5th INTERNATIONAL MEETING OF TE CARPATIAN REGION SPECIALISTS IN TE FIEL OF GEARS POSSIBLE AN REAL POWERFLOWS IN CONNECTE IFFERENTIAL GEAR RIVES WIT η 0 <i <1/η 0 INNER RATIO. Feenc Aó, Levente Czégé Univesity of Miskolc, eatment of Machine Elements Abstact: It s well known [1], that the oweflow of two degee of feedom thee base element tye lanetay dives (diffeentials) can be of two kinds in case of given oeties of moving. So two kinematic attibutes, fo examle i inne atio and k kinematic atio detemine the two ossible oweflows unambiguously. Fom the two degee of feedom thee base element tye lanetay dives we can assemble simle one degee of feedom connected lanetay dives by connection and fixation of base elements, in which one of the thee base element tye lanetay dives woks as a diffeential dive. In case of i <η 0 and i >1/η 0 inne atios the oweflow of the one degee of feedom connected dive is detemined by the oweflow of the diffeential dive (fo a given stuctue). Wheeas in case of η 0 <i <1/η 0 diffeential inne atios some connected dives theoetically can wok with two diffeent oweflows when inut and outut base elements ae constant. This ae shows the oblem by an examle, and looks fo the answe, whethe which of the two theoetically ossible oweflows will be ealised in the given lanetay gea dive. Key wods: lanetay gea dive, diffeential dive, connected lanetay dive The simle one degee of feedom connected lanetay dives can be deived fom two degee of feedom thee base element tye lanetay dives (diffeentials) by connection and fixation of base elements. enoting the numbe of thee base element tye diffeential dives by m, the numbe of moving base elements in connected dive by n 0, the degee of feedom of the connected system (s): s= n 0 -m, (1) 19
so in case of m=2 and n 0 =2 thee must be n 0 =s+m=2+2=4 moving base elements in the connected dive. Since befoe connecting the two diffeential dives had n 0 =6 moving base elements, the connected two degee of feedom lanetay dive must have 2 joined and 2 nonjoined base elements. Fixing an element we can get the one degee of feedom dive. If we fix a connected element, we get a seial-connected dive, in which both diffeential woks as a one degee of feedom dive (Fig.1.a.). At the same time by fixing a nonjoined element we get a diving-side- o a divenside-connected simle one degee of feedom dive (Fig.1.b.-c.), in which one of the thee base element lanetay dives woks as a diffeential dive (). 0 0 0 a.) b.) c.) Fig.1. The othe () can be a one degee of feedom lanetay dive, a nomal dive, a steed-, o an infinitely vaiable dive. We mention, that thee ae 3!=6 diffeent ossibilities to aange the base elements of a diffeential dive. So the numbe of the ossible kinematic schemes: 3*6 m =3*6 2 =108, fom which 36 seial connected, 36 diving-side-connected and 36 diven-side-connected dive. The and toothed base elements of the diffeential dive ae connected by a gea chain (lanet gea o lanet gea chain), the base element is the am (bidge). We use this notations, with the comment, that, and can be single and also joined base element, but if it is joined, the base element of the one degee of feedom dive () can diffe fom the one of the diffeential dive (fo examle: the joined base element units the diffeential s toothed base element and the dive s base element). Afte it let s examine a simle one degee of feedom connected dive built fom a diffeential () with i <η 0 inne atio, and a one degee of feedom dive. The diffeential can wok with, and, (e.g. i =-1, k =0,5)([1], Fig.1. ). As we mentioned, in case of i <η 0 and i >1/η 0 inne atios the oweflow of the connected lanetay dive is detemined exactly by the oweflow of the diffeential dive. When the oweflow is,, the ossible connected dive-vaiations can be seen on Fig.2.. 20
, Fig.2. And fo the case when the oweflow of the diffeential dive is,, the connected dives and thei oweflows ae develoed as seen on Fig.3.. Fom Fig.2.-3. it s obvious, that by same stuctue and same osition of, and base elements (e.g. Fig.2.a. and 3.a. o Fig.2.b. and 3.b.), the diving and the diven base elements ae evesed (e.g. the diection of the oweflow on Fig.2.a.is, while on Fig.3.a. it s )., Fig.3. Thus evey connected lanetay dive has vaious oweflow, so fo a given stuctue and given inut and outut elements the oweflow can be one kind only. Examinations show, that this establishment is tue geneally, if we build in diffeentials with i <η 0 and i >1/η 0 inne atios. It s necessay to mention, that the lanetay dives ae suosed to be non-selflocking in the illustated diections. If the built-in diffeential has η 0 <i <1/η 0 inne atio (e.g. η 0 =0,98, i =1,01, k =0,5)[2], the oweflow can be, o,. The ossible connected dives concened to, can be seen on Fig.4.. To, oweflows we can assign the connected systems seen on Fig.5.. 21
, Fig.4., Fig.5. It can be seen, that in Fig.4.a.- 5.a. and Fig.4.f.- 5.f. the dives have same stuctues and the inut and outut base elements ae also identical, only the closed oweflows ae inveted. The geneal examination of the connected lanetay dives shows, that the theoetical ossibility of two diffeent oweflows develoment exists only when the inne atio of the diffeential is η 0 <i <1/η 0 and when the two oweflows ae,;, o, ;,. These conditions ae satisfied by connected lanetay dives seen on Fig.6.. Thei common oeties ae that am can t be a joined base element, and the oweadding o owe-distibuting base element of the diffeential always must be cental toothed element ( o ). The connected systems on Fig.6.a.-b. have always k >1 kinematic atio and the outut element is. In case of dives seen on Fig.6.c.-d. the kinematic atio is k <1 and is the inut base element. It also can be seen fom Fig.6., that these connected dives have closed oweflow always. 22
,,,,,,,, a.) b.) c.) d.) Fig.6. Examle The owe ates of a diffeential gea dive with i =1,01 inne atio, k =0,5 kinematic atio and η 0 =0,98 efficiency (η 0 efficiency when the diffeential woks as a simle gea dive) have been detemined in ae [2] (Table 1.). Fo the sake of simlicity let s suose, that the dive is without loss (η =1) and P inut owe of the connected system is a unit (P =ϕ =1). By such assumtions the oweloss of tooth-fiction of the connected dive seen on Fig.4.a. is P V(, ) ϕ V =-0,3333 and the efficiency of the diffeential dive is η, =0,9807. Consideing only the fiction losses, the efficiency of the connected system in case of, diffeential oweflow is:, ) P + Pv(, ) 1 0,3333 η = = 0,6666. P 1 ( = If the oweflow is,, the oweloss in the diffeential is P V(, ) ϕ V =-0,9804, so the connected dive s efficiency is (Fig.5.a.):, ) P + Pv(, ) 1 0,9804 η = = 0,0196. P 1 ( = The uestion occus, whethe which oweflow will be ealised in the lanetay dive, so how high the efficiency will be. Based on the ules and exeiences in the ocesses of natue and othe fields, it can be told with high likelihood, that in this case also the theoy of least esistance will edominates. So the oweflow with the lowe loss will exist, and the connected dive will wok with the highe efficiency. 23
It can be laid down geneally, that the altenative woking ossibilities of connected lanetay dives can occu only in cases of η 0 <i <1/η 0 diffeential inne atio, and when am is loaded by extenal moment. So e.g. in Wolfom lanetay dives case this tye of altenative woking is not ossible. Refeences: [1] APRÓ, F.: Egyszeű egy- és kétszabadságfokú bolygóművek tevezése. Kinematikaiés nyomatékviszonyok, teljesítményfolyam és hatásfok. Sajtó alatt. [2] APRÓ, F. : Egyszeű egy- és kétszabadságfokú bolygóművek tevezése. Teljesítményaányok veszteségmentes és veszteséges bolygóműben. Sajtó alatt. [3] MÜLLER,.W.: ie Umlanfgetiebe. Singe-Velag Belin eidelbeg New Yok. Zweite neubeabeitete und eweitebe Auflage: 1998.. 260. 24