Kinetostatic and mechanical efficiency studies on cam-controlled planetary gear trains (Part I) Theoretical analysis

Transcription

1 Indian Journal o Engeerg & Materials Sciences Vol. 0, June 013, pp Ketostatic mechanical eiciency studies on cam-controlled planetary gear tras (Part I) Theoretical analysis Wen-Hsiang Hsieh* Department o Automation Engeerg, National Formosa University Huwei, Yunl 63, Taiwan, R. O. C. Received 13 April 01; accepted 1 February 013 Cam-controlled planetary gear tras (CCPGTs) are planetary gear tras with cam pairs. One o its ma advantages is that it can produce a wide range o non-uniorm output motion. The vestigations on the ketostatic mechanical eiciencies o CCPGTs are perormed. The purpose o the work is to propose a systematic approach o ketostatic mechanical analyses or CCPGTs, it aims at conductg theoretical analysis to obta all the equations or the related analyses the succeedg design. First, kematic analysis is perormed, kematic equations are derived. Then, ketostatic analysis is conducted, the orced actg on each member the equations o riction orces couples are derived, the solvg procedure is also presented. In addition, shakg orces moments, mechanical eiciency, required put power are obtaed. Keywords: Cam-controlled, Planetary gear tra, Ketostatic, Mechanical eiciency, Variable speed mechanism Rothbart 1 illustrated three CCPGTs, called them epicyclic gears movg (or ixed) cam, one o them is shown Fig. 1. Chironis has also discussed a CCPGT his work. Although they have dierent names, both o them are structurally identical. Figures 1 show, respectively, the exploded view the structural sketch, it consists o a cam groove (the rame, member 1), an arm (the put, member ), a planetary gear (member 3), a sun gear (the output, member 4). In general, the planetary arm rotates at constant speed, drives the planetary gear to revolve around the sun gear to sp around itsel simultaneously. At the same time, the planetary gear produces an oscillatory motion through the contact o the attached roller the cam groove. Thereore, the sun gear can produce a non-uniorm motion by engagg with the planetary gear. The ma advantage is that it can produce a wide range o non-uniorm output motion. In addition, it has the advantages o higher reliability, lower cost, aster response, higher power transmission due to its mechanical nature. It is now at work ilm drives, e.g., the eedg o paper sheet an automatic die cuttg creasg mache, shown Fig. 3, etc. However, the design the analysis o CCPGTs are rarely ound the literature. Especially, the topic on * allen@nu.edu.tw how to crease their perormance has hardly been vestigated earlier. Although Rothbart 1 Chironis illustrated their books, but the analysis design or CCPGTs are not addressed. Hsieh et al. 3-6 itiated a series o vestigation on CCPGTs, cludg kematic experimental studies 3, structural synthesis 4, kematic synthesis 5, modelg control 6. The easibility o employg CCPGTs as a drivg mechanism or variable speed mechanisms is veriied, the Fig. 1 Exploded view

2 19 INDIAN J. ENG. MATER. SCI., JUNE 013 systematic approaches or kematic design analysis are proposed by these studies. As ar as the research on ketostatic analysis are concerned, Chiou Chen 7 proposed a theoretical ketostatic analysis on variable lead screw mechanisms. Sheu et al. 8 presented a ketostatic analysis on a roller drive, its results show that the roller drive is o high eiciency. A considerable amount o research 9-1 had been done on the ketostatic study o parallel maches robots. Yong Lu 13 reported the ketostatic model o a lexure-based 3-RRR compliant micro-motion. Norton 14,15 vestigates various manuacturg methods on the idelity o the theoretical acceleration by experimental studies, the results show that some o the manuacturg techniques are signiicantly better than the others. Although, the ketostatic dynamic analysis design are crucial when a mechanism is applied high speed transmission, there is hardly any study on CCPGTs has been reported the literature. Thereore, this topic needs to be vestigated. In this study, a systematic approach o ketostatic mechanical analyses or CCPGTs has been proposed. This study aims to conduct a theoretical analysis proposed a design. The kematic analysis is perormed kematic equations are derived. A theoretical model o ketostatic analysis is ormulated based on Newton s second law o motion, the numerical procedure or solvg the nonlear equation is also presented. Kematic Analysis The kematic analysis o the CCPGT shown Figs 1 was carried out elsewhere 3. For the readers convenience the consistency o symbols, the analysis is repeated with some modiications here, extended to d the velocity o signiicant pots. The geometry o a CCPGT is shown Fig. 4, where O 4, O 3, O are the centers o the sun gear, the planetary gear, the roller, respectively. By applyg the Cose law to the triangle O 4 O 3 O, ormed by the three centers, we have r = c + lr + c lr cosδ... (1) Fig. Structural sketch Fig. 3 Application paper eedg Fig. 4 CCPGT geometry

3 HSIEH: CAM-CONTROLLED PLANETARY GEAR TRAINS THEORETICAL ANALYSIS 193 lr = c + r - c r cosλ... () where δ = O 4 O 3 O ( ABC), λ= O 3 O 4 O, lr = O3O (the length o the rocker). Moreover, the center distance between the sun gear the planetary gear is c m( T3 + T4 ) =... (3) where T 3 T 4 are, respectively, the number o teeth or the planetary gear the sun gear, m is the modulus o both gears. Also, the position vector o the roller center is r = O4O = O4O3 + O3O... (4) In addition, the length sum o two sides a triangle must be greater than the length o the other. That is lr + c > r... (5) I lr + c = r... (6) Then an uncerta state exists such that the rocker will possess two dierent paths. The uncerta state o a mechanism is called the change pot, the coniguration is called the uncertaty coniguration. The CCPGT is then called a change pot mechanism, it should be avoided mechanism design general. Furthermore, θr = θ + λ... (7) where θ r λ are, respectively, the angular positions o r the arm relative to the positive x-axis. Moreover, θ is the angular position (or angular displacement) o the arm. All the above angles are positive i counterclockwise. A CCPGT with an angular displacement θ is shown Fig. 5. Initially, the arm is position 0, the two pitch circles o the sun the planet gears are meshg at pot P. Let the arm rotate an angle θ about O 4, then it will drive both the sun gear the planetary gear to position. Sce the kematic relation o a CCPGT is lear, this displacement can be resolved to two displacements based on the theorem o superposition. First, let the sun gear the planetary gear be ixed to the arm, the constrat between the cam the roller be temporarily cancelled. Then i the arm turns around O 4 by an angle θ, all the members except the cam will move to position 1. Furthermore, let the arm be ixed, both the sun gear the planetary gear reely rotate around their axle. Sce there is a meshg between the roller the cam real situation, it will orce the planetary gear to sp around O 3 through an angle η move to position, then the contactg pot P belongg to the planetary gear, position 0, will move to P 3. At the same time, the sun gear will sp about O 4 by an angle ξ due to the meshg with the planetary gear, the contactg pot P belongg to the sun gear will move to P 4. Addg these two displacements yields θ3 = θ + η... (8) θ4 = θ + ξ... (9) δ = δ 0 + η... (10) where θ 3 θ 4 are the angular displacements o the planetary gear the sun gear, respectively, η (negative clockwise) is the spng angular displacement o the planetary gear produced by the meshg between the roller the cam, δ 0 denotes δ position 0. Sce θ 4 is measured rom the positive x-axis, it is also the angular position o the sun gear. From Fig. 5, the angular position o the planetary gear is θ3 = π δ 0 + θ3 (11) Fig. 5 CCPGT displacement

4 194 INDIAN J. ENG. MATER. SCI., JUNE 013 Moreover, ξ can be obtaed by T 3 T4 ξ = η... (1) Substitutg Eq. (1) to Eq. (9), we have T3 4 T4 θ = θ η... (13) Also, rearrangg Eq. (8) yields η = θ3 θ... (14) then substitutg Eq. (14) to Eq. (13), the equation o angular displacement between the sun gear, the arm, the planetary gear can be derived as: T3 4 T 3 θ = θ ( θ θ )... (15) 4 By dierentiatg Eq. (14) Eq. (15) with respect to time, respectively, we have the equations o angular velocity 3 dη dθ ω = ω (1 + )... (16) T3 4 T 3 ω = ω ( ω ω )... (17) 4 where ω 4, ω, ω 3 are the angular velocities o the sun gear, the arm, as well as the planetary gear. By substitutg Eq. (16) to Eq. (17), it yields T3 dη 4 ω T4 dθ ω = (1 )... (18) I ω, T 3, T 4, dη/dθ are speciied, then ω 4 ω 3 can be ound rom Eq. (18) Eq. (16). Dierentiatg Eq. (16) Eq. (18) with respect to time, respectively, the equations o angular acceleration are d η dη 3 (1 ) dθ dθ α = ω + + α... (19) T 3 dη T3 d η 4 (1 ) T4 dθ T4 dθ α = α ω... (0) where α 4, α, α 3 are, respectively, the angular accelerations o the sun gear, the arm, the planetary gear, Similarly, α 4 α 3 can be ound, respectively, rom Eq. (19) Eq. (0) i α d η/dθ have been additionally speciied. In general, the put speed is uniorm, thereore α is zero. To simpliy the derivation, the contactg pots (p 3 p 4 ) between the planetary gear the sun gear are assumed to be at the pitch pot o two gears. The velocities o contactg pots between the planetary gear the sun gear can then be calculated by v 3 = v + ω 3 O P... (1) p O v = ω 4 O P p where v = ω O O O ()... (3) By deition, the slidg velocity V s between the contactg pots is v = v v s p4 p3... (4) Substitutg Eqs (1) () to Eq. (4), it yields v = ω O P ω O O ω O P... (5) s where ω3 = ω3k ω4 = ω4k O P = O P (cosθ i + s θ j) O O = O O (cosθ i + s θ j) O P = O P [cos( θ + π ) i + s( θ + π ) j] (6)... (7)... (8)... (9)... (30) In addition, ω 3 ω 4 can be computed rom Eq. (16) Eq. (18), respectively. Fally, Let G i (i = -4) denote the mass center o member i, then the accelerations o center o mass a Gi or each member can be ound by a = ω ω O G + α O G... (31) ( ) G 4 4 ag 3 = ab + ag 3/ B = ω ω α O4O3 + ω ω + α O G ( O O ) ( O G ) (3)

5 HSIEH: CAM-CONTROLLED PLANETARY GEAR TRAINS THEORETICAL ANALYSIS 195 a G4 = 0 where α = α k α = α k 3 3 α = α 4 4 k O G = O G (cosθ i + s θ j) 4 4 O G = O G (cosθ i + s θ j) (33)... (34)... (35)... (36)... (37)... (38) Ketostatic Analysis Ketostatic analysis exames the behavior o a mechanism under the action o the applied static orces the ertial orces produced by motion. It aims at dg the reaction orces, riction orces, riction torques between each member, can be used or computg riction power, mechanical eiciency, shakg orces moments, the required put power under various loadgs, etc. The ketostatic model o CCPGT considerg riction orces developed this section is based on Newton s second law o motion. The material properties o each part are assumed to be homogeneous, hence the center o mass will cocide with the center o geometry. Moreover, the normal orce F nij the riction orce actg on each revolute jot can be equivalently expressed as a resultant orce F ij passg through the p center a couple t ij 16, where subscript ij denotes the orce or the couple exerted by member i to member j, as shown Fig. 6, that is, t = µ r F D... (39) ij ij j ij ij F ij makg with the positive x-axis. In addition, g is the gravitational constant, m, is the mass o member i. The ree body diagrams their equations o motion or each member are irstly drawn ormulated. Then, the riction orces torques between each member the procedure or solvg the system o nonlear equation are presented. Fally, the equations or determg mechanical eiciency, shakg orce shakg moment, peak put power are derived. Forces actg on the arm Figure 7 depicts the ree body diagram o the arm, which an put torque Q is applied to the arm, O O 3 are the centers o the revolute jots a b, l (or c) l G are the distances o the O to O 3 O to G, respectively. By applyg Newton s second law o motion, it yields Σ Fx = mag x F1 x F3 x = mag x +... (41) Σ F = y mag y F1 y F3 y mg = mag y +... (4) Where µ ij is the coeicient o ketic riction between i j, r j is the p radius o member j, D ij is a direction dicator that can be determed by D ij 1, i ωij is counterclockwise = + 1, i ωij is clockwise... (40) Similarly, the orces between a cam jot with a roller ollower can be ormulated as those o a revolute jot, except µ ij is replaced by a coeicient o rollg riction. In what ollows, ij is the rictional orce exerted by member i to member j, φ ij is the direction o the Fig. 6 Forces actg on a revolute jot

6 196 INDIAN J. ENG. MATER. SCI., JUNE 013 Σ M = I α Q + F l cosθ F l sθ O O 3 y 3x m gl cosθ + t + t = I α... (43) G 1 3 O where a Gx a Gy are the x y components o the acceleration o mass center G, respectively, IO is the moment o ertia o the arm with respect to O. Forces actg on the planetary gear Figure 8 shows the ree body diagram o the planetary gear, where O is the center o roller, l 3 l G3 are the distances o the O 3 to O O 3 to G 3, respectively. Sce the spng o the roller is a kd o redundant degree o reedom, its motion reaction torque cannot be determed by the kematic dimensions, thereore it is assumed that the roller is attached to the planetary gear, the equivalent coeicient o rollg riction (µ 13 ) is used when computg the riction orce acted on it. Similarly, by applyg Newton s second law o motion, we have Σ Fx = m3a π G3x F3 x + F43cosφ cos( φ43 ) + F cosφ = m a G x (44) Fig. 7 Arm Σ Fy = m3a π G3y F3 y + F43 sφ s( φ43 ) + F13 sφ 13 m3g = m3a G 3 y (45) Fig. 8 Planetary gear Σ M = I α + O G ma O3 G G3 F R cosφ + R sφ F l cosφ sθ 43 3 p 43 3 p F l sφ cosθ m gl cosθ t + t G = I α + m a l sθ + m a l cosθ G G 3 x G G 3 y G (46) where a G3x a G3y are the x y components o the acceleration o mass center G 3, respectively, I G3 is the moment o ertia o the planetary gear with respect to G 3. In addition, φ p is the pressure angle o the planetary gear the sun gear, R 3 is the radius o pitch circle o the planetary gear. φ 43 can be ound as φ 43 = θ + φ p 3 π (47) Forces actg on the sun gear Figure 9 depicts the ree body diagram o the arm, which an output torque Q out is applied to the sun gear. G 4 (or O 4 ) is the centers o the revolute jots d, l (or c) l G are the distances o the O to O 3 O to G, respectively. θ is the angular displacement o the arm. By applyg Newton s second law o motion, it yields Σ F = m a x 4 G4 x π F43cos( + θ + φ p ) + F14 x = m4a... (48) G4x

7 HSIEH: CAM-CONTROLLED PLANETARY GEAR TRAINS THEORETICAL ANALYSIS 197 E 43 = v v v v (56) Similarly, D ij shown Eqs. (51)-(54) can be expressed as ω ω i j D ij =... (57) ω ω i j Σ F = m a y 4 G4 y Σ M = I α F R cosφ π F s( θ φ ) F m g m a =... (49) 43 p 14 y 4 4 G4 y G 4 G p Q R sφ + t = I α... (50) out 43 4 p 14 G4 4 where a G4x a G4y are the x y components o the acceleration o mass center G 3, respectively, I G4 is the moment o ertia o the planetary gear with respect to G 4, R 4 is the radius o pitch circle o the sun gear. Friction orces couples Based on Eq. (39), the riction couples t 1, t 3, t 1, t 14 are t = µ r F D... (51) t = µ r F D... (5) t µ 13r F13 D13 =... (53) t = µ... (54) 14 14r4 F14 D14 where µ 13 is the coeicient o rollg coeicient, O is the radius o roller. And the riction orce o relative velocity between the two gears can be ormulated at the same way by = µ... (55) F43 E43 Fig. 9 Sun gear Where E 43 is the directional dicators o the relative velocity between the contactg pots o the two gears, can be determed by Procedure or solvg the nonlear equations I Q out is given, then Eqs (41)-(55) orm a nonlear system o equations with 14 unknowns, cludg F 1x, F 1y, F 3x, F 3y, F 43, F 13, F 14x, F 14y, 43, t 1, t 3, t 13, t 14, Q. Sce the nonlear system cannot be solved easily, the ollowg iteration approach shown is employed or solvg the system: (i) The riction orce 34 the couples t ij are assumed to be zero, then the nonlear system o 14 equations is reduced to a lear system o ne equations with ne unknowns. (ii) Solve the lear system o equations, by Gaussian elimation approach, or the unknown orces F 1x, F 1y, F 3x, F 3y, F 43, F 13, F 14x, F 14y, the put torque Q. (iii) Substitute the orces obtaed Step (ii) to Eqs (51)-(55), solvg or the riction couples t 1, t 3, t 13, t 14, 34. (iv) Substitute the riction orces couples, computed Steps (ii) (iii), to Eqs (41)-(50), new solutions o F 1x, F 1y, F 3x, F 3y, F 43, F 13, F 14x, F 14y, Q are ound. Moreover, by substitutg the results to Eqs (51)-(55), new solutions or the riction orces the riction couples are obtaed. (v) Repeat Steps (ii)-(iv) until the dierences o the riction orces couples between two consecutive iterations are less than the speciied tolerances. Mechanical eiciency Based on the prciple o energy conservation, the net put work is equal to the sum o changes o ketic energy potential energy, that is, W ( W + W ) = T + U... (58) out where W, W out, W are the works o put, output riction, respectively; T U are the net changes o ketic energy potential energy stored the mechanism, respectively. Dierentiatg Eq. (58) with respect to time yields

8 198 INDIAN J. ENG. MATER. SCI., JUNE 013 P P + P ) = + (... (59) out dt dt du dt where P, P out P are the powers o put, output, riction, respectively; dt/dt du/dt are the rate o changes o stored ketic energy potential energy, respectively. I dt/dt du/dt are neglected here, then we have P out = P P... (60) Moreover, P, P can be expressed as P = t ω + t ω + V t ω + t ω (61) P Q ω... (6) = By deition, the mechanical eiciency is P = 1 p η... (63) Substitutg Eqs (61) (6) to Eq. (63), it yields t ω + t ω + v η = 1 Q ω t ω + t ω Q ω... (64) Shakg orces moments To atta higher dynamic eiciency longer atigue lie, shakg orces shakg moments have to be mimized. Shakg orces or shakg moments are the unbalanced orces or moments transmitted by a mechanism to its rame or base. By deition, the shakg orce is the resultant o the ertia orces, neglectg the gravity, that is, F = ΣmiaGi... (65) The shakg moment is the resultant moment o the ertia orces with respect to a reerence pot, that is, M = Σ ( r m a + I α )... (66) Gi i Gi Gi i where r Gi is the vector rom the reerence pot to the mass center G i. Peak put power required The peak put power is used to determe the required power o the put motor. By deition, it is the maximum required power put among a revolution, that is, P = Max{ Q ( θ ) ω ( θ ), peak or 0 θ π}... (67) Conclusions In this study, a systematic approach or ketostatic mechanical eiciency analyses has been vestigated. Kematic equations or each part have been derived. Moreover, the equations o the orces actg on each part, riction orces couples are obtaed, their solvg procedure has also been proposed. In addition, mechanical eiciency power analyses have been carried out. The above derived equations will be dispensable or the succeedg design o CCPGTs. Acknowledgement The ancial support o the National Science Council, Republic o Cha (Taiwan), under Grants NSC 91-1-E NSC 9-1-E is grateully acknowledged. Reerences 1 Rothbart H A, Cams: Design, Dynamics Accuracy, (Wiley, New York), Chironis N P, Mechanisms & Mechanical Devices Sourcebook, (McGraw-Hill, New York), Hsieh W H, Mech Mach Theory, 4 (5) (007) Hsieh W H & Chen S J, Int J Eng Technol Innovat, 1 (1) (011) Hsieh W H, Mech Mach Theory, 44 (5) (009) Hsieh W H & Lee I C, Int J Model, Identi Control, 1 (3) (011) Chiou S T & Chen F Y, Math Comput Model, 7 (1) (1998) Sheu K B, Chien C W, Chiou S T & Lai T S, Mech Mach Theory, 39 (8) (004) Khan W A & Angeles J, ASME J Mech Des, 18 (1) (006) Jun S K, White G D & Krovi V, ASME J Dyn Syst Meas Control, 18 (1) (006) Xi F, Zhang D, Mecheske C M & Lang S Y T, Mech Mach Theory, 39 (4) (004) Zhang D, et al., Robot Comput-Integrated Manu, 5 (4-5) (009) Yong Y K & Lu T F, Mech Mach Theory, 44 (6) (009) Norton R L, Mech Mach Theory, 3 (3) (1988) Norton R L, Mech Mach Theory, 3 (3) (1988) Hall A S, Notes on mechanism analysis, (Balt, Laayette, Ind.), 1981.