شمارش بر اساس چارچوب‌ها: ریاضیات برای کمک به طراحی سازه‌های سخت (توضیحات ریاضیاتی دولچیانی)

Rigidity theory is a body of mathematics developed to aid in designing structures. Consider a scaffolding that is constructed by bolting together rods and beams. The ultimate question is: "Is the scaffolding sturdy enough to hold the workers and their equipment?" There are several features of the structure that have to be considered in answering this question. The rods and beams have to be strong enough to hold the weight of the workers and their equipment, and to withstand a variety of stresses. Also important in determining the strength of the scaffolding is the way the rods and beams are put together. And, although the bolts at the joints must hold the scaffolding together, they are not expected to prevent the rods and beams from rotating about the joints. So, ultimately, the sturdiness of the scaffolding depends on the way it is braced. Just how to design a properly braced scaffolding (or the structural skeleton of any structure) is the problem that motivates rigidity theory.The purpose of this book is to develop a mathematical model for rigidity.Three distinct models are developed. The structures studied in these models, are represented by a framework: a configuration of straight line segments joined together at their end points. It is assumed that the joints are flexible and a question is posed that asks if the configuration can be deformed without changing the lengths of the segments. A triangle is a simple rigid framework in the plane while the square is not rigid - it deforms into a rhombus The first model presented here is for rigidity, the degrees of freedom model, is very intuitive and very easy to use with small frameworks. But, at the outset, this model lacks a rigorous foundation and actually fails to give a correct prediction in some cases. The next model is based on systems of quadratic equations: each segment yielding the quadratic equation which states that the distance between the joints at the end of the segment must equal the length of the segment. It is easy to see that this model will give the correct answer to our question in all cases. Hence, this is the standard model of rigidity. Unfortunately, this model is not so easy to use even with small frameworks. The third model constructed is equally accurate but is based on a slightly different definition of rigidity, called infinitesimal rigidity. In this model, the quadratic equations are replaced with linear equations and it is, therefore, much easier to use. Ultimately we show that all three models agree except for very few very special frameworks. The final chapter of the book is devoted to using these models to understand the structure of linkages, geodesic domes and tense grity structures. Consider a scaffolding that is constructed by bolting together rods and beams. The ultimate question is whether the structure is strong enough to support the workers and their equipment. This is the problem that motivates the area of mathematics known as rigidity theory. The purpose of this book is to develop a mathematical model for the rigidity of structures. In fact the author develops three distinct models in which the structure under consideration is modelled as a framework. These models are the degrees of freedom model and two models based on quadratic equations and linear equations respectively. The author shows that all three of these models agree except for a very small class of specially constructed frameworks. This is a theory with significant practical applications and will be of interest to a wide range of people including those studying graph theory or mathematical modelling. Jack Graver. Includes Bibliographical References (p. 175-178) And Index.