Issue 21

D. Benasciutti et alii, Frattura ed Integrità Strutturale, 21 (2012) 37-45; DOI: 10.3221/IGF-ESIS.21.05 40 support elastic deformation will be included into the analysis. Temperature and pressure effect (with rigid components) The Reynolds Eq. (1) is solved by using the finite difference method based on a central difference scheme. The numerical algorithm is sketched in Fig. 2. The input data are: journal bearing geometry, applied force F , bearing rotational speed N , inlet oil temperature T in and pressure-viscosity coefficient α, other than the Vogel-Barus model μ ( T , p ). The unknown function in Eq. (1) is the pressure p ( θ ) that, upon integration, must equal the resultant applied load F . However, the problem at hand is actually non-linear for several reasons. Although the pressure p ( θ ) is the unknown function, Eq. (1) does not explicitly depends on the input load F (that is, the pressure resultant), but on the eccentricity e through the lubrication gap h ( θ )= c − e ·cos( θ ). Therefore, at the beginning of the analysis a guess value of eccentricity e 0 (not of the force F ) must first be imposed to obtain a tentative lubrication gap and the resultant first-attempt pressure distribution. The Newton-Raphson rule is then applied to compute (based on the resultant of pressure F i ) an eccentricity increment, δe i , that is next used to update the eccentricity value for next iteration, e i+1 = e i + δe i , and to compute an updated lubrication gap geometry h i+1 ( θ ) for solving again Eq. (1). At each iteration, the numerically calculated pressure distribution p ( θ ) must also be checked for negative values, which have no physical meaning and are set to zero. This adjustment inevitably modifies the overall pressure resultant that balances the applied force, giving rise to another source of non- linearity in problem solution. Several iterations are required to converge at the correct eccentricity value (a tolerance is checked for both eccentricity and force), which gives the correct pressure distribution p ( θ ) that solves Eq. (1) and also balances (as a resultant of pressure) the applied input force F . START h i ( θ )= c - e i ·cos( θ ) p i ( θ ) with p (θ)>0  θ      d cos )( i i p F NO YES Output p ( θ ) , e, h ( θ ) END Fluid-dynamic analysis (Reynolds eq.) (Newton-Raphson) c=R-r e i =e 0 F i e 1i i        FF ee i=i+ 1 Input F, μ ( T,p ) , N, T in , R, r, α   i 1i i 1i i i e e F F FF e        Figure 2 : Sketch of the numerical algorithm for fluid-dynamic analysis of journal bearing (rigid components) To evaluate the effect of temperature on viscosity (and then on pressure), the journal bearing in Tab. 1 was studied at two operating conditions (JB1, JB2) characterized by two different inlet temperatures ( T in =40, 70 °C). Two hypotheses were then adopted to compute the pressure-independent viscosity term μ 0 as a function of oil temperature: in the first, using an average constant temperature T m resulting by a thermal balance inside the oil film (as in R&B approach) [1], in the second using, as a first approximation, a linear variation from inlet T in to outlet temperature T out (that has been calculated by previous thermal balance); note that in both cases the same average oil film temperature T m is obtained.