Finite line contact stiffness under elastohydrodynamic lubrication considering linear and nonlinear force models

Highlights

• Study of stiffness of finite line contact under elastohydrodynamic lubrication.

• Comparison with experimental results of optical interferometry in literature.

• Different roller profiles evaluated: non-profiled, crowned, dub-off and logarithmic.

• Recommendations of linear and nonlinear reduced force models application.

Abstract

The aim of this study is to analyze the stiffness of finite line contact under elastohydrodynamic lubrication (EHL) with linear and nonlinear force approximations. The numerical solution of the finite line contact is compared with experimental results of optical interferometry. Four types of profiles are evaluated: non-profiled roller, crowned roller, dub-off profile and logarithmic profile. For each type of roller, the geometry values vary in order to identify the behavior of the reduced contact force parameters. The results of the force model showed good correspondence with the numerical solution of the EHL contact.

Introduction

Elastohydrodynamic lubrication (EHL) occurs when the pressure is high enough to deform elastically the contacting surfaces, influencing directly the lubricant film thickness. EHL contacts of finite length are common in many machine elements and mechanisms, as cam-follower systems, gear teeth and rolling element bearings. In order to relieve high edge stress concentrations, rollers are usually axially profiled or the corners are rounded off.

Finite line contact studies began in the late 1960s and 1970s, once setting up experiments with line contacts is complex since high loads are required to achieve oil pressures that characterize elastohydrodynamic lubrication. Gohar and Cameron [1] showed the effects on the edges in line contact experimentally, despite the limitation of the apparatus in applying force. Wymer and Cameron [2] published an experimental study measuring the thickness of the contacting oil film in line contact using optical interferometry and hydrostatic force application equipment that allowed higher loads to be used. They showed experimental evidence that the film thickness of EHL finite line contacts are different in the central region and near the edge. In addition, they concluded that the minimum oil film condition occurs near the edge of the roller. This work shows the influence of line contact geometry under different load and speed conditions.

As the influence of the edge on film thickness became more evident with experimental evidence, the first theoretical works of finite line contact appeared in the 1970s. Bahadoran and Gohar [3] and Hooke [4] modified the infinite line solution to obtain the approximation for finite line contact. Mostofi and Gohar [5] presented the numerical solution for finite line contact through finite difference method solving two-dimensional Reynolds equation. A qualitative comparison was made with the results obtained by Ref. [2] and showed similarity in oil film thickness behavior, although the authors could not reproduce the comparison with the same experimental parameters.

Park and Kim [6] numerically solved the EHL problem for the finite line case with finite difference method and Newton-Raphson algorithm. However, as occurred in Ref. [5], the convergence of the results was limited to moderate material loads and parameters. The authors concluded that pressure and film thickness are very close to infinite and finite line contact in the central region of the roller. Near the edge contact, infinite and finite results begin to diverge.

Chen et al. [7] studied the effect of logarithmic geometry of the EHL finite line contact. The logarithmic profile is known to decrease stress concentration at the edges of the contacts. Experimental results from numerical simulations were compared using the Jacobi direct method. Kushwaha et al. [8], using the Newton-Raphson method, analyzed the influence of misalignment between roller and race. Kushwaha and Rahnejat [9] investigated the time domain EHL finite line contact and inserted variations in roller inclination in the axial direction.

In the early 2000s, multilevel techniques were employed in the solution of finite line contact, which allowed the study under higher load conditions and even thermal effects, as seen in Liu and Yang [10] and Sun and Chen [11] 2006. Zhu et al. [12] studied mixed elastohydrodynamic lubrication considering realistic geometries and roughness surface.

Najjari and Guilbault [13,14] were the first to study the edge contact effect on EHL finite line contact using Boussinesq-Cerruti half-space equations and they compared different axially profiled rollers geometries. Shirzadegan et al. [15] studied the transient effects on lubricated contact between cam-follower mechanism. Guo et al. [16] analyzed the edge contact effect evaluated by Ref. [13], but calculating the elastic deformation of bodies considering quarter-space and comparing with half-space. Hultqvist et al. [17] investigated the dynamic of EHL of finite line contact using the equation of motion proposed by Ref. [18] for point contact.

Wijnant's work [18] was one of the pioneers of contact dynamics with an approximation of stiffness and EHL contact damping. Wijnant [18] proposed dimensionless expressions of stiffness and damping for point contact from an analysis of transient contact behavior under a harmonic excitation force. Subsequently, the time domain results of the circular contact were experimentally validated [19]. From the contact equation of motion suggested by Ref. [18], Nonato and Cavalca [20] proposed an expression for the elliptic contact force using a polynomial approximation. Nonato and Cavalca [21] later suggested a nonlinear load-displacement relationship for the contact between rolling element and raceway of ball bearing. The model was applied to a rotor dynamic test rig.

Wiegert et al. [22] proposed an EHL infinite line contact model combining hydrodynamic and Hertzian theories. From the hydrodynamic stiffness and the dry contact stiffness, an equivalent EHL contact stiffness was calculated in order to obtain the vibration response of the EHL contact. Qin et al. [23] theoretically studied the infinite line contact under EHL and proposed an approximation to calculate the oil film stiffness. Qin et al. [23] analyzed the cam-follower mechanism contact dynamic. For the problem of infinite contact under free vibration, Zhang et al. [24] estimated the EHL stiffness and damping.

Tsuha et al. [25] proposed an explicit load-displacement relationship for EHL infinite line contact based on [21]. The EHL contact model was applied to cam-follower and compared with classic Hertzian dry contact approach. The EHL modeling decreased the vibration amplitude of the mechanism, which shows the impact of the oil film on the dynamics of mechanic systems. Later, Tsuha et al. [26] applied the modeling to contacts between rolling element and ring of a needle roller bearing.

Zhou et al. [27] used the model proposed by Ref. [23] to calculate the stiffness in the normal direction of contact on gear teeth. The researchers proposed a stiffness model in the tangential direction, which was calculated from the shear stress of the infinite line contact. Zhou et al. [28] expanded their analysis by proposing the contact damping model in both normal and tangential directions. The effects of contact force, rotation speed and number of gear teeth on the combined coefficients were investigated. Recently, Hu et al. [29] modeled a valve train combining stiffness of oil film and dry contact between cam and tappet and compared the theoretical results with the experiment.

Despite all the increasing effort to study point and infinite line contact stiffness under EHL, finite line contact is still underexplored. As the finite line contact can have different axial profiles, the geometry of the roller can influence the characterization of the contact stiffness. Thus, the main objective of this study is to evaluate the parameters of reduced EHL contact force considering the roller with four types of geometry: non-profiled, crowned, dub-off profile and logarithmic profile.