A_GENETIC_ALGORITHM_APPROACH_TO_MINIMIZE_TRANSMISSION_ERROR_OF_AUTOMOTIVE_SPUR_GEAR_SETS.pdf

A GENETIC ALGORITHM APPROACH TO MINIMIZE TRANSMISSION ERROR OF AUTOMOTIVE SPUR GEAR SETS D. J. Fonseca and S. Shishoo Sundersan et al. 1990; 1989). There is a lack of robust and effective optimization schemes to realistically model the gear noise phenomenon and relate it to gear design parameters. The gear optimization problem admits many design solutions, and often, the one to be selected is that which adapts best to the working environment. This paper focuses on the use of genetic algorithms to address such a design problem. LITERATURE REVIEW Harris (1958) was one of the first researchers to effectively model the spur gear teeth and recognize the importance of transmission errors. Optizs survey in 1969 entitled Noise of Gears provided data relating various gear parameters to the resulting noise. His curves for gear noise are widely used for acceptance testing of gears, and thus, the implicit assumptions in them are worth examining. They suggest that if two gear- boxes are designed so that one transmits twice the power of the other, then the more powerful gearbox would be 3 dBA noisier. Inspection of Opitzs global data on gears suggests that the noise produced in dBA is directly proportional to the manufacturing transmission error (Optiz 1969). Niemann and Baethge (1970) investigated the relationship between transmission error (TE) and acoustic noise. They showed how tooth bend- ing changes the contact ratio for both spur and helical gears. Following on Harriss work, they measured, using seismic equipment, the gear TE in a back-to-back test rig under the influence of increasing loads. Using long tip relief, they confirmed how a spur gear could be designed to have its minimum TE at the design load. Minimize Error of Gear Sets155 Remmers (1972) developed spur gear transmission error models, which approximate the gear tooth as a cantilever beam. The models included deflections due to bending, shear, and base rotation of the tooth, as well as Hertzian deflection at the contact zone. Estrin (1980) suggested an optimization method for a gear mesh based on the application of nonlinear programming techniques. The main tooth proportions are calculated to obtain the optimal properties, which satisfy the conditions linked to the strength, durability, noise, and manufacturing of the gears. The nonlinear model involved objective functions for minimiz- ing compressive stresses and maximizing contact ratio and tip thickness subject to various design and manufacturing constraints. Houser et al. (1984) developed a mathematical procedure to compute static transmission errors, and tooth load sharing for low- and high- contact ratio for internal and external spur gears. A suitable optimization algorithm based on the Complex Method of Box was used to minimize any combination of the harmonics of gear mesh frequency components of the static transmission error. The ability to predict the effects of off- optimum loads, or nonoptimum manufacturing errors on the trans- mission error was shown to be an important feature of the procedure. Two years later, Houser (1986) also developed a load distribution program (LDP) which uses the Simplex algorithm to solve a set of linear elasticity equations. These equations described the deflections of discrete points along the gear meshing points. He used his program to compute trans- mission errors for a 1:1 ratio set of helical gears. Since that time, his program has been applied widely to the study and design of numerous types of gear pairs. Carroll et al. (1989) constructed a dimensionless design solution to the spur gear problem. They introduced a new quantity called the material properties relationship factor. Through this approach, they concluded that the interaction of the bending stress constraint and one of the two contact stress constraints defines the optimum solution. They showed that, in a dimensionless space defined by the new parameters, the optimal gear geometry can be found independently of the load and speed requirements of the gear set. In cases where gear mesh is designed on American Gear Manufacturers Association (AGMA) rating equations, the method yields the smallest gear set that would achieve the desired bending and contact stresses. Sundersan et al. (1989) at the Gear Dynamics and Gear Noise Research Laboratory at Ohio State University developed a procedure that helps design spur gears with minimum transmission error, while being insensitive to manufacturing variance. The procedure uses Taguchis concept of parameter design. The parameter design concept seeks performance variation reduction by decreasing the sensitivity of an engineering 156D. J. Fonseca et al. design to sources of changes rather than controlling such sources. The pro- cedure initially generates candidate gear designs that meet the require- ments of a fixed center distance, reduction ratio, and design torque. These designs also satisfy geometrical constraints, such as avoiding under- cut, having a minimum root clearance, and working depth. These designs are then optimized for minimum transmission error by determining the optimum profile modification, and at the same time, making them least sensitive to manufacturing variance. The statistical optimization was carried out based on Boxs complex method. A unique approach presented by Chandrasekaran et al. (1998) was used in conjunction with a gear design preprocessor constructed by Regalado to perform design simulations of a large number of gear design cases. In each case, over 65,000 designs were evaluated, and the dominance filter resulted in 200 to 900 successful designs, depending on the tolerances applied. Further sorting with the viewer usually ends with 5 to 20 designs of similar performances. It is clear that various optimization methods have been employed in order to achieve the best possible set of variables to minimize gear noise. Optimization of gear design is a complicated task, and conventional optimi- zation tools would have real-life difficulties in achieving a global optimum. Some of the more advanced optimization schemes, such as genetic algorithms, have also been applied, although in a very limited, way to this problem. Yokota et al. (1998) used genetic algorithms to address the gear weight problem. He formulated an optimal weight design problem for a con- strained gear bending strength, shaft torsional strength, and individual gear dimensions as a nonlinear integer-programming problem. Balic et al. (1996) developed an integrated CAD=CAM expert system named STATEX for dimensioning, optimization, and manufacture of gears and gearing. To determine the optimum dimensions of gearing, the system uses a genetic algorithm model. The algorithm minimizes the sum of volumes of gear pitch cylinder by optimizing on parameters such as mod- ule, number of teeth, tooth width, etc. From the reported literature, it is obvious that the problem of gear opti- mization with relation to gear noise is highly complex, and that there are significant limitations for the application of traditional optimization techni- ques to gear noise reduction. Mathematical optimization models are diffi- cult to implement due to the discrete nature of the variables involved, and the complexity of the objective function. The use of more robust and reliable optimization techniques is needed to achieve better solutions. Recently, genetic algorithms have been applied successfully to gear design. Based on the literature findings, the use of genetic algorithms for gear noise reduction appears to be a promising approach. Minimize Error of Gear Sets157 MATHEMATICAL MODELING Most gear design problems present the following requirements and constraints: 1. Requirements: Fixed center distance, reduction ratio, and design torque. 2. Constraints: Maximum outer diameter, maximum face width, and maximum stress. Apart from these requirements and constraints, there is a whole array of discrete and continuous variables, which have to be taken care of while developing the optimization model (Houser 1984). In order to develop an objective function for optimization, a series of calculations had to be carried out through a formal methodology such as the one given by Tavakoli (1983). These steps were: . Determination of ideal involute profile. . Profile modification curve-fit. . Mesh compliance analysis of the gear pair. . Transmission error and load sharing computations. The various assumptions that were taken in development of the mathemat- ical model embedded in the GA-based algorithm were as follows: . Contact is always along the line of action during the duration of a mesh cycle. . The tooth is assumed to be nonuniform cantilever beam in both bending and shear analysis. . The load is assumed to be uniformly distributed along the tooth face width. . Spur gears are made of carbon steel. . The starting positions of profile modifications were fixed for this study. Involute Profile Generation The first step in Tavakolis (1983) method involves the calculation of the true involute profile of the tooth. Figure 1 is used here as a guide for determining the coordinates of the calculation points on a gear tooth. 158D. J. Fonseca et al. Knowing the thickness at the pitch point Tp, and the operating pressure angle /p, the angle c is given by: c /p? Tp=2 ? Rp1 where Rpis the pitch radius. Using the definition of an involute function, angle bjis given by: bj Tan?1Tan/p aj2 where ajis the position angle of the jth calculation point defined as the difference between the roll angle of the jth point and the roll angle of the pitch point. Also: Rj RpCos/p=Cosbj3 where Rjis the radius at point j. It follows from the figure: kj bj? c aj4 FIGURE 1 Coordinates of jth calculation point on gear tooth. Minimize Error of Gear Sets159 Finally, with the origin at the center of the gear, the coordinates of the jth calculation point along the range of contact of a tooth are given by: Xj RjCoskj5 Yj RjSinkj6 In order to represent the gear tooth by a numerical model, a finite number of calculation points are distributed along the tooth profile. Let w define the tooth spacing angle according to: w 2p=N7 where N is the number of teeth. Angle w is divided into NJ equal parts, so that a total of 2NJ calculation points are defined on a tooth profile and its extension. Each of these points is identified by an angle aj, called the position angle, according to: aj jw=NJ j2p=NJN8 where ? NJ1?j?NJ. This angle is illustrated in Figure 2. For points inside the pitch diameter, j is negative, and for the points outside the pitch circle, j is positive. Also, the pitch point is marked by aj 0 for which j 0. Figure 2 also shows that contact between the pairs of teeth is theoretical possible only when: aa? aj? as9 where aaand asare the angles of recess and approach, respectively. Profile Modification Curve-Fit Profile modification (or error) is defined as the amount by which the actual profile deviates from the true involute. If the material is removed from a tooth, the modification is specified as a negative quantity. The design variables that need to be optimized are the magnitudes of modifica- tions specified at four locations along a tooth profile. The calculation points on the tooth profile are determined using tooth geometry, and these same points are used as contact points during the mesh cycle (Tavakoli 1983). Four specific locations along the range of con- tact were chosen as shown in the Table 1. 160D. J. Fonseca et al. If the jth calculation point is located between modification points 1 and 2, knowing its position angle, aj, the modification at that point can be calculated by: Zj Z1 aj? a1Z2? Z1=a2? a110 Equation (10) is adjusted accordingly and coded if the calculation point is located between two other modification points (Tavakoli 1983). TABLE 1Location of Calculation Points PointContact range locationSymbol 1The tooth tip.Z4 2The high point located on the tooth profile between the tip and the operating pitch point. Z3 3The low point located on the tooth profile between the operating pitch point and the first point of contact. Z2 4The first point of contact.Z1 FIGURE 2 Theoretical contact criterion. Minimize Error of Gear Sets161 Mesh Compliance Analysis The tooth compliance is computed for those calculations that theoreti- cally come into contact during a mesh cycle. The displacements are calculated in a direction normal to the tooth profiles. The total static compliance of a pair of contacting teeth is assumed to be contributed by three sources, namely, cantilever beam deflection due to both bending and shear forces, rigid body tooth rotation at its base, and contact or Hertzian deflection. The gear teeth are modeled as nonuniform short cantilever beams in both bending and shear with an effective length of Le, which extends from the tip to the effective base, MM, as shown in Figure 3. The range of the contact is divided into a sequence of transverse seg- ments of rectangular cross section. Each section is denoted by the index i. For each segment, the height, the cross-sectional area (Ai), and the area moment of inertia (Ii) are taken as the average of these values at both faces. The total deflection is obtained by superimposing the contributions of indi- vidual segments. Each segment itself is considered as a cantilever beam with its left face as the fixed end, and the remainder of the tooth adjacent to the right face as a rigid overhang. According to Tavakoli (1983), the total deflection at the load position due to bending is: D11i WjCosbj2L3 i 3L2 iSij=6EeIi 11 D21j WjCosbjL2 iSij=2EeIi 12 FIGURE 3 Spur gear tooth modeled as a non-uniform cantilever beam. 162D. J. Fonseca et al. The deflection of the section due to moments is given by: D12i WjSijCosbj ? YjSinbjL2 i 2LiSij=2EeIi13 D22i WjSijCosbj ? YjSinbjLiSij=EeIi14 where Wjis the normal load at point j on the tooth profile, Sijis the moment arm, Liis the tooth segment thickness, Iiis the mean area moment of inertia, Yjis the y-coordinate of the contact point, and Eeis the effective Youngs modulus of elasticity given as: Ee E=1 ? v215 where E is the Youngs modulus of elasticity and v is the Poissons ration. It is assumed that the gears in question are made of carbon steel. The detailed description of the deviation of the above equations is given in Tavakoli (1983). The deflection of the tooth due to shear for a rectangular cross section is calculated from the following equation: Dsi 1:2 WjLiCosbj=GAi16 where G is the shear modulus of elasticity. Once the deflections due to bending and shear are computed, the total deflection of a tooth at the load position and in the direction of the applied load (normal to the profile) is determined by: Dbj RD11i D21i D12i D22iCosbj17 The total compliance coefficient due to tooth bending as a cantilever beam for the jth calculation point, Qbj, is thus given by: Qbj Dbj=Wj18 Due to fillet geometry and the flexibility of the tooth support material, the tooth acting as a rigid member rotating in its deflection contributes additional deflections. The foundation deflection is a function of not only the fillet geometry, but also the load position and direction, as indicated by Eq. (19). The foundation compliance at the position of the applied load, Minimize Error of Gear Sets163 Qfj, is given by: Qfj Cos2bj FE ( 5:306Lf=Hf2 21 ? vLf=Hf 1:534 1 0:4167tan2bj 1 v !) 19 where Lfand Hfare defined in Figure 4: Lf Xj? XM ? Yjtanbj20 Hf 2YM21 where XM and YM are the coordinators of the point, which identifies the start of the effective tooth base (Tavakoli 1983). The gear tooth also has contact deflections due to compression of the tooth between the point of contact and the tooth centerline. A semi-empiri- cal equation developed by Palmgren for contacting cylinders in roller bear- ings was used to calculate contact deflections for gear teeth. The Palmgren equation for contact compliance of gear teeth is: Qh 1:37=E0:9 12eF 0:8W0:1 n 22 FIGURE 4 Deflection of tooth due to foundation flexibility. 164D. J. Fonseca et al. where Wnis the total normal load, F is the face width, and E12eis the com- bined effective Youngs modulus of elasticity given by: E12e 2E1eE2e=E1e E2e23 E1eand E2eare the effective Youngs modulus of elasticity for the pinion and gear, respectively. Finally, the total tooth compliance at a point ident- ified by j is given by: Qj Qbj Qfj Qh24 Transmission Error and Load Sharing Computations The transmission error for a pair of meshing teeth depends on two main factors: the combined compliance of the pair of teeth and the manufactur- ing errors (i.e., tooth spacing error, tooth profile error, and run out error). A conventional set of three equations for load sharing and total trans- mission error for a low contact ration mesh is presented next (Houser 1984), where Wnis the total transmitted normal load and Qjis the com- bined compliance of the pinion and gear teeth at the jth calculation point along the range of contact, Q0jW0j Etj Ep0j25 Q1jW1j Etj Ep1j Es26 W0j W1j Wn27 Wjis the normal load shared by the same pair of teeth at the same calcu- lation point, and Etis the total transmission error, which is expressed as a linear displacement along the line of contact (Houser et al. 1984). The 0 and 1 subscripts refer to the leading and lagging gear pair, respectively. The transmission error is negative if the gear laps the pinion, and positive if it leads the pinion. The load is assumed to be uniformly distributed along the tooth face width. Esis the tooth spacing error, which is positive if the adjacent teeth are too close to each other, and negative if they are too far apart. Epis the tooth profile-error, which is positive if material is added to the tooth, and negative if it is remov