RADIAL STIFFNESS OF A BICYCLE WHEEL AN ANALYTICAL STUDY

Transcription

1 ADIAL TIFFE OF A BICYCLE WHEEL - A AALYTICAL TUDY José aría ínguez Dpto. de Físca Aplcada II Facultad de Cenca y Tecnología Unversdad del País Vasco Aptdo Blbao PAI Emal: osemara.mnguez@ehu.es (correspondng author) Jeffrey Vogwell Department of echancal Engneerng Faculty of Engneerng and Desgn Unversty of Bath U..

2 Abstract A theoretcal analyss s carred out to study the behavour of the structural components of a spoked bcycle wheel when radally loaded. Ths has been done so as to establsh convenent mathematcal relatonshps for quantfyng the ndvdual contrbutons whch the spokes and the rm make to the radal stffness of the wheel. The effect of spoke pretenson s consdered and also the nfluence that ths has on the effcent dstrbuton of load and upon the strength of the wheel components. eywords: radal stffness, bcycle wheel, spokes, rm, pretenson Introducton Although the wheel has been around for mllenna and the bcycle wheel for over a century, there are stll radcal changes beng made to the desgn of the modern spoked bcycle wheel (Chandler, ). acng cycle wheels, as used n the Tour de France for example, have changed substantally from the mult-spoke wheels wth open cross secton wheel rms as used on the tradtonal racng bke (chraner, 1999; Okama et al. ; uraoka et al. 1). Leadng bcycle wheel manufacturers, such as avc and hmano, have developed wheels whch effcently use few metal spokes that attach to wheel rms whch comprse part closed and open cross sectons. odern materals, such as carbon fbre, and developments n formng technques have resulted n wheels manufactured as sngle tems, effectvely havng even fewer spokes, and such wheels are wdely used n speed racng on velodrome crcuts The requrements of a modern bcycle wheel are many. The wheel must transfer the weght of the cyclst (va the frame, forks and axle) radally from the wheel hub to the rm and thus to the ground (va the tyre). In order to acheve tracton, and thus wheel rotaton, torson must be transmtted from the chan sprocket (attached to the hub) to overcome rollng resstance at the tyre (whch fts wthn the rm). In addton, a wheel must be suffcent strong to wthstand shock loads when rdng over bumpy terran and wthstand cornerng loads (Gordon,

3 3 4). Also the rm must run true and the spokes play a maor role n achevng ths together wth the selecton of approprate materals (cahon and Graham, 199). Understandng how a spoked wheel works s clearly essental but perhaps not mmedately obvous as pre-tensonng of the spokes s fundamental to a wheel functonng effcently, as shown by Detrch (1993, 1999). Wthout pre-tghtenng of the spokes, for example, the weght would only be supported by those spokes n tenson as the spokes n the lower half of the wheel would be n compresson and, because of the slender shape of a spoke, t means that they wll buckle rather than support any apprecable compressve load. Tradtonally wheels have used many spokes and ths s because the open secton rm lacked rgdty and would dstort too much f the spacng between successve spokes were large (Brandt, 1993). However, modern rms are much more rgd n bendng and torson because of the part closed cross-secton and so fewer spokes can be used, accordng to Hed and Haug (1989) and uraoka et al. (1). An optmum desgn, therefore, seeks a compromse between acceptable rm and spoke strength and stffness (Gordon, 4). Although fnte element analyss can be used to good effect to model a wheel assembly and also mechancal tests exst confrmng performance (nard, ), there are advantages also n takng an analytcal approach (Hull et al., ) as the sgnfcance of the many varables can more readly be seen. o the prmary obectve of ths paper s to establsh an analytcal expresson for determnng the radal stffness of an deal spoked wheel n terms of the maor defnng parameters. Idealsed wheel geometry The followng analyss s based on an deal spoked wheel as shown n fgure 1. The wheel conssts of the crcumferental rm, the hub and a number of spokes, whch connect as pn onts between the rm and the hub. The rm has a radus, and the wdth of the hub s d and so the angle, between the centre axs s gven by : d tan (1)

4 4 To determne the wheel radal stffness a downward load P s appled at the hub and ths s reacted by an equal force, P at the rm where the rm contacts wth the ground. The deformaton of the wheel s the shortenng of the dstance between the hub and the contact pont of the rm wth the floor. The orgnal length of the spokes when not under load s L and the length when secured at the rm and hub and tghtened and thus subected to pre-tenson s L o. cos Force equlbrum ow consder radal force equlbrum for half a wheel whereby all spokes are pre-tensoned wth tensle load, T o and C s the crcumferental rm force (see fgure ).. For number of spokes (and wth the top spoke algned wth vertcal datum) the equlbrum condton for the half wheel s / C T cos cos () / n whch the sum s extended over all the spokes anchored to the half rm. o, f the total number of spokes of the wheel s, the number of spokes per angle unt s / and the sum may be evaluated as follows / / / cos / n cosd (3) whch s acturate for most practcal cases. For example, f there are 16 spokes, the sum has a value of 5.7 whereas the an error of ust 1.3 %. the spokes s / approxmaton gves 5.93 ndcatng Consequently, the hoop compresson of the rm caused by the pretenson of

5 5 T cos C (4) from whch the relatonshp between the extenson of the spokes pretenson T and the contracton of the radus compresson can be easly found. L due to ther of the wheel rm due the hoop Effectvely, L beng the orgnal length of a spoke, elastcty and A ts cross secton, E the modulus of L L L L cos T L A E (5) and the radus of the wheel rm, wth a cross secton E, wll be reduced by an amount A and a modulus of elastcty C A E T Lcos A E cos A E A E L (6) As the rm cross sectonal area s typcally about 3 tmes greater than the spoke cross sectonal area, because the rm s usually made from an alumnum alloy whereas the spokes are from steel, n the case of a wheel havng 16 spokes and takng cos. 97 means that :. 5L (7) Equaton (7) descrbes the stffness of the wheel rm wth respect to the spoke tenson and ndcates that the varaton of the radus of the rm and the extenson of the spokes are of the same order of magntude. However, ths reducng of the radus of the rm must not be accounted for when analysng the stffness of the wheel, for t takes place when the wheel s beng bult, prevously to ts use and loadng. evertheless t was studed to llustrate how the spokes and the rm become a ont structure.

6 6 poke contrbuton To study the radal stffness of the wheel t wll be assumed that the hub supports an n-plane load P, whch causes a radal dsplacement a of the hub. The am of ths analyss s to establsh a relatonshp between a and P n order to deduce the stffness. In ths secton only the dsplacement a of the hub resultng from the stran and the deflecton of the spokes s gong to be calculated. The dsplacement a of the hub due to the dstorton of the rm s calculated n the followng secton and then added to a. Fgure 3 shows the hub loaded and dsplaced a at the centre of the rm, wth only one spoke shown for clarty but the calculatons account for all spokes. When the load P s appled on the hub t s supported by all the spokes, each of them beng subected to a tenson T and stretched to a length L a cos (8) cos accordng to fgure 3. Then the fracton P of the total load P assumed by each spoke s P cos cos (9) T the tenson of the spoke beng worth T A E a cos A E a cos L T (1) L cos L cos after substtutng the pretenson T from equaton (5). Consequently the load assumed by each spoke s gven by A E P T cos cos a cos (11) L

7 7 wth whch the total load may be expressed as A E P P T cos cos a cos (1) L 1 In ths equaton both sums are extended over all the spokes anchored to the rm and to the hub. o, f the number of spokes s, there wll be / spokes per angle unt, and the two sums can be calculated as follows: cos cosd (13) and cos cos d (14) When these results are put nto equaton (1) t s obtaned that a L P (15) A E whch s the spoke contrbuton to the radal dstorton of the wheel. Equaton (15), shows effectvely the proportonalty between the dstorton a and the load P and defnes the stffness of the wheel relatve to the spokes as A E (16) L where the nfluence of the count of spokes and of ther cross secton and modulus of elastcty as well as the nfluence of the sze of the wheel are quantfed. At ths stage t may seem surprsng that a does not depend on the pretenson of the spokes. However, ths s so due to the fact that all the spokes anchored to the hub, and pullng from t all around, experment a tenson equally ncreased by such a pretenson T. Ths s shown f the result of equaton (15) s taken nto equaton (1) yeldng that

8 8 T P T cos cos (17) and explans why the magntude of the pretenson of the spokes does not nfluence the dsplacement a of the hub. However t s noted that the above calculatons have been carred out on the assumpton that the pretenson of the spokes s such that they are kept n tenson all the tme. Equaton (17) also shows that the spokes n the upper half of the wheel, for whch / / (see fgure 3), are tensoned beyond the pretenson when the load P s appled, whereas the spokes n the lower half of the wheel have ther net tensle load reduced. The amount by whch ths occurs s due to spoke and rm relatve stffnesses and s studed n the next secton. m contrbuton The wheel rm, when loaded, wll experment bendng due to the reacton of the floor together wth all the spoke tensons, whch brngs about another contrbuton a to the shortenng of the dstance between the hub and the contact pont of the wheel wth the floor. Evdently, ths must be taken nto account when calculatng the stffness of the wheel. The approachng of the hub, upon whch the load P s appled, to the lowest pont of the rm, where the reacton P exerted by the floor acts, s due to the bendng of the lower half part of the rm. Ths part of the rm s supported by the upper half and s subected to the tenson T of the spokes anchored to t and to the reacton of the floor P, as represented n fgure 4. Then the deflecton a of the rm at the lowest pont may be calculated by dervng the stran energy of the bent half part of the rm, by usng the Castglano s theorem (Hearn, 1999, and yder, 1983). Furthermore, accountng for symmetry reasons, the bendng of the half rm s gong to be studed by consderng only a quarter of the wheel rm, as shown n fgure 5, whch supports half the reacton of the floor P / P, the tenson T of

9 9 the spokes anchored to t and the hoop compresson C as well as the bendng moment exerted by the rest of the wheel rm at the base. Consequently, the uprght deflecton usng Castglano s theorem from a at the base of the rm s obtaned a U (18) P where U s the stran energy of a quarter of the rm. The stran energy stored n a quarter of the rm can be calculated from U E I ds (19) where s the bendng moment n each secton of the rm and I ts moment of nerta, and the ntegraton s computed between the lmts of the quarter rm /. Ths calculaton can readly be performed for a wheel havng number of spokes for the followng condtons: 1.- the quarter of the crcle s dvded nto /4 segments (see fgure 5) wth spokes specfed from the bottom (as 1,,..., etc) each havng an arc length / between consecutve spokes;.- n each segment the bendng moment value s taken to be the value at the central pont. Based on these assumptons the stran energy for the quarter of the rm can be expressed as U E I / 4 1 () In ths smplfed expresson, every moment can be wrtten as P, C,, T ) (1) (

10 1 from fgure 5, and thus equaton () represents the total stran energy of the quarter of the rm as a functon of the external loads P, C and couple T and of the external U U P, C, T, ) () ( Ths now enables Castglano s theorem to be appled. Effectvely, the uprght deflecton at the lowest pont of the rm wll be gven by a U P E I / 4 1 P (3) whereas the horzontal deflecton and the angular rotaton at ths pont, whch must be null for symmetry reasons, wll be gven by U C E I / 4 1 C (4) / 4 U and E I 1 (5) ow, f each bendng moment s substtuted from equaton (1) and the tenson of every spoke s taken as T, the last three equatons (3), (4) and (5) allow to express the uprght deflecton at the lowest pont of the rm as a functon of the load P and of the pretenson of the spokes T a a ( P, T ) (6) The deflecton a determned n ths way accounts for the bendng of the rm due to both the load P and the spoke pretenson T. It s obvous that only the term correspondng to the load P must be accounted for when determnng the stffness of the wheel rm, snce the deflecton caused by the spoke pretenson wll occur

11 11 when the rm s beng mounted as a ont structure and wll have already been performed when the loadng of the wheel takes place and the stffness s beng defned. The whole procedure to calculate the rm contrbuton to the deformaton of the wheel s gong to be throughout llustrated n a partcular case. 16 spoke wheel rm In ths secton the rm contrbuton to the dsplacement of the hub towards the contact pont of the rm wth the floor, n a wheel wth =16 spokes, s worked out, as an example of how t can be carred out n any case and n order to analyse how t depends on the dfferent varables whch conform the rm. The bendng moments for the respectve segments between consecutve spokes of the consdered quarter wheel, as shown n fgure 5, are as follows: 1 P sn T1 cos sn C1 cos (7) 3 3 P sn T1 cos sn T cos sn C 3 1 cos P sn T1 cos sn T cos sn T3 cos sn 5 C1 cos 7 P 7 5 T T T 3 sn 1 cos sn cos sn 3 cos sn 7 T4 cos sn C1 cos ubsttutng these values nto equaton (3) gves the deflecton a a sn sn 3 sn 4 sn E I (8)

12 1 whereas equatons (4) and (5) reduce to cos cos 3 cos 4 cos 1 (9) and (3) ow, f = 16 and T T, equatons (7) are smplfed as follows : P.1959T cos. 191 C.55557P.7566T cos C P T cos C (31) P.569T cos C Then equatons (9) and (3) represent a smple system from whch the hoop compresson C and the bendng moment are drectly obtaned n terms of the load P and of the pretenson of the spokes T as follows: C T.93771P.9847 cos (3) and.3384p.118t cos (33) Afterwards, takng values (3) and (33) to expressons (31) gves P.6338T cos.937p.4684t cos (34)

13 P.5545T cos 4.778P.3888T cos obtaned that Fnally, by substtutng values (34) n expresson (8) and P P /, t s a 3.16P.163T cos (35) 8E I where the two terms of the deflecton of the rm appear. As t was sad before only the term due to the load P s accountable when determnng the stffness of the wheel. Consequently, the stffness relatve to the rm, when t has 16 spokes s gven by 8E I (36) adal stffness of the wheel ow the total dsplacement a of the hub caused by the n-plane load P may be expressed by addng a and the correspondng term of a a P P a a P (37) n whch the stffness of the wheel (as a structure comprsng rm and spokes) s computed as W (38)

14 14 In the case of a wheel havng 16 spokes, substtutng equatons (16) and (36) nto equaton (38) yelds the result 8 (39) L.6786 A E E I W 3 Ths equaton quantfes the contrbutons of both the spokes and the rm, n terms of ther geometry and materals, to the stffness of the wheel. If, for example, the spoke stffness s sgnfcantly greater than that of the rm, such that / s small then the overall wheel stffness s domnated by the less stff rm. Ths can be seen from equaton (38) by elmnatng the / rato and leavng W --->. Pretenson of the spokes When studyng the spoke contrbuton to the stffness of the wheel, the length of every spoke, after the wheel s loaded, was named equaton (8) relatve to L and expressed n / cos, whch s the length the spoke had reached after beng anchored and fxed between the hub and the rm, wth a pretenson T. The spokes stuated below the hub, for whch / (see fgure 3) and L / cos, experment a shortenng when the wheel s loaded and so a reducton of ther tenson, from the pretenson T to T T, accordng to equaton (17). The bggest shortenng corresponds to the spoke at the lowest poston and s worth, from equaton (15), a L P P (4) cos A E 8 A E 8cos when the number of spokes s 16. Also, the bendng of the low half part of the rm due to the reacton of the floor and to the tenson of the spokes anchored to t causes an addtonal shortenng

15 15 of these spokes and the subsequent reducton of ther tenson. Ths shortenng s maxmum at the lowest poston where t s worth [(see equaton (35)] 3 a.6786 P (41) cos E I 8cos agan, when the number of spokes s 16. It s mportant to recognse that the developed theory assumes that spokes reman n tenson at all tmes. Ths s because a spoke s ncapable of supportng a sgnfcant compressve load due to ts hgh slenderness (that s length to dameter rato) thus makng t prone to buckle. Ths means that t would have been ncorrect to have assumed that a spoke could support any apprecable compressve load and so means that spokes must reman n tenson at all tmes. Ths leads to the requrement that the axal stran extenson resultng from pre-tensonng needs to exceed the greatest shortenng that a spoke wll experence whch s when passng through ther lowest poston. Therefore, based on the 16 spoke wheel examples, the followng condton must be met [see equatons (4) and (41)]: T L A (4) A E EI 8cos P E from whch the mnmum pretenson the spokes need s deduced T E A P (43) E I 8 Ths wll ensure that all the spokes are always tensoned, whatever ther poston wthn the wheel may be, whch s the most convenent condton for the stablty and strength of the wheel. axmum tenson of the spokes Consderng that the spokes n the upper half part of the rm are not as hghly nfluenced by the bendng of the lower half part, equaton (17) s vald for

16 16 them and may be used to calculate ther tenson, whch depends on ther poston wthn the wheel. Evdently, the maxmum tenson occurs n the hghest spoke, for whch, and s worth T max P P T T (44) cos 8cos f the number of spokes s 16. Then the necessty for the tenson never beng negatve makes condton (43) unavodable and therefore T max E E A I 1 P cos 8 (45) Ths result does not gve the tenson of the hghest straned spoke but ts mnmum value as long as the pretenson ensures that no spoke wll be n compresson. In fact, the maxmum tenson the spokes wll be subected to depends very much on the pretenson T they are provded wth under condton (43). umercal applcatons It s nterestng to apply the theoretcal results so far developed to some specfc cases so that the sgnfcance of ndvdual contrbutons can be seen. The examples consdered are taken from Chandler () and correspond to three commercally avalable wheel models, namely the avc Open Pro, the hmano WH-65 and the olf Vector Pro as shown n fgure 6. Ther structural propertes are gven n table 1. Although the actual wheels are ftted wth dfferent numbers of spokes and wth dfferent mountng arrangements, n ths work the three rms were assumed to have 16 spokes wth a length equal to the radus of the rm, L 35 mm ( cos 1), accordng to the dealsed model of fgure 1. In ths way the results wll enable drect comparsons to be made. Also, n the three cases the rm materal was alumnum alloy ( E 7 G/m ), whereas the spokes are from steel

17 17 ( E G/m ). The dameter of the spokes n all cases s taken as mm ( A mm ). Usng ths data the radal stffnesses for spokes and rms were obtaned usng equatons (16), (36) and (38) and results are gven n table. These results emphasze the sgnfcant dfferences n the magntudes of the spoke and the rm stffnesses and thus ther contrbutons to the wheel as a structural assembly of the two elements. Table 3 presents the mnmum pretenson loads of the spokes for the three cases, as calculated from equaton (43), alongsde wth the maxmum tenson the spokes wll be subected to, as calculated from equaton (45). Both the pretenson and the maxmum tenson are gven relatve to the load appled on the hub. General dscusson Although ths analytcal study of a spoked bcycle wheel s not exhaustve, the analyss has establshed some mportant fndngs regardng the contrbutons made by the rm and the spokes to the structural behavour of a wheel. In ths work the effect of axal load dstrbuton n the spokes and the bendng of the rm are combned to establsh the radal stffness of the wheel. It s found that the results of ths analyss are of the same order as those publshed by Chandler () by Hopkns and Prncple (199) and by Grgnon (), whch suggests that the values are of the rght order of magntude. The study has found that, n general, the system of spokes s much stffer than that of the rm and so the rm stffness largely dctates the wheel radal stffness. Ths s partcularly evdent when studyng equaton (38) whch shows how the lower value of rm stffness domnates the wheel stffness when the system of spokes have a stffness an order of magntude greater. It s nterestng to observe that modern optmal wheel desgn s movng towards wheels havng much stffer rm cross sectons wth fewer spokes. For the radal stffness of such wheels, whch have a wder spoke spacng dstance, to compare wth tradtonal mult-spoke wheels, t s only possble when the cross secton s much stffer n bendng. In table, and wth the help of equaton (38), t can be deduced that sgnfcantly changng the stffness of the spokes wll have lttle effect on the general stffness of the wheel. Consequently, reducng the number of spokes s

18 18 consdered desrable as long as the wheel remans stable and so rm bendng rgdty becomes of crtcal mportance. It s nterestng to observe how the olf Vector Pro rm despte havng a smaller cross secton wth respect to the hmano WH-65 rm, and thus less weght and less materal usage, has an mproved stffness of the wheel due to ts ncreased moment of nerta (see tables 1 and ). In addton, f the rm has a greater bendng stffness, spoke pretenson need not be so great thus permttng a lower maxmum tenson. Ths s because the stffer the rm, the lower the bendng deflecton n the bottom half of the rm and the smaller the spoke contracton necessary wth a lower pretenson [see equaton (43)]. Consequently wth a lower pretenson load when the wheel rotates half a turn and the lower spokes come to the upper half part of the wheel they wll not be placed under such hgh tenson [see equaton (45)]. However, the rm moment of nerta cannot be ncreased ndefntely as there exsts a lmt to the number of pre-tensoned spokes possble and so a compromse s necessary for an optmum desgn to be acheved. Conclusons Ths analytcal study has helped quantfy how the spokes and the rm nteract wth each other and work together when assembled to form a wheel and subected to a radal load. The work has taken nto account both the bendng stffness of the rm secton and the axal tensle stffness and preloadng of the set of spokes n determnng a wheel s radal stffness. It was deduced that achevng an optmum wheel desgn, n terms of wheel radal stffness, necesstates makng a compromse between maxmsng rm crosssecton bendng stffness (whch permts greater dstance between successve spokes) and havng fewer spokes. Because there are many spoke and rm defnng parameters, many havng conflctng effects, achevng an optmum wheel s a complex process and so analytcal equatons have been developed to help smplfy the desgn optmsaton process. Pretenson of the spokes s an essental part of achevng an effcent wheel and the developed theory has enabled the mnmum possble magntude to be determned so that spokes always reman under tenson throughout the wheel rotaton load cycle.

19 19 eferences Brandt, J., The Bcycle Wheel, 1993, Avocet, Palo Alto, Calforna. Chandler,., An Investgaton of Bcycle Wheel m Desgn, Fnal Year Dssertaton, Department of echancal Engneerng, Unversty of Bath.U Detrch,., Cycle, Tensoned poked Wheel Assembly and m, 1993, U.. 5,445,439. Detrch,., Cycle and Tensoned poked Wheel Assembly, 1999, U.. 5,931,544. Gordon, D., Bcyclng cence, 4, The IT Press, Cambrdge. Grgnon, F., Aero Wheels under crutny,, Hearn, E. J., echancs of aterals, 1999, Butterworth Henemann, usselburgh, U.. Hed,. A. and Haug,. B., Bcycle m and Wheel, 1989, U.. 5,61,13. Hopkns,. W. and Prncple, E.., The Du Pont Aerodynamc Bcycle Wheel, 199, Cyclng cence, U.. Hull,. L., Wang, E. L. and oore, D. F., An Emprcal odel for determnng the adal Force-Deflecton Behavour of Off-oad Bcycle Tres,, cahon, C. J. Jr and Graham, C. D. Jr, Introducton to Engneerng aterals: The Bcycle and the Walkman, 199, eron Books, Phladelpha. uraoka, Tsutumo, Okama and hnpe, Bcycle Wheel wth enforced m, 1, hmano Inc., U ,56. uraoka, Tsutumo, Okama and hnpe, Bcycle Wheel wth enforced m, 1, hmano Inc., U ,58. Okama, hnpe, uraoka and Tsutomu, Bcycle Wheel,, hmano Inc., U.. 6,68,348. nard, D., Wheel tffness Test,, yder, G. H., trength of aterals, 1983, The acllan Press Ltd, London. chraner,g., The Art of Wheel Buldng, 1999, Buonpane Publcatons, Denver, Colorado.

20 Fgure captons Fgure 1.- Fgure.- Idealsed spoked wheel: the spokes are anchored to the rm and to the hub. The load s appled on the hub. Half wheel rm supportng the tensons of the spokes fxed to t and the hoop compresson. Fgure 3.- When the external load P s appled on the hub t s dsplaced a, and so a spoke, whose pretenson was T 1, s deflected and enlarged a cos, whle ts tenson becomes T. Fgure 4.- Fgure 5.- The wheel rm s bent by the reacton P of the floor. The deflecton at the contact pont a s the rm contrbuton to the radal dstorton of the wheel. Quarter of the rm supported by the rest of the rm at secton A and subected to the tenson of the spokes anchored to t and to half the reacton of the floor. The acton of the rest of the rm at the bottom C and ) s consdered also as an external load. ( Fgure 6.- Dfferent rm cross sectons.

21 1 m avc Open Pro hmano WH-65 olf Vector Pro Cross-ecton Area A (mm ) oment of Inerta I (mm 4 ) Table 1: m structural propertes.

22 m avc Open Pro hmano WH-65 olf Vector Pro tffness due to spokes (/m) tffness due to rm (/m) esultant adal tffness (/m) Table : adal stffness of wheels.

23 3 m avc Open Pro hmano WH-65 olf Vector Pro Pretenson T.41 P 1.34 P 1.17 P axmum Tenson T.53 P 1.47 P 1.9 P max Table 3: poke pretenson and maxmum tenson, P beng the spoke total load.