International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 14 (2019) pp. 3211-3218 © Research India Publications. http://www.ripublication.com 3211 Calculating Optimum Gear Ratios of Mechanical Driven Systems Using Worm-Helical Gearbox and Chain Drive Tran Thi Hong 1 , Nguyen Van Cuong 2 , Le Hong Ky 3 , Nguyen Thanh Tu 4 , and Vu Ngoc Pi 4, 1 Nguyen Tat Thanh University, Ho Chi Minh City, Vietnam. 2 University of Transport and Communications, Ha Noi city, Vietnam. 3 Vinh Long University of Technology Education, Vinh Long City, Vietnam. 4 Thai Nguyen University of Technology, Thai Nguyen City, Vietnam.  Corresponding author (E-mail address: vungocpi@tnut.edu.vn ) Abstract This paper presents a study on the determination of the optimum gear ratios of mechanical driven systems using a worm-helical gearbox and a chain drive. In the study, an optimization problem was performed to find the optimum gear ratios. In addition, the length of the mechanical system was selected as the objective function of the optimization problem. The effects of the input factors including the total transmission ratio of the system, the calculation coefficient of the worm diameter coefficient, the coefficient of wheel face width and the allowable contact stress of helical gear unit, and the output torque were investigated. To evaluate the influences of these factors on the optimum gear ratios, a simulation experiment was designed and conducted by computer programs. Significantly, several models to determine the optimum gear ratios were proposed. Using these models, the determination of optimum gear ratios is accurate and simple. Keywords: Optimum gearbox design; gear ratio; optimum gear ratio; worm-helical gearbox. 1. INTRODUCTION In optimum design of a gearbox or a mechanical driven system, the determination of the optimum gear ratios has a very important role. This is due to the large dependence of the dimention, the mass and, therefore, their cost on the gear ratios of the gearbox of the system. Heretofore, several studies have been done to identify the optimum gear ratios. The optimum gear ratios can be found by using different methods. The methods can be the graph method [1, 2], the practical method [3] or the model method [3-15]. Besides, the gear ratios have been found for different types of gearboxes. Concerning helical gearboxes, the optimum gear ratios were determined for two step gearboxes [3-7], three step gearboxes [8-14] and four step gearboxes [12-18]. In addition, the gear ratios have been found for different objectives such as the minimum gear mass [9, 12, 13, 15, 17], the minimum area of gearbox cross section [8, 11] and the minimum gearbox length [7, 10]. Regarding worm gearboxes, considerable attention has been paid to this area. For a two step worm gearbox, the gear ratios can be determined by a pratical equation [3] or calculated to get the reasonable housing structure by the following equation [22]:   1/ 2 1 2 h u u u   (1) Furthermore, for this type of gearbox, the optimum gear ratio of the second step was determined by 2 30.97 u  [23]. For worm-helical gearboxes, the optimum gear ratio of the helical gear unit is found by the following practical model [22]:   2 0.03 0.06 h u u   (2) Also, to obtain a good condition for oil lubrication for both steps, the gear ratio of the worm gear unit is determined by a graph [2] or the optimum gear ratio of the helical gear unit 2 u is calculated by [22]: 1/ 2 2 2 6.86 ba u    (3) In which, 2 0.3 0.4 ba   is the wheel face width coefficient of the helical gear unit. For worm-helical gearbox, the optimum gear ratio of the second step 2 u is found as   2 2max 8 10 u u   [24]. In regard to mechanical driven systems, several studies have been proposed on calculating the optimum gear ratios of systems containing a gearbox and a V-belt drive [19, 20] or a chain drive [21]. In this paper, an optimization study on obtaining the optimum gear ratios of mechanical driven systems using a worm-helical gearbox and a chain drive was introduced. The objective of the optimization problem in the study is to attain the minimum system length. Besides, the influences of the input factors including the total gearbox ratio, the calculation coefficient of the worm diameter coefficient, the coefficient of wheel face width and the allowable contact stress of helical gear unit, and