#format jsmath 
= Track Tension =

The 2017 upgraded transformer-track launch loop will distribute the vehicle acceleration force between the rotor and the track, proportionally to the velocity of the vehicle.   The following analysis assumes launch at the maximum cadence, and ignores longitudinal resonances in the track (which may be detrimental or beneficial, hopefully the latter with clever design). 

A transformer track can be very efficient because slip will be a tiny fraction of the huge motor speed, perhaps 99% or better.  For this analysis, we will assume 100% to simplify the math.

==== Parameters used ====

|| 6000 kilograms || Vehicle mass including the launch sled ||
||   45 seconds   || launch cadence                         ||
||   30 m/s²      || launch acceleration                    ||
||   472 m/s      || Earth rotation velocity                ||

=== Likely destination apogees and loop relative launch velocities ===

|| Destination ||   Apogee  || V,,perigee,, ||  V,,launch,, ||
||             ||           ||     m/s  ||   m/s    ||
|| 422 km LEO  ||   6800 km ||  7938.74 ||  7466.36 ||
|| Server Sky  ||  12790 km ||  9038.17 ||  8565.78 ||
|| GEO         ||  42164 km || 10328.29 ||  9855.91 ||
|| Moon        || 384600 km || 11001.06 || 10528.68 ||
|| Escape      ||  Infinity || 11093.37 || 10620.98 ||

=== Track Tension Versus Vehicle Number ===

|| Vehicle || Velocity || Trackforce || downrange || Summed || Tension ||
|| Number || m/s || Newtons || km || Newtons || kg/m ||<-2|2> With strain relief<<BR>>at 200 km ||
|| 0 ||     0 || 180000.00 ||    0.00 || 180000 || 0.26 ||
|| 1 ||  1350 || 162642.86 ||   30.38 || 342643 || 0.50 || Summed || Tension ||
|| 2 ||  2700 || 145285.71 ||  121.50 || 487929 || 0.71 || Newtons || kg/m ||
|| 3 ||  4050 || 127928.57 ||  273.38 || 615857 || 0.90 || 127929 || 0.19 ||
|| 4 ||  5400 || 110571.43 ||  486.00 || 726429 || 1.06 || 238500 || 0.35 ||
|| 5 ||  6750 ||  93214.29 ||  759.38 || 819643 || 1.19 || 331714 || 0.48 ||
|| 6 ||  8100 ||  75857.14 || 1093.50 || 895500 || 1.31 || 407571 || 0.59 ||
|| 7 ||  9450 ||  58500.00 || 1488.38 || 954000 || 1.39 || 466071 || 0.68 ||
|| 8 || 10800 ||  41142.86 || 1944.00 || 995143 || 1.45 || 507214 || 0.74 ||

As vehicles (launched at a 45 second cadence) travel down the track, the tension in the first section will cycle between 490 kN and 310 kN, while the tension in the second section will cycle between 510 kN and 380 kN.  With proper design and careful attention to vehicle spacing, the system can probably be designed to resonate, and "store" some of the tension as inertia rather than longitudinal cable stress.

The track tension and lateral relief cable could be inconveniently large, but the first 200 km is climbing from the station at 50 km to the main launch path at 100 km, and there will be strain relief cables to the surface to relieve incline forces anyway; they can relieve the track launch tension of the first three payloads as well.

This page describes aspects of an advanced 400 tonne-to-escape-per-hour launch loop using perhaps 7 GW of powerplant.  

Earlier, this system will have a smaller power plant and lower maximum launch cadence.  Before that, a smaller, higher acceleration launch loop will send small low cadence payloads to server sky and GEO.  

In the longer term, a much larger launch loop will launch 40 tonne payloads (fully loaded 40 foot intermodal shipping containers) at lower acceleration to highly inclined orbits for the assembly of enormous interplanetary spacecraft (such as Aldrin cyclers), while others launch supplies and passengers for multiple docking intercepts; I'll let someone else calculate that.

[[ attachment:Tracktension.ods | libreoffice spreadsheet ]]

=== Continuum Calculation ===

Rather than calculate discrete payloads, assume a continuous flow of launch material