized spacecraft can be stipulated.

Early studies indicated that inclination effects tended to correspond to averaged magnetic inclination, since the the earth-fixed (but skewed) magnetic axis presented a varied incident angle with respect to the satellite’s plane over a day’s rotation, disregarding nodal regression. With a nominal dipole 11.5o offset from the rotational axis, effective magnetic inclination varies above and below equatorial inclination by the same amount. Positions near the magnetic field equator reduce the damper’s effectiveness.

It is possible to do only limited analysis of forcing functions without digital simulation. Atmospheric density variations in LEO can be attributed to latitude, altitude, solar extreme ultraviolet flux and geomagnetic variations in response to solar flares. If satellite cross sections change significantly with deflections, the effect of density variations can have differing effects on drag. Initially, for Space Station, we simulated spacecraft with symmetric and simple geometric forms; we later discovered protuberances such as radiators broke the symmetry about the yaw axis, resulting in significant yaw-roll coupling motions generated by aerodynamic torque variations. Significant offsets of inertia axes from external geometries due to internal mass distributions introduced moment arms about the center of mass and asymmetric centers of pressure for port and starboard, zenith and nadir cross-sections of the vehicle. If there are significant discrepancies between the location and shape of the inertial ellipsoid (gravity gradient torques) and the surface contours (aerodynamic torques), a new balance of forces must be struck (e.g., Torque Equilibrium Attitude -TEA ) or disturbance torques will continually toss the vehicle about.

### Hardware Descriptions

Application of magnetic dampers to spacecraft depends on hardware issues: the technology for manufacturing the appropriate dampers, and the mass properties of the overall spacecraft to be stabilized. The nature of an ET application is largely conjecture, but initially the mass properties of an empty ET separated from the Orbiter will dominate dynamical considerations.

The ET is 154.2 ft tall and 27.5 ft in diameter and contains propellant for the Shuttle engines (SSMEs). The main components of the ET are a liquid O2 tank, located in the forward position, an aft liquid H2 tank and the intertank connecting the two. The intertank provides space for instrumentation plus attachment structures for the forward ends of the solid rocket boosters (SRBs). The ET carries 1,589,000-lbs of propellant. The H2 tank is 2.5 times larger than the O2 tank, but the O2/H2 load is 6:1 by weight. Tank structure is skin-stringer; thermal protection covering the tank exterior maintains propellant at cryogenic temperatures during the launch sequence and the 8.5 minute powered ascent. Feed lines to the SSMEs trail down the ET’s side from the O2 tank and protude form the base of the H2 tank. Attach structures connect the Orbiter to the ET near the base feed lines and at the intertank. Active control mechanisms are located on the Orbiter.

In the Table-1 are mass properties for STS-44, a flight in which the nominal light weight tank design was used rather than the initial test flights or the anticipated aluminum-lithium designs (several tons heavier or lighter respectively). These mass properties were used for mission planning16. Note that Shuttle weight on this mission is 334,376.2-lbs prior to separation and 255,811 just after. This indicates an expected 12,037 lbs of fluid to remain in the propellant tanks and lines, based on physical pumping constraints and margins to assure nominal targets. The IXZ product of inertia for the ET is high relative to the longitudinal IXX value.

LDEF was a 12-sided cylindrical, gravity-stabilized spacecraft host to 57 experiments, several character-izing the meteoroid and orbital-debris environment for the nominal 9-month mission (see Table-2). As a result of LDEF’s 5.7 year exposure time and heightened interest in the debris collisional threat, the entire spacecraft on retrieval was studied as a meteoroid and orbital-debris detector. Due to gravity-gradient orbital stabilization, the same general surfaces pointed into the velocity vector during the entire mission; its large exposed surface area (~130 m2) provided LEO particulate environment data, including directionality effects for both natural and man-made particles.

The spherical dome of LDEF’s viscous magnetic dam-per was fabricated from 1/32 inch thick 6061-T6 aluminum alloy sheet attached to a cylindrical base with aluminum screws. The cylindrical portion of the housing is a fiberglass structure, covered with an aluminum tape, both inside and outside, to meet thermal control requirements. The mounting plate material is 6061-T6 aluminum alloy, with the top and bottom surfaces also covered by aluminum tape. A thermistor mounted in the top center of the dome provides house keeping data. The assembled damper housing, with the damper inside, is mounted to the space end frame with stainless steel fasteners.

LDEF magnetic damper hardware exhibited a 220,000 pole-centimeter magnetic dipole moment. Measures of the mechanical characteristics are approximate, but correspond to 1 ft-lb-sec damping capability in English units, if entered into a 2nd order mass-spring-damper system (one source estimates 5 Nms or .37631 slug ft2 secs). This coefficient is dimensionally similar to angular momentum, but corresponds to torque divided by angular rate in radians/sec. The manufacturer investigated a “super LDEF” damper with a factor of ten increase in performance. Damper gain in a deployed system can be controlled by the number and quality of the dampers. For an individual damper

Kd = (8p/ 3)mro4/eo (1.)

where m is the fluid viscosity between spheres, ro is the radius of the outer damper sphere, and eo is the difference in radii between outer and inner spheres. To first order, increases of viscosity, outer sphere radius and proximity to inner sphere all tend to increase the damper gain, yet the strength of the magnet would be obliged to increase accordingly to maintain its orientation against the increasingly dissipative medium surrounding it. Ten-fold increase in damper gain (Kd) came with ten-fold increase in dipole strength ( to 2.2×106 pole-cm) and volume, but mass increased from 15 to 300-lbs. Extreme temperature variations could either alter the viscosity or freeze the working fluid and magnet. As a result, unless further testbed information is provided, there is still some uncertainty remaining in proportionalities between Kd, mechanical and magnetic properties.

For the baseline LDEF configuration and the two candidate spacecraft shown in Table-1 (the ET and Resource Node attached to the Russian FGB vehicle), damping time constants ( td ) can be compared as a function of damper gain (Kd) in ft-lb-secs. The ratios of the time constants are proportional to the principal inertias on the pitch axis, but higher orbital inclination for the International Space Station applications (51.6o vs. 28.5o) reduces the time constants to 63%. To obtain the same response from the ET as obtained by LDEF, the gain should be proportionately higher, depending on orbital inclination and principal inertias. It is suspected that an intermediate (e.g., 10 or 20x) gain would be adequate to prevent tumbling or excessive rate buildup. The natural frequency of any pendular motions are hardly effected by damping. The principal moments of inertial ratios dominate these oscillations vs. the influence of any attainable damping mechanism.

### Dynamics and Coordinate Systems

Inclusion of a relatively simple device on board a spacecraft justifies extension discussion of space mechanics and environmental forces, but unlike some systems (e.g., tethers) the mechanics are not intractable. It is well worth questioning whether the mechanics of flight can be linearized in meaningful fashion, especially in matters of stability. In the case of an attitude stabilized spacecraft, the pendular motions in the LVLH can be linearized much like those of a pendulum in terrestrial conditions. Our experience with the magnetic model indicates that the averaged magnetic latitude describes the effectiveness of the magnetic damper since the likely dynamic time constant will be much longer than the period of LEO revolution, and the variations of magnetic inclination are +/-11.5o of the nominal.

Coordinate systems for US manned spacecraft frequently borrow from aircraft conventions with the x along the longitudinal axis, the z toward the nadir in level flight and the y axis extended out the starboard wing to the pilot’s right. The XYZ LVLH system in circular orbit coincides with this system, but these definitions are opposite in sign with the orbital normal N and the radius vector R. Definitions of stability orientation depend more on the size of the principal axis inertia rather than its nomenclature, and in the case of the ET, gravity gradient stability is achieved by aligning the X axis along the radius vector. Unless significant additional hardware is deployed to distinguish the IZ from IY principal inertia, aligning the larger of the two with N will provide only marginal inherent stability. Nonetheless, the body axes are now skewed with respect to LVLH, but we will refer for convenience to roll motion along the velocity vector, pitch about the N, and yaw about the R vector, renaming the principal axes IT, IN and IR accordingly. Additionally, re-defining the roll, pitch and yaw deflections for the minimum axis alignment along the radial, roll and pitch deflections exceeding 90o will signify uncontrolled tumbling.

For a rigid body in circular orbit, the gravity gradient torque is in terms of defection of the principal inertia axes.

Tgg = 3 wo2 Rb x {I } Rb (2a.)

(IZ-IY)a23 a33

= 3 wo2 {(IX-IZ)a13 a33} (2b.)

(IY-IX)a13 a23

(IR – IN) s2f c2q

= 3 wo2{ (IT – IR) }cfs2q = (2c.)

(IN – IT) sfs2q

where wo is the orbital angular rate, {I} the inertia tensor, Rb the unit vector based on the rotation

Rb = {A}ijk RLVLH (3.)

The matrix {A}ijk is a rotational tranformation sequence i-j-k from the LVLH unit vector frame. Resulting unit vector expressions depend on the rotational sequence selection; this is demonstrated by various formations in the space mechanics literature of the deflection angles for gravity gradient torques (discussed elsewhere17). Here we introduce the 1-2-3 yaw, pitch, roll sequence. Note that y does not appear in this formulation, but f and q deflections induce yaw moments.

In order to determine the magnetic forces acting on a spacecraft, it is necessary to establish the position of the spacecraft R and that of the Earth’s magnetic polar axis Nm(t) in inertial space. In the celestial sphere, both vectors can be defined by the angles analogous to terrestrial latitude and longitude, declination d and right ascension a, with subscripts m and r denoting magnetic pole and radius vectors respectively. Below, c and s are used to denote cosine and sine.

R(t) = (cdr car, cdrsar, sdr)T, (4.)

Nm(t) = (cdmcam, cdm sam, sdm )T (5.)

The damper motions are affected by the size of the magnetic torque term DTd. This term originates in the interaction of the local terrestrial magnetic field strength with the intrinsic magnetic dipole of the bar magnet.

eR eT eN

Td = m x B = mR mT mN; (6a.)

BR BT BN

Td = abs( m B) sin qm (6b.)

where qm is due to the displaced orientation of the bar magnet from the terrestrial field lines. A small angle assumption in the pitch plane provides the approximation DTd.

The transformation from an LVLH orbital position frame to the inertial Cartesian coordinate frame with which eq. 3 corresponds ( a 3-rotation matrix based on ascending node W, orbit inclination i, and circular arc along the orbital path from the ascending q*) is shown in the following equation. Thus, unit vectors of the radius, horizontal/tranverse velocity and normal to the orbital plane (eR , eT, eN) can be defined in terms of the transformation matrix.

I (cWcq*-sWci sq*) ( -cWsq*-sWcicq*) (sWsi) eR

[ J] = { (sWcq*+cWcisq*) (-sWsq*+cWcicq*) (-cWsi) } [eT]
K (si sq*) (si cq*) (ci) eN (7.)

sinfm(t) = R(t) � Nm(t) = c(W-am) cdm cq* – c(W+am)ci cdm sq* + si sdm sq* (8.)

Transformation of inertia matrix {I}to principal body axes and inertias with deflections involves solution of the inertial tensor eigenvalue problem. Stability alignment at IR, IT , IN with initial deflections of coordinate axes from LVLH to yo, fo and qo depends on the selection of matrix {A}ijk as will the nature of the linearized equations of motion.

For the homogeneous linearized equations of angular motion, there is a (possible) uncoupling of the roll-yaw motions from the pitch plane.

[IN s2 + sin fm Kd s + 3 wo 2 (IT – IR)] q(t) = 0 (9a.) [IT s2 + 4 cos2 fm Kd s + 4 wo 2 (IN – IR)] f(t) + [ (IT – IN + IR )wo – 1/3 sin2 fm Kd ]s y(t) = 0 (9b.) [(IT – IN + IR ) wo – 1/3sin2 fm Kd ] s f(t) + [ IR s2 + 2/3sin2fi Kd s + wo 2 (IN – It) ] y(t) = 0 (9c.)Transformations between the time dependent and Laplace s- domain functions give specific solutions to the equations of motion in terms of initial angular positon and rate.

[IN s2 + sin fm Kd s + 3 wo 2 (IT – IR)] Q(s) = IN(s qo – dqo/dt) + sin fm Kd qo (10a) [ IT s2 + 4 cos2 fm Kd s + 4 wo 2 (IN – IR)] F(s) + [ (IT – IN + IR )wo – 1/3 sin2 fm Kd]sY(s) =IT(sfo – dfo/dt) + 4 cos2 fm Kd fo + 1/3 sin2 fm Kd yo + [(IT – IN + IR)wo -1/3 sin2 fm Kd ] yo (10b.)

[(IT – IN + IR ) wo – 1/3sin2 fm Kd ] sF(s) + [ IR s2 + 2/3sin2 fm Kd s + wo 2 (IN – It) ] Y(s) =IR (s yo + d yo /dt) + 2/3sin2 fm Kd yo + [ (IT – IN + IR) wo – 1/3sin2 fm Kd] fo (10c.)

These linearized equations of rotational motion for a rigid body on-orbit allow uncoupling of the 2nd order pitch [Q(s) or q(t)] equation from the 4th order roll-yaw equations. Solutions of 2nd and 4th order systems in the s-plane have roots of the form

(s + si + j wni) (s + si – j wni) = s2 + 2ziwni s + wni2 (11.)

where j = � -1; wni, the natural frequency mode i; and zi, the characteristic damping of the resulting system. For oscillatory systems with 0 < zi<1, the homogen-eous roots are of the form s12 = – zi wn i +/- j wni (1 – zi2 )0.5 (12.) Thus, the pitch plane roots for non-zero damping have a real part, and an imainary part, if IT>IR.

s12 = – sin fm Kd / IN +/- j[3(IT – IR)/IN ]0.5 x { 1- (sin fm Kd)2 /[(IT – IR) IN ]}0.5 (13.)

The pitch angle becomes a function of the following form

q(t) = f( t, fm, Kd, qo, dqo/dt, (IT – IR)/IN ) (14.)

Assuming that the principal inertia about the radius vector is the minimum and that the gain Kd is a quantity smaller than the inertias, the pitch motions are oscillatory, damped only by the magnetic gain Kd.

Q(s) = { IN(s qo – dqo/dt) + sin fm Kd qo}/[IN s2 + sin fm Kd s + 3 wo 2 (IT – IR)] (15a.)

Q(s)/ IN = qo s/ {[ s+ z wn + j wn(1 – z2 )0.5 ] [ s+ z wn – j wn(1 – z2 )0.5 ] } +….

[sin fm Kd qo/ IN ]/{ [ s- z wn + j wn(1 – z2 )0.5 ] [ s+z wn – j wn(1 – z2 )0.5 ]} (15b.)With wd = wn(1 – z2 )0.5, converting back to the time domain…

q(t) = qo [wd]exp(- z wn t){cos(wd t) +[ (dqo/dt) – sin fm Kd qo/ IN ] sin(wd t) /wd} (16.)

The roll-yaw axis formulation is more complex, but amenable to the same analytical techniques adapted to simultaneous linear differential equations. Conversion from the time domain to the Laplace domain establishes the role of the initial positions and rates. The resulting transfer function determinant in combination with the substitutions employed with Cramer’s Rule define Laplace s-domain functions for F(s) and Y(s). Conversion of the two sets of expressions to the time domain from the series of terms with the common 4th order denominators results in expressions of the form

b11 b12 F(s) c1 ( fo, yo, dfo/dt, dyo/dt )

{ } [ ] =

b21 b22 Y(s) c2 ( fo, yo, dfo/dt, dyo/dt ) (17.)

f(t)= L -1 [(c1 b22 – c2 b12) / (b11 b22 – b21 b12) ] =

f( t, fm, Kd, fo, yo, dfo/dt, dyo/dt, IT , IN , IR ) (18a.)

y(t) = L -1 [(b11 c2 – b21 c1 ) / (b11 b22 – b21 b12) ] =

f( t, fm, Kd, fo, yo, dfo/dt, dyo/dt, IT , IN , IR ) (18b.)

Forcing functions which are independent of attitude (e.g., a pounding device, thermal excitations, effects of orbital eccentricity) can be represented more easily, simply as functions of time. Paradoxically, the damping in the linearized equations exists only if there is a magnetic field forcing function with time dependence related to orbital revolution and nodal regressions. It is difficult to include aerodynamics in this linearized equations, unless the geometry of the vehicle is extremely simple. Deflections of q,f and y from the LV-LH result in changed projections of plates or cylinders in the velocity vector, even if panels do not articulate. For a cylinder resembling the ET near its vacuum stability point, aero-moments on the cylindrical cross sections above and below the center of gravity are likely to be unequal. LEO aerodynamics generally are modeled without damping forces based on free molecular flow dynamics. If the aerodynamics are largely in the pitch plane, gravity gradient torques will counter deflections, and the aero-moments themselves will decrease with the cosine of q which is not satisfactory for linear analysis. The presence of solar panels or radiators on ET applications could complicate the aerodynamics depending on their arrangement. At this point 6-DOF simulation of on-orbit dynamics are in order for most spacecraft.

Linear stability limits should be considered as well. Methods of analytical dynamics can also provide indicators of bounds for angular rates and attitudes at which tumbling or spinning would commence. Owing to space limitations, we can only mention that rotating frames conserving energy can be mapped out in phase space and that the 2nd order partials of the characteristic Hessian matrix must provide a positive definite matrix (see 18,17). For yaw axis stability, and zero deflections, for example,

( IT wT2 + IN wN2 + IR wR2 ) < wo2 (IN – IT ) or

[ IT (df/dt – sqdy/dt) 2 + IN (cf dq/dt + sfcq dy/dt )2+ IR (-sf dq/dt + cfcq dy/dt )2] < wo2 (IN -IT ) (19.)

The most straight-forward example of position and velocity offsets from stability point is that of 90o rotation of the 1st and 2nd largest principal inertias about the radial axis. This tends to reverse the signs of real roots…

### Simulation Results

Our 6-DOF simulation includes rotational rate and position initial conditions, orbital elements, varied gravity models, varied atmospheric density models and a rudimentary aerodynamic model comprised of cylindrical and flat plate elements. For the nominal ET separation (initial conditions), the Y and Z body axes are transposed so far as stability positioning is concerned; plus there is an additional offset 35o about the longitudinal axis to align the principal axis The flat plates in the aerodynamics model represent solar panels with the option of tracking the sun. The surface of the ET could contribute significantly to any application aerodynamics. Consequently, we divide this surface into two cylindrical centers of pressure located “above” and “below” the c.g. along the longitudinal axis. The moment arms of these surfaces, plus the optional solar panels generate the aerodynamic moments at the end of moment arms projected into the velocity vector. Provision for unsymmetric radiators is achieved with paneling that minimizes its solar projection 90o out of phase of solar array rotation on one axis.

For our discussion we provide several ET test cases, isolating particular disturbance inputs and illustrating the attitude history in the absence or presence of magnetic dampers. In essence, this is a preliminary damper design exercise. The simplest case is the radial orientation of the ET in the Keplerian gravity field with the vehicle coordinates X,Yand Z aligned with the R, N and T providing a 35o deflection about yaw for the inertia axis and 1-2 o for pitch and roll. This results in yaw axis (Fig.-x) oscillations that couple with roll and pitch. Introduction of significant damping (50 ft-lb-secs) results in yaw stabilization about 90o deflection (Fig.-y) and reduced pitch-roll deflections. The yaw time constant is on the order of 104 secs, and about 105 secs on the other axes.

Testing the damper gain performance in the most benign enviroment: principal axes aligned with LV-LH, no initial rates and no series expansion (J2, J4, etc.) of the gravity field, we note that damper gains induce steady state pitch and roll oscillations, although both 20 and 50 ft-lb-sec dampers tend to capture the drifting yaw axis and stabilize it at 90o rotation. The steady state oscillations could be confused with orbital perturbations, using expanded gravity models. To reduce steady state oscillation, hardware complexity and mass, Kd=20 appears preferable to Kd =50. Further reductions (e.g., Kd=10) could be benificiial.

The large product of inertia IXZ relative to IXX tends to couple rotation about all three axes, counter to the linear theory we discussed. In application, this could be countered by attaching payloads opposite to the side cluttered with propellant feed lines. Without any payload solar arrays, radiators or antennas, ET aerodynamics do not pose significant problems at high altitude or unexcited atmosphere. Addition of tracking solar arrays would tend to be destabilizing due to changing cross sections and rotor moments, but then the electrical power supplied could power an active control system such as reaction wheels or magnetic torquers which would supercede magnetic dampers.

### Summary

In the course of discussion, we have reviewed LEO placement of Space Shuttle External Tank and how its attitude can be stabilized using a previously applied technique, dissipation of disturbance torques with passive magnetic dampers. We have enumerated expected disturbances that could be encountered in deployment and during orbital motions, providing equations of motion and analytical tools for their description. Finally, based on hardware character-istics, we provided a preliminary damper design, attempting to minimize mass and magnet field strength by evaluating damping time constants and steady state errors.

### References

1.) Pagano,R. Enright, E.,”Ground Integrated ET-derived Space Platforms for On-Orbit Applications”, Space Manufacturing 9, AIAA/Space Studies Institute (1993).

2.) Anon., “Company Plans Conversion of Shuttle Tanks to Orbiting Research Platforms”, Aviation Week, 29 Feb. 1988, p 38.

3.) Heinbockel, J., Breedlove, W., “An Analysis of the Passive Stabilization of the Long Duration Exposure Facility (LDEF)”, Old Dominion Univ., Norfolk, VA, TR 74-M5, 1974.

4.) Breedlove, W. , Heinbockel, J., “A Formulation of the Equations of Motion for LDEF – Final Technical Report”, NASA CR-132586, July 1974.

5.) Das, A., Foulke, H., Siegel, S., “Passive Stabili-zation of the LDEF”, General Electric Co., Valley Forge, PA, GE 74SD4264, Nov. 1974.

6.) Lerner, G., “18.3 – Attitude Stabilization Systems”, in Spacecraft Attitude Determination and Control, J. Wertz, ed., D. Reidel, 1984, New York.

7.) Kelly, W., “Delivery and Disposal of a Space Shuttle Tank to LEO”, Journal of the Astronautical Sciences, July-September 1984, v. 32, n. 3.

8.) Kelly, W., “Passively Stabilized Buildup – Activity Report”, Lockheed (LEMSCO), October 1986.

9.) Ellis, G.., “CETF Flight-1 Magnetic Passive Damper Study “, LEMSCO Memo, May 1987.

10.) Kelly, W., “Passive Damper Performance with CETF Flight -1″, LEMSCO Memo, 27 July 1987.

11.) Haymes, R., Introduction to Space Science, Wiley, New York, 1971.

12.) Wade, J., “Passive Attitude Damping of Alterna-tive Assembly Configurations of SSF”, Guidance, Control and Dynamics, 1991, v. 14, n. 1, pp. 36-43.

13.) Ellis, G., “SES Passive Plume Effects Analysis for PIT(Truss) MB-1″, LESC, Houston, TX, Memo, 30 Sep. 1991.

14.) Kelly, W., “Modeling Both Spacecraft and Passive Damper Magnet Motions”, Lockheed-ESC, Houston, TX, Tech. Report 28111, NAS 9-17900, April 1990.

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16.) Shuttle Operational Databook, NSTS-08934, Vol. II, 25 July 1996, DCN, Amend. 206.

17.) Kelly, W., “Calculation of Gravity Gradient Torques”, LESC, Houston, TX, Memo, 17 Dec. 1992.

18.) Meirovitch, L. Methods of Analytical Mechanics, pp. 466-470, McGraw Hill, New York, 1970.

**Table- 1 Mass Properties for Candidate Magnetic Damper Spacecraft**

wgt -lbs ET empty, STS-44 LDEF SSF MB-1 CETF (ca. 1987) FGB/Node-1 (1996)

66,528 21,393 33,518 68,243

c.g. (inches – Shuttle System) ( Orbiter System (ft – SSF ) (ft – SSF ) (ft ISS)

x 1354.8 897 0.19 -40.71

y 2.7 0 117.45 .02

z 424.3 400 0.31 14.26

Moments of Inertia (slug-ft**2) (principal)

IXX 354,005 24,090 215,000 4,707,760 66,269

IYY 3,931,111 71,220 59,500 4,890,066 940,049

IZZ 3,929,108 73,270 227,000 339,673 951,860

Products of Inertia (slug-ft**2)

PXY 3,692 3 26,800 -50,062

PXZ 157,498 -4 -4,060 -293,556 (42,537)

PYZ 7,002 -6 -269 292,036

Deflection Angles from LVLH

( assumes longitudinal coordinate axis aligned with radius vector, etc.)

fo -1.5 – – .13

qo 2.02 – – 2.76

yo -35.12 .16 .01

orientation remarks 1,2,3

**Table-2 LDEF Specifications**

Length: 30 feet (9.14 m)

Width: 14 feet (4.27 m)

Empty weight: ~9,000 pounds (3,629 kilograms)

Launch weight: 21,393 pounds (9,724 kilograms)

Experiment bays: 86 (72 peripheral & 14 end)

Number of experiments: 57

Deployment: April, 1984; Challenger (STS-41C)

Retrieval: January, 1990; Columbia (STS-32)

Orbital altitude: 216 – 154 nautical miles (400 – 286 kilometers); at deployment and retrieval, respectively

Exposed surface area: ~130 square meters

Addendum:

Company Plans Conversion of Shuttle Tanks to Orbiting Research Platforms

Aviation Week, p. 38, 29 Feb. 1988

The commercial space initiative approved in February by President Reagan offers to deliver to the private sector used space shuttle external tanks in orbit for no charge for research, storage or space manufacturing.

Privatization of the ETs is more complicated for many reasons than other space platform service propositions and is expected to take subtantially loner to materialize. Although many space experts dismiss the reuse of external tanks as interesting but unrealistic, a company based in Boulder, CO, has been working on an ET privatization plan for more than a year.

External Tanks Corp. (Etco), raised $480,000 last year to support its plan to convert used Ets into habitable laboratories in space. About half the investment, $225,000, came from an entrepreneurial software company, Autodesk, Inc. Etco’s business plan is based on taking control of the tanks in orbit and converting them to usable, pressurized volume, which then could be leased to user’s (AW&ST, Jan 12, ‘87p. 102).

NASA last summer signed a preliminary memorandum of understanding (MOU) with the University Consortium for Atmospheric Research (UCAR), which owns 80% of Etco. Under the non-binding agreement, NASA and UCAR will explore the feasibility of the project and related issues. NASA also agreed to identify possible government uses of the orbiting tanks.

The technical challenge presented by carrying the tank to orbit and maintaining it there is “within the current state of the art of shuttle operations, support systems and technology,” NASA told Congress last year.

However, NASA pointed out several areas of concern with the plan:

o Reduction of the Shuttle payload capability.

o Propellant requirements to prevent reentry of the external tank.

o Accessibility of the orbiting tank.

o Probability of micrometeoroid or space debris damage to the tank or potential impact of the tank with useful satellites.

o Cost of tank modifications and operations.

Etco needs to raise much more money than it has to date to realize its plan. The company will try to raise 5 million in the next round of financing to perform design studiesthat are necessary before building the hardware to convert the tanks to the laboratories. Etco officials believe they will be able to raise the fundas after obtaining an MOU from NASA covering the use of the tanks. They believe this memorandum represents a $200-million asset.

In table-1, among the orbital spacecraft configurations provided is the FGB/Node-1 combination characterizing the International Space Station at completion of its 2nd assembly stage. This configuration is expected to be oriented in a spin stabilized mode supplemented by attitude control thrusters correcting precession. Preliminary analyses of an 8-month delay in arrival of subsequent components indicated an 800-kg increase in the propellant budget for this type of operation, implicitly requiring an additional re-supply flight. Magnetic dampers could perhaps perform the same function, assuming that thermal or other operational constraints can be overcome.

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