هندسه بهینه چرخ دنده برای تنش فیلت حداقل با استفاده از BEM و صحت سنجی آزمایشی با فتوالاستیسیتی Optimum Gear Tooth Geometry for Minimum Fillet Stress Using BEM and Experimental Verification With Photoelasticity

۱ Introduction The advances in the field of computational mechanics and structural optimization have led to the development of numerical modeling techniques, which have been used in gearing applications to produce optimized designs of specific gear pairs Litvin
۱. However, it is admitted Ciavarella
۲ that since the design parameters of each individual gear are many, true real-time optimization requiring numerical stress analysis at every iterative step is practically impossible. In order to cope with this discrepancy, researchers resort to the use of empirical formulas Pedrero
۳, or further simplify the problem by assuming loading at the tip of the tooth Rogers
۴, yielding unreliable results when nonstandard teeth are studied. This paper introduces a new concept in gear modeling by using the contact ratio of a gear pair for the determination of the point of load application. Instead of using the standard design parameters, the problem is simplified by using dimensionless teeth and by incorporating all the geometrical characteristics of the mating gear in the contact ratio of the pair, thus reducing the total number of parameters from seven to three. Each gear is thus geometrically modeled and consequently loaded at different points corresponding to different values of the contact ratio
 and subsequently B.E.A. follows to calculate the maximum root stress. The resultant values are tabulated in a “stress table” characterizing a given number of teeth, which can be readily included in an optimization algorithm, where all the required intermediate values can be quickly calculated by interpolation of the tabulated ones. The new modeling technique offers improved accuracy and significantly smaller calculation times as opposed to the standard techniques employed. Moreover, owing to the concept of the “stress table,” it can be readily synthesized in a modular way in any problem requiring the calculation of the maximum fillet stress. Finally the results are verified using two-dimensional photoelasticity on polycarbonate plastic gear-tooth models. ۶ Conclusions In this paper the concepts of the nondimensional gears and the stress tables have been introduced and used for gear stress optimization with the Complex algorithm. The nondimensional gears are used in order to decrease the total number of the optimization parameters by introducing the 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 nondimensional gear with a given number of teeth for different values of addendum modification, pitch thickness and contact ratio using BEM. During the iterative 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 in the order of 8000 times compared to the standard approach without any effect on the accuracy. Therefore, the proposed modeling provides the design engineer with a fast, productive, reliable, and easy to implement tool for solving gear stress optimization problems and calculating gear stresses at cases not covered satisfactorily by the existing standards.