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ABSTRACT
Tests were run comparing seamed and unseamed inflatable concentrators. Measurements of the surface profiles showed that the effect of seams on surface accuracy was minimal. Surface errors were generally slowly varying across the membrane, exhibiting a "W" or "M" shape. When the membrane included seams, the result was a slight bumpiness in the error profile which retained the "WI' or "M" shape at nearly the same level as without seams. Recent studies indicated that the major source of the surface error was more likely the variation in material modulus as a function of position and stress. We used a special finite-element computer code to quantify the effect of seams; the results show again that the contribution of seams to the error in surface shape is negligible compared to the observed surface deviations in typical reflectors. The significance of these findings is that inflatable reflectors can be made inexpensively using seamed construction with no appreciable degradation of surface accuracy when compared to more expensive, continuous, formed membranes.
INTRODUCTION
NASA sponsored the building and testing of 3-m-diameter paraboloids in 1979. Figure 1 shows the shape of the deviation of a typical such paraboloid from the perfect paraboloid shape. This is the deviation for a scan about 67Omm below the center of the symmetric 3-m-diameter inflated paraboloid. This reflector was made of many pie-shaped gores, seamed together, and the scan shown in Figure 1 crossed over a dozen or so such seams. If there are any slope discontinuities or error sources caused by scanning over the seams, they are insignificant compared to the obvious, systematic distortion. A gradual "W" shaped distortion is obtained from one side of the reflector to the other. This gradual distortion is typical of what is seen for these and other such reflectors built and measured [ 1,2].
During the NASA study, we postulated that the cause of the error seen is the change in the effective modulus of the film used caused by the presence of the seams. Since the seams are covered with a tape, they are thicker than the rest of the film. Since the relative density of tape is higher near the center of the reflector, where the pie-shaped gores are at their narrowest, the seams cause a systematic variation in effective modulus across the surface of the paraboloid. At the time this work was done, there did not exist accurate analysis tools that could prove the effects existed. In this paper we present some of the test data earlier obtained, and show supporting analyses recently performed using a new finite-element code designed for inflatable membranes (FAIM). The determination of the role of seams in surface accuracy is very important to quantify because construction of large paraboloids will most likely need to use seams. The alternatives are relatively expensive and have their own sets of problems. For instance, the modulus of the thin plastic films used for these reflectors varies considerably depending on how the film is manufactured. DuPont and other plastic film producers have spent much research and effort in developing methods for making rolls of flat film with relatively uniform modulus. A uniform modulus is necessary, especially for solar concentrators where the surface must be stretched smooth. Off-the-shelf films such as Kapton have substantial space-flight experience. They can survive even in Low-Earth orbits when properly protected. On the Inflatable Torus Solar Array Technology Program, the Kapton substrate was protected from atomic oxygen attack and space charging by coating with Iridium Tin Oxide - - a relatively inexpensive process.
HAIR TEST PROGRAM
A series of tests were run on Phase 1 of the Highly Accurate Inflatable Reflectors (HAIR) program in 1983 [3]. The analysis of the effect of seams on shape showed that there was a significant difference in the shape of gores once the seams were considered. Figure 2 shows the calculated gore edge dimensions (deviation from a straight line) for the cases of a uniform film and one with seams. The seam correction was made by taking a variable film thickness along the length of the gore corresponding to the extra thickness caused by the addition of the tape used on the seams.This variable thickness changes from a value nearly equal to the sum of the gore film and the seam tape at the narrow point of the triangular (pie shaped) gore, to a value just slightly greater than the gore film alone at the broad end.
To test out this analysis a series of inflation tests were made on films. Table 1 shows the matrix of tests. For each type of specimen, a test was planned both with and without seams.
The measured profiles of the inflated test specimens were compared to the theoretical ones (best fit) to get the shape of the deviation. The inflated surface was measured using a laser mounted on an optical bench, and finding the positions of the spot reflected from the surface as a function of laser position (see [2]).
TEST RESULTS
Figure 3 shows a typical mounting of a test membrane on the face of vacuum chamber used to create the pressure difference for inflation. The test membranes were roughly 1 meter in diameter. Figure 4 shows the deviations found for two paraboloids - - using the two gore shapes shown in Figure 2. For the shape assuming a uniform material, we got the familiar "W" shaped curve found on the earlier NASA tests. For the paraboloid where a seam correction was assumed, the deviation reversed into a "M" shape. The implication is that if a proper correction factor can be found, somewhere in between these two cases, we should have a more accurate shape.
To check the hypothesis further, tests on flat membranes were run. Figures $6, and 7 show results for three flat membranes which were tested first without seams. Then, similar flat membranes were sliced into pie-shaped gores, seamed, and retested. The startling result was that in all three cases, there was little difference between the seamed and unseamed membranes. The effect of the seams could be seen in some of the membranes as small bumps riding on the main surface distortion. The major source of error in these membranes was clearly not the seams.
We explored this further with two additional specimens. First we examined a flat membrane that was then cut into a lattice (Figure 8) where seams were made running not only radially but also circumferentially. The measured surface deviations are shown in Figure 9; they look very similar to those measured earlier. We then formed a spherical membrane by heat-shrinking the mylar membrane over a mold. The resulting deviations from the theoretical shape is shown in Figure 10; they also have the familiar shape - - with little effect of seams.
Some later work has shown that the modulus of thin films typically varies strongly with strain [2]. Since paraboloids have varying radii of curvature, there is a systematic variation of strain as one travels from the center of a paraboloid to its edge. This variation in elastic modulus can be very large and may be the culprit causing the surface inaccuracies. The FAIM computer code discussed in the next section is currently being modified to include these effects.
These test data show that until we understand better the source of surface inaccuracies in inflated membranes, there is no increase in accuracy obtainable by going to unseamed membranes. Once we understand and control the sources of the deviations seen above, then perhaps research into unseamed membranes will provide even higher accuracy. The current state of the art, with deviations on the order of those seen above, will suffice for high quality solar concentrators (concentration ratios above 12,000) and microwave antennas (to mm-wavelengths).
Faim Description
In the late 198O's, L'Garde developed a finite element modeling program, FAIM (Finite Element Analysis of Inflatable Membranes), for the analysis of the large deformations of inflatable parabolic membranes. FAIM is a specialized, geometrically nonlinear finite element program. It is iterative in nature; an initial prescribed, uninflated structural shape is input to be analyzed for stress and deformations.
In its current version, FAIM assumes a linear isotropic material and a geometrically nonlinear deformation behavior. The loadings available are (a) internal/external pressure, (b) concentrated nodal forces, and (c) nodal temperatures. Displacement, skew, and spring boundary conditions may be specified on any number of nodes of the finite element model. A number of enhancements have also been made to FAIM over the years in order to increase its analysis capabilities as well as its computational efficiency. First, a preprocessor and postprocessor with pull-down type menus were coded which makes the input and output interaction user-friendly. Secondly, two special analysis capabilities were implemented in the FAIM code in order to corroborate the stress and deformation behavior of inflatable parabolic shell membrane analytical models with their laboratory prototypes. A "spring boundary condition" mentioned above was coded in anticipation of, the need to characterize the outer radius mounting surface of the parabolic shell as a flexible, rather than an intinitely rigid support. Also, a "prestress" capability was implemented in the code in order to characterize the prestress state of the gore membranes due to the mounting procedure. Currently, a node renumbering routine to reduce the bandwidth of the coefficient matrix is being written. This will reduce both the memory and CPU requirement of the code.
NASA has recently awarded L'Garde, Inc. a Phase II SBIR contract to incorporate nonlinear material and dynamics capability to FAIM. By the end of 1994, FAIM should have nonlinear material capability. And by the middle of 1995, the dynamics capability should be in place.
An S-node isoparametric quad, a 6-node isoparametric triangle, and a 3 node isoparametric cable are the basic elements of the code. A conscious decision was made early to discard the computational advantage of the 4 node quadrilateral and the 3-node triangle membrane elements since these elements were deemed too crude to provide the requisite deformation information. A 3-node triangle for instance can only approximate the smooth surface of an inflatable parabolic shell surface with a collection of flat triangular facets. These facets, each of which has a constant slope are simply not adequate when one needs to determine the surface slopes throughout the continuum.
FAIM Validation Examples
FAIM was run on different computing platforms including a VAX, SUN, and PC486's using the Microsoft and Lahey FORTRAN compilers. Although the convergence process and the number of iterations varied from one computer to the next, the tinal result, once the convergence criterion has been satistied, was the same for all computers and compilers within the specified tolerance. The very first validation test FAIM was subjected to was the & deflection of a seamless paraboloid. For this test, one of the authors (M.T.) solved analytically, the inverse problem of calculating the initial uninflated shape of a surface of revolution so that on inflation to the design pressure, the surface takes on a prescribed paraboloidal shape. [S] The uninflated shape was input to FAIM, and the same internal pressure applied to see if FAIM gives back the correct inflated shape. The results of FAIM against the analytical solution are shown in Figure 11. Figure 11 shows the results for three different initial uninflated shapes corresponding to three values of the Poisson ratio. For the case shown, the edge of the paraboloid shape. [S] The uninflated shape was input to FAIM, and the same internal pressure applied to see if FAIM gives back the correct inflated shape. The results of faim against the analytical solution are shown in Figure 11. Figure 11 shows the results for three different initial uninflated shapes corresponding to three values of the Poisson ratio. For the case shown, the edge of the paraboloid was fixed.
In the figure, r is the radius rz =x* + yz in the paraboloid equation z = (1/4F)r*. F is the focus of the paraboloid, E is the material modulus, p is the inflation pressure, and t is the material thickness. One other test used was that of the deflections of a circular membrane. [6] Excellent agreement was found between the analytical solutions and FAIM calculated results. The result for the deformation of an initially flat circular membrane, clamped at the edge is shown in Figure 12.
The Effect of Seams on RMS Surface Accuracy
In order to assess the effect of seamed-construction on the rms surface accuracy of inflatables, 3 cases with and without seams were examined: (i) the deflection of a one-meter diameter six-gore, flat circular membrane, (ii) the deflection of a one-meter six-gore, pre-formed spherical membrane, and (iii) the deflection of an axisymmetric 64-gore, 14-meter diameter on-axis parabolic reflector.
The resulting deflections for the flat membranes, cases(i) and (ii) above were fitted to the equation describing the shape of a pressurized membrane constrained at a circumference (see eq. 1).
CONCLUSIONS
Tests of inflated membranes have shown that the presence of seams causes surface errors of about 0.1 mm rms in l-meter diameter membranes. On top of this rides a systematic error which is about an order of magnitude greater. Theoretical calculations using finite element analysis confirms that the presence of seams can not explain the observed surface errors; these calculations show that the seam effect should be less than 0.1 mm rms error. Seams can be included in membrane design using a variable thickness of the membrane corresponding to the average thickness including the seam tape.
The advantage of the seamed construction is that large membranes can be made inexpensively, using off-the-shelf films that are relatively well characterized. Seamed construction is currently feasible for inflatable reflectors for microwave antennas to 5 GHz and for optical concentrators for concentration ratios up to 12,000. When the systematic error found in inflated membranes is characterized and removed, these limits may be extended considerably before seams significantly affect the results.
ACKNOWLEDGMENTS
The testing of inflated membranes was sponsored by NASA and the USAF, and was carried out under the direction of Gordon Veal of L'Garde. The FAIM computer code was developed by L'Garde, with some assistance from NASA. The FAIM calculations reported here were sponsored by L'Garde.
REFERENCES