SpinLaunch Secured NASA Funding For A 200kg / 8,047kmh Launch System And I Reevaluated My Own Foundational Numbers With Mounting WFIO Terror.
Orbital Mechanics, The Inertia Problem & A Founders (No Good, Very Bad) Day In The Life.
Podcast of this article: https://rss.com/podcasts/h-industries/463448/
- A Day In the Life
- Orbital Mechanics Basis
- The Inertia Problem
- Steel Bending Moments
- Closing Remarks On All That
A Day In The Life — One Of My Favourite Absurdist Beatles Tracks
SpinLaunch announced the other day that they have secured NASA funding and assistance to model a wild idea that I hope works gloriously. It’s a centrifuge, winding up a ruggedised satellite package to over 2000 meters per second then launching it skywards into orbit. That is pretty damn cool. It was also perfect timing as I’ve just finished an intense period of work on magnets and have surfaced for air but the news did give me a feeling I should go back over my own numbers around orbit, speeds and travel time.
I have spent most of the last three years trying to generate marketing, crowdfunding interest and resolving numerous problems will protecting my IP. April 3rd I submitted a 7500 word revised academic paper on the superconductive electromagnetics that was possibly the hardest thing I have ever done (and I still want to re-edit it). And more than ever I am utterly convinced the design generates the right kind of force required. But then April 9th I found a problem in the foundational assumptions that threatened to derail everything, is this the real WFIO moment — from an excel error? For a solid day I felt like I had failed. How had I not seen this most simple element of force application and geometry? This is the end, might have to get another job [redacted]. What a day. The founders journey is an emotionally rough ride.
Nah hold on, this is the fifth WFIO moment in as many months and every problem so far has had a solution (…so far) so chin up. The TLDR is: I don’t have a Simulink numerical model ready yet but the satellite design does work and there are options to make the orbital mechanics reasonable, so maybe forge on, find funding so I can get fresh eyes on the many many problems. Then I found out my 11yo dog has a golfball size tumour. What a week, when it rains it pours. It’s time to own my mistake and find funding or a job asap. A little advice came from Pokemon Arceus (which I lost a solid month of downtime to) that I felt arrived at just the right time when writing the revised paper: ‘This will be hard. But the plateaus we must climb are the platforms we will stand on.” #foundersjourney
Orbital Mechanics Basis — Arrive On The Next Orbit
The goal is (was) to launch 2t cargo loads for next orbit (1.88y / 687 day) arrival using a swarm of satellites setup in a pyramid with superconductive magnets. The calculator said: 1s pulse x 50G x layer# — the target acceleration rate applied one for each layer. But that assumption and both those numbers are wrong. So very geometrically wrong. I guess thats the mark of progress when inspecting work you did years ago with a lot less knowledge and spotting a silly mistake. At the moment I felt like that mistake had cost me the world.
If you are applying force as an impulse for time, say a second, its easy to multiply in an excel sheet but it doesn’t really match the reality. Electromagnetic fields lose effect according to the inverse square law but the magnetic force dissipates to the inverse cube so the force application drops rapidly as distance grows. We theoretically have to put a cubed increase of impulse to accelerate evenly against an inverse distance cube function for active pulse time as the object moves away. Even if we have the stored energy available to do so, the physical geometry just doesn’t line up at 50G or whatever the actual result may be (tldr: 97.8G)
One whole second is a really long time. When considering SpinLaunch’s proposal, they are doing a centrifugal wind-up for a while before reaching the exit velocity of 8047km/h, or ~2235m/s. At 36 km/h, an object travels 10 meters in one second, probably putting it well beyond the effective range of the electromagnetic field while still being far too slow. The goal is to reach Mars within one orbital cycle — preferably a low multiple of it so we are aiming at the right spot to arrive in orbit on time as Mars travels past again.
Let’s start with the constants.
What is Mars’ average orbital period and distance from Earth?
687 days (59.36*10⁶ seconds) and 225m km (225*10⁶*10³ meters)
What is G in m/s & km/h (1:3.6) and how much is needed to get up to speed in a second?
The standard G = 9.81 m/s² is a measure of acceleration so the G-force measurement will give a quick eyeball verdict on whether the launch plate and cargo can structurally survive the forces involved by considering it as a beam problem.
G = 9.81m/s = 35.3km/h
From the average Martian distance and orbit, if we shot for arrival on the next orbital pass, how fast does that cargo need to move for a 1s pulse? We will return to the cubed expansion in a second because we have already established most of the thrust has to happen in that first fraction of a second.
Next Orbit Arrival in G Speed = (Orb Distance/Period) / (G * Pulse Time)
= (229*10³/59.36) /(9.81*1) = 3,857 m/s = 393.25 G
Assuming it is a 1 second pulse is now looking utterly impossible when you consider that the 393 G goal speed means the cargo would be 3,841m from the launch point when the pulse finishes. Assuming instant acceleration is again unrealistic but illustrates just how far that cargo travels at the target speed in one second. Between travel distance and inverse cube drop off, it really is looking like the shortest feasible pulse is the only answer. Lucky Nb3Sn/Cu magnets can achieve 100T field pulses in fractions of a second sustainable for 10k-200k pulse lifetimes. For a trusted source on the veracity of all this solenoid magnetic propulsion force stuff, check the last paragraph on page 7 — NASA found solenoid magnetic fields can apply as much lift as a helicopter rotor in 1960. A list of our other references can be found here. The first superconductive cable had barely been made in 1960 let alone optimised to todays standards and the braided Rutherford cable design still remain the prefered accelerator cable du jour (with modern materials).
For the structural analysis, a 393 G (13,888 km/h) acceleration of a 2t payload for 1 second creates a point load and bending moment larger than most lovecraftian eldritch horrors (or so I thought). Whether it is a central single application point and the edges bend; or ends are supported and centre bends, neither look good. The question becomes are we breaking the steel beam, or if not, would the acceleration compression force on the cargo crush the contents?Are we able to change the base equation so the forces look a bit less terrifying? Double checking my foundational math was supposed to be quick and reassuring, not an exercise in unveiling ever more cosmic horror.
Doubling the orbital pass time is the key to reducing the acceleration required and considering the force application time point made above we must halve the applied pulse time (or less) though this does give the same force result as above. The cargo could takes 2 orbits, 1374 days, or 3.76 years with the applied acceleration time at amore realistic half a second. Performing the same process over again gives a quarter second pulse time and 4 orbit / 7.5y arrival time but still doesn’t alter that bending moment of mass destruction. Doubling the orbital period again to 4O arrival (7.5y) is getting less pitchable but maybe physically easier since the force application is divided between the 2x2 square of satelittes at the top of the swarm. The only way this sort of delivery timeframe works is if we transport big cargo.
~ (229*10³/(2*59.36)) /(9.81*0.5) = 393 G
~ (229*10³/(4*59.36)) /(9.81*0.25) = 393 G
With bigger satellites that lift more, the containers can be heavier steel and ruggedised against the forces. This includes scaling the plate thickness up to increase the stiffness & moment of inertia to resist bending under the load. 20 ft full scale shipping containers are quite wide could fit a 20t load with plenty of padding. To be lifted into orbit and able to survive the proposed force application, it wont be a standard shipping however the generalised terms and mass values are helpful for first pass metrics. All loads and forces are calculated using the 20t freight goal though actual capacity may be less as this figure will encompass the delivery mechanism mass. The cost efficiency of a full container load will take a few rocket trips back and forth to balance so the lengthy arrival time may just be feasible.
The acceleration of 391G is split between four satellites, each contributing:
F = ma = 20t * (393/4)G = 20.96x10⁶ N per satellite.
The nine satellites below then provide a correspondingly lower offset per satellite to this momentum.
The H.Mk0 design might be functional for pushing 2t but 20t is a very different story. Nb3Sn magnets are absurdly powerful but I need to scale up with a thick yoke rod, few more coils and a couple extra capacitors could do it for a H.Mk1 design. Each Mk0 satellite is approx ~1200 kg (not that bad, about 1/6th of the heaviest ones in orbit) which gives the 2 square layer almost 5t inertial balance - fine for Mk0 2t loads but the goal looks like it should be 20t containers @ 7.5Y not 2t @ 1.88Y so a Mk1 design at 5t each would be ideal for adding more power.
The Inertia Problem —Throwing Tennis Balls At A Bowling Ball?
It is the square pyramid shape that allows the swarm to harness the pulses of the satellites in each layer below; every higher layer object has four satellites underneath it. Each layer of tethered satellites acts as a singular inertial mass, with failure of the electromagnetic tethering determined by the same beam bending mechanics: is the strength of the neighbouring satellites magnetic attraction able to overcome the tensile load acting on the satellite under the higher layer object? Square division of the tethering force from each satellite being linked to four neighbours in a mesh distributes the impact quite reasonably. The force model of the pyramid being launched outwards from its fixed orbital base location (using a thruster offset) is partly reliant on the base inertia equalising that of the smaller structure above. The physical reality of this function may differ somewhat due to spring motion and reaction mechanics in orbit but for now it is an adequate representation until a Simulink model can be built.
Consider the number of satellites in each successive layer of a square grid. Where does the base layer squared size eclipsing the sum of all previous squares stop?
(2*2) > (1*1) ;
9 > (4+1) ;
16 > (9+4+1) ;
25 < (16+9+4+1)
Somewhere between layer 4 and 5 a thruster assembly offset is needed to balance that inertia — again these are all first pass assumptions. If they don’t make sense, there isn’t much point doing a detailed design.
Returning to our satellites being kinda small and the cargo needing to be bigger to be worth it, run the numbers for Mk0 then make the call on Mk1.
L4: 1200 * 16 = 19.2t vs L3,2,1: 1200*14 = 16.8t
So clearly that 2t container is good provided you can solve the bjillions of other problems and maybe accept a long orbital arrival time.
20t on top stacks it at: L3,2,1,0 = 36.8/19.2 = 192% of L4 mass.
Thus a doubling of the base layer is needed to equalise at first pass. This might not be realistic as the wider base itself doesn’t add more satellites directly under the cargo, thus scaling up those units under there is necessary. Maybe a reduction of the layer count to establish a greater margin % is needed too. Considering the potential for shear failure and the bending moment from eldritch hell, having four satellites support the steel plate is the most sensible decision — not a single sat applying point load and all four corners of the plate bending.
Run the mass numbers on a 5t Mk1 design of 3 layers total:
L4: 5000 * 16 = 80t vs L3,2: 5000 * 13 = 65t
That leaves a 15t freight load to be inertially balanced and with a bit of a thruster offset, we could probably get that freight number higher. The target is now 20t freight on a 7.5y trip time — giving more than enough room to reinforce structures and ruggedise the design against the incredible stresses.
Steel Bending Moments
The ‘Success Criterion’ white paper is focused on the design of the superconductive solenoid to enable the electromagnetic propulsion but it also proposes a sample launch plate design. A 2x2m steel square 5cm thick is 1,570kg and will be the landing pad for cargo containers and acts as both the ferry and shield against the electromagnetic propulsion pulsewave. As the intense electromagnetic field spike is applied in a short burst, the energy does not penetrate beyond a surficial skin depth in the plate. The induced current within the plate skin is what generates the weak magnetic field that opposes the strong pulsewave, effectively bouying the cargo outwards on the electromagnetic wavefront. To reduce the burden of propulsion and inertia on any one satellite, a 2x2 grid of satellites arrayed under the plate is used to apply a propulsion pulsewave to each of the plates four corners. The swarm geometry gives granular control of the propulsion vectors and a degree of orbital targetting from minute variances in the applied power. This division allows the 391G target speed to be achieved by four satellites acting in concert, however the bending of the steel plate must be addressed.
With the launch of all four pulsewaves, targeting 391G for the 20t cargo, the container is effectively pressed into the steel plate. In reality this is distributed across the footprint of the container area but at first we treat it as a point load on the centre with the edges of the steel beam being held up by the pulsewave. The pulsewaves must equal a point force requirement of F=ma=20,000*(393/4)*9.81 = 20.96 MN per satellite and 83.85 MN total to exceed theoretical point load of the cargo and meet the required exit velocity. That is a pretty big ask.
If the cross section of the plate is 200x12cm with a steel elasticity modulus of E=210 GPa, the cross sectional moment of inertia (w*h³/12)=28,800 cm⁴. There are numerous great online engineering calculators available and for the inputs above, the beam is deflected 23 cm. Which is not exactly flat anymore or viable. Dropping the beam span to 1.5m gives a deflection of 9.7 cm and adjusting the thickness further then lowers the deflection but maybe additional support structures can be incorporated to the design so it becomes a non-issue. The first pass equations always give clues as to where to look for optimisation and find simplification errors, it is impossible to build a great model if the ground level working is wrong. There are still hundreds of problems with this idea to figure out but it does look like this one is fixable and it isn’t a true WFIO despite what I thought.
Not a bad outcome bad despite the earlier concerns that the bending may just be the ideas undoing even if force application got solved.
Closing Remarks On All That — Mistakes Can Always Be Fixed
This is me owning my mistake on at first thinking 15 seconds of force application was possible before realising even 1s is a huge challenge and making improvements to take the next step forwards knowing my calculations are that bit closer to reality. That problem/solution /low/high is where the reward is in the founders journey and what makes it all worthwhile (and I wont lie, it is a journey through eldritch terror beyond comprehension sometimes). Simulink finite element modelling of all parts is about to begin and the applied force / load balancing is the first problem to do. Elements of the material above will feature in an orbital mechanics paper submitted to a suitable journal once the electromagnetic force application concept paper is peer reviewed.
Back to Bear, our head of security goes in for surgery Thursday 21/4/22 so fingers, paws, toes and tails crossed for a good recovery. He has been my constant companion through the past year and The Great Plague. After a year of operation self funded I dont have much left to give but what I’ve got I’ll give to him. Now it’s time to find funding or get a job, there is no silver bullet — it is just going to take a lot of lead.
And this probably wasn’t my last mistake, when I do find the next one I will adjust, improve and continue on.
Thanks for reading,
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