A refined analytical model for the mesh stiffness calculation of plastic gear pairs

Highlights

• A novel method for the gear mesh stiffness calculation of plastic gear pair is established.

• The influences of meshing deformation on the mesh behavior of plastic gear pair are investigated.

• An iterative algorithm is utilized for the accurate calculation of gear mesh stiffness.

• The proposed method is verified by the finite element method.

Abstract

The accurate calculation of the mesh stiffness for a plastic gear pair is significant for analyzing its transmission performance. During the meshing process, factors such as gear tooth meshing deformation and thermal effect will cause variations of the meshing position and tooth load particularly for plastic gears. These increase the difficulty for the accurate calculation of the gear mesh stiffness for plastic gear pair. To address this issue, the instantaneous gear tooth meshing deformation is obtained based on the potential energy method. Then, the effects of the above factors on the contact condition of a meshing tooth pair are analyzed, based on which the actual mesh position is accurately obtained via an iterative algorithm. Under the commonly-used elastic contact assumption, the contact stiffness at the new meshing position is calculated based on the Hertzian contact theory, and the algorithm of the mesh stiffness calculation for a plastic gear pair is derived by analyzing the influence of the elastic approach of meshing tooth on load distribution mechanism of gear pair. Compared with previous analytical methods, the proposed algorithm for mesh stiffness calculation shows higher accuracy, and it can also properly analyze the effect of load and other parameters on the mesh behavior of plastic gear pair, which can provide a theoretical basis for the performance analysis of plastic gear transmission system.

Introduction

As a new type of gear, plastic gears are gradually replacing traditional metal gears in low and medium load areas [1] due to advantages such as light weight and low cost [2]. The key problem of plastic gears, which restricts its large-scale application, is their lower load-bearing capacity than metal gears. Therefore, studying the load carrying capacity of plastic gears is significant as it ensures the transmission performance of plastic gears at different conditions. Hasl et al. [3] considered the influence of tooth deflection under load on the contact ratio of plastic gear pair, and proposed a method for calculating the nominal tooth root stress based on existing analytic guidelines. They [4] also proposed a method for calculating the root stress of oil-lubricated plastic gears, and verified the rationality of the method through experimental results. Past researches show that the contact analysis of plastic gears is the basis for the load carrying capacity analysis, such as the contact temperature modeling. Based on the contact analysis, Mao [5], Fernandes et al. [6] and Černe et al. [7] proposed a thermal model to predict the contact temperature, which consists of flash temperature, ambient temperature and bulk temperature.

The variation of the gear mesh stiffness (GMS) is an important excitation source to gear systems [8]. Accurate calculation of GMS is the key step for the tooth contact modeling and load carrying capacity analysis. Thus, it has received intensive attention in the literature. Tsai et al. [9] proposed an algorithm to calculate the static transmission error of plastic gear pair based on the Houser's method, which takes the influence of corner contact into account. Based on the potential energy method, Jabbour et al. [10] proposed a mathematical model to calculate the GMS of plastic helical gears, and the premature engagement of tooth pairs is considered. Wan et al. [11] improved the traditional model based on the potential energy principle and studied the evaluation method of the GMS when the base circle radius is larger or smaller than the tooth root circle radius. Ma et al. [12] proposed an improved analytical method by introducing the fillet-foundation deformation correction coefficient to account for the overestimation of the tooth foundation induced stiffness during the meshing of multiple tooth pairs. In addition, the effect of tooth tip modification on GMS was studied. Wang et al. [13] proposed an improved model for the GMS of helical gears, which considered not only transverse-meshing stiffness but also axial-meshing stiffness, and the effect of helix angle on meshing stiffness was furtherly analyzed. Cao et al. [14] improved the potential energy method by analyzing the influence of gear eccentricity error on the center distance and mesh position of gear pair, and investigated the influence of eccentricity errors of sun gear and planetary gear on the dynamic performance of planetary gear transmission system. Based on finite element method, Wang [15] proposed a model to calculate the GMS and analyzed the effect of ambient temperature and external load on the GMS. However, since the plastic is a thermal-sensitive material, the change of tooth contact temperature will affect the analysis accuracy of the load carrying capacity. Gou et al. [16] considered the influence of tooth surface contact temperature on GMS by treating the effect of thermal expansion on the GMS as an additional stiffness component superimposed on the original stiffness components (i.e. the Hertzian contact stiffness, the tooth flexibility induced mesh stiffness and the tooth foundation induced mesh stiffness). Luo et al. [17] proposed a thermal dynamic model of gear system by considering the effect of heat on GMS. The results show that the thermal effect causes an additional thermal load during the meshing process.

There are many studies towards the GMS calculations by using analytical methods, which are applicable to both metal gears and plastic gears. However, due to the lower elastic modulus and poor thermal conductivity of plastics, a lower external load can cause large tooth deformation and higher additional thermal load. These factors will inevitably cause the meshing position to deviate from the theoretical position. In addition, the elastic modulus of plastics is extremely susceptible to the effect of ambient temperature, which will further affect the meshing process of the gear pair. As far as we known, these effects have been rarely considered for the GMS calculation of plastic gears in previous work.

In order to address this issue, this study first investigates the meshing process of plastic gear pair, and studies the influence of tooth deformation and thermal effect on the meshing position. A novel GMS calculation method for plastic gear pairs via an iterative algorithm is introduced. The proposed method overcomes the deficiency of traditional method in the meshing analysis of plastic gear pair, and provides a theoretical reference for further research and analysis on the load carrying capacity. This constitutes the major contribution of this work to the literature. It should be noted that only elastic deformation is considered in this paper, which is the commonly-used assumption.

The paper is organized as follows. Section 2 elaborates on the origin of the problem to be studied in this article, and clarifies the influence of gear tooth deformation and temperature variations on the mesh behavior of a plastic gear pair. Section 3 introduces the modeling process of the GMS of a plastic gear pair, and illustrates load distribution mechanism among meshing tooth pairs. Section 4 first presents a comparison study to demonstrate the rationality of the proposed model, and then analyzes the effects of ambient temperature, load and friction coefficient on the meshing point, GMS and load sharing ratio (LSR) of a plastic gear pair. Section 5 concludes this study.