Improved analytical models for mesh stiffness and load sharing ratio of spur gears considering structure coupling effect

Highlights

• Revised fillet foundation stiffness calculation method considering structural coupling effect is proposed.

• Improved mesh stiffness model is established.

• Improved load sharing model is established.

Abstract

Due to the lack of efficient formulas, fillet foundation stiffness under double tooth engagement has long been a challenging problem for mesh stiffness calculation of spur gears by analytical method. Direct summation of the mesh stiffness of each meshing tooth pair overestimates the total mesh stiffness, which may lead to a large calculation error. An improved fillet foundation stiffness calculation method is proposed considering the structure coupling effect, namely, one gear body is shared by two meshing teeth simultaneously. On the basis of the proposed method, a mesh stiffness model and a load sharing model are analytically established. In the mesh stiffness model, fillet foundation stiffness correction factor is introduced to improve the calculation accuracy. Based on the minimum potential energy principle (MPEP), the load sharing ratio is determined by the proposed load sharing model. The accuracy of the proposed models is validated by comparing with finite element method (FEM).

Introduction

As a key component in various mechanical transmissions, spur gear is an important research object for many researchers. For spur gear system, Time-varying mesh stiffness (TVMS) is considered as an important internal excitation source, which has a key influence on gear dynamics.

Generally, TVMS can be evaluated either by rectilinear mesh stiffness or torsional mesh stiffness. Rectilinear mesh stiffness is defined as the meshing force needed for unit deformation along the line of action [13]. Torsional mesh stiffness is defined as the input torque needed for unit rotation of driving gear hub (driven gear hub being fixed) [11]. The relationship between the two kinds of mesh stiffness can be expressed by the following equation [33]:

$$k_{\mathit{torsional}}=R_{b1}^2{k}_{\mathit{rectilinear}}$$

where $k_{\text{torsional}}$ represents the torsional mesh stiffness; $k_{\text{rectilinear}}$ represents the rectilinear mesh stiffness; $R_{b1}$ is the radius of driving gear hub.

Many calculation methods are developed to study the TVMS of spur gear pairs, i.e., the experimental methods [1], [2], [3], [4], the FEM [5], [6], [7], [8], [9], [10], [11], the analytical method [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25] and the analytical FEM [26], [27], [28], [29].There is no doubt that the experimental methods are accurate and reliable. However, special devices and measurement methods are needed for all the reported experimental methods [1], [2], [3], [4]. Hence, more finite element (FE) models, analytical models and analytical FE models are established to calculate the TVMS of spur gears.

FEM is widely accepted and used by many researchers for its convenience and accuracy. For example, Wang et al. [5] studied the torsional mesh stiffness of spur gear pairs using FEM. Rectilinear mesh stiffness of spur gear pair with tip relief was calculated by Ma et al. [6] with a FE model. Using FEM, Lin et al. [7] investigated the influence of backlash on dynamic properties of gear drives. Based on FEM, the rectilinear mesh stiffness of modified gears with tooth root crack is determined by Pandya et al. [8]. In general, FEM is powerful and efficient. However, it is time-consuming and computationally expensive. In order to reduce the computational cost, analytical FEM was also adopted by some researchers. Using a 2D FE model, the combined torsional mesh stiffness of spur gears was determined by Timo et al. [26], and some empirical formulas were derived based on the FE results. A FEM based procedure was presented by Fernandez del Rincon et al. [27] to investigate the loaded transmission error (LTE) as well as the rectilinear mesh stiffness and load sharing ratio of spur gear transmissions. Chang et al. [28] developed a robust analytical FE model to calculate the rectilinear mesh stiffness of cylindrical gears. Pedersen et al. [29] proposed a rectilinear mesh stiffness calculation model for spur gears, where gear rim size and contact zone size were found to have obvious influence on mesh stiffness.

Different from FEM and analytical FEM, analytical method studied the TVMS of gear pairs based on Timoshenko beam theory [34]. The analytical method, which is also known as the potential energy method, was first proposed by Yang and Lin [13] with the assumption that the total energy stored in a meshing tooth consists of three parts: Hertzian contact energy, bending energy and axial compressive energy. Later, shear energy was also considered by Tian et al. [14]. In order to calculate the fillet foundation deflections, Sainsot et al. [15] derived an analytical formula based on the theory of Muskhelishvili. Chen and Shao et al. [16] established a general analytical rectilinear mesh stiffness model, where tooth deviations and spacing errors were considered. Chaari et al. [17] studied the effect of tooth spalling and breakage on the rectilinear mesh stiffness of spur gear pairs. Liang et al. [18] analytically evaluated the rectilinear mesh stiffness of a planet gear set with tooth crack. Using the analytical method, Saxena et al. [19] investigated the influence of tooth crack on the rectilinear mesh stiffness of geared rotor system. Pedrero et al. [20] proposed a load sharing model based on the potential energy method. In Pedrero’s model, MPEP was adopted. Later, Pedrero’s [20] model was further improved by Miryam et al. [21] with the Hertzian contact energy being considered.

Actually, traditional analytical method may lead to a large calculation error for evaluating the TVMS in double tooth engagement region, i.e., the TVMS is directly calculated by the summation of the mesh stiffness of each meshing gear pair [16], [17], [18], [19], [20], [21], [22], [23]. However, it should be noted that the analytical formula in Ref. [15] is derived under single tooth engagement situation. For double tooth engagement situation, the rectilinear fillet foundation stiffness must be reconsidered. Unfortunately, little work has been done to analytically evaluate the rectilinear fillet foundation stiffness in double tooth engagement region.

Having noted the deficiency, many researchers proposed various revised models using the FEM and the analytical method [5], [25], [26]. Such as Wang et al. [5] treated the gear body stiffness (torsional) as a constant value in double tooth engagement using FEM. Timo et al. [26] proposed a revised mesh stiffness model assuming that the gear body stiffness (torsional) in double tooth engagement region is proportional to that in single tooth engagement region with a fixed ratio using FEM. Ma et al. [25] also proposed a revised mesh stiffness model with similar assumption using analytical method. In Ma’s [25] model, a constant fillet foundation stiffness (rectilinear) correction factor was introduced, which was determined by a specially designed FE model.

Generally, methods proposed in Ref. [5], [25], [26] try to revise the gear body stiffness (or fillet foundation stiffness for rectilinear stiffness situation) in double tooth engagement region by a mathematical way, and are surely effective to some extent. However, some important points are neglected and may need further investigation:

• The gear body stiffness (or fillet foundation stiffness for rectilinear stiffness situation) during double tooth engagement is considered as load-independent for a given mesh position, which is only determined by geometry parameters. This means the meshing force of one gear pair has no influence on the other gear pair theoretically, i.e., the structure coupling effect is ignored.

• The analytical load sharing model is still based on the traditional mesh stiffness model, which may lead to some calculation errors inevitably.

The remaining part of this paper is organized as follows. In Section 2, a revised fillet foundation stiffness calculation method and an improved mesh stiffness model are proposed. Section 3 presents an improved load sharing ratio model based on MPEP. Finally, general FE models are established to prove the correctness of the proposed method in Section 4.