PLANETARY PRECESSIONAL TRANSMISSIONS: GENERATION TECHNOLOGIES

Transcription

1 Planetary preessional transmissions: generation tehnologies 91 PLANETARY PREESSIONAL TRANSMISSIONS: GENERATION TEHNOLOGIES Ph r.s.prof. I. Bostan, Ph r.s.prof. V. ulgheru Tehnial University of Moldova 1. INTROUTION A problem for engineering ompanies (espeially in the metalworking industry, automotive, hemial and metallurgial industries) is to satisfy the ever-inreasing requirements to the transmissions used in majority of industrial mahinery and tehnologial equipment related to bearing apaity, ompatness, mass and dimensions, low ost of prodution, et., and, in partiular, to kinematial harateristis, strutural ompatibility with other aggregates of the equipment, et. Gearings are onsidered the most sophistiated omponents of mahines. Mahine reliability depends very muh on the gearing mehanial transmission operation, in general. The quality indies of traditional gears were inreased largely by hanging involute gearings, and by reating new gearings, suh as Novikov-Hlebanija Symark, et. In the field of planetary transmission it was onsidered properly to follow the way of developing new types with inreased performanes. Sientifi analysts onsider that in the field of tehnial sienes worldwide an essentially new type of mehanial transmission is being invented every 0-5 years. Thus, the German engineer L. Braren developed the yloid planetary transmission "YLO" in 19. The Russian engineer A. Moskvitin invented the harmoni frition transmission in 1944 and in 1959 the Amerian engineer.w. Musser developed the harmoni gear transmission. In the late 70ies a new type of mehanial transmission has been developed at the Polytehni Institute of hisinau (now the Tehnial University of Moldova). The new type of mehanial transmission entered into international terminology iruit as planetary preessional transmission (PPT). The first patent was issued under this name in 198 [1,,,4]. Planetary preessional transmission differs from the lassial one by the new priniple of motion and load transformation and transmission, i.e. by using sphere-spatial motion of the satellite and variable onvex onave profile. ue to these innovative features gearing multipliity in planetary preessional transmission reahes 100% (in lassial transmissions - -7%) whih provides inreased bearing apaity, redued dimensions and weight, extended kinematial range ± ± 599 (in harmoni transmissions ), high kinematial auray, et. The researh team involved in researh on preessional planetary transmissions published over 800 sientifi artiles, obtained about 170 patents, implemented about 0 pratial ahievements in the field of fine mehanis and speialized tehnologial equipment, in roboti omplexes for the exploration of ferro - manganese onretions from the World Oean bottom (USSR onept), in spaeflight tehnique, et. Know-how in the elaboration of multiouple preessional gear, manufaturing tehnology and ontrol methods, and a range of preessional transmission diagrams belong to the researh team from the Tehnial University of Moldova. The speifi harater of sphere-spatial (preessional motions of the preessional transmissions pinion make impossible the utilisation of lassial involute teeth profiles. This fat requires the elaboration of new profiles adequate to the sphere-spatial motion of pinion, whih would ensure high performanes to the preessional transmission. arrying out on the priniple of the transfer funtion ontinuity and gear [1,] based on the priniples of the transfer funtion ontinuity and gear multipliity whih aims to: the elaboration of the gear mathematis model with aount of the peuliarities; the analytial desription of teeth profiles by a system of parametri equations on spherial surfae and normal teeth setion for inner and plane gear; A determination of geometrial and inemati parameters influene of the gear upon the teeth profiles shape and the justifiation of their rational limits of variation; the elaboration of the theoretial basis evaluation of teeth gear multipliity in preessional transmissions; area definition of gear multipliity existene by 100% teeth ouples. the prodution of non-standard teeth profiles requires a new manufaturing tehnology. In the

2 9 Planetary preessional transmissions: generation tehnologies omplexity of problem gear-synthesis-profile study- manufaturing the elaboration of effiient methods of teeth manufaturing whih ensures a maximum produtivity and redued ost while satisfying the requirements related to the gear with preessional motion plays an important role. To solve this problem the following has been done: we elaborated the mathematial model of teeth generation whih shows the interation of teeth in preessional gear; we investigated the kinematis of the mehanism of method realisation for teeth generation; we determined the tool path of motion and the family envelope of the generating surfae by using the omputer; we elaborated and manufatured from metal milling and tooth grinding tools, inlusively their longitudinal modifiation.. GENERATION TEHNOLOGIES.1. General remarks evelopment of mehanial transmission with gear, different from the lassial one, requires omplex researh in various fields. This finding refers to planetary preessional transmission with multiouple gear, whih is haraterized by essential onstrutive-kinematial features. In solving omplex problems related to gear synthesis - profile researh fabriation an important role belongs to developing effiient methods of teeth manufaturing, whih would ensure maximum produtivity, redued ost and quality. Manufature of preessional gear wheels with onvex-onave and variable tooth profile annot be ahieved by existing generation tehnologies, but through fundamentally new tehnology. Generation tehnology of preessional wheel teeth must ensure ontinuity of motion transformation funtion with the following onditions: non-standard and variable tooth profile, and satellite arrying out spherespatial motion with a fixed point. To ahieve the above, a new proedure for teeth proessing is proposed by self-generating method with preessional tool against rotating blank [4]. To develop the theoretial basis for generating tooth profile by running the preessional tool it is neessary to determine the harater of ontinuous ontat of the tool utting edge and profile of the proessed wheel tooth for a omplete tool-blank preessional yle. In this onnetion a mathematial model of tooth self-generating method by running the preessional tool was elaborated, whih fully reflets the atual interation of teeth in preessional transmission. For this purpose the following was desribed: kinematial onnetion of the preessional tool with the blank that ensures ontinuity of motion transformation funtion in the linkage tool-blank ; path of motion of the tool entre in the fixed system of oordinates; path of motion of the tool entre in the movable system of oordinates, onneted with the rotating blank; the generating ontour of the tool in the movable system of oordinate and the system envelope of generating surfaes of the tool for a yle of preession; projetion of tool ontour envelope in the plane system of oordinates... Tehnologial system for teeth generation by sphere-spatial motion of tool as trunated one..1. Kinematis of gear generation system To ahieve teeth generation method a tool arrier devie was developed, whih diagram is shown in Fig. 1. In the designed devie the node, whih involves the tool in sphere-spatial motion, is stopped from rotating around the ommon axis of the main shaft-blank - a kinemati joint. Rotation of blank and main shaft 1is oordinated by the Main shaft Mahin e-tool mass Figure 1. Prinipled spatial diagram of teeth proessing method by preessional tool running division kinemati hain of mahine tool. Kinemati joint of tool with the body must be built

3 Planetary preessional transmissions: generation tehnologies 9 so as to ensure ontinuity of the transmission funtion of rotation motion, i.e. ω 1 ω = onst. ontinuity of transmission funtion of rotation motion is determined by path of motion of point, belonging to the movable system of oordinates. To researh the kinematis of the devie that involves the tool in preessional motion: Imaginary satellite gear (profile generating tool) with an imaginary number of teeth (determined by the mahine-tool kinematis) gears with the blank, fixed on mahine-tool table, with the number of teeth = ±1. At a turning of the main shaft 1 the blank rotates at angle ψ, that orresponds to the angle between the differene of the wheel teeth: ψ = π ( ). (1) To define the position funtion of the given devie ψ = f ( ψ ) it is neessary to determine beforehand the equations of the tool motion in the fixed OY and movable O 1 Y 1 1 systems of oordinates. The link between the mentioned systems of oordinates is determined by the Euler angles. Sphere-spatial motion of tool (imaginary wheel) at uniform rotation of the main shaft 1 ω 1 is desribed by the system of equations ψ = ω t, θ = onst., ϕ = ϕ(t), 1 () esign of the working devie for teeth generating tehnology should provide limitation of tool rotation around the main shaft of the toolmahine by a ertain tehnial solution, for example by kinematial oupling bolt-gutter. In this ase the oordinates of the bolt ontat point (fig.) with the groove in the movable system of oordinates O1Y11 will be: 1 = 0, Y 1 = R, 1 =0, () where R is the radius of point loation. At sphere-spatial motion of tool, the motion of point loated in plane O is limited by the groove walls, i.e. the ondition is realised for eah value of ψ : = 0. (4) Using the transition matrix of the movable system of oordinates O 1 Y 1 1 onneted with the tool and the bolt limiting its rotation around the shaft in the fixed system of oordinates, ondition = 0 o an be written in the form: = Y = 0. (5) Or in extended form> =a 11 +a 1 Y +a 1 =0 By replaing a 11, a 1 and a 1 in (4) we obtain: ( ψ ϕ ψ ϕ θ) ( ) = os sin sin sin os Y osψsin ϕ+ sin ψosϕosθ + sin ψsin θ= 0 (6) For the ontat point of the bolt with the groove oordinates (5) we have: = R ( os ψ sinϕ + sinψ osϕ osθ ) = 0 Thus, ( os sinϕ + sinψ osϕ osθ ) = 0. (7) ψ. (8) By solving equation (8) we determine the linkage equation between the angle of tool selfrotation and the angle of rotation of the main shaft: ( θtgψ ) ϕ = artg os. (9) In suh ase the equations of tool preessional motion take the form: 1 ( ) ψ = ω t, θ =onst. ϕ = - artg os θ tg ψ. (10) To establish the dependene of the angle of rotation of blank ψ on the angle of rotation of the main shaft ψ we desribe the blank motion omposed of the involved rotational motion with the rank of the main shaft ψ e and the relative motion of rotation with regard to the rank of the main shaft ψ r. In the ompound motion of blank ψ e =ψ, and ψ r represents a ertain funtion f(ϕ) of the angle of rotation of tool ϕ, that is: ψ = ψ + f ( φ). (11) For ideal preession of the drive mehanism of mahine/tool funtion f(ϕ) will take the form: ψ = ψ + ϕ. (1) 1 By onsidering equation (9) we obtain the position funtion of the kinematial linkage mehanism of the devie: ψ = ψ artg(os θ tg ψ ). (1) 1 Momentary gear ratio of the kinematial linkage mehanism of the devie is obtained deriving (1) after ψ: dψ ω 1 osθ i 1 = = =.(14) dψ ω os ψ + os θ sin ψ 1

4 94 Planetary preessional transmissions: generation tehnologies Average gear ratio for a rotation of the main shaft will be π med 1 i1 = iψdψ π = 0 (15) 1 z π = ψ artg(osθtg ψ ) 0 =. π z Analysis of dependene (15) demonstrates that for the ratio of teeth z <z the diretion of main shaft rotation of gear utting mahine and blank (imaginary wheel) oinides, and for the ratio of teeth z >z is different. ivision kinematial hain of mahine tool must provide the following kinematial link: at full rotation of the main shaft the blank (imaginary wheel) should rotate under angle ψ = π ( ). This kinematial link defines the average gear ratio of the manufatured gear. Given the fat that the kinematial link tool blank is done by the mahine - tool dividing hain under onditionω 1 ω = onst., the angular veloity variation aused by the kinemati link mehanism of the tool with the frame will transpose on the tooth profile, therefore, it will introdue a diagram error Δψ in the tooth profile. The diagram error Δψ an be identified by angular positioning error med of the blank ψ relative to position ψ of the same blank, whih onditionally would rotate uniformly med with the gear ratio i1 = ( ). in this ase the diagram error will be: med Δψ = ψ i1 = [ ψ artg(osθtg ψ]. (16) So, the kinematial link of the tool with the frame introdues some diagram error in the tooth profile. Fig. shows the graph of diagram error of tool position error ψ at one rotation of the main shaft and motion of point in OY plan. If point makes a motion in OY plane the error is transmitted intat to the tool, and the last generates the tooth profile with the same error. To ensure ontinuity to motion proessing funtion it is neessary to modify the tooth profile by diagram error value Δψ by ommuniating additional motion to the tool. Figure. ependene of tool position error on the angle of rotation ψ of the main shaft for various angles of nutation θ. orretness of additional motion of the tool was established using a omputer alulation program. It was found that generation preision of the manufatured wheel teeth depends on the ontinuity of its angular speedϕ of the tool. Funtion analysis (1) shows that for ϕ = ψ the instantaneous transmission ratio i 1 = onst. For ondition ϕ = ψ from equation (1) we have: ψ = ψ ϕ = ψ = ω t. 1 From this analysis we find that any tehnial solution to eliminate the influene of diagram error of tooth profile preision generation with preessional tool would be profiling of the ontat surfaes of the groove of kinematial link mehanism, whih supports the bolt (delimiter of rotation). The bolt ontat with the shaped surfaes

5 Planetary preessional transmissions: generation tehnologies 95 of groove transmits also the reation torque from the node, on whih the tool is installed to the frame. To ahieve the proposed tehnial solution to exlude the error of profiling of supporting surfae of the link hannel with the bolt it is neessary to desribe the profile of ontat surfaes with parametri equations. In this ase we take an arbitrary point on the tool axis with oordinates, Y, (Fig. 1), and identify the path of motion in the fixed system of oordinates OY to satisfy the ondition i 1 =onst. Using the matrix form for the transition from the oordinate system O 1 Y 1 1 to the fixed system OY we get : Y or by omponents: = A Y (17) =a 11 +a 1 Y +a 1 ; Y =a 1 +a Y +a ;... (18) =a 1 +a Y +a. where a i,j, i,j =1 are osines of angles between the axes of oordinates. onsidering that instantaneous gear ratio i 1 =onst. when ϕ = ψ then equations (18) are transribed as: = 1 (os ψ + os θsin ψ) + Y1 ( 1 os θ) os ψsin ψ+ 1 sinθsin ψ; Y= Y1 ( 1 os θ) os ψsin ψ+ Y1 (sin ψ+ os θos ψ) 1 sinθos ψ; 1 = 1 sinθsinψ+ Y1 sinθos ψ+ 1 os θ. (19) For the ase when point is plaed on axis OY 1 its position is defined by oordinates 1 = 0, Y 1 = R, 1 =0, and equations (19) take the form: = R = R ( 1 osθ ) osψ sinψ ; ( sin ψ + osθ os ψ ) Y ; (0) = R sinθ sinψ. Equations (0) represent parametrial equations of groove lateral surfaes, by whih the limiting bolt of tool rotational motion around the fixed axis O, form a kinematial oupling, and provides the ondition i 1 =onst. Thus, the shape of groove lateral surfaes by whih the bolt forms the kinematial oupling of tool with the asing, desribed by parametrial equations (0), exludes the influene of diagram errors on tooth profile generated with preessional tool.... Analytial desription of the preessional tool path of motion Aording to the priniple of teeth generation by proposed method the tool should opy with ertain auray the shape and path of motion of the pin tooth in the real gearing (involving the entral wheel with teeth satellite gear with pin teeth). In this onnetion it was neessary to researh the tool path of motion with the angle of position to the blank δ 0. For this purpose, a point was identified on the tool axis (fig. ) with oordinates 1,Y 1, 1 in the movable system of oordinates O 1 Y 1 1 and parametrial Figure. etermination of the tool surfae family envelope. equations of its motion in the movable system of oordinates were desribed. For i 1 -onst. after a number of transformations we obtain: = α 11 = α 1 = α α Y α Y 1 + α Y 1 + α ; ; α 1 + α 1 (1) With ondition ϕ = ψ and onstant instantaneous gear ratio i 1 -onst we have: = (os ϕ+ osθsin ϕ) + Y (1 os Θ)osϕsin ϕ+ sin Θsin ϕ; Y = (1 os Θ)sin ϕosϕ+ Y (sin ϕ+ osθos ϕ) osϕsin Θ; = sinθsinϕ+ Y sinθos ϕ+ os Θ () For δ = 0 tool oordinates will take the form: 1 =0, Y 1 =-R u, 1 =0. () In this ase the equations of tool motion depending on the angle of rotation ψ of the main shaft will be: = R(1 os Θ )osϕsin ϕ, u Y = R (sin ϕ+ osθos ϕ), u = R sinθos ϕ. u (4)

6 96 Planetary preessional transmissions: generation tehnologies In the ase of toothed wheels with angle δ >0 the tool should be loated under the same angle. Then point will have the following oordinates: 1 =0, Y 1 =-Rosδ, 1 =-Rsinδ, (5) And the equations of the path of motion of tool in the fixed system of oordinates OY have the form: = R os δ (1 os Θ )osϕsinϕ R sinδ sinθ sin ϕ; u u Y = R os δ (sin ϕ + osθ os ϕ ) + R sinδ sinθ os ϕ; u u = R osδ sinθ osϕ R sinδ os Θ. u u (6) Exat performane of the tool path of motion was taken into aount in the proess of elaboration of the tool-arrier devie, shown in fig Tehnologial equipment for generating teeth with sfero-spae motion of trunated one shaped tool The profile of entral wheel tooth of preessional gear is variable depending on the values of onial axoid angle δ, taper angle of the rollers β, the nutation angle θ, the number of teeth of gears 1, and the orrelation between them. Fabriation of these profiles using traditional methods is pratially impossible, beause for eah orrelation value of all parameters δ, β, θ and tooth profile hanges shape, whih requires the design and manufature of the tool with the respetive profile. Therefore a new generating tehnology was proposed, whih arries out a set of profiles of the teeth, using a tool with the same geometrial parameters. The method onsists of the following: a series of motions oordinated between them against the rotating blank is ommuniated to the tool (milling utter or grinding wheel with trunated one-shaped geometry). The kinemati link of the blank with the tool provides rotation of one-toothed blank in a losed yle of the motion ommuniated to the tool. The tool is given suh a shape and motion that allows the proessing of any possible profile of the set, inluding longitudinal and profile modifiation. The desribed surfae on the peripheral side of the tool against the rotating blank reprodues a ertain oneivable body, alled the imaginary wheel (generators). Using the kinemati hain of gear utting mahine-tool running, gear blank and the tool are brought in a oordinated motion running motion, whih reprodues the imaginary wheel gearing with the blank. Part of metal is removed at eah elementary hange of tool position in spae in relation to the blank. Therefore the working surfae of the wheel teeth proessed is obtained as envelope of a onseutive series of positions of rotating tool profile generator ontour against the blank. A tool holder devie was developed to realize the demanded motions of the tool (Fig. 4, a), whih an be adjusted to gear utting mahines models 5KP5, 50P, 5A50, 5A60, 54, with auray lass GOST To ompensate error diagram of the satellite at its sphere-spatial rotation, a kinematial joint onneting the ross-rail with the body is introdued into the teeth grinder, ensuring ontinuity of the transformation funtion of rotational motion ω 1 / ω = onst. in the kinemati hain main shaft - tool blank. In other words, at teeth proessing by proposed method, their profile is orreted by an amount equal to the kinematial diagram error introdued by the sphere-spatial motion of tool with regard to the asing (bed). It was defined that in real preessional transmissions K-H the link of preessional satellite with the body introdues an error in the driven shaft position. This fat provokes non-uniformity of its rotation at uniform turning of the drive shaft. rawbak is eliminated by transposition of driven shaft position error on the proessed tooth profile. iagram error elimination is ahieved through the onstrution of ross-rail onnetion joint to the body, whih through a am installed on the rank shaft ommuniate auxiliary motion to the tool. The joint ensures ontinuity to the transformation funtion of rotational motion along the linkage shaft-rank-tool-blank. At tooth proessing by proposed method their profile is orrelated to value of the shift angle of driven shaft introdued by preessional satellite link in the real transmission. In the developed tool-arrier devie the point of intersetion of the fixed axis O with the movable axis O 1 of the rank (entre of preession) is on the axis of rotation of the gear utting mahine table. To researh the features of interation between the tool and the wheel proessed tooth (δ > 0), whih axis oinides with axis O of the devie rank-shaft: Fig. 4, a shows the omputer model of the proessing devie for gear wheels with non-standard profile, designed in AutoeskInventor and simulated in MotionInventor. Fig. 4, b shows the piture of gear utting mahine-tool endowed with the devie for profile generation by preessional tool. Fig. 4, presents samples of gear wheels with non/standard profile, worked out on this mahinetool.

7 Planetary preessional transmissions: generation tehnologies 91 am Know- How Kinematial joint Know-How a b Figure 4. Generation devie for gears with non-standard profile (a), mahine-tool with fabriated devie (b) and samples of fabriated gears () 4. ONLUSIONS Among the harateristis of the estimated results of researh in the field of new and effiient drive development we an enumerate the following: the elaborated preessional gears ensure: high bearing apaity; high kinematial effiieny; high kinematial auray; low noise level and vibrations; generation proedure for variable onvexonave teeth profiles provide high effiieny and proessing auray. Strutural optimization of the preessional transmissions will allow synthesis of new diagrams of preessional transmissions with onstant and variable transmission ratio and elaboration of new diagrams of preessional transmissions for speifi running onditions. Referenes 1. Bostan, I. Preessional transmissions with multiouple gearing. hisinau, Ştiinţa Publ. House, Pp Bostan, I., ulgheru, V. et all. Anthology of inventions. Volume 1. Planetary Preessional Transmissions (in Romanian). Bons Offies SRL, (011). pp.594. ISBN: Bostan, I., ulgheru, V. et all Anthology of inventions. Volume. Planetary Preessional Transmissions (in Romanian). Bons Offies SRL, (011). pp.54. ISBN: Bostan, I., ulgheru, V., Ţopa M., Vauleno M. Planetary preessional gear and method for its ahievement. Patent M no BOPI nr. /00. Reommended for publiation: