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

30 Pages·2012·0.65 MB·English

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: [email protected]

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.
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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
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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]
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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
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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
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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.
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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 =1.580 g, and the arrow
1
point mass was m =7.776 g. The arrow components were modelled as 3D solids in
2
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 = 0.034 kgmm2 and I = I = 0.590 kgmm2 applied at x =15 mm from the
x,1 y,1 z,1 1
nock end of the arrow. The arrow point mass moments of inertia I = 0.0198 kgmm2
x,2
and I = I = 4.42 kgmm2 applied at x =703 mm from the nock end of the arrow.
y,2 z,2 2
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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=2800 kg/m3. The aluminium had isotropic material properties with
al
Young’s modulus of E =72 GPa and Poisson ratio of ν=0.33 [15]. The density of the
al al
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=1590 kg/m3 was used for the carbon and epoxy
c
resin layer, a value comparable to similar carbon fibre and epoxy matrix composites
noted in the literature [16].

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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
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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
EI(ξ)= (E I + E I (ξ)+E I (ξ)) (1)
al al c,1 c i i
where EI(ξ) is for the total arrow flexural rigidity at a distance ξ from the rear end of the
arrow, E I is the aluminium layer flexural rigidity, E I (ξ) is the carbon and epoxy
al al c c
layer and EI(ξ) extra flexural rigidity of the arrow components inserted at the rear and
i i
front of the arrow. For an isotropic beam of hollow circular cross section the moment of
inertia is given by
π( )
I(ξ)= r (ξ)4 −r4 (2)
4 o i
where I(ξ) is the 2nd moment of inertia, r (ξ) is the outer radius and r is the inner
o i
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 =222 GPa for the carbon and epoxy layer in the fibre
c
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
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