## 1 Introduction
To decrease the noise and vibration of a meshing helical and spur gear pair, [PERSON] et al. [12; 13; 14; 15] suggested an equation system with a five degree of freedom to modify their geometry in order to lower TE and avoid contact at the edge of gear teeth. Later, they applied this approach to study the contact of the misaligned spur and the helical gear pairs by applying the double crowning of tooth surfaces with parabolic TE function. Moreover, [PERSON] et al. [16; 17] used this novel topology of tooth surfaces to localize the bearing contact. Misalignments of gears can be divided into two types: angular and linear, both of which have an adverse effect on the contact of gears and engagement. Researchers [PERSON] and [PERSON] [18] proposed a more detailed separation of gear misalignment, thus distinguishing them into four types: axial and radial misalignments (linear) and yaw and pitch misalignments (angular), where radial, axial, yaw (in-plane) misalignments, and pitch (out-of-plane) misalignment [19] influence the gear transmission and surface contact. The results of studies of spur gears have shown that the surface contact is more sensitive to angular misalignments than to linear ones and also that the contact area between two gears is significantly reduced in the case of angular misalignment compared to linear ones [20]. [PERSON] and [PERSON] [21] discovered that high-contact-ratio gears under angular misalignment produce more significant stress compared to linear misalignment. Several solutions to gear misalignment have been suggested by various researchers. A representative example is the tooth modification proposed by [PERSON] [22], who found that tooth modifications such as crowning can substantially reduce the stress concentration caused by misalignments. The gear tooth modifications, also called \"corrections\", have been described in various guidelines [23; 24; 25; 26]. The goal of such modifications is to remove the risk of edge contact and related failure during the tooth contact action. It must be mentioned that modifications are errors in themselves. In particular, unnecessary corrections lead to increased TE, overloads, and vibration [27]. The tooth modification focused on the crowning along the tooth flanks and the profile modification along the tooth height. The spur gear with modified tooth surface creates a relatively smooth distribution of pressure and prevents the edge contact effect, making it a capable solution for misalignment. [PERSON], in his work [28], also discussed misalignment of the gears and recommended modification of the tooth flank as an appropriate solution. Furthermore, [PERSON] highlighted the significance of finding optimum criteria for crowning based on static loads and the amount of misalignment. The risk of micro-pitting in gears is lowered by optimum tooth surface adjustment. As has been mentioned earlier,excessive modification of the tooth creates vulnerability to micro-cracks and micro-pitting, which lowers the life of the gear and can potentially aggravate the dynamic response, whereas too little crowning creates issues with localization of contact [29]. Thus, modification of the gear tooth surface is an effective way to improve the meshing properties of gears in order to reduce the sensitivity of the alignment errors; however, there is a necessity to determine the proper amount of modification in order to obtain a smooth transmission of the gear system. Presently, in a pursuit to design higher transmission quality, many studies have focused on the selection of optimal tooth modifications designed to minimize TE by crowning with various types of profiles. Longitudinal modification of gears is traditionally expressed as a parabolic curve. The idea of the predesigned parabolic function approach has been described by [PERSON] [30], who showed that double-crowned gears enable the reduction of vibration and noise of gear drives and provide a parabolic function of TE by absorbing the linear function of TE caused by gear misalignment. [PERSON] et al.