Earth-orbit satellites are vital to weather prediction, data/phone/radio/TV links, aerial reconnaissance, mapping, GPS, etc.
Benefits of space exploration, and possible future outposts on the Moon or Mars, can be profound.  Several websites present general news, of past, present, and proposed space programs.  NASA has compiled data on all failure modes of spacecraft equipment: Chemical battery failures, depleted jet thruster fuels, sticking jet thruster valves, and mechanical bearing wearout of momentum wheels, cause $500M satellites to become space junk.

Space versions of RPM's UPS (Uninterruptible Power Supply) can provide a far superior alternative to chemical batteries; by affording ultra-high reliability, no degradation over time, and virtually unlimited service life, with no need for maintenance.

They can efficiently store and regenerate power -- while 2 or more RPM flywheels affixed to a spacecraft, in addition to providing spacecraft UPS, can also provide CMG (Control Moment Gyro) inertial attitude control.

RPM flywheel UPS + CMG can provide an attractive alternative to electrochemical batteries (which supply stored power from photovoltaic panels during times in orbit when Earth shades them from sunlight) and momentum wheels (which provide inertial attitude control), in applications like the Hubble Space Telescope (shown in Earth orbit in photo at right).

Hubble Space Telescope has provided extraordinary images and insights to planetary science.  Its images are received by laboratories via radio links.

The photovoltaic panels can be seen attached at each side, of this nearly cylindrical space-craft.  A partial view of Earth is seen in the upper background.  The darkness of space is seen in the lower background.

This image is courtesy of the NASA website spacecraft photo gallery, which displays a variety of spacecraft images at various stages (from preliminary concept to orbit), and descriptive narratives.
 

From Ancient Science to Modern Technology

By the early 1600s, celestial orbits were observed and described by Galileo, Kepler, and others.   Newton developed a unified theory, which could predict celestial orbit dynamics.  Of course, it still applies, even to launching and maintaining a space-craft in orbit.

This webpage focuses on quantitative considerations for interested engineers, of spacecraft launch and orbit dynamics, and how RPM flywheel UPS + CMG that would include 2 or more RPM flywheel assemblies like the image seen at left, affixed to a spacecraft as described below, can provide far more reliable, far longer service life, power storage and regeneration, plus inertial attitude control for satellites.

This image shows 4 of the 8 radial electromagnets, 2 seen above the flywheel rotor rim and 2 seen below it. They center the rotor (working with passive centering from axial ring magnets that are obscured by surrounding parts in this image) and can provide precision precession torque to effect spacecraft attitude control.
 

 Launch Velocity to Escape Earth Gravity

Practical approximations, for a spacecraft and Earth model, greatly simplifies orbital dynamics equations:  A circular orbit around a perfectly spherical Earth, negligible air drag, negligible forces from magnetic fields in space, and neglecting other gravity forces (from the Sun and other planets) on the spacecraft, permits reasonably accurate predictions, of spacecraft escape velocity, orbit velocity, orbit stability, etc.

To launch a spacecraft of mass (m), to a great distance (r) in space, where it will be essentially free from Earth gravity (g), it is accelerated by rocket thrust to a very high speed (v_o), called the escape velocity, where its kinetic energy is mv_o²/2.

Earth’s atmosphere has a negligible influence over most of the distance traveled. So v_o can be estimated, by equating mv_o²/2 to the work done against Earth’s gravitational force (mg) which decreases as the spacecraft travels through space at increasing distance (r), where

g  =  g_o ( r_o² / r ² )            r_o  =  Earth’s radius ~ 4000 miles                 [1]

g_o  =  9.8 meters/sec² at distance r_o  from Earth’s center of mass.

These relationships can be expressed by equating the kinetic energy at launch to the 3 integrals in the next line:

Integrating and simplifying to solve for (v_o) yields the result:

v_o  = ( 2 g_o r_o )^1/2  = 11.2 km/sec  =  25,000 miles/hour escape velocity.
 

Velocities and Period of Orbit vs. Height of Earth Satellites

Velocity (v) of a satellite, to travel in a circular orbit, radial distance (r) from Earth’s center, can be computed by equating the centrifugal force (f_c) acting on the satellite in this orbit, to the gravitational force (f_g) which opposes it:

f_c  =  mv²/r  =  f_g  =  mg_or_o²/r²                                                                    [2]

Simplifying and solving for (v) yields:

v = r_o (g_o/r)^1/2                                                                                              [3]
 

Low-Earth-Orbit Satellites

At relatively low orbits (e.g., for r = r_o + 300 km  = r_o + 190 mi = r_o + 990,000 feet)
v ~ 7.8 km/sec ~ 17,000 miles/hour
and distance traveled per revolution in this orbit = 2 (pi) r.

So the period of this orbit ~ 1.5 hour.  In this low orbit, the satellite is in sunshine for nominally 0.8 hours and in darkness for 0.7 hours, each revolution. Photovoltaic (PV) solar panels are normally used to power payloads of such satellites, while in sunshine. Chemical batteries normally store excess power from the PV during sunlight, and supply power to the satellite’s payload, during darkness.  Otherwise, the payload is not operational in darkness.

If only a minimum battery bank, to supply the payload, were provided, the batteries would be deeply discharged each 1.5 hour cycle.  Experience has shown that battery lifetimes would be nominally only 1000 cycles, when deeply discharged. Such a battery lifetime might be only 1500 hours (about 2 months).  After that, the satellite payload is not operational, on the dark side of Earth. That’s unacceptable here.  A substantially larger battery bank, that can supply 10x or so, the payload, is now commonly used.  It significantly increase battery lifetime, because the batteries are not deeply discharged.  It can also provide higher reliability, by redundant parallel connections.  But short battery life is still a factor limiting the useful life of satellites costing hundreds of $millions to construct and launch.

Finite onboard fuel for attitude control jets is another factor limiting useful satellite life.

RPM's flywheel batteries can afford solutions to both limitations, by providing virtually unlimited UPS and attitude control system service life.  In addition to significantly lower cost per year of service, it can reduce accumulating space debris.
 

Geo-synchronous and higher orbit satellites

At a higher orbit, like r = (6.5)(r_o), Eq. [3] yields:  v = 3.1 km/sec.  This nearly geosynchronous orbit would have a period of 24 hours.  And the satellite would be in sunshine almost all the time, where its power can be supplied by photovoltaic panels.  So its power storage capacity can be relatively small.
 

Stability of Orbit and Actions that can Change Orbits

We have thus far neglected perturbations that could change satellite orbits (e.g., atmosphere that would cause drag and resulting kinetic energy loss, micro-meteorite collisions, and radial boost or retro rockets).  So it would be useful to determine trajectory stability (i.e., radial restorative forces that would oppose possible perturbations that would alter the orbit).

An infinitesimal radial movement (dr) would affect the satellite’s kinetic energy (mv²/2) and its centrifugal force (mv²/r) and opposing gravitational force (mg_or_o²/r²).  Note that for positive dr, kinetic energy, gravitational force, and centrifugal force, all decrease.

We will first differentiate kinetic energy with respect to r, which equals mv(dv/dr), and from the above 3 integrals, is seen to be simply -f_g.

This yields:  dv/dr = -f_g/mv.

So:  dv = -(f_g/mv)dr                                                                                [4]

Then differentiate centrifugal force, with respect to both (v) and (r):

df_c  = (2mv/r)(dv) – (mv²/r²)(dr)                                                             [5]

Combining Eq. [4] and [5] yields:   df_c/dr  =  -2f_g/r – mv²/r²                   [6]

Combining Eq. [2] and [6] yields:  df_c/dr  =  -2mg/r – mg/r  =  -3mg/r    [7]

Then differentiate gravitational force. This yields:  df_g/dr = -2mg/r          [8]

Comparing df_c/dr and df_g/dr in Eq. [7] and [8] shows that, for positive (dr), df_c/dr decreases significantly more than df_g/dr. That means a satellite’s orbital trajectory is unconditionally stable, with small or no perturbations.

But if an orbit is too low,  energy loss from atmospheric air drag,  and resultant speed decrease,  would cause f_g to increasingly exceed f_c in Equat. [2],  eventually causing a crash to Earth.

A radially outward rocket thrust on an orbiting satellite (below that needed to achieve 25,000 miles/hour escape velocity) would boost a spacecraft to a higher orbit.  In accordance with Eq. [3], orbital speed (v) would decrease.  Conversely, a radially downward rocket thrust would move the spacecraft to a lower orbit, with correspondingly higher orbital speed.  To the extent that energy expended to alter orbits does not result in large angular momentum change, the inverse relationship of speed and orbit height noted here, is also seen to be consistent with conservation of angular momentum.
 

RPM's UPS + CMG for satellite power + orientation

Power storage for existing UPS (Uninterruptible Power Supply) which supplies spacecraft power during its orbit when Earth shades it from the Sun, is provided by the best chemical batteries available.  But their failure rates cause serious concerns.

Spacecraft attitude control is currently provided by jet thrustors (where, for each axis, a pair at opposite sides of the spacecraft exert torque to accelerate and decelerate its angular speed), momentum wheels (which are bi-directionally driven to exchange angular momentum with the spacecraft, about axes parallel to the wheel's spin axes; and are sometimes called reaction torque wheels), and CMGs (Control Moment Gyros, which may be torqued to cause spacecraft precession, or may be precessed to cause precession torque).

RPM's UPS can do double duty as CMG.  Since UPS service must not conflict with CMG function, and advantages of CMG without mechanical bearings and gimbals would be profound,  RPM's combined UPS + CMG clearly do not function as momentum wheels.  RPM's UPS + CMG can be configured, so that torquers are not needed.  RPM's radial servos, which normally maintain rotor-to-stator centering and alignment, can provide torquing, to control satellite precession angles.

Minimal idling loss, zero maintenance, virtually unlimited life, lower weight, flywheel battery system, achieved by RPM's proprietary regenerative motor and magnetic bearings (see US Patents 6566775 and 6794777), can provide superior power storage and regeneration having virtually unlimited lifetime, with far greater reliability, over existing chemical batteries.  It also can provide inertial attitude (angular orientation) control for the satellite, without need for customary gyros, torquers, gimbals (most with mechanical bearings), and especially the jet thrusters commonly used for attitude control.  RPM's UPS + CMG would be far more reliable than all these conventional parts.  Moreover, the spacecraft version would not need a vacuum enclosure, since the vacuum of space suits it ideally, and is consistent with its virtually zero idling losses.

A 2-flywheel system, exerting matched motor/generator torques (with zero net reaction torque along their spin axes), at matched spin rates, is shown below.  It can provide 2-axis inertial attitude control.
 

Equal and opposite torquing, that tends to tilt the 2 spin axes, with no net reaction forces or torques on the Flywheel Assembly platform, can be provided by RPM's radial servos, exerting radial forces at each end of the flywheel assembly rotors.  For example:

Forces represented by the 4 blue arrows, which tend to tip Flywheel Assembly #1 into the page at the top and out of the page at the bottom, while tipping #2 out of the page at the top and into the page at the bottom, will cause CW precession in the plane of this page, represented by the dashed blue semicircle.

Opposite forces represented by the 4 green arrows will cause CCW precession, represented by the dashed green semicircle.

Forces represented by the 4 red arrows, which tend to pull the 2 rotors together at the top, and push them apart at the bottom, will cause precession around a horizontal axis in the plane of this page, as represented by the dashed red arc.

Forces represented by the 4 purple arrows,  which tend to pull the 2 rotors together at the bottom, and push them apart at the top, cause opposite precession around the same horizontal axis, as represented by the dashed purple arc.

This 2-flywheel UPS + CMG can provide inertial attitude control in 2 orthogonal planes.  Ideally, any constant rotation should be parallel to their spin axes.

Satellites with affixed UPS + CMG platforms, that must constantly rotate about an axis that is not parallel to the 2 spin axes, would require constant rotor torquing.  That would consume power, far in excess of the RPM UPS virtually zero idling loss plus the very low housekeeping power of its power electronics interface.

With the 2 spin axes parallel to the roll axis, the precession torques are related to pitch and yaw rates by:

Yaw Precession Torque = [Flywheel Inertia] x [Flywheel Spin Speed] x [Pitch Rate]          [9]

Pitch Precession Torque = [Flywheel Inertia] x [Flywheel Spin Speed] x [Yaw Rate]          [10]

Clearly, RPM's flywheels, besides providing ultra-efficient power storage and regeneration, with far higher reliability and far longer life than chemical batteries; can additionally provide inertial attitude control, that can be a very attractive, far more reliable, longer life alternative to gyros and attitude control jets, reaction wheels, and existing CMG:

To effect a pitch maneuver, the radial servos can exert variable torque in the yaw plane, equal and opposite, on the two rotors. This will cause a pitch precession rate, related to the yaw torque, flywheel inertia, and flywheel spin speed, as expressed by Eq. [9] above.  It should be noted that angular acceleration, to achieve a given pitch precession rate, requires torque in the pitch plane, which acts as a negative pitch precession torque, tending to cause rotor axis compliance to the applied yaw precession torque.  The total yaw compliance, to achieve a desired pitch rate, is proportional to the maximum pitch rate needed to accomplish a desired maneuver.  If RPM's radial servos apply the yaw precession torques, (i.e., act as torquers) the spin axes will tilt in compliance to it, relative to their stators, during acceleration.  At constant spin speed, at the ends of maneuvers, this tilt is canceled during pitch deceleration, as the yaw precession torque is reduced to zero.

Likewise, to effect a yaw maneuver, the radial servos can exert pitch torque, equal and opposite, on the two rotors, as expressed by Eq. [10].  Spin-axis tilt (i.e., compliance to precession torque along axis of torque), of each rotor relative to its stator’s adjacent components, should not exceed a degree or so, to maintain rotor-to-stator alignment required for normal flywheel battery performance.  This tilt, from alignment, accumulates during the spacecraft’s angular acceleration.  It is proportional to maximum angular speed needed for a pitch or yaw maneuver and the spacecraft’s total angular inertia. It can be conveniently  reset to zero by opposite precession, during end-of-maneuver deceleration.

This compliance tilt is equal to
[(max angular speed)(total angular inertia)] / [(flywheel spin speed)(flywheel inertia)] radians     [11]

From Eq. [11], suppose a yaw maneuver at a maximum angular speed of 1 rpm is required, and spacecraft angular inertia about the yaw axis is 400x that of the flywheel rotor assembly, spinning at 40,000 rpm.  Then the flywheel spin axis compliance tilt would amount to 0.01 radian = 0.6 degree, during the yaw maneuver, and would be inherently reset to zero, during the end of maneuver deceleration. That can be accommodated.

A 3-flywheel system, also with matched torques and spin rates, is shown below.  It can provide 3-axis inertial attitude control.  But precession is not conventional pitch, yaw, and roll.  Instead, it controls precession around a first axis, and around 3 axes each 120^o apart, orthogonal to the first axis.  So control algorithms transform maneuver commands to 4 combinations of applied torques, to achieve desired pitch, yaw, and roll attitude.

Flywheel Assembly #1, #2, and #3 each have a Rotor Spin Axis in a plane parallel to their Mounting Platform.

These spin axes are 120^o apart.  Each has carefully matched angular inertia and spin torques (having spin directions shown in the illustration at left, which increase speed to store solar power and decrease speed when their stored power is regenerated).

Note that net reaction torque, from accelerating and decelerating the spin speed of the 3 flywheels, on the Mounting Platform, is essentially zero.

When equal torques are applied, tending to tilt the spin axes of #1 and #2, by forces represented by the red arrows, the entire assembly will precess, as represented by the dashed red semi-circle, to reach a precession speed proportional to these torques.

When equal torques are applied, tending to tilt the spin axes of #1 and #2, by opposite forces represented by the blue arrows, the assembly will precess in the opposite direction, as represented by the dashed blue semi-circle.

The red and blue maneuver results in additive precession torques from or to each of the 3 flywheel rotors.

When equal torques are applied, tending to tilt the spin axes of #1 and #2, by forces represented by the green arrows, the assembly will precess around the spin axis of #3, as represented by the dashed green circle.

Clearly, 3 combinations of torquing, like that shown by the green arrows representing torquing of #1 and #2, and including (not shown) like torquing of #3, will result in precession around the spin axes of #1, #2, or #3, in responce to maneuver commands.

All rotor torquing, by the green maneuver, results in essentially zero net reaction torque on the platform, whereas torques that accelerate and decelerate the spacecraft rotation, for precession maneuvers, are additive, from or to the 2 flywheel rotors which are torqued.

Although the flywheel assembly platforms shown above will not exert motor reaction torques on an attached spacecraft, nor precession torques due to spacecraft maneuvers by other means (because these torques are self-canceling), considerable precession torques will occur between the flywheel assemblies and their respective platform, if and while its angular orientation is changed.  Precession torques applied by the flywheel rotors' radial servos consume significant power.  At times when solar power is not available, regenerated power would cause spin speeds to decrease.  So platforms that do not require constant turning are recommended.

A 4-flywheel system, also with matched torques and spin speeds, is shown below.  Note that its UPS function can provide power to a dc bus from all 4 systems, connected in parallel.  Load and power sharing is inherent, with their current-controlled power electronics.  A 4-flywheel system might be selected, over a 2- or 3-flywheel system, to use more standardized systems when more power and energy storage is needed.

When all 4 rotors are torqued, tending to tilt the arrow end of their spin axes down (into the Mounting Platform) and the opposite end of each spin axis up (away from the platform), the platform precesses CCW (counter clockwise).  If torqued in the opposite direction, the platform precesses CW (clockwise).  This can be regarded as a roll maneuver.

Torquing forces and resulting precession, to achieve this roll maneuver, are not illustrated here.  This roll maneuver is equivalent to the first maneuver illustrated for the 3-flywheel system described above.  Precession torques, from this maneuver, are additive and equal, from and to each of the 4 flywheel rotors.

When radial forces are applied, to the ends of #2 and #4 rotors, as represented by the red arrows, platform precession around the spin axes of #1 and #3 will result (down at the left and up at the right) as represented by the dashed red arc segment.

When opposite forces are applied, as represented by the blue arrows, reverse precession around the spin axes of #1 and #3 will result, as represented by the dashed blue arc segment. This can be regarded as a pitch maneuver.

Identical torquing of #1 and #3 will result in precession around the #2 and #4 spin axes.  This can be regarded as a yaw maneuver.  Forces and resulting precession, to achieve this yaw maneuver, are not shown here.

All rotor torquing forces produce zero net torque on the 4-flywheel mounting platform.  Precession torques resulting from the pitch and yaw maneuvers, are additive, from and to the selected pair of flywheel rotors which are torqued to effect the maneuver.

Equations [9], [10], and [11] show that, if flywheel rotor spin speed is constant, the tilt compliance causing rotor-to-stator misalignment when the RPM UPS + CMG radial servos provide torquing, will be corrected by end-of-maneuver spacecraft angular deceleration.  Moreover, these equations show that flywheel rotor-to-stator alignment can be accurately adjusted, by judicious spin speed changes, during maneuvers.
 

Miscellaneous UPS + CMG considerations

Another flywheel size consideration, for inertial attitude control, arises from the need for its radial electromagnets to exert adequate pitch and yaw torques, to effect required yaw and pitch angular speed precession rates. Maximum torque from the radial electromagnets is proportional to their area and axial distance apart. This also relates to the flywheel assembly’s volume.

Basic flywheel physics shows that energy storage capacity is proportional to (rim volume) x (rim tensile strength).  Rim volume needed also affects the flywheel assembly’s volume.

For inertial attitude control, maximum (flywheel inertia) x (flywheel spin speed) is desired. Flywheel inertia is proportional to (volume) x (radius squared).

For an excellent comprehensive, illustrated general description, applications, status, background and history of satellites, see: http://www.howstuffworks.com/satellite.htm
 

RPM's UPS for possible future permanent base on Mars

Like Earth, Mars orbits our Sun. A day on Mars lasts roughly as long as an Earth day.  Gravity on Mars' surface is about 1/3 that on Earth.  Temperature on Mars' surface ranges from +15^oC to -140^oC.  Mars' atmosphere is mostly carbon dioxide, at roughly 0.1 psi.

Human, animal, and plant life could be supported on Mars, within a sealed enclosure, that is semi-transparent to sunlight. That enclosure's skin could be covered with semi-transparent photovoltaic panels, and transparent thermal insulating panels that temper the wide extremes between day and night there, hold heat within this habitat so it approximates ideal earth climate, hold oxygen and nytrogen for healthful life-sustainable air at 1 atmosphere,  and provide electric power.  Plants within this habitat could help maintain an ideal oxygen and carbon dioxide balance.

My illustration of such a possible future permanent base on Mars, to provide a permanent life support habitat there, is shown below.

The main enclosure would likely resemble the geodesic dome structures created by Architect Buckminster Fuller.

Construction of such domes involves straightforward repetitive tasks, well suited to a robotic machine that might have many characteristics of the Mars Rovers created at NASA's Jet Propulsion Laboratory.

The dome  would need to be sealed, so that interior air at 14.7 psi would not be lost to the 0.1 psi exterior.

Also, the dome would need to include, preferably interior, and preferably transparent to sunlight during daylight hours, a thermal "blanket" having selectable thermal conductivity.

If provided light, water, and bio-degradables, partly from its human and animal occupants, plants should thrive in Mars soil.  A liner below plant root level could prevent water loss from this closed system habitat, if the deep thermafrost believed to be present does not prevent water seepage.  A cutaway view of the dome shows plants growing in it.

Entrance to the main enclosure could be through a similar but much smaller sealable entry, shown at the main structure's left side.

Although scarcely visible in this illustration, a single cutaway view shows one of several RPM UPS flywheel enclosures, which freely hang from the entry floor.  Hanging by a rigid cylinder, which contains power and signal conductors between the submerged flywheel assembly and a readily accessible power electronics interface in the entry, this power storage and regeneration system would be in a benign environment.

Occupants possibly above it are absolutely safe, in the unlikely event a flywheel explodes in its submerged site.

The floor provides a safety barrier, between the flywheels in their individual submerged sites (which can absorb a maximum adiabatic energy release, without significantly disturbing the other flywheels in their sites), and the enclosed spaces above these sites.

Another benefit from this configuration, is that RPM UPS is not subjected to Mars' extremely cold night temperatures.  With a long and rigid hanging cylinder, that can swing but not rotate,  and a flywheel rotor having a length-to-diameter ratio about 3 times that of terrestrial versions,  Mars' gravity can readily stabilize the flywheel rotor tendency to precess due to Mars rotation.  That is essential to maintaining RPM's proprietary magnetically levitated rotor and stator alignment, without power dissipation, in their sealed vacuum enclosure.
 

A Similar Habitat on our Moon

The moon rotates in its earth orbit once each 656 hours.  Lunar gravity is about 1/6 Earth's.  Thus, maintaining flywheel spin axis verticality on the Moon, by magnetics that don't consume power, would be less challenging than worst-case terrestrial stationary sites near Earth's equator.  And the perfect vacuum surrounding the Moon would eliminate need for housing RPM's flywheel assembly in a vacuum enclosure.

However, the very long lunar days and nights would make temperature control far more challenging than on Mars.  And the Moon's very wide temperature extremes (+127^oC to --173^oC), plus occasional micrometeorite hits, would limit photovoltaic panel life.

Comments regarding this webpage are most welcome.  Please send them by email to Dick Fradella:

You might also like to visit my website index.
It summarizes key features of RPM's UPS, and contains links to my entire website.
 

 
 

Regenerative    Power  and      Motion

 
July 1, 2007