Optimization of distributed axial dynamic modification based on the dynamic characteristics of a helical gear pair and a test verification

Highlights

• A slicing distributed dynamic model of helical gears.

• Study the influence of modification on the load distribution of helical gears.

• The law of modification is verified by experiments.

Abstract

In this paper, a method for the dynamic modification of tooth width is used to optimize the problem of uneven loading of helical gears. Considering the meshing error caused by gear manufacturing, installation, time-varying meshing stiffness, clearance, and shaft bending, the distributed meshing transverse-torsion-pendulum nonlinear dynamic model of a helical gear pair is established. A method to modify the tooth width of the helical gear pair is proposed to optimize the uniform load characteristics of the tooth surface. Through the research of dynamic tooth width modification, the optimal modification parameters are obtained so that the dynamic load distribution of the tooth surface is uniform and the load distribution centre is close to the centre of the tooth surface, which reduces the swing phenomenon. During experimental testing, the swing vibration response of the modified gear is reduced by a maximum of 26.15%, the dynamic load coefficient is reduced by a maximum of 20.84%, and the stress distribution on the tooth surface is uniform.

Introduction

The helical gear transmission includes helical gear teeth, bearings, drive shafts and casings. Under the action of dynamic loads, bending deformation and torsional deformation occur. These phenomena cause the helical gear to deviate from the theoretical design position, resulting in edge contact of the teeth at the end of the tooth surface. The dynamic load distribution is uneven, and an eccentric load occurs. In addition, this load unevenness is further exacerbated by errors in manufacturing, installation, and wear.

Tooth profile modification technology is a key technology used to reduce the vibration, impact and noise of a gear transmission system and improve the working efficiency and service life of the gear transmission. Researchers have gradually carried out many theoretical and technical studies and experimental analyses on gear tooth modification technology. Parag et al. [1] considered the contact state of the gear tooth surface and established a contact model including modification parameters such as the modification amount, modification length, and tooth direction modification. Li [2,3] studied the mesh stiffness and stress of a spur gear with the finite element (FE) method, in which tooth profile modification, lead crown relief, manufacturing error, misalignment error and other factors were considered. Wang and Zhang [4] developed a helical gear time-varying meshing stiffness (TVMS) model by the slice method with tooth profile error. TVMS, transmission error and stress were studied considering the tooth profile error. Wan et al. [5] calculated the TVMS of helical gears by the accumulated integral method. Litvin et al. [6,7] used gear geometric contact analysis technology to reduce the transmission error amplitude of an involute helical gear by performing modifications on the tooth surface in three dimensions. Jiang [8] established a six-degree-of-freedom helical gear dynamic model to study a TVMS calculation method that considered axis direction deformation. Lin [9] proposed a method to analyse the transmission error of a helical gear system considering machining, assembly, and modification errors, and the coupling dynamic model of bending and torsional axis was established for the analysis and control of vibration and noise of the gear system. Hu [10] established an FE node dynamic model of a high-speed gear rotor bearing system with a TVMS, tooth gap, gyro effect and transmission error excitation and verified that the dynamic characteristics of the system can be improved by modifying the profile slightly under some working conditions.

Through continuous research, it has been indicated that modifying the tooth profile can not only compensate for the basic pitch error but also improve the load distribution while reducing the meshing stiffness and meshing shock to reduce vibration and noise. The modification curve, the modification amount and the modification length are the three key elements. Previous studies [11], [12] have suggested that tooth profile drum modification can effectively reduce the maximum tooth root bending stress and maximum tooth surface contact stress of helical gears and transfer the load centre from the edge of the tooth to the centre of the tooth. The corresponding modification curve usually adopts a circular arc curve, quadratic curve or polynomial function curve [13]. To more accurately reflect the tooth surface deformation caused by an uneven load across the tooth width direction, a polynomial function modification method is used in this article.

The analysis method of the helical gear system usually adopts slice theory. The helical gear is regarded as a slice along the tooth width direction, and the cutting tooth can be regarded as a spur tooth [14]. Feng et al. [15] proposed an improved TVMS model of a helical gear pair, considering the basic coefficient of the fillet, nonlinear Hertz contact, correction coefficient and friction coefficient. Wan et al. [16] calculated the TVMS of a helical gear by the cumulative integral method and analysed the influence of cracks and the helix angle on the TVMS. Wang et al. [17], [18], [19], [20] creatively improved the calculation method of the TVMS of helical gear by using the slice method, and calculated the axial deformation of gear teeth, which improved the actuarial accuracy of the meshing stiffness of the helical gear. In the above-mentioned analysis process, the axial vibration of the helical gear is not considered; however, due to factors such as uneven load of the gear and the flexibility of the shaft, the study of helical gears in the actual vibration process must fully consider the axial vibration [21], [22], [23], [24].

With the development of gear system dynamics, dynamic theory has been widely used in gear modification, and various contact analysis models have been established to evaluate the load distribution of helical gears. Liu et al. [25] established a nonlinear analytical model that considers the dynamic load distribution between the individual gear teeth and the influence of variable mesh stiffness, profile modification, and contact loss to study the effects of tooth profile modification on multi-mesh vibration. Tooth contact analysis (TCA) and load tooth contact analysis (LTCA) can accurately simulate the force on gear teeth during the gear meshing process and help analyse this force. Wang and Shi [26] and Wang et al. [27] used the contact analysis method to accurately calculate the stiffness excitation, error excitation and impact excitation and then established a dynamic analysis model of helical gear pairs and double helical gear pairs. Wang [28] proposed a method of tooth profile modification using tooth surface contact and a helical gear dynamic model.

Most of the above studies were based on static analysis and focused on the modified TVMS calculation or contact analysis. Dynamic models that mostly focus on transverse-torsion coupling have also been established in the literature, ignoring the flexibility of the support shaft and the swinging vibrations of helical gears under high-speed and heavy-load conditions. None of these studies can fully reflect the dynamic modification effect of helical gears. This paper establishes a six-degree-of-freedom dynamic model of a helical gear pair considering the flexibility of the supporting shaft. Based on distributed slice theory, the load distribution characteristics between teeth are studied, and experiments are designed to verify the above theories.

In this paper, the gear tooth profile error, gear assembly error, tooth width error and drum shape modification deviation are considered, and the distributed meshing nonlinear dynamic model of the helical gear is established. To alleviate the uneven distributions of the tooth surface dynamic load and its centre of gravity, the three-dimensional dynamic drum shape modification method is studied. The tooth width vibration response and dynamic load distribution of the gear system are analysed. On this basis, the Hertz contact stress is used as the calculation basis of the tooth surface contact stress, and the variation in the tooth surface contact dynamic stress of the gear system with the drum shape modification parameters is analysed.

This paper includes 4 parts. In Section 2, a distributed transverse-torsion-pendulum dynamic model with six degrees of freedom is introduced. In Section 3, dynamic loads and contact stresses are used to verify the results of dynamic gear modification, the relationship between the modification parameters and the optimization index is obtained through the dynamic modification simulation calculation, and the optimization modification scheme of a helical gear is preliminarily determined. In Section 4, the vibration response and stress distribution are taken as the research objects to verify the gear modification results by experiments. Finally, some conclusions are given in Section 5.