Modelling the 3D vibration of composite archery arrows under free-free boundary

Modelling the 3D vibration of composite archery arrows under free-free boundary conditions Authors M Rieckmann School of Mechanical Engineering, The University of Adelaide, SA 5005, AUS. email: marianne@targetplot.com J L Park Department of Mechanical and Aerospace Engineering, Monash University, AUS J Codrington School of Mechanical Engineering, The University of Adelaide, AUS B Cazzolato School of Mechanical Engineering, The University of Adelaide, AUS Abstract Archery performance has been shown to be dependent on the resonance frequencies and operational deflection shape of the arrows. This vibrational behaviour is influenced by the design and material of the arrow and the presence of damage in the arrow structure. -1-

In recent years arrow design has progressed to use lightweight and stiff composite materials. This paper investigates the vibration of composite archery arrows through a finite difference model based on Euler-Bernoulli theory, and a three-dimensional finite element modal analysis. Results from the numerical simulations are compared to experimental measurements using a Polytec scanning laser Doppler vibrometer (PSV- 400). The experiments use an acoustically coupled vibration actuator to excite the composite arrow with free-free boundary conditions. Evaluation of the vibrational behaviour shows good agreement between the theoretical models and the experiments. Keywords finite difference modelling, finite element modelling, composite archery arrow, modal response, laser vibrometry 1 INTRODUCTION Archery is one of the oldest sporting activities undertaken by humans and has long been an area of great interest for engineers. The first documented investigation of the dynamics of arrow flight appears in a sketch by Leonardo da Vinci which shows that an arrow shot backward from a crossbow will rotate about its centre of gravity until pointing forward, and that the centre of gravity follows the same path as if the arrow had been launched normally [1]. In recent years the use of predictive modelling using a finite difference method has been applied to the bow and arrow system to better -2-

understand the dynamic performance of shooting an arrow [2, 3, 4, 5]. Finite Element (FE) modelling is a method that has been successfully applied to predict impact behaviour and the modal response of sports equipment, for example in soccer balls [6], golf clubs [7] and tennis rackets [8]. The FE modelling of a composite archery arrow as used in top level competition is a new area of research that builds on previous research conducted to identify the dominant modes of vibration that exist in a composite cylindrical structure [9]. The focus of the current paper is on the vibration of three different complete composite archery arrows with free-free boundary conditions analysed in three dimensions (3D) using the FE method, and compared to existing finite difference models. Understanding the mechanical system of the bow and arrow is one of the factors noted by Axford [10] that is required in archery performance. The dynamic behaviour of interest includes the two phases of the shot, that is the power stroke as force is imparted to the arrow by the bow, and the free flight of the arrow influenced by aerodynamic drag. It is inevitable that an arrow will flex both during the bow s power stroke and then also as it moves from the bow towards the target as documented by Zanevskyy [11]. Aside from the substantial forces from the string accelerating the arrow up to speed, the bow s string and nocking point (where the arrow is attached during the power stroke) moves both in the vertical and horizontal planes during the power stroke, applying substantial lateral forces to the arrow. Pekalski [2] and later Kooi and Sparenberg [3] -3-

modelled the arrow flexing during the power stroke of a recurve bow, solving the equations of motion using the implicit backward Euler and Crank-Nicholson methods. Park [4, 5], used the Kooi and Sparenberg technique, but with an explicit finite difference scheme, to model the behaviour of an arrow both during the power stroke of a compound bow and then the arrow s behaviour in free flight. The finite difference methods capture the two dimensional (2D) vibrations of the arrow in the plane of maximum arrow flex due to the lateral forces applied to the arrow nock. Novel use of FE modelling to capture the 3D vibrational behaviour of composite arrows broadens previous 2D finite difference modelling and can be used to investigate a wider range of applications. Three-dimensional modelling can analyse all modes of vibration including beam, bar and cylindrical modes of arrow shafts as detailed by Rieckmann et al. [9]. Vibrational behaviour in 3D can answer questions such as the influence of varying stiffness around the arrow shaft that occurs due to damage and analysis of 3D flight dynamics. Predicting the arrow resonance frequencies by the FE method can be used for arrow selection and optimisation as well as evaluating the structural health after an arrow is damaged. The operational deflection shape of many bending modes calculated by the FE method can be used to better define 3D flight dynamics. The selection of arrows and optimisation of archery equipment is directly applicable to the wider sporting community. Archery performance has been shown to be dependent -4-

on the resonance frequencies and operational deflection shape of the arrows. It is necessary to have the arrow flex in order for it to pass the support on the bow s handle section without the rear of the arrow making contact and thus disturbing the path of the arrow. The flexural rigidity of the arrow should allow one full cycle of oscillation by the time it leaves the bow as demonstrated by Klopsteg [12] using high-speed flash photography. This first complete cycle will include the power stroke as well as the start of free flight as the arrow detaches from the bowstring. Klopsteg [12] noted that an arrow which is too stiff oscillates at high frequency with little amplitude such that it does not clear the bow handle, while an arrow that is too compliant will oscillate slowly such that the rear of the arrow fails to move out of the way as it passes the bow handle. Following Klopsteg s work, Nagler and Rheingans [13] provide a mathematical basis for selecting the stiffness of an arrow to match a particular bow. Current methods used to match the arrow to the bow are limited and involve a generalized arrow selection chart supplied by the arrow manufacturers [14] followed by a process of trial and error to refine the size of the arrow groups on the target. Predictive modelling that removes the need for the difficult and time consuming experimental tests will improve the selection and optimisation of arrows. This paper presents two alternative methods for accurately modelling the vibratory response of archery arrows in free flight, as well as the experimental validation. The desired level of accuracy for the models is based on the difference between selecting -5-

one type of arrow rather than a stiffer or more compliant arrow, that is a difference of less than 3% in the fundamental bending mode. The approach used in this research is to compare the results from the finite difference method and the FE technique to experimental measurements for three different types of composite archery arrows. The aim is to validate the theoretical models to give greater confidence in using models to predict the resonance frequencies and operational deflection shapes of composite arrows. The arrow models may then be used to provide a precise match of arrow to bow for a range of different arrow types, lengths and arrow point masses. Such models may also be used as a basis for investigating specific performance requirements such as the aeroelastic behaviour of arrows in flight. 2 ARCHERY ARROW SPECIMENS 2.1 Archery arrow construction Arrows most commonly used in major competition are made from tubular carbon fibre composite material bonded to an aluminium inner core and are constructed with arrow components including nock, fletching and arrow point. This research investigates three different composite arrow types; namely the ProTour380, ProTour420 and ProTour470 made by Easton Technical Products. These arrows were constructed to the same shaft length of 710 mm, but have a different nominal outer diameter as shown in Figure 1. Note that the outer diameter, and hence the mass and stiffness, vary along the arrow s length. -6-

Fig. 1 Measured outer diameter of arrow specimens verses length 2.2 Arrow component properties Arrow components including nock, fletching and arrow point must be accounted for in the theoretical models. The nock of the arrow made from polycarbonate plastic, which fastens the arrow to the bowstring, is attached to a small aluminium pin inserted into the arrow shaft. Three fletches made from soft plastic are glued to the rear of the arrow shaft. The point of the arrow made from stainless steel has a long shank that fits inside the arrow shaft as shown in Figure 2. The measured mass of the arrow components showed the arrow nock and fletches had a combined mass of m 1 =1.580 g, and the arrow point mass was m 2 =7.776 g. The arrow components were modelled as 3D solids in ANSYS Workbench 12.1 to find the centre of gravity as seen in Figure 2, and the mass moments of inertia. The arrow nock, nock pin and fletches combined had moments of inertia I x,1 = 0.034 kgmm 2 and I y,1 = I z,1 = 0.590 kgmm 2 applied at x 1 =15 mm from the nock end of the arrow. The arrow point mass moments of inertia I x,2 = 0.0198 kgmm 2 and I y,2 = I z,2 = 4.42 kgmm 2 applied at x 2 =703 mm from the nock end of the arrow. -7-

Fig. 2 Composite arrow detailing arrow components, with mass and moment of inertia loading used in FE model applied at the point mass centres of gravity (CoG) 2.3 Arrow shaft material properties The arrow shaft composite material was made of a core tube of aluminium with an outer layer of carbon and epoxy resin. The core material AL7075-T9 aluminium had an outer diameter of 3.572 mm (9/64 inch), a thickness of 0.1524 mm (6/1000 inch) and a density of ρ al =2800 kg/m 3. The aluminium had isotropic material properties with Young s modulus of E al =72 GPa and Poisson ratio of ν al =0.33 [15]. The density of the outer layer of carbon and epoxy resin was estimated from physical measurements. The thickness of the carbon fibre composite material varies along the length of the arrow and was determined by measuring the outer diameter of the arrow shaft. The mass of the arrow can also be measured and hence the density of the carbon fibre composite obtained. An effective density of ρ c =1590 kg/m 3 was used for the carbon and epoxy resin layer, a value comparable to similar carbon fibre and epoxy matrix composites noted in the literature [16]. -8-

3 METHODS 3.1 Estimation of unknown material properties An estimation of Young s modulus was required to model the carbon and epoxy layer of material since the exact specification of fibre volume fraction and constituent material properties was not available from manufacturers specifications. The arrow shafts under test had a high proportion of carbon fibre to matrix with all of the fibres running longitudinally along the arrow shaft. Young s modulus in the fibre direction along the arrow shaft was estimated using the static spine test measured as the deflection of the arrow in thousandths of an inch when an 880 gram (1.94 lbs.) mass was suspended from the centre of the arrow supported at two points 711 mm (28 inch) apart [4]. For example, the ProTour380 had a deflection of 9.652 mm (380/1000 inch) when tested for static spine [14]. The elastic modulus of the carbon fibre layer was determined using the finite difference method that modelled a beam of varying thickness as detailed in Section 3.2. Young s modulus of elasticity was assumed to be constant along the length for carbon and epoxy layer. This was a valid assumption as the fibre-volume fraction varies very little along the length, just the thickness or amount of the material changes. The static spine test was performed mathematically using the finite difference method to solve the equations of motion and the value of Young s modulus for the carbon fibre composite material -9-

adjusted such that the correct static deflection was obtained. The carbon fibre composite total flexural rigidity for a composite arrow of varying thickness is given by ( ) ( ξ ) E I + E I ( ξ ) E I ( ξ ) EI + = al al c, 1 c i i (1) where EI(ξ) is for the total arrow flexural rigidity at a distance ξ from the rear end of the arrow, E al I al is the aluminium layer flexural rigidity, E c I c (ξ) is the carbon and epoxy layer and E i I i (ξ) extra flexural rigidity of the arrow components inserted at the rear and front of the arrow. For an isotropic beam of hollow circular cross section the moment of inertia is given by 4 4 ( ) π I( ξ ) = r o ( ξ ) r i (2) 4 where I(ξ) is the 2nd moment of inertia, r o (ξ) is the outer radius and r i is the inner radius. This formula was used for each layer, aluminium and carbon epoxy, and the arrow components with the respective outer and inner radii. Note that the outer radius of the carbon epoxy layer and the arrow components vary along the length and the inner radius of the arrow components is zero. The one unknown in Eqn 1 determined a constant Young s modulus of E c =222 GPa for the carbon and epoxy layer in the fibre direction along the arrow shaft. For 3D modelling as used in the FE technique, orthotropic elastic carbon epoxy materials require properties defined orthogonal to the fibre direction. These properties -10-

were estimated from similar materials that are transversely isotropic and the rule of mixture calculations for carbon fibre and epoxy matrix composites [16, 17]. A Young s modulus of E 2 =E 3 =9.2 GPa, shear moduli of G 12 =G 13 =6.1 GPa, and Poisson s ratios of ν 12 =ν 13 =0.2 and ν 23 =0.4 were used. To complete the required nine constants of the orthotropic elastic material, the shear modulus G 23 was calculated as E 2 /(2(1+ν 23 ))=3.28 GPa [18]. 3.2 Finite difference method Using the finite difference technique of Park [5], the arrow was modelled as an inextensible Euler-Bernoulli beam with point masses distributed along the shaft. These masses include the nock, fletching and arrow point applied as lumped masses at appropriate grid points along the beam. The arrow components that were inserted and glued into the arrow shaft increase the stiffness of the arrow shaft at both ends. The extra stiffness associated with the arrow nock pin and point shank was added to finite difference models such that the stiffness of the arrow shaft increased at both ends; more so at the front, in view of the nock pin and point shank as shown in Figure 3. The arrow model needs to include these additional masses and increased shaft stiffness. -11-

Fig. 3 ProTour420 arrow flexural rigidity along length of arrow shaft The equations of motion for the arrow were obtained from Newton s second law of motion 2 ( ξ, t) M ( ξ, t) 2 ζ m shaft ( ξ ) = (3) 2 2 t ξ and the Euler-Bernoulli beam equation ( ξ, t) 2 ζ M ( ξ, t) = EI( ξ ) (4) 2 ξ where ξ is the distance from the rear end of the arrow, m shaft (ξ) is the arrow shaft mass per unit length, ζ(ξ, t) is the deflection of the arrow as a function of time t, M(ξ, t) is the bending moment, and EI(ξ) is the arrow s flexural rigidity. These equations of motion were then solved using the explicit finite difference method as detailed by Park [5] and for the mathematics in greater detail by Kooi and Sparenberg [3]. The finite difference method only considered the first vibrational mode due to the difficulty of initially deforming the arrow to a shape consistent with the higher modes. -12-

Initially the arrow was flexed as it would be under gravity and at time t=0 it was released from the effects of gravity. The arrow modelled with no damping then flexes at its natural frequency, which can be measured. The location of the nodes of the fundamental mode of vibration can also be readily obtained and the results are shown in Table 1. It is useful to know the locations of the vibrational nodes for better understanding the behaviour of the arrow as it exits the bow. For these calculations the arrow was modelled as 16 segments, each 44.375 mm in length, and with time steps of 0.000025 s (40 khz). Tests for convergence using 32 segments showed that the use of an increased number did not improve accuracy significantly. The time step used in the finite difference method is related to the space step chosen. If the time step is too great for a given space step there is a danger that the explicit finite difference method becomes unstable. Hence the time step was selected to be well away from that danger, although as noted this does mean greater computing time. 3.3 Finite element method Finite element (FE) models were developed to investigate the modes of vibration of the three composite archery arrows. The arrow was modelled using shell elements as it was intended to also investigate the shell modes of vibration. The shell modes are important in relation to the detection of damage in the composite arrows that are prone to splitting longitudinally along the direction of the carbon fibres which is the subject of a future study. Such damage could be detected from the cylindrical modes of vibration -13-

such as circumferential, torsional and breathing modes rather than from the beam vibration modes alone [19, 20]. The computer package used was ANSYS Workbench 12.1 that provided the SHELL281 element, a multi-layered quadratic shell element of 8 nodes. The shell element was defined with two layers to account for the isotropic aluminium inner tube and the orthotropic carbon epoxy outer layer with 10 elements around the circumference as shown in Figure 4. Orthotropic elastic material properties were defined such that the carbon fibre lay unidirectionally along the arrow shaft, with transversely isotropic properties orthogonal to the fibre direction as detailed in Section 3.1. The inner diameter of the carbon epoxy layer was taken from manufacturer specifications, while the outer diameter of each arrow specimen was measured (as detailed in Figure 1) to find the thickness of the carbon epoxy layer. The shell body was mapped meshed (typical element aspect ratio 1:1) to give 549 element divisions along the 710 mm shaft of the arrow. A total of 16490 nodes and 5490 elements were used. Fig. 4 ProTour420 arrow shaft meshed as multi-layered quadratic shell elements. -14-

Three FE models were defined to match the physical properties of ProTour380, ProTour420 and ProTour470 composite arrows. Simplifications that were made to the FE models include approximation of the tapered section and use of point masses (including rotational inertia) for the arrow components. The tapered section of the arrow was defined by dividing the shaft length into 50 mm sections each of a different constant thickness. This section length is reasonable given that the finite difference method achieved convergence with similar length segments. The section length used results in changes of the outer diameter of no more than 5% per section. The mass of the arrow point, nock and fletches were simplified into two point masses with mass moments of inertia as detailed in Section 2.2 and were applied at the point mass centre of gravity, one at the front and one at the rear of the arrow as seen in Figure 2. These point masses and mass moment loadings were applied with a region of influence over the arrow shaft to replicate the contact area of the arrow components and account for the increase in stiffness caused by the shank inserts. It is noted that the small step discontinuities in the thickness and simplified point masses may cause minor discrepancies in the results of the FE models when compared to finite difference models and experimental results. Eigenvalue modal analyses were conducted using ANSYS to obtain the natural frequencies and mode shapes. The finite element analysis results for the natural frequencies are listed in Table 1. This FE modelling confirmed that the low frequency -15-

vibrational behaviour was dominated by the bending modes of vibration. Other cylindrical modes of vibration such as circumferential, torsional and breathing modes were insignificant to the overall deflection of the arrow. The FE technique calculates the minimum and maximum deflection locations at the surface of the shell element. The node locations for the fundamental bending mode were found by observation from the position of the minimum deflection of the arrow shaft are also detailed in Table 1. 3.4 Experimental method Experiments were conducted on the composite arrow specimens using a Polytec PSV- 400 3D Scanning Laser Doppler Vibrometer (SLDV). The objective of these experiments was to measure the 3D vibrations of the arrow specimens with free-free boundary conditions, and through those measurements validate the theoretical models. The measurements included the resonance frequencies and modal loss factors of the first eight bending modes, along with the operating deflection shape of the fundamental bending mode of vibration. Details of the experimental apparatus along with the experimental process are presented in this section. The experimental apparatus shown in Figure 5 included a frame to support the composite arrow and two acoustically coupled vibration actuators. The support frame consisted of an optical breadboard with many mounting positions to accommodate the different test specimens. The arrow was suspended horizontally by light elastic cords -16-

positioned at the first bending node locations to minimise the effects of additional mass or damping in the connections to the specimen. The vibration actuation was designed to be non-contact by using an acoustically coupled source. The acoustic actuators were a compression driver from a 50 W TU-50 horn speaker with outlet diameter of 25 mm, and a 3 W loudspeaker with an 80 mm diameter diaphragm. The limitation of the acoustic vibration actuators was the diameter of the sound outlet, where a small diameter will tend to excite higher frequency vibrations in the specimen, while a large diameter sound source will tend to excite low frequency vibrations. The coherence between the source signal and the measured vibration indicated that the compression driver was suitable for the frequency range of 1 khz to 4 khz, while the loudspeaker produced better coherence for the first four bending modes in the range of 50 Hz and 1 khz [9]. The Polytec PSV-400 was applied in a 3D configuration using three laser scanning heads to calculate the velocity of vibration in three-dimensional space. The PSV-400 3D SLDV was aligned with high accuracy [21] to obtain precise measurements for a predefined single line of 25 points along the length of the composite arrow. Three test specimens of each arrow type were prepared and tested. -17-

Fig. 5 Experimental apparatus with acoustic vibration actuators To enable comparison of the predictive models calculated natural frequency to the experimental results the modal loss factor, a measure of the damping in each mode, was calculated. The modal bandwidth at a point 3 db down from the peak of the bending mode frequency and was related to the modal loss factor by η b = f / f b (5) where η b is the modal loss factor, f is the modal bandwidth and f b is the mode centre frequency for bending mode b. The resonance frequency can be found by η = b f b f n 1 2 2 (6) -18-

where f b is the mode resonance frequency, f n is the natural frequency and η b is the modal loss factor. The measured results show a modal loss factor of less than 0.012, thus for these composite arrows each measured resonance frequency is approximately equal to the corresponding natural frequency calculated by the predictive models. 4 RESULTS The results obtained from the theoretical models and experimental methods were compared using the frequency and the deflection shape. The natural frequencies from the finite difference and FE predictive models were compared to the resonant frequency of the Experimental method as shown in Table 1, with the fundamental mode of vibration compared in Figure 6. Then the mode shape was compared to the operational deflection shape of the fundamental mode of vibration as shown in Figure 7. Table 1 Predictive modelling results of natural frequency bending modes calculated using finite difference (FD) and FE methods compared to experimental measurement of resonant frequency f b and modal loss factor η b of the tested specimens. Frequency listed as mean ± standard deviation (Hz). Fundamental node locations at distance from nock end of arrow shaft are also noted. ProTour380 FD method FE method Experimental f b η b 1 st bending mode (Hz) 82.3 82.7 83.3±0.4 0.010 Rear node location (mm) 145 143 142-19-

Front node location (mm) 643 642 636 2 nd bending mode (Hz) 249 248±0.1 0.005 3 rd bending mode (Hz) 498 490±0.7 0.005 4 th bending mode (Hz) 819 795±2.3 0.007 5 th bending mode (Hz) 1218 1185±2.9 0.006 6 th bending mode (Hz) 1700 1663±2.9 0.008 7 th bending mode (Hz) 2256 2230±2.8 0.007 8 th bending mode (Hz) 2875 2856±16.5 0.010 ProTour420 FD method FE method Experimental f b η b 1 st bending mode (Hz) 80.3 81.8 80.2±0.4 0.007 Rear node location (mm) 144 144 144 Front node location (mm) 645 640 640 2 nd bending mode (Hz) 247 241±1.3 0.003 3 rd bending mode (Hz) 495 478±1.8 0.009 4 th bending mode (Hz) 815 775±4.2 0.004 5 th bending mode (Hz) 1210 1152±3.6 0.004 6 th bending mode (Hz) 1687 1618±5.0 0.004 7 th bending mode (Hz) 2237 2165±7.0 0.006 8 th bending mode (Hz) 2849 2777±12.5 0.007 ProTour470 FD method FE method Experimental f b η b 1 st bending mode (Hz) 78.4 77.6 78.3±0.4 0.008 Rear node location (mm) 143 139 139 Front node location (mm) 647 645 642 2 nd bending mode (Hz) 236 238±1.7 0.004 3 rd bending mode (Hz) 475 471±4.5 0.012-20-

4 th bending mode (Hz) 783 768±6.4 0.003 5 th bending mode (Hz) 1165 1144±7.5 0.005 6 th bending mode (Hz) 1624 1607±7.9 0.005 7 th bending mode (Hz) 2153 2150±8.7 0.008 8 th bending mode (Hz) 2741 2761±11.5 0.009 Note: shaded rows show measured results using the compression driver, and unshaded rows show results from the loudspeaker experiments. Fig. 6 Comparison of measured and calculated frequencies for the fundamental frequency using both finite difference (FD) and FE methods. The numerically derived mode shapes and operational deflection shape of the fundamental vibration mode for three arrow types were compared with the results for the ProTour420 shown in Figure 7. The mode shapes were calculated by the finite -21-

difference method and the FE technique, where the finite difference method calculated 17 points along the centre line of the arrow, and the FE technique was calculated at 550 points and is shown as a solid line. The magnitude of the calculated lateral displacement is a relative value that has been normalised to unity. The experimentally measured operational deflection shape of the fundamental vibrational mode was also normalised and is shown as 25 measurement locations on the arrow. These locations were offset a small distance from the nock end of the arrow shaft to allow a good reflective surface for the lasers. The last measurement point was defined on the arrow point just beyond the length of 710 mm arrow shaft. Fig. 7 ProTour420 normalised lateral displacement of first bending mode of vibration for theoretical models and experimental measurements. 5 DISCUSSION The models presented in this paper were developed to predict the resonance frequencies and operational deflection shapes of composite archery arrows. This information could -22-

be used to provide a better match of arrow to bow for a range of different arrow configurations. The fundamental vibration predicted by both the finite difference and the FE models were within 2% of the measured fundamental frequencies for the three different composite arrows as shown in Figure 6. A comparison of the FE natural frequency and the measured resonant frequency for the first eight bending modes of vibration also shows good correlation. When comparing the predicted mode shapes to the operational deflection shape of the ProTour420 for the fundamental vibration mode, the finite difference lateral displacement diverges up to 3% toward the point end of the arrow. A small difference of 5 mm is also seen in the node locations as listed in Table 1 and Figure 7. The comparison between the models and the experimental results indicates a close correlation and achieves the desired level of accuracy of less than 3% difference in the fundamental bending mode frequency between the predictions and actual values. The validation of these predictive models has implications for the wider archery community in that these accurate models can be used to assist archers in optimal equipment selection. The application of the theoretical models should take into account the level of detail required in the results, such as the number of dimensions (two or three) or number or type of modes required. The finite difference method provides a two dimensional analysis of the fundamental bending mode of vibration, while the FE method provides three dimensional analysis of many bending modes and other -23-

cylindrical modes of vibration. The modelling methods presented in this paper can be used to predict the resonance frequencies of a range of different arrow types, lengths and arrow point masses to assist archers in optimal equipment selection. The predicted frequency of vibration can be used to compare and contrast a wide range of arrow types with arrow selection charts provided by manufacturers [14]. The finite difference method is sufficient for this application when the flexural properties of the arrow are axisymmetric. When the flexural stiffness is not symmetrical around the shaft the FE method can predict the 3D vibrational behaviour. A development of the models used in this paper could be used to predict the operational deflection shape and dynamic performance of an arrow shot from a compound or recurve bow. By extending the finite difference method the arrow can be modelled in 2D to include the power stroke as well as the aeroelastic behaviour of arrows in free flight [4, 5]. For greater precision the FE method may also be extended to model the dynamic performance using many modes of vibration in 3D. 6 CONCLUSION This paper has compared the results from the finite difference method and the FE technique to experimental results for three different composite archery arrows. The finite difference method was used to solve equations of motion and find the fundamental mode of vibration for an arrow flexing in free space. The FE technique investigated the modes of vibration up to the eighth bending mode in 3D. Experiments -24-

were conducted to measure the three dimensional vibrations of the arrow specimens and determine the resonance frequencies and operational deflection shapes of the bending modes of vibration. The theoretical models of these composite archery arrows with freefree boundary conditions have shown excellent correlation to the experimental measurements. The validation that has been performed by this research gives greater confidence in applying these theoretical modelling methods to predict the resonance frequencies and operational deflection shapes of composite arrows during free flight. ACKNOWLEDGMENT The authors acknowledge the support of Easton Technical Products, who donated arrow products for this research. The assistance from Dorothy Missingham who provided technical writing advice is acknowledged. Authors 2012 REFERENCES 1 Foley, V. and Soedel, W. Leonardo's contributions to theoretical mechanics. Scientific American, 1986, 255(3), 108-113. 2 Pekalski, R. Experimental and theoretical research in archery. J. Sports Sci., 1990, 8, 259-279. -25-

3 Kooi, K. W. and Sparenberg, J. A. On the mechanics of the arrow: the archer s paradox. J. Engng Math., 1997, 31(4), 285-306. 4 Park, J. L. The behaviour of an arrow shot from a compound archery bow. Proc. IMechE, Part P: J. Sports Engineering and Technology, 2010, 225(P1), 8-21. DOI:10.1177/17543371JSET82. 5 Park, J. L. Arrow behaviour in free flight. Proc. IMechE, Part P: J. Sports Engineering and Technology, 2011, DOI:10.1177/1754337111398542. 6 Price, D.S. Computational modelling of manually stitched soccer balls, Proceedings of the Institution of Mechanical Engineers. Part L, Journal of materials, design and applications, 2006, 220(4), 259-268. 7 Hocknell, A. Hollow golf club head modal characteristics: Determination and impact applications, Experimental mechanics, 1998, 38(2), 140-146. 8 Allen, T., Haake, S. and Goodwill, S. Comparison of a finite element model of tennis racket to experimental data. Sports Engineering, Springer London, 2009, 12, 87-98 -26-

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21 Cazzolato, B. Wildy, S. Codrington, J. Kotousov, A. and Schuessler, M. Scanning laser vibrometer for non-contact three-dimensional displacement and strain measurements, Proceedings of the Australian Acoustical Society Conference, Australia, 2008. APPENDIX Notation b E E al E c E i EI f f b f n G I I al I c I i bending mode number Young s modulus Young s modulus for aluminium layer Young s modulus for carbon and epoxy layer Young s modulus for inserted arrow components flexural rigidity resonant frequency in hertz mode centre frequency or mode resonance frequency natural frequency shear modulus 2 nd moment of inertia of arrow shaft 2 nd moment of inertia of aluminium layer 2 nd moment of inertia of carbon and epoxy layer 2 nd moment of inertia of inserted arrow components -29-

I x, I y, I z L a m m shaft M P r i r o x y z f δ max ζ η η b ν ξ ρ mass moments of inertia arrow length mass of arrow components arrow shaft mass bending moment force exerted in static spine test inner radius of the arrow shaft outer radius of the arrow shaft t time abscissa of the fixed coordinate system ordinate of the fixed coordinate system applicate of the fixed coordinate system modal bandwidth maximum deflection of simply supported beam deflection of the arrow modal loss factor bending mode loss factor Poisson s ratio distance from the rear end of the arrow shaft density -30-