Arrow design

14.5.3 Arrow design

Successful discharge of an arrow from a bow involves

a careful balancing of three arrow characteristics; namely, length, mass and stiffness (‘spine’). Subse- quent flight depends on the aerodynamic qualities of the design of head, shaft and fletching. The length of the arrow is determined by the geometry of the human body; typically, lengths range from 71 to 76 cm. The mass chosen depends initially upon the type of archery,

e.g. maximum range, target shooting, etc. The prod-

s ⊳ gives the

kinetic energy of the arrow, hence:

D 0.5 m v 2 o

where m D mass of the arrow and v o

D its initial

velocity. Thus velocity increases with bow efficiency and decreasing arrow mass.

Finally, an arrow must possess an optimum, rather than maximum, stiffness (‘spine’) which must be matched to the bow. Bending stiffness of an arrowshaft is measured in a three-point bend test (Figure 7.6). Central to the design of an arrow is the phenomenon known as the Archer’s Paradox (Figure 14.5). At the loose, with the arrow pointing slightly away from the target, the arrow is subjected to a sudden compressive force along its length which, together with the deflect- ing action of the archer’s fingers, generates lateral vibrations in the moving arrow. Correct matching of the dimensions, stiffness and vibration characteristics (frequency, amplitude) of the arrow enables the arrow to clear the bow cleanly. In addition to dependence on the arrow’s dimensions, the frequency of flexu- ral vibration is proportional to the square root of the

of 60 Hz. Reasonable agreement has been obtained between theory and high-speed cin´ephotographic stud- ies. 4

Of the 15 kinds of wood used as arrows for long- bows in medieval times, ash was generally regarded as the best. Nowadays, arrowshafts are tubular and made from (i) drawn and anodized aluminium alloy (7075- T9, 7178-T9), (ii) similar alloys bonded to a smooth outer wrap of unidirectional CFRP and (iii) pultruded CFRP. (Early CFRP arrows were unpopular because they tended to develop splintering damage.) Most arrowshafts are constant in diameter along their length but have the disadvantage that their bending moment varies, increasing from zero at the ends to a maximum at the centre. Tapering (‘barrelling’) the tubular shaft from the middle to the ends reduces this undesirable flexing characteristic: barrelled arrows are used by top professionals.

Feathers are the traditional fletching material and still used but are fragile and suffer from the weather. They rotate the arrow and give stability during flight

4 At the Royal Armaments Research & Development Establishment (RARDE), UK.

Materials for sports 413 beam deflecting under load is:

⊲ 14.3⊳ where M D bending moment, I D moment of inertia of point from neutral axis of beam, E D modulus of

elasticity and r D radius of curvature of loaded beam. In particular, we are concerned with (i) the maximum

max in the convex surface of a tube subjected to a nominal bending moment and (ii) the

max

should bear a good relation to the yield strength of the material and r should be maximized.

A tubular cross-section offers special advantages. Within a bent beam, it locates as much material as possible in the highly stressed regions which lie distant from the neutral axis: this axis lies in the plane mark- ing the transition from tension to compression. Being symmetrical in section, a tube can be loaded trans- versely in any direction and can withstand torsion. Its

64, where D and d are the outside and inside diameters of the tube, respec- tively. From relation (14.3) it can be seen that, for a given bending moment, increasing the moment of iner- tia reduces stress and increases the radius of curvature.

Figure 14.5 The Archer’s Paradox (after Pratt, 1976) . In similar fashion, it can be reasoned that reducing the max ⊳ . Sometimes it is beneficial to raise the moment of iner- but consume kinetic energy. Smooth polymeric vanes

tia by changing from a circular cross-section to a more made from polyethylene terephthalate (Mylar) are

expensive elliptical cross-section. Thus, in front wheel strong, weather resistant and, because of their lower

forks, which are subjected to severe bending stresses, aerodynamic drag, give greater range; the same poly-

an increase in the major diameter of the ellipse reduces mer is also commonly used in stranded form (Dacron)

stress in the crucial plane.

for bowstrings. Table 14.1 compares the bending characteristics of tubes made from four typical materials used for cycle frames; that is, from plain carbon steel, 0.3C–Cr–Mo

14.6 Bicycles for sport

alloy steel (AlSl 4130), 6061 (T6) aluminium alloy and Ti –3Al –2.5V alloy. Calculated values for bend

curvature and maximum stress, which are the crite- The modern bicycle is a remarkable device for con-

14.6.1 Frame design

ria of stiffness and permissible loading, are compared. verting human energy into propulsion. The familiar

The frames of mountain bicycles must sustain sudden diamond frame, with its head, top, seat and down

impact shocks; accordingly, larger-diameter (D) and/or tubes, evolved in the late nineteenth century. When

thicker-walled tubing is used for certain frame mem- in use, it distorts elastically; this compliance provides

bers in order to reduce stress levels. The specific elastic rider comfort. Compliance absorbs energy and frame

stiffness is accordingly maximized in racing machines. minium alloy offers weight saving but, because of its The stress distribution in a working frame is complex,

relatively low E value (70 GN m ), at the expense being in-plane as well as out-of-plane. Sudden impact

of greater flexure of the frame. Titanium alloy allows stresses must be withstood. Poor design, workman-

reductions in tube diameter and wall thickness; its spe- ship and/or maintenance can lead to component failure

is about two and a half times which, because of the fluctuating nature of stressing,

greater than that of Cr–Mo steel. often has fatigue characteristics.

Although not included in Table 14.1, cold-drawn McMahon & Graham (1992) have provided a

seamless Mn–Mo tube steels have a special place in detailed comparison of typical tube materials for

the history of competitive cycling. e.g. Reynolds 531. frames. Beam theory is used to identify the key

Their nominal composition is 0.25C–1.4Mn –0.2Mo. design parameters. The basic linking formula which

Introduced in the 1930s, they have been used for the expresses the stresses and strains at points along a

frames of many Tour de France winners and are still

414 Modern Physical Metallurgy and Materials Engineering

max calculated for tubes subjected to a bending moment of 100 N m (from McMahon & Graham, 1992)

Material

D d Moment

Mass per

Radius of

of inertia I

unit length

curvature r

(MN m ) Racing cycles

48 138 0.58 Cr–Mo steel

C steel

67 190 0.39 Al alloy

Mountain cycles (Top tube) Al alloy

57 76 0.30 (6061-T6) Ditto

49 61 0.24 Ti–3Al–2.5V

widely used for racing cycles. 5 mixture of dispersed alloy carbides, pearlite and possi- bly bainite forms in the weld fillet as they air cool from

14.6.2 Joining techniques for metallic frames

temperatures above 850 °

C. The latest type of low- alloy steel for frames, available in either cold-drawn or

The above guidelines provide a general perspective heat-treated condition (Reynolds 631 and 853), is air but do not allow for the potentially weakening effect

hardening. Although inherently very hard (400 VPN), of the thermal processes used for joining the ends

TIG-welding increases its hardness in the HAZ. It pos- of individual frame tubes. Such joints often coincide

with the highest bending moments. In mass produc- sesses better fatigue resistance than other alloy steels tion, the cold-drawn low-carbon steel tubes of standard

and its strength/weight ratio makes it competitive with bicycle frames are joined by brazing. Shaped rein-

Ti –3Al –2.5V alloy and composites. forcing sockets (lugs) of low–carbon steel, together

Aluminium alloy tubes, which are solution treated with thin inserts of solid brazing alloy, are placed

and artificially aged (T6 condition), present a problem around the tube ends, suitably supported, and heated.

because heating during joining overages and softens

A 60Cu–40Zn alloy such as CZ7A (British Stan- the structure, e.g. 6061, 7005. The high thermal con- dard 1845) freezes over the approximate temperature

ductivity of aluminium worsens the problem. Titanium range of 900–870 °

C as the frame cools and forms a alloys, such as the frame alloy Ti –3Al –2.5V, absorb strong, sufficiently ductile mixture of ˛ and ˇ phases

gases and become embrittled when heated in air, e.g. (Figure 3.20). Butted tubes are commonly used to

oxygen, nitrogen, hydrogen. Again, it is essential to counteract softening of the steel in the heat-affected

prevent this absorption by shrouding the weld pool zones (HAZ); they have a smaller inside diameter

with a flowing atmosphere of inert gas (argon). (d) toward the tube ends. For limited production runs of specialized racing frames made from butted alloy tubes, fillet brazing with an oxy-acetylene torch at a

14.6.3 Frame assembly using epoxy adhesives

lower temperature is more appropriate, using a silver These joining problems encouraged a move toward brazing alloy selected from the AG series of British

the use of epoxy adhesives with sleeved tube joints. 7 Standard 1845, such as 50Ag–15Cu–16Zn –19Cd

As well as helping to eliminate the HAZ problem, (melting range 620–640 ° C). Cadmium-free alloys are adhesives make it possible to construct hybrid frames

advocated if efficient fume-extraction facilities are not available because CdO fumes are dangerous to health.

from various combinations of dissimilar materials

(adherends), including composites. Brake assemblies and has tended to replace brazing, e.g. lugless Cr–Mo

Tungsten-inert gas (TIG) welding 6 is widely used

can be glued to CFRP forks. Adhesive bonds also steel frames for mountain bicycles. Unlike oxyacety-

damp vibrations, save weight, reduce assembly costs lene flames, heating is intense and very localized. The

and are durable. Extremes of humidity and tempera- hardenability of Cr–Mo steels is such that a strong

ture can cause problems and care is essential during adhesive selection. Adhesives technology meets the

5 ‘531’ tubes were used for the chassis of the jet-powered Thrust 2 vehicle in which Richard Noble broke the one-mile

7 Adhesive-bonded racing cycles, sponsored by Raleigh land speed record (1983), achieving a speed of 1019 km h

Cycles of America, were highly successful in the 1984 6 Patented in the 1930s in the USA., where argon and

Olympic Games. Subsequently, Raleigh made mountain helium were available, this fluxless arc process is widely

cycles from aluminium alloy tubes (6061-T8) bonded with used for stainless steels and alloys of Al, Ti, Mg, Ni and Zr.

Permabond single-part ESP-311 epoxy adhesive.

Materials for sports 415 stringent demands of modern aircraft manufacturers 8

and makes a vital contribution throughout the world of sport.

The structural adhesives most widely used in gen- eral engineering are the epoxy resins; their ther- mosetting character has been described previously (Section 2.7.3). Normally they are water resistant. They form strong bonds but, being in a glassy state, are brittle. Accordingly, thermoplastic and/or elastomeric constituents are sometimes included with the ther- mosetting component. When using the two-part ver- sion of a thermosetting adhesive, it is important to control the proportions of basic resinous binder and

catalytic agent (hardener) exactly, to mix thoroughly Figure 14.6 High-performance Zipp bicycle with monocoque frame (courtesy of Julian Ormandy, School of and to allow adequate time for curing. In single-part

Metallurgy and Materials, University of Birmingham, UK) . epoxy adhesives the resin and hardener are pre-mixed: rapid curing is initiated by raising the temperature above 100 °

C. Thermoplastic adhesives, used alone, are has a smaller resistance to corrosion fatigue. The latter weaker, more heat sensitive and less creep resistant.

property is boosted by plating the carbon steel with a Elastomeric adhesives, based on synthetic rubbers,

sacrificial layer of zinc or cadmium. Both types of steel are inherently weak. Meticulous preparation of the

respond well to the strain-hardening action of wire adherend surfaces is essential for all types of adhesive.

drawing through tungsten carbide dies. Wheel rims should be strong, stiff, light and corrosion resistant.

14.6.4 Composite frames

They are often formed by bending strips of extruded, precipitation-hardenable aluminium alloy to shape and

Epoxy resins are also used to provide the

joining e.g. 6061-T6.

matrix phase in the hollow, composite frames of Conventional multi-spoked wheels generate energy- high-performance bicycles. Carbon fibre reinforced

absorbing turbulence during rotation. The distinctive polymers (Section 11.3.2.1) combine high strength and

CFRP front and rear wheels of highly specialized time- stiffness; their introduction facilitated the construction

trial machines, made by such firms as Lotus, Zipp, of monocoque (single shell) frames and led to the

RMIT-AIS and Ultimate Bike, have a much lower appearance of a remarkable generation of record-

aerodynamic coefficient of drag. They are the prod- breaking machines. 9 Typically, they feature a daring

ucts of extensive computer-aided design programmes, cantilevered seat, a disc rear wheel and three-spoke

wind tunnel simulations and instrumented performance open front wheels, all of which are made from CFRP.

testing.

An example is depicted in Figure 14.6.

14.6.5 Bicycle wheels

14.7 Fencing foils

The familiar array of wire spokes between axle and rim normally uses hard-drawn wire of either plain 0.4%

A typical steel foil is about 0.9 m long and tapers to carbon steel (AISI 1040) or austenitic 18Cr–8Ni stain-

a rectangular cross-section of 4 mm ð 3 mm. This less steel (McMahon & Graham, 1992). Each spoke

design gives a low resistance to buckling under the is tangential to the axle, thus preventing ‘wind-up’

large axial stress produced when an opponent is struck displacement between axle and rim, and is elastically

directly, an action which can bend the foil forcibly pretensioned (e.g. 440 MN m ) so that it is always in

into a radius as small as 20 cm. Traditionally, sword- tension during service. During each wheel revolution,

makers use medium-carbon alloy steels of the type the stress on a given spoke is mostly above the preten-

employed in engineering for springs. The extensive sion stress, falling once below it. Under these cyclic

range of elastic behaviour that is associated with a conditions, carbon steel has a greater nominal fatigue

high yield strength is obviously desirable. The foil is endurance than the corrosion-resistant 18/8 steel but

formed by hot working 10 mm square bar stock and then oil quenching and tempering to develop a marten-

sitic structure with a yield strength in the order of Urea-formaldehyde resins (Beetle cements) revolutionized

aircraft building in the 1940s when they were used for 1500–1700 MN m 2 . On occasion, during a fencing

bonding and gap-filling functions with birchwood/balsa bout, the applied stress exceeds the yield strength and composites and spruce airframes e.g. De Havilland

the foil deforms plastically: provided that the foil is Mosquito, Airspeed Horsa gliders.

defect free, the fencer can restore straightness by care- 9 The prototype was the Lotus bicycle on which Chris

ful reverse bending. In practice, however, used foils Boardman won the 4000 m individual pursuit in the 1992

are not defect free. During bouts, repeated blows from Olympic Games at Barcelona.

the opposing blade produce small nicks in the surface

416 Modern Physical Metallurgy and Materials Engineering of a foil. In time, it is possible for one of these stress-

and resistance to prolonged contact with snow and raising notches to reach a critical size and to initiate

moisture. More specifically, in cross-country (Nordic) fatigue cracking within the tempered martensite. Final

skiing, lightness is very important as it makes striding failure occurs without warning and the buttoned foil

less tiring. From a commercial aspect, it is desirable instantly becomes a deadly weapon.

that materials for individual items of equipment should One research response to this problem was to con-

be able to display vivid, durable colours and designer centrate upon improving fracture toughness and resis-

logos.

tance to fatigue failure, thus eliminating instantaneity of failure. 10 In this alternative material, a steel –steel

14.8.2 Snowboarding equipment

composite, lightly tempered fibres of martensite are aligned within a continuous matrix phase of tough

The bindings which secure a snowboarder’s boots to austenite. 10 mm square feedstock of duplex steel for

the top surface of the board are highly stressed dur- the blade-forging machine is produced by diffusion

ing a downhill run. Good binding design provides a annealing packs of nickel-electroplated bars of spring

sensitive interaction between the board and the snow- steel at a temperature of 1000 °

boarder’s feet, facilitating jumps and turns. Modern working. While the bars are at elevated temperatures,

C, extruding and hot

designs are complex and usually employ a variety of nickel interdiffuses with the underlying steel. Nickel is

polymers. Thus, the recent snowboard design shown in

a notable austenite ( )-forming element, as indicated Figure 14.7 includes components made from an acetal previously in Figure 9.2. The optimum volume frac-

homopolar (Delrin), a nylon-based polymer (Zytel) and a thermoplastic polyester elastomer (Hytrel). tion of tough austenite is about 5%. This duplex mate- 11

rial has the same specific stiffness as the conventional Highly crystalline Delrin is tough, having a low glass- steel and has greater fracture toughness. In the event

transition temperature ⊲T g ⊳ , and strong and fatigue of a surface nick initiating a crack in a longitudinal

resistant. It is also suitably UV resistant and moisture filament of brittle martensite, the crack passes rapidly

resistant. Zytel is tough at low temperatures, can be across the filament and, upon encountering the tough

moulded into complex shapes and can be stiffened by interfilamentary austenite, abruptly changes direction

glass-fibre reinforcement. The third polymer, Hytrel, and spreads parallel to the foil axis, absorbing energy

has properties intermediate to those of thermoplas- as the austenite deforms plastically and new surfaces

tics and elastomers, combining flexibility, strength and are formed. In practical terms, if the fencer should fail

fatigue resistance. Both Hytrel and the nylon Zytrel to notice marked changes in the handling character-

can be fibre reinforced. Thus, some snowboard blades istics of a deteriorating foil, the prolonged nature of

are made from Zytrel reinforced with fibres of either final fracture is less likely to be dangerous and life

glass or aramid (Kevlar). Colourants mixed with the threatening. Although safer, the duplex foil involves

resins give attractive moulded-in colours. increased material-processing costs and has a yield

strength about 5–10% lower than that of heat-treated spring steel.

Highly alloyed maraging steels (Section 9.2.3) are used nowadays for top-level competition fencing. By combining solid solution strengthening with fine pre- cipitation in low-carbon martensite, they provide the desired high yield strength and fracture toughness.

A typical composition is 0.03C (max)–18 Ni –9 Co –5 Mo –0.7 Ti –0.1 Al.