THE MATHEMATICAL MODELLING OF BALL-JOINTS WITH FRICTION. R. M. Sage

Transcription

1 THE MATHEMATICAL MDELLING F BALL-JINTS WITH FRICTIN R. M. Sage Thesis submitted fr the degree f Dctr f Philsphy at the University f Leicester. June 1987

2 UMI Number: U All rights reserved INFRMATIN T ALL USERS The quality f this reprductin is dependent upn the quality f the cpy submitted. In the unlikely event that the authr did nt send a cmplete manuscript and there are missing pages, these will be nted. Als, if material had t be remved, a nte will indicate the deletin. Disscrrlatin Publishing UMI U Published by PrQuest LLC Cpyright in the Dissertatin held by the Authr. Micrfrm Editin PrQuest LLC. All rights reserved. This wrk is prtected against unauthried cpying under Title 17, United States Cde. PrQuest LLC 789 East Eisenhwer Parkway P.. Bx 1346 Ann Arbr, Ml

3

4 ACKNWLEDGEMENTS The authr wuld like t express his gratitude fr the assistance prvided by Prfessr G. D. S. MacLellan during the curse f this wrk. Thanks are due t the Central Electricity Generating Bard fr their financial assistance which made the cmpletin f the prject pssible, and t Dr. C. sgd at the Berkeley Nuclear Labratries. The authr wuld als like t thank the staff f the drawing ffice, labratries and wrkshp f the Department f Engineering; in particular Mr. D. Pratt, Mrs. K. P Baglin, Mr. C. Mrrisn and especially Mr. P. Preem.

5 Nmenclature Unless therwise stated at the apprpriate pint in the text the ntatin used in this thesis is as fllws: a,b,c,d,e,g,k =i Dimensins within the spherical jint l-vi ^ Yung's mduli Resultant frictinal frce F N rn M. _R rn 2sin*cs#cs( %^)dc< Directin csines f a vectr x Mj M. R M N P P P ^1 ^2 r X Mment f slip frictin Mment f rtatinal frictin Ttal frictinal mment Nrmal reactin Applied lad Pressure Maximum pressure Radii f scket and ball Radius f jint f.

6 c<, p ( - Angular c-rdinates f a pint n the cntact area - Angle defining the extent f the cntact area Angle between the axis f rtatin and the line f actin f the nrmal reactin 0 - Angle between the axis f rtatin and the directin f the applied lad p V w - Cefficient f frictin - Pissn's rati - Angular velcity

7 CNTENTS Acknwledgements Page Nmenclature Chapter 1 Intrductin 1 Chapter 2 Backgrund AMP2D 2.1 The Frictin Mdel fr AMP2D 2.3 AMP3D-ADAMS Chapter 3 Sme Frictinal Effects in a Spherical 14 Jint 3.1 The Expressins btained by Yu.P.Smirnv 3.2 The Pressure Distributin 3.3 The Nrmal Reactin 3.4 The Frictinal Frce 3.5 The Frictinal Mment The Mment f Slip Frictin The Mment f Rtatinal Frictin 3.6 The Evaluatin f the Expressins 3.7 The Characteristics f the Frictinal Frce and Mments The Frictinal Frce The Mment f Slip Frictin The Mment f Rtatinal Frictin Chapter 4 The Frictinal Effects in Relatin t the Applied Lad The Frictinal Frce 4.2 The Mment f Slip Frictin 4.3 The Mment f Rtatinal Frictin 4.4 The Evaluatin f the Expressins 4.5 The Characteristics f the Frictinal Frce and Mment The Frictinal Frce The Frictinal Mment Chapter 5 The Mathematical Mdel The Angle (, Defining the Area f Cntact

8 5.2 The Reactin t the Effects f Frictin 5.3 The Directin f the Frictinal Mment Relative t a General C-rdinate System 5.4 The Displacement and Velcity Dependence f Frictin in a Spherical Jint 5.5 Apprximate Expressins 5.6 Straddle-type Spherical Jints Chapter 6 Experimental Apparatus, Prcedure and Results The Apparatus 6.2 The Prcedure 6.3 Interpretatin f Results 6.4 The Results Chapter 7 Cnclusins 140 Appendices References

9 CHAPTER NE Intrductin The Central Electricity Generating Bard relies heavily n mechanical systems in the cntrl and peratin f its generating plant. mechanisms perate In its nuclear plants many f these in the hstile envirnment f the reactr cre where lubricatin in the cnventinal sense is nt pssible. As a result, frictinal frces can be relatively high and s must be taken int cnsideratin in design assessments. T aid in the design and assessment f its plant items the CEGB has develped (in cllabratin with Mechanical Dynamics Inc. f Ann Arbr, Michigan) tw cmputer prgrams AMP2D (Advanced Mechanisms Prgram fr 2- Dimensins) and AMP3D-ADAMS (Advanced Mechanisms Prgram fr 3-Dimensins - Autmatic Dynamic Analysis f Mechanical Systems). AMP2D was develped t simulate a tw-dimensinal mechanical system f links, pin-jints, pulleys, etc., with a minimum f input data and AMP3D- ADAMS was develped t extend the range f simulatins t three-dimensinal mechanisms. A mdel f the effects f frictin within a tw- dimensinal mechanism has been incrprated int AMP2D

10 (Threlfall, 1978). This mdel is based n the assumptin that the frictinal frce prduced is directly prprtinal t the nrmal reactin, althugh, in rder t vercme the cmputatinally undesirable effect f a step change in the frictinal frce n the reversal f mtin, the mdel als assumes that the frictin prduced is displacement and velcity dependent at velcities clse t er. Unfrtunately, the frictin mdel used in AMP2D cannt be simply adapted fr use in AMP3D-ADAMS. The reasn fr this can be seen when cnsidering the characteristics f the additinal jints that have t be mdelled in rder t simulate three-dimensinal mechanisms, particularly thse f the spherical jint. In such a jint the resistance t the relative mtin due t frictin is determined nt nly by the cefficient f frictin and the magnitude f the lading, but als by the nature f the pressure distributin within the jint which results frm that lading. Cnsequently the frictinal effects prduced can n lnger be readily determined by the simple expressins used in the frictin mdel incrprated int AMP2D. There is a further difficulty in adapting this mdel in that the expressins fr the velcity and displacement dependence f the frictinal frces prduced cannt be simply extended t jints where the sliding velcity is nt cnstant ver the surface f cntact, as is the case with the spherical jint.

11 Thus, in rder t include frictinal frces in the simulatins prvided by AMP3D-ADAMS r any ther three- dimensinal mechanism simulatin package, it is necessary t develp a mathematical mdel f the effects f frictin in a spherical jint. The basis f this thesis has been the develpment f such a mdel and its assessment by cmparing the results that it prvides with thse btained experimentally. Chapter tw f this thesis gives the backgrund t the prject, describing the tw prgrams AMP2D and AMP3D-ADAMS and in sme detail the frictin mdel incrprated int AMP2D. An extensive literature survey revealed that little wrk had been dne in the area f the mathematical mdelling f frictin in spherical jints. The ne significant paper that did cme t light, as a result f a search dne by the CEGB n the DAILTECH database, was that by Smirnv (1981) which gave expressins fr the frictinal frce and mments prduced in a spherical jint. These expressins were fund t be incrrect but when crrected they became the basis fr the required mathematical mdel. The derivatin f these expressins is described in chapter three. Chapter fur extends the analysis t take int accunt the interactin between the frictinal frces prduced and the pressure distributin in the jint. ther refinements necessary t prvide an adequate mdel are described in chapter five.

12 The thery indicates that the angle between the directin f the applied lad and the axis f rtatin f the jint is a majr factr in determining the magnitude and directin f the frictinal mment prduced in a spherical jint. In rder t verify this predictin an experimental rig was designed t allw the frictinal mment n a scket, prduced by a ball rtating in it, t be measured fr a range f values f this angle. The apparatus and the experimental prcedure, tgether with a summary f the results btained, are described in chapter six. The results were btained fr varius values f the cefficient f frictin in the range frm 0.05 t 1.1. A cmparisn f these results btained experimentally with thse given by the theretical analysis cnfirms the assertin that the angle described abve is a significant factr in the determinatin f the frictinal mment prduced in a spherical jint. It als indicates that the mdel develped prvides a satisfactry means f determining the frictinal effects prduced in a spherical jint.

13 CHAPTER TW Backgrund 2.1 AMP2D AMP2D, the Advanced Mechanisms Prgram fr 2-Dimensins", (sgd, Threlfall, 1983) is the third in a series f cmputer prgrams designed t simulate tw dimensinal mechanical systems. This series began with the prgram DAMN (Smith, 1971) which was develped at the University f Michigan under the guidance f Prfessr M. A. Chace. DAMN was then refined and develped t prduce the prgram DRAM, the 'Dynamic Respnse f Articulated Machinery' (Chace, Angell, 1975). An early versin f this prgram was bught by the CEGB in 1977 and used as the basis fr AMP2D. Meanwhile, the team at the University f Michigan frmed their wn cmpany called Mechanical Dynamics Inc., and cntinued t develp DRAM alng different lines frm AMP2D. AMP2D was develped in rder t prvide cmputer mdels f a wide range f tw dimensinal mechanical systems s that their respnse t impsed frces r mtins culd be simulated and the resulting jint frces and mtins f the parts within the system determined. Mst tw dimensinal mechanical systems which can be described in

14 terms f rigid parts cnnected by jints and acted upn by frces and mtin generatrs can be mdelled by AMP2D, including thse with many degrees f freedm and thse with nne. AMP2D allws the parts t be jined t frm clsed lps r pen chains r a mixture f bth. ne part in every system mdelled by AMP2D must be assumed t be mtinless. This is called the grund part and acts as a reference fr the system. The ther parts may have mass r inertia as required by the mdel. The shape f the parts and ther pints f interest such the centre f mass and the psitin f the jints are defined using markers. In AMP2D the jints available t cnnect the parts are either pin jints r slides althugh ther frms f cntact such as cams can be simulated. The relative mtin between the parts can either be impsed directly using a mtin generatr r can result frm the frces acting between them. AMP2D prvides mdels fr frces such as springs and dampers as well as mtin generatrs which may have a cnstant rate r may vary harmnically. It can als mdel impact and mre cmplex frces can be described t the prgram via the user expressins. Unusual generatr functins r particularly cmplex frces can als be input using Frtran rutines. Jint reactin frces and trques are handled autmatically by the prgram and s d nt have t be specified in the definitin f the mechanism.

15 The infrmatin n the structure f the mechanism and its drives is given t AMP2D in the frm f a series f statements. Each statement begins with a keywrd that identifies the item such as a part, marker, frce, etc., which is defined in that statement. Further statements specify the infrmatin required as utput. Each f these statements requests either the relative linear and angular displacement, velcity r acceleratin between any tw pints in the system r the frce and trque in a jint. ther statements cntrl the duratin f the simulatin, the number f measurements t be made during that simulatin and the frm in which the results are t be presented. It is als necessary t define the system f units t be used, the errr levels and the gravitatinal frce. 2.2 The Frictin Mdel fr ÂMP2D The cmputer simulatin f mechanisms was develped primarily fr the autmtive industry where it is desirable t keep frictinal frces at a lw level and as a result frictin was nt included in these simulatins. Hwever, as the CEGB are interested in mechanisms in nuclear systems where frictinal frces can be relatively high, a mdel f the effects f frictin was develped which culd be incrprated int AMP2D (Threlfall, 1978).

16 This mdel was based n the assumptin that, when sliding takes place, the frictinal frce prduced, F, is prprtinal t the nrmal reactin, R, between the surfaces in cntact, i.e.: F = pr (2.1) f frictin. where the cnstant p is knwn as the cefficient The frictinal frce acts t ppse the mtin f the sliding surfaces and s its directin depends upn the relative mtin f these surfaces (Fig.2.1). Thus, this simple mdel has the disadvantage that there is an instantaneus change in the frictinal frce frm +F t -F as the relative velcity f the sliding surfaces passes thrugh er upn a reversal f their mtin. This is cmputatinally undesirable as it wuld cause severe prblems fr the integratin rutines used in AMP2D. This prblem was vercme by incrprating int the frictin mdel the displacement dependence f frictin during very small mtins. This bserved characteristic f frictin is prbably due t the elasticity and plasticity f the surfaces in cntact (Bwden and Tabr, 1964) and results in the frictinal frce behaving fr very small mtins like a spring with hysteresis (Fig.2.2). Initially, the frictinal frce is er fr 8

17 FRICTINAL F r c e + F R e l a t iv e V e l c it y -F F i u rc 2,1 S im p le F r ic t i n a l Frce - V e l c i t y R e l a t i n

18 F r ic t i n a l F r c e D is p l a c e m e n t ^ R F ig u r e 2.2 F r ic t i n a l F r c e D is p l a c e m e n t R e l a t i n S h w in g H y s t e r is is L p

19 er displacement. As the displacement increases the frictinal frce rises rapidly at first and then mre slwly until F = pr. If the velcity then changes sign, the frictinal frce rapidly changes t F = -pr. Hwever, if the velcity drps t er the frictinal frce remains at its current value which means that the mdel is capable f dealing with static frictin and self-jamming. This displacement dependence f frictin was included in AMP2D using a mdel prpsed by Dahl (1968) where: df 2 -- = axf- pr s)^ (2.2) dx where : X is the displacement s is the sign f the sliding velcity # is a cnstant chsen by the user The CEGB derived an equatin frm this mdel (sgd, 1983) which enabled a factr, f, t be calculated fr each integratin time step in the prgram where f is defined as : f = --- (2.3) pr and is determined by: X f,-19s Ax(f, -s) f ^--- (2.4) Xg-19 Ax(f^-s)

20 where : f-^ is the value f f at the last time step Ax is the displacement during the current time step X is the sliding distance (measured frm the pint at which the velcity was last er) at which f =0.95. A secnd bserved characteristic f frictin which is significant enugh t require being incrprated int the frictin mdel used in AMP2D is the decrease in the frictinal frce frm a high "static" value during initial mvements t a lwer "dynamic" ne as the velcity increases. T mdel this characteristic f frictin the prgram calculates an instaneus frictin cefficient u^ using the equatin; Pj = (Pgf-sp^)e-^+ sp^ (2.5) where : f is the factr defined abve p is the cefficient f static frictin Pj is the cefficient f dynamic frictin V is the velcity f the sliding surfaces V is the velcity at which the changever frm the static t the dynamic cefficient f frictin is 95% cmplete The tw equatins 2.4 and 2.5 enable AMP2D t mdel the displacement and velcity dependence f frictin, the user f the prgram prviding the values fr the cefficients f static and dynamic frictin and fr x_ and v_ in rder t fit the mdel t the situatin being simulated. 10

21 2.3 AMP3D-ADAMS Mechanical Dynamics Inc., f Ann Arbr, Michigan, wh had develped the prgram DRAM which was the basis fr AMP2D, went n t develp a further prgrm called ADAMS, the 'Autmatic Dynamic Analysis f Mechanical Systems' (rlanea, 1973) t simulate three dimensinal mechanical systems. This prgram is als used by the CEGB, althugh because there are slight differences between the versin run by the CEGB and that sld by Mechanical Dynamics Inc., the CEGB versin is knwn as AMP3D-ADAMS (Advanced Mechanisms Prgram in 3-Dimensins - Autmatic Dynamic Analysis f Mechanical Systems). AMP3D-ADAMS (sgd and Threlfall, 1984) has been develped t d in three dimensins what AMP2D is capable f in tw dimensins. Althugh its methd f wrking is different, the use and applicatins f the prgram are virtally identical t thse described fr AMP2D. The main difference, apart frm the bvius fact that AMP2D is limited t the simulatin f tw dimensinal mechanisms, is that AMP3D-ADAMS can mdel a greater number f different types f jint. As well as the pin-jints and slides mdelled in AMP2D, AMP3D-ADAMS can als simulate spherical, universal, cylindrical, screw, rack and pinin, and planar jints. It is the simulatin f these additinal jints that 11

22 prevents the straight-frward adaptin f the frictin mdel used in AMP2D fr incrpratin int AMP3D-ADAMS. The prblem lies in the assumptin, used by the frictin mdel in AMP2D, that the resistance t mtin due t frictin in a jint is directly prprtinal t the transmitted lad in that jint. While this is a reasnable assumptin fr the simple pin-jints and slides mdelled in AMP2D, it des nt hld fr several f the ther jints simulated in AMP3D-ADAMS. This can be clearly seen when cnsidering a spherical jint (Fig.2.3). The frictinal resistance in such a jint is dependent nt nly n the cefficient f frictin and the magnitude f the lading but als n the nature f the pressure distributin ver the cntact area between the ball and the scket in the jint which is prduced by that lading. The variatin in the pressure distributed ver the cntact area results in a similar variatin in the frictinal frces prduced which means that the ttal frictinal effects in the jint cannt be simply determined. In additin, the sie f the cntact area and its psitin relative t the axis f rtatin f the jint will als affect the frictinal resistance prduced. The expressins fr the velcity and displacement dependence f frictin used in the mdel incrprated int AMP2D can als nt be simply adapted t the case f a jint such as a spherical jint. This is due t the fact 12

23 Rxis F RTRriH BALL 77 R u 6L DeriNlNfr THE Ex t e n t f t h e C n t a c t AREA S c k e t C rss- s e c t i n f THE P r essu r e Dis tr ib u ti n F ig u r e 2.3 C rss - s e c t i n f a S p h e r ic a l J in t

24 that as the jint rtates the velcity and displacement f pints n the cntact area will vary with their distance frm the axis f rtatin, shwn in Fig.2.3, which means that the effect n the magnitude f the frictinal frces prduced will als vary ver the cntact area. Thus, the develpment f a mathematical mdel f the frictinal effects in a spherical jint was necessary if a frictin mdel is t be prduced fr AMP3D-ADAMS r any ther three-dimensinal mechanism simulatin package. 13

25 CHAPTER THREE Sme Frictinal Effects in a Spherical Jint 3.1 The Expressins btained by Yu. P. Smirnv The starting pint fr the analysis f the effects f frictin in a spherical jint was the paper written by Smirnv (1981) which gave a series f expressins t be used in the determinatin f the frictinal frce and mment prduced in a spherical jint. These expressins are given belw in the ntatin used by Smirnv. In rder t derive the expressins Smirnv began by assuming that the cntact between the ball and the scket f a spherical jint under lad wuld ccur ver a circular area with an axisymmetric distributin f pressure ver that area abut its centre. He defined the sie f this cntact area by the angle (Fig.3.1) and assumed that the pressure, n, was distributed alng the radial c-rdinate ck, accrding t the csine law: n = n^cs(-f^) (3.1) where n^ is the maximum pressure atc=0. Smirmv then stated that the magnitude f the nrmal 14

26 flxis F R t a t i n Edg-E p S c k e t, E d E f CNTACT C l?06s -S E C TI N F Pkessuaf DtsTRtdur/M Figure 3.1 C r s s- s e c t i n f A 5 p h e r ic a u J in t IN b l iq u e Pr t e c t i n S h w in g t h e N t a t i n U s e d by Yu. R S m ir n v

27 reactin, N, resulting frm the distributed pressure was given by the expressin: pû(a N = 27Tn^ cs( ^ ^ ) d (3.2) U Assigning the symbl h t the integral in equatin 3.2, Smirnv gave the fllwing expressin fr the magnitude f the vectr 5, which he defined as the sum f the frictinal frces acting ver the cntact area: $ f ècs ( (c&xsin#-sin#cspcs*)dadp N Trh 7l - (csdcs* + sincx sin^cs/3)^ c/ u (3.3) where he als defined: f as the cefficient f slip frictin f the materials in cntact p as the aimuthal c-rdinate f a pint n the cntact area as the angle between the axis f rtatin f the jint (the angular velcity vectr w) and the radius vectr r, drawn frm the centre f the jint t the centre f the cntact area, which is cincident with the nrmal reactin N. Smirnv derived the frictinal mment prduced in the jint in the frm f tw cmpnents - the mment f slip frictin, M, and the mment f rtatinal frictin, He gave the fllwing expressins fr the magnitudes f these tw cmpnents: 15

28 Zn n*' M f 0^ cs(-^^) (sin%-sin^xsin%cs^3-sinacs*cs#csp)d*dp rn TTh J w /l - (csacs# + sincxsinycsp)^ (3.4) ^ ^ i rn TTh 0 V C(0 sing(cs(-^ ^ ^ )(sinacs*-csd^in#csp)d*dp >/l - (csacs# + sinasindcsp)^ (3.5) Smirnv als gave three expressins which defined the directins f the frictinal frce and the tw cmpnents f the frictinal mment. These expressins are; i = -f^n w X r w X r (3.6) S where f-, = ^ N M = w X r r X w X r (3.7) M where f«= ^ rn = -f^nr sgn(w x r) (3.8) Mr where fq = rn Equatin 3.6 indicates that the frictinal frce prduced in the jint acts in a directin perpendicular t the 16

29 plane in which the axis f rtatin and the nrmal reactin N bth lie. Equatins 3.7 and 3.8 indicate that the frictinal mment lies within this plane and that the mment f slip frictin and the mment f rtatinal frictin are perpendicular cmpnents f this mment, the mment f rtatinal frictin being cincident with the nrmal reactin, N. Unfrtunately, Smirnv gave n details f hw he btained the expressins given abve. Thus, it was necessary t determine hw they were derived in rder t fully understand them and t be able t develp frm them an adequate mathematical mdel f the effects f frictin in a spherical jint. 3.2 The Pressure Distributin In rder t derive expressins fr the frictinal effects prduced in a spherical jint it is first necessary t cnsider the nature f the cntact between the ball and the scket in the jint and t determine the frm f the pressure distributin between them. The nature f the cntact between tw elastic bdies with spherical surfaces is well knwn (Burr, 1981) and can be seen t apply in the case f a spherical jint. As the radius f the scket cavity is necessarily greater than 17

30 that f the ball, cntact will ccur at a sin^/c pint when there is n lad being transmitted thrugh the jint (Fig.3.2a). When a lad is applied t the jint the ball and the scket will elastically defrm arund the pint f cntact t prduce an area f cntact (Fig.3.2b). Frm the symmetry f the spherical surfaces it can be deduced that this area will be bunded by a circle, whse radius will be defined by the angle. The extent f this cntact area depends upn a number f factrs including the magnitude f the lad, the clearance between the ball and the scket, and the elastic prperties f the materials frm which the jint is made. As a result there is a wide range f pssible sies f the cntact area, ver which it is unfrtunately nt pssible t readily determine the pressure distributin. Hwever, there are tw specific situatins in which it is pssible t determine the pressure distributin; ne when the dimensins f the cntact area are small cmpared t thse f the ball and scket and the ther when cntact ccurs ver a full hemisphere. In the first f these situatins the pressure distributin was determined by Heinrich Hert (Timshenk and Gdier, 1970). Hert's analysis shws that in this case the pressure distributin is hemispherical in nature with the maximum pressure at the centre f the cntact area. Althugh this analysis des assume that there is n 18

31 777: 777 BALL S cket F i (xur.e 3. 2 r C n t a c t in a n {Julrdcd S p h e r ic a l J in t (c l e a r a n c e EXA&ERATEpj L ad 777 S c k e t F ig u r e 3. 2 b C n t a c t in A L a d e d S p h e r ic a l J in t (c l e a r a n c e EXAGGERATED^

32 frictin between the surfaces in cntact, Gdman (1962) has extended the thery t include the effects f frictin and he states that it is acceptable t assume that the nrmal stress distributin and thus the pressure distributin remain unchanged by the effects f frictin. Under the hemispherical pressure distributin the pressure p at a pint n the cntact area is given by the expressin; p = P c s ( - f ^ ) (3.9) where : is the angular distance frm the centre f the cntact area p^ is the maximum pressure at = 0. The secnd situatin fr which an apprximatin t the pressure distributin can be determined is that when cntact ccurs ver a full hemisphere. If the ball and the scket are assumed t have the same elastic prperties the situatin is apprximately equivalent t that when a cncentrated frce, P, is acting rthgnally t the bundary plane f a semi-infinite bdy (Fig.3.3). The radial stress distributin abut the pint f lading in such a halfspace has been determined and is given by Lur'e (1964) as: P u = [-( 4m-2 )CS0 + (m-2)] (3.10) ^ 2TrmR 19

33 FiGüRE 3.3 F rc e A ctin 6 n the B unrry Plane F A S e MI- INFINITE B Y

34 where : m is Pissn's number, rati, V the inverse f Pissn's R is the radial distance frm the pint f lading 0 is the angle measured frm the vertical. Taking the Pissn's rati fr steel, v=0.3, this equatin gives : P Cr = rri.2-1.7cs0) (3.11) ^ ttr"^ Thus the pressure distributin ver a hemisphere f radius r fitting perfectly in a hemispherical cavity in the halfspace can be taken as being apprximately given by: P p = ---- ^(1.7cs0-0.2) (3.12) TTr Determining the pressure distributin relative t the maximum pressure p^ which ccurs at 0 = 0 and nting that the angle 0 is equivalent t the angle c defined previusly, equatin 3.12 becmes: p - Pd.133cs(X ) (3.13) Extending the hemispherical pressure distributin defined 20

35 previusly t cntact ver a full hemisphere, i.e. ^=90, equatins, 3.9 becmes: p = PgCS n The cmparisn f the tw equatins, 3,13 and 3.14, shws that the pressure distributin can be cnsidered t be apprximately hemispherical when cntact ccurs ver a full hemisphere. There is a discrepancy between the values that wuld be given by these tw expressins when the angle C is clse t 90^ at which pint equatin 3.13 gives negative values fr the pressure. Hwever, this can be seen t be a cnsequence f the assumptin that the jint culd be represented by a cntinuus bdy. Fr an actual jint the discntinuity between the ball and the scket means that the pressure cannt be less than er, i.e. there can nly be cmpressive frces between the ball and the scket nt tensile nes. Having determined that the pressure distributin can be cnsidered t be hemispherical in nature in the tw situatins cnsidered, it wuld seem reasnable t assume that the pressure distributin is als hemispherical in nature when the extent f the cntact area lies between these tw extremes. This is particularly s as this type f distributin has the necessary characteristics f being axisymmetric, f having the maximum pressure at the centre f the cntact area, and f having n pressure at the 21

36 edges. The validity f this assumptin can be seen in the cmparisn f the theretical values prduced using it with thse btained experimentally. 3.3 The Nrmal Reactin The next stage in the analysis f the effects f frictin in a spherical jint is the determinatin f the nrmal reactin, N, resulting frm the pressure acting ver the cntact area. The distributin f this pressure is assumed t be given by the expressin in equatin 3.9 where : p = As the radii f the ball and the scket in a spherical jint are usually very similar, the lcal defrmatin required t prduce the cntact area will nrmally nt be s great that this area cannt be assumed t be a spherical cap with a radius f curvature r, where r is the radius f the jint. Thus it is pssible t determine the nrmal reactin resulting frm the pressure acting ver the cntact area by first cnsidering the reactin due t the pressure n a ring element f this area and then extending this result t the entire cntact area. The area da^, f a ring element f width rdcx, as shwn in 22

37 Fig.3.4, is given by: da^ = 2nrsin*.rd* 2 = 2Trr sin*df Frm the symmetrical nature f the ring element, it can be seen that the resultant reactin dn^ ver the whle ring, due t the pressure p, will act alng the line frm the centre f the cntact area Z t the centre f the jint 0 and is given by: dnr = 2Trr^sin^xd<.p^cs(-^^).cs< = 2TTr^p^sinfcs^>fcs ( Extending this result t the entire cntact area, the ttal nrmal reactin is given by: N = P( 2., ^TT 2-rrr p^sin(cs^cs, WLC -r-)d<?< J Pfi( Tfr p^ 2sinc<cse(cs(-^^)di>^ (3.15) Representing the integral in this expressin by the symbl I, the nrmal reactin is given by: N = TTr^p^I (3.16) where: pi I = 2sin(cs(cs ( ^-~)d(j< (3.17) Cmparing the expressin btained fr the nrmal reactin 23

38 'View frm rue D irectin f. THE Nrmal \ R eactin N rsin Figure 3.4- The C ntact Rrea in A S p h e r ic a l J in t

39 in equatin 3.15 with that given by Smirnv in equatin 3.2: n^ N = 2ttp cx^cscxcs ( di 1/ there can be seen t be several discrepancies between the tw expressins. Similar discrepancies were als fund between the expressins derived fr the frictinal frce and mments and thse given by Smirnv. Smirnv was cntacted in an attempt t discver the reasn fr these discrepancies and he indicated that the errrs were in his calculatins. 3.4 The Frictinal Frce Having determined the nrmal reactin resulting frm the pressure distributed ver the cntact area between the ball and the scket in a spherical jint, it is then pssible t derive an expressin fr the frictinal frce prduced ver this cntact area when the ball is rtated in the scket. The spherical jint, with the ball rtating at an angular velcity w abut an axis inclined at an angle t the line f actin f the nrmal reactin, N, is shwn in Fig.3.5. The frictinal frce ver the whle cntact area can be btained by first determining the frictinal frce n an 24

40 Axis F R t a t i n r<fc Directin^ F View Shwn IN Fi V iew FRM D ir ecti n F the N r m a l R e a c t i n N F i& u re 3.5 S p h e r ic a l T i n t with rue Axis f R t a t i n AT AN ANGLE X T THE NRMAL REACTIN N

41 elemental area, whse psitin at the pint A is defined by the angles C and p, and then extending this result t cver the entire cntact area. The area JA f the element can be seen frm Fig.3.5 t be given by: da = rdcx.rsincdp = r sine(d(3ccfp The pressure distributin is given by equatin 3.9 as: p = Thus, the nrmal reactin n the element, dn, due t the pressure is given by: dn = r^p^sin(xcs(-^^)dkdp and s the magnitude f the frictinal frce n the element, df, which is prprtinal t the nrmal reactin, is given by: df = pdn = pr^p^sin(cs( Y^)d(dp (3.18) where p is the cefficient f frictin. The frictinal frce will act t ppse the mtin f the element. Thus as the element is n the surface f the ball rtating abut the axis represented by the vectr w, the 25

42 frictinal frce will act in the plane that passes thrugh the pint A and is als perpendicular t the axis f rtatin. In this plane, which is shwn in Fig.3.6, the frictinal frce acts rthgnally t the line drawn frm the pint A t C, the pint at which the axis f rtatin passes thrugh the plane, in the directin which ppses the rtatin. The pint A lies n the arc SUT prduced by the intersectin f the plane described abve and the cntact area between the ball and the scket. Because f the symmetrical nature f the cntact area and the pressure distributin ver it, the pint D, which lies ppsite A n this arc as shwn in Fig.3.6, will have a frictinal frce acting n it f equal magnitude t that n A. The frictinal frces at these tw pints can each be divided int tw cmpnents; ne f which is cincident with the line drawn frm A t D, the ther being perpendicular t this line. Frm Fig.3.6 it can be seen that the tw cmpnents perpendicular t the line AD are equal in magnitude but act in ppsite directins. Thus it is nly the cmpnents cincident with the line that need t be taken int cnsideratin in determining the ttal frictinal frce prduced in the jint. Fig. 3.6 shws that the magnitude f this cmpnent at the pint A, (ÏFCS0, depends upn the value f the csine f the angle 0, which is given by: 26

43 The B u n d a r y THE CKTACT f f^rea 6fcs <ff CS F kture 3.6 A Vie w f t h e S p h e r ic h l J in t A ln- THE Axis f Rtatin

44 b b cs = = -p=k===^ (3.19) c }la^ + b^ where a, b and c dente the lengths f the sides f the triangle ABC. Frm Fig.3.5 the distance a frm A t B can be seen t be given by: a = rsincsinp (3.20) The distance b frm C t B can be determined gemetrically frm the dimensins f the jint shwn in Fig.3.7. Frm this diagram it can be seen that: b = (d + e) sin# - g where : d = rcsc# e = ktan # g = k/cs# Frm Fig.3.5: k = rsinc<csp Thus the distance b is given by: b = (rcscx + ktan#)sin# - k/cs# = r(cs*sin# + sinacsptan#sin# - sindsp/cs#) 27

45 Figure 3.7 The D imensins within a S pherical J in t

46 2 = r(csdsin# + sinfcsp x (sin # - l)/csif) b = rccsdsin# - sindcspcs#) (3.21) Substituting the expressins btained fr a and b int equatin 3.19 the csine f the angle 0 is given by: csdsin# - sindcspcs# CS0 =, 2 2 ' ' 2 vsin dsin p + (csdsin# - sindcspcs#) Appendix 1 shws that the expressin: sin dsin p + (csdsin# - sindcspcs#) is equivalent t: (csdcs# + sindtspsin*) Thus the csine f the angle 0 can be given as: csdsin# - sindcspcs# CS0 = ^ (3.22) /l - (csdcs# + sindcspsin#) as : and s the frictinal frce cmpnent dfcs0 is given jf(csdsin# - sindcspcs#) dfcs0 = y_... (3.23) vl - (csdcs# + sindcspsin#) Substituting the expressin btained in equatin 3.18 fr df, the cmpnent dfcs0 is given by: 28

47 ctfc s 0= pr p^sindcs(-^-^) (csdsin#-sin(cspcsx)(fexcfp (3.24) - (csdcs# + sindcspsin#)^ Extending this expressin t cver the entire cntact area, the ttal frictinal frce, F, is given by: 2rr C F=pr^p, sindcs(-^-~ )(csdsin# - sindcspcs#)dddp (3.25) Vl - (csdcs# + sindcspsin#) 0 U Taking the expressin btained fr the magnitude f the nrmal reactin, which is given in equatin 3.16 as: N = TTr p^i where I is given in equatin 3.17 as: I = 2 s i n dc s dc s ( ^ ^ ) d d the magnitude f the resultant frictinal frce, F, can be fund in terms f its rati t that f the nrmal reactin. This remves the term fr the maximum pressure, which is unlikely t be knwn, frm the resulting expressin given by: 2ir dg F _ N TTl sindcs( ^ ^ )(csdsin#-sintxcspcs#)dddp Vl - (csdcs# + sindcspsin#) (3.26) The equivalent expressin given by Smirnv in equatin

48 i s : Zir F p Cl?cs(-y ^ ) (csdsin*-sindcspcs#)dddp N nh U J Jl - (csdcs# + sindcspsin#) where frm equatin 3.2: 1d h = 2<??csdcs( j ^ ) d d J As the ttal frictinal frce ver the cntact area is cmprised slely f cmpnents acting parallel t the line AD, shwn in Fig.3.6, its directin will als be parallel t this line. Frm Figs.3.5 and 3.6 it can be seen that the line AD is perpendicular bth t the vectr w representing the axis f rtatin and t the directin f the nrmal reactin, N. This means that the directin f the frictinal frce can be btained frm the vectr prduct f w and N. As the directin f the cmpnents f the frictinal frce is frm A twards D, the vectr F representing the resultant frictinal frce ver the cntact area, and shwn in Fig.3.8, can be given by: w X N F = f.n -- (3.27) Iw X N F where the factr f, represents the rati given in N equatin

49 FIGURE 3.8 T h e D i r e c t i n s f th e F r i c t i n a l Frce a n d M m e n t s Pr d u c e d IN A SPHERICAL T i n t

50 3.5 The Frictinal Mment As well as the resultant frictinal frce described in the previus sectin,the frictinal frces acting ver the cntact area f the spherical jint als prduce a mment acting abut the centre f the jint. this frictinal mment is nt knwn, As the directin f it is necessary t determine the mment in the frm f cmpnents acting abut mutually perpendicular axes. In rder t simplify the methd f deriving evaluate these cmpnents, the expressins required t the mst cnvenient axes t cnsider are as fllws: 1) the axis cincident with the line f actin f the nrmal reactin N, shwn by the line Z in Fig.3.5, 2) the axis shwn by the line PQ in Fig. 3.5 which is perpendicular t the first axis and lies in the plane cntaining the nrmal reactin N and the axis f rtatin shwn by the vectr w, 3) the axis, perpendicular t the plane described abve, which passes thrugh the centre f the jint, the pint 0 in Fig A cnsideratin f the frictinal frces acting n the cntact area, as given belw, shws that there is n 31

51 cmpnent f the frictinal mment acting abut the third axis described abve. In the previus sectin it has been shwn that the frictinal frces acting n the elemental areas at the pints A and D shwn in Figs.3.5 and 3.6 can bth be represented by tw cmpnents cifcs0 and cîfsin0. The dfcs0 cmpnents have been defined as being cincident with the line AD and Fig.3.5 shws that this line is parallel t the axis being cnsidered. Thus the cîfcs0 cmpnents f the frictinal frce prduce n mment abut this axis. It has als been shwn in the previus sectin that the cffsin0 cmpnents f the frictinal frces at A and D are equal in magnitude and act in ppsite directins. Frm Fig.3.5 and 3.6 it can be seen that the lines f actin f these dfsin0 cmpnents are f equal distance frm the third axis described abve. Thus, when taken tgether, these cmpnents als prduce n mment abut the third axis. As A and D can be taken t represent any pair f pints n ppsite sides f the cntact area, it can be seen that there is n cmpnent f the ttal frictinal mment acting abut this third axis. Thus the ttal frictinal mment acts abut an axis lying in the plane described abve as cntaining the line f actin f the nrmal reactin, N, and the axis f rtatin f the jint. Fr cnvenience the tw perpendicular cmpnents f the frictinal mment lying in this plane 32

52 are described by the same terms as used by Smirnv. Thus the cmpnent acting abut an axis cincident with the line f actin f the nrmal reactin is called the mment f rtatinal frictin and the ther cmpnent is called the mment f slip frictin The Mment f Slip Frictin The mment f slip frictin has been defined as the cmpnent f the ttal frictinal mment acting abut the axis represented by the line PQ, shwn in Fig.3.5. This cmpnent can be determined by cnsidering the mment prduced abut the axis by the frictinal frce acting n the elemental area at A and then integrating t btain the ttal mment prduced abut this axis by the frictinal frces acting ver the entire cntact area. It has been shwn that the frictinal frce n the element at A can be divided int tw cmpnents, dfcs0 and dfsin0, f knwn directins as shwn in Fig.3.6. The expressin fr the cmpnent dfcs0 is given in equatin 3.23 as; dfcs0 = df(csasin#-sindcspcs*l Jl - (cs#cs^ + sinddspsindj^ where frm equatin 3.18: 33

53 cff = p r p ^ s in < x c s (-Y ^ )d ((fp T determine the ther cmpnent, cffsin0, it is necessary t find an expressin fr the sine f the angle 0. Frm the gemetry f the triangle ABC in Fig 3.6 it can be seen that ; sin0 = + b Substituting the expressins btained in equatins 3.20 and 3.21 fr a and b, this expressin becmes: sincsinp sin0 = 2 "2 sin (Xsin p+ (csasin* - sinacspcs#) Appendix 1 shws that: sin a sin p + (csasin# - sinacspcs 2^) is equivalent t: (csacs# + sinaspsin*) Thus the magnitude f the frictinal frce cmpnent, jfsin0 can be given by: dfsinasinp dfsin0 =. - (3.28) yl - (csacs# + sinacspsin#)^ As stated befre, the cmpnent f the frictinal frce 34

54 dfcs0 at A acts alng the line AD. Fig.3.7 shws that the perpendicular distance frm this line t the axis represented by PQ is given by the distance d where: d = rcsa Thus the mment, which acts abut the axis represented by PQ and is due t the frictinal frce cmpnent dfcsq, is given by: df(csasin*-sinacspcs*).rcsa r' ' '~2~ Vl - (csacsü+sinacspsin#) dfrccs asina-sinacsadspcs#) Vl - (csacsf+sinacspsin#)^ (3.29) Fig.3.6 shws that the ther cmpnent f the frictinal frce, dfsin0, acts in a directin parallel t the line CB at a distance a which frm equatin 3.20 is given by: a = rsinasinp Fig.3.7 shws that the line CB lies in the same plane as the axis represented by PQ but inclined by the angle "6 t it. Thus the mment, dmgg, which acts abut the axis represented by the line PQ and is due t the cmpnent dfsin0 f the frictinal frce n the element, is given by; 35

55 dfsinasinp. sin*.rsinasinp dmgg = 7-.j--- yl - (csacs*+sinacspsin*) dfrsin asin psin* /l - (csacsd+sinacspsin*)^ (3.30) Frm Figs.3.5 and 3.6 it can be seen that bth f the mments dmg^ and dmgg act in the same directin abut the axis represented by PQ. Thus the ttal mment dmg abut this axis due t the frictinal frce n the element at A can be fund by taking the sum f these mments and is given by; dmg (fmgç + dmgg dfr(cs asin*-sinacsacspcs*+sin asin psin*) y1 - (csacsa+sinacspsin*)^ dfr(sin*-sin^xsin*-sinacsacspcs*+sin^asin* c s ^ p s \r\i) Jl - (csacs*+sinacspsin*)^ dfr(sina-sinacsacspcs*-sin acs psin*) Jl - (csacs*+sinacspsin*)^ Substituting the expressin given in equatin 3.18 fr the frictinal frce Jf, this expressin fr the mment dmg becmes : 3 CTT pr p^sinacsc%g^ (sin*-sinacsacspcs* f - 51n^cs^psm*) (^ctp dmg= i' -- "2 yl - (csacs*+sinacspsin*) 36

56 Extending this result t cver the whle f the cntact area, the mment f slip frictin Mg is given by; sin«cs(-j ^ (sin*-sinacsacspcs* Mg=pr p^ s m ^ c c s ^ p s m * ) c L c d p Jl - (csacs*+sinacspsin*)^ (3.32) Then, taking the expressin given fr the nrmal reactin N in equatin 3.16 as; N = Tir p^i where I given by equatin 3.17 as; I = ncc att 2 s inac sac s ( 2 "^^) d a the magnitude f the mment f slip frictin can be fund in terms f its rati t the magnitude f the nrmal reactin, as the magnitude f the frictinal frce was befre. The rati is given by; rn tti U c _._2 sinacs (*^^) (sin*-sin^xsin*cs^p - Sinacsacs pcsyjdajp Jl - (csacs*+sinacspsin*) (3.33) The equivalent expressin given by Smirnv in equatin 3.4 IS ; 2n C a 2 /air cs( N, (sin*-sin <Xsin*cs 2, p -sinacsacspcsddadp rn Trh J Jl - (csacs*+sinacspsin*) 37

57 where frm equatin 3.2: n^ h = 2^csacs ( )da 0 The mment f slip frictin has been defined as the cmpnent f the ttal frictinal mment acting abut the axis which is perpendicular t the line f actin f the nrmal reactin N and als lies within the plane cntaining bth that line f actin and the axis f rtatin f the jint represented by the vectr w. Thus the directin f the mment f slip frictin can be btained frm the vectr prduct f the vectr N and the vectr prduct w x N. As shwn in Figs.3.5 and 3.6 the directins f the cmpnents f the frictinal frces acting n the cntact area indicate that the vectr. Mg, representing the mment f slip frictin is directed alng the line PQ frm P twards Q. Thus the vectr Mg, shwn in Fig.3.8, is given by: M = N X (w X N) (3.34) -S w X N Mg where the factr f«represents the rati given in ^ rn equatin

58 3.5.2 The Mment f Rtatinal Frictin The mment f rtatinal frictin has been defined as the cmpnent f the frictinal mment acting abut the axis cincident with the line f actin f the nrmal reactin, which is shwn by the line Z in Fig.3.5. As with the mment f slip frictin, this cmpnent f the ttal frictinal mment can be determined by cnsidering the mments prduced abut the axis by the tw cmpnents f the frictinal frce n the elemental area, dfcs0 and dfsin0, and then integrating t extend this result t the entire cntact area. Frm Figs.3.6 and 3.7, it can be seen that the perpendicular distance frm the line f actin f the cmpnent dfcs0 t the axis represented by the line Z is given by the distance k, while Fig.3.5 shws that: k = rsindcsp Cmbining this expressin fr k with that given fr the cmpnent dfcs0 in equatin 3.23, the expressin fr the mment abut the axis represented by Z and due t the cmpnent dfcs0 is given by: df(cs6(sin*-sinc(cspcs*).rsindcsp dmr RC 2' 1 - (CSTCs*+sin^cspsin*) 39

59 2 dfrsinf(csfcspsin*-sinfcs p cs*) =. : Y=---- (3.35) Jl - (csacs*+sinxcspsin*) As shwn in the previus sectin the cmpnent dfsin0 acts parallel t the line CB in Fig.3.6 at a distance a given by equatin 3.20 as; a = rsinasinp and Fig.3.7 shws that the line CB lies in the same plane as the line Z but inclined at an angle f (90 -*) t it. Thus using this expressin fr the distance a and that given fr the cmpnent dfsin0 in equatin 3.28, the mment,, abut the axis represented by Z and due t the cmpnent cîfsin0 f the frictinal frce n the elemental area is given by: dm, dfsinasinp.cs*.rsinasinp ^ Jl - (csacs*+sinacspsin*)^ 2 dfrsin asin pcs* Jl - (csacs*+sinacspsin*) (3.36) It can be seen frm Figs.3.5 and 3.6 that the tw mments dmrc and dm^^g due t the frictinal frce cmpnents dfcs0 and dfsin0 act in ppsite directins abut the 40

60 axis represented by the line Z. Thus the resultant mment acting abut this axis and due t the frictinal frce n the elemental area at A, is given by the difference between dmj^^ and dm^^g. The directin f this mment can be defined as being psitive when it acts in ppsitin t the rtatin f the jint. As it is the mment dmj^g due t the cmpnent dpsin0 which acts in this directin, this means that the resultant mment, dm^^, is given by: dmr dm^g - dmp_ç dfrsina(sinasin^pcs*-csacspsin*+sinacs^pcs*) Jl - (csfcs*+sinacspsin*) dfrsinac sinacs*-csacspsin*) Jl - (csacs*+sinacspsin*)^ Substituting the expressin given in equatin 3.18 fr the magnitude f the frictinal frce df, this expressin fr the mment dm^ becmes: pr p^sin acs(-^^) (sinacs*-csacspsin*) dadp dm R = r --- ' "2* Jl - (csacs*+sinacspsin*) (3.37) Integrating t extend this result t the entire cntact area, the mment f rtatinal frictin, Mj^, is given by: 41

61 Mj^=pr 2ir ^ sin^cs (sinccs*-cs(cspsin*)d«i^dp (3.38) Jl - (csccs*+sin<cspsin*) w J As previusly, if the expressin fr the nrmal reactin N is taken as given by equatin 3.16: N = TTr p^i where the term I is given by equatin 3.17 as: n< I = 2 s incc s<c s (-^-~) d < the magnitude f the mment f rtatinal frictin can als be given in terms f its rati t the magnitude f the nrmal reactin. This rati is given by: rn tti v _._2. sin^g^cs( 2 ( s in txc s* CSecspsin*)cbidp Jl - (cs*cs*+sinxcspsin*) (3.39) This expressin can be cmpared t the equivalent ne given by Smirnv in equatin 3.6 as: rn nh w Jp ^sinr/cs(-^ ^ )(sinacs*-csdcspsin*)d*dp Jl - (csdbs*+sindbspsin*j where frm equatin 3.2: h = 2 0^csacs ( ^ ^ )d( 42

62 The vectr representing the mment f rtatinal frictin has been defined as being cincident with the line f actin f the nrmal reactin, N, shwn by the line Z in Fig.3.5. As the mment has been defined as being psitive when it acts in ppsitin t the rtatin f the jint, the vectr ^ is directed frm 0 twards Z prvided the angle between the vectr N representing the nrmal reactin and the vectr w representing the angular rtatin f the jint is less than 90. If this angle is greater than 90 the rientatin f the vectr ^ is reversed because the directin f the rtatin relative t it is reversed. T shw this characteristic f the mment, the expressin fr the vectr shwn in Fig.3.8, is given by: = -f^rn sgn(w.n) (3.40) where the factr f- represents the rati given in rn equatin 3.39 and where the term sgn(w.n) is either +1 r -1 depending upn the sign f the scalar prduct f the vectrs w and N. 3.6 The Evaluatin f the Expressins T summarie the results btained in the preceding analysis, the frictinal effects prduced in a spherical 43

63 jint have been shwn t cnsist f a frictinal frce F and a frictinal mment given in the frm f tw perpendicular cmpnents, the mment f slip frictin Mg and the mment f rtatinal frictin M^^. Expressins fr the frce F and the tw mments M^ and M«have been K derived and are given, in equatins 3.27, 3.34 and 3.40 respectively, as: w X N w X N -fr Mg = N X (w X N) w X N M^ = -f^rn sgn(w.n) where the expressins fr the factrs f^, f2 and f^ are given by equatins 3.26, 3.33 and 3.39 as: sin^cs(-j^) (csasin*-sinacspcs*) d^^dp = TTl Jl - (csdcs*+sin#cspsin*)^ sin(cs(-^;) (sinv -sin «sinîtcs^ -8inacsaCspcs#)d*dp Jl - (csdcs*+sindcspsin*) '\(e P TTl sin4xcs(-"2 ^) (sin*cs*-cs#cspsin*) d^dp Jl - (cs#cs*+sinacspsin*) 2- V V 44

64 and where the term I is given by equatin 3.17 as: f)0(c I = 2sin<xcsacs( ^^)d( *0 These expressins indicate that the magnitude and directin f the frictinal frce and mment prduced in a spherical jint are dependent upn the fllwing factrs: 1) the directin f the axis f rtatin represented by the vectr w 2) the magnitude and - 3) the directin f the nrmal reactin N 4) the cefficient f frictin p 5) the angle defining the extent f the cntact area 6) the radius, r, f the spherical jint and 7) the angle *, the angle between line f actin f the nrmal reactin N and the axis f rtatin. f these factrs, three - the radius f the jint, the cefficient f frictin and the directin f the axis f rtatin - are independent variables which have t be 45

65 supplied in rder t determine the frictinal frce and mment. The angle * can be determined frm the directins f the nrmal reactin and the axis f rtatin. The angle is dependent upn a number f additinal factrs as well as the magnitude f the nrmal reactin N which in turn results frm the sum f the applied lad n the jint and the frictinal frce prduced. The methds f determining bth the angle and the magnitude and directin f the nrmal reactin N are described in the fllwing chapters. As well as these factrs the determinatin f the frictinal frce and mment prduced is als dependent upn the evaluatin f the integrals cntained within the expressins given abve. The integral given in equatin 3.17 as : 1 I = 2sin(cs(cs ( has been evaluated in Appendix 2 and is given as: -27Tsin2f ) ^ I = y fr ( = ~ 4 A Unfrtunately the evaluatin f the duble integrals in 46

66 equatins 3.26, 3.33 and 3.39 is nt as simple. As these integrals are nt cmprised f elementary functins, they have t be evaluated numerically, which is dne using Simpsn's rule fr tw-dimensins (Salvadri and Barn, 1952). The three duble integrals cntained within the expressins are f the frm: Zir f (f, p)df(dp p «, These integrals can be evaluated by first integrating with respect t d while regarding p as a cnstant and then integrating the resultant functin with respect t p. Thus the duble integral can be cnsidered as: Zn 1 J f ((/, p)d( ( dp (3.41) Simpsn's rule fr a single integral evaluates that integral by the fllwing frmula : f(x)dx= [f (x^)+4f (x. )+2f (Xg)+4f (X;,) f (x «) I a 2 ^ n- where : h = b - a n 47

67 and where : n is the number f subintervals ver the range f integratin and must be even. h is the width f the subintervals. f(x.) is the value f the functin being integrated at the pint x = x. n the range f integratin. This apprximatin is extended t the duble integrals being cnsidered by dividing the range f the 'uter' integral, as shwn by the expressin in 3.41, int an even number f subdivisins and then, using the given value f ne<c p, evaluating the 'inner' integral - f(a\ p)d( - at each subinterval by Simpsn's rule. J These values fr the 'inner' integral at each subinterval can then be used t calculate the 'uter' integral by the same frmula. A cmputer prgram was written t evaluate, by this methd f numerical integratin, the expressins given fr the factrs f2, f2 and f^, which cntain the duble integrals as shwn in equatins 3.26, 3.33 and The results btained are shwn in Fig.3.9. As the factrs are directly prprtinal t the cefficient f frictin p, the results are given in the frm f the ratis f the three factrs t the cefficient f frictin. These ratis are pltted against values f the angle *, the angle between the line f actin f the nrmal reactin and the axis f rtatin, fr varius values f the angle, which defines the extent f the cntact area. 48

68 >Q C I K M. Ul < Uj in '<3' H V ) Z a: c Uj -J u <3 a: (M k c Z Ul J G L cj cr C UJ c v!) LT

69 >0 CD m II! > u. X, Uj -J Z ce Uj Xh ir> H v> % ëc C ce U V- Vō-J CL. «j h k c c UJ CD < r " c lu c D lz

70 G VÛ II c li. X Uj -u Z ce Vu X h m <r H (/I Z X ce a Uj V- V ō-1 d. r < 3 Z X CNJ k X c vu C L cr CT) lu ce3

71 X) 03 fn. II ^1 C 2; ^ I Uj ~j Cj % q: Uj Xh in VI % te c Ui V- V-'! -J Cl_ N3 *$* 2 te J k c C UJ N fnj C > in -d tr 01 Ui ce 31 Ll

72 The results shw a number f characteristics which crrespnd with the expected behaviur f the frictinal frce and mments and these are described in the fllwing sectin. 3.7 The Characteristics f the Frictinal Frce and Mments The Frictinal Frce The first thing that can be seen frm the characteristics f the factr f^, as shwn in Fig.3.9, is that there is n resultant frictinal frce prduced when the axis f rtatin is aligned with the directin f the nrmal reactin ( = 0 ). This is due t the fact that in this psitin the axis f rtatin passes thrugh the centre f the cntact area and s the axisymmetric nature f the pressure distributin ver the cntact area means that the sum f the frictinal frces prduced is er. As the angle increases, the cmputed results shw that the frictinal frce als increases until the pint is reached when the axis f rtatin n lnger passes thrugh the cntact area (Ï > At this pint the magnitude f the fractinal frce, as shwn by the factr f2, is clse t its maximum value which is finally reached when the axis f rtatin is perpendicular t the 49

73 directin f the nrmal reactin (^ = 90 ). This behaviur results frm the fact that the frictinal frce prduced at a pint n the cntact area acts tangentially t the perpendicular line frm the axis f rtatin t that pint, in the directin that ppses the rtatin. Thus, when the axis f rtatin passes thrugh the cntact area, a prprtin f the frictinal frces, prduced ver that area will act in ppsing directins; and, as the pint at which the axis f rtatin intersects the cntact area mves twards the edge f the area, this prprtin decreases and s the resultant frictinal frce increases. nce the axis is beynd the cntact area the frictinal frces act largely in the same general directin and s prduce apprximately the maximum resultant frce. Further increases in the angle then nly prduce slight increases in the unifrmity f the directins f the frictinal frces, resulting in a slight increase in the magnitude f the resultant frictinal frce. The cmputed results presented in Fig.3.9 als shw that the rati f the factr f^ t the cefficient f frictin \x has a maximum value f apprximately 1.0 when the angle (^ is small which increases slightly fr larger sies f the cntact area. This is due t the fact that when the cntact area is small it is als rughly flat. The magnitude f the frictinal frce prduced at a pint n 50

74 this area is given by the prduct f the nrmal reactin at that pint and the cefficient f frictin. Thus if the cntact area is rughly flat and the frictinal frces ver it act largely in the same directin, as when the axis f rtatin des nt pass thrugh it, the rati f the resultant frictinal frce t the resultant nrmal reactin, which is the definitin f the factr f^, is apprximately equivalent t the value f the cefficient f frictin p. As the sie f the cntact area increases it can n lnger be cnsidered t be apprximately flat. Thus, there are cmpnents f the frictinal frces at the leading and trailing edges f cntact area as the jint rtates which act in ppsing directins. This shuld result in a reductin in the magnitude f the resultant frictinal frce prduced. Hwever, the increasing curvature f the cntact area als means that there are cmpnents f the nrmal reactins at pints diametrically ppsite each ther n the cntact area acting in ppsing directins with the result that, fr the same resultant nrmal reactin, the reactin at these pints is higher prducing a higher frictinal frce. These tw effects cunteract each ther, and s the maximum value f the factr f^ des nt vary greatly until the cntact area is very large. It then increases because the secnd effect is significant ver all but the centre f the cntact area and nt just at the leading and trailing edges. 51

75 3.7.2 The Mment f Slip Frictin The mment f slip frictin exhibits characteristics very similar t thse f the frictinal frce as can be seen by cmparing the tw factrs f^ and 2 in Fig.3.9. The similarity results frm the definitin f this mment as the cmpnent f the ttal frictinal mment acting abut the axis, perpendicular t the line f actin f the nrmal reactin, which lies in the plane cntaining bth this line f actin and the axis f rtatin, as shwn in Fig.3.8. Thus, when the axis f rtatin is aligned with the directin f the nrmal reactin (IT = 0 ) it is perpendicular t the axis the mment is defined as acting abut and s the magnitude f the mment is er. It fllws that the mment reaches its maximum value when the axis f rtatin is aligned with the axis which the mment is defined as acting abut (2T = 90 ). Fig.3.9 shws that when the angle is small the values f the factrs f^ and 2 are practically identical. This is due t the fact that, when the cntact area is small, the mment f slip frictin can be cnsidered t be prduced slely by the resultant frictinal frce. As described previusly, when small cnsidered t be a flat surface. the cntact area can be This means that, because f its psitin relative t the axis abut which the mment f slip frictin is defined as acting, the sum f the mments prduced abut this axis by the cmpnents f 52

76 the frictinal frces that are nt part f the resultant frictinal frce is er. The perpendicular distance frm the cntact area, when small and rughly flat, t the axis abut which the mment f slip frictin acts is apprximately equal t the radius f the jint r. Thus, in this situatin, the magnitude f the mment f slip frictin Mg is apprximately equal t the prduct f the radius r and the resultant frictinal frce F. Then cnsidering the definitins f the factrs f^ and ±2 which are : it can be seen that they will be rughly equal. When the cntact area is larger, the increased curvature f the surface means that there are cmpnents f the frictinal frces, in additin t thse included in the resultant frce, which cntribute t the mment f slip frictin and this results in the values f the factr f2 increasing relative t the crrespnding values f f^ as shwn in Fig.3.9. At the same time the increased curvature als means that the perpendicular distance frm the axis abut which the mment f slip frictin is defined as acting t the line f actin f the resultant frictinal frce prduced ver the cntact area is reduced and s the increase in the magnitude f the mment and 53

77 hence the factr ±2 is nt as great as it therwise wuld have been The Mment f Rtatinal Frictin As shwn by the factr f^ in Fig.3.9 the mment f rtatinal frictin has cmpletely different characteristics frm thse described fr the mment f slip frictin and the frictinal frce. Its maximum value ccurs when the axis f rtatin is aligned with the directin f the nrmal reactin = 0 ), which is as expected because this mment is defined as acting abut the axis cincident with the directin f the nrmal reactin. Thus it fllws that the magnitude f this mment is er when the axis f rtatin is perpendicular t the directin f the nrmal reactin (ï = 90 ). The cmputed results als shw that the maximum value f the mment f rtatinal frictin is very dependent n the sie f the cntact area as given by the angle c^. The mment is very small fr a small area f cntact and increases rapidly with the sie f the cntact area. This is due t the fact that the axis abut which this mment is defined as acting passes thrugh the centre f the cntact area. Thus the greater the sie f the cntact area, the greater is the mment prduced by the frictinal frces abut that axis. 54

78 When cntact ccurs ver a cmplete hemisphere = 90 ), Fig.3.9d shws that the maximum value f the factr f^ is equal t the cefficient f frictin p. This is a result f the defined hemispherical pressure distributin acting ver a cmplete hemisphere. The results pltted in Fig.3.9 shw that there is a change in the way the mment f rtatinal frictin varies with the angle at apprximately the pint when the axis f rtatin n lnger passes thrugh the cntact area. This is due t the fact that, as described in Sectin 3.7.1, when the axis f rtatin intersects the cntact area a prprtin f the frictinal frces prduced ver that area act in ppsing directins. It is these ppsing frces that largely prduce the mment abut the axis thrugh the centre f the cntact area and as the pint f intersectin f the axis f rtatin mves twards the edge f the cntact area the prprtin f ppsing frces, and thus the mment, is reduced. When the axis f rtatin n lnger passes thrugh the cntact area the remaining mment is prduced by the curvature in the directins f the frictinal frces which are still tangential t the perpendicular lines frm the axis f rtatin. This remaining mment is steadily reduced as the axis f rtatin cntinues its shift away frm the cntact area. 55

79 CHAPTER FUR The Frictinal Effects in Relatin t the Applied Lad The previus chapter has shwn hw expressins were derived that enable the frictinal frce and the cmpnents f the frictinal mment t be determined relative t the magnitude and directin f the nrmal reactin within a spherical jint. Hwever, as the nrmal reactin results frm a cmbinatin f the applied lad acting n the jint and the frictinal frce prduced within it, the means f determining the frictinal effects relative t the magnitude and directin f the applied lad cannt be btained by a straight-frward extensin f these expressins. It is necessary t cnsider the effect f the frictinal frce n the nrmal reactin if that frce and the cmpnents f the frictinal mment are t be determined relative t the applied lad. 4.1 The Frictinal Frce Fig.4.1 shws a spherical jint viewed in the plane cntaining the applied lad P and the resulting frictinal frce F. The nrmal reactin N is given by the sum f these tw frces. Thus: 56

80 Rxis F R tptj n E & e r S c k e t C r s s - s e c t i n f P r e s s u r e D is t r ib u t i n FieuRE 4-.1 C r s s - s e c t i n f A S p h e r ic a l T i n t IN b liq u e Prtectin THRUGH THE P l a n e C n t a in in q t h e APPLIED L ad AND THE F r ic t i n a l F r c e

81 p + F + N = 0 (4.1) Fig.4.1 als shws the axis f rtatin which is represented by the angular velcity vectr w and which des nt lie in the plane cntaining the frces mentined abve. The angle 0 is the angle between the axis f rtatin and the directin f the applied lad P while the angle between the axis f rtatin and the line f actin f the nrmal reactin has already been dented as the angle ^. By selecting a particular c-rdinate system fr the jint as shwn in Fig.4.2, the vectrs w, P and F, representing the axis f rtatin, the applied lad and the resultant frictinal frce respectively, can be expressed as: w = w(-sin0i + cs0k) (4.2) P = -Pk (4.3) F = F dpi. + mpj + npk) (4.4) where Ip, mp and np are the directin csines f the frictinal frce F. Then, by substituting the expressins given fr the ap'plied lad P and the frictinal frce F int equatin 4.1, the nrmal reactin N is shwn t be given by: 57

82 F ig u r e 4.2 T he r ie n t a t i n f the S pherical J in t R e l a t iv e t t h e C h s e n C- R D i N A T G S y s t e m

83 N = -Flpi - Fmp2 + (P Frip)k (4.5) These equatins shw that in rder t determine the frictinal frce F it is necessary t derive expressins fr its magnitude F and directin csines Ip, mp and np in terms f knwn quantities. Fr a frictin mdel which is t be incrprated int a simulatin package such as AMP3D-ADAMS, the knwn quantities are the magnitude and directin f the applied lad, the directin f the axis f rtatin and frm these the angle 0 between them. All f these quantities wuld be supplied by the prgram at each step f the simulatin. The required expressins can be derived frm thse btained fr the frictinal frce relative t the nrmal reactin in sectin 3.4, where equatin 3.27 gives: w X N Iw X N and where the factr f^, representing the rati F/N, is given in equatin 3.26 as: F _ JJ sinccsc-(csasin*-sinacspcs*) dcdp N nl Jl - (csdcsa+sinacspsin*)^ By substituting the rati F/N fr the factr f^ in equatin 3.27, the frictinal frce F can be given as: 58

84 w X N F = F -- (4.6) w X N Frm this expressin it can be seen that the directin f the frictinal frce F is perpendicular t that f the nrmal reactin N. Thus the magnitude f the scalar prduct N.F f these tw vectrs is er. Using the expressins fr the frictinal frce F and the nrmal reactin N which are shwn in equatins 4.4 and 4.5, the scalar prduct is given by: N.F = [-Flpi-Fmpi+(P-Fnp)k].[F(lpi+mp2+npk)] = -f H / - - p2np2 + Fn^P Frm the definitin f directin csines:? 7 9 Thus, the scalar prduct N.F is given by N.F = FnpP - F^ = 0 The slutin t this equatin is either that F = 0 which is the case when n frictinal frce is prduced r that the directin csine n^ is given by: n = (4.7) ^ P 59

85 The ther tw directin csines Ip and nip can be btained frm the directin f the frictinal frce which can be determined frm the expressin in equatin 4.6. Using the expressins given in equatins 4.2 and 4.5, the vectr prduct w X N is given by: w X N = [w(-sin0i.+cs0k) ]x[-flpi.-fmp2 +(P-Fnp)k] = w[fmpsin0k^(p-fnp)sin0j-flpcs0j+fmpcs0i] = w(fmpcs0i+[(p-fnp)sin0-flpcs0]j+fmpsin0k) (4.8) The magnitude f this vectr prduct w x N is given by: ^N = wjf mp CS 0+[(P-Fnp)sin0-FTpCs0] +F^mp^sin 0 = w Jf mp^+[ (P-Frip)sin9-FlpC0se]^ (4.9) Substituting the expressins btained fr the vectr prduct w X N and its magnitude w x N int equatin 4.6, the frictinal frce F is given by: F = F (FmpCs0i+[(P-Fnp)sin0-FlpCs0]j+FmpSin0k) y?^mp^+[(p-fnp)sin0-flpcs0] Equating the i^, j_ and k cmpnents f this expressin fr the frictinal frce with thse given in equatin 4.4, the fllwing expressins are btained fr the three directin csines Ip, mp and np: 60

86 Fm cs0 ^F ^ -- /l'^mp^+[(p-fnp)sine-flpcse] (P-Frip) sin0-flpcs0 ^ /P^^+[(P-Fnp)sine-Flpcs0]^ FnipSin0 ^ "12 T J f nip +[ (P-Fnp)sin0-FlpCs0] Equating this expressin fr the directin csine np with that given in equatin 4.7, the fllwing expressin is prduced: F P FnipSin0 (P-Fnp)sine-Flpcse]^ which gives: ^F^nip^+L (P-FUp)sin0-FlpCS0]^ = PnipSin0 (4.10) Using the relatin given in this equatin, the expressins fr the directin csines Ip and nip can be reduced t: F CS0 Ip, = (4.11) ^ P sin0 f = (P-FUp)sin0 FlpCS0 Pnipsin0 61

87 Then substituting the expressins given fr the directin csines Ip and np in equatins 4.11 and 4.7 int this expressin fr nip gives: (P-F^/P)sine - F^cs^e/Psine Pnip sin0 which gives: 2 F^ F^cs^e Thus the directin csine nip is given by: F^ = Il - R---R- (4.12) ^ ' P^sinie Equatins 4.7, 4.11 and 4.12 give expressins fr the directin csines f the frictinal frce F in terms f its magnitude F. the factr f^, This magnitude can be determined using given in equatin 3.26, where frm the definitin f the factr: F = f^n (4.13) where N is the magnitude f the nrmal reactin N, given in equatin 4.5. Substituting the expressins given fr the directin csines in equatins 4.7, 4.11 and 4.12 int this equatin, the nrmal reactin N is given by: 62

88 -F^cse / F^ / F^ N = i -F/1-7T 1 + P -----Ik (4.14) P sine i P^sin^e V I Then the magnitude f the nrmal reactin will be given by: jp^cs^e ~ ( f2 "\ ' ~ N = - 5 r + f 1-2 T + p - 2 F^ J p ^ s i n ^ e V P ^ s i n ^ e y P ^ 'F^(cs^e-l) P^sin^e F^ + -* + P - F P^ = /"2.2 P^ - F^ (4.15) Substituting this expressin fr nrmal reactin int equatin 4.13, the magnitude f the the magnitude f the frictinal frce F can be expressed in terms f the factr f2 and the magnitude f the applied lad P as fllws: F^ f 2 = f^2(p2_p2) f.p F =, ^.1 (4.16) This expressin fr the magnitude f the frictinal frce can then be substituted int equatins 4.6, 4.11 and

89 t give the directin csines 1^, nip and np in terms f the factr f^, the magnitude f the applied lad P and the angle 0 between the axis f rtatin and the directin f the applied lad. The expressins fr the directin csines becme: f-, CS0 4 = t u (4-17) in0 Jl+f 2 (l+f^ )sln^e (4.18) np =. ' 2 (4.19) 4.2 The Mment f Slip Frictin Expressins fr the magnitudes and directins f the mment f slip frictin and the mment f rtatinal frictin relative t the applied lad can be derived in the same way as thse fr the magnitude and directin f the frictinal frce were. The mment f slip frictin Mg can be expressed in terms f its magnitude and directin csines as: % ^ (4.20) The expressins fr the magnitude and directin csines 64

90 can then be derived using thse btained fr the mment f slip frictin relative t the nrmal reactin in sectin 3.6 where equatin 3.34 gives: -f-r Mç = ---- (NxwxN) IwxNl and where the factr f2 representing the rati Mg/rN is given in equatin 3.33 as: CC sin«cs(-j^ (siny-sin^xsin#cs^3-sinacs*cspcs#)d*dp rn 7tI WU Vl - (csacs^+sinacspsin*) Substituting the rati Mg/rN fr the factr f2 in equatin 3.34, the mment f slip frictin can be given as: -Mn(NxwxN) N_ = (4.21) ^ NlwxNl Frm the definitin f the factr f2 the magnitude f the mment f slip frictin Mg is given by: Mg = rnf2 Substituting the expressin fr the magnitude f the nrmal reactin N, given in equatin 4.15, int this equatin gives: Mq = rf Jp^-F 2 Then, using the expressin given fr the magnitude f the 65

91 frictinal frce F in equatin 4.16, the magnitude f the mment f slip frictin Mg can be expressed as: rf^p l.fl 2' (4.22) The expressins fr the directin csines f the mment f slip frictin can be derived using the expressin fr the directin f the mment btained frm equatin This invlves determining an expressin fr the triple vectr prduct Nx t N. The determinatin f this expressin can be simplified by replacing the term by the symbl x. The expressins fr the magnitude and directin csines f the frictinal frce F given in equatins then becme: F = Px (4.23) XCS0 Ip, = ---- (4.24) sin0 x2 Dip, = / (4.25) ^ striq np = X (4.26) Substituting the expressin given in equatin 4.23 fr the 66

92 magnitude f the frictinal frce F int equatin 4.14, the nrmal reactin N can be expressed as; N = P "-X CS0 sin0 i - X /I - sin^0 2 + (1-x )k (4.27) Then, substituting the expressins given fr the magnitude and directin csines f the frictinal frce F in equatins int equatin 4.8, the vectr prduct can be given as: \f^n = wp^xcs0/l - X sin^0 1 + (1-x )sin0-2 2 X cs 0 sin0 X + xsin0/l - j- k sin 0 sin^0 1+ = wp^xcs0/lsin^e-x^ sin0 2+ xsin0/l- R- ki ^ s±n9 ^ X The triple vectr prduct N x ^ N can then be btained by taking the vectr prduct f the nrmal reactin N given in equatin 4.27 and the vectr prduct given abve. This gives: NxwxN = wp2 '-X CS0 I X i - x/lsin0 sin^0 1 + (l-x2 )k 67

93 X XCS0 /l- X sin^0 1 + sin^0-x^ sin0 i + xsin0/l- ---7^ ki / sin^0 = wp' -x^cs0(sin^0-x^) sin^0 k + X CS0 11- y- 2 sin 0 + X CS0 1 - X \ 2 I k - X sin0 1 - ^ sin^0 s i n V - r) ) 1 X (1-X )(sin 0-x ) + X (1-x )cs0/l - Tj sin 0 sin0 = wp x^-sin^0 sin0 i + XCS0/1 - yp- 2 sin 0 (4.28) The ther factrs in the expressin in equatin 4.21 giving the directin f the mment f slip frictin are the magnitudes f the vectr prduct and the nrmal reactin N. By substituting the expressin in equatin 4.10 int equatin 4.9, the magnitude f the vectr prduct is given by: = wpmp sin0 Substituting the expressin given in equatin 4.25 fr the directin csine m^ int this equatin gives: l^nj = wpsin0 1 - X' sin^0 (4.29) The magnitude f the nrmal reactin N can then be 68

94 btained by substituting the expressin fr the magnitude f the frictinal frce F in equatin 4.23 int equatin 4.15 t give: N = P - X (4.30) Then, substituting the expressins, given fr the triple vectr prduct Nx t N in equatin 4.28, the magnitude f the vectr prduct in equatin 4.29 and the magnitude f the nrmal reactin N in equatin 4.30, int equatin 4.21, the mment f slip frictin Mg is given by: MgWPx^-sin^0 sin0 i + XCS0 / p- 2 sin^0 1 - x/ wpsin0 / 1 - sin^0 = M, in 0-x sinsv/lx^ - XCS0 s i n e v / n ^ Equating the i., 2 h cmpnents f this expressin fr the mment f slip frictin with thse given in equatin 4.20, the fllwing expressins are btained fr the directin csines f the mment f slip frictin, 1 MS ^MS and n MS' Sin^0-x2 "MS sin0 1-x' 69

95 XCS0 sin0/l-x "^MS - Replacing the symbl x by the term f,/ / 1+f, ^ the expressins fr the directin csines becme: f-,^cs^0 = / (4.31) sin 0 ms -f2 CS0 sin0 /(l+f2 ^) ^ " -f2 CS0 sin0 (4.32) n^g = 0 (4.33) 4.3 The Mment f Rtatinal Frictin The mment f rtatinal frictin M^^ can be expressed in terms f its magnitude and directin csines as: % = M R < W + m rj. + (4.34) 70

96 The expressins fr this magnitude and these directin csines can then be btained in the same way as thse fr the magnitude and directin csines f the mment f slip frictin were. The expressins fr the mment f rtatinal frictin relative t the nrmal reactin N were derived in sectin 3.7 where equatin 3.40 gives: = -fgrn sgn(w.n) and where the factr f^ representing the rati M^/rN is given in equatin 3.39 as: Zff f>0< sin4xcs( 2 Z )(sinacsa-csacspsin*) dadp rn tti J V \/l - (csas*+sin*cspsin*0^ Substituting the rati M^/rN fr the factr f^ in equatin 3.40, the mment f rtatinal frictin can be given as: N = -Mp -R R N sgn(w.n) (4.35) The magnitude f the mment f rtatinal frictin M^ is btained frm the definitin f the factr f^ as: Using the expressin fr the magnitude f the nrmal reactin N which can be deduced frm equatin 4.22, the 71

97 expressin fr the magnitude f the mment f rtatinal frictin FL becmes: rf,p Mp = ^, (4.36) The expressins fr the directin csines f the mment f rtatinal frictin can be derived using the expressin fr the directin f the mment which can be btained frm equatin Using the expressin given fr the nrmal reactin N in equatin 4.27 and that given fr its magnitude N in equatin 4.30, the term N/N is given by: N -x^csg / sin^0-x I ^ / 9 9 N sin0vl-x J sin 0(l-x ) X + / H r k (4.37) where x represents the term 1 ^ 1 The scalar prduct w.n can be determined using the expressin fr the nrmal reactin N in equatin 4.27 and that given fr the vectr w in equatin 4.2. It is given by: w.n = wp -x^cs0 / x^ «(-sin0 2 +cs0 k).i i-x 1-2 +( 1-x )k, \ sin0 J sin G ' 9 9 = wp[x CS0 + (1-x )cs0] = wpcs0 72

98 Thus the sign f the scalar prduct w.n is given by the sign f the csine f the angle 0, i.e.: sgn(w.n) = sgn(cs0) (4.38) Substituting the expressins btained in equatins 4.37 and 4.38 int equatin 4.35, the mment f rtatinal frictin can be expressed as: Mr = MṘ 2 X CS0 sin^0-x^ sine/ït: 2 - sin^0(l-x^) 2 - y ï -? k sgn(cs0) Equating the 1, 2 and k cmpnents f this expressin fr the mment f rtatinal frictin with thse given in equatin 4.34, the fllwing expressins are btained fr the directin csines f the mment 1 ^, m ^ and n ^: MR 2 X CS0 sin0 = sgn(cs0) mr sin^0-x^ sin Z7J2 sgn(cs0) 0(l-x ) MR = -/l-x^ sgn(cs0) Replacing x by the term f the expressins fr the directin csines becme: 73

99 f-,^ CS0 i m j f 2 sgn(cse) (l+fi^)sin0 /l+ i W f1^cs0 -. % sgn(cs0) (4.39) sin0 Jl+fj m r f 1 sin0 sin 0 1+f sgn(cs0) 1+f fl / 2 sin 0 - J sgn(cs0) (4.40) sin0 V l+f^ f / "m r = -/I - sgn(cse) -1 2=sgn(cs0 ) (4.41) 4.4 The Evaluatin f the Expressins The results btained in this chapter can be summaried as fllws. The c-rdinate system in a spherical jint was defined s that the vectr w representing the axis f rtatin and the vectr P representing the applied lad 74

100 are given by equatins 4.2 and 4.3 as: w = w(-sin9i + csgk) P = -Pk where 0 is the angle between the axis f rtatin and the line f actin f the applied lad. This meant that the frictinal frce F culd be given by equatin 4.4 as: F = F d p i + mpi + npk) where its magnitude and directin csines were given by equatins as: F = ft CS0 1_ = ^ ^ -Inejl+f ^ (l+f2^)sin^0 /dfj2 and where the factr f^ representing the rati F/N 75

101 was given by equatin 3.26 as: lit singecs( (csasinf-sinaspcsf) d(dp /l - (csas^^sinaspsiny)^ JJ The frictinal mment was then given in the frm f tw perpendicular cmpnents, the mment f slip frictin Mg and the mment f rtatinal frictin These mments are expressed in equatins 4.20 and 4.34 respectively as: -S m S^- ^MS-^ -R ^R^^MR- m r J- ^MR-^ where their magnitudes and directin csines are given by equatins 4.22, , 4.36 and as: rf,p «S = 7 = 2 = 1 ^ 1 f / c s ^ m MS -fj CS0 sin0 "ms rfgp I W 76

102 MR f2 CS0 sgn(cs0) m MR sin^0 - sin0 1+f. sgn(cs0) -1 n MR 1 ^ 1 ^ sgn(cs0) and the tw factrs f2 and f^, representing the ratis Mg/rN and M^/rN respectively, are given by equatins 3.33 and 3.39 as: 9 9 s i n f c s(sin2f-sin #sin*cs p- sinfcs(csj3csy)d(dp /1 - (cs^cszt + sin^<cspsin2() sin^gxcs( 2 ^ ) (sin*cs*-cs*cspsin#ldxdp ^3 tti 1/1 -(csacs# + sindcspsin#) J The frictinal mment is given in the frm f tw perpendicular cmpnents because it is cnvenient t derive expressins fr these cmpnents rather than fr the ttal mment. This ttal frictinal mment M can then be btained by taking the sum f the cmpnents and is given in the frm: M = M(l^i+m^2+n^k) (4.42) 77

103 where its magnitude and directin csines are given by; M = /i JMgZ + (4.43) Ijj = (4.44) M ^S^MS + M 2^ (4.45) + = (4.46) M The results given abve shw that the determinatin f the magnitudes and directins f the frictinal frce and mment prduced in a spherical jint depends upn the evaluatin f the three factrs f^, f2 and f^. In chapter three it was shwn that these three factrs can be evaluated numerically prvided that the values f the angles and are knwn. The methd f determining the angle (^ is d e s c n b c c i. in t h e fllwing c h a p t e r. T h e, a n g l e Ü h a s b e e n defined as the angle between the axis f rtatin, represented by the vectr w, and the line f actin f the nrmal reactin N, as shwn in Fig.4.1. Thus, in rder t determine the effects f frictin relative t the applied lad, it is necessary t derive an expressin fr the angle ^ in terms f the angle 0, which is the angle between the axis f rtatin and the line f actin f the applied lad. 78

104 Frm the definitin f the angle 7^, this expressin can be btained using the equatin fr the vectr prduct which can be re-arranged t give; I^N sin# = ---- wn Using the expressin fr the magnitude f the vectr prduct given in equatin 4.29 and the expressin fr the magnitude f the nrmal reactin N given in equatin 4.30, the abve expressin becmes; sin # = - 4 r wpsing/ sin 0 wp /l-x :2- where x represents the rati Replacing the term x by this rati, the expressin becmes; = Jsin^0-f2 ^cs^ 0 i.e. sin^# = sin^0-f^^cs^0 (4.46) As the factr f^ has been shwn t cntain a cmplex duble integral invlving the angle, which can nly be evaluated numerically, it fllws that the value f the angle cannt be determined analytically frm equatin 79

105 4.46. Hwever, this equatin can be used t determine the value f the angle fr several particular values f the angle 0. Firstly, when the axis f rtatin is cincident with the line f actin f the applied lad, i.e. the angle 0 is either 0 r 180, there is n resultant frictinal frce prduced because the axis f rtatin is passing thrugh the centre f the axi-symmetric pressure distributin ver the cntact area, and s the factr f^ has a value f er. Equatin 4.46 then becmes; sin^y = 0 i.e. 2f= 0 r 180 The defined rientatins f the angles 0 and shwn in Fig. 4.1 indicate that when 0 = 0, 2T= 0 and when 0 = 180, y=180. The tw ther situatins in which the angle 2T can be determined frm equatin 4.46 ccur when the angle 0 is 90 and when it is 270. In bth f these psitins the equatin becmes; sin^# = 1 i.e. Zr= 90 r 270 Again frm the rientatin f the tw angles as shwn in 80

106 Fig.4.1, it can be seen that when 0 = 90, = 90 and when 0 = 270, = 270. Equatin 4.46 als shws that the angle 2T nly has the same value as the angle 0 at these particular values, which means that the angle will always be in the same quadrant as the angle 0. This enables the intermediate values f the angle t be determined by iteratin. The iterative rutine used is shwn in Fig.4.3. Having determined the apprpriate quadrant frm the value f the angle 0, the value f the angle is initially set at the mid pint f that quadrant. Using this value f the angle, the factr f^ is evaluated and the errr btained when these values f the angle and the factr f2 are substituted int equatin 4.46 is determined. If this errr is belw a set value, the value f the angle is taken as crrect. If it is nt the value f ^ is increased r decreased as apprpriate and the prcedure repeated until the crrect value f the angle is btained. This iterative rutine, derived fr tgether with the expressins the magnitudes and directins f the frictinal frce and mments, was incrprated int the cmputer prgram used t evaluate the factrs f^, f^ and fg numerically. The results btained using this prgram are shwn in Figs Fig.4.4 shws the magnitude and directin f the 81

107 n = I n t ( - ^ ) i.e. n IS S e t» t t h e VRLUE f T rucatsd T inrecer FRM Xt = D e t e r m in e f j f r m X E r r r = 0 + s in ^ ^ - s in ^ Is r r r < 0,00001? Y e s N U se ^ VflUTE f G /y e n I s E r r r > 0. 0? N Yes F ig u r e 4.3 T h e I t é r a t iv e R u t in e U s e d t D etermine the V a lu e f t h e A n g le 2^

108 frictinal frce F pltted against values f the angle 0. As the magnitude f the frictinal frce is directly prprtinal t that f the applied lad P it is shwn in the frm f the rati F/P. The directin is given in the frm f the three directin csines Ip, mp and np. Fig.4.5 shws the magnitude and directin csines f the ttal frictinal mment pltted against the values f the angle 0. The magnitude f the mment is directly prprtinal t the magnitude f the applied lad and t the radius f the jint r, s it is given in the frm f the rati M/rP. The magnitude and directin csines f the nrmal reactin N are shwn pltted against values f the angle 0 in Fig.4.6, the magnitude being given in the frm f the rati N/P. The three sets f graphs shwn in Figs all shw results btained fr varius values frictin p in the range frm 0.2 f the cefficient f t 1.0 fr a particular value f the angle, which defines the sie f the cntact area, f 30. Fig.4.7 shws the magnitude and directin csines f the frictinal mment pltted against values f the angle 0 fr varius values f the angle in the range frm 5 t 90 and fr a particular value f the cefficient f frictin f p = 1.0. Thus Figs shw the characteristics f the frictinal frce and mment prduced fr a range f values f the cefficient f frictin with a cntact area f an extent that is likely t ccur and Fig.4.7 shws the characteristics f 82

109 CD C vj C U :5 u. Ut (C p SD ce IL b. L <r (b lu -I VÛ cc U1 X h >- «0 tt V>J <r k w IL b. U1 UJ X H IL bj V - i - _l Q. J CL ut c UJ C ' L U_ CL cf -t* UJ a UL

110 CD CD r^ u ce Cv Z K lï ce il CD w _j L h Z UJ ce s ; u. u. lu <r r </) Z ce VÎJ : w e tu v> W 3 a: > c\j tu 2 % h LU CC UJ C ' c L r\j < r JÛ 4-4-' UJ c D

111 CD c r^ w vj : C : u. Z U u SD in lu -ICJ Z ce H Z W C I I W 'T r J u hh Cl u E lu V) * w 3 5 lu X )- PvJ U. E C <r rv- C5 CD L C - H ---- r CD CD rsj CD CD CD 4- lu ûc 3

112 es c r\j CD c > L r sj U lu ce (V ce li. CD w _j ce w I h h V) r cc vî> CH û LU V-h CL u. C bj Z h LU C (5 Z P le L \- 2 UJ u. U. Ul w e u ȯ V» lu _l a: > UJ H c L LL C

113 C CM 00 «-* Ui -I r^ L W (C (Z CD lu c Z p vj lï U. k u u. u. u u I I- ' f r «Z C 0 C lu X H IJL r\i CL CC ce lu c LA < r h cf m It

114 CD CD W va a : % P 5 Ll ce u CD w _i a: k k lü a. u. lu C ; <r M h V) a va ce lu ü_ V) Ul D _) CH > rvj tu Z p Ul cc r\j c n Ul _Û lf> -t- tu cc 3sa

115 CD laj VÜ Z a: (X p u in J c a u. (D w -j c UJ I h < Z d UJ V-H CL u. P w iz u. Ul U g ü.» UJ 3u U ȯ LU Z J CD Z p w c (5 lu r h- I J d> r I i_n I I CD I CD lu c D

116 U ) un r CL CNJ W Ul c ' "0 in

117 r\j <r CD C CD 0 CD C Ut DJ L 'J r % ut 2<c P û: if : u. Q> UJ _i cc u I h y- «0 Z cr v5 (C Ui t-»ō Ll k 2 W il UUl U i X H u. CL J Cl IZ <x CC U i L < r f\j w a 3 LL

118 u Ui cc es: ce u Z P lï (D L U) ce )- Z lü in c, lu I h H V) Z CC v5 a: Ui V-H u. u. Ul lu f u. V) lu 3 a: > CNJ CM tu Z ô p lu K Ui I I I CNJ I r I L I c I es I _ > lu a: D

119 c c u vil ce (X a u. (D li. P G: U u. w v5 Z ce w Xh»- Z u VJ IL II U L < 2 «C c w hh w 3_l 5 Cl II w f\j h 0 w 01 6 LU r ) I r\j I r I I L I c I I I I v LU QC3 lz

120 r\j <r CD CD CD C CD CD CD CD CD CD CD <r CD r CD r\j Ui ct CV CC u. CD w _i a: h r cr VÎJ ^ U i V- H D. C Ui Z ^ y- Ul ; :L p 5 Ll l H Z Ui u. u. Ul Ui I K U. V) Ui D <C > CD c CD i_n < r CD DJ s UJ cc D

121 CD c u. 0 Ul Z cr : û: b. (D lü s «X W a ; lu % H iô ^ 5sbet v/> W 3_l cr > r rsj Ll H H _J CL CL h cc W un c un <T CL d r~tü c 3 ÜL

122 (D CD LD r^ LT> >T D: 11 CD w _l ce Ul p v~ V» 5 cc Q lu hh CL Ul _) ce Ul X H u. V) lu 3 -I 5 bu Ul CJ X c r U) 2 CNJ V- w c 111 CD LTI -Q r- U i c: D u

123 c a u. CD Ul _j v3 Z ex Ul _J ür a: lü X h > in Ul p V- w Z 5 v5 C Q lu V- h u u ȯ V) Ul 3 -J 5 lu Ul 0 Z d: œ Z E r LD w w J Z V- U ce Lü J I r I I LD I I I CD I E CD W (C3 Lu

124 c : II CD Vu c cr vu _) ce vu X H Ul es vu f K- V) Z 5 cr Q VU hh u. U) vu 3 > ü. lu v5 Z a: CZ r Z c w Z</) Z LD V ō vu u I fm I U l I I I ' I I c ' c -ci r- Ul et 3vb u _

125 the frictinal mment fr varius sies f the cntact area when the frictin is high t shw the effect f varying the area clearly. The characteristics shwn are described in the fllwing sectin. 4.5 The Characteristics f the Frictinal Frce and Mment The Frictinal Frce As a result f the interactin between the frictinal frce and the nrmal reactin prduced in a spherical jint, which has been analysed in this chapter, the characteristics f the magnitude and directin f the frictinal frce can nly be described with reference t thse f the nrmal reactin and even then they cannt always be described in a straight-frward manner. The basic characteristics f the magnitude f the frictinal frce, shwn in Fig.4.4a, are largely thse described fr the factr f^ in sectin and seen in Fig.3.9b. When the axis f rtatin is aligned with the line f actin f the applied lad i.e. 0 = 0, there is n resultant frictinal frce prduced because in this psitin the axis f rtatin passes thrugh the centre f the cntact area ver which the pressure prducing the frictinal frces is axisymmetrically distributed. As the 83

126 angle 0 increases the pint f intersectin f the axis f rtatin mves twards the edge f the cntact area and s the resultant frictinal frce steadily increases as described in sectin When the axis f rtatin n lnger intersects the cntact area the magnitude f the frictinal frce is clse t its maximum value which is finally reached when the axis f rtatin is perpendicular t the line f actin f the nrmal reactin i.e. = 90. As shwn in sectin 4.4, this ccurs when the axis f rtatin is perpendicular t the applied lad i.e. 0 = 90. Fig.4.4a shws that as expected the magnitude f the frictinal frce increases as the cefficient f frictin rises. Hwever, it als shws that the magnitude f this increase is steadily reduced with the rise in the cefficient f frictin. This reductin is due t the fact that the directin f the frictinal frce is perpendicular t that f the nrmal reactin while the nrmal reactin N is prduced by the sum f the applied lad P and the frictinal frce F. Thus, as shwn in Fig.4.1, the directin f such that it acts against the frictinal frce is always the applied lad t reduce the magnitude f the nrmal reactin. This is cnfirmed by cmparing the magnitudes f the frictinal frce and the nrmal reactin in Figs.4.4a and 4.6a. The result is that as the cefficient f frictin rises the magnitude f the nrmal reactin is reduced and s the magnitude f the 84

127 frictinal frce, which is dependent n that f the nrmal reactin, is nt as great as wuld be expected. Fig.4.4a als shws that as the cefficient f frictin rises the magnitude f the frictinal frce reaches the pint at which it is clse t its maximum value at higher values f the angle 0. This characteristic results frm the effect f the frictinal frce n the directin f the nrmal reactin as described belw. When the cefficient f frictin is lw, the frictinal frce is small and s the nrmal reactin has rughly the ppsite directin t the applied lad which acts vertically. This is shwn in Fig.4.6d by the directin csine n^ apraching unity as the cefficient f frictin is reduced. The directin f the frictinal frce has been shwn in sectin 3.4 t be perpendicular t bth the axis f rtatin and the directin f the nrmal reactin. Thus,when the cefficient f frictin is lw the frictinal frce acts largely in the j directin as given in Fig.4.2, which is shwn in Fig.4.4c by the directin csine m^, appraching unity as the cefficient f frictin drps. As the cefficient f frictin rises, the increased in frictinal frce results^a shift f the directin f the nrmal reactin frm the vertical as shwn by the directin csine n^ in Fig.4.6d. There is a crrespnding shift in the directin f the frictinal frce twards the vertical, due t the fact that it is perpendicular t the 85

128 directin f the nrmal reactin, which is shwn by the increase in the directin csine n^ in Fig.4.4d. When the angle 0 is small the axis f rtatin lies clse t the vertical in the i-k place as shwn in Fig.4.8. Thus when the directin f the frictinal frce shifts twards the vertical it als shifts twards the ± directin t remain perpendicular t the axis f rtatin. This is shwn in Fig.4.4b by the increase in the directin csine Ip as the cefficient f frictin rises. As the nrmal reactin is prduced by the sum f the frictinal frce and the applied lad which acts in the vertical r k directin, the shift in the frictinal frce twards the ± directin prduces a crrespnding shift in the directin f the nrmal reactin. This is shwn in Fig.4.6b by the increase in the directin csine 1^. Fig.4.6b als shws that as the angle 0 increases the shift in the nrmal reactin twards this directin als at first increases because f the increase in the magnitude f the frictinal frce prducing it as shwn in Fig.4.4a. The shift brings the directin f the nrmal reactin clser t the axis f rtatin with the result that the axis f rtatin cntinues t intersect the area f cntact at higher values f the angle 0 and it is this which causes the magnitude f the frictinal frce t reach the pint clse t its maximum at a higher value f the angle 0. The higher the cefficient f frictin is, the greater the frictinal frce with the cnsequent 86

129 rm FigüRÊ 4-.ga Th F r i c t i n a l F rce AND Mr\EHT PRDUCED WHEN THE CEFFICIENT F Frictin is Lu (/J.-Û.2) and THE Angle 0 is Small

130 np Figure A-.5b T h e F r i c t i n a l F rc e and M m e n t P r d u c e d when th e C e ffic ie n t f prictrn IS High [jj.^i.0) and th e A n g le 0 is S m a ll

131 Fi&üre 4.8 & T he F r ic t i n r l Frce a n d H m ew t prucep WHEN THE C EFFICIENT F F r ic t i n is L w ( / i ) a n d t h e A n g le G is L r r g ê

132 Figure h.sd. The FRICTINAL Frce AND M m nt PRDUCED WHEs THE CEFFlCtENT f F ricti n is High (/u= i.) and th e A n g le 9 is L a r g e

133 increase in the shift in the directin f the nrmal reactin twards the axis f rtatin, resulting in the magnitude f the frictinal frce reaching the value clse t its maximum at a higher value f the angle 0. When the value f the angle 0 reaches this pint at which the magnitude f the frictinal frce n lnger increases, the shift in the directin f the nrmal reactin away frm the vertical and the resulting shift in the directin f the frictinal frce twards the vertical bth cease as shwn by the directin csines n^ and n^ in Figs.4.6d and 4.4d respectively. As the angle 0 increases, the axis f rtatin rtates twards the ± directin and s t remain perpendicular t this axis the directin f the frictinal frce shifts away frm the i. directin as shwn by the decrease in the directin csine Ip in Fig.4.4b. The cmbinatin f these tw directin f the frictinal cnditins means that the frce suddenly shifts twards the j_ directin as shwn by the characteristics f the directin csine mp in Fig.4.4c The Frictinal Mment The basic charateristics f the magnitude f the frictinal mment as shwn in Fig.4.5a are largely a cmbinatin f thse described fr the mment f slip frictin and the mment f rtatinal frictin in sectins 87

134 3.7.2 and which are shwn by the factrs 2 and f^ respectively in Fig.3.9. At small values f the angle 0, when the angle ^ is als small, the frictinal mment is cmprised mainly f the mment f rtatinal frictin. Then at larger values f the angle 0 the mment f slip frictin predminates. As with the frictinal frce the magnitude f the mment increases with a rise in the cefficient f frictin and fr the higher values f the angle 0 this increase is reduced as the cefficient f frictin rises due t the actin f the frictinal frce reducing the magnitude f the nrmal reactin. Hwever, when the angle 0 is er, the magnitude f the frictinal mment is directly prprtinal t the cefficient f frictin because there is n resultant frictinal frce t effect the magnitude f the nrmal reactin. Fig.4.5a shws that the magnitude f the frictinal mment reaches its maximum value at higher values f the angle 0 as the cefficient f frictin rises in the same way that the magnitude f the frictinal frce did. This is due t the effect f the frictinal frce n the directin f the nrmal reactin as described in the previus sectin. Fig.4.7a shws the variatin f the magnitude f the mment with the value f angle c^ which results frm the fact that the mment f rtatinal frictin, being the cmpnent f the mment prduced abut the axis passing thrugh the centre f the cntact area, is very dependent n the sie f the area, while the mment f slip 88

135 frictin, being the cmpnent prduced abut an axis perpendicular t the ne passing thrugh the centre f the cntact area, is nt. The characteristics f the directin f the frictinal mment, as shwn in Fig.4.8 and by the directin csines 1^, m^ and n^ in Fig.4.5, are als largely a cmbinatin f thse described fr the mment f slip frictin and the mment f rtatinal frictin. As stated abve, at lw values f the angle 0 the mment f rtatinal frictin is predminant and this mment acts abut an axis cincident with the directin f the nrmal reactin which, as shwn by Fig.4.6, is very clse t the vertical in this situatin. Thus fr small values f the angle 0 the frictinal mment largely acts abut the vertical directin as shwn by the directin csine n^ in Fig.4.5d. As the cefficient f frictin rises, the increased frictinal frce prduces a shift in the directin f the nrmal reactin twards the axis f rtatin as described befre. This means that the angle remains lw and s the mment f rtatinal frictin cntinues t predminate at higher values f the angle 0. As the directin f the nrmal reactin is still mainly vertical, this results in the directin csine n^ shwn in Fig.4.5d remaining at a high value fr larger values f the angle 0. When the cntact area is larger the mment f rtatinal 89

136 frictin remains significant fr higher values f the angle 0 and s the ttal mment cntinues t be directed twards the vertical at these higher values f the angle 0. This is shwn in Fig.4.7d by the directin csine n^ remaining at a high value, fr greater values f the angle 0 as the angle (X^rises. Fr higher values f the angle 0 the mment f slip frictin becmes the majr cmpnent f the frictin mment and s the directin f the ttal mment tends twards that f the mment f slip frictin. The directin f the mment f slip frictin has been defined as being perpendicular t the line f actin f the nrmal reactin and lying in the plane cntaining this line f actin and the axis f rtatin. Thus, when the cefficient f frictin is lw with the result that the directin f the nrmal reactin is clse t the vertical, the directin f the mment tends twards the t directin as shwn by the directin csine 1^ in Fig.4.5b. As the cefficient f frictin rises the nrmal reactin is deflected twards the 1 directin, bringing it clser t the axis f rtatin, fr the lwer values f the angle 0 as described befre. Thus, at thse values f the angle 0 which are high enugh that the mment f slip frictin predminates and lw enugh that the shift in the directin f the nrmal reactin described abve ccurs, the directin f the mment is deflected twards the j_ 90

137 directin as shwn by the directin csine in Fig.4.5c. Then, as als described previusly, at higher values f the angle 0 the directin f the nrmal reactin shifts away frm the t directin and twards the j_ directin and this is reflected in the frictinal mment being deflected frm the j directin twards the i directin. When the cefficient f frictin is higher the shift ccurs mre rapidly at a higher value f the angle 0 as is shwn in Figs.4.5b and 4.5c by the directin csines 1^ and m^^. Fig.4.7 shws that when the cntact area is small this shift in directin is mre prminent which is a result f frictinal frce reaching its maximum value at much lwer values f the angle 0 as shwn in Fig.3.9. Then fr larger cntact areas the shift is less prminent as the mment f rtatinal frictin becmes mre significant ver a larger range f the angle 0. 91

138 CHAPTER FIVE The Mathematical Mdel This chapter describes the areas in which the theretical analysis was extended in rder t develp, frm the expressins derived fr the frictinal frce and mment in the preceding chapters, a mathematical mdel that will adequately simulate the effects f frictin in a spherical jint. 5.1 The Angle Defining the Area f Cntact In sectin 4.4 f the previus chapter it was shwn that the expressins fr the frictinal frce and mment culd be evaluated prvided that the value f the angle c^, which defines the sie f the cntact area, was knwn. Hwever, the extent f the cntact area depends upn several factrs as described in sectin 3.2 and s it is necessary t derive an expressin fr the angle which, if the expressins fr the frictinal frce and mment are t be used in a mathematical mdel, must be given in terms f quantities whse values are likely t be knwn by a user f the mdel. The expressin derived fr the angle (^ is based n that 92

139 prvided by the thery develped by Hert fr the radius f the cntact area between tw spheres (Timshenk and Gdier, 1970), which is given in terms f the transmitted lad, the relative radii f the spheres and their elastic prperties. Unfrtunately, Hert's thery is based n the assumptin that the radius f the cntact area is very small in cmparisn with the radii f the tw spheres, which is unlikely t be the case in a spherical jint under a nrmal lad. Hwever, Gdman and Keer (1965) have extended Hert's analysis in the case f an elastic sphere indenting a spherical cavity in an elastic slid by aviding the assumptin f a small cntact area. Their results shw that the expressin given by Hert's thery fr the radius f the cntact area can be extended t relatively large areas f cntact prvided that the sphere and cavity are f apprximately equal radius. Hert's thery gives the fllwing expressin fr the radial arc f the cntact area, a, shwn in Fig.5.1: f3prtr«(ct+c«) a = I--- - (5.1) 4(R^-R2> where P R^ R2 is the lad is the radius f the scket is the radius f the ball 93

140 L a d S c k e t C l e a r a n c e EXAGGERATED Ri R% Figure 5.1 D im e n s i n s f t h e C n t a c t A rea

141 and the factrs Ct and c«are given by: 1 - c. = (5.2) El 1 - v / c, = ---- (5.3) ^2 where and are the Pissn s ratis and and E2 are the mduli f elasticity fr the scket and ball respectively. The radii f the ball and scket in a spherical jint are apprximately equal. Fr example, the Rse Bearings catalgue (1983) gives the radial clearance, & nrmally fitting spherical bearing with a diameter f 25.4mm as being between and 0.015mm, which means that the rati f the radius f the scket t that f the ball lies in the range frm t Thus the angle defining the area f cntact can be determined using the radius R which is the mean radius between thse f the ball and the scket. It can be seen frm Fig.5.1 that the angle is given, in radians, by the expressin: 94

142 Substituting the expressin given fr a in equatin 5.1 int this expressin gives: [3PR^R2 (C2 +C2 ) ^ = R''(R^-R2> As the radius R is the mean f the tw radii R^ and R 2 whcih are themselves apprximately equal, this expressin is practically equivalent t: 3f 3P(c.+c,) (5.4) 4R(Rj-R2) where the tw factrs c^ and C2 are given by equatins 5.2 and 5.3 respectively. This expressin gives the angle in terms f quantities whse values are likely t be knwn. clearance and the elastic prperties The radius, radial f the materials within the jint will have t be supplied by the user f the mdel. The lad will be given by the nrmal reactin determined fr the jint. As the magnitude f the nrmal reactin depends upn the magnitude and directin f the frictinal frce prduced which in turn depends upn the extent f the cntact area, the actual value f the angle fr a particular spherical jint can nly be determined by iteratin. The iterative rutine used t determine the value f the angle in the cmputer prgram used t 95

143 predict the effects f frictin in a spherical jint is shwn in Subrutine Alpha f the flw diagram f the prgram cntained in Appendix 3. The results prduced by Gdman and Keer indicate that an accurate value fr the angle will be btained frm the expressin given in equatin 5.4 while the angle is 20 r less. Cnsidering these results and the assumptins riginally used in determining the expressin, it wuld appear safe t assume that reasnably accurate values will cntinue t be given as the angle (^ rises up t at least 30, prvided that the radii f the ball and the scket are as similar as has shwn t be the case with a nrmally fitting spherical jint. increases further it is As the value f the angle (^ likely that the expressin will becme increasingly inaccurate, giving a value fr the angle higher than it actually is. Hwever, a cnsideratin f the effects f the sie f the cntact area n the frictinal mment prduced, which can be seen in Fig.4.7, shws that when the value f the angle 0 is high, large variatins in the value f the angle have little effect n the frictinal mments. The angle 0 is the angle between the directin f the applied lad and the axis f rtatin in a spherical jint, which tends t have a high value in many applicatins f spherical jints, particularly when straddle type spherical jints are used. Thus the accuracy f the value determined fr 96

144 the angle C^ will nt be significant particularly if it is less than 70. Fig.5.2 shws values fr the angle btained using the expressin in equatin 5.4 pltted against the lad fr several values f the radial clearance btained in a nrmally fitting jint with a radius f 12.7mm. The Rse Bearings catalgue indicates that the maximum dynamic lad n a straddle-type spherical jint f this sie will be apprximately 25kN and s it can be seen frm Fig.5.2 that under nrmal lading the value f the angle is likely t be less than 70. When the value f the angle 0 is smaller Fig.4.7 shws that it will be necessary t determine the value f the angle mre accurately. Hwever, in this cnfiguratin the lad transmitted by the jint is likely t be much less. Fr example the maximum axial lad that can be safely applied t a straddle-type jint is abut 20% f its maximum radial lad, depending n the design f the jint. This means that when the value f the angle 0 is lw the sie f the cntact area is likely t be in the range where it can be mre accurately determined by the expressin in equatin 5.4. Thus, althugh the expressin derived t determine the value f the angle is nt likely t be very accurate when the angle is large, it can be taken as being 97

145 % w \ Ln\ l 0 \ f\j\ L \ \ \ N1 i_n <r <t 0 -J J J a U) 3 tl CL q; q : J n U_1 J C W w Z c a: \IS u q : CL CD a: 5 c QC w % h 0\J CL 6 IL V) W 3 5 W C C) cc </) 3 cc ce > N K il w _) cc > CC ÜJ J <n / Lfi LA LH LD LH in L - L L N J rj " ' l w n j

146 sufficient fr the purpses f the mathematical mdel. 5.2 The Reactin t the Effects f Frictin It is necessary t cnsider the effects the frictinal frce and mment will have n the mechanism which cntains the jint and hw the reactins t these effects might alter the cnditins at the jint. Cnsidering first the effect f the frictinal frce, the previus chapters have shwn hw the frictinal effects ver the surface f the ball in the jint can be reduced t a single resultant frce acting thrugh, and a single resultant mment acting abut, the centre f the jint. The frictinal effects prduced n the surface f the scket are equal in magnitude and ppsite in directin t thse n the surface f the ball and s they can be reduced t a single frce and mment at the centre f the jint which are equal and ppsite t thse acting n the ball. Thus the tw frictinal frces will balance each ther and, althugh they affect the magnitude and directin f the nrmal reactins within the jint as shwn in chapter fur, beynd the jint they have n direct effects. Hence it is nly the frictinal mment that has an effect n the mechanism cntaining the jint. As can be seen 98

147 frm Fig.5.3, the frictinal mment is balanced by a cuple prduced by the reactin at the next jint r supprt f the mechanism and an equal reactin at the centre f the jint. and ppsite This reactin prduced at the centre f the jint affects the resultant lad prduced n the jint which in turn affects the frictinal frce and mment prduced. Hwever, as the directin and magnitude f this reactin depends upn the structure and cnfiguratin f the mechanism, its effect cannt be incrprated int a mdel which represents the jint independently f the mechanism cntaining it. When incrprated int a mechanism simulatin package such as AMP3D-ADAMS the main prgram will calculate the reactin prduced and s determine the resultant lad n the jint at each stage f the simulatin. 5.3 The Directin f the Frictinal Mment Relative t a General C-rdinate System In rder t btain the expressins fr the frictinal mment prduced in a spherical jint as described in chapter fur it was necessary t select a particular crdinate system fr the jint s that the applied lad P and the vectr w, representing the axis f rtatin, were given by equatins 4.2 and 4.3 as: 99

148 F r ic t i n a l FecES n FRICTIN Al MMENTS B a l l a n d S c k e t -. La N BALL AND S l k ET: LUS i-e ~Rm S ck et Rm R eactin aû-a in s t Fr ic t i n a l M m e n t -. F ig u r e 5.3 R e a c t i n t t h e E f f e c t s f Frictin

149 w = w(-sin0i + cs0k) P = -Pk The frictinal mment M was then given in terms f this c-rdinate system by equatin 4.42 as; M + MÎ + where M and 1^, m^ and n^ are the magnitude and directin csines f the frictinal mment given by equatins 4.43 t Fr the mdel f the frictinal effects in the spherical jint t be f value in the simulatin f a larger mechanism the expressin fr the frictinal mment has t be given in terms f a general c-rdinate system which can be represented by the unit vectrs i', j', k'. In this general c-rdinate system the applied lad P and the vectr w can be expressed as: w = w(l 'i' + m 'i' + n 'k') (5.5) w w w P = P(l_'i' + m 'j ' + n^'k') (5.6) p P P while the frictinal mment M can be given as: M = (5.7) 100

150 where 1*', and n^'; Ip', nip' and Hp'; and 1%', m,, and n^,' are the directin csines f the three vectrs M M relative t the general c-rdinate system. In the simulatin f a mechanism the directin f the applied lad and the axis f rtatin at a jint will be knwn by the prgram and s the relatinship between the tw c-rdinate systems can be btained by cmparing the equatins in which they are given in relatin t each crdinate system. Frm equatins 4.3 and 5.6 it can be seen that; k = -(l_'i' + m^ i' + ni 'k') (5.8) p P P and frm equatins 4.2 and 5.5; -sin9i+cs0k = 1 'i +m 'i'+n 'k* (5.9) w w w By cmbining equatins 5.8 and 5.9 the unit vectr i can be given as ; -1 i = --- [(1^ '+l^'cs0)i'+(m^ '+m 'cs0)i' - sine P - w p +(n '+n 'cs0)k'] (5.10) w p An expressin fr the third unit vectr j can then be btained frm the vectr prduct ^ i which gives; - 1 j = -(1 'i'+m 'i'+n 'k')x--- [(1 '+1 'cs0)i' ^ P - P ^ P - sin0 ^ P +(m '+m^'cs0)i'+(n '+n^'cs0)k'] w p w p 101

151 1 sin0 [1 '(m '+m 'cs0)k'-l '(n '+n 'cs0)i' p w p p w p -m '(1 *+l 'cs0)k'+m^'(n '+n^ cs0)i' p w p p w p +"p' (lw '+lp'cs0)l'-np' (m^'+nip'cse)! ] sin0 +(lp'mw'-mp'lw')k'] (5.11) Expressins fr the sine and csine f the angle 0 in terms f the unit vectrs f the general c-rdinate system can be btained using the scalar prduct f the vectrs w and P which frm Fig.4.2 can be seen t be given by: w.p = w P c s (tt 0) = -wpcs0 (5.12) In terms f the directin csines the scalar prduct is given by: w-z = wp(lw'lp'+mw'mp'+"w'"p') (5-13) Frm equatins 5.12 and 5.13 the csine f the angle 0 can be seen t be given by: and s the sine f the angle 0 is given by: 102

152 sine = /l-(lw'lp'+*w'*p'+nw'np')^ (5.15) The directin csines f the frictinal mment relative t the general c-rdinate system, 1^', m^' and n^', can then be fund by taking the sum f the cmpnents f the directin csines 1^, m^ and n^ that act in the relevant directin given by the unit vectr d., j' r k'. Using equatins 5.8, 5.10 and 5.11 the directin csines are given by: V - «V ^m '+m_ 'cs0\ /n *1 '-1 'n «' ' L e " ' )- v p ' <= > V =»» v - > where cs0 and sin0 are given by equatins 5.14 and These expressins will give the directin f the frictinal mment relative t the general c-rdinate system. The directin f the frictinal frce r nrmal reactin can be btained relative t the general crdinate system, if required, by using the same equatins after substituting the relevant directin csines fr 1^, 103

153 5.4 The Displacement and Velcity Dependence f Frictin in a Spherical Jint The displacement and velcity dependence f frictin was riginally incrprated int the frictin mdel develped fr AMP2D t avid the cmputatinally undesirable effect f an instantaneus change in the frictinal frce frm a psitive t a negative value n the reversal f the directin f mtin. Hwever, it als allwed the change frm a higher 'static' value f the cefficient f frictin during initial mtin t the lwer 'dynamic' value at higher velcities t be mdelled as is described in sectin 2.2. When the directin f rtatin is reversed in a spherical jint there is an instantaneus change in the magnitude and the directin f the frictinal mment prduced and s it can be seen that the displacement and velcity dependence f frictin needs t be incrprated int any mdel f frictin in a spherical jint. Hwever, as described in sectin 2.3, the mdel used in AMP2D cannt be simply adapted because the relative velcity f the surfaces in cntact and thus the cefficient f frictin varies ver the area f cntact within the jint. 104

154 This prblem can be vercme using the mdel develped fr frictin in a spherical jint in chapters three and fur by cnsidering an instantaneus cefficient f frictin at each elemental area n the cntact area. The value f this instantaneus cefficient f frictin varies with its psitin n the cntact area and s is cntained within the integrals in the expressins fr the frictinal frce and mment. The expressins fr the factrs f^, f and fg given by equatins 3.26, 3.33 and 3.39 then becme: Itr PjSin«cs(*-^~) (csaainy-sinacspcsdjd^dp (5.19) v/l - (csacs^+sinacspsinf)^ 0 <J pjsinacs(-^^ (sin#-sin^/sin*cs^; -sinacsacspcs&()dadp V - (csacsü+sindcspsin#) 2 2n p^sin^xcs(-^-^) (sinacsÿ -csasinycsp)dadp Jl - (csacsa+sinxcspsin#) (5.20) (5.21) where I is given by equatin 3.17 as: I = PC C^TT 2 s inac sac s ( 2 "^) da The instantaneus cefficient f frictin pj can be determined using the expressins that were derived fr the frictin mdel incrprated int AMP2D as described in sectin 2.2. Equatin 2.5 gives: 105

155 -3v = (Pgf - spj)e V. + spj where is the cefficient f static frictin p^ V is the cefficient f dynamic frictin is the sliding velcity f the surfaces in cntact s is the sign f that velcity V is the sliding velcity at which the changever frm 'static' t 'dynamic' frictin is 95% cmplete f is a factr calculated fr each integratin time step f the prgram and given by equatin 2.4 as: X f, - 19sAx(f,-s) ^ JL» Li x^ - 19Ax(f^-s) where f^ is the value f f at the last time step Ax is the displacement during the current time step x^ is the sliding distance (measured frm the pint at which the velcity was last er) at which If 1=0.95 Within the expressins the nly terms that are dependent n the psitin f the elemental velcity v and the displacement Ax. area are the sliding These tw terms can be given by the fllwing expressins: V = r'w Ax = r'a0 106

156 where w is the speed f rtatin f the jint A0 is the angular displacement f the jint during the current time step f the simulatin r ' is the perpendicular distance frm the element t the axis f rtatin. Frm Fig.3.6 it can be seen that the perpendicular distance r' is the distance frm A t C which is dented by the symbl c where : c = /a^ + b^ Expressins fr the distances a and b have been determined in sectin 3.4 and are given by equatins 3.20 and 3.21 as : a = rsinxsinp b = r(csxsinf-sinxcspcs*j Thus the perpendicular distance r ' is given by; 2 2 T sin Xsin p + (csxsin2^ -sinxcspcs20 which frm Appendix 1 can be seen t reduce t: r ' = r /l - (csxcsa+sinxcspsina^ Substituting this expressin int thse given fr the sliding velcity v and the displacement A x prduces the 107

157 fllwing equatins: V = rw y1 - (csacs%+sin#cspsinf)^ (5.22) X = ra0 f1 - (csacsa+sinxcspsin#)^ (5.23) Substituting these expressins int thse fr the instantaneus cefficient f frictin pj, the factrs f^, f2 and f^ can be evaluated by numerical integratin as befre and s the frictinal frce and mment can be determined using this adaptin f the frictin mdel already develped, prvided the user f the mdel supplies the values f the cefficients f static and dynamic frictin, p and p,, and the values fr v and x. ^s Fig.5.4 shws the effect f the velcity dependence f frictin n the frictinal mment determined using the mdel described abve. The graph shws the variatin in the frictinal mment, as the cefficient f frictin changes frm a high 'static' value t a lwer 'dynamic' value with the increasing angular velcity, fr varius values f the angle 0, the angle between the axis f rtatin and the directin f the applied lad. It can be seen that when the angle 0 is small with the axis f rtatin passing thrugh r clse t the cntact area the variatin in the frictinal mment is mre gradual than when the angle 0 is large. This is due t the fact that when pints n the cntact area are clse t the axis f 108

158 V i K ^ 9 CD r^ > ô -J ÜJ > ; a: u3 > -- i n Z ce U] f v~t % E VÎJ Œ CD W -Itiï GC r û U) H h _l Cl U %h u. vn u m f\j 3 I I \jj ^5 - j- ir> CD L <r fm w C 3 vt> CL L

159 rtatin, their sliding velcities remain small and s within the range where the frictin is velcity dependent fr much higher values f the angular velcity. It will be shwn in the fllwing sectin that it may be useful t btain sme relatively simple and apprximate expressins fr the values f the frictinal frce and mment. This is due t the fact that the numerical integratin and the iteratins used in the mdel require a cnsiderable amunt f cmputer time and als that abslute accuracy is nt required by a cmputer simulatin which aims t be a guide t what happens rather than a precise methd f predictin. This means that if numerical integratin is t be avided the velcity and displacement dependence has t be determined by a mre apprximate methd than described abve. Frm the equatins given abve, it can be seen that the nly factr in the expressins determining the instantaneus cefficient f frictin which is dependent upn the psitin f the element is its perpendicular distance r' frm the axis f rtatin. Thus if an apprximately average value can be taken fr the perpendicular distances frm the axis f rtatin f all the elements n the cntact area, the cefficient f frictin des nt need t be included in the integratin. This can be demnstrated using a simple apprximatin fr 109

160 the average perpendicular distance r'. When the cntact area is small and the angle 0 is large the mean value fr the distance r* is given by the perpendicular distance frm the centre f the cntact area t the axis f rtatin. Frm Fig.3.5 it can be seen that this distance is given by: r'= rsin* As the cntact area increases in sie the mean psitin f the elements n it will mve frm its centre twards the centre f the jint. Taking this effect int accunt, as well as the pressure distributin which results in greater frictinal effects at the centre f the cntact area, a suitable apprximatin fr the distance r ' can be given by: r' = rsin# cs( ^ ) Hwever, this apprximatin is based n the assumptin that the distance r' is equivalent t the perpendicular distance frm the mean psitin f the elements n the cntact area t the axis f rtatin, which is nt valid when the axis f rtatin passes thugh the cntact area. Thus, the apprximatin has t be adapted t cver this situatin. A pssible apprximatin that culd be used is : 110

161 r' = rsin2rcs(--^ -)+rsin(-^ ^) Fig.5.5 shws a cmparisn f the values f the frictinal mment btained using this apprximatin with thse btained previusly. It can be seen that the apprximatin gives results clsely cmparable with thse btained previusly when the angle 0 is large. This is due t the fact that when the angle 0 is large there is little variatin in the sliding velcity ver the cntact area. When the angle 0 is smaller there is a greater difference between the values btained using the apprximatin and thse btained previusly. This is due t the greater variatin in the sliding velcity ver the cntact area, althugh the difference culd prbably be reduced by determining a mre accurate apprximatin. 5.5 Apprximate Expressins As stated briefly in the previus sectin, the numerical integratin and the iteratins necessary t determine the frictinal effects in a spherical jint require cnsiderable cmputer time. Thus if the frictin mdel based n these rutines is t be incrprated int a cmputer simulatin prgram such as AMP3D-ADAMS, it will greatly increase the time and therefre the cst f the 111

162 Q_ CL en 3 w ie + c 1 X g (L ce W X h 2 H c2 Lu W U lu û Z Ui a u Q LU V u ȯ b ÜJ t w V H <4 Ui > L w P e 3 X w I H Z Z E ce w c Q IS ( M CD -h us w (X 3 v5 ü. X V- w C l C3; Ln in L Ui x3 CL L

163 simulatin. Hwever, the purpse f such a prgram is nt t prvide precise results but t give an apprximate simulatin f events. Thus reasnably accurate apprximatins t the expressins used t determine the frictinal effects in a spherical jint wuld prvide adequate results and save n time and expense. As the frictinal frce has been shwn t have n direct effect beynd the spherical jint, it is nly the expressins fr the frictinal mment M fr which apprximatins have t be btained. In equatin 4.42 this mment is given by; M = "(1%! + + rij^k) while its magnitude and directin csines are given by t 4.46 as : - / S 7 R ^S^MS M ^s ms M % m r M ^r m r 112

164 The magnitudes and directins f the tw perpendicular cmpnents f this mment, the mment f slip frictin Mg and the mment f rtatinal frictin are given by equatins 4.22, and 4.36 and as; M = rf-p f1 ^CS^0 -f^csg «s = T " m S - 0 rf,p ^ % IT77~2 MR ~ f,cs^9 2 / "2 ' sgn(cs0) sin 0 F h / 2 m ^ = /sin 0 - % sgn(cs0) ^ sin0 J 1+f^^ -1 ^MR = "/ ' "ÿ sgn(cs0) yr^ 113

165 Cnsidering these expressins it is clear that they culd easily be evaluated if simple apprximate expressins were determined fr the factrs f^, f2 and f^. Fig.3.9 shws hw the rati f each f these three factrs t the value f the cefficient f frictin varies with the values f the angle 7$, which is the angle between the line f actin f the nrmal reactin and the axis f rtatin, and the angle which defines the extent f the cntact area. It can be seen frm these graphs that, when the value f the angle ÿ is high and that f the angle is lw, the values f the factrs f^ and f2 tend twards the value f the cefficient f frictin and the value f the factr f^ tends twards er. It can als be seen frm equatin 4.46 that the value f the angle 7^ is high when the value f the angle 0, the angle between the axis f rtatin and the directin f the applied lad, is als high. Thus it can be seen frm Fig.3.9a that fr a wide range f the higher values f the angle 0 when the angle is small the factrs f^, f2 and f^ can be taken as being given as: f-i = f«= p fg = 0 114

166 Substituting these values fr the three factrs int the equatins given previusly, the frictinal mment M can be given apprximately, when the angle 0 is large and the angle C^ is small, by: M + m 1 by: while the magnitude and directin csines are given prp M = Mg = -r 2 p^cs^e ms -pcs0 sin0 ^M ^MS Fr slightly higher values f the angle (^ the magnitude f the factr f^ has t be taken int cnsideratin as can be seen frm Fig.3.9b. Hwever, a gd apprximatin t the frictinal mment can be btained in this case by assuming a unifrm variatin f the factr f^ ver the higher values f the angle 0. Thus the three factrs culd be given by: 115

167 f-, - - p which when substituted int the expressins fr the frictinal mment give a gd apprximatin ver a range f the smaller values f the angle cx'^. This apprximatin may be f sme value in the case being cnsidered in the next sectin - that f a straddle-type jint (where the angle 0 is always high) under lw lads, s that the cntact area is small and thus the value f the angle <^ is lw. Hwever, in general, where the angle 0 is likely t be small r the cntact area large a mre wide-ranging apprximatin t the three factrs will be required. As can be seen frm the expressins in equatins 3.26, 3.33 and 3.39, the magnitudes f the factrs f^, f2 and f^ depend upn the values f the angles (^ and and the cefficient f frictin p. The angle '7$ is shwn by equatin 4.46 t be dependent n the angle 0. Then, as the factrs are directly prprtinal t the cefficient f frictin p, gd apprximatins fr them can be fund by fitting surfaces t the variatins f the ratis f these factrs t the cefficient f frictin against the angles 0 and. There are cmputer packages available t 116

168 d this but it has nt yet been attempted and is ne f the utstanding requirements if the mdel develped is t be incrprated int a cmputer simulatin prgram such as AMP3D-ADAMS. 5.6 Straddle-type Spherical Jints Spherical jints are cmmnly used in the frm f the straddle-type jint, als knwn as a spherical bearing, which is shwn in Fig.5.6. The frictin mdel develped in the previus chapters can be applied directly t this type f jint when the lad n the jint is f lw magnitude and is largely radial in directin i.e. it acts rughly perpendicular t the axis f the straddle as shwn in Fig.5.6. The direct applicatin f the mdel is limited t these cnditins because it is necessary fr the cntact between the ball and the straddle supprt t ccur ver a circular area, which is ne f the assumptins the mdel is based upn. Thus the extent f the circular cntact area predicted by the frmula in sectin 5.1 must remain within the limits given by the edges f the straddle supprt. The range f lads applied t a straddle-type jint fr which the mdel is valid can be determined by taking the dimensins f a typical jint such as supplied by the Rse 117

169 Ball C F ig u re 5.6a S tr a p p le -ty p e S p h e r ic a l J in t (6= q*j St r a d d l e Supprt Figure 5.6 b St r r d l e-ttpe Spherical Jimt

170 Bearings catalgue (1983). Fr example, a jint with a radius R f 12.7mm and a radial clearance (R^-R^) in the range mm has a cntact area defined by the angle (^ which is given by equatin 5.4 as: SNCc^+c^) = I radians ' 4R(R2-Rj ) The factrs c^ and C2 are given by equatins 5.2 and 5.3 as : I-V.2 l-vgz The Pissn's ratis v^, V 2 and the Yung's Mduli and E 2 can be taken fr a stainless steel jint as: v^ = V 2 = 0.3 = E2 = 2.05x10" N/m^ while N is the resultant lad prduced by the sum f the applied lad and the frictinal frce in the jint. Then, assuming an average radial clearance f (R^-R^) = 0.01mm, the expressin in equatin 5.4 can be evaluated t give: <^ = radians The extent f the straddle, which is shwn by the angle Y 118

171 in Fig.5.6, is given fr this jint as 32. The frictin mdel applies prvided the predicted cntact area is cntained with the cnfines f the straddle supprt. Thus the sum f the angle (^ and the angle ip, which is the angular mvement f the centre f the cntact area twards the edge f the straddle, must be less than the angle Y. Fr the particular jint being cnsidered, the maximum angle f misalignment, shwn by the angle X in Fig.5.6, is 11, which gives a minimum value f the angle 0 fr the jint f 79. Taking this value f the angle 0 and a high value fr the cefficient f frictin f yn =1.0, the value f the angle y is predicted by the frictin mdel, using a cmputer prgram, as being rughly 12 ver a wide range f values f the angle (^. Thus the maximum pssible value f the angle is given by: C( = Y - IfJ = = 20 The magnitude f the resultant lad N can then be determined, frm the expressin given fr the angle ( abve, as : = 813 Newtns Using the cmputer prgram, the frictin mdel shws that 119

172 in this situatin the rati f the resultant lad N t the lad applied t the jint P is given by; N/P = Thus in the jint described abve the applied lad P can have a magnitude f up t 1,000 Newtns while the frictin mdel can still be applied. If the cefficient f frictin is lwer the pssible magnitude f the applied lad P is higher. Fr example, when the cefficient f frictin has a value f p = 0.2 the applied lad can be as high as 3,000 Newtns and the frictin mdel can still be applied. Hwever, when the applied lad is higher than that described abve r there is a significant degree f axial lad n the jint the cntact area reaches the edge f the straddle-supprt and s is n lnger circular. In these cases the frictin mdel will have t be revised and adapted if it is t be used t predict the effects f frictin within a straddle-type spherical jint. 120

173 CHAPTER SIX Experimental Apparatus, Prcedure and Results 6.1 The Apparatus Having develped a theretical mdel f the effects f frictin in a spherical jint, experimental results were then required t determine whether the mdel prvides a reasnable accurate estimatin f the frictinal mment prduced in an actual jint. Thus it was necessary t be able t measure the frictinal mment prduced in a jint while varying the cnditins at that jint s that the mdel culd be cnfirmed t be reasnably accurate ver a range f the factrs predicted t affect the frictinal mment. The previus chapters have indicated that there are three main factrs which affect the frictinal mment prduced in a spherical jint. The first is the angle 0, which is the angle between the directin f the applied lad and the axis f rtatin in the jint. The secnd is the cefficient f frictin and the third main factr is the angle cx^ which defines the extent f the area f cntact and which depends upn a number f ther factrs including the magnitude f the lad and the radial clearance in the 121

174 jint. An investigatin fund n evidence f any previus experimental wrk dne t measure the frictinal mment prduced in a spherical jint while these factrs are varied and s the apparatus had t be specially designed fr this task. There were tw particular prblems t be vercme in the design f this apparatus. The first was that the directin f the frictinal mment varied with the changing cnditins which meant that the mment had t be measured in the frm f three perpendicular cmpnents. The secnd prblem was that the frictin prduced in the apparatus had t be negligible when cmpared t that prduced in the jint if reasnable results were t be btained. A number f pssibilities were cnsidered befre the design f the apparatus shwn in Figs.6.1 t 6.3 was decided upn. This design invlved the measuring f the frictinal mment prduced n the was laded vertically frm belw. scket f a jint which The ball was rtated in this scket by means f an attached shaft whse angle t the vertical culd be altered t vary the angle 0 between the directin f the applied lad and the axis f rtatin. The cefficient f frictin and the extent f the cntact area culd then be varied by using a series f different balls and sckets. 122

175 > m À F igure 6.1 The GriMBflL Mechanism

176 9 F i &ure 6.2 T he THE SUPPRTING SHAFTS B R AC K ET R T A T ln G FR THE B a LL