In this paper, a method for optimal design of structure parameters of gears in order to reduce the vibration of the car gearbox during the work process is presented. The model of a pair of interlocking gears was simplified by the two pairs of useful volume and elastic springs. From this model, it is established the formulas in order to determine elastic stiffness of gear, synthetic hardness a pair of interlocking gears, useful volume of gears, private frequency and the speed limit of the gear.
THE INTERNATIONAL CONFERENCE ON MARINE SCIENCE AND TECHNOLOGY 2016 An optimal gear design method for minimization of transmission vibration Nguyen Tien Dung1, Nguyen Thanh Cong2 VietNam Maritime University, dungnt@vimaru.edu.vn University of Communications and Transport Abstract In this paper, a method for optimal design of structure parameters of gears in order to reduce the vibration of the car gearbox during the work process is presented The model of a pair of interlocking gears was simplified by the two pairs of useful volume and elastic springs From this model, it is established the formulas in order to determine elastic stiffness of gear, synthetic hardness a pair of interlocking gears, useful volume of gears, private frequency and the speed limit of the gear Selection of minimization of transmission vibration is objective function in order to optimize structural parameters of gears transmission The technical parameters of the car is chosen, the optimal results show that deviation of speed limit of gear with gear rotation speed when the preliminary design is 9688 rad/s, after calculating the design values increased 34440 rad/s This method is used to improve the quality of gearbox and minimize the time for design of gearbox Keywords: Gears, transmission vibration, matlab, gearbox design, Structural parameters Introduction The criterion of noise and vibration is one of the criteria to appreciate quality of automobile gearbox The ratio of transmission system and the torque were changed by the pair of gears in the gearbox Thus, the transmission gears are main causes of noise and vibration of the automobile gearbox The cause of the noise and vibrations of transmission gears can by itself, due to structural or manufacturing error when assembly the gears The design aims to determine the gearbox’s feature and size parameters These parameters are chosen by experience before, however it is hard to achieve the best conditions In the scope of this paper, the author introduces a method to design optimal basic parameters of gearbox structure via the multivariate extreme value analysis with nonlinear constraints using Sequential Quadratic Programming (SQP) Fmincon function in the Matlab program Using this method is to improve the quality, as well as minimizing the time gearbox design Establishing dynamic modelling of spur gear pairs Modelling of spur gear pair is shown in Figure In a short time, contact points between a pair of gear teeth is deformed elastically rc1: base radius of driver gear; rc2: base radius of driven gear; r1: pitch radius of driving gear; r2: pitch radius of driven gear; w: pressure angle Figure Diagram of a pair of interlocking gears HỘI NGHỊ QUỐC TẾ KHOA HỌC CÔNG NGHỆ HÀNG HẢI 2016 207 THE INTERNATIONAL CONFERENCE ON MARINE SCIENCE AND TECHNOLOGY 2016 Thus, model gearboxes has the property that inertial properties J (Kg.m2) and elastic properties is characterized by stiffness K (N/m) When vibration analysis of gears uses line of action AB to calculate 2.1 Dynamic modelling of spur gear pairs Modelling of spur gear pairs in figure can be simplified as shown in figure The gear train describes similar pair of disks, their mass are M1 and M2, they is associated with a pair of spring in series has the individual stiffness respectively K1, K2 Figure Modelling of elastic oscillations of spur gear pairs The effective mass of driving gear and driven gear M1, M2 are determined of formula: J1 J1cos M1 2 r1 mn z1 cos w (1) J2 J cos r22 mn2 z22 cos w (2) M2 Where: J1, J2 - moment of inertia of driving gear and driven gear; Reference radius r1 mn z1cos mn z2cos , r2 ; - Tooth taper angle; mn - Normal module 2cos 2cos The individual tooth stiffness of apair of teeth in contact is obtained by assuming that one of the mating gears is rigid and applying load to the other The individual stiffness Ki at any meshing position i can be obtained by dividing the applied load by the deflection of the tooth at that point Characterizing the elastic property of driving gear and driven gear is the individual stiffness K1, K2, are determined by formula: K1 K2 P1 y1max P2 y2max 3E1I1 2 b1E1 hf 125 (3) 3E2 I 2 b2 E2 h3f 125 (4) Where: P1, P2 - Tangential force on the gear pairs; y1max, y2max - Maximum shear deformation of the teeth: 3 Ph Ph Ph f1 f1 f1 y1max E1 I1 E1I1 3E1I1 y2max P2 h3f 2 E2 I P2 h3f E2 I P2 h3f 3E2 I HỘI NGHỊ QUỐC TẾ KHOA HỌC CÔNG NGHỆ HÀNG HẢI 2016 (5) (6) 208 THE INTERNATIONAL CONFERENCE ON MARINE SCIENCE AND TECHNOLOGY 2016 Figure Components of the applied load I1, I2 moment of inertia of tooth cross-section: 3 m m b1 n b2 n 3 bs bs 3 I1 1 b2 mn3 b1 mn3 ; I 2 12 12 12 12 96 96 s1, s2- Normal pitch: s1 s2 mn (7) Whole depth: h f h f 1, 25.mn At any position in the mesh cycle,apair of teeth in contact can be modelled as two linear springs connected in series The system stiffness against the applied load, called the combined mesh stiffness Kth at contact point P At the moment, modelling of elastic oscillations is provided the oscillation system with a effective mass Mth and a spring with stiffness Kth can be calculated by the following equation: M1 M J1 J cos M th M1 M J1 z22 J z12 mn2 cos (8) K1 K 2 3b1b2 E1E K1 K 125 b1E1 b2 E (9) Kth Own oscillation frequency of a pair of interlocking gears: fn 2 K th m cos n M th 100cos 10 J1 z22 J z12 b1b2 E1 E b1E1 b2 E J1 J (10) 2.2 The cause of vibration of a pair of interlocking gears Excitation frequency of driving gear: f n1 z1 60 (11) Resonance occurs when excitation frequency f = fn, coincides with very strong oscillation of a pair of interlocking gears On the contrary, when f