[机械模具数控自动化专业毕业设计外文文献及翻译]【期刊】余弦齿轮传动的传动特性分析-外文文献

余弦齿轮传动的传动特性分析 Wang jian Luo shanming Chen lifeng Hu huaring School of electromechanical engineering Hunan university of science, and technology, Xiangtan 411201,china Abstract: Based on the mathematical model of a novel cosine gear drive, a few characteristics, such as the contact ratio, the sliding coefficient, and the contact and bending stresses, of this drive are analyzed. A comparison study of these characteristics with the involute gear drive is also carried out. The influences of design parameters including the number of teeth and the pressure angle on the contact and bending stresses are studied. The following conclusions are achieved: the contact ratio of the cosine gear drive is about 1.2 to 1.3, which is reduced by about 20% in comparison with that of the involute gear drive. The sliding coefficient of the cosine gear drive is smaller than that of the involute gear drive. The contact and bending stresses of the cosine gear drive are lower then those of the involute gear drive. The contact and bending stresses decrease with the growth of the number of teeth and the pressure angle. Key words: Gear drive Cosine profile Contact ratio Sliding coefficient Stress 0 introduction Currently, the involute, the circular are and the cycloid profiles are three types of tooth profiles that are widely used in the gear design1 . All of these gears used in different fields due to their different advantages and disadvantages. With the development of computerized numerical control (CNC) technology, a large amount of literature is presented in investigations on mechanisms and methods for tooth profile generation. ARIGA, et al2 , used a cutter with combined circular-arc and involute tooth profiles to generate a new type of Wildhaber-Novikov gear. This particular tooth profile can solve the problem of conventional W-N gear profile, that is, the profile sensitivity to center distance variations. TSAY, et al3, studied a helical gear drive whose profiles consist of involute and circular-arc. The tooth surfaces of this gearing contact with each other at every instant at a point instead of a line. KOMORI, et al4, developed a gear with logic tooth profiles which have zero relative curvature at many contact points. The gear has higher durability and strength then involute gear. ZHAO, et al5, introduced the generation process of a micro-segment gear. ZHANG, et al6, presented a double involute curves, which are linked by a transition curve and form the ladder shape of tooth. LUO, et al7, presented a cosine gear drive, which takes the zero line of cosine curve as the pitch circle, a period of the curve as a tooth space, and the amplitude of the curve as tooth addendum. As shown in Fig. 1, the cosine tooth profile appears very close to the involute tooth profile in the area near or above the pitch circle, i.e., the part of addendum. However, in area of dedendum, the tooth thickness of cosine gear is greater then that of involute gear. The mathematical models, including the equation of the cosine tooth profile, the equation of the conjugate tooth profile and the equation of the line of action, have been established based on the meshing theory. The solid model of cosine gear has been built , and the meshing simulation of this drive has also been investigated8. The aim of this work is to analyze the characteristics of the cosine gear drive. The remainder is organized in three sections 1, the mathematical models of the cosine gear drive are introduced. In section 2, the characteristics, including contact ratio, sliding coefficient, contact and bending stresses, of the cosine gear drive are analyzed, and a comparison study of these characteristics with the involute gear drive is also carried out. The influences of design parameters, including the number of teeth and the pressure angle, on contact and bending stresses are studied. Finally, a conclusion summary of this study is given in section 3. Fig . 1 1 Mathematical Model of the cosine gear drive According to Ref.8, the equation of the cosine tooth profile, the conjugate tooth profile and the line of action can be expressed as follows 公式 Where m and Z1 represent the modulus and the number of teeth, respectively, h is the addendum, I and a denote the contact ratio and the center distance, respectively, is the rotation angle relative to system as shown in Fig.2, is the angle between -axis and the 1(1,1,1) 1tangent of any point on the cosine profile, is the rotational angle of 1gear 1 which can be given as follows 公式 Fig.2 2 CHARACTERISTICS OF THE COSINE GEAR DRIVE Based on the mathematical model of the cosine gear drive, three characteristics, contact ratio, sliding coefficient, and stresses, are analyzed. In addition, all these characteristics are compared with those of the involute gears 2.1 Contact ratio The contact ratio could be considered as an indication of average teeth-pairs in mesh of a gear-pair and naturally is ought to be defined according to the rotation angle of a gear from gear-in to gear-out of a pair of teeth9 . As shown in Fig.3, the contact ratio of the cosine gear can be expressed as follows 公式 where and are the values of rotation angle as = and = ， respectively, which can be calculated by Eq (3) Fig.3 Contact ratio of the cosine gear drive Three examples as shown in Table 1 have been carried out by using program Matlab The contact ratios of the involute gear drives with the same parameters are also shown in Table 1 for the purpose of comparison. According to Table 1, the contact ratio of the cosine gear drive is about 1.2 to 1.3, which is about 20% less than that of the involute gear drive. According to Refs 10-11, the contact ratio of gears applied in gear pump is about 1.1 to 1,3, therefore, such cosine gear drive can be applied in the field of gear pump. Table 1 2.2 Sliding coefficient Sliding coefficient is a measure of the sliding action during the meshing cycle. A lower coefficient will have greater power transmission efficiency because of the less friction. The sliding coefficient is defined as the limit of the ratio of the sliding arc length to the corresponding arc length in plane meshing. The sliding coefficients U1 and U2 can be expressed as follows12 公式 Where and denote the radius of the pitch circle， respectively， L represents the vertical coordinate of point H in coordinate system ， H is the intersection point of the normal line of the contact point and the line ， as shown in Fig 4 FIG.4 Therefore， slope k of the straight line PH can be expressed as follows 公式 6 Substituting Eq (3) into Eq (6) gives 公式 7 where and are the differential coefficients of and to , respectively, which can be expressed as 公式 Therefore， the vertical coordinate of the point H in coordinate system can be expressed as follows 公式 8 Where (x0， Y0， ) denotes the coordinate of the contact point in coordinate system Substituting Eq (3)and Eq (7) into Eq (8) gives 公式 9 Substituting 0 and Eq (9) into Eq (5)， the sliding coefficients can be obtained The gears are designed to have a module of m=3 mm a number of teeth of Z1=35， and a transmission ratio of i=2 The pressure angle of the involute gear is 20o while it is 22。 for the cosine gear According to Eqs (5)-(9)， a computer simulation to plot the graphs of sliding coefficients for the driving and the driven gears of the cosine gear drive is developed as shown in Fig 5 The sliding coefficients of the involute gear drive 13 are also listed in Fig.5 for the purpose of comparison. According to Fig.5 the sliding coefficients of the cosine gear drive is smaller than that of the involute gear drive. which can help to improve the transmission performance. 图 5 2.3 Contact and bending stresses In general, an FEA model with a larger number of elements for finite element stress analysis may lead to more accurate results. However, an FEA model of the whole gear drive is not preferred, especially considering the limit of computer memories and the need for saving computational time This paper establishes an FEA model of three pairs of contact teeth for the cosine gear drive. Two models of contacting teeth based on the real geometry of the pinion and the gear teeth surfaces created in Pro/Engineer are exported as a IGES file which is then imported into the software Ansys for stress an analysis. The numerical computations have been performed for the cosine drive with the following design parameters： Z1=25， Z2=40。 m=3 mm， a=22。 ， a width of b=75 mm The basic mechanical properties are modulus of elasticity E = 210 GPa and Poissons ratio = 0 29 The torque is 98790 N mm Two sides of each model sufficiently far from the fillet are chosen to justify the rigid constraints applied along the boundaries A large enough part of the wheel below the teeth is chosen for the fixed boundary Areas are meshed by using plane-82 elements The finite element models are shown in Fig.6, and there are 3373 elements and 10053 nodes Two options related to the contact problem. Small sliding and no friction have been selected Fig 7 shows the contour plot of Von-Mises stress The numerical results are listed in Table 2 图 6 Tu7 Table 2 Under the same parameters， stress distribution of an involute gear drive shown in Fig 8 is also analyzed for the purpose of comparison The bending stress obtained in the fillet of the contacting tooth side are considered as tension stresses， and those in the fillet of the opposite tooth side are considered as compression stresses. Tu8 From the obtained numerical results， the following conclusions can be made： the maximum contact stress of the cosine Rear is reduced by about 22 23 in comparison with the involute gear The tension bending stress of the cosine gear is 25 34 less than that of the involute gear, and the compression bending stress is reduced by about 28 67 in comparison with the involute gear An application of a cosine tooth profile allows reducing both， contact and bending stresses 2 4 Influences of design parameters on stresses Based on the finite element models， two examples are used to clarify the influences of design parameters including the number of teeth and the pressure angle on contact and bending stresses Example l： the gears are designed to have a pressure angle of a=22o. at the pitch circle， a module of m =3 mm。 a width of b=75 mm The other main parameters are shown in Table 3 Table3 With the same material parameters as aforementioned， the contact and bending stresses of three sets of cosine gears are analyzed by using program Ansys Results are shown in Fig 9， Fig 7 and Fig 10， and the values of the contact and bending stresses are shown in Table.4 According to Table 4. both the contact and bending stresses decrease with the growth of the number of teeth For instance， the contact stress， tension and compression bending stresses are 569 76 MPa 11 7 5 1 MPa and 124 98 MPa， respectively， as the number of teeth Z1=20， while 410 61 Mpa 64 52Mpa and 74 41 MPa as the number of teeth Z1=30 Tu9 Tu10 Table 4 Example 2： the gears are designed to have a module of m=3mm，number of teeth Zt=25， a width of b=75mm The other main parameters are shown in Table 5 Table5 With the same material parameters as aforementioned, the contact and bending stresses are also computed by using program Ansys Results are shown in Fig 7， Fig 11 an d Fig 12， and the values of the contact and bending stresses are shown in Table 6 Tu11 Tu12 Table6 According to Table 6， the contact and bending stresses decrease with the growth of the pressure angle For instance， the contact stress，tension and compression bending stresses are498 98 M Pa 86 04 MPa and 95 59 MPa， respectively， as the pressure angle of =22。 while 395 43 MPa， 7 1 8 1 MPa， and 86.32 MPa as the pressure angle of =24。 3 CONCLUSIONS A new type of gear drivesa cosine gear drive is investigated which takes a cosine curve as the tooth profile Based on the mathematical model, the characte