Space Elevator Dynamic Response
to In-Transit Climbers
David
D. Lang1
1David D.
Lang Associates, Seattle WA.
Abstract:
This presents findings of time-domain simulation studies of the space elevator
using the Generalized Tethered Object Simulation System (GTOSS). Overview of
mathematical models comprising GTOSS are presented. Physical configuration of
the elevator as it manifests itself within GTOSS is described. GTOSS simulates
climber transit by modeling chains of multiple objects and tethers, the climber
being an object between two adjacent tethers, thus, climbing occurs as the
earth-side tether undergoes deployment and ballast-side tether undergoes
retrieval. Once an element of ribbon enters the domain of the climber, that element is no longer a participant in the free
motion model of tether lying outside the climber (until it emerges). So that
which is true in nature is extant within GTOSS to simulate a climber and should
provide useful insight into the nature of climber design and its impact on elevator
operations. The study characterizes effects on the ribbonÕs transverse,
longitudinal, and libration mode oscillations due to start-up, cruise transits,
and motion arrest.
1. Introduction
This
paper explores the dynamic response corresponding to a variety of climber
operations including: initial lift-off dynamics, nominal transits, transit
resumes, and transit arrests. This study attempts to identify potential problem
areas, and recommend areas for further examination. In particular, this is an
exploration of dynamic attributes that could lead to ribbon failure or present
operational limitations. Due to lack of concrete information on climber design,
such as mass tensor properties and control system strategies, climbers are
simulated as 3 (rather than 6) degree of freedom rigid bodies.
2. GTOSS Overview
The
Generalized Tethered Object Simulation System is a time-domain dynamics
simulation code, developed by the author in 1982 to provide NASA with the
capability to simulate the dynamics of combinations of space objects and
tethers for flight safety certification for the Shuttle Tethered Satellite
System (TSS) missions. Since then, GTOSS has undergone continuous evolution and
validation, being applied at some stage in the formulation of virtually every
US tethered space experiment flown to date; more than 25 aerospace
organizations have employed it. GTOSS was designed for generality, thus
allowing its current use in simulating space elevator behavior. Below is an
overview of its features.
¥
Multiple rigid bodies, with 3 or 6 degrees of freedom, connected in arbitrary
fashion by multiple tethers, all subject to natural planetary environments,
including sophisticated models for earth attributes as well as more rudimentary
models for the other planets.
¥
Tethers represented by either massless or massive models.
The massive tether
model is a Òpoint synthesisÓ approach, each tether employing a constant number
of up to 500 nodes, specifiable by tether (500 being a system configurable
limit).
¥
All tethers can be deployed from, or retrieved into, objects by means of
user-definable scenarios. The deployment/retrieval dynamics model includes
momentum effects of mass entering or leaving the domain of the tether itself,
and produces related forces on objects deploying and retrieving the tether
material.
¥
Tethers
can be defined to have length dependent non-uniform material properties.
Elastic cross section, aerodynamic cross section, and lineal mass density are
independently specified for up to 15 separate regions. Properties at sub-nodal
points within each region are determined by interpolation. Each region can have
its own modulus of elasticity and material damping attributes.
¥
Tethers are subject to distributed external forces arising from the following environmental
effects: Aerodynamics in the subsonic and hypersonic regime; Electrodynamics
due to tether current-flow interaction with the EarthÕs magnetic field using
current-flow models that incorporate the earth magnetic field and effects of an
insulated or bare-wire conductor interacting with the orbital plasma
environment model. Note, with an appropriate ribbon-to-plasma electron contact
model, this could simulate grounding-current in a conducting elevator ribbon.
¥
Tethers experience longitudinal thermal response. Tethers gain heat under the
influence of solar radiation, earth albedo, earth infrared radiation,
aerodynamics, and electrical currents; heat loss occurs through radiative
dissipation.
¥
Tethers can be severed at multiple locations during simulation.
¥
Objects and tethers can be initialized in many ways, including creating
stabilized extremely long tether chains, attached to and rotating with a planet
(a space elevator) with due consideration for non-uniform tether properties and
the concomitant longitudinally varying strain distribution of elastic tether
material.
¥ GTOSS creates a database containing
results of response to the user-defined material configuration, initialization
specifications, and environmental options; this permanent data base can then be
post processed to produce a wide
variety of result displays, from tabular data, to graph plots, to animations.
3. Climber Simulation Rationale
Construction
and operational payload climbers traversing the ribbon will excite transverse
and longitudinal string mode
responses and elevator libration motion (the simple pendulum-like motion of the
elevator about its anchor point). Such responses will reflect all the
potentially non-linear effects related to tapered ribbon design, inverse-square
gravity field, centrifugal forces, Coriolis effects, atmospheric disturbance,
and climber speed modulation.
GTOSS
possesses the ability to simulate a climberÕs transit of the ribbon by modeling
chains of multiple
objects and tethers. The climber would be an object in a chain with two adjacent tethers, the
earth-side tether undergoing appropriate deployment, while the ballast-side tether undergoes
complimentary retrieval
(for upward climbing). An argument in behalf of this approach starts with the fact
that once an element of ribbon enters the "domain of the climber"
(ie. gets clenched-in and/or threaded-through rollers, etc.), and until it
emerges, that element is within the domain of the climber itself, thus, not a
participant in the free dynamic motion of the ribbon lying outside the climber.
Such a state of affairs appears to meet all the pertinent criteria for
application of a tether deployment/retrieval simulation. So that which is true
in nature is extant within GTOSS to simulate climber action. In further
affirmation of this viewpoint, note that ribbon strain distribution internal to
the climber will, in general, be unlike that of adjacent external ribbon
because unique strain states can be imposed upon the ribbon within the climber
mechanism; indeed, exceeding limit-strain within the climber may be a factor in
climber traction designs that engage the ribbon through overlapping roller
schemes to take advantage of capstan effects. Intuitive reasoning opposed to the
above climber simulation rationaleÕ and based on a priori knowledge of ribbon
continuity must be tempered by the fact that as far as external ribbon
dynamics are concerned, there could just as well be a recycling plant within
the climber ingesting ribbon from above, re-synthesizing it to make a new
ribbon, and deploying it out the bottom.
For
climber studies, the GTOSS configuration consists of a climber containing two
reels of ribbon, both of which characterize the ribbonÕs dual taper design.
Earth-side deployment occurs such that the earth-end taper would emerge first,
while for the upward deployed ribbon, the ballast-end would emerge first. In
this way, no matter where the climber is positioned, the ribbon below and above
properly portray the total earth-to-ballast profile.
Conventionally,
deployment conjures up images of a
reel positioned at altitude, with ribbon being dropped down; that is not
what is occurring during climber operations. To clarify, consider two points,
P1 and P2, between which ribbon is to be dispensed. Two distinct processes can accomplish this,
process A and B. In process A, the reel is positioned at P2 remaining
stationed there with the ribbon
spooled-off and dragged to P1. In process B, the reel starts at
P1, and is then transported to point P2, with ribbon being laid-down between P1 and P2. These are dynamically distinct
processes, in that if observed from a location fixed between P1 and P2, the
following would be noted: in process A, there would be a continuous parade of different ribbon particles traversing by, while
for process B, a single particle
of ribbon would appear at the observation point, and remain there throughout
the deployment. Process B is realized by GTOSS climber simulation both above
and below the climber.
4. GTOSS Space Elevator Physical Properties
By
using appropriate definitions of tether properties above and below the interior climber-object, GTOSS is made to reflect the design
profile properties along the entire elevator. Conveniently, since each finite
tether model can have independent properties, assigning a different number of
nodes to each tether can provide dissimilar nodal resolutions at different
regions of ribbon if desired.
Ribbon
characteristics vary with length, and are comprised of: density, elastic-area,
modulus, aerodynamic-area, and damping, all corresponding to baseline ribbon
design as described in References 1 and 2. The ballast mass is 634,000 kg at a
radius of 100,000 km. The climber mass corresponds to the nominal 20 ton
design; attributes for inertia tensor, control effectors, etc., are currently
un-defined, so the climber is simulated with 3 (rather than 6) degrees of
freedom. The elevatorÕs tapered ribbon is initialized by GTOSS to a stabilized
vertical state with ribbon strain distribution in equilibrium with gravity,
centrifugal, and attached climber loads. The dual tapered ribbon is designed to
achieve, over its entire length, a uniform stress distribution of about half of the 120 giga Pascal capability
anticipated for operations. GTOSS confirms this as shown by the unloaded
ribbonÕs stress profile in figure 1 below.
Figure 1. Ribbon Stress vs Altitude
This
uniform stress results from GTOSS simulation of the interplay between planetary
environments and the ribbonÕs tapered
cross sectional profile, modulus, and density (shown in the figures
below). The slight droop in the stress curve near the earth can be attributed
to approximation errors associated with curve-fitting the ribbon profileÕs
taper gradient near the earth.
Figure 2. Ribbon Elastic Cross Sectional Area vs
Altitude
Based
on elastic cross sectional area profile and a nominal value of ribbon
materialÕs bulk density of 1.3 gm/cm3 (CNT), the lineal density
profile shown in figure 3 below was derived for use in GTOSS.
Figure 3. Ribbon Density vs
Altitude
5. Case Definitions and Simulation
Results:
Unless
otherwise specified, the following definitions apply:
(a). Nominal
climber Transit speed is 200 km/hr.
(b). Nominal Lift-off or Transit-resume is
defined as a one hour duration linear ramp-up from zero to
nominal transit speed.
(c). Nominal Arrest is defined as a one hour duration
linear ramp-down from nominal transit speed to zero.
(d). Sudden Arrest is a step change from transit speed to zero speed.
All
Transit and Arrest cases start with a stable transit in progress. All Liftoff or Transit-resume
cases start with zero rate.
Case 1: Nominal Payload Liftoff from Ground (1 hr ramp-up).
Case
2: Limit
Payload Liftoff from Ground (1 hr
ramp-up).
Case 3: Transit Resume from GEO (1 hr ramp-up).
Case
4: Nominal Transit
to Ballast (200 km/hr).
Case 5: Fast Transit to Ballast (400 km/hr).
Case 6: Nominal Transit to GEO (200 km/hr).
Case 7: Sudden Arrest at 2 km (zero ramp-down time).
Case 8: Modulated Arrest at 2 km (60 sec ramp-down time).
Case 1: Nominal Liftoff
from Ground (1 hour ramp up).
This
case shows the mechanism by which nominal climber liftoff can occur. A 20 ton
climber weighing 178,035 N on the launch pad will experience a ribbon tension
from above of 197,000 N and from below of 19,700 N. Before liftoff, the climber
net weight (gravity less centrifugal force) is in equilibrium with this tension
differential across the climber. While this tension differential allows
practical liftoff acceleration to occur, it also represents the maximum
possible liftoff acceleration. The mechanism by which the tension in the lower
ribbon is exploited for liftoff consists of the climberÕs transporting ribbon
from above itself to below. An element of ribbon length, when transported from
the domain of the upper ribbon has very little effect upon the strain in the
upper ribbon due to that ribbonÕs extreme length, thus tension is
correspondingly affected but little
due to the low effective spring rate of the upper ribbon. On the other hand,
that same parcel of ribbon, when introduced into the domain of the lower
ribbon, has a profound effect upon its strain state due to the lower ribbonÕs
short length, thus, a correspondingly greater effect upon the lower tension due
to the lower ribbonÕs high spring rate. Transport of ribbon from above to
below, can cause an immediate and significant drop in lower ribbon tension;
this in turn creates an imbalance on the climber accelerating it upwards. Once
the lower ribbon becomes slack, then the maximum possible immediately
available force imbalance on the
climber will have been realized. Any additional acceleration can only come from
increasing strain in the upper ribbon. Figure 4 below depicts a nominal
liftoff.
Figure 4. Climber Liftoff
vs Time
As
seen in figure 4 above, tension at the ground dips down initially as the
climber starts transporting upper ribbon into the lower ribbon domain. This
tension drop, that can be easily and quickly induced, provides the initial
upward force imbalance that starts the climber in motion. In spite of upper
ribbon continuing to flow into the lower domain, the lower tension starts to
increase due to upward climber motion continually separating the lower ribbon
attach points, thus tending to neutralize strain reduction that caused the
tension to drop initially, as shown in figure 5 below.
Figure 5. Average Strain in Lower Ribbon vs Time
During
the liftoff acceleration period, the upper ribbon is very slowly acquiring a
tension increase due to the strain increase brought about by the climberÕs
continual removal of ribbon from the upper domain; also at work is the general
gravity/centrifugal effect that brings about the characteristic concave-down
tension-versus-altitude profile of the elevator. The oscillations seen in the
these figures is associated with the fact that the liftoff scenario employed
was not designed to control longitudinal oscillations. Since the end-to-end
ribbon transmission time for a stress disturbance is about an hour, with the first
longitudinal mode having a period of about 2 hours, it is evident that stress
disturbances, once initiated, will periodically manifest themselves at the site
of the initial disturbance and longitudinal oscillations will exhibit (2 hr)
periodicity.
Upper
and lower ribbon climber tensions are compared in figure 6 below. It is seen
that upper tension also drops initially, with both tensions following a similar
profile in time. The initial drop in upper tension is because the climber
starts accelerating upward under the force imbalance due the immediate
reduction of lower tension. Even though the climber is transporting equal
increments of ribbon length from the upper into the lower domain, such
increments arenÕt nearly as effective in increasing tension above as it is in lowering tension below. Both tensions, continue to rise
and converge, finally peaking and coalescing to a zero differential at GEO.
Lower Ribbon at Climber Upper Ribbon at Climber
Figure 6. Upper and Lower
Tension at the Climber vs Time
Figure
7 below shows tension observed at 50 km altitude on the ribbon. The
discontinuous change at 0.75 hours is where the climber passes 50 km point.
Tension after Climber
passage Tension prior to
Climber passage
Figure 7. Tension at 50 km
Altitude vs Time
The
climber itself progresses up the ribbon with only slight westward horizontal
displacement as shown as libration angle relative to local vertical in figure 8
below. This libration is due to Coriolis effects, and its steady diminishing is
likely attributable to geometry of increasing altitude combined with steady
tension increase (above and below) that provides some restoring stiffness against horizontal displacement.
Figure 8. Climber Libration
vs Time
Case 2: Limit
Payload Liftoff from Ground.
This
explores repercussions of attempting climber launch at the maximum weight the elevator can support, a situation in
which the lower ribbon has
essentially zero initial tension. Unlike Case 1, the only means to effect
liftoff in this situation is to ÒclimbÓ the upper ribbon, a process equivalent
to increasing upper ribbon tension alone to effect liftoff and
subsequent transit velocity.
Figure
9 below shows a set of related liftoff parameters (note, a somewhat different
set of parameters are presented than shown in figure 4.)
Tension just above
climber Tension just below
climber No tension
40 km of loose ribbon on
Launch Pad
Actual Climber altitude Lower ribbon length Climber altitude
catches up with lower ribbon
length
Figure 9. Liftoff
Parameters vs Time
Since
the lower ribbon tension is initially near zero, the process of transferring
ribbon into the lower domain to effect a tension differential is no longer
effective; only increasing strain in the upper ribbon will act to effect
liftoff. This is clearly not an operationally feasible scheme, since it
operates the elevator at a transiently unstable load condition capable of
pulling down the ballast.
The
lower ribbon is observed to go slack for about 1.5 hours; by the time ribbon
transfer-rate has ramped to its nominal value, the climber has deposited 40 km
of slack ribbon on the ground.
Albeit an inefficient means of creating tension differential, removing this
amount of ribbon from the upper domain has increased upper tension to
eventually accelerate the climber to an altitude equal to the amount of slack
ribbon below; this increase is not
obvious at the tension scale in
figure 9. At the point climber altitude matches the (heretofore slack) ribbon
length below, a severe impact event occurs creating a longitudinal strain disturbance
seen to manifest itself on the order of every 2 hours at the climber,
consistent with the round-trip
stress wave propagation time to the ballast. At first impact, altitude rate is
twice nominal transit rate. The climber quickly acquires nominal rate after
impact transients subside, as shown in figure 10 below.
Stress-wave round trip propagation time
Figure 10. Climber Altitude
Rate vs Time
Figure
11 below shows snapshots of the impact disturbance at various times as it
progresses along the ribbon.
Figure 11. Tension Snapshots
vs Altitude along Ribbon
Note,
the climber progresses up the ribbon less than 1% of the total ribbon length
during the 4 hours of simulated time.
This
case clearly exhibits the fact that creation of immediate and significant climber acceleration cannot exceed a level corresponding to a force
equal to the initial tension at the climber interface with the lower
ribbon; in fact, smooth and timely liftoff operation depends upon this initial
lower tension. Thus, once a design acceleration is specified, then maximum liftoff mass of the
climber is determined. This will likely not constitute a significant
operational constraint since the time to accelerate to nominal transit speed is
likely not a critical design parameter provided a reasonable level is possible,
and it seems that it is.
Case 3: Transit Resume
from GEO (1 hour ramp-up).
This
case examines a climber resuming to nominal transit speed from a parked
position at GEO. Starting from 35,400 km altitude, the climber ribbon rate
ramps up to nominal over a period of one hour. Figure 12 below shows the
resulting altitude and climb rate response, and indicates that a resume to
transit
Figure 12. Climber Altitude
and Rate vs Time
speed
is feasible from GEO. This situation presents a different situation than
liftoff from ground. The mechanism that creates altitude rate, while intrinsically the same at ground or GEO, is not as immediate or as
effective at GEO. It is apparent, from the periodic oscillations in the
altitude rate, that the climber is interacting with the ribbonÕs first
longitudinal natural mode. In equilibrium at GEO, before a transit resume, the
climber would have the same tension above as below. Ribbon is still transferred
from above to below to create a tension differential, but it is unlike ground
liftoff where upper ribbon being transferred to the lower ribbon causes an
immediate and significant tension drop below but little tension increase above.
At GEO, this ribbon transport process has nearly equal effectiveness above and
below for creating tension change. Figure 13 below shows upper and lower
tensions at the climber. Meaning can be extracted from this apparent meandering
behavior if the differential in
these two tensions are examined.
Tension Above Tension Below
Figure 13.
Tension Above and Below Climber vs Time
This
tension differential, shown in figure 14 below, represents the forcing function
that creates vertical climber acceleration. During the initial 1 hour ramp-up
of ribbon transfer-rate, the net vertical tension differential is increasing to
Ramp-up ends
Figure 14. Tension
Differential Across Climber vs Time
accelerate
the climber. At one hour, ribbon transfer rate has arisen to the constant
nominal transit speed and at that point an oscillation manifests itself in the
differential tension, with a period characteristic of the first longitudinal
mode of the ribbon. The ribbon transfer scenario used in this study has made no
attempt to minimize coupling with the first longitudinal mode, as may be
possible with proper climber design. Unlike transverse string modes,
longitudinal modes inherently induce strain rate, thus might be effectively
controlled with internal material damping.
Case 4: Nominal Transit
to Ballast at 200 km/hr.
Figure
15 below presents an overview of a 200 km/hr transit of essentially the entire
length of the elevator from 2,000 to 98,000 km.
Transit Arrest
Figure 15. 2,000 to 98,000 km Climber Transit @
200 km/hr vs Time
The
characteristic oscillations in the altitude rate starting at about 450 hours
represents the longitudinal response of the climber-ribbon combination to a
linear ramp-down transit arrest. This is an artifact of the arbitrary arrest
scenario used in this study, and likely would be suppressed with a well
designed climber-arrest control strategy. The elevatorÕs 5 day in-plane (east-west) libration period
manifests itself throughout transit indicating a progressively westward
retrograde bias under the influence of Coriolis effects on the climber until
climber transit arrest; then, starting from the arrest state-conditions, a new
phase of libration develops, symmetrical about the vertical. So the net overall
effect on the elevator as a whole, due to this particular full length climber
transit and arrest, is to have induced a net libration of about +/- 0.55 degrees.
The
resulting libration appears to be dependent upon when the arrest actually
occurs. From examination of figure 15, up until arrest occurs at 450 hours, the
developing libration oscillation has points of zero libration rate; if arrest
completes at a zero libration-rate-point, the resulting libration of the
elevator simply acquires the corresponding amplitude as its initial condition
(and maximum amplitude). Thus based on time-of-arrest for this transit, it
appears that a net libration between +/- 0.3 and +/- 0.65 degrees could have resulted.
Figure
16 below shows climber altitude rate oscillations at arrest, using a magnified
time scale to expose an approximately 6 hour period of oscillation. The
climber, since it arrests very near the ballast mass, excites the ballastÕs longitudinal
bobbing mode; due to the
dominating ballast mass, this mode can be visualized as a one degree-of-freedom
spring-mass system comprising the ballast mass and an effective end-to-end
ribbon spring rate of 0.04 N/m. The smaller periodic irregularities are likely
related to longitudinal-mode interaction with the
Figure 16. Climber Altitude
Rate after Arrest vs Time
climber
mass now parked very near the ballast.
Compared
to gross elevator libration, the climber is producing insignificant transverse
string-mode displacements as seen in figure 17 below depicting snapshots taken
throughout the 28 days of simulated time. Here, each snapshot simply appears as
a straight line.
Figure 17. Ribbon Snapshot
Envelope vs Distance along Ribbon
Figure 18 below depicts a
shape snapshot with greatly magnified horizontal scale
Indicates gross Libration Climber location
Figure 18. Ribbon
Displacement Snapshot vs Distance along Ribbon
showing transverse displacement at 2 days into the transit (10,000 km).
Though appearing as a solid line, this snapshot consists of dots, each
being a nodal point. At the time of
this snapshot, nodal density is far greater between the climber and ground than between climber
and ballast; by the end of the transit, just the opposite will be true. Figure
19 below shows that normal climbing produces virtually no over-stress. At 450
hrs, arrest maneuver stress oscillations are seen.
Figure 19. Ribbon Stress
Profile vs Distance along Ribbon
Case 5: Fast Transit to
Ballast at 400 km/hr.
Figure
20 below presents an overview of a 400 km/hr transit of essentially the entire length of the elevator from
2,000 to 98,000 km.
Dotted
line is Ballast Solid line is Climber
Figure 20. 2,000 to 98,000 km Climber Transit @
400 km/hr vs Time
It
is notable that this particular 400 km/hr transit induced libration angle about
twice that of the 200 km/hr transit. However, unlike the previous 200 km/hr
case, it appears that this arrest occurs at a more advantageous libration
state; had arrest occurred at the
most inopportune time, then up to +/- 1.3 degrees of libration may have resulted. The
general effect of increasing transit rate appears to be an increase in westward
bias, or slope of the average libration angle while under transit rate; peak-to-peak libration excursions during
transit are more or less invariant at
about 0.3 degree. This mean that regardless of optimal arrest timing, faster
transits will always result in more residual elevator libration.
An
interesting effect is seen in the transit-arrest disturbance induced at 98,000
km as it propagates to lower altitudes on the ribbon. Figure 21 is a stress
time history at 5,000 km and 50,000 km altitude, indicating a disturbance
magnification factor of three. Under close examination, this disturbance is
found to consist of; stress waves being propagated up and down the ribbon,
bobbing mode response of the ballast, and excited longitudinal modes. This
magnification might be attributable to two effects, (a) an increase in tension
level as the strain
Climber passes 50,000 km
Stress at 50,000 km
Climber passes 5,000 km
Figure 21. Ribbon Stress at 2 Altitudes vs Time
energy
encounters ribbon of smaller elastic cross section (similar to the increase in
ocean wave height as wave energy encounters narrowing land-constraints), (b)
for a given disturbance tension
level, an increase in stress due to smaller elastic cross section of the
ribbon.
Case 6: Nominal Transit
to GEO at 200 km/hr.
Figure
22 below presents libration response of a 200 km/hr transit from 2,000 km to
GEO altitude (35,400 km); note, general response was typical of the
Region of interest Dotted
line is Ballast Solid line is Climber Arrest
Figure 22. Elevator
Libration vs Time
other
transits. It may be possible to modulate transit rate to arrive at GEO arrest
with near zero libration (note ÒRegion of interestÓ above), a fact that could
have benefit for elevator operations. Note that the act of climbing induces
transverse vibrations of the ribbon as shown in figure 23 below that depicts a
snapshot of
Ribbon transverse
string mode displacement at Climber near GEO is less than 10
km Ballast Libration displacement
Figure 23. Ribbon snapshot:
Total Displacement from Vertical vs Altitude
ribbon
displacement just before GEO arrest. From figure 23, it can be concluded that
by the time the climber nears arrival at GEO, about 10 km of transverse string
displacement has been induced; this corresponds to about 0.002 deg deflection
(as viewed from ground). Given that gross elevator libration at this point is on
the order of 0.1 degrees, this transverse deflection does not appear to be
operationally significant at this transit speed. Transverse string response can
translate into equivalent librations since, like libration, it is a
manifestation of momentum disturbance in the horizontal direction. Note also
that the restorative effects of ribbon tension toward keeping the climber
aligned between anchor and ballast are operationally insignificant due to the
very small angles that the ribbon makes with the line-of-sight between ballast
and anchor (note axis scaling in figure 23).
Case 7: Sudden Arrest at
2 km.
This
case examines response of a climber at full transit speed of 200 km/hr
experiencing a sudden
arrest. Starting at 1 km,
the climber maintains full speed for 18 sec until, at which point, ribbon
transport rate becomes instantly zero
at 2 km altitude. Figure 24 below shows the response.
Sudden arrest altitude Region of rebound
under influence of upper ribbon
Climber slowly rebounds
under upper ribbon tension Climber is catapulted
downward by high impact tension with the
lower ribbon
Figure 24. Climber Altitude and
Altitude Rate vs Time
The
response is characterized by the climber first engaging against the stiff
spring of the lower ribbon and being catapulted back downward in reverse to
immediately put the lower ribbon in a state of slack; this is followed by a
slower rebound upward under the influence of the softer spring of the upper
ribbon; these two processes then cyclically repeat while attenuating due to
damping. The difference in upper and lower spring rates is clearly seen in the
altitude rate graph above. Figure 25 below shows tension history at two ribbon
points.
Tension @ 20,000 km Tension @ 1.5 km Disturbance finally appears @20,000 km arrest
Figure 25. Ribbon Tension
at 1.5 km and 21,000 km vs Time
Tension
at 1.5 km exhibits load spikes characteristic of impact loading. Note that at
20,000 km, the ribbon does not experience this transient until almost 10
minutes after the sudden arrest has occurred below it on the ribbon.
Figure 26 shows a magnified display of details of the
initial impact.
Initial impact peak load 2nd rebound against
lower ribbon Lower Ribbon is Slack as climber rebounds via upper ribbon tension Sudden arrest Climber passes 1.5 km
Figure 26. Ribbon Tension
at 1.5 km Altitude vs Time
While
this response is certainly not desirable, figure 27 below shows why it could be
disastrous. It is apparent from this stress time history, that the design
strength of the ribbon, including the factor-of-safety of 2, is seriously threatened by the sudden arrest load.
Figure 27. Ribbon Stress at
1.5 km Altitude vs Time
This
is good reason for climber mechanism design and liftoff speed modulation
scenarios to eliminate possibility of such an event.
Case 8: Modulated Arrest
at 2 km.
This
case is identical to Case 7, except, a gentler 60 sec ramp down to zero rate is
employed. Figure 28 shows the response to such an arrest.
Figure 28. Climber Altitude
and Altitude Rate vs Time
The resulting stress time
history at 1.5 km, is shown in figure 29 below; compare this with figure 27
above.
Arrest completes Arrest starts Climber passes 1.5 km
Figure 29. Stress @ 1.5 km
vs Time
There
are many possible arrest scenarios to optimize aspects of climber operations;
the above arrest is arbitrary, primarily illustrating that practical scenarios
can be easily designed to bring about transit arrest in a reasonable time, even
at low altitude, without threatening the ribbon strength.
6. Conclusions
The
major operational effect of a climber transit is seen to be due to the Coriolis
dynamics as the climber ascends. The residual libration resulting from a
transit appears to be a function of both the speed and distance of the transit,
as well as when the transit arrest occurs relative to the libration cycle being
induced throughout the transit. As a rule, faster transit causes greater
libration.
This
study points out the potential effects of simultaneous climber interactions
with both a very low effective end-to-end spring-rate of an elevator ribbon of
full length, and the high spring rates associated with shorter ribbon sections
extant near the ground at liftoff. This manifests itself in a variety of
elevator climbing operations, but most dramatically in the process of both
accelerating and decelerating a climber on the ribbon, specially in near-ground
operations such as liftoff. The longitudinal string modes of vibration were found
to be easily excited under climber acceleration or deceleration. Bobbing mode
frequencies of the ballast mass, as well as climber mass can manifest
themselves in response to climber activity. Stress wave propagation effects are
also seen to manifest themselves.
7. Future Work
Many
areas of new investigation regarding climber transit were identified, but left
unaddressed by this preliminary study. Noting that passive (i.e.,
non-horizontal-thrusting) transits leave a residual libration artifact in the
elevator, there may be significant issues concerning how these artifacts will
accumulate or be controlled over successive transits during long term
operations. Will it be possible to plan transit launch phasing to minimize
residual levels of libration without undesirable impact on transit schedules?
Can transit speed modulation and arrest timing be used effectively to minimize
resulting libration? If elevator traffic models were to include a (roundtrip) shuttling
schedule between earth and LEO, then
how might these trips be phased to take advantage of the reverse-Coriolis
effect on the way down in order to control residual libration of the elevator?
What is an acceptable level of libration? What are optimal lift-off scenarios?
Rigid body response of climbers interacting with: atmospheric loads, ribbon
string modes, and elevator libration may have impact on beamed-power system
design, and should be addressed.
Acknowledgements
Funding
for this work has been provided by the Institute for Scientific Research,
Fairmont, WA.
References
1. Edwards, Bradley C.,
Westling, Eric A. ÒThe Space
ElevatorÓ, published by Spageo Inc, San Francisco, CA, 2002.
2. Edwards, Bradley C.,
unpublished communications with the author.
3. Pearson, Jerome, ÒThe Orbital Tower: a Spacecraft Launcher Using the EarthÕs Rotational EnergyÓ, Acta Astronautica. Vol. 2. pp. 785-799