Perforated Shell Structure Analysis
6.3 The Loading
The stress on the grids due to acceleration while in operation in space is negligible, and is not considered to be one of the limiting factors on the effective lifetime of the spacecraft; the NSTAR engine produces 92.67 mN of thrust at the highest throttle level, TH15 [9]. However, in order for the spacecraft to have an opportunity to successfully fulfill its mission, it must survive the few minutes it takes to place the spacecraft in orbit. Depending on the launch vehicle used, the entire spacecraft will be subjected to acceleration forces, vibrations in a large range of frequencies, as well as possibly impulsive shocks from the ignition of explosive bolts [18]. If the engine grids fracture, either from excessive stress or from a mutual collision, the entire mission could be jeopardized due to an ineffective or inoperable engine.
In order to safeguard against such catastrophic failures, all components of a spacecraft must meet certain random vibration specifications [75, 76]. Ideally for this work, we would like to be able to simulate the effects of this same stress test on a model of any candidate grid geometry, and analyze both the expected stresses and displacements achieved in such a test. If the stress is too great, such as to expect the grid to fracture, or if the displacements are too large, such that the two grids would be expected to collide with each other, the candidate geometry would be deemed inadequate.
The finite-element modeling software used in this study, to be discussed further in Chapter 7, has been shown to be capable of dynamic loading [77]. Once again we face the issue of being able to model the structural response of the grid rapidly. Unfortunately time-dependent loading was deemed to require too much time to be implemented into the current optimization problem, and
it was decided to limit the current study to static loading of the grid under simulated constant acceleration.
Since we have chosen to only simulate static loads, it is unclear as to what an appropriate simulated load would be to, at the very least, provide insight into which grid shapes are optimal.
Not only is the magnitude of the load an unknown quantity, but the direction in which this load should be applied to the grid is unclear. Depending on the specific craft and launch vehicle, the engine grid could have any orientation with respect to the direction of travel during launch. With nothing to give us preference to one loading direction over another, it was decided to apply the load in the direction opposite to the engine thrust vector.
In order to first gain some confidence in the structural analysis code, simulations for circular flat plates, with the same diameter and thickness as the NSTAR accel grid, under a range of uniformly distributed transverse loads were performed. The effective elastic properties were used, assuming the plate was made from molybdenum and perforated in the same hole pattern as the NSTAR accel grid.
The elastic properties were found in the following way: The two effective quanties from Figure 6.2 were interpolated for a thickness-pitch ratio of 0.19, corresponding to the NSTAR accel grid, on all five curves shown. From these five values, corresponding to the five indicated open-area fractions, the quantities were interpolated for an NSTAR accel grid open area-fraction of 0.24. The following values were found:
E∗ E
1 +ν 1 +ν∗
= 0.682, and E∗
E
1−ν 1−ν∗
= 0.540.
Using a value ofν= 0.30 for molybdenum [78], solving these two equations for the effective Young’s modulus and Poisson’s ratio yields
E∗
E = 0.625, and ν∗= 0.201.
The Young’s modulus of molybdenum is E = 329 GPa, which yields an effective modulus ofE∗ = 196.7 GPa.
10−4 10−3 10−2 10−1 100 101
100 101 102 103
Normalized LoadQR4g/E∗t4 NormalizedDeflectionδmax/t
Linear Flat Plate Non-linear Flat Plate FEM Flat Plate Accelt= 0.51 mm Accelt= 1.02 mm Accelt= 2.04 mm Screent= 1.52 mm
Figure 6.3: Theoretical- and FEM-deflection comparison. Comparison of deflections computed from FEM analysis of a flat plate under large deflection is shown to have less than 8% difference from the deflections predicted by non-linear large-deflection plate theory. NSTAR accel and screen grids of various thicknesses were analyzed using the FEM code. The effective moduli were computed by interpolation of the data of Figure 6.2, and summarized in Table 6.1. The radius of the plate was specified to beRg= 15 cm.
The maximum displacements,δmax, predicted by the code for various loads,Q, were compared with those predicted by large-displacement flat plate theory [79]. The normalized results are shown in Figure 6.3. The displacements obtained from the simulations were systematically less than theory, where the difference increased with increasing load. The results differed by no more than 8% over a range of loads spanning three orders of magnitude.
To obtain an idea of the magnitudes of the expected displacements from different loads, the NSTAR grids were modeled in a similar manner to the flat plates above. Grids with 1x, 2x, and 4x the accel grid thickness, and 4x the screen grid thickness were all modeled under a range of transverse static loads. The results are also shown in Figure 6.3. The effective properties used for each are shown in Table 6.1.
Table 6.1: Effective properties for simulated NSTAR grids of different thickness: Obtained from interpolation from Figure 6.2
Thickness Thick-Pitch Open-Area Grid (mm) Ratio p+tt Fraction EE∗
1+ν 1+ν∗
E∗ E
1−ν 1−ν∗
E∗
E ν∗
Accel (1x) 0.51 0.187 0.239 0.682 0.540 0.625 0.201
Accel (2x) 1.02 0.315 0.239 0.634 0.540 0.598 0.226
Accel (4x) 2.04 0.479 0.239 0.576 0.540 0.563 0.281
Screen (4x) 1.52 0.406 0.671 0.102 0.147 0.114 0.457
It was found that as the thickness of the simulated grid decreased, the time required to complete the computational analysis increased, and was less likely to converge to a solution. To avoid possible issues of non-convergence during the optimization procedure outlined in Chapter 7, we decided in some situations to increase the thickness of the simulated grid. The data from Figure 6.3 will be used to obtain scaling factors for the expected displacements. The constraint on our optimization will be that the maximum grid deflection must not exceed a certain value under a specified loading. Since increasing the thickness of the grid makes it more resistant to deflection, the scaling factors obtained from Figure 6.3 will enable us to scale the allowable maximum displacement of the thickened grid appropriately, such that both the grid of correct thickness and the artificially thickened grid will violate the displacement constraint under approximately the same load.