Frank Baginski

Structural Analysis of Pneumatic Envelopes: Variational Formulation and Optimization-Based Solution Process

Large super-light structural systems that for functional reasons require large surfaces are composed at least in part of structural membranes. The underconstrained nature of such structural membranes poses analytical challenges, but also provides design opportunities that are not commonly found in other structural systems that require the arsenal of solid-mechanics analytical tools for the assessment of design validity and performance. Overcoming some of the challenges that are posed by the underconstrained nature of such systems is an important ingredient in the development process for gossamer spacecraft. Our approach is a variational formulation of the analytical problem in conjunction with optimization techniques in the solution process. The optimization-based solution process avoids convergence problems that are encountered in the implicit solution process of finite element formulations of these underconstrained structures. To illustrate our approach, we carry out a structural analysis of a pumpkin balloon. Our formulation incorporates wrinkling of the balloon film and structural lack of fit between the skin and the tendon in the unloaded, that is, unstrained, structure. Our results on pumpkin balloons suggest the possibility of similar success if our methods are applied to other pneumatic envelopes.

Modeling the Shapes of Constrained Partially Inflated High Altitude Balloons

During a balloon mission, the maximum film stresses most likely occur while the balloon is constrained by the launch spool. The focus is on shapes of very small gas volume that are representative of spool shapes and partially inflated ascent shapes where a launch collar is still attached. The balloon is constructed from long, flat, tapered sheets of 20.32-μm polyethylene that are sealed edge-to-edge to form a complete shape. Two caps are assumed to be located on top of the balloon. Load tapes are attached along the edges of the gores. The potential energy of the balloon system includes contributions due to hydrostatic pressure, film weight, load tape weight, film strain, and load tape strain. Solutions are determined that minimize the total energy subject to a volume constraint plus appropriate constraints to model nonfully deployed configurations. To model fine wrinkling in the balloons membrane surface and to avoid compressive states, energy relaxation is employed. A number of numerical solutions are presented to illustrate the approach and provide estimates of the film stresses for these types of configurations.

Existence Theorems For Tendon-Reinforced Thin Wrinkled Membranes Subjected to a Hydrostatic Pressure Load

In this paper, we establish rigorous existence theorems for a mathematical model of a tendon-reinforced thin wrinkled membrane that is subjected to a shape dependent hydrostatic pressure load. We are motivated by the problem of determining the equilibrium shape of a strained high altitude large scientific balloon. This problem has a number of unique features. The balloon is very thin (20-40 μm), especially when compared with its diameter (over 100 meters). The balloon is unable to support compressive stresses and, instead, wrinkles or forms folds of excess material. Our approach can be adapted to a wide variety of inflatable structures, but we will focus on two types of high altitude balloons, a zero-pressure natural shape balloon and a super-pressure pumpkin-shaped balloon. We outline the shape finding process for these two classes of balloon designs, formulate the problem of a strained balloon in an appropriate Sobolev space setting, establish rigorous existence theorems using direct methods in the calculus of variations, and present numerical studies to complement our theoretical results.

A parallel shooting method for determining the natural shape of a large scientific balloon

Large scientic balloons provide a dependable low cost platform for carrying out research in the upper atmosphere. Usually, the design of such a balloon is based on an axisymmetric natural shape dened by the solutions of a mathematical model derived by researchers at the Uni- versity of Minnesota in the 1950s. For a natural-shape balloon, all the tension in the balloon fabric is carried in the meridional direction and the circumferential stress is assumed to be zero. In this paper, we will establish existence results for the model equations and present numerical solutions for a variety of parameters. For the case of a balloon at ∞oat altitude, the model equations can be solved by an ordinary shooting method. To model axisymmetric ascent shapes, one needs to make some crude assumptions on how excess lm is handled. When the volume of the lifting gas is very small, the ordinary shooting method is ineective for computing axisymmetric ascent shapes. In the past, ad hoc assumptions were employed to circumvent this diculty. In the work presented here, a parallel shooting method is used to determine these shapes without the need of additional assumptions.

MODELING ASCENT CONFIGURATIONS OF STRAINED HIGH ALTITUDE BALLOONS

We consider the problem of estimating stresses in the ascent shape of an elastic high-altitude scientie c balloon. The balloon envelope consists of a number of long, e at, tapered sheets of polyethylene called gores that are sealed edge-to-edge to form a complete shape. Because the e lm is so thin, it has zero bending stiffness and cannot support compressions. In particular, the balloon e lm forms internal folds of excess material when the volume is not sufe ciently large. Because of these factors, a standard e nite element approach will have dife culty computing partially ine ated balloon shapes. In our approach, we develop a variational principle for computing strained balloon shapes that incorporates regions of folded material as a part of the geometric model. We can apply our modeltofullyine atedorpartially ine ated cone gurations.Theequilibriumshapeisthesolutionofminimum energy satisfying a given volume constraint. Weapply our model to a design shape representativeofthoseused in scientie c ballooning and compute a family of ascent cone gurations with regions of external contact for a volume as low as 22% of its e oat value.

Numerical solutions of boundary value problems for K-surfaces in R3

A K-surface is a surface whose Gauss curvature K is a positive constant. In this article, we will consider K-surfaces that are defined by a nonlinear boundary value problem. In this setting, existence follows from some recent results on nonlinear second-order elliptic partial differential equations. The analytical techniques used to establish these results motivate effective numerical methods for computing K-surfaces. In theory, the solvability of the boundary value problem reduces to the existence of a subsolution. In an analogous way, we find that if an approximate numerical subsolution can be determined, then the corresponding K-surface can be computed. We will consider two boundary value problems. In the first problem, the K-surface is a graph over a plane. In the second, the K-surface is a radial graph over a sphere. From certain geometrical considerations, it follows that there is a maximum Gauss curvature Kmax for these problems. Using a continuation method, we estimate Kmax and determine numerically the unique one-parameter family of K-surfaces that exist for K E (0,Kmax). This is the first time that this numerical method has been applied to the nonlinear partial differential equations for a K -surface. Sharp estimates for Kmax are not available analytically, except in special situations such as a surface of revolution, where the parametrization can be obtained explicitly in terms of elliptic functions. We find that our numerical estimates for Kmax are in close agreement with the expected values in these cases.

On the Design and Analysis of Inflated Membranes: Natural and Pumpkin Shaped Balloons

Large scientific balloons are used by NASA and the space agencies of many countries to carry out research in the upper stratosphere. Such a balloon typically consists of a thin plastic shell with several external caps. Load tendons run the length of the balloon from top fitting to bottom fitting, dividing the balloon into identical regions called gores. The gores are made from flat panels of 20--30

Modeling Nonaxisymmetric Off-Design Shapes of Large Scientific Balloons

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Modeling the Equilibrium Configuration of a Piecewise-Orthotropic Pneumatic Envelope with Applications to Pumpkin-Shaped Balloons

m polyethylene film that are sealed edge-to-edge to form the complete shape. A typical fully inflated shape can be over 120 meters in diameter and over 1 million cubic meters in volume. To date, the workhorse of NASAs balloon program has been the zero-pressure natural shape balloon, an axisymmetric onion-like design that dates back to the 1950s. The equilibrium equations at float for a natural shape balloon lead to a nonlinear boundary value problem that can be solved to determine the design shape. In recent years, demand for long duration midlatitude balloon flights has led to a design concept known as t...

Stability of Cyclically Symmetric Strained Pumpkin Balloons and the Formation of Undesired Equilibria

The design of a large scientific balloon is usually based on an axisymmetric model, describing its shape at float altitude. However, the shapes observed during ascent are clearly nonaxisymmetric and are characterized by a number of distinctive features, including a periodic lobe pattern surrounding the gas bubble, internally folded balloon fabric, and flat winglike structures in the lower portion of the balloon below the gas bubble. In this paper, we present a mathematical model that captures the complex geometries of these off-design shapes. Real balloons are made from long tapered sheets of polyethylene that are sealed edge to edge. We base our geometric model on this construction, enabling us to define a reference configuration and to estimate the distortion in our computed balloon shapes. We compute one-parameter families of ascent shapes with triangular, square-shaped, and pentagonal symmetries. Our computed solutions possess many nonaxisymmetric features that are observed in real balloons.