William H. Arnold, Stuart Bowen, Kevin Fine, David Kaplan, Margaret Kolm, Henry Kolm, Jonathan Newman, Gerard K., O'Neill and William R. Snow
Various structural and dynamical problems related to both small-scale forces between the drive coils and within the bucket structure as well as the overall combined large-scale dynamical interaction of the bucket stream and MDRE structure are examined. The large-scale dynamics appear weakly stable. Finally, MDRE operation in an inverse-square-law gravitational field is discussed and the required curved shape of the guideway is computed.
INTRODUCTION
In the following sections we discuss various structural and dynamical problems associated with operation of mass drivers. First, because of the very large accelerations (~103-104 m/s2) we are concerned with the various forces and stresses between the drive coils and within the bucket; in particular, the stresses on the super-conducting coils. We then compute the "tear-off point," that point along the MDRE at which the feeder structure alone can no longer support the imposed axial stress, and supplemental strengthening must be incorporated, probably in the form of steel cables.
We next examine the large-scale dynamical interaction of the bucket stream with the primary load bearing MDRE electrical structure. The bucket stream is modeled as a fluid stream whose acceleration along the limp MDRE structure provides the tension force in both the acceleration and deceleration section. The transverse oscillations of this structure are computed. Finally some consideration is given to the problems of steering a MDRE and the requirements for operation in an inverse-square gravitational field. In particular the necessary changes to the nominal straight form of the MDRE in free space are computed for operation in a gravitational field.
SMALL-SCALE FORCES AND STRESSES
Mass Driver Structure
Viewed in the macroscopic scale, the mass-driver (MD) structure experiences an overall tension which is due to the gross acceleration of buckets down the length of the track. Under closer examination, however, the forces that act on the support structure between individual drive coils are seem to behave in a manner such that both local compression and local tension are experienced. To determine the stresses to which the support structure between drive coils will be subjected, and hence must be designed to withstand, it is necessary to first determine a time-dependent description of the small scale forces acting on any individual drive coil. With this in hand, it will then be possible to come to some conclusion concerning the forces exerted on the structure between two adjacent drive coils.
Due to the triggering mechanism in the electric circuit, current is permitted to flow in a drive coil only when a bucket coil is within a distance of 2
m from that drive coil. During the period in which it is on, the drive coil will experience inductive forces not only with the bucket coil, but also with other neighboring drive coils which themselves have current flow. Calculations reveal that the reaction force on the drive coil due to the bucket coil is actually an order of magnitude smaller than the reaction forces due to the other neighboring drive coils.
The force interaction between the bucket coil and the drive coil is described by the equation (ref. 1):
where iB is the constant current in the bucket; iD is the time-varying current in the drive coil; and dM/dx is the gradient of the mutual inductance between drive and bucket coils and is a function of distance separating the coils. Figure 1 plots the reaction force felt by a drive coil due to the passage of a bucket coil. While the force calculations presented are made for the specific case of a mass driver reaction engine of caliber 0.0508 in (ref. 2), they are nonetheless typical of all mass drivers in terms of general behavior (though not necessarily in terms of magnitude). Because of the antisymmetry of both the iD and dM/dx curves, the bucket coil receives twin impulses from the drive coil - first it is pulled toward the drive coil as it approaches, and next it is shoved away from the drive coil as it departs downstream.
The second source of forces acting on the drive coil results from inductive interactions with other active neighboring drive coils. These forces are described equally well by equation (1) if the neighboring drive coil current is substituted for the bucket current and a different dM/dx curve, one applicable to equal radii coils, is used.
Over a short distance (typically several centimeters), it is reasonable to assume that bucket velocity V is constant. With a fixed drive coil spacing (m ), then, the time the bucket coil takes to travel from one drive coil to the next, t, is also constant and equal to m /V. The current flow in each drive coil is sinusoidal, and designed to have a period of 4t. Consequently, any two adjacent drive coils, if active, will be 90o out-of-phase with each other, and at any particular moment, current will be simultaneously flowing in four contiguous drive coils. Figure 2 indicates which drive coils will be active during each of four quarter-wave phases of a reference drive coil. The appendix documents the specific force equations and presents a calculator program to solve those equations. The results from the appendix are plotted in figure 3 .
Through the superposition of the force exerted on a drive coil due to passage of a bucket coil and the force exerted on that drive coil due to its neighboring drive coils, a complete description of the total force exerted on the drive coil may be ascertained. A plot of this is shown in figure 4 , from which it may be seen that the coil experiences forces in both the positive and negative directions.
With a knowledge of the behavior of the force on the individual drive coil, attention may now be focused on the stresses applied to the structure between any two adjacent drive coils. Consider a section of the mass driver track in which the bucket velocity is substantially less than the speed of sound for the support structure material (for aluminum, approximately 5000 m/sec). If the velocity of the bucket is assumed to be constant, then each drive coil will propagate identical stress waveforms with identical periods, and the net force felt by a structural member joining two adjacent drive coils will be the difference in the instantaneous force exerted on each coil individually. Figure 5 graphically displays this difference in force (assuming constant bucket velocity) for the example mass-driver reaction engine. From this figure, it may be observed that the tension to be sustained in the material is roughly twice as great as the compression and is of a magnitude approximately double that of the maximum force acting on an individual drive coil.
If the bucket velocity is equal to or in excess of the speed of sound for the structural material, then the stress wave caused by the passage of a bucket coil will never reach the following sequential drive coil before the time that coil starts to propagate its. own stress wave. Considering the bucket velocity to be constant, all drive coils will experience the same forces within the same length of time. Hence, all coils will propagate the same tension waves in one direction and the corresponding compression waves in the other.
Since these waves all propagate at the same velocity, no wave will "catch up" to the one in front of it. However, tension stress waves traveling in one direction will pass through compression stress waves moving in the other direction. The additive behavior of these complementary waveforms will give the maximum stress in the local structure. This behavior is well understood and maximum stresses are easily predicted.
Unfortunately, the assumption of constant bucket velocity over significant portions of the mass-driver track is not reasonable (indeed, it is contrary to the concept of a linear accelerator). The velocity increases, and the time during which the forces act on the drive coil decreases. Consequently, the stress waves propagated by contiguous drive coils will have increasingly shorter periods. An analytical or numerical analysis of the constructive or destructive interference of these stress waves is beyond the scope of this study and considerations of how their effects behave along the length of the track must be left for future study.
In summary, calculation of the forces exerted on any drive coil lead to the conclusion that the structural members joining two drive coils will be subjected to both local tensions and local compressions. Over small lengths of the track, bucket velocity may be taken as constant and hence the stress waves Propagated by drive coils will be identical. Over these short regions, forces acting on the structural members joining adjacent drive coils can be determined. However, a complete analysis of the structural load forces between drive coils must include changing bucket velocity and the consequent interference patterns caused by nonidentical stress waves. It must be noted that only the first few coil sections will be subjected to a net compression since as one progresses along the MD the intercoil forces shown here are superimposed on the increasing tension,
Bucket Structure
The mass-driver bucket is subjected to various forces which must be evaluated in designing the bucket. Considered here are the forces on the bucket coils: the "hoop stress" which keeps the coil in tension, the forces between the two coils, and the stress associated with connecting the payload to the superconducting coil.
Hoop stress- A radial force in the outward direction is experienced by any current-carrying loop. This force is equal in magnitude to d energy/dR = (1/2)I(2L/R) where I = current, R = radius, and L = self-inductance of the loop. The force per unit length is then I 2L /4
R 2 which results in a tension on the wire of I 2L /4R .
For a bucket coil with thickness 0.1 D and radius 0.26D, L =(6128.46XI0-9) D (henries). Current is 2.5X108) A/m-2) X (0. 1 D )-9 and R is 0.26 D, yielding tension as a function of caliber:
The stress on the loop material is:
= T / (0.1 D) 2 = 1.172X106 D 2 PaEven for a large 0.5-m caliber mass driver, the hoop stress is 2.931X105 Pa.
Force between coils- An attractive force will exist between two coaxial coils carrying current in the same direction. The magnitude of the force is I1 I2 (dM /dx) where I is current and M is mutual inductance. For the specified bucket geometry, in which both bucket coils are of equal radii and the ratio of radius to distance between them is always 0.26 D 1.11 D, dM /dx is a constant 1.3903XI0-8. I = 2.5X108. A/m2 X (0.1 D)2 , So the force between bucket coils is:
For a 0.0508-m caliber model, this force is 0.58 N. For a 0.5-m caliber machine (larger than any currently being considered) the force would be 5430 N, less than one-third the force of the accelerated payload mass on the bottom of the bucket; this force is not a limiting factor in bucket design.
Load attachment forces- In order to minimize heat loss from the superconducting coil, the bucket coils will have to be attached to the payload at the minimum possible number of points with the smallest possible connection. The penalty, of course, is in stress on the coils at the attachment points.
The sections of the coil between attachment points are comparable to curved, double-cantilevered beams which are being loaded with a uniform force normal to the plane of curvature. Calculations for such beams have been done and enable us to deduce that, for a coil attached at eight evenly spaced points, the transverse shear force at the attachment points will be wR (0.3927), where w is the loading in force/unit circumferential length of winding, and R is coil radius. For a coil attached in only four places, the force is wR (0.7854).
Supposing that one coil had to accelerate the total combined mass of payload and bucket at an acceleration a, then F = (M1 + MB)a = W(2R); m1 = (D /0.197)3 and mB = 2.263(D /0.197)3, so wR = (1/2)xax 3.263(D /0. 197)3. The force on the attachment points is then proportional to D3 and to a.
This force will act upon connections which will probably be kept quite cold, and hence brittle, by their proximity to the superconducting coil. Attachment that will distribute the transverse force so as to keep stress within acceptable limits while minimizing mass and heat loss is one of the engineering problems of actual bucket design. These considerations affect mass driver design insofar as that the stress placed on the coils increases proportionally to D. Once the stress limits of the material have been reached, either the proportion of load attachment area must be increased with D for large-scale mass drivers or acceleration must be decreased (held constant with respect to 1/D).
A sample calculation: for a 0.4-m caliber mass driver, accelerated at 1000 g's by one coil, with four attachment points which distribute the load equally over four sections of the coil, each spinning a 10o arc, the stress on the coil tubing is about 2.31 X 107 Pa.
Tear Off Point
We compute the "tear-off" point, that is the position along the MDRE where the stress placed on the feeders alone requires additional strengthening to sustain the axial tension.
The tension is
and the feeder area is
where mBL = payload + bucket mass, and L = acceleration length.
If = ult at the tear off point, then we can write the above as
where Ve = 
.
Since mBL= m1[1 +
(mB/m1)],
= thrust force
If x/L>1, tear-off is beyond the attachment point; if
x/L<1, tear-off occurs prior to attachment point and additional strengthening must be provided.
For A1 2020 T4:
ult = 4.27X108 N/m2Steel:
ult = 1X109 N/m2For OPT-4:
In figures 6 and 7, x/L is shown as a function of the exhaust velocity Ve for various propellant accelerations as computed from the OPT-4 formulas. The tension at the attachment point =3.263 where T is the MDRE thrust.
LARGE-SCALE DYNAMICS AND STABILITY
Bucket Stream in a Curved Tube
To understand the combined effects of the bucket stream on the stability of the mass driver we consider the following sample problem.
We imagine a curved section of tube containing two frictionless ball streams moving in opposite directions within the tube and which elastically rebound from the end walls of the tube. The momentum change of the ball streams at each end provides a tension force T which will tend to straighten the curved tube.
The mass flow of each ball stream is 
= 
BV where B = mass of balls per unit length and V = velocity of each ball. The momentum per ball is p = mV where m = ball mass. The momentum change per ball due to the elastic rebound at the end is 
p = 2mV so that the momentum change per second 
= 2V = 2BV 2 which must equal the tension force T. If the curvature of the section is u", where u is the displacement of the tube, the net downward straightening force on the section is:
Opposing this straightening force are the centrifugal forces of the two ball streams which tend to increase the curvature of the tube. If R = 1/u" is the radius of curvature then the upward centrifugal force due to the curvature of the two ball streams is:
which exactly balances the straightening forces. Therefore, the curved tube-ball stream structure is dynamically neutral, neither straightening itself nor curving itself more strongly.
Additionally, one can easily show that the total downward force on the section given on 2/Tu' is exactly balanced by the total upward force exerted by the ball streams on the tube 4VB2u' where u' is the slope of the curved tube at the ends. Hence, bending the tube does not give a net upward or downward force, which we also know from Newton's Third Law.
Another way to arrive at the neutral curvature stability is to consider the moment of the tension and centrifugal forces taken about the center of the curved tube at A.
The clockwise moment of the tension force about the point A is:
The counterclockwise moment of the centrifugal force due to an element of tube ds = Rd
about the point A.
due to the two ball streams is
So
Since these are the same there is no net moment tending to bend or straighten the tube-ball stream structure.
Large-Scale Dynamical Stability
Mass-driver reaction engine structure- The MDRE structure is typically a very long (3-10 km) and at least for the electrical components, quite thin (
1 m) structure operating in tension. Transverse oscillations will travel very slowly along the structure at velocities of ~10-50 m/sec. Typical payload launch rates are 10 Hz and are considered high frequencies. We therefore consider the following as a useful limiting model to describe the combined mass-driver bucket stream interaction for large-scale, low-frequency dynamical motions.
Continuum treatment of bucket flow- If = the mass flow of buckets and payloads considered as a fluid, = 
AV, V = bucket velocity, then B = A = bucket and payload density per unit length.
For uniformity accelerated motion
Using 
= bucket repetition rate, Hz, = mBL, mBL = bucket and payload mass, = BV, and = B
. Hence
If n(x) = number of buckets between o and x, the mass of buckets between o and x is:
and hence
The tension at x isT(x) = mBLan
(x) = mBL
= mBL
V(x), and T(x) = V(x).
We will ignore longitudinal motion of the rope in the x direction and assume the bucket stream coincides with the rope.
The transverse equation of motion governing the transverse displacement u(x,t) of the rope is derived using the momentum theorem applied to an incremental control volume of length dx enclosing both the rope and the bucket stream.
The momentum theorem can be written as:
 rate of increase of momentum inside volume V |
 rate of momentum gain by convection of momentum inward across the surface S enclosing volume V |
 external forces acting on volume V |
Where is the volume density of the material having velocity v.
We assume small slopes and curvatures so that sin u'~u' where u' = 
u/x and cos u' 
1.
In the above sketch 
and 
refer to the surfaces at x and x + dx, respectively, across which buckets are flowing.
For the rope
(2)
where 
= u/ t and 
and 
are unit vectors in the x and y directions.
For the bucket stream
(3)
is the transverse rope velocity and vu' is the transverse bucket velocity relative to the rope.
The continuity equation for the control volume is
(4)
The total mass inside the control volume is
and is independent of time. Hence the left-hand side of equation (4) is zero and
Now
is the mass flow of the bucket stream across the surface 1 at x. The minus sign occurs at 
and d
are in the opposite directions.
Similarly
It is convenient to write the mass flow of buckets along the rope
External forces
These are only due to tension and if we are in free space
Since (5)
Momentum flow now: Now
On surface in the preceding sketch, = - BV' so
while on surface = BV' so
The total momentum flow is thus
(6)
Momentum inside control
(7)
In the transverse direction we have:
Substituting for My, canceling the common dx noting BV = is a constant independent of x or t, and that v, , B are functions of x but not of t, we finally have
(8)
It is interesting to note several limits to equation (8).
B = 0
which is the wave equation for a variable tension, variable density rope.
B remaining finite
which again is the wave equation on a rope having an increased mass per unit length.
Form of various terms: The rope tension due to the bucket stream can be written as
Hence the terms in equation (8)involving u" cancel. Also since
the terms involving u' v' also cancel in equation (8) leaving
(9)
Since v = = bucket velocity
(10)
For reasons of constant power dissipation per unit length, the rope density is also proportional to 
Thus we let
(11)
where T = mass per unit length of buckets and rope at the attachment pointx = L. If we assume harmonic motion
then equation (9) becomes
(12)
which has the solution
(13)
where
(14)
The eigenfrequencies are obtained by applying the boundary condition that
Hence
or
(15)
The complete solution is
(16)
A plot of the fundamental n = 0 and the first three harmonics are shown in figure 8.
The total mass of the acceleration or deceleration section is:
so that the period of the nth harmonic can be written as
that is, proportional to the mass-driver mass divided by the mass flow through the driver.
Comments regarding MDRE Solution, Equation (16)
1. the eigenfrequencies of the MDRE structure are very small. For an asteroid class MDRE of acceleration length 10.6 km, = 10 X 3.8 = 3.8 kg/sec; = 150kg/m, B = 2.76X10-2 kg/m at attachment point Ve = 5,000 m/sec, the first eigenfrequencies are (OPT-4)
| n |  n rad/sec | period,hr |
| 0 | 1.36X10-4 | 12.83 |
| 1 | 4.09X10-4 | 4.27 |
| 2 | 6.81X10-4 | 2.56 |
| 3 | 9.54X10-4 | 1.83 |
2. Because of these long periods it is clear that a beam term
will need to be included in the right-hand side of equation (9). The inclusion of the fourth-order term will allow satisfaction of the following boundary conditions:
Values appropriate to EI for external stress structures must be obtained to proceed further. The effective transverse moment I~ (transverse dimension)4 while Young's modulus can be changed using active tensioners. Adding springs through an external structure will give a tension in addition to the tension due to accelerated bucket stream. This will leave a u" term in equation (16), as will an initial tension T0 at x = 0 due to the bucket turn around at finite speed either for insertion into the acceleration section or removal from the deceleration leg of the mass driver.
3. The eigenfrequencies of the acceleration and deceleration sections can be made the same. Since the force exerted on the buckets is the same, in each section, the acceleration ratio is
(17)
The requirement for n to be the same in the two sections is
Using
then
(18)
which is the same as equation (17) provided T,decel = T accel, generally, >> B.
4. The solutions n(x) form an orthogonal set on (x) using a new variable z = 
. Thus, given an initial u(x,0) and/or u(x,0) the constants in equation (16) can be determined using the Fourier sieve. Since the higher harmonics are not integral multiples of the fundamental, the overall shape of the MDRE composed of a summation of the initial eigensolutions will not repeat itself. Each normal node has one more node than the preceding one and the nodes crowd together toward the free end where the tension is small. The slope at the attachment point is finite.
S. Since the variable x occurs as in the sine or cosine, transverse waves are dispersive, the wave speed depending on frequency. For moderate period waves (~1-0.1 Hz) the local wave speed can be written as
6. We conclude from the weakly oscillatory nature of the solutions to the free space MDRE electrical structure-bucket interaction that the MDRE is stable. The driving term on the right-hand side of equation (9) is proportional to the bucket mass flow and hence the external truss structure will need to be sized in some manner related to the mass flow.
7. The most important parameter of the external structure is its mass. From a viewpoint of overall efficiency, the external structure should be as light as possible. When this factor is entered into the overall optimizations, it will undoubtedly tend to shorten mass driver length to decrease overall mass. This has not been done at the present time because of ignorance of appropriate structural mass values.
Even if we do not have figures on structural mass, we can still determine how these masses will scale with changes in total length. Assume that our mass driver external structure is made from a long chain of identical sections, so that the value of EI does not change over the length. If we change the total length of the mass driver, how must we change EI so that the deflections are geometrically similar?
We will assume that the forces acting on our mass driver are gravitational tidal forces, which will increase linearly with the length. This means that the applied forces will also be geometrically similar. The basic differential equation describing the deflection is
If we change the length by a factor of 2, the applied force at a point increases as 2, and the applied moment increases as 22. Defining
Then the deflection is described by the nondimensional equation
For a similar deflection 2u0/
x02 must be the same which implies the factor L3/EI must be constant. But EI increases as r (transverse size of external structure) to the fourth power. Thus for similar deflection
But mass and length of the track increase as r2 . This implies that mass and length of the structure will increase as L9/16 and total mass as L25/16 . So total mass increases with increased length not only because a longer structure is needed, but also because this structure must be more solidly built.
This model assumes that the major loads applied to the MDRE are gravitational. It may be that they will in fact be due to internal mechanic motions (mating of pellets, etc.). As long as it is known how these forces scale with total length, this kind of analysis can be applied. If applied moments are constant with length, for example, then mass increases as L17/16; very nearly a power of one.
Steering the MDRE
A requirement exists for changing the thrust vector orientation during both orbital and free space operations. One method for accomplishing this steering is to utilize the torque produced when the thrust vector does not pass through the center of mass of the entire system. It must be recognized that the application of the torque so produced to the very long, basically flexible MDRE to produce the desired overall angular acceleration without undue bending, constitutes a major yet unsolved structural problem, probably requiring active structural control. Clearly there will be upper limits of the overall angular acceleration and bending that will still allow passage of the buckets through the guideway without their scraping the guideway edges.
We dynamically represent the MDRE by a long uniform rigid rod with the propellant and payload masses arranged as shown:
We let
| mD = | electrical and external structural mass of MDRE "uniform rod" |
| 2m = | total propellant and payload mass |
| L = | total length , acceleration and deceleration of MDRE "rod" |
| 2c = | distance between the two separated payload and propellant masses, perpendicular to the rod |
| d = | axial separation between the payload and propellant masses at the attachment point and the center of MDRE |
| = | offset distance, the perpendicular distance between the thrust vector and the center of mass of the propellant and payload mass |
The moment of inertia of the combined system about the center of the rod is
where
the position of the overall system center of mass lies a distance of
1 toward the propellant and payload center of mass and has a perpendicular distance to the rod given by a.
Now
Hence the moment arm between the overall system center of mass and the thrust vector which is assumed coincident with the MDRE axis is
Using the Lagrangian transfer theorem to find the system moment of inertia about the displaced center of mass and noting
we find
The angular acceleration is then
when T = thrust of MDRE. The required offset is
Note that 2 << d2 so 2 can usually be neglected in the expression for Iz.
The required angular acceleration for tangential thrusting in a nearly circular orbit is computed in the following section.
As an example consider a MDRE in a circular orbit at an altitude of 200km requiring 
= l.22Xl0-10 rad/sec2. If T = 560 N, mD = 200X 103 kg, 2m = 1900X103 kg, L = 10.3km, d = 3.537km, c = 10 m then v = 0.0952, Iz = 4.023X 1012 kg/m2 and the required offset = 0.97 m.
Rather than using the entire propellant and payload mass for the active control mass, a smaller mass could be moved through a greater distance to provide the required torque.
As a second example we compute the offset needed for an asteroid retrieval mission utilizing the moon with perilune 87 km above the surface and V
= 0.84 km/sec for which we have computed = 7.62X 10-7 rad/sec2.
For a typical Opt 4.0 asteroid retrieval T= 1.5X104 N, mD = 3.512X 105, d = 345 m, c = 50 m, 2m = 1.832X 108 + 1.500X 108 = 3.332X 108 kg, we find Iz = 1.300XI012 kg/m2 so the offset = 66 m, which is larger than the separation c. Hence, either one needs a larger separation or a larger perilune and/or smaller V which will give a smaller .
MDRE OPERATION IN A GRAVITATIONAL FIELD
Angular Acceleration
We compute the angular acceleration required for operation in the inverse-square gravitational field around a central body.
The angular rate of revolution in a circular orbit a radius r around a central body is
(19)
(20)
To find the angular acceleration, that is the rate of change of the pitching rate, we differentiate equation (19), with respect to time and substitute equation (20) to find
(21)
Since a = aO /(1 - aO t /Ve ) where Ve is the exhaust velocity, aO is the initial acceleration = T /mO, mO is the initial mass, we can integrate equation (21) by separating variables to find
and
or
Another case of interest in asteroid retrieval missions is the angular accelerations required during a hyperbolic encounter with the Moon or a planet. We characterize the hyperbolic orbit by the nondimensional velocity at infinity 
= V /VO and the nondimensional periapsis distance 
p = rp /rO where VO is the circular velocity about the primary at radius rO. The unit of time tO = rO /VO. The nondimensional angular momentum is given by
and the total velocity is
where ( )' = d( )/dtO, and = true anomaly. For constant angular momentum
so that the nondimensional angular acceleration of the true anomaly is
Using the vis-viva relation
During hyperbolic encounter one may simply orient the MDRE along the radius vector so the angular acceleration required is given by the above. Using the Moon (rO = 1738 km, VO = 1.679 km/sec) with = 1.05 , that is, 87 km altitude perilune, = 0.50 km/sec so =0.8395 km/sec, the peak = 0.9217 at = 1.2 and hence:
For a MDRE in circular orbit at an altitude of 200 km above the Earth with T = 560 N, Ve = 8000 m/sec and mO = 2.1X106 kg,
and
so
= 1.22X10-10 rad/sec2with
= 1.18X10-3 rad/secMass-Driver Shape in Orbit
The nominal shape of the mass driver reaction engine in free space will be a straight line. However, when the MDRE is operating in orbit about the Earth or Moon the nominal shape will deviate from a straight line. This arises from the fact that the bucket is moving at a Then different orbital velocity from that of the main structure of the MDRE. If left to itself the MDRE would tend to deflect downward (in a rotating coordinate frame) when Ve is such as to give retrograde elliptical orbits for the released pellets. Thus, the thrust would not be tangential for a straight MDRE and additional forces on the guide way would result in the bucket scraping the walls. Therefore, the nominal shape of the MDRE must coincide with the trajectory of the accelerating bucket with the additional constraint that at release the pellet velocity is tangential to the orbit.
The reference frame used moves in a circular orbit of radius r as shown below.
Hill's equations for the motion relative to a coordinate system in uniform circular motion with the x direction along the radial direction are (ref. 3)
where ax,ay are the applied accelerations. The constraint that the acceleration is tangential to the local shape of the MDRE requires
Since
Then
Now Hill's equations become
with
where
K = GM = gravity parameter of the central body
To obtain the shape of the MDRE as given in figure 9 these two equations must be numerically integrated to obtain x and y. To accomplish this and to meet the constraint that at the Ve point the velocity vector be tangent to the orbit, these equations are integrated backward in the acceleration section with the final launch conditions becoming the initial conditions.
We compute the shape of a MDRE in a 200-km altitude spiral orbit about the Earth with the following performance.
| accel. | a = 5000 m/sec2 | L= 6400 m |
| decel. | a = 8205.1 m/sec1 | L= 3900 m |
The deceleration section shape was found by integrating forward with initial conditions being the final conditions from the accelerator section. A second-order Runge-Kutta program was used to integrate these with a step size of 0.05 sec for a duration of 1.6 sec for the acceleration section and a step size of 0.025 sec for a duration of 0.975 sec for the deceleration section.
