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Modelling the Meshing of Cycloidal Gears DOI 10.1515/ama-2016-0022 acta mechanica et automatica, vol.10 no.2 (2016) MODELLING THE MESHING OF CYCLOIDAL GEARS Jerzy NACHIMOWICZ*, Stanisław RAFAŁOWSKI* *Bialystok University of Technology, Department of Mechanical Engineering, Faculty of Mechanical Engineering, ul. Wiejska 45C, 15-351 Białystok, Poland [email protected], [email protected] received 16 June 2015, revised 11 May 2016, accepted 16 May 2016 Abstract: Cycloidal drives belong to the group of planetary gear drives. The article presents the process of modelling a cycloidal gear. The full profile of the planetary gear is determined from the following parameters: ratio of the drive, eccentricity value, the equidistant (ring gear roller radius), epicycloid reduction ratio, roller placement diameter in the ring gear. Joong-Ho Shin’s and Soon-Man Kwon’s article (Shin and Know, 2006) was used to determine the profile outline of the cycloidal planetary gear lobes. The result was a scatter chart with smooth lines and markers, presenting the full outline of the cycloidal gear. Key words: Cycloidal Drive, Modelling, Profile, Planetary Gear, Gear 1. INTRODUCTION and Song, 2008). Therefore, it was necessary to demonstrate the process of modelling a toothed gear with a cycloidal profile. Analysis of the function of the cycloidal transmission indicates that it is a rolling transmission, in which all elements move along a circle (Rutkowski, 2014). Because of its specific design, cy- 2. EQUATIONS DESCRIBING THE PROFILE cloidal gears have many advantages, such as: a wide range OF A CYCLOIDAL GEAR single stage reductions (even up to 170), making the drive rela- tively small, high mechanical efficiency, and silent operation. Moreover, its minimal moment of inertia results in smooth start Modelling a cycloidal gear is possible with equations (1, 2) and allows fast change of movement direction. The drive also has presented in an article by Shin and Kwon (2006): a significant overload capacity and reacts quickly to change 퐵푥 = 푟 ∙ 푐표푠(휑) − 푞 ∙ cos(휑 + Ψ) − 푒 ∙ 푐표푠(푁휑) of load (Chmurawa, 2002; Bednarczyk, 2014). (1) 퐵푦 = −푟 ∙ 푠푖푛(휑) + 푞 ∙ sin(휑 + Ψ) + 푒 ∙ 푠푖푛(푁휑) Cycloidal gear stands out among all gears with its lower mass, smaller housing and its quietness. These are the advantages where: 푟 – distance between the ring gear rollers [mm]; in machines that demand high gear ratio. In the current planetary 푞 휑 radius of the ring gear roller [mm]; 푁 – number of rollers gears, for getting enough reduction, three drive ratios are needed. of the ring gear; 푒 – eccentricity value [mm]; 훹 – contact angle It leads planetary gears to getting bigger and heavier with every between the roller and the lobe of the cycloid gear [o]; 휑 – angle drive ratio added. In cycloidal gears only two drive ratios are between point B and the x axis in the central point of the needed to get planetary gear reduction with three drive ratios. coordinate system [o]. Moreover, inner clearances helps cycloidal gear to provide super The contact angle between the roller and the lobe of the cy- precise movement in machinery. cloid gear is determined from the following equation: Due to their advantages, cycloidal drives are becoming in- creasingly popular in modern industry, and are used in wood sin[(1 − 푁)휑] Ψ = 푎푡푎푛 [ 푟 ] (2) processing machines, CNC and machine tool workshops, tech- − cos[(1 − 푁)휑] nical head drives, as well as glass and textile machines. 푒푁 The gears are also used in robot arm joints, hoists and turn- (0° ≤ 휑 ≤ 360°) tables. However, the development and spread of cycloidal drives In order to determine the curvature profile of the cycloidal is not as dynamic as in the case of other similar devices. This gear, the coordinates of multiple B points must be determined, is because constructing cycloidal drives is not an easy task – their which form the shape of the gear (Fig. 1). The equations (1, 2) design is highly complicated, and so is the technology of curved make it possible to determine the coordinates of point b for any meshing of planetary gears. Also, producers of machinery which angle φ. The precision of the profile depends on the step value uses these type of drives do not share their designs (Chmuraa, of angle φ the smaller the step is, the corresponding shape 2002; Siczek and Warda, 2008; Yan and Lai, 2002; Feng is more precise. What is characteristic about these equations and Litvin, 1996; Kabaca, 2013; Bednarczyk, 2013; Blagojevic is that there is always one less lobe of the cycloidal gear than et al., 2012; Hwang and Hsieh, 2007; Meng et al., 2007; Figliolinii there are rollers in the ring gear. et al., 2013; Sung and Tsai, 1997; Hong-Liu et al., 2013; Wand 137 Jerzy Nachimowicz, Stanisław Rafałowski DOI 10.1515/ama-2016-0022 Modelling the Meshing of Cycloidal Gears 4. MODELLING THE PROFILE OF THE CYCLOIDAL GEAR Microsoft Excel was used to make the necessary calculations in order to determine the profile of the cycloidal gear. The spread- sheet presents the basic parameters of a cycloidal drive (Fig. 3), which were then used to determine the additional parameters of the drive with the corresponding equations (Fig. 4). Fig. 1. Cycloidal gear parameters: 푟 – radius of the ring gear; 푞 – radius of the roller; 푒 – eccentricity; 휑 – angle between point 퐵 and the 푥 axis in the central point of the coordinate system; 퐵 – point Fig. 3. Basic parameters of the cycloidal drive required on curvature profile of the lobe; 퐵푥, 퐵푦 – coordinates of point 퐵 for the calculations 3. GENERAL PROFILE OF THE LOBE OF THE CYCLOIDAL GEAR Fig. 2 presents the outline of the lobe of the cycloidal gear. In order to form the shape of the lobe, the number of points was set Fig. 4. Additional parameters of the cycloidal drive required at 푘 = 95 with coordinates 퐵 = (퐵푖푥 , 퐵푖푦). After determining for the calculations the positions of all points against the beginning of the coordinate system, the points are connected with a spline. Equation formulas and the values in individual cells are pre- As the profile of the cycloidal gear must be very precise, two sented in Tab. 1. When designing the profile of the gear, it is additional points were added at both ends, whose position was necessary to verify the condition necessary for the drive to work determined for adjacent lobes of the gear. (cell H3), which was set for conditional formatting. When the This assumption is presented in equation: condition is met, the background of the cell is green – when the w ∈ < −2; k + 3 > ∈ C (3) condition is not met, it is red. The next stage of the design is the determination of the lobe where: 푤 – number of point; 푘 – number of points per each lobe profile of the cycloidal gear. To do this, Tab. 2 was created in the of the gear; 퐶 – set of total numbers. spreadsheet which determined the coor-dinates of all points com- prising the lobe outline of the cycloidal gear. The first column of the table lists the indexes of the B points which created the profile of the lobe, beginning in cell A19. The initial value of the index is −2, increased by 1 at each step, and the final value is 푘 + 2, which in this case is 97. The second column contains the φ angle values in radians. The next column calculates the Ψ angle corresponding to angle φ. The coordinates of the points creating the profile of the lobe are presented in columns D and E. Fig. 2. Lobe profile outline: B0, B1, Bk - successive points creating the outline of the lobe; B(x0, x0), B(x1, y1) – coordinates for points B0 and B1 respectively Tab. 1. Formulas and their use in the calculations Cell E3 F3 G3 H3 I3 푄 factor introduced in 푘 number of 퐵 points drive’s functional Shin and Know (2006), 푁 number of rollers Definition per each lobe of the condition 푑휑 angle increase used in the of the ring gear; gear (Shin and Know, 2006) calculations Equation 푄 = 푒 ∙ 푁 (assumed value) 푁 = 푧1 + 1 푟/(푒 ∙ 푁) > 1 푑휑 = 360/(푧1 ∙ 푘) Formula = 퐵3 ∙ 퐺3 = 95 = 퐷3 + 1 = 퐴3/(퐵3 ∙ 퐺3) = 360/(퐷3 ∙ 퐹3) 138 DOI 10.1515/ama-2016-0022 acta mechanica et automatica, vol.10 no.2 (2016) Tab. 2. Equation formulas for cycloidal points A B C D E 18 풌 흋 휳 푩풙 푩풚 퐴푇퐴푁((푆퐼푁(−11 43,64 ∙ 퐶푂푆(퐵19) − 4 −43,64 ∙ 푆퐼푁(퐵19) + 4 푅퐴퐷퐼퐴푁푆 (퐴19 19 -2 ∙ 퐵19))/(1,818 − 퐶푂푆(−11 ∙ 퐶푂푆(퐵19 + 퐵20) − 2 ∙ 푆퐼푁(퐵19 + 퐵20) + ∙ 0,344498 ∙ 퐵19))) ∙ 퐶푂푆(12 ∙ 퐵19) 2 ∙ 푆퐼푁(12 ∙ 퐵19) Tab. 3. Equation formulas for the full outline F G H I J 18 훗 [o] 훗 [rad] 횿 푩풙(흋) 푩풚(흋) 퐴푇퐴푁((푆퐼푁(−11 ∙ 퐺19)) 43,64 ∙ 퐶푂푆(퐺19) − 4 −43,64 ∙ 푆퐼푁(퐺19) + 4 ∙ / 19 0,2 푅퐴퐷퐼퐴푁푆(퐹19) ∙ 퐶푂푆(퐺19 + 퐻19) − 2 푆퐼푁(퐺19 + 퐻19) + 2 ∙ (1,818 − 퐶푂푆(−11 ∙ 퐶푂푆(12 ∙ 퐺19) 푆퐼푁(12 ∙ 퐺19) ∙ 퐺19))) The results of the calculations were put in a scatter chart with smooth lines and markers, presenting the outline of the cycloidal gear lobe (Fig. 5). Fig. 6. Full outline of the cycloidal gear 5. CONCLUSIONS Fig. 5. Lobe profile in a coordinate system Modelling the full outline of the cycloidal gear significantly fa- The last stage of designing the cycloidal gear is determining cilitates the design process of the cycloidal drive. Because its complete outline. Tab. 3 was created analogously to determin- the cycloidal gear is an extremely important element of the drive, ing the position of the cycloidal gear lobe.
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