IMECE OPTIMAL DESIGN OF WORM GEAR SYSTEM USING IN CVVL FOR AUTOMOBILES

Transcription

1 Proceedings of ASME 2013 International Mechanical Engineering Congress & Exposition IMECE 2013 November 15-21, 2013, San Diego, CA, USA IMECE OPTIMAL DESIGN OF WORM GEAR SYSTEM USING IN CVVL FOR AUTOMOBILES Nogill Park Department of Mechanical Engineering Pusan National University Busan, Korea Jonghyeon Sohn Department of Mechanical Engineering Pusan National University Busan, Korea Gyeangmyeang Baek Department of Mechanical Engineering Pusan National University Busan, Korea Chunghan Oh Gasoline Engine Engineering Team Hyundai-Motor Company Hwaseong, Korea ABSTRACT The worm gear is used in the motor drive system of automotive continuous variable valve lift because of its compactness and self-locking ability. A ZK worm and an involute helical gear can be meshed in order to reduce production cost. However, the gearing is not suitable for reliability and the NVH problem. To improve dynamic performance, an optimal design process is considered. Transmission error is calculated theoretically and minimized with several gear design parameters. An inequality condition such as the teeth interference elimination is added. NOMENCLATURE i reference point of pinion (on axis of rotation) o reference point of gear (on axis of rotation) q contact point g gear ratio N i normal vector of pinion surface N o normal vector of gear surface iq position vector of pinion surface oq position vector of gear surface i, j, k unit vectors of global coordinate system 1. INTRODUCTION To improve the fuel efficiency of a vehicle, an attempt at varying the valve lift or compression ratio of an engine has been recently made. Due to reduced vibration, less noise and higher power density than different types of gears, the worm gear is mainly used for the motor driving power steering(mdps) of an automotive steering system, the control system of an automotive continuous variable valve lift (CVVL)[1] or variable compression ratio (VCR). In the design of the gear, combining the tooth surface satisfying the gearing law[2] is a principle. But there are some cases when the gears with the non-conjugate tooth surfaces are combined to reduce production cost. According to the classification of AGMA6022-C93[3], there are at least four cylindrical worm shaft tooth forms, among which the ZK-type tooth form is relatively easy to process. However, it is difficult to process the paired worm wheel. For this reason, there are cases that the involute helical gear easy to process is substituted. In this study, we would like to propose a method for designing a worm gear system combining a ZK-type worm shaft as the non-conjugate tooth surface with an involute helical gear. When the non-conjugate tooth surface is combined, teeth interference(ti) and transmission error(te) are essentially caused. It is possible to evade TI and minimize TE by adjusting the pressure angle and the helical angle as basic design 1 Copyright 2013 by ASME

2 parameters of two gears. For doing this, we will propose the analysis of calculating TE of the gear system with which the non-conjugate tooth surface is combined, and draw the design direction that evades TI by analyzing the correlation between TI and TE. In addition, we verify validity and practicality by applying the proposed designing method to automotive CVVL and showing performance improvement. 2. KINEMATIC ANALYSIS OF GEARED SYSTEM WITH NON-CONJUGATE TOOTH SURFACES If the non-conjugate tooth surfaces are combined together, the geometrical conditions to be satisfied in the contact point are that the directions of the normal vectors of both gears at the contact point must be same, and the location of the contact point must be placed properly at the installed state. Being expressed in mathematical formulas, these are as follows: N i N o = 0 (1) iq = io + oq (2) Where iq is expressed with the functions of curved surface parameters (a, b) and a mechanical parameter (θ 1 ) of the pinion surface as iq(a, b, θ 1 ); oq is expressed with the functions of curved surface parameters (c, d) and a mechanical parameter (θ 2 ) of the gear surface as oq(c, d, θ 2 ); and io as a position vector between reference points of the pinion and gear is a prefixed constant value when the gears are installed. In the worm gear system, i, o are generally the center points, respectively, located on the shortest paths between two different axes and io is set to be the center line of the shortest distance between two axes. N o = oq oq c d Accordingly, N i is expressed with the functions of parameters a, b and θ 1. In short, N i (a, b, θ 1 ). N o is expressed with the functions of parameters c, d and θ 2. In brief, N o (c, d, θ 2 ). From formula (1), two independent algebraic equations can be obtained as shown below F 1 (a, b, c, d, θ 1, θ 2 ) = ( N i N o ) i = 0 (5) F 2 (a, b, c, d, θ 1, θ 2 ) = ( N i N o ) j = 0 (6) From formula (2), three independent algebraic equations can be obtained as shown below. F 3 (a, b, c, d, θ 1, θ 2 ) = (iq io oq ) i = 0 (7) F 4 (a, b, c, d, θ 1, θ 2 ) = (iq io oq ) j = 0 (8) F 5 (a, b, c, d, θ 1, θ 2 ) = (iq io oq ) k = 0 (9) From formulas (5) to (9), when θ 1 is set as a free parameter and curved surface parameters a, b, c, d are eliminated, θ 2 becomes the function of θ 1. By using these, TE is calculated by using the following formula: TE(θ 1 ) = θ 2 (θ 1 ) θ 20 (10) where, θ 20 = gθ 1. When the tooth surface moves in the meshing region, a pair of gears starts to contact (contact-1) and when the surface gets out of the region, meshing is finished. When contact-1 moves as much as the pitch angle (θ 1 p = 2π Z 1 ), a pair of rear gears starts to contact (contact-2). When TE occurs, the phenomenon of teeth interference or separation occurs at contact-2. The detailed explanation is as shown below. (4) Figure 1. Geometrical conditions of non-conjugate surface contact of the gear pair When the surface of each gear is arithmetically defined, the normal vector can be obtained from the following formulas: N i = iq iq a b (3) Figure 2. Teeth interference phenomenon due to transmission error ( case of TE(θ 1 ) < TE(θ 1 θ 1 p ) ) 2 Copyright 2013 by ASME

3 Fig. 2 illustrates the case when contact-1 entered the meshing region and moves further as much as the pitch angle. TI occurs at the time that contact-2 just entered the meshing region. When the front pinion(driving) surface rotates as much as θ 1 from the reference line of the rotation angle, the front gear(driven) surface rotates as much as θ 2 (θ 1 ). Since the rear pinion surface is placed behind as much as θ 1 p, its rotation angle is θ 1 θ 1 p. Accordingly, the rear gear surface moves as much as θ 2 (θ 1 θ 1 p ) from the reference line of the rotation angle. Let s take the rear gear surface at θ 2 (θ 1 θ 1 p ) as the theoretical rear gear surface. Then the rotation angle between the front gear surface and the theoretical rear gear surface θ 2 p is established as follows: Figure 3. Transmission error that induces teeth interference θ 2 p = θ 2 (θ 1 ) θ 2 (θ 1 θ 1 p ) (11) But the actual rear gear surface is placed behind as much as the gear pitch angle ( θ p 20 = 2π Z 2 ). When formula (10) is substitute into formula (11), the difference of the rotation angle of the theoretical rear gear and the actual rear gear is established as follows: θ p 2 θ p 20 = TE(θ 1 ) TE(θ 1 θ p 1 ) (12) where, θ p 20 = gθ p 1. If θ p p 2 = θ 20 (or TE(θ 1 ) = TE(θ 1 θ p 1 ) ), while the surfaces of the front pinion and front gear are being meshed, those of the rear pinion and rear gear start to be meshed smoothly and double contacts are developed without any troubles. For the reason that gears are actually placed at equal intervals as much as the pitch angle θ p 20, if θ p p 2 < θ 20 (or TE(θ 1 ) < TE(θ 1 θ p 1 )), the rear pinion surface placed ahead of the actual rear gear. Accordingly, the interference between the rear pinion surface and the actual rear gear surface occurs. In details, while the gear is inserted to the radial direction, the edge of the gear hits the pinion surface to the radial direction and moves in. Because this directly causes a rattle noise, and even damage of the tooth surface, this must be eliminated upon design. Fig. 3 represents TE occurring in the contact region when the pinion rotates. TE is repeated at a cycle which is the pitch angle because the front tooth surface and the rear tooth surface are placed at the interval as much as the pitch angle. Since the rotation angle in the contact region is greater than the pitch angle, double contacts occur at some region. Figure 4. Actual transmission error inducing teeth interference In the b~c section in Fig. 3 as shown above, meshing is led by contact-1 and the gears are meshed and move by the contact at a single point. When contact-1 is at point c (θ 1c ), contact-2 moves in the meshing region. At the time, TEs at contact-1 and contact-2 are cc 1 = TE(θ 1c ), and ca 2 = TE(θ 1c θ p 1 ), respectively. In Fig. 3, because cc 1 < ca 2 or TE(θ 1c ) < TE(θ 1c θ p 1 ), a tooth impact occurs at contact-2 (Refer to Fig. 2). After the impact, as the rotations of gears increase discontinuously as much as c 1 a 2, teeth are separated at contact-1 and the gears move to d while meshing is led by contact-2. When they go past d, contact-1 gets meshing again and it is developed until e, where the meshing region ends. At the time, contact-2 is the state of the tooth surfaces being separated. At point e, (θ 1e ) the TEs at contact-1 and contact- 2 are ee 1 = TE(θ 1e ), and eb 2 = TE(θ 1e θ p 1 ), respectively, when the gap between tooth surfaces of contact-2 is largest. The maximum value of the gap (G max ) is calculated in the following formula: G max = TE(θ 1e ) TE(θ 1e θ 1 p ) (13) After that, while contact-1 ends out of the meshing region, the gap (G max ) at point e is filled, the front impact occurs between the tooth surfaces. Consequently, Fig. 4 represents TE actually occurring as shown in Fig. 3 with a thick line. 3 Copyright 2013 by ASME

4 To put it shortly, if TE(θ 1 ) < TE(θ 1 θ 1 p ), TI occurs at point c leading the impact between the edge of the gear and the pinion surface to the radial direction and the teeth separation occurs at point e leading the frontal impact between the tooth surfaces. TI at point c causes fatal damage to the tooth surface and creates rattle noise due to the high impact speed to the radial direction. Therefore, TI must be completely removed. Since the frontal impact occurs on the tooth surface due to the gap at point e, the impact speed is not higher than that at point c. TE works as a vibration source leading not only the rattle noise by tooth impact but also the whine noise of the gear system with the form of displacement excitation at the contact region. To minimize the whine noise, the design strategy, therefore, must be made to the direction of minimizing the size of the frequency spectrum of the periodic function of the TE calculated from Fig. 4. separated. Meshing is still led by contact-1, which moves until e where the meshing region ends. At point e (θ 1e ), the gap between the tooth surface of contact-1 and that of contact-2 may be obtained from formula (13). Fig. 7 represents TEs actually occurring in Fig. 6 with a thick line. Figure 6. Transmission error that induces teeth separation Figure 5. Teeth separation phenomenon due to transmission error (case of TE(θ 1 ) > TE(θ 1 θ 1 p )) Fig. 5 shows the case that the teeth separation occurs without TI. In formula (12), when TE(θ 1 ) > TE(θ 1 θ p 1 ), θ p 2 > θ p 20. As the actual rear gear surface is placed ahead of the rear pinion surface, no contact occurs at contact-2, but the gap between the tooth surfaces occurs. While passing through the meshing region, contact-1 continuously leads the contact and contact-2 is developed while the tooth surfaces are separated. At the moment that the tooth contact of the front pinion and the front gear at contact-1 are ended, frontal impact of the tooth surfaces at contact-2 occurs. Fig. 6 shows the distribution of TEs. Contact-1 leads meshing at point b to point e via c without any sudden change. The region of b~c is developed at the contact of a single point. At point c, contact-2 enters the meshing region. The TEs of contact-1 and contact-2 are cc 1 = TE(θ 1c ) and ca 2 = TE(θ 1c θ p 1 ), respectively. In Fig. 6, cc 1 > ca 2 or TE(θ 1c ) > TE(θ 1c θ p 1 ). The gap between tooth surfaces of contact-2 is formed as much as a 2 c 1 (Refer to Fig. 5). Accordingly, moves into the meshing region while the teeth are Figure 7. Actual transmission error inducing tooth separation According to the meshing mechanism as shown above, it can be found that the gear system made in combination with the non-conjugate tooth surface causes TI and TEs. To commercialize the gear system in combination with the nonconjugate tooth surface, TI must be removed and TEs must be minimized. To remove TI, it, therefore, must be satisfied that cc 1 > ca 2 at point c, or TE(θ 1c ) > TE(θ 1c θ 1 p ). When the gap (G max ) between the tooth surfaces is minimized at point e, the TE is minimized. 3. OPTIMAL DESIGN OF WORM GEAR FOR AUTOMO- TIVE C.V.V.L To reduce the production cost, we would like to design the worm gear system by combining the non-conjugate tooth surface as a speed reducer for automotive CVVL. A ZK-type worm easy to manufacture is used for a worm shaft and an involute helical gear is used for a worm wheel. The 4 Copyright 2013 by ASME

5 requirements required to design the speed reducer is in Table 1 as shown below. Table 1. Design requirements of the worm gear system for automotive CVVL Item Requirement Rated Power (W) 185 Input Speed (rpm) 5000 Reduction Ratio 60:1 The design optimization strategy is to minimize G max at point e where TI does not occur at point c in Figs. 4 and 7. Accordingly, the optimization problem may be formulated as follows: Objective function : min G max Inequality condition : TE(θ 1c ) > TE(θ 1c θ p 1 ) Design parameters : Addendum coefficient of worm shaft Normal pressure angle of worm shaft Addendum coefficient of worm wheel Normal pressure angle of worm wheel Helix angle of worm wheel As a design optimization technique, the direct search method was used to define the scope of design parameters. The initial design value and the final design value are in Table 2. The initial design values were decided as the values used when ZI[3] worm and the involute helical gear were combined. While it is set to make the helix angle of the worm wheel and the lead angle of the worm shaft the same, the normal pressure angle was matched to be 20. At the state, when the ZI-type worm shaft is used, it meets the meshing principle of the gear. Therefore, neither TI nor TE occurs theoretically. But in case of a ZK-type worm shaft, TI and TE occur. In case of the initial design values, the TI level at point c and G max at point e in Fig. 4 are 3.18" and 9.94", respectively, as rotation angles of the worm wheel. In case of the final design values, TI was removed and G max was reduced to 3.96". Table 2. Design parameters and meshing performance of the worm gear system for automotive CVVL Design parameters and Meshing Performance Initial Design Final Design Addendum coefficient of worm shaft Addendum coefficient of worm wheel Normal pressure angle of worm shaft ( ) Normal pressure angle of worm wheel ( ) worm wheel helical angle ( ) Maximum interference level ( '' ) TE peak ( '' ) (a) Initial design (b) Final design Figure 8. Transmission error distribution diagram of the worm gear system Fig. 8 represents the transmission error distribution diagram. In case of the initial design, TI is observed as shown at point c in Fig. 4. In addition, G max is also high. In case of the final design, G max is reduced to roughly one third and TI is not observed. Fig. 9 shows the shapes of three-dimensional (3D) surfaces drawn on CATIA V5. The surfaces were obtained from the group of numerically calculated coordinates of the tooth surfaces of the worm shaft and the worm wheel and they were magnified by 1000 times for 3D precision. In case of the initial design, TI occurring at point c in Fig. 4 may be seen in Fig. 9-(a). Even though it is not apparently distinguished because it is a 3D model at the actual ratios, TI may be confirmed by comparing the sizes of the interference areas. (a) Initial design 5 Copyright 2013 by ASME

6 (b) Final design Figure 9. Teeth interference check on the graphic tool The experimental apparatus for the comparison of vibration and noise performance (NVH) regarding the initial design and final design of the design parameters is illustrated in Photo 1. The NVH test was performed after a microphone (B&K: Type 4189-A-021) was installed in front of the gear, and a one-directional ICP-type accelerometer (PCB: M355B15) was attached on the top of the driving motor. To acquire and analyze data, PAK MK-II (Muller-BBM) was used. The test was conducted at 3,000 rpm or lower where the noise of the worm gear system is audible. (a) Initial design Figure 10. Experimental apparatus for NVH of the CVVL worm gear system The test results are represented in Fig. 11. According to the waterfall diagram regarding the initial design, abnormal noise and vibration were synchronized in the ranges between 1 khz and 5 khz, and the severe rattle noises are observed. According to the waterfall diagram regarding the final design, it is shown that the rattle noise disappears and the slight whine noise appears. The validity of the TE analysis proposed from the experimental verification and the practicality of the design results were identified. Figure 11. Results of NVH Test (b) Final design 6 Copyright 2013 by ASME

7 4. CONCLUSION We proposed the transmission error (TE) analysis of the gear system in combination with the non-conjugate tooth surface, and made the following conclusion as a result of designing the worm gear system for automotive CVVL by using it: 1) We identified that the practical results may be obtained when TI is removed and TEs are minimized in the combination of the non-conjugate tooth surfaces by using the ZK worm and the involute helical gear for the speed reducer of the CVVL mechanisms of the automotive engine. 2) It was confirmed that the TE obtained after the rotation as much as the pitch angle of the pinion is utilized as the reference of the existence of TI, and when the value is positive, the edge of the gear is inserted to the radial direction of the following tooth surface of pinion and cause the rattle noise that hits the opposite tooth surface. When the value is negative, the tooth separation occurs between the tooth surfaces and when new meshing is created, the impact occurs between the following tooth surfaces. 3) We could obtain the practical results regarding the worm gear system in combination with the non-conjugate tooth surfaces through optimization that eliminates TI and minimizes the maximum TE. REFERENCES [1] Kyoung-Pyo Ha, Donghee Han and Woo Tae Kim, 2008, Development of Continuously Variable Valve Lift Engine, KSAE [2] Faydor L. Litvin, 2004, Gear Geometry and Applied Theory, pp.7-137, Cambridge [3] AGMA, 1993, 6022-C93, Design Manual for Cylindrical Wormgearing, pp.5-8 [4] I.H. Seol, F.L Litvin, 1996, "Computerized Design, generation and simulation of meshing and contact of worm-gear drives with improved geometry", Computer Methods in Applied Mechanics and Engineering, Vol.138, pp [5] William P. Crosher, 2002, Design and Application of the Worm Gear, pp Copyright 2013 by ASME