Recoil And Vibration In An Archery Bow Equipped With A .
Shock and Vibration 19 (2012) 235–250DOI 10.3233/SAV-2011-0626IOS Press235Recoil and vibration in an archery bowequipped with a multi-rod stabilizerIgor Zaniewski Casimir Pulaski Technical University, Radom, PolandReceived 14 April 2010Revised 13 December 2010Abstract. The aim of this research is to create a mechanical and mathematical model of a multi-rod stabilizer for the sport archerybow and to analyze its capability to damp disagreeable recoil and vibration of the bow during internal ballistic motion. Theresearch methods are based on the Euler-Bernoulli theory of beam bending, Lagrange equations of the second kind, the Cauchyproblem, and the Runge-Kutta method. A mathematical software package was used to analyze the problem. The approach to theproblem of sport-bow stabilization in the vertical plane that is proposed in this paper addresses the practical needs both of appliedengineering mechanics and of the sport of archery. Numerical results from computer simulation are presented in both tabularand graphical form. The common motion of the string and arrow (internal ballistic motion) is accompanied by intense vibrationwhich is caused by disruption of the static force balance at the moment of string release.Keywords: Archery, bow, vibration, recoil, stabilizer, modelingNomenclaturec U , cLccfghU , hLIU , ILIHIIVkllU , lLlalmU , mLMmAmsmastiffness of the limbsstiffness of the rod relative to a normal force at a free endstiffness parameter of the stringacceleration of gravityvirtual lengths of the risermoments of inertia of the limbs relative to their junction with the riser, with the addition of anequal portion of the string massmoment of inertia of the riser relative to the pivot pointmoment of mass inertia of the bow relative to the hinge axismoment of inertia of the bars relative to their center of masseigenvalue of the systemlength of the limbslength of the arrowlength of the rodmass of the limbs with the added mass of the string attached to their tipsmass of a load concentrated at a free endmass of the string attached to the nock pointmass of the stringmass of the arrow Address for correspondence: Malczewskiego 22, Radom 26-600, Poland. Tel.: 48 48 3618809; Fax: 48 48 3617803; E-mail: izanevsky@onet.eu.ISSN 1070-9622/12/ 27.50 2012 – IOS Press and the authors. All rights reserved
236I. Zaniewski / Recoil and vibration in an archery bow equipped with a multi-rod stabilizermVPqrU, rLs U , sLSU , SLTt“U” “L”xV , yVzαq , ακαβχεηικμρψωmass of the stabilizer barspotential energy of a bow-and-rod systemlinear displacement of the free end of a rod in bendingdistances of the mass centers of the limbs from their junction pointslengths of the string branches in the drawn-bow situationlengths of the branches of the free stringkinetic energy of the bow-and-rod systemTimesubscripts for the upper and lower limbscoordinates of the mass centers of the stabilizer barslongitudinal coordinate of a rodindependent constantsinclination angle of a side rodseparation angle between side rodshypothetical bending functiondistributed stiffnesslinear displacement of a roddimensionless value of the moment of inertia of the bowangular displacement of the bow riserdimensionless value of a mass located at the free end of a roddistributed mass of a rodorientation angle of the arrownatural circular frequencies1. IntroductionThe sharp recoil of the handle after the bowstring has been released and the vibration of the bow after an arrowhas been launched from its nock point have a negative effect on results in the sport of archery. Intensive dynamicprocesses during the joint movement of an arrow and a string result in an unstable extension of the archer’s handto the handle of the bow. This causes a significant dispersion of arrows. Furthermore, the disagreeable sensationsof recoil and vibration result in a conditioned counter reflex in the archer just before string release. This causes adisruption of accurate aiming at a target exactly at the moment when an arrow’s direction of flight is determined [3].To reduce this recoil and vibration, modern archery bows are equipped with a special device called a stabilizer.The stabilizer is a system of long and short rods and weights which are mounted on the bow handle [2]. A stabilizerhelps the archer to improve the stability of the bow during aiming and during the joint movement of the arrow andstring. A stabilizer accumulates and dissipates a part of the kinetic energy involved, reducing the recoil and vibrationof the handle.Commonly, a modern sport-bow stabilizer includes up to five rods mounted on a handle inside the bow [1]. Acentral rod and two side rods are fastened to an adjustable uni-bar (Appendix A). The central rod is always fastenedat the same position perpendicular to the handle. The side rods have an opposite axial direction relative to the centralrod and have adjustable space joints. These joints include strong teeth that provide multiple points of adjustmentand ensure that the setup does not slip. The uni-bar is fastened to the handle using an extender bar with the sameaxis as the central rod. The whole system of bars and rods has the main vertical bow plane as its plane of symmetry.Usually, there are one or two additional rods in the stabilizer system. These upper and lower rods are fastened tothe handle like cantilever beams in the main plane of the bow and are symmetric relative to the center of the handle(Fig. 1).The setting and tuning process to adjust the stabilizer parameters to the archer’s anthropometric parameters andto his/her style of shooting is based on “trial and error” experimental methods. Because this method requires toomuch time and effort by archers, a modern training process needs more effective methods and technologies. Thedevelopment of scientifically based methods to improve stabilizer parameters requires the creation of a mechanical
I. Zaniewski / Recoil and vibration in an archery bow equipped with a multi-rod stabilizer237Fig. 1. A modern sport bow in its main plane [6]: 1 – tips; 2, 13 – limbs; 3, 12 – limb-angle setup mechanisms; 4 – upper stabilizer, 5 – sign; 6 –arrow rest; 7 – arrow; 8 – handle; 9 – central stabilizer; 10 – side stabilizers; 11 – lower stabilizer; 14, 16 – string branches; 15 – nock point.and mathematical model of the archery bow stabilizer. This research problem has a substantial theoretical andexperimental basis [5–11].Recoil and vibration in an archery bow equipped with a central-rod stabilizer were studied using mechanical andmathematical modeling methods [5,6]. On this basis, the construction of an advanced three-rod stabilizer was alsostudied [10]. Because modern sport-bow stabilizers include more than three rods and are equipped with concentratedloads at the free ends of the rods, these models must be improved. The aim of this research is to create a mechanicaland mathematical model of a multi-rod stabilizer with concentrated loads at the free ends of the rods and to analyzetheir capability to damp disagreeable recoil and vibration of a bow during internal ballistic motion.2. Basic model of a bow-rod stabilizerEach stabilizer rod is modeled as an elastic cantilever beam in the context of Euler-Bernoulli theory because itslength is much greater than its cross-sectional diameter. Sport archers stretch a bow with a joint motion of the stringand the arrow while trying to maintain a steady body posture. The archer’s body mass is significantly greater thanthat of the bow. Therefore, the point of contact (O) can be assumed to be an immovable pivot point (Fig. 2a). Thedisplacements of points on the handle and stabilizer due to rotation and bending are much less than the length of therods; therefore, a linear model can be used:
238I. Zaniewski / Recoil and vibration in an archery bow equipped with a multi-rod stabilizerFig. 2. (a) Bow-and-rod schematic model; (b) cantilever beam loaded at the free end with a concentrated force F .(1)η κz qχ,where η is the linear displacement of the rod; z is the longitudinal coordinate of the rod; κ is the angular displacementof the bow riser (relative to the hinge, p. O); q is the linear displacement of the free end of the rod in bending; and χis a hypothetical bending function. It is assumed that κ and q are functions of time, but that χ is a function of z.It is further assumed that the deformation of the center line during bending of a cylindrical cantilever beam locallyloaded at a free end by a force normal to the longitudinal axis (Fig. 2b) can be represented by a hypothetical bendingfunction:z 1 z 2 3 ,(2)χ 2 llwhere l is the length of the rod. This function satisfies three of four boundary conditions, i.e., zero displacement,zero angle with a normal to the handle at the fixed end, and one of the two dynamic boundary conditions, i.e., zeroforce moment at the free end.According to the model, the kinetic (T ) and potential (P ) energy of a bow-and-rod system can be written as: l 2 2 2 2 l 2 21 1 η η η η T and P ρdz I Mεdz,(3)2 t z t z 0 t z l2 z 200where ρ is the distributed mass of the rod; I is the moment of mass inertia of a bow relative to the hinge axis; M isthe mass of a load concentrated at a free end (see Fig. 2a); ε is distributed stiffness; and t is time.The model was then evaluated for free vibration of the system. Using the Rayleigh-Ritz method, equations forfree vibration of a system with a cylindrical rod (ρ const and ε const) can be obtained as: 1133m M q̈ m M lκ̈ cq 0;14040(4) 111m M l2 κ̈ m M lq̈ 0,I 340where m is the mass of the rod and c 3εl3 is the stiffness of a cantilever beam with a concentrated force at the freeend (see Fig. 2b). Solution of the equations for free vibration yields:q αq sin ωt;κ ακ sin ωt,(5)where αq , ακ are independent constants and ω represents the natural circular frequency of the system. Substitutingthe solutions of Eq. (5) into Eq. (4) yields a system of two linear algebraic equations relative to the constants: 1133m M ω 2 αq m M ω 2 lακ 0;c 14040 11122m M l ω ακ m M lω 2 αq 0.I 340
I. Zaniewski / Recoil and vibration in an archery bow equipped with a multi-rod stabilizer239Fig. 3. Main eigenvalue of a bow and stabilizer modeled as a cantilever-rod system.The two values of ω for which the determinant of this system of equations is equal to zero are the natural frequenciesof the system. One of these is a zero natural frequency that is related to the circular rotation of the bow and rodaround the hinge (p. O). The second value is nonzero and represents coupled bending and rotational vibration. Thefourth power of this natural frequency (an eigenvalue) can be represented in a nondimensional form as: 11/40 μ)2(4,(6)(kl) 3/ 33/140 μ ι 1/3 μ 3 2where kl 4 mlε ω is the eigenvalue of the system; ι mlI 2 is the dimensionless value of the moment of inertiaof the bow; and μ Mm is the dimensionless value of a mass located at the free end of a rod. Results for themain nonzero eigenvalue for various relationships of mass-inertial parameters are shown in Fig. 3. As expected, thegreater the mass at the free end or the moment of mass inertia at the hinge, the lower was the natural frequency.To verify the stabilizer-rod model based on the hypothetical Eq. (2), a boundary problem using differentialequations and boundary conditions was investigated. Because no results have been reported in the open mechanicaland mathematical literature for a beam with one end attached to a load and the other end free, this problem ispresented here in detail. t2Substituting the two expressions for energy in Eq. (3) into a Hamilton functional, δ (T P ) dt 0, a standardprocedure yields the corresponding differential equation: 2 2η 2 ηρ 2 2 ε 2 0 t z zt1and boundary conditions 2η 3η I;2 z z t2 2 2η 2η ηz l, 2 0;ε 2 M 2. z z z tz 0, η 0; εSolutions in the form of eigenvalues (kl) were obtained using Krylov functions for a cylindrical rod (ρ const andε const). They are the roots of the determinant: 4 ι (kl)4 2klι (kl) sh (kl) ch(kl) cos(kl) sin(kl) (7) μklsh (kl) μkl [ch (kl) cos (kl)] μkl sin (kl) 0. ch (kl) sh (kl) sin (kl) cos (kl)
240I. Zaniewski / Recoil and vibration in an archery bow equipped with a multi-rod stabilizerTable 1Relative error (%) of the main eigenvalue calculated usingthe Rayleigh-Ritz method for solution of a boundary problemι\μ0110 112.200.010.020.03The zero solution of Eq. (7) corresponds with a common rotation of the beam and the load at a fixed end relativeto the hinge axis. When ι 0, μ 0, the result is kl 0; 3,927; 7,069; 10,210 . . . , which are the same as the, where i 2, 3, 4 . . . are the numbers of the naturalwell-known solutions for the beam with one hinged end, π(4i 3)4frequencies. There is no zero solution when μ 0, ι : kl 1,875; 4,694; 7,855 . . . , which are the same as the(i 3, 4, 5 . . .).well-known solutions for a cantilever beam, π(2i 1)2For real ratios of the rod and load masses, (ι 10; μ 1) using the Rayleigh-Ritz method, a very precise estimate(for engineering purposes) of the first natural frequency can be obtained; the relative error of the main eigenvalue isapproximately one percent (Table 1). It is interesting to note that the precision of the method increases with the loadmass at the free end of the rod. For example, the relative errors for a simple cantilever rod are 0.72–0.73%, but in thecase of a cantilever rod with a concentrated load of the same mass (equal to the rod mass) at the free end, the relativeerror is 0.02–0.03%, i.e., the error decreases more than exponentially with increasing mass. This phenomenon canbe explained by the fact that a hypothetical vibration mode presented in a form of a static cantilever rod loaded at onefree end is similar to the real mode. The difference between a cantilever rod with a load at one free end and withoutthe load is in the order of the distribution of inertial forces along the rod. The larger the load, the more similar to astatic problem is the distribution of inertial forces.The hypothetical function given in Eq. (2) satisfies three of four boundary conditions. One of two dynamicboundary condition is not satisfied exactly, i.e., a cross-sectional force at the free end. Despite this, the functionprovides increased precision for the main natural frequency. For example, in the case of a typical sport archerybow with a single-rod stabilizer (ι 10; μ 0.2), the relative error is approximately 0.21% (see Table 1) for an2eigenvalue and approximately 0.42% for a natural frequency; because ω (kl) , the relative error δω 2δ (kl).3. Central-rod modelMovement of the central bar occurs in the vertical plane of symmetry of the bow. Consider a bow with a stabilizerrelative to an immovable Cartesian coordinate system, ξOη, based on the vertical plane of symmetry of the bow(Fig. 4). In addition, in the same plane, consider a movable Cartesian coordinate system, xOy, that is fixed to theriser; the y-axis is parallel to the central-rod axis, but oriented in the opposite direction; the x-axis is oriented upward.Now consider a displacement of points along the longitudinal axis as a sum of two components: along the axis, ξc ,and normal to it, ηc . The longitudinal displacement is the same for all the points of the rod because they dependon the handle angle and the distance of the pivot point (O) from the rod axis. The normal displacement consists oftwo components: a rotational displacement relative to the pivot point (similar to a longitudinal displacement) and abending displacement:ξc xV κ;ηc (zc yV ) κ qc χ,(8)where xV , yV are the coordinates of a common point of projection to the plane of symmetry of the axes of all threerods; zc is the longitudinal coordinate of the rod; qc is the displacement (as a function of time) of the free end of therod caused by bending. According to the results of a previous study of the dynamics of a single-offset stabilizer [10],a form of the bending calculation for a horizontal cantilever beam with a concentrated load at the free end can beused Eq. (2).The kinetic energy of the central rod is: lc 1 ,(9)Tc ρc ξ̇c2 η̇c2 dzc Mc ξ c2 η̇c22zc lc0
I. Zaniewski / Recoil and vibration in an archery bow equipped with a multi-rod stabilizer241Fig. 4. Schematic model of a multi-rod stabilizer: (a) in the arrow plane and (b).in the transverse plane.where lc is the length of the central rod; ρc is the distributed mass of the rod; Mc is the mass of a load concentrated at is the sign of the partial derivative with respect to time. Substituting Eqs (8)the free end of the rod; and (·) tand (2) into Eq. (9), and after integration and intermediate transformations: 1 225 J6 ) q̇c 2 4 (9J2 4 6J11xκ̇2 (l y)cVVTc mc J2 lc x2V yV2 2yV J1 lc κ̇2 Mc,(10)22 2 (lc yV ) κ̇q̇c q̇c2[(3J3 J4 ) lc (3J2 J3 ) yV ] q̇c κ̇where mc lcJi 0 lc0ρc dzc is the mass of the rod andρc zci dzcmc lci(Appendix 2). For a cylindrical rod, Ji 1i 1 .4. Side-rod modelThe position of the side rods relative to the bow handle is determined by two angles. The first, the inclinationangle (α), is measured between the plane of the side rod and the plane of the central rod (Fig. 4). The second, theseparation angle (β), is measured between the axis of the rod and the plane of symmetry of the bow. In the schematicdiagram, the angle α is presented for the case of parallel side rods (β 0). The angular displacement of the handleand the uni-bar (κ) is measured counterclockwise from the axis Oη.The bending displacements of the side rod can be divided into two components. The first (ν) is a componentnormal to the rod axis as an upward displacement in the vertical plane when the side rods lie in the same plane asthe central rod (α 0) or outside the bow when the side rods are inclined (α 0). The second component (τ ) is adisplacement normal to the vertical plane of the rod inside the bow. Coordinates zd1 and zd2 are directed along thelongitudinal axes of the side rods.The spatial geometry of the side rods causes certain spatial characteristics of their bending. Therefore, theirdisplacement must be considered relative to a three-dimensional Cartesian coordinate system, Oξηζ. Because the
242I. Zaniewski / Recoil and vibration in an archery bow equipped with a multi-rod stabilizerside rods are located symmetrically relative to the vertical plane, their displacements are also symmetric. Here theyare presented as a sum of two components similar to those of the displacements of a central rod:ξd (xV zd sin α) κ (qυ sin α cos β qτ sin β) χ;ηd (yV zd cos α cos β) κ qυ χ cos α;ζd (qυ sin α sin β qτ cos β) χ,(11)where qν , qτ are the bending displacements of the free end in the ν- and τ -directions respectively and χ is a bendingform such as Eq. (2). In Eq. (11) for the lateral displacement component, (ζd ), the plus sign in the double sign ‘ ’is associated with the left-side rod labeled as 1 (Fig. 4), and the minus sign with the right-side rod labeled as 2.The kinetic energy of the two side rods together is: ldTd ρd ξ d2 η̇d2 ζ̇d2 dzd ,(12)0where ld is the length of the side rod and ρd is the distributed mass of each of the rods.Substituting Eqs (11) and (2) into Eq. (12), after integration and intermediate transformations: 1 (9J4 6J5 J6 ) q̇υ2 q̇τ2 4 2 xV yV2 2J1 ld (yV cos α cos β xV sin α) J2 ld2 sin2 α cos2 α cos2 β κ̇2T d md [(3J2 J3 ) (xV sin α cos β yV cos α) (3J3 J4 ) ld cos β] κ̇q̇υ [(3J2 J3 ) xV (3J3 J4 ) ld sin α] κ̇q̇τ sin β 22 (xV ld sin α) (yV ld cos α cos β) κ̇2 q̇ν2 q̇τ2 Md 2 [(xV ld sin α) sin α cos β (yV ld cos α cos β) cos α] κ̇q̇ν 2 [(xV ld sin α) sin β] κ̇q̇τwhere
IOS Press Recoil and vibration in an archery bow . of recoil and vibration result in a conditioned counter reflex in the archer just before string release. This causes a . Commonly, a modern sport-bow stabilizer includes up tofive rods mounted on a handle inside the bow [1]. A
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