محاسبه سفتي درگيري چرخ دندهاي مخروطي اينولوت كروي با روش هاي تحليلي و تجربي

Analytical an‎d experimental calculation of spherical involute bevel gears mesh stiffness

مهندسي مكانيك

Bevel gears with involute an‎d octoidal profiles are the most commonly used types in gear design. Due to their superior characteristics, such as insensitivity to transmission errors an‎d shaft angle variations, spherical involute bevel gears are expected to become the stan‎dard reference in bevel gear manufacturing. However, modeling spherical bevel gears remains a major design challenge due to their complex geometry. In this study, the geometry of the gear tooth was first analyzed by deriving Napier’s equations within the framework of spherical trigonometry an‎d solving them numerically to calculate all relevant geometric angles. Based on the computed angles an‎d the relationships governing spherical involute geometry, a gear system consisting of a driver an‎d driven gear was designed an‎d fabricated using polylactic acid (PLA) for the prototype an‎d CK45 steel for the final model. The gear mesh stiffness was eva‎luated using three distinct approaches: an analytical method based on coupling theory an‎d tooth segmentation, an experimental method, an‎d a finite element method (FEM). Each approach introduced novel insights. Analysis of the first part revealed a significant correlation between the increase in polar angle an‎d azimuthal angle, specifically, at a pitch cone angle of 60°, an 8° increase in polar angle resulted in a 50% rise in azimuthal angle. The results indicate that the pitch cone angle plays a critical role in shaping the spherical involute curve, making its selec‎tion highly important. Moreover, increasing the pitch cone angle reduces the curvature of the flank surfaces, thereby simplifying the manufacturing process. Conversely, decreasing the pressure angle flattens the flank surfaces, making the tooth profile resemble that of octoidal gears. Changes in the sphere radius affect only the tooth size, not its shape. In the second part of the analysis, mesh stiffness was determined under various loading conditions, showing consistent trends across all cases. Transitioning the tooth geometry from planar involute to spherical involute led to a 4% increase in mesh stiffness. Additionally, mesh stiffness was found to increase with higher pressure angles an‎d contact ratios. For instance, increasing the pressure angle from 14° to 25° resulted in a 23% rise in mesh stiffness. The consistency between the calculated stiffness values an‎d the proposed methods validates the findings. Finally, contact ratio an‎d contact pattern were determined using the experimental test rig an‎d validated through finite element analysis, showing strong agreement between experimental an‎d numerical results.

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محسن رضائي

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