Question

A paper on the ‘Hyperloop Alpha’ concept for a high speed transportation system was published in 2013. Since then, there has been a lot of hype, time, and money directed toward the concept. Your task this week is to evaluate the hyperloop concept purely from a 2D particle kinetics point of view.

For this assignment, the important specifications from the paper are as follows:

Urban cruise speed: 300 mph (480 kph)

Inter-city cruise speed: 760 mph (1,220 kph)

Axial Acceleration (along direction of travel): 1 g

Lateral Acceleration (normal to direction of travel): 0.5 g

Capsule weight (incl. passengers): 57,000 lb (26,000 kg)

Pylon support spacing: 100’ (30 m)

3. At a minimum, any motion path must be continuous in position, velocity, and acceleration to avoid impacts. Suppose the path deviates from the theoretical line by a distance h, due to a relative displacement of one pylon with respect to its neighbors. If we assume a polynomial path along these two pylon spacings, for a total distance of 2L, the lowest-acceleration shape that meets these end-conditions is given by the function:

y(x) = 64h((x/2L)^3-3(x/2L)^4+3(x/2L)^5-(x/2L)^6)

c. Assuming that the velocity in the nominal motion direction ( ˙x) is constant, so that v = ˙xˆI + ˙yJˆ, and assuming that y˙ ≪ v, so that y/v ˙ ≈ 0, determine an expression for the tangential velocity, the tangential acceleration, and the normal acceleration of the capsule as a function of the amount of deviation (h), the pylon spacing (L), and the nominal velocity ˙x.

d. At what position x does the peak value of the normal acceleration, an occur?

e. At the location of peak normal acceleration, you determined in part ‘d’, determine an expression for the radius of curvature ρ of the motion of the capsule as a function of the amount of deviation (h) of the pylon.

g. Could you allow the same h in a curved section of track? Why or why not?

h. Does this level of precision in straightness sound plausible? Based on the values you calculated, would you expect there to be any significant impact on the cost?

Answer 1

Short of figuring out real teleportation, which would of course be awesome (someone please do this), the only option for super fast travel is to build a tube over or under the ground that contains a special environment. This is where things get tricky. At one extreme of the potential solutions is some enlarged version of the old pneumatic tubes used to send mail and packages within and between buildings. You could, in principle, use very powerful fans to push air at high speed through a tube and propel people-sized pods all the way from LA to San Francisco. However, the friction of a 350 mile long column of air moving at anywhere near sonic velocity against the inside of the tube is so stupendously high that this is impossible for all practical purposes. Another extreme is the approach, advocated by Rand and ET3, of drawing a hard or near hard vacuum in the tube and then using an electromagnetic suspension. The problem with this approach is that it is incredibly hard to maintain a near vacuum in a room, let alone 700 miles (round trip) of large tube with dozens of station gateways and thousands of pods entering and exiting every day. All it takes is one leaky seal or a small crack somewhere in the hundreds of miles of tube and the whole system stops working. However, a low pressure (vs. almost no pressure) system set to a level where standard commercial pumps could easily overcome an air leak and the transport pods could handle variable air density would be inherently robust. Unfortunately, this means that there is a non-trivial amount of air in the tube and leads us straight into another problem.