Research PaperA new mesh stiffness model for modified spur gears with coupling tooth and body flexibility effects
Introduction
In gear dynamics field, time-varying mesh stiffness plays an important role in predicting the dynamic response of the system. Many methods have been proposed and developed to calculate the mesh stiffness, such as the analytical method (AM) [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [17]–18], the FEM [19], [20], [21], [22]–23] and the experimental method (EM) [24], [25]–26].
According to [1], the elastic energy stored in the tooth contains three parts: the Hertzian contact energy, bending energy and axial compressive energy. Later, shear energy was introduced to the mesh stiffness model [1] by Tian et al. [2]. As the gear body is assumed to be rigid in [1] and [2], this kind of model is defined as the ‘rigid gear body’ (RGB) model in this paper. Using the RGB model, Tian et al. [2] revealed the influence of tooth faults on steady-state response of a gearbox; Pandya et al. [3] investigated the mesh stiffness of spur gear pairs with different contact ratios; Liang et al. [4] established a mesh stiffness model for planetary gear pairs with tooth crack, where the tooth part was assumed to be supported by the root circle rather than the base circle; Miryam et al. [5] studied the load sharing ratio (LSR) and mesh stiffness of a spur gears pair without tooth profile errors. Saxena et al. [6] studied the effect of shaft misalignment and friction force on the mesh stiffness of spur gears.
Some researchers [7–13] also developed many mesh stiffness models with the flexible gear body hypothesis, which is defined as the ‘flexible gear body’ (FGB) model in this paper. For most FGB models, the gear body-induced tooth deflection was calculated by the formula that Sainsot et al. [7] proposed in 2004. For example, Chaari et al. [8] analytically studied the meshing characteristics of cracked spur gears and validated the proposed model by FEM; Wan et al. [9] considered the fact that the involute tooth profile starts from the root circle rather than base circle, and proposed an improved FGB model for spur gears with tooth root crack; The influence of gear center distance variation on the mesh stiffness of spur gear pair was revealed by Luo et al. [10] using the analytical formula proposed in [7]; Chen et al. [11] proposed a FGB model to study the mesh stiffness of spur gear pair with modifications and tooth crack; Based on [12], Yu et al. [12] proposed a mesh stiffness model considering the corner contact effect of spur gears; Chen et al. [13] established a mesh stiffness model of locomotive spur gears with tooth root crack.
Among all the FGB models, the mesh stiffness model proposed in [11] is the most typical one, in which the deformation compatibility condition was considered. The deformation compatibility condition, which means the base pitch of each tooth pair that in simultaneously contact should be equal, is an important principle in [11]. However, some researchers [13–15] pointed out that the mesh stiffness is overestimated under double-tooth engagement region in [11]. Actually, the coupling effect of adjacent teeth was ignored in [11]. Besides, effective formula for calculating the fillet-foundation stiffness under multi-tooth engagement situation is lacking.
In order to solve this problem, Ma et al. [15] proposed the concept of ‘fillet-foundation stiffness correction factor’, and the formula proposed in [7] was then revised mathematically. Based on [15], Xie et al. [16] also established an improved mesh stiffness model considering the coupling effect of adjacent teeth. Later, Xie et al. [17] proposed analytical formulas to calculate the so-called ‘local gear body-induced tooth deflections’ and ‘the structure coupling gear body-induced tooth deflections’, which has made further improvement to the model they proposed. Recently, based on the formulas proposed in [17], Chen et al. [18] further considered the influence of tooth profile deviations and made some improvements to [11]. Admittedly, these revised models [15], [16], [17] are surly effective from the perspective of mathematics, but some disadvantages still exist. For example, one has to establish a FE model first to obtain the ‘correction factors’, and this brings some inconveniences. Besides, the deformation compatibility condition was ignored in [16]. For convenience purpose, the mesh stiffness model proposed in [16] is defined as the RFGB (revised flexible gear body) model in this paper.
FEM was also utilized by many researchers [19–23] to calculate the mesh stiffness of spur gears. For example, Wang et al. [19] studied the mesh stiffness and load sharing ratio of high contact ratio spur gears using the adaptive meshing technique. Shao et al. [20] calculated the mesh stiffness variation of a spur gear pair with local faults using a special-designed FEM. Tharmakulasingam [21] revealed the influence of tooth profile modification on the static transmission error and mesh stiffness of spur gear pair by finite element analysis. Base on finite element analysis, Chen et al. [22] studied the meshing characteristics of helical gears with spalling fault. The loaded tooth contact analysis method was used by Huangfu et al. [23] to obtain the mesh stiffness and contact stress of spalled gear pairs. However, fine mesh grids are needed in most FEM, which is computational expensive and time-consuming.
In addition to the AM and FEM, EM [24–26] was also applied by some researchers. For example, Pandya et al. [24] used the photo elasticity technique to calculate the mesh stiffness of spur gears. The photo elasticity technique and strain gauge technique were utilized by Raghuwanshi et al. [25, 26] to obtain the gear mesh stiffness of cracked spur gears. Compared with AM, delicate measurement devices or complex techniques are needed in most EMs, which is inconvenient and lacks of generality.
According to the above literature review, it is clear that such a mesh stiffness model is lacking, in which both the tooth flexibility, the gear body flexibility, the coupling effect of adjacent teeth and the deformation compatibility condition are considered. To solve this problem, an ‘improved flexible gear body’ (IFGB) model is proposed, in which both the four factors have been considered. The remaining part of this paper is organized as follows:
- (1)
Section 2 briefly describes the four models, including the RGB, FGB, RFGB and the proposed IFGB model.
- (2)
To validate the proposed IFGB model, comparisons are made between the mesh stiffness models and the FEM in Section 3.
- (3)
Main conclusions are drawn in Section 4.
Section snippets
The RGB model
In the RGB model, the tooth part is seen as a cantilever beam supported by the rigid gear body (see Fig. 1).
According to the Euler-Bernoulli beam theory, the flexibility of a cantilever beam with rigid support can be expressed as follow [4]:where qa, qb, qs denote the axial compressive, bending flexibility and shear flexibility, respectively.
According to [16], the axial compressive, bending flexibility and shear flexibility can be calculated by the following equations:
Model comparisons and verifications
In this section, the calculation results of the proposed IFGB model will be compared with the other mesh stiffness models and the FEM.
Conclusions
In this paper, a new analytical model is proposed to calculate the mesh stiffness of modified spur gear pair. Compared with those widely used analytical models proposed by the former researchers, both the tooth flexibility, gear body-induced local flexibility, gear body-induced coupling flexibility and the deformation compatibility condition are considered.
Main conclusions are drawn as follows:
- (1)
The RGB model and the FGB model, although widely used, overestimate the gear mesh stiffness. Compared
Acknowledgment
This research is sponsored by K.C. Wong Magna Fund in Ningbo University, the National Natural Science Foundation of China (No. 51975301).
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