Deployable Structures

In this chapter we consider a type of transformable structures, capable of executing large configuration changes in an autonomous way. In most cases, their configuration changes between a compact, packaged state and a large, deployed state: the transformation from the former to the latter state is called deployment. The reverse transformation is called retraction.

One could almost redefine biology as the natural history of deployable structures. An organism is successful partly because it uses the minimum amount of material to make its structure and partly because it can then optimise its use of that material so that it can influence as much of its local environment as possible. The more of its environment it can control and utilise for energy gain (“feeding”) per unit energy expended in growing and moving, the more successful the organism will be since it will have more energy available for reproduction, the ultimate criterion of success. One might even invent a parameter of success based on effectiveness of deployment. Perhaps this would be the least volume fraction of its environment which an organism can occupy. One would then equate the “addressable volume” (i.e. the volume you can entrain by waving your arms and legs around to their maximum extent) with the volume which your body occupies. A similar sort of parameter has been proposed for animals which feed by filtering particles out of the water (e.g. barnacles, sea anemones) or the air (e.g. a spider with its web). The longer, thinner and more mobile the limbs the greater the relative addressable volume but the greater the likelihood of the structure breaking.

The concept of using ideas from nature to further technology has been given a number of names such as “Biomimetics”, “Biomimesis”, “Biognosis” and “Bionics”. In each instance it’s probably fair to adopt the attitude of Lewis Carroll’s Humpty Dumpty and say that the meaning of all four words is whatever I want it to be — and in this instance I shall define the meaning of all four words as the same. Biomimetics is the technological outcome of the act of borrowing or stealing ideas from nature. It is difficult to trace the origins of this approach, since man has looked to nature for inspiration for more than 3000 years (when the Chinese hankered after an artificial silk). In modern times, the word “bionics” was coined by Jack Steele of the US Air Force in 1960 at a meeting at Wright-Patterson Air Force Base in Dayton, Ohio. He defined it as the science of systems which have some function copied from nature, or which represent characteristics of natural systems or their analogues. In 1966 R-G Busnel, of the animal acoustics laboratory in Jouy-en-Josas in France, organised a meeting on the theme “Biological models of animal sonar systems” in which the Office of Naval Research of the USA was involved. They had already funded other work in the general area of biological engineering, such as Torkel Weis-Fogh’s work on resilin (a rubbery type of insect cuticle) and elastin in Cambridge. Busnel’s meeting was one of the first at which these problems were discussed by biologists, engineers and mathematicians in order to discover general principles of technology.

Due to their small thickness, membranes can be easily bent, but are comparatively difficult to stretch. Hence, in studying the packaging of membranes it is normal to model them as inextensional plates of zero thickness.

The current trend in the aerospace industry is towards deployable structures that are simpler, and hence cheaper and more reliable. Therefore, structures that require movable joints are being replaced by flexible structures that can be folded elastically. Thin shell structures are the ideal choice for this because, provided that folding does not involve a significant amount of mid-surface stretching, the stresses due to bending can be made as small as required to avoid yielding of the material, by choosing a sufficiently small thickness for the shell. Two examples of this “new” type of deployable structures will be discussed in this chapter.

Many deployable structures contain bi-stable features. For example, bi-stable latching elements are often used to freeze a mechanism in a particular configuration: the latch snaps when the structure reaches its deployed configuration and no further motion is possible. This chapter will show that it is possible to design complete deployable structures that are bi-stable: although much less common, this approach may lead to a future generation of structures with exciting new capabilities. We begin by explaining the concept of bi-stable deployable structures using a simple structure, and then move on to a more challenging application.

In the design of engineering structures, an important question is whether a structure is rigid. For conventional structures, rigidity is a fundamental requirement. However, there are cases where just the opposite is required. In order to answer the question of rigidity we have to know the static-kinematic properties of the structure. In the forthcoming, these properties will be investigated for bar-and-joint assemblies, that is, for structures composed of straight bars and frictionless pin joints. Firstly, we survey the basic terms to be used in the analysis.

In the theory of mechanisms, many linkages are known whose instantaneous degree of freedom in certain positions of the linkage is greater than their degree of freedom (in our terms: the infinitesimal degree of freedom of a finite mechanism is greater than its finite degree of freedom). The sign of this is that the Jacobian matrix of the constraint functions has a rank deficiency. For a four-bar linkage with equal opposite bars, Litvin (1980) called the attention to the fact that, if one of the bars is fixed in a horizontal position, then the two bars joining to it incline to the horizontal at an angle. Then the relationship between these two angles determines two curves. One is associated to the parallelogram, the other to the anti-parallelogram shape of the linkage. The two curves have a point in common if the four joints lie on a straight line. In this position, the Jacobian matrix has a rank deficiency. Litvin (1980), however, did not attribute any additional significance to this point.

In any structure the equilibrium matrix A relates the vector of generalised stresses σ to the vector of generalised loads l by the linear equations of equilibrium

During an exhibition devoted to construction, held in Moscow in 1921, Johanssen presented a sculpture made with three struts and eight cables. This system had no rigidity, and its mechanisms could be activated with one of the cables. It was a kind of “ proto-form ”. In 1948, Kenneth Snelson worked with Richard Buckminster Fuller, in Black Mountain College. Fuller wanted to realize his idea of “ islands of compression inside a sea of tension ”. In response, Snelson made three models (see Motro, 1996).

Folding is required to reduce the volume of objects in space. This operation allows transportation and storage of folded objects. The use of folding systems has greatly evolved since the itinerant habitation of the first men. Nowadays it covers a wide range of applications, from the simple fisherman’s chair, to architectural projects and satellite components.

The purpose of this chapter is to describe a general methodology for efficient and general computer simulation of deployable mechanical systems.