Bow/Arrow Models and Methods



B.Ponnusamy, Universiti Malaysia Pahang (UMP)

The objective of this journal is to develop a design optimised Recurve bow riser besides analyze the safety factor and maximum deflection. The designed bow riser is simulated for  Finite  element  analysis  at  draw  lengths  of  660mm  and  710mm  which  gives  out  two  set of acting  forces. The  material  to  develop  the  bow  riser  model  is  Aluminium  alloy  6061  T6. The finite  element  analysis  reveals  that  the  design  can  sustain  both  set  of  forces  with maximal deflection  of  0.2934mm  and  safety  factor  of  10.6363.  Preliminary  design  optimisation  also included where removal of 16% of material to promote lighter bow riser. The final mass of the bow riser weighs about 0.92 kg.

Applications of Physics to Archery

H. O. Meyer, Physics Department, Indiana University, 6 November 2015

Archery lends itself to scientific analysis. In this paper we discuss physics laws that relate to the mechanics of bow and arrow, to the shooting process and to the flight of the arrow. In parallel, we describe experiments that address these laws.

An advanced model of the round-wheel compound bow

M. Tiermas, MeccanicaAn International Journal of Theoretical and Applied Mechanics AIMETA2015:262

A mathematical model of an archery twin-round-wheel compound bow is introduced. The force-draw curve of a compound bow is measured and compared to the prediction of the model. Also the path of the limb tip of the compound bow is checked. The model seems to be accurate enough for bow design or adjusting purposes.

Motion of the bow during aiming and releasing the shot

J. Edelmann-Nusser et al, 19 International Symposium on Biomechanics in Sports (2001)

The range of motion of a recurve bow during aiming and releasing the shot is very small. Thus on-target-trajectories are used. To measure the on-target-trajectories of bows a system originally designed for gun shooting was modified. The system can be fitted onto the archer’s personal bow. 15 highly skilled archers of the German National Teams of women and men participated in the study. Two intraindividual typical strategies of aiming were found: 11 archers try to hold the aim on sight for at least 2 seconds whereas 4 archers tend to “slide” on the target and release the shot at once when the target is on sight. For 14 of 15 archers the “hold boxes” disclose an asymmetry in aiming: The horizontal magnitude of the hold box is larger than the vertical dimension. From release to when the arrow leaves the string, the right handed archers tend to move to the left and the left handed archers tend to move right.

Dynamics analysis on reflexed bow

Jihe Zhou et al,      23 International Symposium on Biomechanics in Sports (2005)

Reflexed bow and composite bow competition are two main categories in shooting game. Nowadays, only reflex bow competition is in Olympics game. Therefore, we choose the reflex bow as our test object. According to the literature we collected. Up to now, There is no experiment carried out on dynamics analysis on reflex bow .so we make some efforts on this field. We attempted to have a further understand on it and hope to make some useful suggestion on archery athletics, help them ,to understand bow's feature and improve their results.


Zanevsky I.P.

The aim of the research is to develop a method of mathematical modelling and computer simulation of static and dynamic interaction in the archer, bow and arrow system intending to get practical recommendations for the sport of archery.
Methods: of biomechanics (human body model as a system of mechanical oscillators); theoretical mechanics and mathematics (Lagrange equations of the second kind; the principle of d’Alembert; Cauchy problem, method of iterations and Runge-Kutta method); computing (MathCAD and Mathematica packets); experimental (high speed video analysis; nature mechanical modelling) [14-18].

Compound Bow Asymmetry in the Vertical Plane

Ihor Zanevskyy,  Sports Engineering 2012

The aim of the research was to create a mechanical and mathematical model of a compound bow and analyse its work without any prior hypothesis about the
symmetry of limb deflection. The method of the research was based on the methods of theoretical mechanics, applied and computer mathematics.

The Analysis of" S-Curve Motion" of Arrow in Archery Sports

W. Hui et al Proc. IMAC X1 24/06/2011

In this paper, the multl-rigld body spring hinge system model of an arrow arm is set up, based on which its dynamical buckling characteristic equation and post-buckling
equilibrium equation, when it is sent out, are deduced. Also, its nonlinear dynamic response in motion is analyzed

Physical Laws of Archery

Thomas L. Liston

Book 1989

Fundamentals of the Design of Olympic Recurve Bows

Lieu d.k. et al

Modern materials and fabrication methods offer new opportunities to redesign competition recurve bows. Through improved bow geometry and proper construction methods, designs can be created which propel arrows with greater energy and efficiency, smoothness on the draw, and stability than before. This paper outlines the physics of bow behavior, and how desirable performance characteristics can be quantified. Also examined is how changing the bow geometry, new materials, and construction techniques can lead to improve bow performance. Recommendations are forwarded on how target bows can be redesigned for better performance in the future.

An Experimental Study on the Influence of Compound Bow Draw Cycle Characteristics on Vibratory Response

Chris G. Meyer et al;  Department of Mechanical Engineering, University of Kentucky

The purpose of this study is to compare the vibratory response characteristics of a compound bow as the setup parameters are varied. Changes in response of the bow are explored as the draw length, peak poundage during draw cycle, and arrow mass to peak draw force ratio are adjusted. Several experiments were performed in an effort to determine the combination of parameters that yields the most favorable response characteristics at various accelerometer locations for a given bow design. Natural frequencies were determined through impact tests. Then,
shot tests were performed. The peak accelerations at five different locations were determined for various setups. Also, the Fast Fourier Transforms of the response data for the tests were calculated. Since changing draw length and peak poundage alters the power stroke characteristics of a bow, the spectra of the responses were normalized by the total energy available.

Ballistic Pendulum for velocity measurement - Patent Application

P. Klopsteg 1925

Optimizing Flexible Behaviour of Bow Prototype Using Taguchi Approach

J. Laeng et al.  J. Applied Sciences 6(3) 2006

This study presents the application of the Taguchi method, a powerful tool to process optimization for quality, to find the optimal process parameters for Fused-deposition Modeling (FDM) rapid prototyping machine that was used to produce flexible Acrylonitrile-Butadiene-Styrene (ABS) bow and arrow prototype. In order to achieve optimum performance of the bow so as to obtain the maximum throwing distance of the arrow from it, Orthogonal Array (OA), main effect analysis, signal-to-noise (S/N) ratio and analysis of variance (ANOVA) are employed to investigate the process parameters. Through this study, not only can the optimal FDM parameter combinations to be obtained, but also the main process parameters that affect the performance of the prototype can be found. Experiments were carried out to confirm the effectiveness of this approach. From the results, it is found that FDM parameters, especially air gap, slice height and raster angle have the most significant impact on the elastic performance of the flexible ABS prototype. The optimum levels of parameters at different angle of displacement of the bow are also presented.

Arrow-mounted Ballistic System for Measuring Performance of Arrows Equipped with Hunting Broadheads

John Barton et al;  9th Conference of the International Sports Engineering Association

Measuring an arrow’s ballistic performance is challenging because of the dynamic nature of arrow flight. For example, it is difficult to precisely measure parameters such as arrow velocity on target-impact, total time of flight and arrow shaft oscillation. This challenge becomes increasingly difficult as the distance of the shot increases. It is also of great interest to bowhunters to understand the ballistic performance of arrows that include hunting broadheads. A novel arrow-mounted ballistic sensing system, which is engineered to withstand the high accelerations experienced at launch and impact, enables the precise measurement of arrow ballistics in flight. By continuously recording arrow drag in flight, the system enables measurement of the ballistic performance of an arrow as it travels downrange. The authors have also built an adapter that is connected to the housing of the sensing system to allow for comparative ballistic tests to be performed on hunting broadheads. Here, we present results obtained using the sensing system to perform initial testing on two commercially available broadheads at shot distances of approximately 45.72m.

Compound Archery Bow Asymmetry in the Vertical Plane

Ihor Zanevskyy, Sports Engineering 15(1) March 2012

The aim of the research was to create a mechanical and mathematical model of a compound bow and analyse its work without any prior hypothesis about the symmetry of limb deflection. The method of the research was based on the methods of theoretical mechanics, applied and computer  mathematics. A pair and shift mechanism with cams and levels was used as a basis of the mechanical and mathematical model of a compound bow. The difference between results of modelling and measurement was near 7 %, while the error due to digital image deformation was evaluated to be near 1 % which was considered as acceptable.

A Miniaturised Arrow Ballistic Measurement System

        John Barton et al, Full Flight Technology

A novel miniaturised system for measurement of the in-flight characteristics of an arrow is introduced in this paper.
The system allows the user to measure in-flight parameters such as the arrow’s speed, kinetic energy and momentum, arrow drag
and vibrations of the arrow shaft. The system consists of electronics, namely a three axis accelerometer, shock switch,
microcontroller and EEPROM memory embedded in the arrow tip. The system also includes a docking station for download and
processing of in-flight ballistic data from the tip to provide the measured values. With this system, a user can evaluate and
optimize their archery equipment setup based on measured ballistic values. Recent test results taken at NIST show the
accuracy of the launch velocities to be within +/- 0.59%, when compared with NIST’s most accurate ballistic chronograph.

Archer, Bow, Arrow Behaviour in the Vertical Plane

I.P. Zanevsky, Acta of Bioengineering and Biomechanics Vol. 8, No. 1, 2006

Theoretical and experimental results of research on the problem of archer, bow and arrow behaviour in the vertical plane are presented. The aim of the research is to develop a method of computer simulation of static and dynamic interactions between the archer, bow and arrow system in order to provide archery with practical recommendations. A model of an archer’s body is presented as a mechanical system composed of a few solid bodies, which are connected to each other and to the ground with viscoelastic elements. Mechanical and mathematical model of bow and arrow geometry in vertical plane in braced and drawn situations is investigated.

Hunting Bow Accuracy Killers

        Larry Wise

Bow and Catapult Internal Dynamics

Mark Denny, Eur. J. Phys. 24 (2003) 367–378

A simple model of bow and arrow dynamics is presented, which makes clear the physical principles, and reproduces the features obtained via more detailed,
but less accessible calculations. We apply this instructive model to determine the efficiency of bows and of torsion-spring catapults.

Bow and Arrow Efficiency

Richard Baugh

An Approach to the Study of Ancient Archery using Mathematical Modelling

B.W. Kooi and C.A. Bergman, Antiquity 71:(271) 124-134 (1997)

The archer’s bow is a machine whose purpose is to impart the stored energy effectively and accurately to propel the arrow. A Mathematical modelling of different bow types shows how their engineering characteristics define their performances.

On the Mechanics of some Replica Bows

Bob Kooi, Journal of the Society of Archer-Antiquaries 36:14-18 (1993)

In this paper different approaches to the study of ancient bows are compared. One way of studying ancient bows is to make replica bows for experiments, see Bergman, McEwen & Miller 1–3  and Alrune 4 . Another way is to make mathematical models. Such models enable us to compare the performance of bows used in the past and in our time

On the static deformation of the bow

B.W. Kooi, J.A. Sparenberg, On the static deformation of a bow Journal of Engineering Mathematics 14(1):27-45 (1980)

The storage of deformation energy in a bow with or without recurve is considered. Some numerical examples are discussed. For a simple bow it is shown that theoretically a shooting efficiency of hundred percent is possible.

Bow and Arrow Dynamics

W. C. Marlow, Am J. Phys. 49(4) April 1981 pp 320-333

Past analyses of bow and arrow dynamics have assumed the string to be inextensible. This results in predictions of efficiencies that are  significantly higher than measured values (efficiencies over 90% are predicted versus 70% to 85% for measurements, circa 1960). The present analysis allows for an elastic string. It is found that arrow exit then takes place when the string and bow limbs  still have substantial kinetic energy, and therefore this energy is unavailable for kinetic energy of the arrow.

Archer's Compound Bow-smart use of Nonlinearity

Randall D. Peters

On the mechanics of the bow and arrow

Bob Kooi

The Dynamics of a Bow and Arrow

C. N. Hickman

Short Bows and Long bows: Scaling Effects in Archery

Ugo Bardi

Reverse Engineering the Bow: A Simple Dynamic Model

Ugo Bardi

Reverse Engineering the Bow: A Simple Static Model

Ugo Bardi


A study of projectile motion in a quadratic resistant medium via fractional differential transform method

A.M.A. El-Sayed et al, Applied Mathematical Modelling 30 October 2014

A new approach for modelling the equations that govern the motion of a projectile in a uniform gravitational field is presented. In this study, the fractional derivatives and quadratic-drag term are introduced to make the model more realistic. The formulated mathematical model is non-linear coupled fractional-order system where fractional derivatives are in Caputo sense. These equations are analytically solved by using the recently modified fractional differential transform method. The results obtained from a numerical case study are compared to those of the corresponding integer-order model.

Optimization of design parameters for Turkish Tirkes (war) bow

Sermet Demir et al, Composites Part B: Engineering MAy 2014

The bow and arrow is a projectile weapon system that predates recorded history and is common to most cultures. The Turkish bow is the most efficient one in its category. Despite of its superiority, Turkish bow is the one which is least documented in the literature. Technical drawings for the Turkish bow are missing. Turkish bow is a system consisting of different elements. Each element has its own distinctive feature and serves for a specific purpose. Recent interest in Turkish bow simply involves the replication of museum samples without any consideration about the performance characteristics of the replica. The present work aims at describing the Turkish bow, war bow known as Tirkes in specific. Characteristic shape parameters will be identified and the effect of each parameter on bow performance will be discussed. Parametric optimization to maximize bow efficiency will then be introduced. The bow shape will first be described. Characteristic shape parameters defining the bow geometry will be identified and the range in which they vary will be determined. The bow is drawn in the ANSYS environment. Based on the design drawing a model bow is manufactured. Due to its superior flexing characteristics, E-glass fiber epoxy system is used in the composite structure. The model bow is tested to determine the characteristic draw weight - draw distance behavior of a typical bow. A mathematical model which is a simplified analysis of recurved bow types is used to compare behavior of model and manufactured bow draw weight , draw distance graph.

An Improved archery simulator for objective dynamic tests of bows & arrows

R. Pekalski, Proceedings of the Fifth Meeting of the European Society of Biomechanics, September 8–10, 1986, pp639-644

Presented study deals with the archery shooting technique. The mechanical model of a competitor-bow-arrow system was developed. It can be used to determine the connections between the parameters of this system and the accuracy of hitting the target. It is shown that these connections may be useful in the evaluation and improvement of the equipment as well as the shooting technique.

An archer's perceived form scales the “hitableness” of archery targets.

Lee, Yang; Lee, Sih et al, Journal of Experimental Psychology: Human Perception and Performance, Vol 38(5), Oct 2012, 1125-1131

For skills that involve hitting a target, subsequent judgments of target size correlate with prior success in hitting that target. We used an archery context to examine the judgment–success relationship with varied target sizes in the absence of explicit knowledge of results. Competitive archers shot at targets 50 m away that varied in size among five diameters. Immediately after the arrow's release, its flight and landing were occluded and archers chose which of 18 miniature targets looked most like the distal target. Greater apparent size correlated with higher accuracy. In a second experiment, nonarchers merely aimed the bow (without an arrow) at varied targets. Apparent size was larger when the bow arm was stabilized than when it was not. Archery is seemingly an instance of affordance-based control: For an archer, the affordance of the target is the “hitableness” of its central regions, a property inclusive of his or her momentary, and perceptible, archery form.

J. Barton et Al, Conf. Proc.  IEEE  Sensors 2011 pp1289 - 1292

A novel miniaturised system for measurement of the in-flight characteristics of an arrow is introduced in this paper. The system allows the user to measure in-flight parameters such as the arrow's speed, kinetic energy and momentum, arrow drag and vibrations of the arrow shaft. The system consists of electronics, namely a three axis accelerometer, shock switch, microcontroller and EEPROM memory embedded in the arrow tip. The system also includes a docking station for download and processing of in-flight ballistic data from the tip to provide the measured values.

A review of recent research into aerodynamics of sport projectiles

John Eric Goff, Sports Engineering April 2013

A review of aerodynamics research connected to sport projectiles is presented here. The review’s focus is on work conducted in the current millennium, though deference is made to some classic work still invaluable to modern research. Besides serving as a resource for seasoned scientists and engineers, this article is especially geared toward young investigators who are just beginning careers in sport science. Basic and sophisticated methods are discussed, including vacuum physics, air drag, lift, numerical approaches, trajectory analysis, wind tunnels, and computational fluid dynamics. Eighteen sports are discussed with an eye to future research.

bow Arrow dynamic behaviour

James L Park, Archery Forum 2012

Downloadable software and discussion thread for arrow power stroke behaviour simulation

High Speed Video Analysis
of Arrow Behaviour during the power stroke of a recurve archery bow

James L Park & Oliver Logan, J. of Sports Engineering and Technology, June 1, 2012

The manner in which an archer holds and releases the string of a recurve bow causes the arrow to flex during the bow’s power stroke and on its way from the bow to the target. While an experienced observer can subjectively evaluate an arrow’s pitch and yaw visually after it has left the bow, the arrow’s behaviour during the bow’s power stroke happens too quickly for visual observation. High-speed video is consequently a much more effective tool. Similarly, high-speed video is an effective tool for studying an archer’s technique during the bow’s power stroke.

Arrow behaviour in the lateral plane during and immediately following the power stroke of a recurve archery bow

J. L. Park , J. Sport & Eng. Tech. Nov. 2012

An arrow exiting a recurve archery bow flexes laterally. This is important in ensuring that the rear of the arrow clears the bow without obstruction and is a consequence of the manner by which the string leaves the archer’s fingers. Pekalski and then Kooi and Sparenberg modelled the arrow behaviour during the bow’s power stroke using a flexible beam. In this article, the method used by Kooi and Sparenberg has been extended to remove subjective components, to include the carbon fibre composite arrows now used almost exclusively in competition and to consider the arrow after it has left the bow. The various user-selectable parameters have been examined to ascertain their impact on archer performance. The modelling was tested against measured parameters of a number of expert archers and against the use of the recurve bow in a shooting machine.

Synthesis of a torsional spring mechanism with mechanically adjustable stiffness using wrapping cams

Mehmet Kilic et al;  Mechanism & Machine Theory 57 (212) Nov 2012 pp 27-39

Most adjustable springs that have appeared in the literature are based on the antagonistic actuation of two quadratic springs. When two quadratic springs are coupled antagonistically, the resulting system attains a linear spring behavior enabling the adjustment of the stiffness mechanically. However, a non-linear spring with perfect quadratic behavior is difficult to realize. In this study, the synthesis of a non-linear spring using mechanisms is investigated where the problem is reduced to a function generation problem. A wrapping cam mechanism is proposed as an effective mechanism to be used in non-linear spring synthesis. An analytical solution is developed to calculate the necessary cam profile.

Evaluation of arrow release in highly skilled archers using an acoustic measurement system

Mario Heller, Procedia Engineering v34  2012 pp532-537

During aiming, an archer holds the stretched bow while a small piece of metal - the clicker - presses the arrow lateral against the bow. At the end of the aiming phase, the archer pulls the arrow back until the clicker slips over the arrowhead and causes a click, and then the archer shoots. To investigate the precision of the timing of this motor program, an acoustic measurement system has been developed. The system is composed of a modular microphone (AKG C 480 B comb-ULS/61), and an external FireWire Audio Interface (Focusrite Saffire LE 24Bit / 96 kHz). From the audio signal, the time of the clicker's fall, the release of the shot and the hit of the target can be detected. Seven elite archers (two males, five female) performed 30 shots (ten ends of three arrows without any time limit) at 18 m indoors. Means and coefficients of variation (CV) were calculated from average speed and clicker time, i.e. the time between the click and the release of the shot. Forward stepwise multiple linear regression analysis was done to determine mean scores. CV of the clicker time accounted for 66% of the variability in scoring. A “leave-one-out” cross validation procedure revealed consistent estimate of the model (all corrected R2 were in a range of 0.66 and 0.75 with p<0.03). It has been shown that a highly precise timing in arrow release in terms of small CVs of clicker time is important for high mean scoring.

Their Arrows will Darken the Sun

Book - The evolution and science of ballistics Mark Deny

The behaviour of an arrow shot from a compound archery bow

This paper provides the results of an analysis of an arrow leaving a compound bow in both the vertical and the horizontal planes. Techniques similar to those used by Kooi for a recurve bow were used, together with photographs obtained using a triggered flash light. The paths taken by the bow's nocking point in both the vertical and the horizontal planes had a substantial influence on the arrow behaviour. In the vertical plane this was both measured and modelled for one bow type and for a second bow type it was modelled. In the horizontal plane it was measured and modelled for one bow type but with the locus varied by twisting the bow laterally.

A whole mechanical and mathematical model of an arrow–bow motion system, which accounts for arrow deflection in the lateral plane, has been created. The arrow is modelled as a shaft within the framework of the technical theory of vibration of beams with particles at the ends. The model takes into consideration the mechanical properties of a string, bow limbs and a grip as an oscillator of concentrated elastic and inertial elements connected with the feathered end of the arrow. Experimental methods for measuring arrow stiffness in sport archery practice are discussed with reference to the dynamic stability of the internal ballistics.

Modelling of the archery bow and arrow vibrations

Vibration processes in the compound and open kinematical chain with an external link, as a model of an archery bow and arrow system, are evaluated. A mechanical and mathematical model of bend oscillations of the system during accelerate motion of the external link is proposed.

Experimental and Theoretical Research in Archery

Roman Pekalski,  Journal of Sports Sciences 8 (3) 1990

The aim of this study was to introduce certain methods and research techniques and to present the results of experiments on parameters of archery equipment to optimize the interaction of the archer‐bow‐arrow system's elements. In order to achieve the research requirements, the following were devised:

A mathematical model of an arrow's movement during its interaction with a bow.

A mechanical model of an archer‐bow‐arrow system, which constitutes a device for mechanical loosing of an arrow from the bow strings (DMLA).

Three series of high‐speed filming.

Bow-Arrow interaction in Archery

 ...I present a more accurate model and focus on the biomechanical aspects and the use of the knowledge gained in the sport of archery; the  mathematical aspects are presented elsewhere.

Archery simulation

The problem of bow shooting is considered. Its solution is obtained in the quasistatic approximation when it is assumed that, at the successive   instants of time, the limbs of a bow reach near-equilibrium states. Some dependencies of the initial arrow velocity on all basic physical parameters of the bow are obtained.

Mechanics of the Bow and Arrow

Collection of technical articles on bow and arrow mechanics.  Author: Bob Kooi

Topics on Archery Mechanics

Guide to the bow/arrow system and arrow flight.  Author: Joe Tapley


General Technical stuff on interior, exterior ballistics (from Italian)

Bow and catapult internal dynamics

A simple model of bow and arrow dynamics is presented, which makes clear the physical principles, and reproduces the features obtained via more detailed, but less accessible calculations. We apply this instructive model to determine the efficiency of bows and of torsion-spring catapults.