On the Mechanics of the Bow and Arrow 1 B.W. Kooi Groningen, The Netherlands 1983 1B.W. Kooi, On the Mechanics of the Bow and Arrow PhD-thesis, Mathematisch Instituut, Rijksuniversiteit Groningen, The Netherlands (1983), Supported by ”Netherlands organization for the advancement of pure research” (Z.W.O.), project (63-57) 2 Contents 1 Introduction 5 1.1 Prefaceandsummary.............................. 5 1.2 Definitionsandclassifications . .. 7 1.3 Constructionofbowsandarrows . .. 11 1.4 Mathematicalmodelling . 14 1.5 Formermathematicalmodels . 17 1.6 Ourmathematicalmodel. 20 1.7 Unitsofmeasurement.............................. 22 1.8 Varietyinarchery................................ 23 1.9 Qualitycoefficients ............................... 25 1.10 Comparison of different mathematical models . ...... 26 1.11 Comparison of the mechanical performance . ....... 28 2 Static deformation of the bow 33 2.1 Summary .................................... 33 2.2 Introduction................................... 33 2.3 Formulationoftheproblem . 34 2.4 Numerical solution of the equation of equilibrium . ......... 37 2.5 Somenumericalresults . 40 2.6 A model of a bow with 100% shooting efficiency . .. 50 2.7 Acknowledgement................................ 52 3 Mechanics of the bow and arrow 55 3.1 Summary .................................... 55 3.2 Introduction................................... 55 3.3 Equationsofmotion .............................. 57 3.4 Finitedifferenceequations . .. 62 3.5 Somenumericalresults . 68 3.6 On the behaviour of the normal force T at t =0............... 80 3.7 Acknowledgements ............................... 82 3 4 CONTENTS 4 The static recurve bow 83 4.1 Summary .................................... 83 4.2 Introduction................................... 83 4.3 Theequationsofmotion ............................ 87 4.4 Finite-differenceequations . ... 93 4.5 Thequalitycoefficients ............................. 98 4.6 Threemodels .................................. 100 4.7 Influence mass and stiffness of string . 106 4.8 Fibratorymotion ................................ 109 4.9 Sensitivity analysis for a straight-end bow . ........ 117 4.10 Modelforstatic-recurvebow . .... 130 4.11 Static-recurvebow .............................. 136 4.12 Influence of draw, weight and mass of limb . .... 156 4.13 Another check on our numerical method . .... 165 Chapter 1 Introduction 1.1 Preface and summary The invention of the bow and arrow may rank in social impact with the invention of the art of kindling fire and that of the wheel. It must have been in prehistoric times that the first missile was projected by means of a bow. Where and when we do not know, perhaps even in different parts of the world at about the same time. Then man was able to hunt game and to engage his enemies at a distance. In the 15th century the bow in the ”civilized” world was superseded on battle fields by the fire-arm and became an instrument for pastime. Today, archery is a modern, competitive sport. The mechanics of the bow and arrow became a subject of scientific research after the bow had lost its importance as a hunting and war weapon. In the 1930’s C.N. Hickman, P.E. Klopsteg [6] and others performed experiments and made mathematical models. Their work both improved the understanding of the action of the bow and influenced the design of the bow strongly. In this thesis a mathematical simulation is made of the mechanical performance during the projection of the arrow by means of a bow. Because nowadays fast computers are available, we are able to cope with more advanced models, which are supposed to supply more detailed results. The flight of the arrow through the air and the way it penetrates the target is beyond the scope of this thesis. Characteristic for the bow are the slender elastic limbs. The bow is braced by putting a string shorter than the bow between the tips of the limbs. We distinguish between three different types of bows on the ground of the interplay between string and limbs. For bows of the first type, the ”non-recurve” bows, the limbs have contact with the string only at their tips. The Angular bow used in Egypt and Assyria and the famous English wooden longbow are non-recurve bows. In almost all Asia a bow made of wood, horn and sinew was used. In braced situation the string lies along a part of the limbs near the tips. Along this length and often further these limbs are stiff; they do not deform during the drawing of the bow. These bows are called ”static-recurve” bows. More recently bows are designed 5 6 CHAPTER 1. INTRODUCTION made of wood and man-made materials such as glass or carbon fibres imbedded in resin. For these bows the string also lies along the limbs in the braced situation for a short length. However, the limbs are now elastic along their whole length. These bows are called ”working-recurve” bows. In this thesis we restrict to bows of the mentioned three types which are symmetric, so we do not discuss the interesting asymmetric bow of Japan. In Chapter 2 we deal with bracing and drawing bows of all three types. The limbs are considered as beams for which the Bernoulli-Euler equation holds. In each situation, the equations form a system of coupled ordinary differential equations with two-point boundary conditions. A shooting method is described by which this system is solved. As a result of this, the problem is reduced to the solution of two non-linear equations with two unknowns. Attention is payed to the problem of finding starting points for the secant-Newton method which is used for solving these two equations. After loosing, part of the energy accumulated in the limbs is transferred to the arrow. In Chapter 3 we deal with the dynamics of the non-recurve bow. The bow is assumed to be clamped in the middle. The string is assumed to be without mass and to be inextensible. The governing equations are the equations of motion for the limbs. These equation are derived using Hamilton’s principle, the internal as well as the external damping having been neglected. There are two independent variables, the length coordinate along the bow and the time coordinate and there are six unknown functions. The boundary conditions at the tips contain the equation of motion for the arrow. A numerical solution is obtained by means of a finite-difference method. A Crank-Nicolson scheme is used, then for each time step a system of non-linear equations has to be solved. This has been done by a modified Newtonian method. The solutions of previous times are used to obtain starting points. At the moment of release the solution of the shooting method as described in Chapter 2 is used as starting point. In Chapter 4 the string is elastic and possesses mass. A part of this chapter deals also with the non-recurve bow, of which we consider now also the vibratory motion of bow and string after arrow exit. Then the governing equations form two coupled systems of partial differential equations. Besides time, for one system the length coordinate along the limbs and for the other one the length coordinate along the string is the independent variable. At the tips these systems are linked by the boundary conditions. The main object of Chapter 4 is the dynamics of the static-recurve bow. For this type it is necessary to take into account that the string has contact with the tips during the first part of the shooting and after a beforehand unknown time with the part of the ears between tips and string-bridges. In this case but also when the arrow leaves the string, the boundary conditions change abruptly at a moment which has to be calculated. Because of lack of time we have to leave the dynamics of the working-recurve bow out of consideration. In this thesis we do not deal with proofs of existence and convergence of the numerical methods. We did try, however, to obtain an insight into the accuracy of the developed methods. For example, the analytic solution of a linearized problem is compared with the obtained numerical solution. This is done in Section 4.13. The mathematical simulation is used for theoretical experiments. The aim of these experiments is to get insight into the influence of different quantities which determine the 1.2. DEFINITIONS AND CLASSIFICATIONS 7 action of the bow and arrow. This supplies the possibility to compare several types of bows which have been developed in different human societies. This is also done in Chapter 4. It appears that the static-recurve bow is not inherent better than the long straight bow. The meaning of the word inherent in this context is given in Sections 1.7 and 1.9 of this introduction. When the different properties of the materials, wood, horn and sinew, are deliberately used, more energy per unit of mass can be stored in the limbs of the Asiatic bow than in those of the wooden bow. Further, the static-recurve bow can be made shorter without the loss of much quality. Their shortness makes them handier and suitable for the use on horseback. Chapter 2 and 3 are reprints of published papers and Chapter 4 that of an unpublished one. As a result of this each chapter begins with an introductory section in which we give a short outline of archery and each chapter has a separate list of references; sometimes a reference is a chapter of this thesis. In this introduction we quote from various books and papers given in Lake and Wright [11]. This is an indexed catalogue of 5,000 articles, books, films, manuscripts, periodicals and theses on the use of the bow, from the earliest times up to the year 1973. 1.2 Definitions and classifications In this section we give the nomenclature of the different parts of the bow and arrow and the classification of bows we have used. Characteristic features of the bow are the slender elastic ”stave” and the light string, shorter than the stave, see Figure 1.1.