A model for the study of meshing stiffness in spur gear transmissions

Abstract

This work describes an advanced model for the analysis of contact forces and deformations in spur gear transmissions. The deformation at each gear contact point is formulated as a combination of a global and a local term. The former is obtained by means of a finite element model and the latter is described by an analytical approach which is derived from Hertzian contact theory. Then the compatibility and complementary conditions are imposed, leading to a nonlinear system of equations subjected to inequality restrictions that should be solved once the position of each gear centre is known. A numerical example is presented where the quasi-static behaviour of a single stage spur gear transmission is discussed, showing the capabilities of the methodology to obtain the Loaded Transmission Error under several load levels as well as some other related measures such as load ratio or meshing stiffness.

Introduction

Gear transmissions are critical mechanical components found in a wide range of machinery. The industrial applications are countless and cover fields such as aerospace, agriculture or wind generation among others. As the working speeds for gear transmissions increase [1], the dynamic problems become more important and, as a consequence, the dynamic behaviour of gear transmission has also become a growing subject of concern for manufacturers and final users, involving aspects such as design, condition monitoring, vibration and noise control [2].

The main feature that characterises the dynamic behaviour is the periodic excitation induced by the variable meshing stiffness [3]. This phenomenon is a consequence of the changes in the number of teeth couples contacting simultaneously and it is a function of the angular position over the meshing period. Moreover, particularly in spur gears, the magnitude of the transmitted torque modifies the features of the meshing stiffness and therefore the dynamic behaviour. The characterisation of this periodic excitation is crucial in order to achieve better analysis in the design stage [4], improving durability but also reducing the levels of noise and vibration.

Gear noise and vibration are closely related with the so-called Transmission Error (TE), which was defined by Harris as “the difference between the position that the output shaft of a drive would occupy if the drive were perfect and the actual position of the output” as cited in ref. [5]. According to the previous definition, the TE should be expressed in radians but in order to make its understanding easier, TE in radians is actually multiplied by the base radius and provided in micrometers as a function of the angular position [3]. If gear teeth were perfect and non-deformable there would be no TE unless tooth profiles deviate from their theoretical shape. Nevertheless, real gears have flexible teeth and therefore the TE appears as a magnitude dependent on the load. At this point it should be stated that even though the term TE is widely used indiscriminately, it is possible to distinguish several kinds of TE depending on their source.

Thus, in real gears, profile and pitch errors are unavoidable and as a consequence of this, the so-called manufacturing error (ME) appears, which does not consider deflections. On the other hand, the term used is the static transmission error (STE), which depends on the magnitude of the torque transmitted by the gear pair and is sometimes called loaded transmission error (LTE). Finally from the point of view of dynamics, the suitable term is dynamic transmission error (DTE), which is also affected by the rotational speed.

The LTE is closely related with the meshing stiffness, which is the source of the parametric excitation of the gear pair and plays a fundamental role in its dynamic behaviour. In this sense, LTE can be used as an excitation in dynamic simulation [6], [7] and also as a measure to predict the noise level for a certain gear pair [8].

The meshing stiffness has been included in dynamic models following several approaches ranging from the simple average throughout a meshing cycle to more advanced formulations that take into consideration the periodically time-varying stiffness representing the fluctuation in the number of contacting tooth pairs. The approach based on a constant value for the meshing stiffness is very valuable from the point of view of global dynamic behaviour as it can provide a good estimation of the resonances and their consequences on the operation of the transmission. On the other hand, time-varying (or angular-varying) meshing stiffness is essential when noise and vibration is the subject under study.

One of the first approaches to study the variable compliance throughout a cycle was carried out by Weber [9] that subsequently with Banascheck [10] proposed the superposition of deflections due to teeth bending, contact between tooth surfaces and gear bodies considered as semi-infinite elastic planes. This approach provides a value for meshing stiffness as a function of the angular position throughout the meshing period. Nevertheless, some aspects are neglected, such as the elastic coupling between successive teeth pairs under load. Moreover, the change in the number of active contacts is defined from the gear kinematics without considering that the real contact ratio should be modified by the magnitude of the torque to be transmitted.

Other authors use a numerical approach, usually based on both 2D and 3D Finite Element (FE) models [11]. This approach should have a fine mesh in the contact area and requires a great computational effort as it also includes nonlinear gap elements. On the other hand, the analysis of successive gear meshing positions requires the development of a new mesh for each contact position with the corresponding computational cost [12], [13]. Unfortunately, the centres of the gears can move from their theoretical position leading to modifications in the mounting distance and thus also in the pressure angle and contact ratio. That means a new different FE model depends not only on the transmitted torque but also on the supporting shaft and bearing stiffness.

To solve this problem various alternatives have emerged, such as the use of artificial neural networks [14] or the development of hybrid models combining numerical and analytical formulations [15], [16].

Although the analysis presented in this work is limited to the quasi-static case, the main purpose is to allow the introduction of additional phenomena such as friction and profile errors with special attention to the presence of defects, such as surface cracks and pitting, for their subsequent inclusion in a dynamic model. This type of study requires an accurate knowledge of the contact forces in both bearings and gears in order to be able to develop new tools and predictive maintenance strategies based on vibration measurement.

Pursuing this objective, in this paper a model for the assessment of the contact forces between gear pairs is presented. After this introduction where the reasons and background of the proposed work have been explained, the approach adopted for the formulation of gears will be presented. Then, the model is validated through a quasi-static analysis of an example to visualise and assess how the magnitude of the transmitted torque, the friction and the modified centre distance affect the resultant LTE, meshing stiffness and load sharing ratio (LR). Next, the model developed is compared with some other common formulations for the calculation of meshing stiffness, highlighting the advantages and improvements of the proposed procedure. Finally, the article concludes with a summary of comments about the results and the features obtained with the application of the model.