Arqueria

International Journal of Computer Science in Sport – Volume 10/2011/Edition 2

www.iacss.org

Mathematical and Computer Model of Sport Archery Bow
and Arrow Interaction
Ihor Zanevskyy
Casimir Pulaski Technical University in Radom, Poland

Abstract
The aim of the research was to summarize a mathematical model of bow and
arrow interaction in the vertical plane and to develop anappropriated computer
model. A bow was modeled as a non-deformed riser and hinged to it two nondeformed limbs with Archimedean springs. Their stiffness as well, as an elasticity
of a string was taken into account in a linear approach. An asymmetric bow
scheme was considered. A stabilizer system was modeled as a construction of
multi-rod cantilever flexible bars fixed to the riser. An arrow was modeledas a
rod hinged to the string in a nock point. A mathematical model of bow and arrow
interaction was created using Lagrange method as a system of ten differential
equations and initial conditions. Correspondent Cauchy problem was solved using
Runge-Kutta method and NDSolve programs from Mathematica package. The
initial conditions were determined as solution of a static problem based on thebow model in a drawn situation. Correspondent system of non-linear algebraic
equations were solved using Newton method with FindRoot program. The model
shown its possibility to study main kinematical and kinetic parameters of bow and
arrow interaction in the vertical plane. Results of modeling was presented in
graphs.
KEYWORDS: ARCHERY, BOW, STABILIZER, ARROW, MODELING
Introduction
Anarchery sport bow consists of three main parts. They are a riser, limbs, and string (Figure A
in Appendix 1). Additionally, there are few stabilizers (from one to five rods) and a sight
mounted on the riser. An archery arrow consists of a shaft with a head, fletching, and a nock
(Figure A2). A nock point on a string and a rest and a clicker in a central part of a riser serve to
state an arrowposition relatively a bow.
A bow and an arrow during their interaction are in a common space motion. Their
displacements in the vertical plane (exactly to say, in projection to the vertical plane) and their
dimensions are commensurable quantities. Their displacements in the lateral direction are
significantly smaller. Therefore, it is reasonably to consider bow and arrow interaction in the
verticalplane separately from the lateral motion.
The first attempt to the problem of archery bow modeling was done by Hickman (1937) with
his model of a limb as a non-deformed rod hinged to a riser by an Archimedean spring.
Marlow (1982) developed the model taking into account elasticity of a string. Kooi and
Sparenberg (1980) developed a flexible bar model of a limb. Ohsima & Ohtsuki (2002)investigated static and dynamics of Japanese bow which is not traditionally equipped with a
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International Journal of Computer Science in Sport – Volume 10/2011/Edition 2

www.iacss.org

stabilizer. An arrow was modeled as a mass concentrated in a nock point of a string.
Zanevskyy (2006) considered an arrow as a rod and took into consideration an asymmetry bow
scheme and an angulardisplacement of a riser in the vertical plane relatively a grip point.
Zanevskyy (2008) worked out a model of bow stabilization in the vertical plane. Park (2010)
studded compound archery bow dynamics in the vertical plane assuming symmetrical
deformations of limbs and ignoring stretching of a string. Zanevskyy & Zanevska (2009)
developed a model of a multi-rod stabilizer as a system of cantileverflexible bars and studded
their influence on the bow and arrow dynamics in the vertical plane.
Papers on a complex mathematical and computer model of a modern sport bow and arrow
interaction in the vertical plane has not been published. The aim of the research was to
summarize a mathematical model of bow and arrow interaction in the vertical plane and to
develop an appropriated computer model....