Support Structure Project for the Constellation-X Mirrors

Stress and Vibration Analysis

Overview

The x-ray mirrors to be used for Constellation-X are made of glass instead of aluminum. This improves the performance of the mirrors, but glass is also more fragile than aluminum. Our goal is to help design a support structure that will prevent the mirrors from failing during launch, when they will experience an acceleration of approximately 15g's. The planned launch vehicle for the Constellation-X telescopes is the Atlas V 551, which was also used for the New Horizons spacecraft (below).

Stress Analysis

The mirrors must experience high accelerations in order to get into orbit. Therefore, our support structure must ensure that the forces during launch are distributed as evenly as possible on the mirrors, because any point forces will likely cause failure.

Vibration Analysis

Because a spacecraft under launch conditions undergoes strong vibrational frequencies below 50Hz, we must ensure that any resonant frequencies present in our assembly are above 50Hz. Otherwise, the mirrors could easily resonate, thus building up energy and deforming or shattering. Holding the mirrors as stiffly as possible increases resonant frequency, but a design requirement of the mirror support system is that it cannot be fully constrained along the top and bottom edges.

We have used finite-element analysis using COMSOL, specifically the eigenfrequency and stress modules in Shell mode, to model the vibrations experienced by the mirror as it is launched into space. There are several design tradeoffs that we have considered. For example, holding the mirrors more tightly increases their eigenfrequencies (which is good), but holding the mirrors more softly decreases overall stress through damping. In addition, covering a higher percentage of the mirror edges keeps it more stable; however, this also blocks x-rays, which is highly undesirable.

We began by approximating the support structure for each mirror by placing fixed points on the mirror itself, without considering any damping materials. This allowed us to see certain trends; for example, as we added more fixed points to the mirror, the eigenfrequencies increased, as expected; however, the benefit gained by an additional point was almost negligible beyond about five points (see figure below).

We then moved on to simulating the support structure as a series of damping tabs along the edges of the mirror. These tabs are models of the damping material that exists between the glass mirror and the titanium beams. The outer edges of these tabs were held fixed to simulate the stiffness of the titanium, while the tabs themselves were modeled with nylon's material properties. The figure below shows the geometry that was used in our final simulations of the largest mirror.

Using the basic geometry shown above, we scripted COMSOL with MATLAB in order to run many simulations that changed the geometry slightly. Specifically, the widths of the tabs along the sides, as well as on the top and bottom of the mirror, were altered in each simulation. This gave us several insights. First, changing the mirror coverage along the sides does not affect the eigenfrequencies of the structure very much. Second, the eigenfrequencies of the structure increase dramatically each time an additional edge is held in some fashion. This is shown by the figure below.

As the figure above shows, having all four mirror edges fixed in some fashion appears to be the best way to stabilize the eigenfrequency of the mirror. For instance, if only three edges have fixed pads, it takes over 90% coverage to even approach a 100Hz resonant frequency. However, if all four edges have fixed pads, then it takes less than 10% mirror coverage along the top and bottom to raise the mirror's eigenfrequency above 100Hz. Furthermore, our stress analysis shows that this setup keeps the mirror's stress well below its breaking point. The maximum tensile strength of the mirror is on the order of 10^8Pa, and the maximum von Mises stress that our FEA models experience is on the order of 10^5Pa (see figure below).

Using our finite element analysis models, we have decided on an appropriate support structure for each individual mirror. We will have fixed pads covering 40% of the mirror's side edges, as well as fixed pads covering 10% of the mirrors along the top and bottom. This is consistent with results seen by NASA engineers. Furthermore, we believe that this layout fulfills all the design requirements: the mirror's eigenfrequency is held at well above 50Hz, the stresses experienced by the mirror are orders of magnitude below its breaking point, and the obscuring of x-ray radiation is minimal. Further details are included in our final paper.