Strojniški vestnik - Journal of Mechanical Engineering 60(2014)9, 561-570 © 2014 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2013.1574 Original Scientific Paper Received for review: 2013-11-27 Received revised form: 2014-02-19 Accepted for publication: 2014-03-31 Position-Parameter Selection Criterion for a Helix-Curve Meshing-Wheel Mechanism Based on Sliding Rates Jiang Ding - Yangzhi Chen* - Yueling Lv - Changhui Song South China University of Technology, China The space curve meshing wheel (SCMW) is an innovative gear mechanism mediating transmission between space curves instead of classic space surfaces; the most common type of SCMW is the helix curve meshing wheel (HCMW). In this study, we propose a position-parameter selection criterion of the HCMW based on its slide rates. The sliding rates of the contact curves at the meshing point are defined and calculated; the optimal meshing condition of the HCMW is attained by analyzing the features of the slide rates; the position-parameter selection criterion is subsequently attained as well as corresponding contact curve equations. Both simulation and practical examples using different positionparameters are provided to verify the transmission continuity, and their slide rates are calculated. The calculating result shows that the HCMW coincident with the position-parameter selection criterion has better slide rates and, therefore, can have better tribological performance. The design method proposed in this paper aims to change the current situation so that the position-parameters of the HCMW are determined according to designers' experience, and theoretically provides a foundation for its standardized production in industry. Keywords: gear, space curve meshing wheel, helix curve meshing wheel, position-parameter, parameter selection, sliding rate 0 INTRODUCTION Closely related to the friction and wear performances of meshing gear teeth, the slide rate is an important index for estimating the transmission quality of a gear pair [1]. To reduce wear and to prevent the scuffing of the gears, the slide rates of the two surfaces should be close and minimized in many situations. For example, in precision instruments, cycloid gears with the same slide rates at every meshing point are more commonly used than involute gears, which have different slide rates at different meshing points. As the traditional gear pairs mediate transmission between two conjugate surfaces [2] to [4], the current research is mainly about the slide rates between two surfaces [5] and [6]. Recently, the Space Curve Meshing Wheel (SCMW) was proposed, based on the space curve meshing theory [7] to [10], instead of the classic space surface meshing theory [2]. Mediating transmission between the contact curves on the surfaces of the driving and driven tines, the SCMW possessed the advantages of small size, a large transmission ratio, and high design flexibility. Since its invention, progress has been attained in many aspects, including meshing equations [7] to [11], design criteria [12], contact ratio [13], bending stress [14], manufacturing technology [15] and [16], and practical application [17] to [21]. At present, the most common SCMW is the Helix Curve Meshing Wheel (HCMW), as shown in Fig. 1. The modification coefficients of the transitional gears are important factors affecting the slide rates of the conjugate surfaces; similarly, the position- parameters (a and b in Fig. 1, further details in section 1.2) of the HCMW influence the meshing radii and, subsequently, the slide rates of the contact curves. However, the position-parameters are currently selected according to designers' experience. If they are not selected properly, the slide rates may be so large that the tines will wear out quickly. a) b) Fig. 1. Helix Curve Meshing Wheel; a) non-vertical case and b) vertical case In this paper, a position-parameter selection criterion for the HCMW based on its slide rates *Corr. Author's Address: South China University of Technology, School of Mechanical and Automotive Engineering, Guangzhou, China, meyzchen@scut.edu.cn 561 Strojniski vestnik - Journal of Mechanical Engineering 60(2014)9, 561-570 is proposed. It should be noted that the positionparameters also affect other parameters, such as the maximum radius or the height of the HCMW pair, but their effects on the meshing radii and the subsequent slide rates are direct and primary. Additionally, the space curve meshing skew gear mechanism (SCMSGM) in [11] is the latest SCMW. However, its geometric conditions remain under research, and its slide rates will be studied afterward. According to its current application [17] to [21], this study mainly focuses on the HCMW under operation conditions of self-lubrication and dry friction. In addition, the relative slide speed between the driving and driven tines should be preserved to form a hydrodynamic film with lubricant to reduce the wear under operation conditions. The definitions of the slide rates are still available in that occasion, while the position-parameter selection criterion is slightly different. 1 SLIDE RATES 1.1 Slide Rates between Two Conjugate Space Curves The working part of the SCMW is a pair of conjugate space curves named "contact curves". As shown in Fig. 2, suppose that the contact curves, which are denoted as r : rl = rj(1) (t) and r2: r2 = r2(2) (t), mesh at point M at the given start moment. After a period of At, point Mj on the curve r meshes with point M2 on the curve r2. The corresponding arc lengths are denoted as MM! — 51 and MM2 — S2, while the corresponding chord lengths are | MMl | = |Ar 1| and Therefore, I MM2 I = |Ar2|. lim s2 = lim |Ar Aí^ü 2 Aí^ü1 lim s. = lim Ar, At 1 At ^Q1 11 and = lim — S Ar (2)|-|Ar (i) I 1 = lim1- At^0 Ar (2) (2) The slide directions of r and r2 can be attained from either Eqs. (1) or (2). Take Eq. (1) for example: when a1 > 0, i.e. s1 > s2, the relative slide direction of r2 comparing with r1 is from M to M2 and consistent with the moving direction of the meshing point; when a1 < 0, i.e., s1 < s2, the relative slide direction of r2 comparing with r1 is from M2 to M and reverse with the moving direction of the meshing point. 1.2 Slide Rates of the SCMW Contact Curves As the SCMW is mainly designed to operate in conditions with low loads, the elastic deformation [12] has insignificant impact on the slide rates, and the contact curve equations of the SCMW are used for analysis. The contact curve equations are attained in the space curve meshing coordinates, as shown in Fig. 1. The coordinate o1 - x1y1z1 is stationary with respect to the driving wheel, and the coordinate o2 - x2y2z2 to the driven wheel. In a non-vertical case, the distance from o2 to x1 is denoted as a; in a vertical case, the distance from o2 to z1 as b. As shown in Fig. 2, the driving and driven contact curves of the SCMW are denoted as r/M xM (2) M yM r (2) _ -[ * (2) yM 7(1)]T M j z(2)T in o1 - and M j in o2 - X2y2Z2. Their differential equations are as in Eqs. (3) and (4): dÁ (i) dt .'(i) y '« y m .'(i)" (3) Fig. 2. Slide rates of conjugate space curves The slide rates of space curves are defined as the limit value of the ratio of the lengths difference between two relative arcs divided by the length of the given arc: a, = lim —— — 1 —1^0 s 2 = lim Ar, - Ar2 I ArJ (1) d2r2(2) _r „ ,(2) dt m lArJ y '(2) y m ,'(2)T ' M J lim diri( (i) dt d2 r2 (2) dt d1 r1 (1)1 d2 r2 (2) (4) (5) From Eqs. (1) and (3) to (5), the slide rate of the driving contact curve is as in Eq. (6): a, = 1 - (ff)2 +(y M})2 + ( M})2 +(y +( M)2 (6) From Eqs. (2) to (5), the slide rate of the driven contact curve is as in Eq. (7): 562 Ding, J. - Chen, Y.Z. - Lv, Y.L. - Song, C.H. Strojniski vestnik - Journal of Mechanical Engineering 60(2014)9, 561-570 a2 = 1 - (x 'S) + (y ^)2 + (z x M})2 + (y f)2 + ( M>)2 (7) 1.3 Slide Rates of the HCMW Contact Curves The driving contact curve of the HCMW in ol - xly1z1 is circular helix curve, and its equation is as below [9]: xM = m1 cost y M = misin t (s ^ t ^ tE ), (8) zM = nn + nt where m1 is the helix radius, m1 > 0; n is the pitch coefficient, equaling the ratio of the pitch and 2n [9], and usually n > 0 as the driving cylindrical helix is a right-hand screw; t is an independent parameter indicating the length of the contact curve; ts and tE are the starting and ending values for the meshing point, respectively. In this paper, ts=-n, so the initial value of the driving contact curve height is zM = 0, which means that the driving curve begins from the plane x1o1y1; tE=-n/2, i.e., tE-ts = n/2, so a quarter of a circle is used. The transmission ratio of the HCMW pair is denoted as /'12, and the angle between the angular velocity of the driving and driven wheels is denoted as 0, 0 < 0 < n. The driven contact curves for non-vertical and vertical cases are reflected in Eqs. (9) and (10), respectively [9]: -n(t + n)sin6 Icost + K cos 6 (2) y m = cos6 ^ =-n (t + n)cos6 - n (t + n)sin6 I sin , (9) xM = -(nt + nn -b)cos — t + n •12 . t + n yM = (nt + nn - b)sin — (10) Z(2)= 0 M Eqs. (9) and (10) are a common conical helical curve and a planar Archimedean helical curve, respectively. It can be proved from both Eqs. (9) and (l0) that the initial value of the driven contact curve height is zM = 0, and the driven curve begins from the plane x2o2y2. Differential equations of Eqs. (8) to (10) are as in Eqs. (ll) to (13): 1(1) • < x M = -m1 sin t 1(1) y M = m cos t, z ,(1) = n M '(2) _ 1 í mi -a ' ■ N • ~ ^ • í t + n ^ (11) x y _--1 —:--n (t + n)sin0 I sin i12 V cost 1 I+ '12 J ( +n sin 0 cos t + n y ,(2)_-1 í mi - a y M ~ .i „ i12 V cost - n(t + n)sin0 Icos t + n -,(12) f -n sin 0 sin z M _ -n cost t + n ,(2) nt + nn - b . t + n t + n x M =-sin--n cos- -cos- - n sin - 12 12 12 ,(2) nt + nn - b t + n . t + n y m = t h: Z '(2) = 0 M u (13) From Eqs. (6), (11) and (12), the slide rates of the HCMW contact curve for non-vertical case are as in Eqs. (14) and (15): - 1 ( m1 - a i121 cosd - n(t + ^)sm0 Vm1 2 , 2 n a2 = 1 - - 2 , 2 n (14) - (15) 1 ( m1 - a il21 cosd - n (t + ^)smd From Eqs. (7), (11) and (13), the slide rates of the HCMW contact curve for vertical case are as in Eqs. (16) and (17): 12 i12 V 112 X„ = 12 2 2 n 12 2 2 n 12 Position-Parameter Selection Criterion for a Helix-Curve Meshing-Wheel Mechanism Based on Sliding Rates 563 Strojniski vestnik - Journal of Mechanical Engineering 60(2014)9, 561-570 CT = i A ^ nt + nn - b ^ 2 -I + n \ hi ) u2 = 1- f i V nt + nn - b | 2 + n (16) (17) K v 12 2 POSITION-PARAMETER SELECTION CRITERION BASED ON SLIDE RATE 2.1 Monotonicity and Feasible Region of Sliding Rates The non-vertical HCMW is illustrated as an example. Differentials of Eqs. (14) and (15) are as in Eqs. (18) and (19), respectively: n sin9| ——a - n (t + w)sin9 cos 9 ■2 I 2" '12 V — 1 1 ( m1 - a n (t + w)sin9 (18) The same monotonicity can be attained in the vertical HCMWs. The only difference is that the meshing radius of the driven contact curves (see Eq. 30 in section 2.2) is required as: - (nt + nn - b) > 0. (25) From Eqs. (20) and (25), the maximum limit of t is denoted as tmax uniformly: m - a n cos 6 sin 6 b — n n ■n (6*n/2 ) (6 = n¡ 2 ) (26) As shown in Fig. 3, a{ ^ 0 and a{ ^ 0 if t ^ tmax. Therefore, from Eqs. (14) to (17) and (26), the maximum a1 and the minimum a2 are as in Eqs. (27) and (28) for both non-vertical and vertical cases: u2 = 1 (27) (28) n sin inOyfmî + n a, = -- m, - a , \ ■ n —1--n (t + ^)smO cosd v ' 11 m—a - n(t + ^)sinO I i12 { cos O J J -. (19) +n n > 0 is supposed in section 1.3. The meshing radius of the driven contact curves should be above zero (see Eq. 30 in Section 2.2), so: m - a ■n(t + rc)smd > 0. cosd If 6± 0 and 6± n are supposed, then: sin 6 > 0, (20) (21) 1 j m - a i12 ^ cos B - n (t + ^)sinB hi > 0, yjm1 + n2 > 0. + n2 j > 0, (22) (23) (24) From Eqs. (20) to (24), it can be concluded that a{ > 0 and a{ < 0. Therefore, in the non-vertical HCMWs, a1 is monotonically increasing while a2 is monotonically decreasing. Fig. 3. Monotonicity and feasible region of sliding rate equations It should be noted that 6 ± 0 and 6 ± n are supposed in Eqs. (21) and (22). The non-vertical HCMWs when 6 = 0 or 6=n are also called parallel-axis HCMWs. Their driving and driven contact curves are circular helix curves, and their slide rates equations are constant functions. However, the maximum values of c1 and c2 are the same as Eqs. (27) and (28). The allowable slide rate is denoted as [c], [c] > 0. According the monotonicity of c1 and c2, their feasible regions are shown in Fig. 3 and Table 1. In Fig. 3, the intersection of the two curves is defined as a pitch t = max 2 n i12 ^ cos 9 2 2 12 2 564 Ding, J. - Chen, Y.Z. - Lv, Y.L. - Song, C.H. Strojniski vestnik - Journal of Mechanical Engineering 60(2014)9, 561-570 position, i.e. t = tp . It can be concluded that if and only if t = t„, a, = a2 = 0 (29) In the following section, Eq. (29) will be used to obtain the optimal meshing condition of the HCMW. Table 1. Feasible region of sliding Rates t < tp t = tp ¿p < t < ^max Oj 0 < |aj| < [a] 0 < aj < ï O2 0 < a2 < ï 0 0 < Oil < [a] 2.2 Optimal Meshing Condition of the HCMW and PositionParameter Selection Criterion The driven contact curve is either a common conical helical curve in a non-vertical case or a planar Archimedean helix in a vertical case. Therefore, from Eqs. (9) and (10), the helical radius of the driven contact curve is defined as Eq. (30): m' a -n(t + n)sinO (O ^ n/2) cos O K ' '. (30) -(nt + nn -b) (O = n¡ 2) When t = tp, m2 = m2p is defined as pitch radius of the driven contact curve. From Eqs. (29) and (30), it can be derived that when t=tp, (31) Eq. (31) is defined as the optimal meshing condition of the HCMW. From Eq. (30), the range of m2 depends on the value range of t and thus the range of m2 may not cover mP2. To guarantee that the entire meshing process has appropriate average slide rates, it is recommended that: 1P = ( + V2. (32) From Eqs. (30) to (32), the position-parameter for the non-vertical HCMW case is: a = (1 -i12 cos6)ml -nsin0cos0((ts + tE)/2 + n),(33) and for the vertical HCMW is: b = il2ml + n ( + tE )/2 + n). (34) Eqs. (33) and (34) can serve as the selection criterion of the position-parameters for the HCWM. Substituting Eq. (33) into Eq. (9), or substituting Eq. (34) into Eq. (10), the same equations of the driven contact curve can be attained: xM = (i12m1 + n sin$((ts + tE )/2 -1 ))cos — t + n "12 . t + n yM =-(( + nsin0((ts + tE )/2 -1))sin-7- i12 zM = -n (t + n)cos0 (0 /3 (t + 3n/ 4))sin ^ ZM2)= 3(t + n) (37) From Eq. (31), m2p = 12 mm and from Eq. (33), ap = 14.041 mm. As a comparison, with the same driving contact curve, the previous driven contact curve [9] is derived from Eq. (9) as below: x(2} =(2a -12 - 3yß (t + n}} cos- t + n 2 t + n yM = -(2a -12 - 3^3 (t + n}}sin zM= 3 (t + n} (38) As shown in Fig. 5, these driven wheels can mesh with the same driving wheel. It can be concluded that the value of a influences the intersection point of the center lines of driving and driven wheels. The values of a mainly change the meshing radii of the driven wheels and subsequently the slide rates of the HCMW pairs. Fig. 5. Different driven wheels meshing with the same driving wheels From Eq. (20), a > 10.081 mm is required. Different values of a are selected as Table 3. An equation of the driven contact curve can be derived by submitting each value of a into Eq. (38). It is worth noticing that Eq. (37) is equal to Eq. (38) when a3 = 14.041 mm. Table 3. Values of a [mm] a1 a2 a3 (ap) a4 a5 12.041 13.041 14.041 15.041 16.041 3.2 Simulation As shown in Fig. 4, according to the parameters in Table 2 and the different a values in Table 3, a driving wheel and driven wheels of different sizes can be attained in Pro/Engineering. a3=ap = 14.041 a4 =15.041 a5 =16.041 Fig. 4. Simulation of driving and driven wheels 566 3.3 Transmission Continuity Test As shown in Fig. 6, the driving wheel and the driven wheels were manufactured through selective laser melting (SLM) technology [16] and post-processed with electrochemical brushing process [15]. Within the processing capacity of the SLM, the manufacturing cost decreases as the weight of the HCMW decreases. Mainly operating in conditions with low loads, the HCMW is usually small and light. Therefore, the SLM technology was chosen in this paper and will be recommended for large-scale industry. To verify their transmission continuity, the rotation speeds of the wheels were measured with a test rig developed by our research team [9], which is shown in Fig. 7. The test procedure is shown in Fig. 8, and the test condition in Table 4. Fig. 6. Manufactured driving wheel and driven wheels Ding, J. - Chen, Y.Z. - Lv, Y.L. - Song, C.H. Strojniški vestnik - Journal of Mechanical Engineering 60(2014)9, 561-570 Fig. 7. Test rig: 1) DC power, 2) 4-DOF moveable platform, 3) fixed bracket, 4) driven wheel, 5) driving wheel, 6) DC motor, and 7) cable connected to acquisition card and computer Table 4. Test condition Voltage on motor 1.1 V Currency on motor 0.1 A Angular velocity of motor shaft 36° / s Torque transmitted 0.18 Nm Encoder Data-acquisition frequency 1 Hz 4-DOF Moveable Platform Angular Velocity DC Motor i Driving Wheel Meshing » Encoder - Driven Wheel A Angular Velocity Fixed Bracket Transmission Ratio Fig. 8. Test procedure Table 5. Transmission ratio a ai a2 as a4 as Practical value 1.99 2.00 2.00 2.01 2.00 Theoretical value 2 Relative error [%] 0.5 0 0 0.5 0 Continuous records of the test are shown in Fig. 9. As shown in Table 5, all the driven wheels can continuously commit the transmission within the allowable error range [9]. Fig. 9. Continuous records of the test Position-Parameter Selection Criterion for a Helix-Curve Meshing-Wheel Mechanism Based on Sliding Rates 567 Strojniski vestnik - Journal of Mechanical Engineering 60(2014)9, 561-570 Table 6. Maximum absolute values of a1 and o2 corresponding to different a a [mm] 12.041 13.041 14.041 15.041 16.041 Range of o1 -0.003 to 0.249 -0.090 to 0.202 -0.182 to 0.141 -0.279 to 0.069 -0.379 to -0.012 1 O1 1 max 0.249 0.202 0.182 0.279 0.379 Range of o2 0.012 to 0.275 -0.074 to 0.218 -0.164 to 0.154 -0.253 to 0.083 -0.332 to 0.003 1 O2 1 max 0.275 0.218 0.164 0.253 0.332 a) b) Fig. 10. Slide rates of the driving and driven contact curves; a) a1 vs. t and b) o2 vs. t It should be noted that the accuracy of setting the parameter a in the test rig is 0.1 mm, which is much lower the accuracy of the simulation in the previous section and the theoretical calculation in the next section. Furthermore, only the transmission continuity of the HCMWs is verified in the experiment. 3.4 Slide Rate Discussion As shown in Fig. 10, the slide rates of Eq. (38) with different position-parameters are calculated from Eqs. (14) and (15) and drawn with MATLAB. As shown in Table 6, the ranges of a1 and a2 are measured from Fig. 9. Since the signs of a1 and ct2 indicate the relative slide direction, only the maximums of their absolute values are compared. The result shows that when a = ap = 14.041 mm, both |CTi|max and |o2|max achieve their minimums simultaneously. Although all five pairs of the HCMW can mediate the transmission, the one coincident with the position-parameter selection criterion has the best slide rates. 1. The optimal meshing condition is attained from the analysis of the slide rate, and it shows that the slide rates of the HCMW are zero if and only if the meshing radius of the driven contact curve equals the product of the transmission ratio and the meshing radius of the driving contact curve; 2. The mesh at the midpoint of the contact curves should be coincident with optimal meshing condition, which is defined as the positionparameter selection criterion; 3. Numerical examples show that the HCMWs coincident with the position-parameter selection criterion that possesses the best slide rates. However, some issues remain to be improved: an experimental measure method of the HCMW slide rates has yet to be proposed; the slide rates' effects on the friction and wear of the HCMW remain to be explored; the forming conditions of the hydrodynamic film should be studied to determine whether the HCMW will work with lubricant in the future; furthermore, the allowable slide rates under lubricant condition need to be determined for standardization production. 4 CONCLUSIONS AND PROSPECTS 5 ACKNOWLEDGMENT This paper presents the position-parameter selection criterion of the HCMW based on its slide rates. Specifically, the results can be concluded as below: Funding supports from the National Natural Science Foundation of China (No. 51175180) are gratefully acknowledged. 568 Ding, J. - Chen, Y.Z. - Lv, Y.L. - Song, C.H. Strojniski vestnik - Journal of Mechanical Engineering 60(2014)9, 561-570 6 NOMENCLATURE a Position-parameter, distance from point op to axis z b Position-parameter, distance from point op to axis x ¿12 Transmission ratio M Meshing point MM1 Arc length between M and M1 MM2 Arc length between M and M2 MM1 Contact vector from M and M1 MM„ ■ 2 Contact vector from M and M2 n Pitch parameter of driving curve r1 Driving curve equation in vector form r1 Driving tine radius r2 Driven curve equation in vector form r2 Driven tine radius s1 Arc length on driving curve s2 Arc length on driven curve t Scope parameter of helix curve ts Starting value of t tE Ending value of t tp t when HCMW in pitch position tmax Maximum available value of t Z1 Number of driving tines Z2 Number of driven tines r1 Driving curve r2 Driven curve 8 Included angle between angular velocity vectors 01 Slide rate of driving curves 02 Slide rate of driven curves [o] Available slide rate (superscript) Corresponding coordinate 7 REFERENCES [1] Wu, X.T. 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