Towards Optimum Involute Gear Design by Combining Addendum and Thickness Modifications

Transcription

1 Towards Optimum Involute Gear Design by Combining Addendum and Thickness Modifications Vasilios Spitas and Christos Spitas Abstract Involute gear sets are being produced through a variety of cutting tools and generation methods in a number of pressure angle and whole depth systems with m being the most common. Positive addendum modifications (profile shifts) are also common particularly because the resulting teeth are stronger, although at the expense of the contact ratio, scoring resistance and pitting resistance. This paper discusses the effect of combined addendum modifications and changes in nominal tooth thickness of meshing gears on the minimization of the root bending stress. The tooth mesh-strength problem is treated as non-dimensional, which substantially reduces computational time as well as the total number of design variables. Instead of modeling the loaded gear tooth and running a numerical method (i.e. FEM) to calculate the maximum root stress at every iterative step of the optimization procedure, the stress is calculated by interpolation of tabulated values, which were calculated previously for different combinations of the design parameters. Significant stress reduction was achieved in this way as was confirmed experimentally with photoelasticity. Keywords Spur gears Root stress BEM Optimization Photoelasticity List of Symbols a o a 12 b c c Pressure angle Center distance Tooth width Cutter radius coefficient V. Spitas (&) National Technical of Athens, Athens, Greece vspitas@central.ntua.gr C. Spitas Delft University of Technology, Delft, The Netherlands c.spitas@tudelft.nl G. Dobre (ed.), Power Transmissions, Mechanisms and Machine Science 13, DOI: / _12, Ó Springer Science+Business Media Dordrecht

2 174 V. Spitas and C. Spitas c f c k c s e P N r b r g r k r s o t g x z Dedendum coefficient Addendum coefficient Thickness coefficient factor Contact ratio Normal force Operating pitch radius Involute base radius Outside radius Critical bending stress Tooth thickness at pitch circle Base pitch Addendum modification coefficient Number of teeth 1 Introduction Gear optimisation has always been a difficult task, mainly due to the complex kinematics and the continuously changing load in terms both of point of application, direction and magnitude. Despite the advancement of novel computational techniques gear optimisation was mainly limited to specific gear pairs (Yeh et al. 2001; Litvin et al. 2000) mostly due to the multitude of the gear design parameters (Ciavarella and Demelio 1999). Some researchers deal with this fact by using empirical formulas (Pedrero et al. 1999), or by introducing simplified models (Andrews 1991; Rogers et al 1990). A different approach was followed by Spitas et al. (2006) and Spitas and Spitas (2006) where non-dimensional teeth were considered and the Highest Point of Single Tooth Contact (HPSTC) was defined with respect to the geometry of the tooth in question and the contact ratio of the pair instead of the standard design parameters of the meshing gear, thus reducing the total number of parameters. Each gear was modeled and consequently loaded at different points corresponding to different values of the contact ratio and numerical analysis followed to calculate the maximum root stress. The resultant values were tabulated in a stress table characterizing a given number of teeth, which were used in an optimization algorithm, where all the required intermediate values were quickly calculated by interpolation of the tabulated ones. The benefits of this modeling technique include improved accuracy and significantly smaller calculation times as opposed to the standard techniques employed. Also, due to the stress table concept the method can be readily synthesized in a modular way in any problem requiring the calculation of the maximum fillet stress. The optimum design was verified using two-dimensional photoelasticity on PCB plastic gear-tooth models.

3 Towards Optimum Involute Gear Design Basic Geometrical Modelling In a pair of spur gears let gear 1 be the pinion and gear 2 the wheel. The law of gearing (Townsend 1992) requires that these gears should have the same nominal pressure angle a o and the same module m in order to mesh properly. In the general case the gears are considered to be non-dimensional (i.e. with m = 1, b = 1) and incorporate addendum modifications x 1 ; x 2 respectively. Their pitch thickness is given by the following relationship: s oi ¼ c si p þ 2x i tana o ð1þ where c si is the thickness coefficient of gear i, ði ¼ 1; 2Þ; which in the general case is c s1 6¼ 0:5 6¼ c s2 : The center distance O 1 O 2 of the non-dimensional gear set is calculated using the following formula: a 12 ¼ z 1 þ z 2 2 þ ðx 1 þ x 2 Þ ð2þ The actual operating pitch circle r bi of gear i; ði ¼ 1; 2Þ should verify the law of gearing and therefore be equal to: r bi ¼ z i a 12u ð3þ z 1 þ z 2 Gears 1 and 2 are revolving about their centers O 1 and O 2 respectively and meshing along the path of contact AB as illustrated in Fig. 1. During meshing there are two pairs of gear teeth in contact along the segments AA 0 and BB 0 ; thus sharing the total normal load, while there is only a single such pair when tooth contact takes place along the region A 0 B 0 ; carrying the total normal load. Point B 0 is the Highest Point of Single Tooth Contact (HPSTC) for gear 1 and its position, defining the radius r B 0; is (Spitas et al. 2006; Spitas and Costopoulos 2001): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii r B 0 ¼ O 1 B 0 ¼ r 2 k1 þ ðe 1Þt g ðe 1Þt g 2 r 2 k1 r2 g1 ð4þ From the above equation it is evident that the position of the HPSTC of a gear depends only on its geometry and on the contact ratio of the pair, in which all the characteristics of the mating gear are incorporated in a condensed form. Stresses can also be calculated in non-dimensional teeth r u ðz, x, c s ; eþ with unit loading P Nu ¼ 1 and related to the actual stress r using the following equation: r ¼ r u P N bm ð5þ

4 176 V. Spitas and C. Spitas Fig. 1 Path of contact O 2 Gear 2 r k2 r b2 r g2 A t g t g A B C B E r k1 D r g1 r b1 Gear 1 O 1 3 Constrained Optimisation of Gear Teeth The static analysis results obtained by loading the aforementioned non-dimensional gear sets with various addendum modification coefficients and tooth thickness coefficients were tabulated in stress tables (Spitas et al. 2006; Spitas and Spitas 2006). In order to calculate an intermediate stress value r corresponding to a given number of teeth ðþand z set of parameters z, x, c s ; e not necessarily included in the stress table, linear interpolation was used as described in (Spitas et al. 2006). Analytical optimization methods are not suitable for gear stress optimization problems due to the complex implicit functions that relate the main geometrical variables to the resulting stresses. An efficient method of solving such intricate problems is the Complex algorithm (Box 1965), which calculates the minimum of a function of n variables f(x), where x ¼ ðx 1 ; x 2 ;...; x n Þ T is the variable vector. The independent variables of a stress optimization problem considering nondimensional gears are the following: Addendum modifications: x 1 for gear 1 x 2 for gear 2 Thickness coefficients: c s1 for gear 1 c s2 for gear 2. The objective function without any constraints is defined as minfðx 1 ; x 2 ; c s1 ; c s2 Þ¼maxðr 1 ; r 2 Þ, where r 1 ; r 2 are the maximum tensile stresses developed at the fillets of the conjugate gears 1 and 2 respectively when loaded at their corresponding HPSTC. Also, the optimization must be constrained because the optimum teeth should still fulfill certain operational criteria. There are 7 different constraints described below and in order to include them in the optimization

5 Towards Optimum Involute Gear Design 177 procedure the following form of the objective function is adopted using weighted residuals: min fðx 1 ; x 2 ; c s1 ; c s2 Þ ¼ maxðr 1 ; r 2 Þþ X7 i¼1 w i c i ð6þ The penalty functions c i and the weighting coefficients w i employed in Eq. (6) are defined below. Constraint 1: Allowable addendum modification: The addendum modification coefficient for gear i is restricted between two values x imin and x imax depending on the number of teeth z: These values are dictated by common gear practice and manufacture. The penalty functions are defined therefore: If x i \x imin or x i [ x imax ; i ¼ 1; 2 then w 1 c 1 ¼ 1000; r 1 ¼ r 2 ¼ 1000 If x imin x i x imax ; for every i ¼ 1; 2 then w 1 c 1 ¼ 0: Constraint 2: Allowable thickness coefficients: For technical reasons the cutting tool producing the gears (rack cutter, pinion cutter or hob) cannot have infinitely thick or infinitely thin teeth and this imposes a constraint on the resulting thickness of the produced gear. Therefore the thickness coefficient should range between the values c smin and c smax ; which are the limit values specified in the stress tables. If c si \c simin or c si [ c simax ; i ¼ 1; 2 then w 2 c 2 ¼ 1000; r 1 ¼ r 2 ¼ 1000 If c simin c si c simax ; for every i ¼ 1; 2 then w 2 c 2 ¼ 0: Constraint 3: Minimum radial clearance: In order to ensure that the conjugate gears operate without the risk of seizure, there should be a minimum allowable radial clearance c rmin m,, where c rmin ¼ 0:25: For the dimensionless gear i it is calculated from the equation: c ri ¼ a 12 r kiu r fiu ¼ c f c k ¼ c r ; where r uki ¼ z i 2 þ x i þ c k and r ufi ¼ z i 2 þ x i c f are the tip and root radius of the non-dimensional gear i and c k ¼ 1:0; c f ¼ 1:25 are the addendum and dedendum coefficients respectively. The penalty functions are formulated as follows: If c r \0:25 then c 3 ¼ 0:25 c r and w 3 ¼ 10 Here the penalty function c 3 has been chosen to be a function of the radial clearance in order to help the convergence of the solution at points where the nondimensional coefficient c r approaches its nominal value of The value of the weighting coefficient was chosen equal to 10 in order to improve the convergence of the algorithm. If c r 0:25 then w 3 c 3 ¼ 0: Constraint 4: If the tip radius r ki of gear i revolving about O i exceeds a maximum value r kimax so that the intersection of the tip circle of the gear with the common path of contact at point U defines on the mating gear j a radius which is lower than its form radius r js ; then interference occurs, since the tooth part below

6 178 V. Spitas and C. Spitas the form radius has a trochoidal and not an involute form. Consequently, it should always be r ki r kimax ; where r kimax ¼ O i U: In terms of the corresponding dimensionless gears this results in the following penalty function: If r kiu [ r kimax ; i ¼ 1; 2 then c 4 ¼ maxðr k1u r k1max ; r k2u r k2max Þ and w 4 ¼ 5: If r kiu r kimax for every i ¼ 1; 2 then w 4 c 4 ¼ 0: Constraint 5: Minimum tip thickness: In common gear practice the tip thickness is never below 0.2 times the module or tip fracture would occur. In a nondimensional gear the tip thickness should always be s ku 0:2: If s kui \0:25; i ¼ 1; 2 then c 5 ¼ minðs k1 ; s k2 Þ and w 5 ¼ 10 If s ki 0:25 for every i ¼ 1; 2 then w 5 c 5 ¼ 0: Constraint 6: Allowable contact ratio: In order to ensure smooth and unproblematic running the contact ratio of a gear pair should exceed 1.2. A usual upper limit is 1.8, which in 20 standard or shifted spur gears is never surpassed. Similarly to the constraints 1 and 2 the contact ratio eof the pair should lie in the range defined in the stress tables, therefore big penalties are applied at the boundaries: If e\e min or e [ e max then w 6 c 6 ¼ 1000; r 1 ¼ r 2 ¼ 1000 If e min e e max then w 6 c 6 ¼ 0: Constraint 7: Allowable backlash: The backlash of a gear pair (B) should always be positive and usually optimized designs require that this is kept minimum since the thicker the working teeth are, the less the root stress is. Although zero backslash is never actually desirable for power transmissions, the presence of a minimum backlash does not seriously reduce the tooth thickness, hence the root stresses, and therefore in order to simplify the calculations the optimum backlash can be considered zero. This can be expressed in terms of a penalty function, suitably big beyond the permissible boundaries, as: c 7 ¼ B and w 7 ¼ 100: After the formulation of the objective function, the algorithm described in Fig. 2 is executed. 4 Results and Discussion Various combinations of a 20 involute pinion with 12 teeth meshing with gears having 15, 18, 22, 28 and 50 teeth were modeled and examined. The stress tables were first constructed for all the above-described numbers of teeth and for the optimization algorithm the following parameter values were used: e = 10-4 (tolerance), a = 1.2 (reflection coefficient), b = 1.0 (expansion coefficient), c = 2.0 (contraction coefficient). The parameter values were chosen so as to provide quick convergence and stability of the algorithm.

7 Towards Optimum Involute Gear Design 179 Fig. 2 The optimization algorithm START Physical Problem Stress Table Gear 1 Stress Table Gear 2 Non-dimensional Problem Complex Algorithm Optimum? No Constraints Yes Re-dimensioning of optimum solution Output END In Fig. 3 the values of the maximum root stress for the non-dimensional pinion with 12 teeth and its conjugate gears with teeth are plotted. In this figure it can be observed that the maximum pinion stress (gear 1) is always greater than the maximum stress on the mating wheel (gear 2) in the case of standard gears while Fig. 3 Non-dimensional root stress for z 1 ¼ 12 pinion teeth 4,00 3,50 3,00 Gear 1 (standard Gear 2 (standard 2,50 2,00 Optimum 1,50 1, Number of teeth of gear 2

8 180 V. Spitas and C. Spitas Fig. 4 Photoelastic fringe patterns on sample gear-tooth models 56 lbs. these stresses are equal in the case of optimized gears. The total computational time was 12 s on a 2.4 GHz Intel Quad-Core system. The optimization algorithm used a complex of 1,000 vectors and reached the optimum solution after 29 iterations performing over 40,000 stress calculations. By following the standard methodology performing full gear modeling (run time 0.2 s), mesh generation (run time 0.1 s) and BEA (run time 6 s) at each iterative step, this would result in a total run time of nearly 70 h. The optimum design has been experimentally verified using two-dimensional photoelasticity (Spitas et al. 2006). The optimum gear tooth of a 28-tooth pinion / 50-tooth wheel system is illustrated in Fig. 4. The photoelastic specimen was cut from a 10 mm PSM-1 sheet from Vishay Inc. and placed on a special fixture on a planar polariscope where it was loaded at its HPSTC with progressively increasing normal load until a 3rd order isochromatic fringe appeared at the root fillet. Similar tests on other plastic models indicated that the numerical results are in good agreement with photoelastic investigations (Spitas et al. 2006). The experimental results are in excellent agreement with the numerical predictions (maximum deviation of 3.6 %) and the new design offers a decrease of the maximum fillet stress ranging from 13.5 to 36.5 % depending on the geometrical characteristics of the gear pair. 5 Conclusion Non-dimensional gear tooth modelling in terms of the number of teeth, the profile shift and the tooth thickness coefficient was employed for gear stress optimization with the Complex algorithm. The non-dimensional models were used in order to decrease the total number of the optimization parameters by introducing the

9 Towards Optimum Involute Gear Design 181 contact ratio of the pair as the parameter defining completely the point of loading. This reduction in the number of parameters enabled the tabulation of the maximum root stress developed on each non-dimensional gear with a given number of teeth for different values of addendum modification, pitch thickness and contact ratio using numerical analysis. Throughout the optimization procedure, the stress values for different combinations of the geometrical parameters of the conjugate gears of the pair were calculated from interpolation of the tabulated values at high speed and with satisfactory accuracy. In this way, the run time decreased dramatically without any effect on the accuracy. References Andrews J (1991) A finite element analysis of bending stresses included in external and internal involute spur gears. Strain Anal Eng Des 26(3): Box M (1965) The complex algorithm, Computers 8:42 Ciavarella M, Demelio G (1999) Numerical methods for the optimization of specific sliding, stress concentration and fatigue life of gears. Int J Fatigue 21: Litvin F, Qiming L, Kapelevich A (2000) Asymmetric modified spur gear drives: reduction of noise, localization of contact, simulation of meshing and stress analysis. Comput Methods Appl Mech Eng 188: Pedrero J, Rueda A, Fuentes A (1999) Determination of the ISO tooth form factor for involute spur and helical gears. Mech Mach Theory 34: Rogers C, Mabie H, Reinholtz C (1990) Design of spur gears generated with pinion cutters. Mech Mach Theory 25(6): Spitas V, Costopoulos T (2001) New concepts in numerical modeling and calculation of the maximum root stress in spur gears versus standard methods. A comparative study. In: Proceedings 1st national conference on recent advances in Mechnical Engineering, Patras, No. ANG1/P106 Spitas V, Spitas C (2006) Optimising involute gear design for maximum bending strength and equivalent pitting resistance. Proc IMechE Part C: J Mech Eng Sci 221: Spitas V, Costopoulos T, Spitas C (2006) Optimum gear tooth geometry for minimum fillet stress using BEM and experimental verification with photoelasticity. J Mech Des 128(5): Townsend D (1992) Dudley s gear handbook. Mc-Graw Hill, New York Yeh T, Yang D, Tong S (2001) Design of new tooth profiles for high-load capacity gears. Mech Mach Theory 36: