Slider Crank Mechanism Design with Time Ratio and Minimum Transmission Angle

Transcription

1 Vol. 7, No. (0), pp Slider Crnk Mehnism Design with Time Rtio nd Minimum Trnsmission Angle Hn Jigung, Zhng Chunyn nd Zuo Weiyng Shool of Mehnil nd Eletril Engineering, Jingsu Norml University, Xuzhou, 6, Chin Abstrt The size prmeters of slider rnk mehnism re diretly treted s design vrible. The nlytil synthesis method of the mehnism with time rtio nd the seleting rnge of design vrible re presented. The mehnism synthesis method whih simultneously stisfies the onditions of time rtio nd imum trnsmission ngle is presented. The synthesis problem of the slider rnk mehnism, whih mkes it hve mximum imum trnsmission ngle when time rtio is given, is ompletely solved. This method n not only judge the fesibility of mehnism synthesis, but lso detere the prmeters of mehnism one time. The itertion nd heking re voided, so it mkes the synthesis of mehnism fst nd urtely. Keywords: Slider rnk mehnism; Mehnism synthesis; Time rtio; Trnsmission ngle. Introdution Slider-rnk mehnism is widely pplied in engineering. Offset slider-rnk mehnism hs hrteristis of quik return, nd its design problems generlly ttributed to design size prmeters ording to the stroke of the slide, the time rtio nd trnsmission ngle et. Trnsmission ngle is one of importnt index to mesure the performne of fore trnsmission. The designer must onsider question is how to mke the mehnism get the best trnsmission performne in the preondition of kinemti requirements. There re lot of the literture to reserh synthesize method of the slider-rnk mehnism ording to the time rtio k. But for given time rtio k, synthesize mehnism to stisfy the llowble trnsmission ngle [ ] or to mke imum trnsmission ngle hieve to mximum, is not very good to solve, the mehnism stisfy the requirements n be obtined often need to pss itertive nd heking, or the proof it is no solution. The rnk length is diretly treted s design vrible, nd the nlyti design method of the plne slider-rnk mehnism is given with the time rtio k nd stisfy llowble trnsmission ngle [ ] t the sme time, in this pper. This method not only n judge the fesibility of the mehnism synthesis, but lso n one detere mehnism prmeters, to void the itertion nd hek, mke the mehnism synthesis fst nd urte, hs ertin prtil vlue.. Designing Slider-rnk Mehnism Aording to Time Rtio As shown in Figure, in the slider-rnk mehnism ABC, offset is e, the length of the rnk AB is, the length of the onneting rod BC is b, the slider stroke is s. Points C nd C re the limiting positions of the slider C. The mehnism hs imum trnsmission ngle when the slider on the point C'. is the ngle ISSN: IJCA Copyright 0 SERSC

2 Vol. 7, No. (0) between one position of rnk nd nother when slider lotes on the limiting position C nd C. The formul of time rtio k is E θ b B C C C' s B γ θ C Figure. Slider-rnk Mehnism k () In order to filitte lultion, the reltive size is used, tht is, tke the slider stroke s =, the following, b nd e re the reltive length when s =. At this time, the mehnism hs three size prmeters, the slider storke s = nd the known time rtio k eh provides onstrint, so the mehnism only one size prmeters n be hosen... Bsi Design Formul of Slider-rnk Mehnism In AC C, supposing tht the ngle AC C is β, the length of AC is b+ nd the length of AC is b-. By the osine lw [], we hve Tht is ( b ) ( b ) ( b )( b ) os b ( b ) os () By the sine lw [], we hve A B' e D B b s in s in Tht is e ( b ) sin ( b ) sin () Simultneous equtions () nd (), we n obtin the design formul of Sliderrnk mehnism. Tht is 5 Copyright 0 SERSC

3 Vol. 7, No. (0) b e ( o s ) ( o s ) sin ( o s ) () Beuse of b nd e is the positive of rel Numbers, it must to be This mens 0.5. ( o s ) ( o s ) 0, 0 By the Grshof onditions: b e. Tking formul () into the inequlity, we n obtin the rnge of rnk reltive length is sin θ < (os θ + ) Slider-rnk Mehnism Design with k nd mke to Ahieve Mximum The best trnsmission mehnism is mehnism whih hs mximum imum trnsmission ngle ( ) mx under the onditions tht the time rtio k is given. As shown in Figure. The eqution of imum trnsmission ngle of sliderrnk mehnism is e b os (6) The extreme vlue of neessrily in the ple of (os ) 0, Tking the derivtive of formul (6) to get ( e ) b b ( e ) 0 In whih, e nd b re tking the derivtive of e nd b from the formul (). We obtin ( os ) sin sin (os ) os 0 (7) With the method of undetered oeffiients for times eqution (7) deomposed ftoring to get ( So, the roots for times eqution (7) re ( os ) sin )( sin os ) 0 (5) os sin sin (os ) Copyright 0 SERSC 55

4 Vol. 7, No. (0) sin sin 8 ( (os ) os ) sin sin 8 ( (os ) os ) Beuse of < 0, it nnot be used s the reltive length of rnk. don't stisfy formul (5), it lso nnot be used s the reltive length of rnk. In ft, when = nd b = + e, the imum trnsmission ngle of the mehnism is 0, os get mximum. So the root for three eqution (7) is sin sin 8 ( os ) (8) (os ) When given the time rtio k, we n synthesis slider-rnk mehnism whih hs mximum imum trnsmission ngle ( by the eqution (8) nd (). ) mx Tking formul (8) into the formul (), get b nd e fter nother, tking them into the formul (6) to get ( ) mx ros( d sin os Where sin 8 ( os ) d. sin ) d sin Formul (9) is minly used for the fesibility judgment of the mehnism synthesis, suh s given the time rtio k =., nd sk the llowble trnsmission ngle for [ ] 0. From the formul () lulted to get. 6, nd from formul (9) get ( ) mx = 8.77 < [ ], this shows tht the mehnism synthesis hs no solution... Slider-rnk Mehnism Design with k nd [ ] From formul (6) nd (), nelltion e gets b os Tking b into the formul () get os sin sin ( sin ) 0 (0) 0 (9) Where 0 os ( 6 os sin ( os ) (os 5 ). 8 os ) 56 Copyright 0 SERSC

5 Vol. 7, No. (0) To solve the four times equtions (0) by the method of literture [8], nd the oeffiients re nlyzed nd simplified. We n get ( ( ) ) () 6 6 7, Where ( ) / ; os( ros ) / 6 ; 5 / ( / 6 8 ) / / 08 ; / When the time rtio k nd llowble trnsmission ngle [ ] is given, ngle is lulted by formul (), tking nd [ ] into the formul () detere, from the formul () detere the other two prmeters of mehnism, get sliderrnk mehnism its imum trnsmission ngle trnsmission ngle [ ]. equl to the llowble Usully, if the time rtio k nd llowble trnsmission ngle [ ] is resonble, the mehnism hs the solution, the formul () provides the rnge of reltive length of rnk. If hoose in the setion [, ], imum trnsmission ngle less thn the llowble trnsmission ngle [ ].. Design Exmples is not Design slider-rnk mehnism, the slider stroke s = 00 mm, the time rtio k =., llowble trnsmission ngle [ ] 0. From the formul () obtined From the formul (9) lulted ( γ ) mx = 7. > [ ] 0.The design is fesible. From the formul () : = = = = = = = = = = The rnge of is [0.0597, 0.976]. Tke = 0.5, from the formul () b =.5967 e = From the formul (6) = 5.9 > [ ] 0 The mehnism size is l AB = s = 90 mm l BC = bs = 9.5 mm l AD = es =.8 mm. Copyright 0 SERSC 57

6 Vol. 7, No. (0) The best trnsmission mehnism design From the formul (8), = From the formul (), b =.79056; e = From the formul (6), = 7.. The mehnim size is l AB = s = mm; l BC = bs = 55.8 mm l AD = es = mm.. Conlusion The design method of plne slider-rnk mehnism is proposed with the time rtio nd llowble trnsmission ngle, in this pper, the theoretil nlysis nd design exmples show tht the theory is orret, relible, high preision. This method diretly used to the size prmeters of the mehnism s design vribles, it is kind of prtil strong design method. Aknowledgements This reserh ws supported by the Ntionl Nturl Siene Foundtion of Chin (Grnt No ), nd the Key Progrm of Ntionl Nturl Siene Foundtion of Heilongjing (Grnt No.ZD009) nd Projet supported by the Mjor Interntionl Joint Reserh Progrm of Chin (Grnt No. 0DFA70). First Author is orresponding uthor. Referenes [] J. Hn, Journl of mhine design, vol., no.. (00). [] Y. Lu, Journl of Wuxi University of Light Industry, vol., no. 7, (998). [] Z. Fu, Journl of Zhuzhou Institute of Tehnology, vol., no., (997). [] M. Lun nd Y. Leu, Mehnism nd Mhine Theory, vol., no., (996). [5] J. Lou, Journl of mhine design, vol., no. 0, (00). [6] Y. Chng, Mehnil Siene nd Tehnology, vol., no., (00). [7] Y. Li, Mehnil Siene nd Tehnology, vol., no., (00). [8] Writing Group, Mthemtis hndbook, Beijing: People Edution, (979), pp [9] W. Zhng, Mehnil Priniples, Higher Edution Press, (007). 58 Copyright 0 SERSC