KINEMATIC ANALYSIS OF BEVEL PLANETARY GEARS BY USING THE INSTANTANEOUS AXIS OF ROTATION

U.P.B. Sci. Bull. Series D Vol. 70 No. 008 ISSN 454-58 KINEMATIC ANALYSIS OF BEVEL PLANETARY GEARS BY USING THE INSTANTANEOUS AXIS OF ROTATION Victor MOISE Iulian Alexandru TABĂRĂ În lucrare se preintă o nouă metodă pentru analia cinematică a mecanismelor planetare cu roţi dinţate conice. Metodele tradiţionale de analiă cinematică a mecanismelor planetare cu roţi dinţate cilindrice utilieaă metoda lui Willis sau metoda contururilor ţinînd cont de semnul raportului de transmitere. Totuşi acest lucru nu este posibil pentru un mecanism planetar conic simplu (cu două roţi în angrenare) deoarece nu sunt paraleli toţi vectorii viteă unghiulară şi prin urmare metode adiţionale sunt necesare. În acest ca se utilieaă o relaţie vectorială între viteele unghiulare relative în raport cu braţul port-satelit şi vectorii ataşaţi roţilor care formeaă angrenajul. Pe baa schemei cinematice a mecanismului se realieaă schema vectorilor unitari care pot fi utiliaţi pentru viualiarea poiţiei iniţiale a mecanismului. In this paper we present a new method for the kinematic analysis of planetary mechanisms with bevel gears. The traditional methods for the kinematic analysis of planetary mechanisms with cylindrical gears employ the Willis method or the contours method considering the sign of the gear ratio. This issue is not possible for single mesh bevel gear planetary mechanism (with two meshing gears) because not all angular velocity vectors are parallel to one another and consequently additional methods are required. In this case it is used a vector relationship between the relative angular velocities related to the planet carrier and the vector attached to the gears composing the mesh. Based on the kinematic scheme of the mechanism a unit vector scheme is accomplished which can be used to visualie the initial position of the mechanism. Keywords: planetary mechanisms bevel gears kinematic. Introduction The kinematics and dynamics of the planetary mechanisms requires the knowledge of the analytical expressions of the kinematic parameters. From the kinematic point of view the epicyclic mechanisms with cylindrical gears can be studied by the Willis method [] [5] [6] [7] [9] [4] [5] [7] complying with the sign convention for the transmission ratio depending on the gear type. The Prof. Mechanisms and Robots Theory Department University Politehnica of Bucharest ROMANIA Lecturer Mechanisms and Robots Theory Department University Politehnica of Bucharest ROMANIA

54 Victor Moise Iulian Alexandru Tabără exterior mesh has the sign minus while the interior mesh has the sign plus. The issue can be solved using the contours method too [0]. In papers [] and [] the authors determine the transmission ratio sign using a visual inspection of the kinematic scheme. In the case of conical planetary gears (figure.a) with a single gearing not all angular velocity vectors are parallel so the Willis method cannot be used. Generally in the bevel planetary gears case the transmission ratio can be written in the following form: i = ω ω ω ω = () showing that the transmission ratio sign cannot be determined. If the determination of the element angular velocity is asked the vector relations can be written under the form of: ω = ω + ω () ω = ω + ω on the basis of which the vector polygon in fig. b. can be achieved. Fig. The bevel planetary gears: a) kinematic scheme b) the angular velocities polygon It is to be observed that the method is hard and it is not convenient to analytical calculation fulfilment.. The instantaneous axis method This paper present a relation between the kinematic and the geometric parameters of the spherical planetary gears with a single mesh in order to allow the application of the analytic and numerical-analytic methods. Figure presents: a) the kinematic scheme b) the structural scheme c) the multipole scheme and d) the vector scheme [0] of bevel planetary gears having two degrees of mobility. This mechanism consists of the sun gear the planet gear and the planet carrier.

Kinematic analysis of bevel planetary gears by using the instantaneous axis of rotation 55 The structural relation of the mechanism [0] is Z( 0) + R() + R() + ( R RT )() which means that the mechanism consists of the basis Z(0) the motor groups R() respectively R() as well as ρ -dyad (R-RT)(). The final Denavit-Hartenberg trihedral [4] [] O ( n0 e n0 e ) with the e unitary vector oriented by O O axis is attached to the basis of the mechanism. The unitary vector e of the O B axis and the unitary vector e under a given direction in the planet gear plane are attached too. The unitary vectors e and e have been attached to the principal axes of the mechanism. The two cones polhodiec and herpolhodiec tangent under the commune generating line are rolling down one over the other without slipping. Let s consider a point P situated on the instantaneous axis of the relative movement of the gears and and Q and R its projections on O O and O B axes. In the point P (and in any other one on the instantaneous axis of the relative motion) there is the equality: ω QP ω RP () = or by using the unitary vectors of the basis trihedral the relation () becomes: ω e QP n ω n RP(- ). (4) 0 = 0 e The angular velocities ω and ω are considered positive in the sense of the unitary vectors e and e respectively. For the calculation facility the O ( i j k ) trihedral is superposed over the basis trihedral O n e n ). The relation (4) becomes: ( 0 0 e ω QP j = ω RP j (5) which under the projection on j direction leads to the relative angular velocity determination of the element related to namely: QP ω = ω (6) RP Considering that the angular velocities of the elements and are known as well as the teeth numbers of the two gears the following relation is obtained: where ω = ω ω. ω = ω (7)

56 Victor Moise Iulian Alexandru Tabără In the same way the relative angular acceleration ε can be determined. (i ) n 0 ( j ) e (k ) e n 0 D O C A C A D B P Q e ω B R O con herpolodic ω e n e n θ e e a ) d ) A Z (0) R() con polodic D B R() θ 0 n C n 0 e e R RT () b ) c) Fig. The bevel planetary gear: a) the kinematic scheme gear with the emphasiing of the axis unitary vectors b) the structural scheme c) the multipolar scheme d) the unitary vector scheme. To determine the axial rotation angle θ the equation (7) of relative angular velocities is integrated. There results: θ = ω dt + C (8) or θ = ( θ0 - θ0 ) + C (8 ) where θ 0 is the angle between the unitary vector n 0 and a unitary vector marked on the gear. The determination of integration constant is done by using the unitary vectors scheme from figure.d (for setting the initial position of the mechanism).

Kinematic analysis of bevel planetary gears by using the instantaneous axis of rotation 57 Thus for θ 0 = π / θ 0 = 0 and θ = π there results: C = π (+ ). (9) Because for writing the relation () it was used the instantaneous axis of the relative motion between gears and we named the method after its name. It is mentioned the fact that the method is also applicable to ordinary or planetary mechanism with cylindrical gears. Another method to demonstrate the relation between the relative angular velocities ω ω and the teeth numbers of and gears is obtained by using the chains of relative motions [] [6] [8]. Thus for the mechanism in figure in point D the vector relation can be written as: V D = V D + V D. (0) Taking into account the fact that: V D = V C + V DC V D = V B + V DB V B = V = = 0 C V D the relation (0) becomes: ω x CD + ω x BD = 0 () or ω x CD + ω x BC = 0 ( ) where ω = ω + ω BD = BC + CD. If the unitary vectors system O ( i j k) is considered the relation ( ) becomes: ω k CD( i) + ωi BC( k) = 0. () After accomplishing the vector products the projection on the directionof the unit vector j and expressing the dimensions of BC and CD as functions of the teeth numbers and gears module the same relation (7) is obtained.. Computational example Fig..a schematies an orientation mechanism with bars and gears having three degrees of mobility. In the figure.b c and d the structural scheme the multipolar scheme the unit vectors scheme of the mechanism respectively are presented. The mechanism consists of the basis Z(0) the motor groups R(8) R(9) R(0) and the ρ -dyads (R-RT)() (R-RT)() (R-RT)() (R-RT)(4) (R-RT)(5) (R-RT)(6) and (R-RT)(7). The angles of the axial rotations are emphasied with the help of

58 Victor Moise Iulian Alexandru Tabără the unit vectors scheme. The unit vectors scheme is used to visualie the current position of the mechanism and to specify its initial position. The principal axes of the mechanism are OO' O' O" and O" P. To the principal axes of the mechanism the unit vectors e e and e respectively have been attached. To the manipulated object axis or to a segment of it the unit vector e 4 has been attached..a. Direct kinematic analysis There are known: - the generalied coordinates θ 80 θ 90 θ 0 0 ω 80 ω 90 ω 0 0 ε 80 ε 90 ε 0 0 from the motors joints A B and C - the initial position of the mechanism: 0 π 0 θ 0 = 80 0 θ = θ 0 90 = 0 0 θ 0 0 = 0 0 0 θ = π / θ = π / - the axial displacements: s = OO s = O O s = O P s 4 = PT - the crossing lengths (the distances between the unit vectors e e e and e 4 ): a = a = a = 0 - the crossing angles (the angles between unit vectors e e e and e 4 of the π considered axes): α = α = α =. The following parameters have been determined: - the relative angular velocities: ω 0 ω ω 60 ω 70 ω 5 ω 6 ω 7 ω 4 ω 4 ω - the relative angular accelerations ε 0 ε ε 60 ε 70 ε 5 ε 6 ε 7 ε 4 ε 4 ε - the angles of the axial rotations θ 0 θ and θ about the unit vectors e e and e respectively To determine the relative angular velocities and accelerations the instantaneous axis method is used as follows: - the joint D: ω 80 AD = ω0 ED after the vectors replacement by the correspondent dimensions and unit vectors there results: ω 80 i ADk = ω0 k ED(-i) or ω ( j) = ω ( ). After the projection on the unit vector j direction the 80 0 j

Kinematic analysis of bevel planetary gears by using the instantaneous axis of rotation 59 relative angular velocity ω 0 is obtained namely: ω 0 = ω80 - the joint G: ω 90 BG = ω60 FG whence it results ω60 = ω90 - the joint Q: ω 0.0 CQ = ω70 NQ whence it results ω 70 = ω0.0 - the joint H: ω 4 IH = ω6 FH whence it results ω 4 = ω6 4 where ω6 = ω60 ω0 = ω90 ω80 - the joint M: ω 5 LM = ω7 NM whence it results ω 5 = ω7 where ω7 = ω70 ω0 = ω0.0 ω80 - the joint K: ω JK = ω5 LK whence it results ω = ω5 4 - the joint R: ω SR = ω4 IR whence it results ω = ω4 where ω 4 = ω4 ω = ω6 + ω5. 4 After the replacements it results: ω 0 = ω80 ω = ( ω80 ω0.0 ) ' 5 5 4 4 4 ω = ω80 + ω90 + ω 0.0. 4 4 The relative angular accelerations are determined in the same way: ε 0 = ε80 ε 60 = ε 90 ε 70 = ε0.0 ε 6 = ε 60 ε0 = ε 90 ε80 6 7 5

60 Victor Moise Iulian Alexandru Tabără ε 4 = ε 6 ε 7 = ε 70 ε0 = ε0.0 ε80 ε5 = ε 7 ' 4 7 ' 5 ε = ε5 ε 4 = ε4 ε = ε6 + ε5 ε = ε 4. 4 4 After the replacements the following relations are obtained: ε 0 = ε80 ε = ( ε80 ε0.0 ) 4 4 4 ε = ε80 + ε90 + ε 0.0. 4 4 For the axial rotation angles θ 0 θ and θ determination the transmission functions of the corresponding relative angular velocities are integrated: θ 0 = ω80 dt + C θ = ( ω80 ω0.0 ) dt + C θ 5 5 [ C 6 5 7 8 4 9 4 5 4 0 = ω80 ( ) + ω90 + ω0.0 ]dt + 4 4 After the integration of the above equations the integration constants are determined and it results: π π θ 0 = θ80 + θ = ( θ80 θ0.0 ) + 4 4 4 π θ = θ80 ( ) + θ90 + θ0.0 + 4 5 4.b. Inverse kinematic analyses In the case of the back kinematic analysis two cases can appear namely: ) the positions velocities and accelerations of the tracing point T are known... namely XT YT ZT XT YT ZT XT YT ZT and the kinematic parameters are required in the motor joints A B and C namely: θ 80 θ90 θ0.0 ω80 ω90 ω 0.0 ε80 ε90 ε 0. 0

Kinematic analysis of bevel planetary gears by using the instantaneous axis of rotation 6 ) the relative angular velocities ω 0 ω and ω are known and the mechanism initial position and the kinematic parameters in the motor joints A B and C are required. For the first case the knowledge of the mechanism constructive characteristics is needed as presented at.a. paragraph. Taking into account the mechanism constructive characteristics the position vector of the tracing point T has the expression: r T = se + s e + s e + s 4 e 4 so that the T point coordinates are: X = s A + s A + s A Y T T = s B + s B + s 4 4 B 4 4 ZT = sc + sc + sc + s4c4 where: C = A = sin θ0 B = cos θ0 C = 0 A = cos θ0 sin θ B = sin θ0 sin θ C = cos θ A 4 = sin θ0 cos θ cos θ0 cos θ sin θ B 4 = cos θ 0 cos θ + sin θ0 cos θ sin θ C 4 = sin θ sin θ s = OO s = O O s = O P s 4 = PT ([] []). After the replacements it results: X T = s sin θ0 + s cos θ0 sin θ s4(sin θ0 cos θ cosθ 0 cosθ sin θ ) Y T = s cos θ0 + s sin θ0 sin θ + s4(cos θ0 cos θ + sin θ 0 cosθ sin θ ) Z T = s s cos θ + s4 sin θ sin θ. The above equations constitute a non-linear system with the unknown θ 0 θ θ a system which can be solved by an adequate numerical method. By deriving with respect to the time the position relations of the tracing point T a linear equation system in the unknowns ω0 ω and ω is obtained. By deriving the velocities relations of the tracing point T a linear system in the unknowns ε0 ε and ε is obtained. Further on it follows the determination of the motor joints variables A B and C that is: θ 80 θ90 θ0.0 ω80. ω 80 ω90 ω0.0 ε80 ε90 and ε 0. 0. By using the relations of the forward kinematic analysis the following relations are obtained: θ 80 = ( θ0 C) 8

6 Victor Moise Iulian Alexandru Tabără 4 4' 4' θ90 = ( θ0 C)( ) ( θ C ) + ( θ C) 4 θ 0.0 = ( θ0 - C) + ( θ - C ). The integration constants are settled for the initial position of the mechanism. π π π For the considered initial position there results: C = C = C =. The expressions of the angular velocities and accelerations in the mechanism motor joints are: ω 80 = ω0 8 4 4' 4' ω 90 = ω0 ( ) ω + ω 4 7 ω 0.0 = ω0 + ω respectively ε 80 = ε0 8 4 4' 4' ε 90 = ε0 ( ) ε + ε 4 7 ε 0.0 = ε0 + ε. In the case of the second variant namely when the relative angular velocities ω0 ω şi ω are known the angular velocities ω 80 ω 80 and ω0. 0 of the motor joints AB and C are known. By the integration of the angular velocities equations the angles θ 80 θ 80 and θ 0. 0 are achieved. The integration constants are determined in the same way.. Conclusions The instantaneous axis method facilitates the determination of gears mechanisms kinematic parameters and it simultaneously allows the use of the analytic methods in the study of these mechanisms. The unit vectors scheme allows the visualiation of the relative positions of the mechanisms elements which leads to the accomplishment of a complete image over the orientation of the mechanism.

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64 Victor Moise Iulian Alexandru Tabără b) a) d) c) Fig.. The kinematic structural multipole and unitary vector scheme of an orientation mechanism