# Active Structures

Active structures rely on constant power input, in addition to the material and mechanical properties of their construction materials (active support). This is in contrast to passive structures, which solely rely on the aforementioned properties (passive support). An example of an active structure is the force of a jet of water holding up a tethered lid of a trashcan in the air, versus the passive structure of a concrete pillar.

Nearly everything, from skyscrapers to houses are passive structures. Low-power active structures are in use now, for things like roof support.

The advantage of active structures is that they can be much more massive than passive structures [Footnote 1], enabling structures many kilometers tall without requiring significant tapering. Some proposals for non-rocket launch infrastructure rely on active support, with the advantage of the option for being built by modern, existing materials.

Most known designs of active structures rely on the force of a stream of mass to support them, using an accelerator to drive the mass stream.

1. Passive structures can attain extremely tall heights, however, they require pyramid-like tapering with a significant base area to support the weight.

# Active Support Principles

As gravity[1]is what pulls down objects, active support must counteract gravity. Since it is the acceleration that causes objects to be pulled down, it follows that active support should accelerate in the opposite direction; the acceleration must be equal to gravity to support the structure.

The gravitational acceleration of the planet is given by:

${\displaystyle A_{g}=G(M_{p}/r_{p}^{2})}$

• Where ${\displaystyle A_{g}}$ is the gravitational acceleration.
• Where ${\displaystyle G}$ is the universal gravitational constant, defined to be 6.6743e11 m3/kg/s2.
• Where ${\displaystyle M_{p}}$ is the mass of the planet.
• Where ${\displaystyle r_{p}}$ is the radius of the planet.

On Earth, ${\displaystyle A_{g}}$ equals 9.80665 m/s2, a constant known as ${\displaystyle g}$.

## Mass Streams

Mass streams generally use particle accelerators or similar technology to create the streams. They use a deflector, usually magnetic, to receive the force from the stream and redirect it back towards the ground to create a loop.

Before we can calculate the kinetic energy required for each particle/pellet in the stream, important in determining energy input, the force, or the weight of the active structure must be calculated:

${\displaystyle F=MA_{g}}$

• Where F is the weight of the active structure.
• Where M is the mass of the active structure.

With that information in mind, the kinetic energy of the stream is given by:

Long form

${\displaystyle KE_{s}=(v-{\sqrt {v^{2}-2A_{g}h}})/A_{g}\cdot F/(2{\sqrt {v^{2}-2A_{g}h}})v^{2}}$

Simple form

${\displaystyle KE_{s}=hF(b({\frac {1}{\sqrt {1-1/b}}}-1))}$
${\displaystyle b=v^{2}/2A_{g}h}$

• Where ${\displaystyle KE_{s}}$ is the kinetic energy of the stream.
• Where ${\displaystyle v}$ is the velocity of the stream.
• Where ${\displaystyle h}$ is the altitude of the deflector.

Notes:

1. As a rule of thumb, whenever ${\displaystyle v>{\sqrt {10A_{g}h}}}$ energy costs are almost optimal. Provided that the stream velocity is above that threshold, you can forego the complex equations involved and go with ${\displaystyle KE_{s}\approx 0.6hF}$ instead. The choice of pellet characteristics can then fall under engineering constraints.
2. The functions overestimate the energy content due to the presence of a constant gravitational field - in reality, it would not be constant.

Finally, the kinetic energy of the stream is divided by the number of pellets/particles in the stream to give the kinetic energy of each pellet/particle:

${\displaystyle KE_{p}=KE_{s}/N}$

• Where ${\displaystyle KE_{p}}$ is the kinetic energy of the particle/pellet in the stream.
• Where ${\displaystyle N}$ is the number of pellets/particles in the stream.

Additionally, the mass of the pellet/particle can be calculated with:

${\displaystyle m=2KE_{p}/v^{2}}$

• Where ${\displaystyle m}$ is mass of the particle/pellet.

A calculated example:

Given a 1,000 ton structure on Earth (implying a ${\displaystyle A_{g}}$ of 9.8066 m/s2), a height of 10 km, 1,000 pellets in the stream and pellet velocity of 10 km/s, the force is 9,806,650 newtons and the kinetic energy of the pellet is 49,105.495 megajoules. Secondarily calculating the mass with the KE divided by the number of pellets and given velocity we get 0.982 kg.

# Active Structures

## Existing

• The air-supported fabric roofs of the Tokyo Dome, Japan, and the Silverdome, USA use (and for the latter, used) constant fan pressure to keep the roofs aloft.

## Proposed

• The Lofstrom Launch Loop is a thin 2000+ km long and 80 km tall active structure, and uses its own mass stream to help launch payloads to orbit. It uses attractive magnetic levitation for the mass stream; the mass stream is a solid continuous iron rotor. The loop suffers from some unaddressed instability concerns.
• The Space Cable is a similar concept to the launch loop. It differs from the launch loop in that it uses magnetically interacting bolts instead of a continuous rotor, is smaller in length, and has addressed instability concerns.
• The Orbital Ring, which uses a mass stream travelling faster than the orbital velocity to support a ring above a planet, as the stream keeps it from falling through momentum, and is tethered to the earth for stability.
• The Pneumatic Freestanding Tower, which uses pressurized gas to support large structures such as a space tower. It utilizes compressors to provide pressurized gas and alleviate leaks. The main concerns are buckling due to the height of the tower, though it has mechanisms in place to prevent this.

# Control Systems

Control Systems
Active structures can suffer from stability issues as mentioned before, such as for example in the launch loop unstable attractive magnetic levitation of the mass-stream in the launch-loop requiring active control of the deflector magnet. The unpredictable winds in the atmosphere are also a concern. Control systems are also needed in even just skyscrapers, with devices like tuned mass dampers to deal with vibration[2].

# Safety Engineering

Active structures are subject to the problem of how to ensure that they don't fail, or a bit worse, only fail gracefully, when something in their active systems breaks down. This is not a question of if; entropy breaks everything. All electrical and mechanical systems have a mean time between failure. If an active structure is only supported by a single active support "string", the failure of that "string" will cause a catastrophic failure.

By adding redundancy to our structure, we can ensure it can tolerate the failure of some of its components, and possible "fail gracefully" with a time period allowing for evacuation and response measures to be taken, and/or a "controlled failure" of the structure in which terminal velocity of the falling structure and the production of energetic debris is reduced.

Accepting an increase in mass, we split the support power required between some ${\displaystyle N}$ "strings" operating in parallel. Each string only operates at a partial power, with an oversize factor of ${\displaystyle 1/N}$ added on top. If one or a few of the strings in the parallel system fail, the other strings are ramped to full power, generating sufficient support power to ensure the active structure remains standing despite the failure of some of its "strings". We can also use this to shut down strings intentionally for inspections, maintenance, overhauls, or other work, overall allowing us also to keep the active structure alive over time by incrementally replacing and upgrading its parts.

Usually safety redundancies have at least three systems operating in parallel. You may consider having a larger number of systems.

Note: in terms of safety engineering, no degree of redundancy reduces the chance of a total, catastrophic failure to zero. There is some chance that even a very redundant system may experience the failure of all its critical components at once. But this is given for any system, and you could consider pushing the safety factor of an active structure to the same point (or beyond) any passive structure.

1. https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
Wikipedia article about gravity in classical mechanics.
2. https://en.wikipedia.org/wiki/Tuned_mass_damper
https://en.wikipedia.org/wiki/Active_structure
Wikipedia articles about control systems and the aforementioned tuned mass damper, as well a Practical Engineering video on it.

## Derivation of the Gravitational Acceleration Equation

1. The gravitational force equation[GAE 1] is ${\displaystyle F=G(m_{1}m_{1}/r^{2})}$, where ${\displaystyle F}$ is the force, ${\displaystyle G}$ is the universal gravitational constant, ${\displaystyle m_{1}}$ is the mass of the first object, ${\displaystyle m_{2}}$ is the mass of the second object and ${\displaystyle r}$ is the distance between their centers of mass.
2. ${\displaystyle r}$ becomes the radius of the planet from the frame of reference of a planet.
3. The equation ${\displaystyle F=MA}$, which gives the force needed to accelerate an object is rearranged to give acceleration, thus ${\displaystyle A=F/M}$
4. Since ${\displaystyle A=F/M}$, where ${\displaystyle A}$ is the acceleration, divide by the mass of the second object and therefore cancel out its inclusion in the expression, giving you the gravitational acceleration equation.

Reference for the Derivation of the Gravitational Acceleration Equation

1. https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
Reference for the gravitational force equation.

## Derivation of the Stream Kinetic Energy Equations

### Long Form

1. Given a classical particle accelerating upwards, it will achieve a final velocity of ${\displaystyle v_{f}={\sqrt {v^{2}-2A_{g}h}}}$ where v is the initial speed.[SKE 1].
2. The necessary force for deflecting the particle down is given by ${\displaystyle F=2{\dot {m}}v_{f}}$[SKE 2]. The total energy content is then ${\displaystyle mv^{2}/2}$.
3. The stream must be long enough for one full circulation up and down, so any particle going up the height of the stream to the deflector takes a time ${\displaystyle T}$ where ${\displaystyle v_{f}=v-A_{g}T\Rightarrow \color {green}{T=(v-v_{f})/A_{g}}}$, for a full circulation it takes ${\displaystyle 2T}$.
4. Therefore ${\displaystyle m=2T{\dot {m}}}$, so ${\displaystyle KE_{s}=T{\dot {m}}v^{2}}$. To get rid of ${\displaystyle {\dot {m}}}$ we divide ${\displaystyle F}$ by ${\displaystyle 2v_{f}}$.
5. This gives our stream kinetic energy equation, taking the limit as ${\displaystyle v}$ monotonically approaches infinity ${\displaystyle hF/2}$ from above.

### Simple Form

1. The simple form is just a factorized version of the long form, where ${\displaystyle k}$ is defined as ${\displaystyle 2A_{g}h/v^{2}}$ or ${\displaystyle 1/b}$.

### Regarding the notes

1. The simple form shows that whenever ${\displaystyle v^{2}/2A_{g}h>5}$, the energy content in the stream is less than ${\displaystyle 0.6hF}$. Rearranging the equation for ${\displaystyle v}$ and accounting for 5 gives the rule of thumb.
2. The upper bound nature of the equations means that:
For ${\displaystyle v>{\sqrt {10A_{g}h}}}$, ${\displaystyle 0.5hF for any ${\displaystyle h}$ and any gravitational field that is decreasing.

Reference for the Derivation of the Stream Kinetic Energy Equations

1. https://www.school-for-champions.com/science/energy_potential.htm
The expression is derived from the expression for the velocity of falling objects: ${\displaystyle V={\sqrt {2A_{g}h}}}$. This can be thought as of the amount of velocity that gravity subtracts from the initial velocity.
2. ${\displaystyle {\dot {m}}}$ is the mass of the particle ${\displaystyle \bullet }$ the repetition rate, the repetition rate being analogous to ${\displaystyle N}$, and ${\displaystyle {\dot {m}}}$ analogous to ${\displaystyle KE_{s}/N}$.

# Credit

To Tshhmon for writing the article

• To SOPHONT SIMP and pMXoTJFu for sweeping the article.
• To AdAstraGames for contributing useful information and sweeping the article.
• To Sevoris for writing the safety engineering section.
• To MatterbeamToughSF, Kerr in particular and Favalli for help with the math.