Effects of couple stresses in the cyclic squeeze films of finite partial journal bearings
Introduction
The squeeze film mechanism is of practical significance in many areas of engineering and is commonly observed in the bearings of automotive engines, aircraft engines, machine tools, turbomachinery, and skeletal joints. Conventionally, the prediction of squeeze film motion assumes that the lubricant behaves as a Newtonian viscous fluid. However, experimental results show that the addition of small amounts of long-chained additives to a Newtonian fluid minimizes the sensitivity of the lubricant to change in shear rate and provides beneficial effects on the load-carrying and frictional characteristics [1], [2]. Moreover, a base oil blended with additives can stabilize the behavior of lubricants in elastohydrodynamic contacts and reduce friction and surface damage [3]. To describe the rheological behavior of this kind of non-Newtonian lubricant, many microcontinuum theories have been generated [4], [5], [6]. The Stokes theory [4] is the simplest generalization of the classical theory of fluids which allows for polar effects such as the presence of couple stresses, body couples and non-symmetric tensors. This couple stress fluid is a special case of a non-Newtonian fluid and is intended to take account of particle-size effects. According to the Stokes microcontinuum theory, the field equations of an incompressible fluid with couple stresses are given bywhere the vectors V, B, and G represent the velocity, body force per unit mass, and body couple per unit mass, respectively; ρ denotes the density, p is the pressure; μ is the classical viscosity coefficient; and η is a new material constant responsible for the couple stress fluid property. Application of the couple stress model to biomechanic problems has been proposed in the study of peristaltic transport [7], [8]. In lubrication fields many authors have successfully investigated the couple stress effects on different lubrication problems such as the externally pressurized bearings [9], [10], [11], the slider bearings [12], the rolling contact bearings [13], [14], [15], the squeeze film bearings [16], [17], [18], [19], [20]. In the previous researches [19], [20] the leading author has analyzed the squeeze film characteristics of a long partial journal bearing and a finite journal bearing subject to a steady load, respectively. However, the motion with a squeeze film mechanism often falls into the category that the bearings operate under dynamic loading conditions in which the applied load may vary in magnitude or direction with time. Consequently, the locus of the journal center will undergo fluctuations compatible with the variations in the applied load. Therefore a further study is needed.
On the basis of the microcontinuum theory, this study is intended to predict the effects of couple stresses on the pure squeeze-film behavior of a finite bearing subject to a time-dependent load. The modified Reynolds equation is obtained by using the Stokes equations to account for the influence of couple stresses resulting from the lubricant blended with various additives. The film pressure is numerically solved and applied to derive the nonlinear motion equation of the journal. The dynamic squeeze-film characteristics (such as the velocity of the journal center, the locus of the journal center, and the minimum permissible film clearance) for the bearing lubricated with a couple stress fluid will be compared to the Newtonian-lubricant case.
Section snippets
Analysis
Consider the physical configuration of a pure-squeezing partial journal bearing with length L shown as in Fig. 1. The journal of radius R is approaching the bearing surface with a velocity dh/dt. The lubricant in the system is taken to be an incompressible non-Newtonian couple stress fluid. It is assumed that the body forces and body couples are absent, the fluid film is thin as compared to the radius of journal, and fluid inertia is small as compared to the viscous shear. The velocity
Dynamic squeeze film characteristics
By integrating the film pressure acting on the journal shaft one obtains the dynamic squeeze film force.
The dynamic equation of the journal shaft is thenwhere W(t) represents the dynamic applied load and m is the mass of the journal. Assume that the time-dependent load is of the sinusoidal oscillating form:where the amplitude Wd is smaller than the absolute value of the dynamic squeeze film force. Introduce the Sommerfeld number S
Results and discussion
According to the Stokes microcontinuum theory, η is a material constant responsible for the couple stress property. Since the dimension of μ is that of viscosity and the dimension of η is that of momentum, the ratio (η/μ) has dimensions of length squared, and the dimension of l=(η/μ)1/2 is of length. Consequently, this length l may be considered as some property, depending upon the molecular dimensions of the additives in a Newtonian lubricant. With the aid of the definition l*=l/C, this couple
Conclusions
On the basis of the Stokes microcontinuum theory, the effects of couple stresses on the dynamic squeeze film characteristics of finite partial journal bearings are presented. The dynamic film pressure is numerically solved from the modified Reynolds equation, which takes into account the couple stress effects resulting from the lubricant blended with various additives. Considering the bearing subject to a time-dependent sinusoidal load, the nonlinear motion equation of the journal is then
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