Mechanism and Machine Theory 133 (2019) 127–149 Contents lists available at ScienceDirect Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory Research paper Concave and convex modiﬁcations analysis for skewed beveloid gears considering misalignments Siyuan Liu, Chaosheng Song ∗ , Caichao Zhu, Gaoxiang Ni, Najeeb Ullah The State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing 400030, China a r t i c l e i n f o Article history: Received 9 August 2018 Revised 14 November 2018 Accepted 20 November 2018 Keyword: Skewed beveloid gear Contact characteristics Concave and convex modiﬁcations Misalignments Shaft deformation a b s t r a c t The mathematical models of beveloid surface with concave and convex modiﬁcations were derived. The ﬁnite element and analytical mesh models were developed considering the shaft deformation and misalignments including pinion axial position, center distance and shaft angle errors. Then, tooth contact analysis were conducted to study the impacts of loads and misalignments on contact characteristics. Results show that the higher load ap- plied has stronger effects for enlarging contact pattern and lowering transmission error with the concave and convex modiﬁcations. Moreover, the proposed tooth modiﬁcations can decrease the sensitivity of contact characteristics to misalignments. The shaft angle error has the greatest impact on the mesh behaviors. The contact path and pattern are prone to shift the location along tooth trace direction due to shaft angle and center dis- tance errors. The contact stress, bending stress and transmission error are seen to increase due to these two assembly errors. Pinion axial position error does not affect the location of the contact path and pattern signiﬁcantly. However, it can increase the magnitude and change the location of the maximum tooth bending stress in the axis direction. Finally, it only produces minimal effect on the contact stress, contact area and transmission error. © 2018 Elsevier Ltd. All rights reserved. 1. Introduction Beveloid gears, commonly called as involute helicoid or conical involute gears, are applied to intersected and skewed gearing transmission with small shaft angle, which offers signiﬁcant space and weight savings for angle-drive marine propul- sion system, all-drive automotive drivetrain and Benson’s Swash-Plate engine. However, the main shortcoming for this type of driving is the theoretical point contact that results in the high relative sliding velocity, low surface durability, high noise and vibration. To overcome these shortcomings, the macro-geometry design method with proximate line contact constraints to calculate and control the ﬁrst principle directions(FPD) angle for engaged tooth surfaces were proposed [1–2]. But the problem is that the decrease of the designed angle tends to increase the sensitivity of the gear mesh characteristics to the misalignments due to the manufacturing errors, assembly errors and deformations. In general, tooth modiﬁcation is a typi- cal approach to mitigate this issue. Currently, there were many studies done to investigate the tooth modiﬁcations including crowing and slope modiﬁcations in the design and manufacturing of gears. But, very little investigations have been directed at the concave-convex modiﬁcations and the sensitivity analysis to the misalignments for beveloid gears with a very small skewed shaft angle (lower than 15 °). ∗ Corresponding author. E-mail address: chaoshengsong@hotmail.com (C. Song). https://doi.org/10.1016/j.mechmachtheory.2018.11.012 0094-114X/© 2018 Elsevier Ltd. All rights reserved.