On the non-linear dynamic behavior of elastohydrodynamic lubricated point contact

Abstract

The complex dynamic concepts of mechanical systems are regarded each day as new barriers to be overcome. One of the most complex systems, despite its common construction design, is the rolling element bearings. The interactive dynamic interfaces of such bearings are normally disregarded by engineering analysis on the day to day basis due to its complexities. This paper intends to propose a new approach to the characterization of the elastohydrodynamic lubricated point contacts on such components, in order to fully depict its non-linear dynamic behavior, avoiding the use of rough hypothesis on a systemic procedure. A multi-level method was used to solve the coupled lubrication–deformation problem, alongside a Newmark-ß integrator of the motion equation for the contact system. A range of dynamically similar contacts were evaluated, so as to characterize its nonlinear dynamic behavior. A least-squares method was applied to the multi-level algorithm results, fitting the displacements–force relation to a linear and also to a third order polynomial stiffness. The fitting results were compared, clearly showing the nonlinear behavior of such contacts. Also, the oil film damping was regarded as viscous, leading to good overall response. Some peculiarities of the proposed adjust method are also considered.

Introduction

The time reduction between product updates and new developments in the mechanical industries brings new challenges to the development team. Old assumptions and rough concepts can mislead the project and increase the number of practical experiments needed. Working with prototypes or finished products experimentally is a great engineering tool to understand systemic behavior. Also, methods like DOE and accelerated testing can decrease the total experimentation phase time. However, the only way to overcome the time issue is to reduce the chances of a prototype go wrong. Using the right simulation tools and methods, the development group can provide better parts, or in some cases, even finished parts, without extensive bench tests ever occur. So, the need for better mechanical models is the first to arise.

Dealing with complex dynamic systems is a common procedure in the industries nowadays. For those cases, even now, some rough modeling hypotheses are used, and the problem itself is not very well represented. That is the case of rolling element bearings.

That kind of machine element usually has failure issues related to vibration and noise, which arise from problems such as pitting and spalling. Those events are directly related to the contact forces acting between components and the associated contact fatigue. Thus, it is essential to understand those conditions in order to increase bearing life and quality.

The rolling elements bearings, due to its periodical geometrical nature, are a vibration source themselves. In order to fully understand the complexities of the dynamic behavior of rolling elements bearings, its basic functional characteristics must be studied, the mechanical contacts linking its elements and the raceways. Those contacts are the only vibration transmission points between the shaft inside the bearing and bearing housing.

The first studies on the properties of these contacts were made by H. R. Hertz and published on the work “Über die Berührung fester elasticher Körper”. Due to this work, the general contact mechanics of elastic bodies was named after Hertz. Directly from his work, the nonlinear behavior of the contact can be attained.

The direct use of the dry contact stiffness, as presented in [1], can be a useful approximation to the dynamics of the full bearing, but, doing so, the lubricant effects are neglected. Since the first studies on the lubrication of highly loaded contacts, the damping and stiffness of the oil film are known to be effective over the contact. Due to the influence of the elastic deformation on the oil film thickness this type of lubrication was entitled Elastohydrodynamic (EHL).

The first satisfactory numerical results for the point EHL contact were presented by Hamrock in [2]. In his work, a finite difference method was used for the steady state lubricated problem, using a Gauss-Seidel iterative method. But there was not until great improvement on the computational power and the use of advanced methods that the transient EHL contact could be analyzed.

In the 80s and 90s there were multiple successful attempts to introduce a robust numerical method to evaluate the EHL static condition. Evans and Snidle [3], [4] proposed a quasi-inverse method to solve heavily loaded point contacts, as the previous Gauss-Seidel iterative schemes were not sufficiently robust for that matter. The method was based on the inverse solution employed by Dowson and Higgins for the line contact problem.

Also the substitution of the Gauss-Seidel iterative scheme by a Newton–Raphson method was investigated by Park and Kim [5] to overcome the high computational costs of such procedure. However, even with the low dependence of the convergence on the relaxation factors, the Newthon–Raphson method is highly dependent on the initial guess; in this case, of the pressure and thickness distributions.

In 1991, Venner introduced the multi-level method for the EHL point contact, using the multi-level multi-integration, MLMI, to evaluate the elastic deformation due to the high contact pressure (Venner and Ludbrecht, [6]).

Based on a set of meshes with different grid sizes, this method can greatly reduce computational time by operating the different error frequencies components on different discretization grids. The multiple grid approach reduces dramatically the computational costs and to overcome the high load convergence problem, a hybrid relaxation method is used, adding the Jacobi relaxation method to evaluate high pressure zones. Anyhow, a finite difference method is used to evaluate the Reynolds equation on those grids.

Most of the developments on the transient EHL contacts afterwards were in the surface discontinuities field. In [7] and [8], the effect of surface topology was evaluated as a moving transverse ridge through the contact or as waviness of the surface.

Using this improved method, Wijnant first demonstrate the transient contact response due to harmonic excitation and free vibration in [9]. In his work, the influence of the transient response is observed over the film thickness and most of all the first linear fit of the dynamic response is introduced, achieving the first simulated values for the oil damping on EHL contacts.

Using these fitted values of damping and stiffness coefficients, Wensing [10] observed the influence of the rolling element bearing on a simple rotor system. Also in Wijnant and Wesing [11], some comments on the contact dynamics can be found. Some improvements on the transient EHL algorithm were also proposed by Goodyer [12], focused on an algorithm optimization and studying some surface topology problems. However, no present models for the contact force of lubricated point contacts is robust enough to be incorporated in a full bearing model, as the use of linear springs can be misleading and misinterpret some of the oil film behaviors. For instance, the predicted mutual approach has an asymmetric behavior with respect to the equilibrium position; hence no linear spring can contemplate such peculiarity.

The first attempts to simulate a transient non-linear model of the EHL contacts were made by Nonato and Cavalca [13], using a least square method to fit the transient response of a circular EHD contact. In this work, the same approach will be taken to evaluate the transient and harmonic responses of elliptic EHL contacts. Both results will be compared, in order to fully understand the behavior of the fitting methods. Also, the use of transient and harmonic responses from identical contacts parameters should give a quite trustful method to verify the EHL dynamic simulations. At the end, both methods are supposed to have similar behavior and shall dynamically describe the predicted lubricant film behavior.

A trustful EHL contact dynamic force model is a significant step on having the full rolling element bearing model, regarding the lubricant forces equilibrium. Consequently, the bearing model could be validated against real rolling element bearings on a straightforward Jeffcot rotor test bench, as the validation of such transient contacts are of great complexity. Therefore, this first-hand methodology complies with the necessities for a robust contact model applied to the full bearing dynamic system, avoiding any misleading interpretation of the predicted film, due to a direct linearization of the contact.

The main objective of this paper is to introduce a more reliable lubricated contact force model, based on the predicted film behavior of the EHL transient multi-level algorithm. Hereafter, allowing a future validation of the full lubricated rolling element bearing model on rotor dynamics.