Analytical and Numerical Solutions for a Rotating Annular Disk of Variable Thickness

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1 Appled Mathematcs do:0.436/am Publshed Onlne Novembe 00 ( Analytcal and Numecal Solutons fo a Rotatng Annula Ds of Vaable Thcness Abstact Ashaf M. Zenou Daoud S. Mashat Depatment of Mathematcs Faculty of Scence Kng AbdulAzz Unvesty Jeddah Saud Aaba Depatment of Mathematcs Faculty of Scence Kafelsheh Unvesty Kaf l-sheh gypt -mal: zenou@sc.fs.edu.eg Receved July 9 00; evsed Octobe 9 00; accepted Octobe 00 In ths pape the analytcal and numecal solutons fo otatng vaable-thcness sold ds and numecal soluton fo otatng vaable-thcness annula ds ae pesented. The oute edge of the sold ds and the nne and oute edges of the annula ds ae consdeed to have clamped bounday condtons. Two dffeent cases fo the adally vayng thcness of the sold and annula dss ae gven. The numecal soluton as well as the analytcal soluton s avalable fo the fst case of the sold ds whle the analytcal soluton s not avalable fo the second case of the annula ds. Both analytcal and numecal esults fo dsplacement and stesses wll be nvestgated fo the fst case of adally vayng thcness. The accuacy of the pesent numecal soluton s dscussed and ts ablty of use fo the second case of adally vayng thcness s nvestgated. Fnally the dstbutons of dsplacement and stesses wll be pesented and the appopate compasons and dscussons ae made at the same angula velocty. Keywods: Rotatng Annula Ds Sold Ds Fnte Dffeence Method. Intoducton The poblems of otatng sold and annula dss have been pefomed unde vaous nteestng assumptons and the topc can be easly found n most of the standad elastcty boos [-5]. Most of the eseach wos ae concentated on the analytcal solutons of otatng dss wth smple coss-secton geometes of unfom thcness and especally vaable thcness [6-4]. The analytcal elastcty solutons of such otatng dss ae avalable n many boos of elastcty. As many otatng components n use have complex coss-sectonal geometes they cannot be dealt wth usng the exstng analytcal methods. Numecal methods such as the fnte element method [5] the bounday element method [6] and Runge-Kutta s algothm [7] can be appled to cope wth these otatng components. In ths pape we wll pesent the analytcal soluton fo the otatng sold ds wth abtay coss-secton of contnuously vaable thcness. In the followng a unfed govenng equaton wll be fst deved fom the basc equatons of the otatng dss and the poposed stess-stan elatonshp. Next fnte dffeence method (FDM) s ntoduced to solve the govenng equaton. A compason between both analytcal and numecal solutons s made. The accuacy of the numecal soluton s used to fnd the dsplacement and stesses of otatng vaable-thcness annula ds whose analytcal soluton s not avalable. Fnally a numbe of numecal examples ae gven to demonstate the valdty of the poposed method.. Basc quatons As the effect of thcness vaaton of otatng dss can be taen nto account n the equaton of moton the theoy of the vaable-thcness dss can gve good esults as that of the unfom-thcness dss as long as they meet the assumpton of plane stess. Afte consdeng ths effect the equaton of moton of otatng dss wth vaable thcness can be wtten as whee d d and h h h 0 () ae the adal and ccumfeental Copyght 00 ScRes.

2 43 A. M. ZNKOUR T AL. stesses s the adal coodnate s the densty of the otatng ds s the constant angula velocty and h s the thcness whch s functon of the adal coodnate. The elatons between the adal dsplacement u and the stans ae espectve of the thcness of the otatng ds. They can be wtten as du u () d whee and ae the adal and ccumfeental stans espectvely. Fo the elastc defomaton the consttutve equatons fo the otatng ds can be descbed wth Hooe s law. (3) Usng () nto quaton (3) one can obtan the consttutve equatons fo and as: du u d u du. d Let us consde a symmetc thn ds wth espect to the md-plane ts pofle vayng n the adal decton accodng to the fomula see Fgue : h h 0 n (5) b whee h 0 s the thcness at the axs of the ds n and ae geometc paametes h () (0n 0) and b s the oute adus of the sold ds. A unfom-thcness ds s obtaned by settng n = 0 and a lnealy deceasng thcness s obtaned by settng =. Futhemoe f < the pofle s concave and f > t s convex. The thcness of the ds s assumed to be suffcently small compaed to ts damete so that 3. Fomulaton and lastc Soluton fo Sold Ds The substtuton of quatons (4) and (5) nto quaton () poduces the followng confluent hypegeometc dffeental equaton fo the adal dsplacement u (): d u d ( ) u n d b d n b (6) n 3 b u 0. n b (4) (a) (c) Fgue. Vaable-thcness sold ds pofles fo (a) =.5 and n = 0.8; = 0.5 and n = 0.8 and (c) =.5 and n = 0.5. R b b UR u b. (7) Then quaton (6) may be wtten n the followng Copyght 00 ScRes.

3 A. M. ZNKOUR T AL. 433 smple fom d U R dr nr n R 3 U R nr n R R du d R 0. (8) The geneal soluton of the above equaton can be wtten as U R C F R C F R P R (9) whee C and C ae abtay constants and F and F ae gven by: F R RH nr (0) F R H nr () R n whch whee 4 () 4. The functons H ([ ] [ ] z) ae the genealzed hype-geometc functons q q q H z z q0 q! (3) q 0 z whee ( ) q s the Pochhamme symbol gven by... q q (4) q n whch epesents Gamma functon. The tem PR ( ) n quaton (9) s the patcula soluton of quaton (8) whch can be wtten as RFR RFR (5) PR4 FR drf R d R n R FF 4 F Fn R F 3 (6) n whch F3 R RH nr (7) F4 R H 3 nr (8) R The substtuton of quaton (9) nto quaton (4) wth the ad of the dmensonless foms gven n quaton (7) gves the adal and ccumfeental stesses n the foms of (9) and (0): nr nr dp P RC F F3C F F4 R R dr R nr nr d P P R C F F3C F F4. R R dr R (9) (0) 4. Analytcal Soluton fo the Rotatng Sold Ds The analytcal elastc soluton fo the sold ds wth vaable-thcness s completed by the applcaton of the bounday condtons. Snce the adal dsplacement should be vanshed and the stesses should be fnte at the cente of the ds then the constant C vanshes. The adal dsplacement s vanshed at the oute edge of the ds = b o (R = ) hence P C. () F So one can easly obtan the soluton fo the pesent otatng vaable-thcness sold ds by the substtuton of quaton () nto quatons (9) (9) and (0). 5. Fnte Dffeence Algothm fo Sold Ds The esoluton of the elastc poblem of otatng ds wth vaable thcness s to solve a second-ode dffeental equaton quaton (8) unde the gven bounday condtons. Ths equaton can be wtten n the followng geneal fom: U p R U q R U s R 0 R U 0 U 0 () whee the pme (') denotes dffeentaton wth espect to R and Copyght 00 ScRes.

4 434 A. M. ZNKOUR T AL. n R pr nr R n R qr nr R s R R.. (3) It s clea that the above poblem has a unque soluton because pr ( ) qr ( ) and s( R ) ae contnuous on the gven nteval ]0] and qr ( ) 0 on ]0]. The lnea second-ode bounday value poblem gven n quaton () eques that dffeence-quotent appoxmatons be used fo appoxmatng U and U. Fst we select an ntege N 0 and dvded the nteval ]0] nto ( N ) equal subntevals whose end ponts ae the mesh ponts R R fo 0... N whee R /( N ). At the nteo mesh ponts R... N the dffeental equaton to the appoxmated s U R p R U R q R U R s R (4) If we apply the centeed dffeence appoxmatons of U ( R ) and U ( R ) to quaton (4) we ave at the system: R p R ( ) U R q R U (5) R p R U R s R fo each... N. The N equatons togethe wth the bounday condtons U0 0 (6) U 0 N Ae suffcent to detemne the unnowns U 0... N. The esultng system of quatons (5) s expesses n the t-dagonal N N -matx fom: whee AU B (7) A R q R... N R A pr... N R A pr 3... N A j Aj 0... N j N j B R s R... N. (8) The soluton of the fnte dffeence dscetzaton of the two-pont lnea bounday value poblem can theefoe be found easly even fo vey small mesh szes. 6. Numecal xamples and Dscusson fo Sold Ds Some numecal examples fo the otatng vaablethcness sold dss wll be gven accodng the analytcal and numecal solutons ( 0.3). Accodng to quaton (7) the followng dmensonless esponse chaactestcs u U R u b (9). R R detemned as pe the analytcal soluton ae compaed wth those obtaned by the numecal FDM soluton. The esults of the pesent nvestgatons fo the adal dsplacement u ae epoted n Table. Fo ths example N = and 79 so R has the coespondng values and 0.05 espectvely. If we use the Rchadson extapolaton method wth R and 0.05 we obtan esults lsted n Table. The fst extapolaton s 4U R 0.05 U R 0. xt ; (30) 3 the second extapolaton s 4U R 0.05U R 0.05 xt ; (3) 3 the thd extapolaton s 4U R 0.05U R 0.05 xt 3 ; (3) 3 the foth extapolaton s 6xt xt xt 4 ; (33) 5 and the fnal extapolaton s 6 xt3 xt xt 5. (34) 5 All of the esults of xt4 and xt5 ae coect to the decmal places lsted. In fact f suffcent dgts ae mantaned these appoxmatons gve esults that agee wth the exact soluton wth maxmum eo of and espectvely at the mesh ponts. Copyght 00 ScRes.

5 A. M. ZNKOUR T AL. 435 The dstbuton of the adal dsplacement adal and ccumfeental stesses ae pesented n Fgue. The numecal FDM soluton s compaed wth the analytcal soluton fo the otatng vaable-thcness sold ds wth = 3 and vaous values of n. It s clea that the FDM gves dsplacement and stesses wth excellent accuacy wth the exact analytcal soluton. It can be seen fom Fgue that the FDM can descbe the dsplacement and stesses though-the-thcness vey well enough. 7. Fomulaton and Numecal Soluton fo Annula Ds Hee we consde a thn annula ds vaes contnuously n the fom of a fom of a geneal paabolc functon (see Fgue 3): a h h 0 n (35) b a (a) (a) (c) Fgue. Dmensonless adal dsplacement u adal stess σ and ccumfeental stess σ fo the vaable-thcness sold ds ( = 3): (a) n = 0. n = 0.5 and (c) n = 0.8. (c) Fgue 3. Vaable-thcness annula ds pofles fo (a) =.5 and n = 0.8 = 0.5 and n = 0.8 and (c) =.5 and n = 0.5. Copyght 00 ScRes.

6 436 A. M. ZNKOUR T AL. whee a s the nne adus of the annula ds. The equlbum equaton coespondng to quaton (35) may be easly gven but ts analytcal soluton s not. The substtuton of quatons (4) and (35) nto quaton () wth the help of the dmensonless foms gven n quaton (7) poduces the followng confluent hypegeometc dffeental equaton fo the adal dsplacement U( R ) of the annula ds: Fgue 5. Dmensonless adal dsplacement u of the vaable-thcness annula ds fo d eent values of and n. (a) Fgue 4. Dmensonless adal dsplacement u adal stess σ and ccumfeental stess σ fo the vaable-thcness annula ds ( = 3): (a) n = 0. n = 0.5 and (c) n = 0.8. whee d U nrr du R R d R R A nr d R 4 n RR R U 0 RAnR R A (36) a R A R A. (37) b A Mang analogous steps as gven fo the sold ds quaton (36) can be wtten n the followng geneal fom: U pruqru sr (38) A R U A U 0 whee the pme (') denotes dffeentaton wth espect to R and nrr pr R R A nr n RR qr R R AnR R sr. R A (39) So FDM gves easly the adal dsplacement of the otatng vaable-thcness annula ds. Usng the cuve fttng and least squae method one can obtan the adal Copyght 00 ScRes.

7 A. M. ZNKOUR T AL. 437 and ccumfeental stesses. Tang n = and 0.8 and = and 3 n the vaable thcness functon gven n quaton (35) a otatng annula ds wth such vaable thcness s studed. The nne and oute ad of the ds ae taen to be a = 0. b (R = A = 0.) and b (R = ) and the esults ae gven n tems of the otatng angula velocty. The esults calculated wth the FDM fo dsplacement and stesses of a otatng vaable-thcness annula ds ae gven n Fgues Conclusons Ths pape pesents a unfed numecal method fo the elastc calculaton of otatng dss wth a geneal abtay confguaton. The govenng equaton was deved fom the equlbum equaton and the stess- stan elatonshp. The analytcal soluton was gven fo the otatng vaable-thcness sold ds. The calculaton of the otatng sold and annula dss was tuned nto fndng the soluton of a second-ode dffeental equaton unde the gven condtons at two bounday fxed ponts. Fnte dffeence method algothm was ntoduced to solve the govenng equaton fo both sold and annula dss and a numbe of numecal examples wee studed. The esults fom the analytcal and FDM solutons wee compaed. The poposed FDM appoach gves vey ageeable esults to the analytcal soluton. 9. Acnowledgements The nvestgatos would le to expess the appecaton to the Deanshp of Scentfc Reseach at Kng AbdulAzz Unvesty fo ts fnancal suppot of ths. 68/48 /ع No. study Gant 0. Refeences Fgue 6. Dmensonless adal stess σ n the vaablethcness annula ds fo dffeent values of and n. Fgue 7. Dmensonless ccumfeental stess σ n the vaable-thcness annula ds fo d eent values of and n. [] S. P. Tmosheno and J. N. Goode Theoy of lastcty McGaw-Hll New Yo 970. [] S. C. Ugal and S. K. Fenste Advanced Stength and Appled lastcty lseve New Yo 987. [3] U. Game Tesca s Yeld Condton and the Rotatng Ds ASM Jounal of Appled Mechancs Vol. 50 No. 983 pp [4] U. Game lastc-plastc Defomaton of the Rotatng Sold Ds Ingeneu-Achv Vol. 45 No pp [5] A. M. Zenou Analytcal Solutons fo Rotatng xponentally-gaded Annula Dss wth Vaous Bounday Condtons Intenatonal Jounal of Stuctual Stablty and Dynamcs Vol. 5 No pp [6] U. Güven lastc-plastc Stesses n a Rotatng Annula Ds of Vaable Thcness and Vaable Densty Intenatonal Jounal of Mechancal Scences Vol. 34 No. 99 pp [7] U. Güven On the Stess n the lastc-plastc Annula Ds of Vaable Thcness unde xtenal Pessue Intenatonal Jounal of Solds and Stuctues Vol. 30 No pp [8] U. Güven Stess Dstbuton n a Lnea Hadenng Annula Ds of Vaable Thcness Subjected to xtenal Pessue Intenatonal Jounal of Mechancal Scences Vol. 40 No pp [9] U. Güven lastc-plastc Stesses Dstbuton n a Rotatng Hypebolc Ds wth Rgd Incluson Intenatonal Jounal of Mechancal Scences Vol. 40 No. 998 pp Copyght 00 ScRes.

8 438 A. M. ZNKOUR T AL. [0] A. N. aslan Inelastc Defomaton of Rotatng Vaable Thcness Sold Dss by Tesca and Von Mses Ctea Intenatonal Jounal fo Computatonal Methods n ngneeng Scence Vol pp [] A. N. aslan and Y. Ocan On the Rotatng lastc-plastc Sold Dss of Vaable Thcness Havng Concave Pofles Intenatonal Jounal of Mechancal Scences Vol. 44 No pp [] A. N. aslan Stess Dstbutons n lastc-plastc Rotatng Dss wth llptcal Thcness Pofles Usng Tesca and von Mses Ctea Zuch Altenatve Asset Management Vol pp [3] A. M. Zenou and M. N. M. Allam On the Rotatng Fbe-Renfoced Vscoelastc Composte Sold and Annula Dss of Vaable Thcness Intenatonal Jounal fo Computatonal Methods n ngneeng Scence and Mechancs Vol. 7 No. 006 pp. -3. [4] A. M. Zenou Themoelastc Solutons fo Annula Dss wth Abtay Vaable Thcness Stuctual ngneeng and Mechancs Vol. 4 No pp [5] O. C. Zenewcz The Fnte lement Method n ngneeng Scence McGaw-Hll London 97. [6] P. K. Banejee and R. Buttefeld Bounday lement Methods n ngneeng Scence McGaw-Hll New Yo 98. [7] L. H. You Y. Y. Tang J. J. Zhang and C. Y. Zheng Numecal Analyss of lastc-plastc Rotatng Dss wth Abtay Vaable Thcness and Densty Intenatonal Jounal of Solds and Stuctues Vol. 37 No pp Copyght 00 ScRes.