12 th National Congress on Theoretical and Applied Mechanics September 2013, Saints Constantine and Helena, Varna, Bulgaria

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1 th Natonal Congress on Theoretal and Appled Mehans 3-6 September 03 Sants Constantne and Helena Varna Bulgara APPLICATION OF GEARING PRIMITIVES TO SKEW AXES GEAR SET SYNTHESIS PART : MATHEMATICAL MODEL FOR SYNTHESIS AND APPLICATIONS EMILIA ABADJIEVA Gfu Unversty Mehanal Engneerng Department - Yanagdo Str. Gfu Japan e-mal: abadjeva@gmal.om VALENTIN ABADJIEV Insttute of Mehans Bulgaran Aademy of Senes Aad. G. Bonhev blok 4 3 Sofa Bulgara e-mal: abadjev@mbm.bas.bg HARUHISA KAWASAKI Gfu Unversty Mehanal Engneerng Department - Yanagdo Str. Gfu Japan e-mal: h_kawasa@gfu-u.a.jp TETSUYA MOURI Gfu Unversty Mehanal Engneerng Department - Yanagdo Str. Gfu Japan e-mal: mour@gfu-u.a.jp ABSTRACT. Spatal skew-axes gears are used to transform rotaton between shafts wth nonparallel and non-ntersetng axes. Prnples of mathematal modellng of a tooth ontat synthess for ths general gearng ase are dsussed n ths paper. The researh shows that synthess of the gear drves an be realzed by applaton of a pth ontat pont mathematal model. Thе approah requres applaton of gearng prmtves (pth onfguratons) to the synthess of Sprod and Helon gear sets. These gears are alternatve of a bevel gear wth straght teeth that s norporated n the drvng of the robot s hand onstruted n Mehnal Engneerng Department at Gfu Unversty Japan. KEY WORDS: Mathematal model synthess pth onfguratons. Introduton The study s an extenson of what s started n Part. The am s to fnd out a soluton of one of the bas tasks of the synthess of skew-axes gears - synthess of pth onfguratons (pth rles and pth surfaes) whh are prmtves of ther synthess. The mathematal model and algorthms that ontans geometr

2 E. Abadjeva V. Abadjev H. Kawasak T. Mour haratersts of pth onfguratons are llustrated n Fg. of Part of the urrent study. In Fg. s shown only one alternatve of pth onfguratons - pth rles.. Synthess usng a pth ontat pont The the mathematal model for synthess based on a pth ontat pont ensures solvng of two bas problems: synthess of the pth onfguratons; synthess of the atve tooth surfaes. Pth onfguratons: essene and defnton. The present paper treats the knematal theory of the spatal transformaton of rotatons n the ontext of defnng the knemat-geometr essene of bas elements of the mathematal models for synthess of hyperbolod gears and ther role n solvng the gears synthess. These elements have been alled pth onfguratons (pth rles and pth surfaes) and treat one atual but stll dsputable (as a ontent and termnology) part of the meshng theory. From the performed analyss of the publatons [] and the omments above we an onlude that: - If the law of transformaton of rotatons = ω ω = ons tant ( ω - angular veloty of the rotatng lnk ) between fxed and rossed axes and (the shortest dstane between them s a w = ons tant and the angle between them s δ = ( ω ω ) = ons tant ) s gven and f the poston of a pont Р (treated as a pont of ontat of onjugate tooth surfaes Σ and Σ ) n the fxed spae s known then the dameters and the mutual poston of the rles H ( = are ompletely and unquely determned. The rumferental veloty vetors ( = of the ommon pont Р and the relatve veloty vetor V at the same pont (.е. the plane T m where the oplanar vetors V ( = and V le) as well as the normal m m to T m at the pont Р are determned n a unque way too. - It s suffent to know the mutual poston of the rossed axes of rotaton and and the poston of the pont P (as a ommon pont of the tooth surfaes Σ and Σ ) n the fxed spae n order the rles H ( = to be ompletely and unquely determned (as dameters and mutual poston). The plane T m formed by the tangents to the rles H ( = at the pont Р as the normal m m to T m at the pont Р are unquely determned too. Mathematal model for synthess of pth onfguratons. Let two rossed axes and representng the axes of rotatons of the movable lnks of a three-lnk tooth mehansm be gven n the fxed spae. Тher mutual poston s defned by the angle δ = onstant (the angle between the angular veloty vetors ω and ω of the movable lnks ( = ) and the shortest dstane a w = ons tant. The onrete study s performed when δ ( 0 π ). Eah pth rle V H les n a plane perpendular to

3 Applaton of Gearng Prmtves Part. Mathematal Model for the axs of rotaton of the movable lnk and has a radus equal to the dstane from the pont Р to the axs. The pth plane T m s determned by the tangents to the rles H and H at the pth ontat pont Р. Besdes the pth normal m m to T m s determned at Р. The study performs by means of the notatons and the oordnate frames S O x y ) and S O x y ) ntrodued n Fg. ( z ( z (Part ). The dmensons and the mutual poston of H and H are ompletely determned by the ylndral oordnates a r θ ( = of the ontat pont Р n the systems S ( = and by the angles δ ( = between the planes of H ( = and the normal m m. The graphal model shown n Part s orented to the synthess of pth rles. Ths type of pth onfguratons are harater for the tradtonal onstrutve types of spatal gears (hypod ones Sprod and Helon ones et.). We represent the radus-vetor O P and the unt vetor m of the normal m m by a r θ and o a r θ and usng a w and δ = 90. Thus we get the followng set of equatons: (.) osθ = a r a = r osθ sn δ = osδ osθ r snθ = a snθ osδ snθ = osδ snθ w r osδ osθ = sn δ. Later the study of the system (.) wll be performed takng nto aount the π π π π followng geometr ondtons θ 0 θ 0 δ 0 δ 0 a w > 0 r > 0 a 0 ( =. Eah of the last three equatons n (.) s a onsequene of the other two.e. (.) s a set of 5 ndependent equatons wth 9 unknowns: a w δ r a θ δ r a θ. Therefore eah soluton of (.) s a funton of 4 of them (we wll onsder them as free ones). We suppose that the ndependent (free) parameters are a w δ r and a. We wll look for analytal relatons that they have to be fulflled so that the set (.) to have a soluton. Synthess of orthogonal ontatng pth onfguratons. Let us pay attenton only to the ases orented to the synthess of Sprod and Helon gears but also to suh essental for the prate ases of orthogonal hyperbolod gears. (a). δ = 0 and a 0. Ths ase refers to the orthogonal hyperbolod gears wth external meshng when the oaxal surfaes of the movable lnk = - referene root and tp ones are of ylndral form.

4 E. Abadjeva V. Abadjev H. Kawasak T. Mour Fg.. Pth onfguratons for gears of type Helon: а) pth surfaes pth rles H ( = ; a w = 3 5 mm; = 90 ; a = 4 mm; r = 0 04 mm o δ = 90 ; System (.) has the followng unque soluton = 0 ; 4 s H ( = ;b) r = mm; a = 9 5 mm; aw θ = 0 tanθ = a = r a (. aw π r = δ = snθ n an arbtrary hoe of a w r and a. The parameters n (. defne the dmensons and the poston of the pth onfguratons orrespondng to spatal hgh reduton gears of type Helon (Fg. ) [ ]. (b). δ > 0 and a 0 In ths ase the ondton for an exstene of pth onfguratons s a ( aw r ) tanδ and the soluton of the system s: Fg.. Pth onfguratons for orthogonal Sprod gears: а) pth surfaes pth rles H ( = ; a w = 3 5 mm; = 84 ; a = mm; r = mm o δ = 90 ; s H ( = ; b) = 5 ; a = 9 5 mm; r = 4 mm;

5 Applaton of Gearng Prmtves Part. Mathematal Model for (.3) r tan a ot δ + θ = a a w aw snθ = r + a otδ snδ = r os os θ δ = r osθ a = w r snθ. snθ The parameters of the pth onfguratons of orthogonal hypod gears and gears of type Sprod [ ] are alulated by the relatons gven n (.3). The pth surfaes and rles are vsualzed n Fg.. The pth onfguratons shown n Fg. and Fg. are gearng prmtves and they are appled to the synthess of Helon and Sprod gear drves. These transmssons are desgned as alternatve mehanal drvng of robot s hand [3]. 3. Tehnal applaton The future humanod robots wll exeute varous omplated task by ommunatng wth human users. Suh robots wll be equpped wth anthropomorph mult-fngered hands whh are smlar to the human hands. The man future purpose of suh humanod robot s hand s to replae the human presene n dangerous tasks n felds suh as: ndustral manufaturng spae sea-bed and so on. One other future applaton of the anthropomorph robot hand s ts use as a prosthet for the handapped people. Hene the requrements to suh hands are to obtan haratersts as auray smoothness exatness. The purpose of the realzed study s to replae bevel gear wth straght teeth: ( m = 0 5 mm; = 4 ; z = 0 ; z = 40 ) wth knematally and strength equvalent hyperbolod gear of type Helon and Sprod whh s haraterzed wth nreased number of smultaneously ontatng teeth (see Fg. 3a b). Fg. 3. CAD desgn of spatal hyperbolod gear drves (pnon teeth z = 8 ; gear teeth - z = 3 ; tooth module - m 0 5 Sprod = mm; offset - a w 3 5 = mm) of type: a) Helon; b)

6 E. Abadjeva V. Abadjev H. Kawasak T. Mour 4. Conluson The present paper gves a bref survey of the pth ontat pont approah used to desgn mathematal models of spatal transmsson synthess. Ths approah has been appled n the Insttute of Mehans of Bulgaran Aademy of Senes to model dfferent types of hyperbolod gears wth arbtrarly skewed axes. The applaton of the pth ontat pont model of spatal gears synthess n Sprod and Helal gears s also llustrated. Algorthms and omputer program for desgn of Sprod and Helon gears are elaborated. The above gear pars are pratally realzed as gear drves. Ths onfrms the model plausblty and enables spef skew axes gear sets to synthess. The realzed study s an llustraton of the applaton of the approah to the synthess of hyperbolod gear pars upon a pth ontat pont. Here s presented an attempt to reate an adequate gear drve applable to the desgn of the robot s hand. Ths robot s an nnovaton produt of Mehanal Engneerng Department at Gfu Unversty Japan Aknowledgement Fnanal supported by the Japan Soety for the Promoton of Sene (JSPS) s gratefully aknowledged. The frst author s ndebted to hs olleagues from Mehanal Engneerng Department at Gfu Unversty Japan for ther onstant support. R E F E R E N C E S [] ABADJIEV V. Gearng Theory and Tehnal Applatons of Hyperbolod Mehansms. D. S. Thess Bulgaran Aademy of Senes Insttute of Mehans (007) 309 (In Bulgaran). [] DUDLEY D. Gear Handbook. The Desgn Manufature and Applaton of Gears MGraw-Hll Eds. New York ( p. [3] MOURI T. KAWASAKI H. NAKAGAWA S. T. MIURA. Hgh Power Humanod Robot Hand Pro. of RSJ0 No RSJ0AC0-3 Sapporo Japan (n Japanese).