February 20, 2013

Geodesic Vacuum Sphere Materials

Using the equations described in the previous post regarding geodesic vacuum spheres it is possible to determine the utility of materials to building one.  To simplify this a spreadsheet was written to find the optimal frequency for a given material and proposed altitude.  This sheet uses a simplified version of the US standard atmosphere model to estimate temperature and pressure given altitude, then finds density using the ideal gas equation.  Sheet two calculates the left side of the equation derived in the previous post which is compared to the right hand side calculated in sheet one:
$\sum_{i=1}^{f}(\frac{i-1}{f}) < \frac{1}{2}(0.113 \frac{\sigma_C}{P_i} \frac{\rho_i}{\rho_{mat}} - 1)$
Based on the results two materials are selected as potentially useful: unidirectional carbon-epoxy and boron-epoxy composite stringers.  It is observed that even under the maximum possible load pressure, full vacuum at sea level, both materials are still serviceable.  At higher altitudes the pressure difference decreases and the member thickness is allowed to drop, resulting in higher frequency spheres.
At sea level only a frequency of 1 is permissible, however for an airship optimized for cruise above 20km the boron-epoxy is capable of supporting any desired frequency.  A functional limit of 12 is in place to avoid localized buckling.
This sheet also calculates the excess lift produced by a vacuum sphere of the chosen material, in terms of lifted kg per cubic meter.  Lifted mass is possibly the most important result, as it allows further development of the proposal.  A rough layout becomes possible once one can calculate the size of vacuum spheres necessary to lift a given payload or vehicle.
For obvious reasons given the data presented boron/epoxy has been selected moving forward.  For this material the following has been found:
• sea level: f=1, lift=3.9kg/m3, r/R=0.07
• 10km: f=2, lift=1.5kg/m3, r/R=0.04
• 15km: f=4, lift=0.8kg/m3, r/R=0.018
• 20km: f=12, lift=0.4kg/m3, r/R=0.006
From this point a reference vehicle optimized for cruise at 15km is to be described.  However this information should also be useful for construction of a sea level test article, should materials and facilities become available.