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余弦齒輪傳動(dòng)的傳動(dòng)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 design[1] . 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 al[2] , 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 al[3], 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 al[4], 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 al[5], introduced the generation process of a micro-segment gear. ZHANG, et al[6], presented a double involute curves, which are linked by a transition curve and form the ladder shape of tooth.
LUO, et al[7], 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 investigated[8]. 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, θ is the rotation angle relative to system 1O1,x1,y1 as shown in Fig.2, β is the angle between x1-axis and the tangent of any point on the cosine profile, φ1 is the rotational angle of gear 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 teeth[9] . 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 follows[12]
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(chǎn) 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.
圖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．a(chǎn)nd Poisson’s 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．
圖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，F(xiàn)ig．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，F(xiàn)ig．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 drives—a cosine gear drive is investigated．which takes a cosine curve as the tooth profile．Based on the mathematical model, the characteristics including the contact ratio．the sliding coefficient and stresses are studied．The effects of gear design parameters．such as the number of teeth，pressure angle at pitch circle，on stresses of cosine gears have also been analyzed．The results of performed research allow the following conclusions to be drawn．
(1) The contact ratio of the cosine gear drive is about 1．2 to1．3．which is about 20％ less than that of the involute gear drive according to Table 1．
(2)The sliding coefficient of the cosine gear drive is smaller than that of the involute gear drive according to Fig．5．
(3)The contact and the bending stresses of the cosine gear drive are lower than that of the involute gear drive．For instance，under the given parameters as shown in section 2, the maximum contact stress of the cosine gear is reduced by about 22．23％ in comparison with the involute gear, and the compression bending stress is 28．67％ less than that of the involute gear．
(4) Both the contact and bending stresses decrease with the growth of the number of teeth and the pressure angle according to simulation results of the example FE mode1．
(5)The cosine gear drive is a new type of gear drives．Therefore．other characteristics such as inspection，sensitivity of center distance error of this drive and its manufacturing should be researched further.