FINITE DIFFERENCE SOLUTION OF MIXED BOUNDARY-VALUE ELASTIC PROBLEMS

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1 4 t Internatonal Conference on Mecancal Engneerng, December 6-8, 1, Daa, Banglades/pp. V FINITE DIFFERENCE SOLUTION OF MIXED BOUNDARY-VALUE ELASTIC PROBLEMS S. Reaz Amed, Noor Al Quddus and M. Waa Uddn Department of Mecancal Engneerng, Banglades Unversty of Engneerng and Tecnology, Daa 1, Banglades Abstract A new numercal metod of soluton as been developed for te analyss of deformaton and stresses n elastc bodes subected to mxed -condtons. Te program s capable of dealng wt bot regular and rregular sapes of boundares approprately. An deal matematcal model, based on te dsplacement potental functon, as been used n te fnte dfference soluton. Te present paper demonstrates te applcaton of te newly developed computatonal sceme to a wdely used body wt curved boundares, namely, nvolute profle spur gear teet. Keyword: arbtrary-saped elastc body; dsplacement potental functon; fnte-dfference tecnque. INTRODUCTION Te rapd growt n te use of computers n te past decade gave rse to te developments of advanced computatonal sceme. But someow te elastcty problems are stll sufferng from a lot of sortcomngs. Actually, te management of condtons and sapes as remaned as te bggest urdle n te soluton process of te problems of sold mecancs. Te necessty of te management of sape as lead to te nventon of te fnte element tecnque wt ts overwelmng popularty. Te oter factor of mpedment to qualty soluton of elastc problems s te treatment of te transton n condtons. Te dffcultes nvolved n tryng to solve practcal stress problems usng te exstng approaces, for example, te stress functon approac [1] and te two dsplacement functons approac [], are clearly ponted out n our prevous reports [-6] and also by Durell [7]. In an attempt to crcumvent te dffcultes, Dow et al [8] ntroduced a new modelng approac for te fnte dfference applcatons n sold mecancs. Tey reported tat te accuracy of te fnte dfference metod n reproducng te state of stresses along te was muc ger tan tat of fnte element metod of analyss. Te dsplacement potental functon formulaton of two dmensonal elastc problems used ere was frst ntroduced by Uddn []; later Idrs [] used t for obtanng analytcal solutons of mxed value elastc problems and Amed [4-6] developed te computatonal sceme to extend ts use n obtanng te numercal solutons of a number of mxed value problems. Te ratonalty and relablty of te formulaton s ceced repeatedly by comparng te results of mxed value elastc problems obtaned troug ts formulaton wt tose avalable n te lterature. Recently, our wor as been extended to nclude te problems of arbtrary sapes[1]. Te present paper demonstrates te applcaton of te newly developed computatonal sceme to a wdely used body wt curved boundares, namely, nvolute profle spur gear teet. FORMULATION OF THE PROBLEM Wt reference to a rectangular co-ordnate system, te dfferental equatons of equlbrum for twodmensonal problems n terms of dsplacement components are [1] u 1 u u + + = (1) u 1 u u + + = () Tese two omogeneous ellptc partal-dfferental equatons, wt approprate condtons, ave to be solved for te case of a two-dmensonal problem wen te body forces are assumed to be absent. In te present approac, te problem s reduced to te determnaton of a sngle functon ψ [] nstead of fndng te two varables u and v, smultaneously, satsfyng te equlbrum equatons (1) and (). Te potental functon ψ(x,y) s defned n terms of dsplacement components as u = 1 v = ( 1 ) + () Secton V: Appled Mecancs 171

2 ICME 1, Daa, December 6-8 Wen te dsplacement components n equaton (1) and () are replaced by ψ(x,y), equaton (1) becomes an dentty and te only condton tat as to be satsfed becomes = (4) 4 4 Terefore, te problem s formulated n suc a way tat a sngle functon ψ as to be evaluated from te barmonc equaton (4), satsfyng te condtons tat are specfed at te. Now, te condtons, at any pont on an arbtrary-saped, are nown n terms of te normal and tangental components of dsplacement, u n and u t and of stresses, σ n and σ t. Tese four components are frst expressed n terms of u, v, σ x, σ y, σ xy [6] te components of dsplacements and stress wt respect to te reference axes x and y and fnally, n terms of te functon ψ, as follows: m m ( ) ( 1 ) u n x, y = + l (5) σ u t σ n ( x y) ( 1 ) l l, = m (6) ( x, y) ( x, y) = = E ( ) lm + ( l m m ) ( l + m ) ] + lm (7) E ( ) ( l m ) lm( + ) t ( l m ) + lm( ) ] + 1 (8) y Te computatonal wor n solvng any problem remans te same n te present case as t was n te case of φ-formulaton, snce bot of tem ave to satsfy te same b-armonc equaton. But te ψ-formulaton s free from te nablty of te φ-formulaton n andlng te mxed condton. SOLUTION PROCEDURE Fnte-dfference tecnque s used to dscretze te governng b-armonc equaton and also te dfferental equatons assocated wt te condtons. Te dscrete values of ψ(x,y) at te mes ponts of te doman concerned (see Fg. 1), s solved from te system of lnear algebrac equatons resultng from te dscretzaton of te governng equaton and te assocated condtons. Te present sceme nvolves evaluaton of te functon ψ at te nodal ponts of rectangular grds over a geometrcally rregular regon, were te may not always pass troug te rectangular mes ponts, as sown n Fg.1. Te governng b-armonc equaton, wc s used to evaluate te functon ψ only x/ Pyscal σ n Involute toot Root secton Gear stoc Reference False 1 y/ Fg.1 Dscretzaton sceme for te gear toot doman. (m = 1 mm, a = m, b = 1.5 m, θ p =, d p =5 mm, load σ n /E = e 4). at te nternal mes ponts, s expressed n ts correspondng dfference equaton usng central dfference operators. Te complete fnte-dfference expresson of te b-armonc equaton (4) s gven by R 4 {ψ(-,)+ψ(+,)}-4r (R ){ψ(-1,)+ψ(+1,)} -4(R ){ψ(,+1)+ψ(,-1)}+(6r 4 +8R +6)ψ(,) +R {ψ(-1,-1)+ψ(-1,+1)+ψ(+1,-1) +ψ(+1,+1)}+ψ(,-)+ψ(,+)= (9) Te correspondng grd structure of te governng equaton for an arbtrary nternal mes pont (,) s sown n Fgure. It s tus clear tat wen te pont of applcaton (,) becomes an mmedate negbour of te pyscal, te equaton wll nvolve mes ponts bot nteror and exteror to te pyscal Fg. Grd structure for te governng b-armonc equaton. An magnary (false), exteror to te reference of te doman s ntroduced as sown n te Fg.1. As te dfferental equatons assocated wt te condtons contan second Secton V: Appled Mecancs 17

3 ICME 1, Daa, December 6-8 and trd-order dervatves of te functon ψ, te use of central dfference expressons ultmately leads to te ncluson of ponts exteror to te false. Te dervatves n te expressons of -condtons are tus replaced by ter bacward- or forwarddfference formulae, eepng te order of te local truncaton error te same. Any -value at a -pont, not matcng wt te feld grd ponts, s replaced by te lnear nterpolaton of ts value at te two or four negbourng grd ponts, one of wc s desgnated as te reference pont, wc s unque to te pont. Te dfference equatons of te condtons for any arbtrary pont on te, not matcng wt te feld grd ponts, can readly be obtaned by nowng te actual poston of te pont wt respect to ts reference grd pont. Referrng to Fg., te dfference expressons for condtons at any arbtrary pont P, te locaton of wc s defned by Fg. Locators c and s of te pont P wt respect to ts reference pont R1. Pyscal - -1 c and s, s obtaned by usng lnear nterpolaton of te expressons of te -values at te ponts R1, R, R and R4. Te fnal form of te grd structure so obtaned for te stress components for ponts not matcng wt te feld grd ponts s llustrated n Fg.4. Snce tere are two condtons to be satsfed at an arbtrary pont on te pyscal of te elastc body, two dfference equatons are assgned to a sngle pont on te. Out of tese two equatons, one s used to evaluate te functon ψ at te reference pont correspondng to te pyscal pont and te remanng one for te correspondng pont on te false. Te dscrete values of te potental functon, ψ (x,y), at mes ponts are solved from te system of algebrac equatons resultng from te dscretzaton of te governng equaton and te assocated condtons, by te use of drect metod of soluton (Colesey s trangular decomposton metod). Fnally, te same dfference equatons are organzed for te evaluaton of all te parameters of nterest n te soluton of te body at every nteror as well as ponts from te ψ values at mes ponts of te doman. RESULTS AND DISCUSSIONS R1 R c s P +1 R4 R Pyscal Gear toot as been taen as te arbtrary-saped elastc body, made of ordnary steel (=., E=9 GPa), to obtan te numercal values of stresses n t as well as te deformed sape of t under load. Te conugate loadng on te toot s approxmated ere by te dstrbuton of normal load over a small regon at te contact pont. It sould be mentoned ere tat, because of te Sant-Venant s prncple, te dstrbuton of loadng at te contact pont does not affect te stresses n te gear at te crtcal regon of te fllets or at any oter crtcal secton of te toot, far away from te loadng, so long te total loadng remans te same. Obvously, te stresses at and near te contact pont s very muc dependent on te approxmaton n te assumed dstrbuton of loadng at te contact pont. 4e-4 e-4 x/ = 17 x/ = 18 x/ = 19 x/ = σ x / E +5 -e Fg. 4 Grd structure for σ n or σ t at top-left ( pont not matcng wt feld grd pont) e Grd poston (y/) Fg. 5 Radal stresses (σ x /E ) at sectons x/ =17,18,19 and 1 of te gear toot Secton V: Appled Mecancs 17

4 ICME 1, Daa, December 6-8 Te stresses at dfferent sectons of te tp-loaded spur gear toot, sown n Fg.1 (referred to as type-a ere), are presented n Fgs 5-6. Overall, te magntudes of te stresses σ y and σ xy are smaller tan tat of te radal stress σ x. Maxmum stresses are found to occur around te root secton of te toot, and te fllet zone s te most crtcal secton n terms of stresses. Te magntude of stresses ust below te root secton s lower and gradually decreases towards te center of te gear but does not become zero on te bottom, wc s consdered rgdly fxed. Te load appled normally to te toot face near te tp as bascally two components, one of bendng, wc s compressng te load-free sde and tensonng te loaded sde of te toot, wle te oter s compressng over te wole secton. Tus, te combned effect of bendng and compresson gves lower value of tenson n te load-sde fllet and ger value of compresson n te load free-sde fllet (Fg.5). σ xy / E 1e-4-1e-4 -e-4 -e-4 x/ = 17 x/ = 18 x/ = 19 x/ = 1-4e Grd poston (y/) Fg. 6 Searng stresses (σ xy /E ) at sectons x/ = 17,18,19 and 1 of te gear toot Te deformed toot profle of te gear wt respect to ts ntal undeformed profle under te loadng at dfferent regons are llustrated n Fg.7. Te profles sow tat maxmum dsplacement occurs under te tp loadng. Te toot deflects n te drecton of load and te deflecton s maxmum at te tp and gradually decreases towards te root. Anoter spur gear toot (profle: nvolute, m = 1 mm, θ p =, a = m, b = m p, d p = mm ) s consdered ere for comparng te present ψ-soluton wt te publsed FEM results[9]. Te values of radal stress at te root secton, due to te tp, ptc and root loadng are presented n Table-1 along wt te FEM results. Te magntudes of te stresses at te extreme end of te root secton, especally at te tenson sde, are found to be n good agreement. However, te FEM soluton [5] predcts lower value of te compressve stress at te bottom surface of te root secton tan tat Table 1 Comparson of present soluton for stresses wt tat of FEM soluton Load Radal stress at te Root secton (MPa) Loaded surface Load-free surface FDM FEM FDM FEM Tp Ptc Root at ponts above t. Ts s gly unlely, as te bendng stress s te gest at te bottom, wc s furter enanced due to te effect of stress concentraton at tat pont. In our present approac, a sngle varable s evaluated at eac pont nstead of solvng for two varables smultaneously as n te case of FEM, wc reduces our computatonal wor drastcally. It s noted tat te tme taen to solve a gear toot (9 X 7 mes) on a personal computer (45 MHz) was found to be.91 s. Te accuracy as well as te relablty of te present metod as been verfed repeatedly by comparng te results wt avalable solutons [4,1]. CONCLUSIONS Te dffcultes of managng te sapes n fnte dfference tecnques of analyss for wc te fnte-element metod of soluton of elastc problems was nvented wt a manfold ncrease of computatonal wors and a lot of loss of sopstcaton s overcome ere by a novel computatonal tecnque n te management of sapes n te fnte-dfference metod of soluton. Bot te qualtatve and quanttatve results of te spur gear toot, and ter comparson wt tose of FEM soluton establs te relablty and sutablty of te present numercal model. NOMENCLATURE load-free ptc-loadng tp-loadng Fg.7 Toot profles under root loadng (dsplacements are 1 tmes enlarged) E elastc modulus of te materal Posson s rato u,v dsplacement components n te x- and y- drectons σ x, σ y normal stress components n te x- and y- drectons φ Ary s stress functon Secton V: Appled Mecancs 174

5 ICME 1, Daa, December 6-8 ψ dsplacement potental functon u n, u t normal and tangental components of dsplacement on te σ n, σ t normal and tangental components of stress on te l, m drecton cosnes of te normal at any pont on te, mes lengts n te x- and y- drectons R / a, b addendum and dedendum of te gear m,,d p,θ p module, ptc dameter, pressure angle of gear toot c, s locators n y- and x- drectons, wt respect to te reference grd pont 1. Tmoseno S, Gooder JN. Teory of Elastcty. ( rd edn.). McGraw-Hll, New Yor, N. Y. (1979).. Uddn, M.W., "Fnte Dfference Soluton of Two- Dmensonal Elastc Problems wt Mxed Boundary Condtons", M. Sc. Tess, Carleton Unversty, Canada, (1966).. Idrs, A.B.M., Amed, S.R. and Uddn, M.W., "Analytcal Soluton of a -D Elastc Problem wt Mxed Boundary Condtons", Journal of te Insttuton of Engneers (Sngapore), Vol. 6, No. 6 (1996), pp Amed, S.R., Kan, M.R., Islam, K.M.S. and Uddn, M.W., Analyss of Stresses n Deep Beams Usng Dsplacement Potental Functon Journal of Insttuton of Engneers (Inda), Vol. 77, (1996), pp Amed, S.R., Idrs, A.B.M. and Uddn, M.W., "Numercal Soluton of Bot Ends Fxed Deep Beams", Computers & Structures, Vol. 61, No. 1 (1996), pp Amed, S.R., Kan, M.R., Islam, K.M.S., and Uddn, M.W., "Investgaton of stresses at te fxed end of deep cantlever beams", Computers & Structures, Vol. 69, (1998), pp Durell, A.J. and Ranganayaamma, B., "Parametrc soluton of stresses n beams", Journal of Engneerng Mecancs, Vol. 115, No. (1989), pp Dow, J.O., Jones, M.S. and Harwood, S.A., "A new approac to modelng for fnte dfference applcatons n sold mecancs", Internatonal Journal for Numercal Metods n Engneerng, Vol., (199), pp Ramamurt, V. & Rao, M.A., Dynamc Analyss of Spur Gear Teet, Computers and Structures, Vol. 9, No. 5, (1988), pp Aanda, M.A.S., Amed, S.R., Kan, M.R., and Uddn, M.W., "A fnte-dfference sceme for mxed -value problems of arbtrarysaped elastc bodes", Advances n Engneerng Software, Vol. 1, No. (), pp REFERENCES Secton V: Appled Mecancs 175