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{{short description|Part of a wheel extending radially from the hub to the rim}}
{{other uses}}

A '''spoke''' is one of some number of rods radiating from the center of a [[wheel]] (the [[Bicycle hub|hub]] where the [[axle]] connects), connecting the hub with the round traction surface.
[[Image:Wheel Iran.jpg|thumb|right|A spoked wheel on display at the [[National Museum of Iran]], in [[Tehran]]. The wheel is dated to the late 2nd millennium BCE and was excavated at [[Choqa Zanbil]].]]
[[Image:Cartwheel and axle assembly.jpg|thumb|right|The remains of a pair of [[cart]] wheels with metal axle assembly.]]
[[Image:Ox wagon at Aliwal North.jpg|thumb|right|An [[ox-wagon]] in [[Aliwal North]], [[South Africa]]. Note the three missing spokes and the metal tire.]]
[[Image:Wooden Spokes.JPG|right|thumb|Wooden spoke wheel with metal rim from antique truck on display in [[Underground Atlanta]].]]
[[Image:Bicycle wheel.jpg|right|thumb|Metal [[tension-spoked wheel]] from a bicycle.]]
The term originally referred to portions of a log that had been [[riving|riven]] (split lengthwise) into four or six sections. The radial members of a wagon wheel were made by carving a spoke (from a log) into their finished shape. A [[spokeshave]] is a [[tool]] originally developed for this purpose. Eventually, the term spoke was more commonly applied to the finished product of the [[wheelwright]]'s work than to the materials they used.

==History==
The spoked [[wheel]] was invented to allow the construction of lighter and swifter vehicles. The  earliest physical evidence for spoked wheels were found in the [[Sintashta culture]], dating to {{circa}} 2000 BCE.<ref>Lindner, Stephan, (2020). [https://www.cambridge.org/core/services/aop-cambridge-core/content/view/C4FDF8C5E7D1D20A28BEB7F6C50A9AF4/S0003598X2000037Xa.pdf/chariots_in_the_eurasian_steppe_a_bayesian_approach_to_the_emergence_of_horsedrawn_transport_in_the_early_second_millennium_bc.pdf "Chariots in the Eurasian Steppe: a Bayesian approach to the emergence of horse-drawn transport in the earlysecond millennium BC"], in Antiquity, Vol 94, Issue 374, April 2020, p. 367: "...The 12 calibrated radiocarbon dates belonging to the Sintashta horizon range between 2050 and 1760 cal BC (at 95.4% confidence; Epimakhov & Krause 2013: 137). These dates correlate well with the seven AMS-sampled Sintashta graves in the associated KA-5cemetery, which date to 2040–1730 cal BC (95.4% confidence...)".</ref> Soon after this, horse cultures of the [[Caucasus]] region used horse-drawn spoked-wheel war [[chariots]] for the greater part of three centuries. They moved deep into the [[Greek peninsula]], where they joined with the existing Mediterranean peoples to give rise, eventually, to [[classical Greece]] after the breaking of [[Minoan civilization|Minoan]] dominance and consolidations led by pre-classical [[Sparta]] and [[Athens]]. 

Neo-[[Chalcolithic]]/proto-historic period (1800-1200 BCE){{refn|group=note|Neumayer 1800-1200 BCE;<ref name="Neumayer_Chariots_slideshow">{{cite web | title=Chariots in the Chalcolithic Rock Art of Indian | last=Neumayer| first=Erwin | url=https://www.harappa.com/sites/default/files/pdf/WHEELS%20in%20Indian%20Rock%20Art%20Erwin%20Neumayer.pdf}}</ref><ref name="Harappan_Neumayer">harappa.com, [https://www.harappa.com/content/chariots-chalcolithic-rock-art-india Chariots in the Chalcolithic Rock Art of India]</ref> Celeste Paxton 2300-1000 BCE).<ref name="Paxton">Celeste Paxton (2016), in [https://www.horsetalk.co.nz/2016/07/22/voices-past-researcher-mystery-40-centuries/ Voices from the past: Researcher explores a mystery across 40 centuries ].</ref> See Bradshaw Foundation, [http://www.bradshawfoundation.com/india/pachmarhi/index.php The Prehistoric Paintings of the Pachmarhi Hills], for proto-historic/chalocolithic.}} paintings in various regions of India such as Chibbar Nulla, Chhatur Bhoj Nath Nulla, Kathotia, etc. depict the usage of chariots with spoked wheels.<ref>{{cite book |title=Prehistoric Rock Art of India |first1=Erwin |last1=Neumayer |date=28 March 2013 |isbn=978-0198060987}}</ref><ref>{{Cite web |last=Neumayer |first=Erwin |title=The Chariot in Indian Rock Art |url=https://www.harappa.com/sites/default/files/pdf/WHEELS%20in%20Indian%20Rock%20Art%20Erwin%20Neumayer.pdf |website=Harrapa.com}}</ref> 

[[Celt]]ic chariots introduced an [[iron]] rim around the wheel in the 1st millennium BCE. The spoked wheel was in continued use without major modification until the 1870s, when [[Wire wheel|wire wheels]] and rubber [[tire]]s were invented.<ref name="Herlihy">{{cite book
  | last = Herlihy
  | first = David
  | title = Bicycle: the History
  | publisher = Yale University Press
  | year = 2004
  | page = [https://archive.org/details/bicyclehistory0000herl/page/141 141]
  | url = https://archive.org/details/bicyclehistory0000herl/page/141
  | isbn = 0-300-10418-9
  | access-date = 2009-09-29
  }}</ref>

== Construction ==

Spokes can be made of wood, metal, or synthetic fiber depending on whether they will be in [[Tension (mechanics)|tension]] or [[Physical compression|compression]].

===Compression spokes===
The original type of spoked wheel with wooden spokes was used for [[horse]]-drawn [[carriage]]s and [[wagon]]s. In early motor cars, wooden spoked wheels of the [[Artillery wheel|artillery]] type were normally used.

In a simple wooden wheel, a load on the hub causes the wheel rim to flatten slightly against the ground as the lowermost wooden spoke shortens and compresses. The other wooden spokes show no significant change.

Wooden spokes are mounted [[Radius|radially]]. They are also dished, usually to the outside of the vehicle, to prevent wobbling. Also, the dishing allows the wheel to compensate for expansion of the spokes due to absorbed moisture by dishing more.<ref name="Hansen">{{cite web
| url = http://www.hansenwheel.com/Wagon%20Wheels.htm
| access-date = 2006-08-22
| title = Hansen Wheel and Wagon Shop
| year = 2006
|archive-url = https://web.archive.org/web/20060814231954/http://www.hansenwheel.com/Wagon%20Wheels.htm <!-- Bot retrieved archive --> |archive-date = 2006-08-14}}</ref>

===Tension spokes===
{{main|Wire wheel}}
For use in [[bicycle]]s, heavy wooden-spoked wheels were replaced by lighter wheels with spokes made of tensioned, adjustable metal wires, called [[wire wheels]]. These are also used in [[wheelchair]]s, [[motorcycle]]s, [[automobile]]s, and early [[aircraft]].

====Types====
Some types of wheels have removable spokes that can be replaced individually if they break or bend. These include bicycle and wheelchair wheels. High quality bicycles with conventional wheels use spokes of [[stainless steel]], while cheaper bicycles may use [[galvanized]] (also called "rustless") or [[chrome plated]] spokes. While a good quality spoke is capable of supporting about 225 [[kgf]] (c. 500 [[pounds-force]] or 2,200 [[newton (unit)|newton]]s) of tension, they are used at a fraction of this load to avoid suffering fatigue failures. Since bicycle and wheelchair wheel spokes are only in tension, flexible and strong materials such as synthetic fibers, are also occasionally used.<ref name="spinergy">{{cite web
 |title       = PBO Spoke Technology
 |year        = 2006
 |url         = http://www.spinergy.com/catalog/tech_special_feature_v2.php
 |access-date  = 2011-10-21
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20111030122350/http://www.spinergy.com/catalog/tech_special_feature_v2.php
 |archive-date = 2011-10-30
}}</ref> Metal spokes can also be ovalized or bladed to reduce aerodynamic drag, and butted (double or even triple) to reduce weight while maintaining strength.

A variation on the wire-spoked wheel was Tioga's "Tension Disk", which appeared superficially to be a solid disk but was in fact constructed using the same principles as a normal tension-spoked wheel. Instead of individual wire spokes, a continuous thread of Kevlar ([[aramid]]) was used to lace the hub to the rim under high tension. The threads were encased in a translucent disk for protection and some aerodynamic benefit, but this was not a structural component.

{| class="wikitable"
|+ usual spoke diameters<ref>{{cite web |url= https://www.sapim.be/spokes/basic/leader |title= Basic spokes |publisher= Sapim N.V. }}</ref>
|-
! [[wire gauge]] !! diameter !! section area
|-
| 15G || 1.8&nbsp;mm || {{#expr:1.8^2}} mm<sup>2</sup>
|-
| 14G || 2.0&nbsp;mm || {{#expr:2^2}} mm<sup>2</sup>
|-
| 13G || 2.3&nbsp;mm || {{#expr:2.3^2}} mm<sup>2</sup>
|-
| 12G || 2.6&nbsp;mm || {{#expr:2.6^2}} mm<sup>2</sup>
|-
| 11G || 2.9&nbsp;mm || {{#expr:2.9^2}} mm<sup>2</sup>
|-
| 10G || 3.2&nbsp;mm || {{#expr:3.2^2}} mm<sup>2</sup>
|}

====Tangential lacing====
Wire spokes can be radial to the hub but are more often mounted [[tangent]]ially to the hub. Tangential spoking allows for the transfer of torque between the rim and the hub. Tangential spokes are thus necessary for the drive wheel, which has torque at the hub from pedalling, and any wheels using hub-mounted brakes such as disk or band brakes, which transfer torque from the rim to the brake in the opposite direction—(via the hub) when braking.

====Wheelbuilding====
Constructing a tension-spoked wheel from its constituent parts is called [[wheelbuilding]] and requires the correct building procedure for a strong and long-lasting end product. Tensioned spokes are usually attached to the rim or sometimes the hub with a [[spoke nipple]]. The other end is commonly peened into a disk or uncommonly bent into a "Z" to keep it from pulling through its hole in the hub. The bent version has the advantage of replacing a broken spoke in a rear bicycle wheel without having to remove the [[Cogset|rear gears]].

Wire wheels, with their excellent weight-to-strength ratio, soon became popular for light vehicles. For everyday cars, wire wheels were soon replaced by the less expensive metal disc wheel, but wire wheels remained popular for [[sports car]]s up to the 1960s. Spoked wheels are still popular on [[motorcycle]]s and [[bicycle]]s.

== Spoke length ==

When [[Wheelbuilding|building a bicycle wheel]], the spokes must have the correct length, otherwise there may not be enough threads engaged, producing a weaker wheel, or they may protrude through the rim and possibly puncture the inner tube.

=== Calculation ===

For bicycle spokes, the spoke length is defined from the flange seat to the thread tip. For spokes with bent ends, the nominal spoke length does ''not'' include the width of the spoke at the bent end.

For wheels with crossed spokes (which are the norm), the desired spoke length is
:<math>l = \sqrt{ d^2 + {r_1}^2 + {r_2}^2 - 2 \, r_1 r_2 \cos(a)} - r_3</math>
where
* '''''d''''' = distance from the center of hub (along the axis) to [[flange]], for example 30&nbsp;mm,
* '''''r'''''<sub>1</sub> = spoke hole circle radius of the hub, for example 19.5&nbsp;mm,
* '''''r'''''<sub>2</sub> = half of effective Rim Diameter (ERD), or the diameter the ends of the spokes make in a built wheel (see 'Discussion' attached to this article for explanation) of the rim, for example 301&nbsp;mm,
* '''''r'''''<sub>3</sub> = radius of spoke holes in the ''flange'', for example 1.1&nbsp;mm,
* '''''m''''' = number of spokes to be used for one side of the wheel, for example {{Sfrac|36|2}}=18,
* '''''k''''' = number of crossings per spoke, for example 3 and
* '''''a''''' = 360° ''k''/''m'', for example {{Sfrac|360° ⋅ 3|18}} = 60°.

Regarding '''''d''''': For a symmetric wheel such as a front wheel with no disc brake, this is half the distance between the flanges. For an asymmetric wheel such as a front wheel with [[disc brake]] or a rear wheel with chain [[derailleur]], the value of '''''d''''' is different for the left and right sides.

'''''a''''' is the angle between (1) the radius to a nipple hole in the rim to which a spoke is attached, and, (2) the radius to the flange hole holding the spoke. The spoke crosses either 1, 2, or 3 oppositely pointing spokes depending on the lacing design. On the flange, the angle between the radii of adjacent holes is 360°/''m'' (for equally spaced holes). For each spoke crossed, the hub is rotated with reference to the rim one "angle between adjacent flange holes". Thus, multiplying the "angle between adjacent flange holes" by ''k'' gives the angle '''''a'''''. For example, a 32 spoke wheel has 16 spokes per side, 360° divided by 16 equals 22.5°. Multiply 22.5° ("angle between adjacent flange holes") by the number of crossings to get the angle '''''a'''''—if 3-crosses, the 32 spoke wheel has an angle '''''a''''' of 67.5°. Regarding '''''r'''''<sub>3</sub>: The size of the spoke holes in the flange does not matter for the needed spoke length. This term removes the effect of the hole size. Since the holes are usually small (just over 2&nbsp;mm in diameter), the effect is small and in practice matters little.

For [[Wheelbuilding|radially]] spoked wheels (zero crossings), the formula simplifies to the [[Pythagorean theorem]], with spoke length '''''l''''' plus '''''r'''''<sub>3</sub> being the slope, '''''r'''''<sub>2</sub> minus '''''r'''''<sub>1</sub> being the base, and '''''d''''' being the rise:
:<math>(l + r_3)^2 = (r_2 - r_1)^2 + d^2</math>; or solving for the length
:<math>l = \sqrt{d^2 + (r_2 - r_1)^2} - r_3.</math>

[[Image:spoke-length.png|thumb|300px|right|A flat view of a crossed wheel with one spoke visible]]

=== Derivation ===

The spoke length formula computes the length of the [[space diagonal]] of an imaginary [[cuboid|rectangular box]]. Imagine holding a wheel in front of you such that a nipple is at the top. Look at the wheel from along the axis. The spoke through the top hole is now a diagonal of the imaginary box. The box has a depth of ''d'', a height of ''r''<sub>2</sub>-''r''<sub>1</sub>cos(''α'') and a width of ''r''<sub>1</sub>sin(''a'').

Equivalently, the [[law of cosines]] may be used to first compute the length of the spoke as projected on the wheel's plane (as illustrated in the diagram), followed by an application of the [[Pythagorean theorem]].

== See also ==
* [[Bicycle wheel]]
* [[Spoke nipple]]
* [[Spoke wrench]]
* [[Spokeshave]]
* [[Wheelbuilding]]
* [[Wire wheel]]

== Notes ==
{{reflist|group=note}}

==References==
{{reflist}}

==External links==
{{Wiktionary}}
*[http://www.rst.mp-all.de/splaenge.htm Online spoke length calculator (German), uses slightly different measures than the above formula] {{in lang|de}}

{{Bike equipment}}
[[Category:2nd-millennium BC introductions]]
[[Category:Bicycle parts]]
[[Category:Wheels]]
[[Category:Horse-drawn vehicle parts]]
[[Category:Sintashta culture]]