16 th European Mechanical Dynamics User Conference Distribution of loads in Cycloidal Planetary Gear (CYCLO)including modification of equidistant

Transcription

1 'LVWULEXWLRQRIORDGVLQ&\FORLGDO3ODQHWDU\*HDU&<&/ LQFOXGLQJPRGLILFDWLRQRIHTXLGLVWDQW Author: Manfred Chmurawa (*), Adam Loiec (**) (*) Sileian Univerity of Technology, Intitute of Tranport, Poland, Katowice, Krainiego 8, (**) MESco, Biuro Ulug Inzynieryjnych, Poland, Tarnowie Góry, Zagóra 167, $EVWUDFW The complex contruction of planet wheel in cycloidal planetary gear (Cyclo) practically mae impoible it optimal deign. In planetary gear (Cyclo), toothing of planet wheel ha hape of equiditant of hortened epicycloid. Till now, there are applied nominal toothing of wheel in which only one equiditant occur identical for planet wheel and co-operating wheel. In the paper it i preented original modification of inide cycloidal mehing baed on diverification of equiditant of planet wheel toward co-operating wheel. Two method and comparion between analytical and ADAMS model are preented. Fig.1: Cycloidal Planetary Gear (CYCLO) - 1 -

2 ,QWURGXFWLRQ Currently, in heavy indutry, high peed engine are ued for many type of machine that require application high ratio mechanical gear. Relatively, the mallet mechanical gear i the cycloidal planetary gear nown a cyclo gear (CPG). Cycloidal gear implement inide, out of centre mehing to obtain high ratio at one tage, high coefficient of teeth in contact and low diipation of energy reulting from occurring only roller friction in the gear and coaxiality of haft [,3,4]. Figure how the inematic cheme. The cyclo gear conit of planetary gear Fig. and traight-line mechanim Fig.b in erie connection. Becaue of that ind of connection we get compact gear with tationary central gear (), which i mehing with one or two planet wheel (1, 1 ) driven by the eccentric yoe (3), Fig. c. In cae of immovable tationary wheel (), a inematic ratio i given a follow: where: z1 z1 i = = (1) z z z z 1 =z i a number of teeth of planet wheel 1 or 1, z =z i a number of teeth (roll) of tationary gear. 1 When a high ratio i required within the limit of i=11:87 - a difference z = 1. Outline of planet wheel i an equiditant of hortened epicycloid abbreviation ESE [1,,3]. Central gear conit of et of roll. Torque from the planet wheel i tranmitted by the bolt and di of traight-line mechanim 4 Fig.. The main element of the cyclo gear connecting other element i the planet wheel 1. For balancing body force and lowering of mehing force two identical planet wheel 1, 1' are applied, with revere angle between them equal π. Planet wheel ( 1 or 1' ) i a flat di which perimeter ha a hape of equiditant of hortened epicycloid. In the centre of the wheel there i a big round hole for high effort roller bearing of eccentric and round about it maller hole for bolt of traight-line mechanim. Fig. Kinematic cheme of planetary cycloidal gear - -

3 Inide toothing i realied by two planet wheel and immovable central wheel, Fig.. Planet wheel or wheel have inide teeth and curvilinear hape of equiditant of hortened epicycloid in abbreviation ESE []. Till now it i applied unmodified inide cycloidal mehing which i baed on occurring one and the only one ideal ESE a well for planet wheel a for cooperating wheel. That ind of mehing can be called QRPLQDO and it characteritic feature i abence or caual configuration of clearance between teeth being the reaon of dicontinuity or even topping of revolving movement. Parameter of equiditant: Module pitch of epicycloid coefficient of hortening of epicycloid where e eccentric of gear a radiu of contant wheel of epicycloid (Fig. 1 (ra)) b radiu of rolling wheel (Fig. (rb)) r = a + b um of radiu a and b D 0 = E () H H]. P = = E U (3) In thi paper i hown difference between model of CYKLO with nominal and modificated equiditant parameter applied a numerical method in higher level approximate calculation model to condition in real gear. To viualize difference there i alo the analytical method a well a numerical one preented and obtained reult ha been compared with each other. 0RGLILFDWLRQRILQVLGHF\FORLGDOPHVKLQJ Baic idea of modification of inide cycloidal mehing i diverifying the equiditant of planet and co-operating wheel. It will occur two different equiditant, Fig.4: Nominal equiditant SQT characteried by nown parameter r, e, q, m repreenting tationary co-operating wheel (et of roll) Corrected equiditant S 1 Q 1 T 1 with unnown parameter r, e, q, m Corrected equiditant will caue clearance between teeth { r i } and pitch play δ. When equiditant are co-linear then initial configuration of clearance between teeth occur { r oi } enabling aembling of the element of the gear, Fig.3 and 4. After revolution of planet wheel with angle δ modified equiditant will be tangent to nominal in point S and clearance between teeth would create other aymmetric configuration { r i } with infinitely low clearance in active part of toothing and higher clearance in idle part of toothing, Fig.3. A meaurement of ditribution of load in active part of toothing can be factor of contact (or peudo-contact) deignating et of clearance lower than aumed criteria value r: α α α α 1 p 1p ε = (4) α pc α pc where: α, α 1 - generating angle of nominal ESE, in extreme point 1 and at the contact arc; α p, α 1p - angle of poition of extreme point of contact arc 1 and, Fig. 4; α pc - angular pitch of toothing in gear

4 Fig. 3. The inide cycloidal mehing in planetary gear with configuration of clearance between teeth modified planet wheel for ε = ε max = 0,45 (90 % teeth in active contact) Fig. 4. Ditance of equiditant { r i } and contact arc α p α 1p after revolution of planet wheel with angle δ > 0-4 -

5 0DWKHPDWLFDOPRGHORIPRGLILFDWLRQRIWRRWKLQJZLWKLPSOHPHQWDWLRQRIGLYHUVLILFDWLRQ RIHTXLGLVWDQW The goal of modification i modelling of clearance (ditance) between teeth by mean of correction of equiditant. For nown parameter of nominal equiditant r, e, q, m in gear with ratio i = z = z -1 it i earched corrected equiditant fulfilling criteria of optimiation : ufficiently high factor of contact ε, criterial value of clearance between teeth r, r o, ufficiently low angle of pitch play δ < δ min, equal value of equiditant eccentric e = e. After introduction tationary and movable co-ordinate ytem, Fig. 4: co-ordinate of nominal equiditant in tationary co-ordinate ytem Oxy ( z ) α co( z α) in x( α) = r co( α) + e co(z α) q co α + arc tg (5) 1 m + ( z ) α co( z α) in y( α) = r in( α) + e in(z α) q in α + arc tg (6) 1 m + co-ordinate of corrected equiditant in movable co-ordinate ytem Ox o y o ( ) in z αo x o( αo, r,q ) = r co( αo) + e co(z αo) q co αo + arc tg (7) r ( ) + co z αo e z ( ) in z αo y o ( αo, r, q ) = r in( αo ) + e in(z αo ) q in αo + arc tg (8) r ( ) + co z αo e z co-ordinate of corrected equiditant in tationary co-ordinate ytem Oxy x ( α o, r, q, δ) = x o ( α o, r, q ) co( δ) y o ( α o, r, q ) in( δ) (9) y ( α o, r, q, δ) = x o ( α o, r, q ) in( δ) + y o ( α o, r, q ) co( δ) (10) where: r, e, q, m, z, z nown parameter of nominal equiditant [], r, q, m - ought parameter of corrected equiditant, α, α o, δ, e =e ought additional parameter, connected with modification, Fig.4 m, m - coefficient of hortening of epicycloid. Modification of mehing require earching of value of 9 variable: parameter of corrected equiditant r, q, m angle of revolution δ, (compenating pitch play), additional parameter, decribing poition angle α 1, α, α o1, α o of extreme point 1, 1,, on contact arc and poition angle α, α o of tangent point S, Fig.4-5 -

6 Configuration of clearance can be determined from et of ditance of two point poitioned on the equiditant and normal line to nominal equiditant, Fig.4. Unnown parameter of corrected equiditant and additional parameter of modification reulting from optimiation criteria can be calculated from the following et of equation: 1 1 ( α o1, r, q, δ) + x ( α o1, r,q, δ) y( α1) + x( α ) 0 (11) y ( α1) y ( α1) y 1 = 1 1 ( α o, r,q, δ) + x ( α o, r, q, δ) y( α ) + x( α ) 0 (1) y ( α ) y ( α ) y = 0,5 {[ x( 1) x ( α o1, r, q, δ) ] + [ y( α1) y ( α o1, r, q, δ) ] } = r 0,5 {[ x( ) x ( α, r,q, δ) ] + [ y( α ) y ( α, r,q, δ) ] } = r α (13) α (14) o o x( α ) = x ( α, r, q, δ) (15) o y( α ) = y ( α, r, q, δ) (16) o y( α x( α ) y = ) x ( α ( α o o, r, q, r,q, δ), δ) (17) y( α ) y o ( α o, r, q ) arc tg arc tg = δ (18) x( α ) x ( α, r, q ) o o 0,5 0, 5 [ x( ) + y( α ) ] = [ x ( α, r, q, δ) + y ( α, r, q, δ) ] o α (19) o 1 + m 1 (q 1 1 m e z q) + m = r (0) α α α pc 1 = ε (1) Equation (11), (1) are for normal line traniting through point 1, 1 and,. Equation (13), (14) repreent ditance of equiditant between thee point. Equation (15)-(19) decribe condition of tangency of equiditant in baic ytem Oxy. Equation (0) reult from including corrected equiditant in nominal one and (1) connect ought angle α 1, α with coefficient of contact ε. Sytem of equation (11) : (1) i olved by iterative method of Levenberg-Marguardt which i a variation of gradient method in MathCAD 6.0 oftware. Preciion of calculation (error vector) wa et on level A the reult of olving ytem of equation for aumed value ε j we can get 9 ought value, which can be component of vector of mehing modification: z1j ( α1) j z j ( α o1) j ] j ( ε j) = z 3j = ( α ) j for j = 1,,..., m () M M δ z 9 j - 6 -

7 For m-criterial value of vector of contact ratio: ε = [0,4; 0,7; 0,30; 0,33; 0,36; 0,39; 0,4; 0,45], we get matrix of modification (matrix of poible olution): $ = [] 1 (ε 1 ), ] (ε ),..., ] m (ε m )] (3) Exemplary matrix A for cycloidal gear with ratio L ] =19 i hown in Table 1. Element of matrix ha been calculated for the following nominal equiditant: r = 96mm; e = 3mm, q = 8,5mm; z = z +1 = 0; m = 0,65; α pc = 18, and criteria of optimiation: r = 0,01m; max r io = 0,03mm; δ min = 0,05 Table 1:Matrix of optimal olution for cycloidal gear with ratio i = 19: &RHIILFLHQWRIFRQWDFWε M ε 1 = ε = ε 3 = ε 4 = ε 5 = ε 6 = ε 7 = ε 8 = 1XPEHURIWHHWKLQFRQWDFW 3, , , , , , , , $ = 5, , , , , ,4474 9, , , , , ,6101 3,044094, , , , , ,8635 8, , ,63141,5988, , , , , , , , ,63744,147577, , , , , , , , , , , , , ,9969 6, ,108 8, , , , , , , , , , , , , , , , , , , ,0647 8, , ,64594 α 1 [ ] α RN [ ] α [ ] α RN [ ] α V [ ] α RNV [ ] UN>PP@ TN>PP@ δ [ ] P>@ 3DUDPHWHUVRIPRGLILFDWLRQDQG FRUUHFWLRQRIPHVKLQJ 'LVWULEXWLRQRIORDGVDFWLQJRQSODQHWZKHHOV Dependence between torque in CPG i [,6,7 ] : M1 = M c = M h i η (4) M h = R e co α R (5) M1 M + M h = 0 (6) where: M 1 = M c torque, ariing in planet wheel 1 i 1, M torque giving load on interacting central wheel, M h input torque (driving) on eccentric haft (yoe haft), i, η - inematic ratio and efficiency of gear; η 1, R eccentric reaction force, α R incline angle of force R, e eccentric of gear, e = O a O b

8 Torque M c arie in planet wheel a a conequence of loading the gear of drive torque M h and can be calculated from that wheel auming that load i ditributed equally on both wheel. Relation (7) reult from balance of moment in planetary gear with poitive bae ratio i 0 >0 acc. to [ 7 ]) and CPG i that ind of gear. So yoe haft i differential haft and central wheel haft i aggregating haft. Torque acting on planet wheel produce in CPG three unnown load ditribution, a follow: load ditribution in mehing,ditribution of force P i between teeth, load ditribution (Q j ),acting on bolt of traight-line mechanim load ditribution of eccentric R on Q ri loading roller element (roll) in bearing hole. Fig.5 Sytem of force, train and moment and rule of balancing force acting on planet wheel Figure 5 how how to balance the force acting on the planet wheel 1 or 1. Force between teeth P i and force Q j are function of diplacement δ i and δ j which arie in point of application of force. And force Q ri depending on reolving of force R are the function of geometrical feature of roller bearing and mainly depend on radial clearance. [3]. To calculate force between teeth P i and reaction force Q j there i applied analytical method, maing ue of implifying aumption [ 5,6,7 ]. $QDO\WLFDOPHWKRGIRUGHWHUPLQLQJORDGVLQF\FORLGDOJHDU Following aumption in analytical method ha been done, Fig.5 : load are ditributed equally on both planet wheel and each tranmit torque M c being half value of output moment M 1 load i tranmitted only by one (active) ide of planet wheel and direction of mehing force action mae pencil of line with common tarting point in roller point of mehing O, ; - 8 -

9 diplacement δ i in place of acting of mehing force P i reult from light angular diplacement β of planet wheel a rigid plate coming from hift of roll of tationary wheel and local mutual train of roll and teeth; load from planet wheel i tranmitted onto one (active) ide of traight line mechanim and direction of action of force Q j are parallel toward line O a O b (of eccentric) ; diplacement δ i in point of action of force Q j reult from light angular diplacement φ of traight line mechanim di and are created by deflection of bolt and mutual train of bolt and hole of planet wheel ; eccentric reaction force R i a concentrated force and i ditributed into component Q ri and reult from the value of input torque M h and condition of equilibrium. Method of calculation of mehing force P i and bolt reaction force Q j analytically i decribed in reference [ 5, 6 ]. 0HWKRGRIFDOFXODWLRQVDQGDSSOLHGQXPHULFDOPRGHOV In ADAMS ha been created model of cycloidal planetary gear (Cyclo) with planet wheel and cooperating element (bolt and roll). The geometry of interacting element i imple (circle with different radiu) and it i no problem to create the geometrical model of them. The more complicated i the hape of external edge of the planet wheel. It i decribed by parametric equation of equiditant [, 6 ] : x y ee ee = r coα + e co(z α) q co( α + γ) (7) = r in α + e in(z α) q in( α + γ) where: α generating angle of equiditant, r, e, q, ] parameter of mehing, Fig.; γ overtaing angle, depending on coefficient of hortening equiditant m and gear ratio. In the ADAMS model of Cyclo, wa implemented two different ind of equiditant with nominal and corrected parameter a profile of teeth of planetary wheel. The profile of teeth wa implemented a generation of the curve point (equiditant) on the bai of parametric equation (5,6) with given tolerance every 0,05 0 with parameter of nominal equiditant: r = 96,0mm; e = 3mm, q = 8,5mm; m = 0,65; z = z +1 = 0, ; α pc = 18, and corrected equiditant : given in Table 1 for ε=0.7, 0.39 and 0.45 Value of force in both of method ha been calculated for given ize of cycloidal gear a in Fig.5 with ratio i = 19, power N = 6.4[W] and rotational peed n h = 750 [rpm]. For thi gear M 1 = M c = 880 [Nm], and force R = 10,3 [N]. Reult from analytical calculation wa precie preented in [,3]. The max value of Pi=1711 [N] and Qi = 3776 [N] In numerical method, the calculated force are in max level about for Pi=00[N], and Qi=3800[N] (Fig.7). (thi i only an example for modificated equiditant with parameter where ε=0.7 (Table 1)) - 9 -

10 Fig.6. Model of Cycloidal Planetary Gear (CYCLO) built in ADAMS Fig.7. An example of nonfiltered trace of mehing force between planetary wheel and tationary wheel and reaction between bolt and planetary wheel. ( ε = 0.7 )

11 &RQFOXVLRQV Preented reult concern cycloidal gear with nominal and modificated mehing with tranmiion ratio i =19. It hould be alo executed calculation of entire range of applied ratio i =11:89 and baing on them to try to generalie ditribution of force of mehing force P i and reaction force Q j The baic apect of modification of mehing in cycloidal planetary gear i diverification of equiditant. Matrix of modification A (Table 1) preent example field of olution of inide cycloidal mehing for aumed criteria of optimiation. Matrix A enable viualiation of clearance { r i } aigned to individual teeth that enable and mae eaier election of efficient parameter of correction of planet wheel. Calculated parameter of corrected equiditant r, e, q, m can be ued while producing toothing that will aure co-operation from 50 : 90% of teeth of planet wheel in it active part of mehing. Ditribution of mehing force P i and reaction force Q j calculated in ADAMS have a little different trace comparing it with ditribution determined analytically. It reult from omitting in analytical method inertion of moving part of gear. Thi model wa modeled in ADAMS a a rigid body, o to get more detail about behavior of thi ind of gear, it hould be alo modeled a a flexible bodie. To mae thi reult more authentic, alo wa built a real model of CYCLO gear which i actuall in teting tage and thi wor will be continue and develop. 5HIHUHQFHV [1] Chmurawa M., John A., Koot G.: The influence of numerical model on ditribution of load and tre in cycloidal planetary gear. 4 th International Scientific Colloquium Cax Technique, Bielefeld Sept. 1999, Germany. [] Chmurawa M.: Ditribution of load in cycloidal planetary gear. International Conference Mechanic 99, Kauna Univerity, 8 9 April 1999, Lithuania. [3] Chmurawa M., Warda B.: Wyznaczanie rozladow obciazenia na czeci toczne w lozyu atelity przeladni planetarnej cyloidalnej. Praca nauowa (niepubliowana) Pol. Sl. Intytut Tranportu, Katowice [4] Hamera K.: Da Cyclogetriebe eine geniale Idee und ihre techniche Verwirlichung. Techni Heute. Verlag Chritiani, nr 6, Bonn, Juni [5] Kudriavcev V.N.: Planetarnyje peredaci. Mainotroenije, Mova Leningrad [6] Lehmann M.: Berechnung und Meung der Krafte in einen Zyloiden Kurvencheiben Getriebe. Diertation.Techniche Univerität. München [7] Lehmann M.: Sonderformen der Zyloidenverzahnung. Kontrution 31 nr 11, [8] Müller L.: Przeladnie obiegowe. PWN.Warzawa KLVODERXUKDVEHHQPDGHLQWKHIUDPHVRI6WDWH&RPPLWWHHIRU6FLHQWLILF5HVHDUFK.%1SURMHFW1R7&