Trapezoid Bolt

attachment:BoltCross.png

The launch loop rotor is composed of separable bolts, mostly made of thin laminations of transformer steel, along with carbon fiber stiffeners, embedded aluminum induction motor conductors, and a central spine. The bolts are normally used in two modes; assembled into a multilobed rotor for the main tracks and inclines, and separated into separate bolts for minimum radius deflection.

The number of lobes in the rotor (and the number of bolts in parallel) remains to be determined; probably between 4 and 6. The length of the bolts may also change. The size and cross section is constrained by the linear density of the rotor.

For a 4.32 kg/m rotor with six bolts arranged in a hexagon, each bolt will mass 7.2 grams per centimeter of length. If 30% of the bolt mass is non-iron mass m₂, and 70% of the bolt is m₁ then m₁ is approximately 5 g/cm = 5e-3 kg/m.

The density of 4% silicon transformer steel is 7.65 g/cm³, so A₁, the cross section of the area with mass m₁, is A₁ = 0.65 cm² = 6.5e-5 m². ~~~ A₁ = (X-C) × Y ~ so ~ Y = A₁ / (X-C) .

Deflection Mode

In deflection mode, an electromagnet pole (shown at the bottom of this diagram) pulls the rotor towards it, with near-saturation flux density B_{max} , perhaps 1.8 Tesla. The material will be strongly saturated and with considerable hysteresis at this field strength; that actually helps regulate the deflection force.

The deflection field "entering" at the bottom exits at the two diagonal pole faces, with less flux density. The (atttracting) deflection force at the bottom is F_X ~=~ X ~×~ B_{max}^2 / 2 \mu_0 so FX = X × 1.29 MPa . The flux at each diagonal face is B_D = ( X / 2 D ) B_{max} so the force at each face is F_D ~=~ D ~×~ B_D^2 / 2 \mu_0 ~=~ D ~×~ (X/2D)^2 B_{max}^2 / 2 \mu_0 ~=~ X^2/D × B_{max}^2 / 8 \mu_0 so FD = X²/D × 0.16 MPa .

Note that the flux B_D cannot exceed B_{max} , so D > X/2 . The bolt cannot be very thin in the Y direction.

A fraction C/D of that force is directed upwards for each face; the total upward force through both faces is FC = 2 C (X/D)² × 0.16 MPa. Since D² = Y² + C² = (A₁/(X-C))² + C² , the total downward deflection force is:

\Large F_y ~=~ X \left( 1 - { { X ~~ C } \over { 2 \left( \left( {\LARGE { A_1 \over { X-C } } } \right)^2 + C^2 \right ) } } \right) { { B_{max}^2 } \over { 2 \mu_0 } }

This deflects mass m₁ + m₂, so the total radial acceleration is ay = Fy/( m₁ + m₂ ), which we hope to maximize by optimizing X and C, given A₁.

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