Mesh stiffness calculation of helical gears with profile modification

Abstract

Time-varying contact line and mesh stiffness are important parameters for dynamic investigation of helical gear pair with profile modification; however, the calculation methods of the time-varying mesh with high accuracy and high efficiency are also very limited currently. A new mesh stiffness calculation method for helical gear pairs with profile modification is proposed. The new method is conducted by cutting the helical gear tooth into slices to obtain the change of the contact line and the time-varying mesh stiffness with considering the change of each slice's engagement performance introduced by the profile modification. Then, the mesh stiffness of the assumed three different modification cases is calculated by using the proposed method. The calculated results indicate that the proposed method can be used to calculate the mesh stiffness of helical gear pairs, especially of those with tooth profile modification. 11Introduction Helical gears have been widely used in all sorts of mechanical equipment as a motion and power transmission device. Tooth profile modification is frequently used to improve their dynamic performance and reduce the impact when teeth start to engage or quit engagement. Profile modification causes a lot of changes to gear pairs, such as engagement situation [1], tooth stress [2], and transmission error [2-4]. Researchers conducted many studies to figure out the influence of profile modification on gear characteristics. Ozturk et al. [3] studied the dynamic transmission error and loaded static transmission error of a spur gear pair with profile modification using different models. The result showed that profile modification can be used to reduce dynamic transmission error. Bruyère and Velex [4] proposed a simplified analysis method to evaluate the effects of tooth profile modification on minimising transmission error variations for the spur and helical gear pairs, and they also proposed a unique equation which leads to a nearly constant transmission error. He and Singh [5, 6] studied the relationship between normal loads and friction forces for modified spur gear pairs. Velex et al. [7] worked on the possible reduction of contact line length introduced by profile modification. They thought this contact length reduction is symmetric with regard to the engage and exit process, and can be simply quantified by a single parameter which can be obtained through the reasonable approximations they proposed. Liu and Parker [8] presented an analytic model for the vibration analysis of multiple gears considering the influence of profile modification as profile modification changes the load distribution between tooth pairs. As mesh stiffness is an important parameter of gear pairs, a lot of studies about mesh stiffness were published. Chen and Shao [9] presented an analytical mesh stiffness calculation model for spur gear to calculate their mesh stiffness after profile modification considering that the tooth profile error changes the engagement performance of tooth pairs. The change in the contact line of the helical gear pair and the corresponding mesh stiffness follows a similar pattern. The mesh stiffness per unit length of the contact line was assumed to be constant in [10, 11]. Based on the constant mesh stiffness per unit length of contact line assumption, Jiang et al. [11] used the polynomial fitting method to smoothen the mesh stiffness curve. The tooth pair stiffness curve is extended to be parabola, and the mesh stiffness at the start and end of engage are symmetric in [12]. Based on the potential energy method of the spur gear, Wan et al. [13] proposed the accumulated integral potential energy method to study the time-varying mesh stiffness of helical gear pairs by dividing tooth into pieces, where each piece was regarded as a spur gear with no elastic coupling with each other. As for the existing mesh stiffness calculation of the helical gear, the methods proposed by Jiang et al. and Bruyère et al. [11, 12] are rough: the mesh stiffness per unit length of the contact line is not unchangeable and the contact line length after modification is hard to obtain through geometry study, thus making it hard to get the mesh stiffness after modification. Accumulated integral potential energy method did not take profile modification into consideration and the interaction between pieces was not included. Finite element analysis (FEA) can be used to calculate the mesh stiffness and contact line length of the modified helical gear, but it cost a lot of time to set up a model and analysis. The main work of this paper is trying to set up an analytical method to study the contact line length and mesh stiffness of helical gear with profile modification. The paper is arranged as follows. The presented mesh stiffness calculation method for helical gear pairs with and without profile modification is described in Section 2, and then the mesh stiffness and contact line length simulation of helical gear pairs with and without modification are discussed in Section 3. Finally, conclusions are given in Section 4. 22Proposed mesh stiffness calculation method After profile modification, some tooth segments keep engaged, some tooth segments lose contact, making the contact line length and mesh stiffness changed. Contact lose can be distinguished through comparing the modification amplitude with the deformation amount of the unmodified gear. If we compare the modification amplitude with the deformation amount of small tooth segments along the contact line, the contact line length of the modified helical gear pair can be obtained, and its mesh stiffness can be obtained subsequently. Thus, the first step of the proposed method is to cut the teeth into slices, each slice in a small tooth