SOLID MECHANICS TUTORIAL FRICTION CLUTCHES

Transcription

1 SOLI MECHANICS TUTORIAL FRICTION CLUTCHES Ths wrk cvers elements f the syllabus fr the Edexcel mdule 17P HNC/ Mechancal Prncples OUTCOME. On cmpletn f ths shrt tutral yu shuld be able t d the fllwng. escrbe a cncal and a flat plate clutch. escrbe a multplate clutch. Explan the cnstant wear thery. Explan the cnstant pressure thery. Slve prblems nvlvng pwer transmssn wth clutches. It s assumed that the student s already famlar wth the fllwng cncepts. Frctn thery. Angular mtn. Pwer transmssn by a shaft. Basc ntegral calculus. All these abve may be fund n the pre-requste tutrals..j.unn 1

2 FRICTION CLUTCHES 1. INTROUCTION Frst let's revse the bascs f dry frctn. Cnsder a small blck sldng ver a surface. The frce pressng the tw surfaces tgether s R (the nrmal frce). When the surfaces slp, the frce F requred t prduce mvement s equal and ppste f the frctn frce between the surfaces. F and R are related by Culmb s Law f Frctn. F where µ s the ceffcent f frctn. Fgure 1. WEAR THEORY Research has shwn that the wear between tw rubbng surfaces depends upn the pressure between the surfaces and the speed at whch they rub. There are tw theres cncernng the trque requred t prduce slp between the surfaces f a clutch. One thery assumes the pressure s even ver the surface f cntact n whch case the wear s greater at the utsde due t the greater velcty f rubbng. The ther thery assumes that the wear s unfrm n whch case the pressure s nt evenly dstrbuted.. CONICAL CLUTCHES The pcture shws a typcal cncal clutch fr larger pwer transmssn applcatns. There are tw cnes cvered n frctn materal and when they are frced apart they rub aganst the steel uter casngs and lck them tgether thus engagng the tw halves. Fgure.J.UNN

3 .1 GEOMETRY A cncal clutch transmts rtatn frm ne shaft t anther thrugh frctn frces n the cncal face. The cne has a half angle f β and the tw halves are frced tgether wth a frce R. Fgure Cnsder an elementary rng n the face f the cne at radus r and radal wdth dr. Fgure The length f the rng alng the slpng surface s dr/snβ. The area f the rng (da) s apprxmately the crcumference (πr) tmes the wdth dr/snβ. π rdr da...(1) snβ.j.unn

4 . UNIFORM PRESSURE THEORY The frce pressng the surfaces tgether prduces a unfrm pressure between them f p N/m. The frce nrmal t the surface s R' and the frce n the element s dr' π prdr dr' p da substtutng equatn (1) fr da we have dr' snβ The ttal frce R' actng n the cncal area s gven by ntegratng. R' π prdr snβ R'snβ p π [ ] π p snβ π p r rdr snβ...() πp snβ [ ] When the clutch slps, the frctn frce actng n the rng s µdr'. Ths frce dr prduces a small trque d µ rdr' µ p π r snβ The ttal trque s btaned by ntegratng between the nsde and the utsde. ' [ ] [ ] π p µ snβ Substtute equatn () fr p π p µ r r dr snβ π p µ 1snβ [ ] In ths dervatn, R' s the ttal frce actng nrmal t the surface. If ths s reslved t gve the axal frce R R'snβ and s snβ [ ] [ ]...().J.UNN

5 . UNIFORM WEAR THEORY Cnsder the elementary rng agan. dr' pda The velcty f any pnt s v m/s and the angular velcty s ω rad/s. Unfrm wear thery assumes that the wear s cnstant everywhere and t s drectly prprtnal t pressure x velcty (when slppng). Wear p v Snce v ωr, then wear pωr Fr cnstant ω, wear pr p wear/r The wear s cnstant s t fllws that p cnstant/r c/r As befre the nrmal frce s dr' pda Substtute equatn (1) fr da dr' p π r dr' Integratng between the nsde and utsde we get R' dr cπ snβ cπ snβ dr snβ dr...() cπ snβ and substtutng p [] r R'snβ c...(5) π( ) cπ ( snβ c r When the clutch slps, the frctn frce actng n the rng s µdr' Ths frce prduces a small trque f dt. c µ π r dr d µ rdr' and substtutng equatn () fr dr' we have d snβ Next we ntegrate. ' ' [ ] ' ( + )( ) ( ( + ) c π µ c π µ r rdr snβ snβ ) ( ) c π µ snβ ) [ ] R'snβ Equatn (5) was c and substtutng t n t the equatn gves π( ).J.UNN 5

6 Agan, reslvng R' t gve the axal frce R we get: snβ ( )...(6) + WORKE EXAMPLE N.1 A cncal clutch has an ncluded angle f 10. The uter and nner dameters are 80 and 0 mm respectvely. Calculate the frce requred t press the tw halves tgether f t s t transmt 00W at 600 rev/mn. The ceffcent f frctn s 0.. Use bth the unfrm wear thery and the unfrm pressure thery. SOLUTION Identfy the varables and cnstants. N 600 rev/mn β 10/ m 0.0 m µ 0. Pwer P 00 W UNIFORM PRESSURE 60P π N snβ 60 x 00 π 600 [ ] 0.R [ ] [ ] sn60 [ ] ) x [ ] 0. x [ ].181 x x sn(60 R UNIFORM WEAR snβ ( + ) ( + ).181 Nm N T x sn(β.18 x x sn(60) R N µ 0. x ( ).J.UNN 6

7 SELF ASSESSMENT EXERCISE N.1 1. The fllwng data s fr a cncal clutch. Insde dameter 0 mm Outsde dameter 110 mm Ceffcent f frctn 0. Axal frce 800 N. Included angle 80. Speed 1000 rev/mn Calculate the trque and pwer that can be transmtted wthut slppng usng a) The unfrm pressure thery. (11.11 Nm and 116 W) b) The unfrm wear thery. (10.0 Nm and 109 W). The fllwng data s fr a cncal clutch. Insde dameter 0 mm Outsde dameter 10 mm Ceffcent f frctn 0. Included angle 100. Speed 000 rev/mn Calculate the axal frce needed t allw the transmssn 800 watts wthut slppng usng a) The unfrm pressure thery. (0.1 N) b) The unfrm wear thery. (8.8).J.UNN 7

8 . FLAT CLUTCH PLATES The dagram shws a basc flat clutch. A dsc wth frctn materal s pressed aganst a secnd dsc thus engagng them by frctn and makng bth dscs rtate tgether. Fgure 5 A flat clutch s a specal case f a cncal clutch wth an ncluded angle f 180. It may be dealsed lke ths. Fgure 6 Cnsder a rtatng shaft wth a dsc at the end that presses up aganst anther s that rtatn s transmtted frm ne t the ther by frctn. Ths s the specal case f the cne clutch when β 90 and snβ 1. Ths prduces the results:.1 UNIFORM PRESSURE THEORY [ ] [ ] per surface f cntact. UNIFORM WEAR THEORY ( ) per surface f cntact +.J.UNN 8

9 . MULTI-PLATE CLUTCHES Fgure 7 These are cnstructed wth ne set f plates attached t the nner shaft and the ther plates t the uter case. The plates are frced tgether wth a mechansm. On the dagram, there are fve surfaces n cntact and ths allws a greater trque t be transmtted befre slp ccurs. If there are n surfaces f cntact then the maxmum trque s ncreased n tmes. Values f pressure p vary frm 50 kpa t 800 kpa dependng n the materal. The ceffcent f frctn s typcally 0.5 fr dry materals and 0.05 when mmersed n l..j.unn 9

10 WORKE EXAMPLE N. The fllwng data s fr a multplayer clutch. Number f Cntact surfaces. 5 Speed rev/mn 000 Outsde dameter mm 150 Insde dameter mm 80 Ceffcent f frctn 0.5 Axal frce R s 600 N Calculate the maxmum pwer that the clutch can transmt wthut slppng based n cnstant wear thery. Calculate the maxmum pwer that the clutch can transmt wthut slppng based n cnstant pressure thery. SOLUTION Identfy the fllwng n 5 N000 rev/mn 0.15 m 0.08 m µ 0.5 R 600 N Unfrm Pressure Pwer ( ) ( ) ( ) ( ) π NT π x 000 x.57 Pwer 911 Watts Unfrm Wear 0.5 x 600 x 5.15 Nm π x 000 x Watts 60 ( + ) n ( ) π NT 60 n 0.5 x 600 x 5.57 Nm.J.UNN 10

11 SELF ASSESSMENT EXERCISE N. 1. A mult-plate clutch must transmt 0 kw f pwer wthut slppng at 000 rev/mn. The ceffcent f frctn s 0.8. The nner and uter dameters are 80 and 160 mm respectvely. The axal frce appled t the plates s 60 N. etermne the number f plates requred usng:. The unfrm pressure thery. (5.958 rund up t 6). The unfrm wear thery. (6.178 rund up t 7). A mult-plate clutch must has three cntact surfaces and transmts pwer at 1500 rev/mn. The ceffcent f frctn s 0.. The nner and uter dameters are 0 and 150 mm respectvely. The axal frce appled t the plates s 00 N. Calculate the trque and pwer that can be transmtted wthut slppng usng:. The unfrm pressure thery. (.8 Nm and 896 Watts). The unfrm wear thery. (1.6 Nm and 9 W).J.UNN 11