Curved Surface Thinsats
The first thinsats will probably be flat - this makes them easier to make on existing flat panel assembly lines. However, the substrates are aluminum, and they can be made with any shape.
If second generation thinsats are also uniformly thin, but have a spherical curvature like a shallow spherical shell capping a tank or a domed stadium, a few degrees of curvature may bring some significant advantages.
Advantage: Vibration/Flexing
A flat thinsat can vibrate or flex, like rolling a sheet of paper, at frequencies on the order of 1 Hertz. A curved thinsat can still flex, but the rolled surface has a much higher bending moment, which means the flex frequencies will be much higher as well. That means that bending force perturbations from turning on and off the electrochromic thrusters will cause less vibrations.
The thrusters will turn on and off slowly, perhaps 100 milliseconds from clear to reflective. The frequency spectrum of these slowly changing forces will drop off rapidly at higher frequencies - they will not deposit significant vibration energy at the higher resonant frequencies (hundreds of hertz? TBD) of a curved thinsat.
Advantage???: Thermal expansion
If there is differential thermal expansion between the front and backside of a thinsat, it can curl as it moves into eclipse, and the temperature drops by almost 100C. While we can expect the curvature of a thinsat to change as well, a temperature change will not bend a thinsat in unpredictable ways, and probably won't change the radial symmetry. This needs more study - it is possible that the shape change could be worse. If so, we've eliminated this geometry, but learned an interesting way to build a thermostat spring or a temperature valve.
Advantage: Phased Array Endfire
The antenna arrays on a thinsat will be surrounded by a complex stew of conducting objects, especially the thin strips of solar cell that cover most of the surface. If a thinsat is absolutely flat, then endfire transmission - beams broadcast in the plane of the thinsat - will be difficult. Thinsats will have trouble talking to neighboring arrays in the noon orbital position. Thinsats will have trouble talking to some ground locations when they are between the 6 o'clock and 8 o'clock morning positions (and the 4'oclock and 6 o'clock afternoon orbit positions). And if we minimize night sky light pollution by orienting night-side thinsats edgewise to the day/night terminator on earth, then the terminator will get poor coverage from all night-side thinsats.
On the other hand, a slightly curved thinsat will always have some antennas exposed to edgewise targets. The coverage will not be good, but it will be better than an edgewise flat thinsat.
Advantage: Parabolic mirrors
We can have specialized thinsats working in groups, with many thinsats acting as a distributed parabolic receiver dish for a single thinsat in front of them. This will be good for radar reception, perhaps even optical imaging. Note that the off-axis curved thinsats will produce radial light pressure thrust, so we may need additional complications and optical reflectors to hold them in place relative to the rest of the array.
Advantage: Deployment
One of the problems with flat thinsats is deploying them in orbit - they will probably be somewhat "sticky" and not want to come apart. If we make the even and odd thinsats in a stack with slightly different curvatures, then their natural tendency is to spring apart, like this: (|(|(|(|(| in vacuum, while air pressure will tend to hold them together on the ground. We do not need much curvature difference to cause this to happen. Rather than need dispensing machinery, a thinsat launch stack will be naturally self-deploying.
Disadvantage? Design Complexity
A curved surface phased array is difficult to analyze with classic formulas and out-of-the-box opaque commercial software for phased array design. OTOH, it is just as amenable to numerical simulation as its flat cousins. The conductors and solar cells on the thinsat surface (flat or curved) will require numerical simulation anyway.
Accommodating Curvature in Manufacturing
One drawback of curved thinsats is that existing planar imaging systems won't work. However, large thinsat manufacturing volumes will soon justify the production of imagers with slight spherical distortion, designed to focus and project imaging patterns on a curved surface. Wafer chucks, photoresist spinners, silk-screen stencils, and other manufacturing tools can also be modified or produced specially for working with a standard curved surface.
Analysis
Key words:
- shallow dome ( saucer dome )
- membrane analysis
- hoop stress
- Roark's formulas for stress and strain / Warren C. Young, Richard G. Budynas
- Table 13.3 - 3b Page 611 in Edition 7
- Table 13 case 2 (Edition 7) "Partial spherical shell, load P concentrated on small circular area of radius r0 at pole; any edge support".
- 6th OHSU SCI and ENG REFERENCE TA407.2 .R6 1989 6th edition, page 547 Table 30 Case 3b Shallow spherical shell, point load at pole,edge fixed and held.
- gives a table for $ \alpha $, credited to Reissner, which runs out of gas for ultra thin but significantly curved shells.
- Reissner, E.: Stresses and Small Displacements of Shallow Spherical Shells, II, Journal of Mathematics and Physics, vol. 25, No. 4, 1947.
PSU TA658.3.G47 1984 Gere & Krawinkler Structural and Construction ...
- PSU TA660.S5 BS2 1983 Billington Thin Concrete Shell Structures