Success Criterion Isolation
The force experienced at the top of the solenoid relative to the plate
distance of ρ is created by the interaction of the pulsed and the
induced electromagnetic fields. This force is acting on both the
solenoid top and the surficial ERC in the under-side of the object
above. The force must exceed the inertia of the object above to create
maglev thrust and initiate propulsion. Any object’s inertia is a product
of the mass and change in acceleration or local gravity. In orbit, a
degree of gravity is present, anchoring satellites and the moon to their
respective orbits. To determine the contextual object above’s inertia,
the standard G = 9.8 m/s2 is used to generate the
upper bound value for force required to move that object from it’s
resting orbit.
With the inertial force for a chosen mass and gravity set as the left
hand side of (17) and all other factors except the currents \(I_{s}\)and \(I_{e/p}\) being determined by coil winding geometry, the equation
can be simplified to optimise the power supply system. The reduction of
all coil design and resulting elliptical factors into a single
multiplier ϖ of the plates induced current \(I_{P}\) allows designs to
be quickly inspected for validity in the same manner as (10) above. Any
inertial force requirement can be set then a required current found for
comparison to the induced current, if the induced current is larger than
that required, maglev propulsion is a success. Alternatively, if the
induced current far exceeds the required current, (18) could be
rearranged to find the largest accelerable mass for any design. The
derivation path is applicable to any two coil context however the
interaction formulae do change based on winding geometry categories.
\(F_{\text{sz}}\left(\rho\right)=\ \varpi\ I_{P}\ \)=\(\text{ϖ\ }\frac{\text{L\ }\frac{dI_{S}(t)}{\text{dt}}\ }{M}=ma=\ F_{\text{ma}}\)
Magnetic force interaction simplification (18)
The satellite propelled cargo containers will be accelerated at tiered
rates according to their contents. Construction materials and
non-sensitive bulk cargo could potentially be launched at up to 50 G
pulsed acceleration however sensitive equipment will be limited to 20 G
acceleration change in line with NASA’s 2018 Mars Rover orbital entry
speed.
- Cargo mass – \(m_{c}\) = 2,000 kg
- Gravity Constant – G = 9.8 m/s2
- 20 G Acceleration – \(a_{20}\) = 196 m/s2
- 50 G Acceleration – \(a_{50}\) = 490 m/s2
- Scenario \(F_{c20}\) Force - 20 G for 1 s = 392 kN
- Scenario \(F_{c50}\) Force - 50 G for 1 s = 980 kN
- Carbon steel density = 7.85 g/cm3
- Cargo plate mass - \(m_{p}\) = 1,570 kg
- Total object mass - \(m_{t}\) = 3,570 kg
- Scenario \(F_{t20}\) Force - 20 G for 1 s = 700 kN
- Scenario \(F_{t50}\) Force - 50 G for 1 s = 1,749 kN
While the scenarios above discuss a steel plate’s decomposition for
electromagnetic analysis, from the presented dimensions the calculated
mass exceeds that of a 20-ft shipping container with a cargo mass
capacity of 25,400 kg. This leads to the conclusion that a variety of
container designs can be substituted within that representative plate
mass \(m_{p}\) and then optimised to achieve success. To assess the
viability of the overall system, both the cargo mass \(m_{c}\) and
container mass \(m_{p}\) must be totalled \(m_{t}\) then later optimised
with respect to the solenoid strength and power storage capacity.
To accelerate the proposed cargo and plate at 50 G for a 1 second pulse
requires a force \(F_{t50}\) of 1749 kN to overcome inertia while\(F_{c20}\) = 392 kN is the local minima at 22% of \(F_{t50}\).
Considered in reverse, \(F_{t50}\) is 446% of \(F_{c20}\) resulting
from the 178.5% increase from \(m_{c}\) to \(m_{t}\) and increase of
acceleration by 250% from 20G to 50G. With the force requirement being
met by a sum of propulsion pulse vectors, the required force output per
individual satellite is lower however the composition of this function
must be investigated specifically due to the multitude of mutual
inductances. Swarm force distribution function aside, achieving the
minima of 392 kN force in \(F_{c20}\) for scenario 1\(a_{e}\) requires a
pulse induced current \(I_{1e}\) of 23.5x106 A and\(I_{1x}\) of 1.2x106 A when considering the minimum
ERC scenario 1\(a_{x}\). While the peak current and minimum ERC size
scenario is optimal, no presented scenario achieved the required current
induction for successful maglev propulsion. Division of the induction
requirement between four tethered satellites under the plate does not
achieve the requisite current with the presented design either. Despite
this, the investigation of the problem context and construction
delivered valuable design conclusions.