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Unformatted text preview: Technical Paper Wear Predictions for a Simple-Cam Including the Coupled Evolution of Wear and Load© Predictions of wear based on theforces and slip t-‘eiocities caicuiated using the unworn first cycie geometry will not accurateiy predict a tnechanisrn. ’s useful! life. This is because there is a conpiing be— tween the contact conditions and the geometry of the components, it"i’tic'i’t are changing as they wear. The abiiity to accurateiv predict how a mechanism t-L-‘iii perform over an extended number of cycies requires knowiedge of how the components are changing during operation. Using an Archard’s wear constant, a CiOSflifiJt'ftt expression describ- ing the coupled evolution of the contact toads and wearfor a t'rircuiar—cam with aflat-facedfoiiower is developed. Further, these closed firrm expres— siottsfor food and wear, which are given in terms of the number of com—cycles n, are nort— dimensionaiized by the cam eccentricity, width, and the Archarri’s wear constant. The non—dintensionai wear and ioad are functions ofthe number ofcycies and the non-dimensional group termed wear—com— pliance, which is the product oftheArchard 's wear constant andr the spring constant over the cam width. Non-dimensional ciosedform equations are oiso developed for an uncoupled evolution oj‘ge- otnetry and wear; and the predictions in asefiti i i f'e are comparedfor the coupieti and uncoupied equa— tions. The unconpied i-vearpredictions always over predict wear, and greatly over predict wear under conditions of high spring sttflitess, iow wear re— sistance, narrow cam widths and iarge nutnhers ofcycies. It is suggested that a coupling ofthe evo- iution of geometry and contact conditions must be accounted for when making iije predictions of wearing mechanisms based soieiy on wear: Final manuscript approved February ‘t, 2001 Review led by Barry Riddle W. GREGORY SAWYER (Member, STLE) University of Florida Department of Mechanical Engineering Gainesville, Florida 32611 KEY WOFIDS Life Prediction Methods; Cams; Wear and Failure INTRODUCTION In many mcchanical systems, it is desirable to design components for inl‘initc lil‘c. Unfortunately, such designs are practically impossible to real— i7.c, and subsequently engineers are constrained to design for finite life. While there are many established life prediction theories for material fail— ure in mechanical designs (cg. fatigue. fracture, and creep) there are no available theories that include the effects of wear. Thus, designers must rely on experience and available rankings of material wear resistance to guide the designs and material sclcctions. Predictions of wear based on the forces and slip velocities calculated for an unworn mechanism may greatly under predict or. worse yet. greatly ovcr prcdict the useful life, This is a result of the basic coupling between thc contact conditions and the geom— etry of the components, which are changing as they wear. if the effects of wear could be accounted for in a simple and formulaic method. material sclcctions could be based on function and expected life, and the mechanism’s components could be designed to enhance function. expected life, or. po— tentially, both function and life. The ability to accurately predict how a mechanism will perform over an extended number ofcyclcs requires knowledge of how the components are changing during operation. The geometric evolution of these componcnts as a result of wear requires, at a minimum, knowledge of the complete history of contact locations. pressures, and component wear—rates. As wear occurs, the changes in the geometry change the kinematics, which influ— ence the contact loads, contact locations, slip velocities, and sliding dis— tanccs. For many simple mechanisms. closed form solutions can be devel— oped to predict displacements, sliding velocities, and contact loads (texts on the subject have been written by Shigley and Uicker (1995) it), Sncck (1991) {2), and many others); for more complex mechanisms, many com- nicrcially available software packages numerically evaluate the kinematics and dynamics (cg. ProMechanica, Adams, Working Model, and Autolcv). Over the past decade, efforts have been made to include wear within numerical and finite element methods. These models use an Archat‘d’s wear (Continued on next page) Journal of the Society of Tribologists and Lubrication Engineers 31 (Continued from previous page) Fig. 1—A schematic drawing of a circular-cam with a flat- faced follower, the angular coordinate 9, and the vectors for the radius R, eccentricity E, and po- sition ,5 are shown. As the circular cam rotates, the inital centroid described by the eccentricity vector sweeps out a circular path as shown by the dashed line. model for the incremental wear predictions. Podra and Andersson {3) used a finite element analysis to predict the wear ofa hetnispherically tipped pin sliding against a plane, and found good agreement with pin—on—disk experiments. Podra and Andersson (4) performed a similar finite element analysis for a conical spinning contact and compared these results to an analytical model with favorable agreement. Podra and Andersson (5} also simulated the wear for a sliding sphere-on—flat and cylinder—on—flat configuration, compar- ing the results of a finite element simulation to a numerical simulation, which used an elastic foundation or Winkler model to calculate the contact pressures. The wear predic— tions and contact pressures calculated using the Winkler model were in good agreement with the finite element analy— ses. with the greatest deviation occurring during the early sliding distances where the contacts were under their most concentrated conditions. Flodin and Andersson (6) used the Winkler model and numerically evaluated the mild wear oc— curring in spur gears. allowing the pressures to evolve as the surfaces wear. Hugnell andAndersson {7) and later Hugnell, et al. {8) simulated the mild wear in a cam—follower contact. allowing the shape and pressures to evolve as a function of the number of cam rotations; these models had a non—rotat~ ing and rotating cam follower, respectively. Maxian. et al. (till—(H) used finite element analysis with coupled evolutions in shape and pressures for the polyethylene components in total hip replacements. The calculated worn geometries and the rates of material loss were in good agreement with clini— cal observations for a variety of designs. Kurtz. et a]. H2) extended the work of Maxian, et al. (9)41!) to include wear on both the articulating and backside of the polyethylene component. Sui, et al. {)3} studied the wear of Po]ytetrafluoroelhylene lip seals using an iterative rezoning 32 procedure and finite elements to predict the worn geometry of such seals. Barecki and Scieszka {14) numerically simu- lated the wear process and changes in the pressure distribu- tion of the friction linings in a post-type brake system. Closed form analytical solutions that allow geometries and load to evolve for simple mechanisms are rare. The only such solution that the author is aware of is by Blanchet U5), who modeled the coupled evolution in geometry and load for a simple Scotch—Yoke mechanism. This mechanism has a crank with a rigidly attached pin, which is located some distance away from the line of rotation, and slides within a vertical slot connected a mass. The resulting mass position. mass velocity. and pintslot reaction forces are sinusoidal funes tions. As wear occurs, the slot evolves into an ellipse result- ing in larger displacement, higher velocities, and greater pint slot reaction forces. Thus, the cyclic rate of change of the depth of wear into the slot is a continuously increasing func— tion with number of cycles. Through recursive application of the equations for depth of wear and contact force, Blanchet ()5) developed a closed—form solution capable of predicting the evolution in the pin and slot geometries as well as the kinematics and dynamics of the Scotch Yoke as a function of the number of crank revolutions. ANALYSIS A closed form expression describing the coupled evolu— tion ofthe contact loads and the wear for a circular cam with a flat-faced follower (Fig. l) is developed. In this two—di— mensional mechanism of width w, the cam is a circle of ini— tial radius R that rotates about an origin point located a dis— tance e away from the cam’s center. Wear of the cam due to sliding against the flat-faced follower is considered; wear on the follower is not included. An angular coordinate (9) rela— tive to the initial center of the circular cam is used to define the point of contact; 8 is also a cam—attached angular coordi- nate. The depth of wear along the cam surface is denoted by Ami. where the subscript (8) denotes that it is a function of the angular coordinate. A closure equation for the circular cam with a flat-faced follower is given by Hq, [1]. In this equation ,6 is the posi— tion vector from the origin of cam rotation to the contact— point on the flat—faced follower. e is the eccentricity vector from the origin of cam rotation to the initial center of the circular cam. and fr is the radial vector going from the ini— tial cam center to the contact—point. The ttnit vectors :3 and j are the horizontal and vertical unit vectors respectively, as shown in Fig. l. p=E5+RzecosQi+esin6j+Rj [I] in this simple cam mechanism. the contact location on the follower face is described by the 9' component and the compression in the spring is described by the } component. The angular coordinate 8 describes both the rotation of the cam (see Fig. l) and the corresponding angular location of September 2001 LUBRICATION ENGINEERING Fig. 2—A schematic drawing of a circular-cam with a flat- faced follower, after a certain number of cycles the evolution in shape of the cam is shown. The corresponding angular coordinate 8, which de- scribes the location of contact on the cam, and vectors for the radial distance R, eccentricity £3, and position ,6 are shown. The dashed circle corresponds to the nth cycle while the solid circle is the profile for the (n+1) cycle. contact on the follower Fig. 2). As shown in Fig. l, 9 = 0, 21c when the eccentricity vector is horizontal and the cor— responding contact point on the cam is at 8 + in“? (see Fig. 2). Using an Archard’s wear constant. Sawyer {16} developed an expression for the incremental change in wear depth (dh) for a differential element moving through a two—dimensional linc contact with a pressure distribution given by PM. where s is the coordinate axis through the contact. This expression is given in Eq. [2]. and the total accumulation of depth of wear from the entrance to the exit of the contact is given by Eq. [3], where K is the Archard’s wear constant= L is the length of contact, w is the width, and FD is the normal load. dh=xega or L h. = K Pmds = —” [3] {J ' For the circular—cam mechanism, an initial contact force F is defined for the 6 = 0 location. As the cam rotates= the eccentricity causes this force to vary according to Eq. [4], where k (Ntrn) is the spring constant. Using Eq. [3], the re— sulting depth of wear along the cam surface after the first cycle of loading is given by Eq. [5]. F Egg:r+amw m TABLE luv-ANALYTICAL EXPRESSIONS FOR WEAR DEPTH AS A FUNCTION or ANGULAR COORDINATE 8, SPRING CONSTANT k. ARCHARD'S WEAR CONSTANT K, WIDTH w, INI'HAL CONTACT Ponce F . AND ECCENTRLCITY e FOR THF. FIRST EIGHT CYCLES. n . iq) ......._____.__ . . .. . 0 . l K! +eksmq! i W Klf.'l'ek5i"qlt—tx+2w] 2 W! Mtge —3tht-+.1s-"l 3 w .- F- - ' . . . . Wt—t—‘x‘Hrs-summit» p4..-'| _4_.__ W “Lem...” . ' .__..___ AGE .t. , . , . . . wow —5t K'\-.'+]Jr K'n" —]U.lt-Kn.'+5u': 5 a. . . WFFK‘warn-user“; +2nt=x3u-"—15ixu-‘ + F:ny 6 W . I”: k- _ ..___... .. I \ I wrac- -—tt’K‘w-+ Ell-‘K'w' - iii-"KM" a 351mm — Mix-.3 + 7311'": 7 W mpk-‘K‘aar"x‘.u—2sx‘x‘w‘+sct‘x‘w'_itu'x‘.n I stirs-1t" llama is” 8 “ A =§tf+wsnm m 9'. —| [ ’1” ‘H; As shown. greatly exaggerated, in Fig. 2, the compres- sion in the spring and the resulting contact forces are re— duced in the following cycle (n+1) because of the wear in— curred during the current cycle (It). The cumulative depth of wear Am” and contact force Fm” for an arbitrary number of cycles are given in Eqslfi] and [7]. respectively: K AND.” : A[_3}.n—| +;(F[‘_H}Ji—|) [email protected]—Amfl) in fin.” These equations can be evaluated numerically cycle—by— cycle or analytically starting from the expression for anzl and the initial condition Atom] and marching forward in cycles applying Eqs. [6] and [7] recursively. The resulting depths of wear as a function of q for the first eight cam revolutions are shown in Table 1. The resulting depth of wear Aim and the contact forces Fm” can be described exactly by the sum— mations given in Eqs. [8] and [9], respectively. K F+eksint9i -"" t _,t-K ""‘HJ am— [ )2 H ( l m w M, Ma —i]l w (Continued on next page) Journal of the Society of Tribologists and Lubrication Engineers 33 (Continued from previous page) [r i n - w-T—wwv-rsm —- anh-__ M. n M _ M '-__\II ‘. _ M "-._ \ ‘ s. y M I w 'x . -.‘ -,l ._ " H E I": ‘3‘ a. si. '3 '- '- I I_ R. I‘ - ~ 5‘ xi, ': . -- \I‘ 'I ll III .gl ~3- ,l E . ' 't_ \-._ , i 3 ': 'i '. 's I'. -. i - .' 1': - r -..--r._~m ;.- - I Fig. 3—A plot of Eq. [16], which is the ratio of the non- dimensional wear depth predictions of the coupled Eq. [13] and the uncoupled Eq. [15], for values of ,8" [ram 10-1-1 0'8 over a range in cycles from 1-10la is shown. (a) linear (b) logarithmic = (r + saksin9)[l _ " '1_’..“ _I[ 4"" 19] w Emilia—i). w The group kK/w is a non—dimensional product. of the lin— ear spring constant and the linear wear rate; for the purpose ofthis manuscript, it is termed “wear-compliance“ and given the symbol The other non—dimensional groupings are wear—depth. Eli-10m: Awmle and load, F*: FKr‘ew. Thus, Eqs. [8] and I9] can be written in a non—dimensional form, as shown in Eqs. [I0] and [I I ]. n—I (Sr-“HM : (Fe: + Sin flfijii (_fis]l-’|—]—EJ] [IUJ i:l_l as = {F + small — flat—H)” ' I I u The series g_ll_lf"_ly:__.-i-;""' is recognized as the function 1'5"“ '["’r"3"-I_ This greatly simplifies Eqs. [IO] and I I ll, and pro— vides closed form expressions for the depth of wear and load as shown in Eqs. ll2l and ['13]. 34 65;.)_”=(F*+fiistn9) 1* l—(l—fi"=)” |l2j [3 2U?“ +fi*sin6)(l —l3*]" 113] DISCUSSION As visible in Eqs. [12] and Hill= the sinusoidal depen— dence on contact force and wear predicts a maximum depth of wear at 8 = 7:32 and a minimum at 8 : 3nf2. Sinusoidal dependencies are also predicted if no coupling between the wear and the subsequent loading are assumed. and, instead, the initial loads are used to make predictions ofthe changes in geometry and performance. The resulting equation pre— dicts a linearly increasing depth of wear with number of cycles: this is shown in Eq. [14] and in a non—dimensional form in liq, [15]. Flip = (I? +t3'isin8)(I 43*)” Ml tim'uripir-ri _ HKLIE 8k sin 6) IV I'15l Atoll.” The fractional difference between the non—dimensional wear predictions of the coupled and uncoupled equations, respectively, is shown in Ed [16] and plotted vs. number of cycles in Fig. 3 for various values of non—dimensional wear compliance 55%» :(FO—fiir) l16| ma-aiwmi "‘8 3 m;- [6M The meaningful limits on are from zero to one. The wear compliance [3* must be greater than zero, which corre- sponds to the condition of infinite wear resistance K : 0, no spring stiffness k = 0, or infinite width. The upper limit on comes from the condition where the flat-faced follower loses contact. with the circular cam, a condition described by F* < 0. This condition would occur first at the location 8 : 3nl2. and as shown in Eq. [ 1?]. if the cam does not lose con— tact on the first cycle (Ff > b*)._ then the available ranges of [W that ensure a continuous contact between the follower are for [3* less than one. 1:" =(1~"“" —B'“)(1—fi"‘]" 20 In] Ill] 3 El and [4* > then qus. | l2] and [I3] predict positive values ofcontact force 17* and wear 6* for all values of 8 at any number of cycles n. Both the contact force and the cyclic rate of change of depth of wear (db‘i‘ldn) asymp— totically approach zero as the number ofcycles approaches September 2001' LUBRICATION ENGINEERING L'L‘ Fig. 4—Plots. (a) a plot of Eq. [21], which gives the life at which the cam wear proceeds through the center of rotation in number of cycles nmx, over a range of [5* from 1-10"3 and various values of the ratio {so-1)l{s(o-1)) from 104-10“ lb) an overlay-plot oi the lite prediction on the ratio of the non-dimensional wear depth pre- dictions of the coupled and uncoupled equa- tions. no. There is a physical limitation on the amount of wear than can be accommodated by the circular cam mechanism. The wear depth can’t penetrate through the center of rotation for the cam: the earliest occurrence would be at the 9 = 3m“? location. This condition is described dimensionally and non— dimensionally by Eq. [l8] where c is the ratio of the eccen— tricity over the radius (8 : er‘R] A ,. <R—e; <——1 [181 'Jr [Q—'—-],n 8 Substituting 8 2 Ball into Eq. [ 12], simplifying and then substituting this result into Eq. [18] for at“ ,_,|_ gives Eq. [19']. which if satisfied ensures that. the depth of wear has not pro- ceeded through the center of rotation. (Fi—fi’i);(t—(l—fii)“)<%—l [19] As previously described. F ’-‘ must. be greater than and [3* must be less than one in order for the cam to remain in continuous contact with the follower. Describing in terms of lit“ by introducing a new parameter ot, where : 0t B’k and 0t > 1, simplifies [it]. [19] as shown in Eq. [20]. Addi— tionally, this equation can be easily solved, yielding expres— sions in terms of 0'. and E. for the critical maxi mum wear com— pliance [if for a cam design that will survive a given num— ber of cycles n, or the maximum number of cycles nm“ that can be survived for a given value of wear compliance Because [if must be greater than zero and less than one, and rim< must be greater than one, it is evident from Eq. [2]] that the ratio (E.0t-l}t(e(Ct-l)) must be greater than zero and less than one. [a—l)(t—[l—fii]")<l—l 120} 1 SUI—l 505—1 " 8(05—1] = [ — ; plum l ix) 2 Figure 4(a) shows the maximum life nmax over a range of values of and varying values of the ratio (smudge.- l)); the maximum life is an extremely strong function of this ratio and a much weaker function of As shown in Fig. 3 the uncoupled wear predictions can greatly over—predict the amounts of wear realized under conditions of high l3* and large numbers ofcyclcs. Equation [l6]. which is the ratio of the fractional difference in the non—dimensional wear pre— dictions of the coupled and uncoupled equations, is a func— tion of wear compliance and number ofeycles, and the maxi— mum number of cycles before failure is a function of the ratio (ecu Uttefa— l )) and wear compliance. Thus, ratio of the non-dimensional wear prediction of the coupled and un— coupled equations can be plotted for various values of (80:— llttefot— l )) and cycles to failure nm. Figure 4(b) is an over— lay plot of the fractional difference in the non-dimensional wear predictions ofthe coupled and uncoupled equations us. both 1) the number of cycles at various values of B41, and 2) the cycles to failure rim“x at various ratio’s of (see 1 )t(e(0t- 1)). As the ratio (err—l )tfetot— l )) decreases greater life and greater differences between the coupled and uncoupled predictions are realized. While the variable n‘m represents a nurnberofcycles when wear proceeds all the way through the rotation point of the cam. it is almost certain that the limit of acceptable perfor— mance of the cam system will occur before this. lfthc maxi— mum allowable amount of wear for a specific system 5*W is defined. Eq. I |2l c...
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