## 1 Introduction  
Hyopid gears are widely applied in power transmissions for intersecting and crossed axes, respectively. Stress concentration is likely to occur at the edge of the tooth due to assembly errors and deformations, which could affect the fatigue life of gears. As a result, the technology of tooth surface modification is adopted in processing. Tooth surface design and processing is always the key point of research for hypoid gears because of geometry-complicated tooth surface and difficult processing techniques. TCA [1] and LTCA [2] are applied to control the contact quality and dynamic performance for hypoid gear. Many works are concerned with tooth design and processing to eliminate edge-loading contact conditions and improve the contact performance of hypoid gears transmissions, such as local synthesis tooth [3], high-order transmission error tooth [4; 5; 6], real tooth [7; 8], global synthesis tooth [9], hypoid gear transmission (HGT) cutting technique [10], etc. These methods are focused on correcting the tooth with parabolic transmission error based on a cradle-type machine, and can effectively reduce stress concentration at the edge of the tooth, but meanwhile could cause excessive mismatch of gear pairs. Therefore, the methods cannot fundamentally improve the dynamic meshing performances. Recently, in [11], a novel method of conjugated action for crowning in a face-hobbed spiral bevel and hypoid gear were driven through the spirac system to improve gear pair contact degree and meshing quality. Besides, in Ref. [12], a method of grinding a stick blade profile to cut hypoid gears was used to improve the machining accuracy of gear surface.  
High-order tooth correction technology of spiral bevel gear [13; 14; 15; 16; 17] based on multi-axis computer numerical control (CNC) machine and machine-tool settings contribute to the processing tooth with complex modification, such as ease-off flank modification. Ease-off flank modification, which can directly demonstrate various mismatch relations between the modified tooth surfaces and conjugate tooth surface, was applied to tooth correction [18; 19] for hypoid gear and spiral bevel gear. Ease-off flank modification approaches were developed according to given transmission error and contact area for spiral bevel gear in [20; 21; 22]. Besides, in [23; 24; 25], multi-objective tooth optimization was carried out for spiral bevel gears and hypoid gear with modification-based LTCA to improve the overall meshing quality. According to [26], an approach of ease-off flank modification with three-segment parabolic transmission error tooth are developed for bevel gear. In [27], an ease-off modification approach for hypoid pinions based on a modified error sensitivity was proposed to improve the meshing performances. In [28], a penetration-based gear contact model for accurate and numerically efficient TCA of spiral bevel gears was used to investigate the simulation of gear alignment errors. In [29], a procedure to determine the extreme relative curvatures between conjugate gear flanks was presented, and ease-off topography and unloaded transmission error were used to assess conjugate action between mating gear flanks in direct contact.  
The above studies are focused on TCA. LTCA simulation was used for spiral bevel gear and hypoid gear with ease-off flank modification. In order to improve the meshing quality of the hypoid gear, this paper proposes a method of tooth design and aims to analyze the pinion tooth with free ease-off flank modification expressed by both predesigned transmission error function and tooth profile modification curves, and the influences of the actual coincidence degree on ALTE under loads is discussed.