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Integral transform solutions for the analysis of hydrodynamic lubrication of journal bearings
Tribology International 52 (2012) 161–169
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Integral transform solutions for the analysis of hydrodynamic lubrication of journal bearings E.N. Santos a, C.J.C. Blanco b, E.N. Macˆedo c, C.E.A. Maneschy a, J.N.N. Quaresma c,n a
´, UFPA, Campus Universita ´rio do Guama ´, Rua Augusto Corrˆea, 01, 66075–110 Bele´m, PA, Brazil School of Mechanical Engineering, Universidade Federal do Para ´, UFPA, Campus Universita ´rio do Guama ´, Rua Augusto Corrˆea, 01, 66075–110 Bele´m, PA, Brazil School of Environmental and Sanitary Engineering, Universidade Federal do Para c ´, UFPA, Campus Universita ´rio do Guama ´, Rua Augusto Corrˆea, 01, 66075–110 Bele´m, PA, Brazil School of Chemical Engineering, Universidade Federal do Para b
a r t i c l e i n f o
abstract
Article history: Received 2 December 2011 Received in revised form 6 March 2012 Accepted 22 March 2012 Available online 6 April 2012
This work deals with analysis of hydrodynamic lubrication of radial journal bearings. The Reynolds equation was treated in order to obtain a hybrid numerical–analytical solution through the Generalized Integral Transform Technique (GITT) for the problem. A parametric analysis is done to investigate the influence of typical governing parameters for such a physical situation. Numerical results for engineering parameters such as pressure field, friction coefficient, axial flow rate and dimensionless load capacity were thus produced as functions of such parameters. Comparisons with results presented in the literature were also performed in order to verify the present results, as well as to demonstrate the consistency of the final results and the capacity of the GITT approach in handling journal bearing problems. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Hydrodynamic lubrication Journal bearings Lubrication theory Integral transforms
1. Introduction The field for application of radial journal bearings is enormous, and includes shipbuilding, industrial machinery and equipment, transportation industry, among others. Research and simulation of physical phenomena involved in the operation of these bearings are very important technologies because they allow evaluation of the phenomena involved with their performance, such as the workload and coefficient of friction. The analysis of journal bearings is probably the most important part of the classical hydrodynamic lubrication theory and, it is also most difficult and complex due to integration of the Reynolds equation. The journal bearings are used mainly for decreasing the friction existing between solid parts of rotating machines and weakening the load variations supported by them. The journal bearing must support the load carried with minimal lost energy and low wear [1,2]. Hydrodynamic lubrication has been studied by many researchers with different numerical techniques for solving the Reynolds equation either for isothermal or non-isothermal flows. Sivak and Sivak [3] obtained a numerical solution of the Reynolds equation by modified Ritz method. Tayal et al. [4] investigated the effect of nonlinearity on the performance of journal bearings with finite width by using the finite element method (FEM). Chandrawat and Sinhasan [5] presented
n
Corresponding author. Tel.: þ55 91 32017837; fax: þ55 9132017848. E-mail address: [email protected] (J.N.N. Quaresma).
0301-679X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.triboint.2012.03.016
a comparison between the Gauss-Seidel iterative method and the linear complementarity approach for determining the pressure field in the analysis of plain and two-axial groove journal bearings in laminar flow operation. Williams and Symmons [6] analyzed a procedure, based on the finite difference method together with a technique known as SIMPLE, to solve the Navier–Stokes equations for the steady three-dimensional flow of a non-Newtonian fluid into the journal bearing with finite-breadth. Sinhasan and Chandrawat [7] presented an elastohydrodynamic study of two-axial-groove journal bearings. Similarly, Sinhasan and Chandrawat [8] have included thermoelastohydrodynamic effects in journal bearings and the FEM has been used to solve the governing equations. Steady performance of a wedge-shaped hydrodynamic journal bearing was analyzed by El-Gamal [9], in which a method of perturbation was used to solve the Reynolds-like equation governing the pressure inside the bearing. Banwait and Chandrawat [10] analyzed non-isothermal plain journal bearing problems by using the linear complementarity and FEM approaches to solve the related governing equations. Blanco and Prata [11] have used the finite volume method (FVM) to simulate and optimize the thrust bearings. Stefani and Rebora [12] applied a finiteelement approach to thermoelastohydrodynamic lubrication analysis applied to the problem of steadily loaded journal bearings. Their results proved be consistent with those from experimental and numerical works in the literature. Various techniques for performance analysis of journal bearings also are presented in the literature. Among them, interesting approximations consider infinitely long bearing for simplifying the solution of the Reynolds equation. Warner [13] used a side
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Nomenclature Ai Bi c Cf Ci D Di e Ei f F~ gi h h~ hi L Li NT Ni p P Pmax Pi Q~ s R u U u~ v w ~ w ~ W ~ 1 W
Coefficient defined in Eq. (13a) Coefficient defined in Eq. (13b) Radial clearance Friction factor Coefficient defined in Eq. (13c) Journal diameter Coefficient defined in Eq. (13d) Eccentricity Coefficient defined in Eq. (13e) Coefficient defined in Eq. (A.1g) Dimensionless friction force Coefficient defined in Eq. (8d) Film thickness Dimensionless film thickness Coefficient defined in Eq. (A.2e) Bearing length Coefficient defined in Eq. (A.2f) Truncation order for the pressure expansion Normalization integral Pressure field Dimensionless pressure field Dimensionless maximum pressure Transformed potentials Dimensionless side leakage flow Journal radius Circumferential velocity component Journal velocity Dimensionless circumferential velocity component Radial velocity component Axial velocity component Dimensionless axial velocity component Dimensionless load carrying capacity Dimensionless load component along the line of centers
flow leakage factor to improve the solution accuracy of long bearing approximation. In similar fashion, Ritchie [14] introduced the short bearing solution by the Galerkin method to improve the accuracy of short bearing approximation at high eccentricity. A simple and precise solution for the infinitely long and infinitely narrow bearings is presented by Reason and Narang [15]. This technique shows good results when compared to those from the FEM approach. Chandan [16] derived a generalized Reynolds equation to include couple stress effects in the analysis of short journal bearings. Sharma et al. [17] solved the Reynolds equation for a non-Newtonian lubricant for a finite width journal bearing through a finite difference scheme and showed that the lubricant non-Newtonian behavior has an improved beneficial effect for the case of relatively short bearings. An analytical solution for a second order model was developed by Capone et al. [18]; this model reduces infinitely long and infinitely narrow bearing theory in limit cases characterized for a parameter pair (L/D, e). Hirani et al. [19] have modified the analysis of Reason and Narang [15] to find a rapid method for evaluating the significant design parameters of journal bearings. Mokhiamer et al. [20] have analyzed the performance of finite journal bearings lubricated with a fluid with couple stresses taking into account the elastic deformation of the liner and concluded that the couple stress influence is significant. Recently, Vignolo et al. [21] obtained an approximate analytical solution to the Reynolds equation for finite length journal bearings by using a regular perturbation method.
~ 2 W x y Yi z
Dimensionless load component perpendicular to the line of centers Circumferential bearing coordinate Radial bearing coordinate Vector solution for the transformed potentials Axial bearing coordinate
Greek letters
a bi
e Z y yL l
m mi x f
j ci c~ i
Cavitation angle Parameter defined in Eq. (A.2d) Dimensionless eccentricity Dimensionless axial bearing coordinate Dimensionless circumferential bearing coordinate Angle that characterizes the lubricant film length Aspect ratio Viscosity Eigenvalues defined in Eq. (6a) Dimensionless radial bearing coordinate Normalized dimensionless circumferential bearing coordinate Attitude angle Eigenfunctions defined in Eq. (6b) Normalized eigenfunctions
Subscripts and superscripts i L max s 1, 2 –
Expansion index Related to angle that characterizes the lubricant film length Maximum value Related to side leakage flow Related to components of the load carrying capacity Integral transformed quantities Related either to dimensionless quantities or to normalized eigenfunctions
Regarding the solution methodology, the so-called Generalized Integral Transform Technique (GITT) [22–29] has been successfully employed in the solution related the mathematical modeling of several problems in the field of heat and fluid flow. However, specifically for applications involving moderate and lower Reynolds number flows, one may cite the works of Pe´rez Guerrero ~ et al. [33], and Cotta [30,31], Pe´rez Guerrero et al. [32], Castelloes Monteiro et al. [34] and Silva et al. [35]. In this context, the present work aims at applying the ideas in the GITT solution methodology to solve a general formulation of the Reynolds equation. Such a hybrid numerical–analytical approach is an eigenfunction expansion methodology for solving linear or nonlinear in multiphysics problems, especially those not a priori transformable by the classical approach. An extensive parametric analysis is done in order to investigate the influence of typical governing parameters for such physical situation. Comparisons with results for typical situations are performed to demonstrate the consistency of the final results and to show the capacity of the GITT approach for handling journal bearing problems.
2. Mathematical formulation Fig. 1 shows a schematic representation and nomenclature used for the mathematical formulation of a radial journal bearing problem. In it, c represents the radial clearance, which is the
E.N. Santos et al. / Tribology International 52 (2012) 161–169
163
Fig. 1. Schematic representation for the problem of radial journal bearing: (a) geometric configuration; (b) coordinates system; (c) load components.
difference between the bearing and journal radii. The center of the journal and bearing are located at points O and O0 , respectively. Also, e is the eccentricity, the distance between these two centers, while ho is the minimum film thickness and occurs at the centerline. The film thickness at any point is represented by h. Therefore, the formulation of radial journal bearings problems is done through the classical Reynolds equation, which is developed by considering mass and momentum balances. It is also considered steady and incompressible laminar flow of Newtonian fluid with constant viscosity. Forces due to lubricant inertia are neglected and it is not taken account pressure variations in axial and in the clearance directions. Within the simplifying hypotheses the classical Reynolds equation in dimensionless form is written as: 0 yL l 2 @ ~ 3 @PðZ, fÞ @ ~3 @PðZ, fÞ h ðfÞ h ðfÞ ð1Þ þ ¼ 6yL h~ ðfÞ @Z @Z @f @f 2
3. Solution methodology
subjected to the following boundary conditions:
3.1. Eigenvalue problem
Pð0, fÞ ¼ 0;
Pð1, fÞ ¼ 0 and PðZ, 0Þ ¼ 0;
@PðZ, 1Þ ¼ PðZ, 1Þ ¼ 0 @f ð2a2eÞ
– Choose an appropriate auxiliary eigenvalue problem; – Develop the integral transform pair for performing the integral transformation and inversion operations; – Integral transformation of the original partial differential equation; – Numerically solve the resulting coupled ordinary differential equations system for the transformed potentials; – Recall the analytical inverse formula to construct the hybrid solution of the desired original potential.
In accordance with the steps stated above, the following choice for the eigenvalue problem is made: 2
where, ~ fÞ ¼ 1 þ ecosðfy Þ; hð L
In the application of the Generalized Integral Transform Technique (GITT) [22–35] for solving the problem given by Eqs. (1) to (3), the following basic steps are considered:
0 h~ ðfÞ ¼ eyL sinðfyL Þ
ð3a; bÞ
The following dimensionless groups were used to obtain Eqs. (1) to (3):
y ¼ x=R; x ¼ y=c; Z ¼ z=L; f ¼ y=yL ; h~ ¼ h=c; e ¼ e=c; ~ ¼ w=U; P ¼ pc2 =ðmURÞ l ¼ 2R=L; u~ ¼ u=U; w
ð4a2jÞ
In Eq. (4d) yL is the angle that characterizes the lubricant film length, which depends on the cavitation boundary, and is taken into account in the Reynolds equation by considering Reynolds or Swift–Steiber boundary conditions, such as evidenced in references by Cameron [1,2], Sharma et al. [17] and Liu [36]. In the present work, such boundary conditions are given by Eqs. (2), where in this case, yL ¼ p þ a, with a being the cavitation angle.
d ci ðZÞ þ m2i ci ðZÞ ¼ 0 dZ2
ci ð0Þ ¼ 0;
ð5aÞ
ci ð1Þ ¼ 0
ð5b; cÞ
Problem (5) is readily solved analytically, to yield the eigenvalues, mi, and the eigenfunctions, ci(Z), as
mi ¼ ip, i ¼ 1,2,. . .;
ci ðZÞ ¼ sinðmi ZÞ
ð6a; bÞ
It can be shown that the eigenfunctions, ci(Z), enjoy the following orthogonality property with their corresponding normalization integral, Ni: ( Z 1 Z 1 0, for i aj 1 ; Ni ¼ ci ðZÞcj ðZÞdZ ¼ c2i ðZÞdZ ¼ ; Ni , for i ¼ j 2 0 0 ð6c; dÞ
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Table 1 ~ and Pmax for the case of e ¼ 10 5 Convergence behavior of the potentials yL, j, W and l ¼ 10 5. NT
yL (degrees)
j (degrees)
~ 103 W
Pmax 104
10 30 50 70 90 110 130
257.5 257.5 257.5 257.5 257.5 257.5 257.5 257.5
70.91 70.91 70.91 70.91 70.91 70.91 70.91 70.91
0.130 0.134 0.135 0.135 0.135 0.135 0.135 0.136
0.871 0.836 0.829 0.826 0.825 0.824 0.823 0.819
a
a Results obtained from an approximate solution for e-0 given in the Appendix.
~ ðZÞ, integrated over the domain [0,1] in Eq. (1), is multiplied by c i the Z direction, and the inverse formula, Eq. (7b), is employed in place of the pressure field, P(Z,f), as well as the orthogonality property when applicable. The same operations can be performed over the boundary conditions given by Eqs. (2c) and (2d), to furnish the following system: " 0 # 2 d Pi ðfÞ 3h~ ðfÞ dP i ðfÞ yL l 2 2 þ mi Pi ðfÞ ¼ g i ðfÞ ð8aÞ 2 ~ fÞ df 2 hð df Pi ð0Þ ¼ 0;
dP i ð1Þ ¼0 df
ð8b; cÞ
The coefficient that appears in Eq. (8a) is defined as g i ðfÞ ¼
0 0 Z 6yL h~ ðfÞ 1 ~ 6yL h~ ðfÞ ½1cosðmi Þ pffiffiffiffiffi c ð Z Þd Z ¼ i 3 3 mi Ni h~ ðfÞ 0 h~ ðfÞ
ð8dÞ
The ODE system defined by Eqs. (8) depends on yL itself, which is calculated with the additional boundary condition given by Eq. (2e). Therefore, using the inverse formula, Eq. (7b), in such additional condition, one obtains the following transcendental equation for the yL calculation: 1 X
c~ i ðZÞPi ð1Þ ¼ 0
ð9Þ
i¼1
Fig. 2. Algorithm of the solution procedure.
3.2. Integral transform pair The above orthogonality property allows the definition of the following integral transform pair: Z 1 P i ðfÞ ¼ c~ i ðZÞPðZ, fÞdZ, transf orm ð7aÞ 0
PðZ, fÞ ¼
1 X
c~ i ðZÞP i ðfÞ, inverse
ð7bÞ
i¼1
pffiffiffiffiffi ~ ðZÞ ¼ c ðZÞ= N are the normalized eigenfunctions. where c i i i 3.3. Integral transformation of the original partial differential equation To obtain the transformed ordinary differential system for the transformed potentials, P i ðfÞ, the partial differential equation,
Eq. (9) is solved through the DZREAL subroutine from the IMSL Library [37] with a prescribed tolerance of 10 9. This subroutine ¨ finds the real zeros of a real function using Muller’s method. In this computation, it is necessary to truncate the infinite expansion in a sufficiently large number of terms (e.g., NT), so as to achieve the user-prescribed relative error target for obtaining the angle that characterizes the lubricant film length. One can see from Eq. (9) that the calculation of the yL depends on the variable Z; however, the present approach has shown that the same values are obtained for yL even for different values of Z. This can be explained by the fact that the boundary conditions given by Eqs. (2a) to (2e) also must be satisfied in this computational process. Once the yL is determined, the transformed potentials, Pi ðfÞ, can be obtained. For this purpose, in order to solve the transformed ODE system, efficient numerical algorithms for boundary value problems with stiff characteristics are to be employed, such as the subroutine DBVPFD also from the IMSL Library [37], which offers an automatic adaptive scheme for local error control of the numerical results for the transformed potentials. The features on this subroutine are focused on solving a (parameterized) system of differential equations with boundary conditions at two points, using a variable order, variable step size finite difference method with deferred corrections. It is then necessary to rewrite the transformed ODE system as a first order
E.N. Santos et al. / Tribology International 52 (2012) 161–169
one, by introducing the following dependent variables: dY i dPi ðfÞ ¼ Y i þ NT ¼ ; df df
Y i ¼ Pi ðfÞ;
Therefore, by making use of Eqs. (10), the transformed system can be rewritten as: " 0 # dY i dY i þ NT 3h~ ðfÞ yL l 2 2 ¼ Y i þ NT ; ¼ mi Y i þg i ðfÞ, Y i þ NT þ ~ fÞ df df 2 hð
2
dY i þ NT d P i ðfÞ ¼ , 2 df df
i ¼ 1,2,3,. . .,NT
ð10a2cÞ
i ¼ 1,2,3,. . .,NT Y i ð0Þ ¼ 0;
Table 2 ~ and Pmax for the case of e ¼ 10 Convergence behavior of the potentials yL, j, W and l ¼0.5.
ð11a; bÞ
Y i þ NT ð1Þ ¼ 0
ð11c; dÞ
5
NT
yL (degrees)
j (degrees)
~ 104 W
Pmax 104
10 30 50 70
233.0 233.0 233.0 233.0 233.0
80.57 80.60 80.60 80.60 80.60
0.560 0.560 0.560 0.560 0.560
0.490 0.489 0.489 0.489 0.489
a
165
1x10-4 Lines - GITT solution Symbols - Approximate analytical solution
9x10-5
=10-5;
=10-5
8x10-5 7x10-5 6x10-5
Results obtained from an approximate solution for e-0 given in the Appendix.
P
a
=10-5 ; =0.5
5x10-5 4x10-5
Table 3 ~ and Pmax for the case of e ¼ 0.5 and Convergence behavior of the potentials yL, j, W l ¼10 5. NT
yL (degrees)
j (degrees)
~ W
Pmax
10 30 50 70 90 110
219.7 219.7 219.7 219.7 219.7 219.7
58.30 58.30 58.30 58.30 58.30 58.30
6.19 6.37 6.40 6.42 6.43 6.43
4.76 4.57 4.53 4.52 4.51 4.51
3x10-5 2x10-5 1x10-5 0 0
20 40 60 80 100 120 140 160 180 200 220 240 260
(degrees)
Table 4 ~ and Pmax for the case of e ¼ 0.5 and Convergence behavior of the potentials yL, j, W l ¼0.1. NT
yL (degrees)
j (degrees)
~ W
Pmax
10 30 50 70 90
219.5 219.7 219.7 219.7 219.7
59.63 60.22 60.26 60.26 60.27
5.75 5.75 5.75 5.75 5.75
4.59 4.48 4.48 4.47 4.47
Fig. 3. Comparison of the GITT and approximate analytical results for the dimensionless pressure field in the circumferential direction for e ¼10 5 and l ¼ 10 5 and e ¼ 10 5 and l ¼0.5.
Table 5 Comparison of the dimensionless load carrying capacity with the literature results for different eccentricities e and aspect ratios l. Aspect ratio (l )
1
4
Authors
Reason and Narang [15] FEM [15] Sharma et al. [17] Hirani et al. [19] GITT – present work Reason and Narang [15] FEM [15] Sharma et al. [17] Hirani et al. [19] GITT – present work
Eccentricity (e) 0.1
0.4
0.5
0.6
0.8
0.228
–
1.722
–
6.924
0.228 – 0.228 0.239
– 1.2386 – 1.22
1.584 – 1.62 1.77
– 2.7206 – 3.91
5.964 7.5788 6.204 8.08
0.0192 –
0.1782 –
1.2216
0.0198 – 0.0192 0.0196
0.1740 – 0.1770 0.287
1.1304 1.1855 1.1616 1.24
– 0.1123 – 0.196
– 0.2967 – 0.429
Fig. 4. Comparison of the dimensionless pressure field in the circumferential direction for e ¼0.4 and l ¼1.
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E.N. Santos et al. / Tribology International 52 (2012) 161–169
At this point, since the transformed potentials, P i ðfÞ, are obtained, the inverse formula given by Eq. (7b) is recalled to calculate the bearing and flow parameters. Therefore, by introducing Eq. (7b) in the usual definitions of the dimensionless load components, the friction force and the side leakage flow, one obtains the expressions for computation of such parameters, respectively, as Z 1Z 1 1 X ~ 1 ¼ yL W PðZ, fÞcosðfyL ÞdfdZ ¼ yL Ai C i ð12aÞ 0
~ 2 ¼ yL W
Z
0
1Z 0
F~ ¼ yL
1
PðZ, fÞsinðfyL ÞdfdZ ¼ yL 0
0
2i¼1
yL Q~ s ¼ 6
1 0
1 1X
Z
Bi C i
"
ð12bÞ
# Z 2p=yL ~ fÞ @P hð 1 þ df dZdf ¼ yL ~ fÞ ~ fÞ 2yL @f 0 hð hð 1
C i Di
1 0
1 X i¼1
Z 2p=yL Z
þ
i¼1
where the various coefficients that appear in Eqs. (12) are calculated from: Z 1 Z 1 Ai ¼ Pi ðfÞcosðfyL Þdf; Bi ¼ P i ðfÞsinðfyL Þdf; 0
Ci ¼
Z
0
Di ¼
0
1
c~ i ðZÞdZ;
Z 2p=yL 0
~ fÞ dP i ðfÞ df; hð df
Ei ¼
Z
1 0
3 h~ ðfÞP i ðfÞdf
ð13a2eÞ
Fig. 2 shows the algorithm of the scheme for the solution procedure. Now, from Eq. (12) above, the dimensionless load ~ , the attitude angle, j, and the friction factor carrying capacity, W Cf are also calculated from their usual definitions, respectively, as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ 2 þW ~ ¼ W ~ 2 ; j ¼ tan1 ðW ~ 2 =W ~ 1 Þ; C f ¼ F~ =W ~ W ð14a2cÞ 1 2
ð12cÞ 4. Results and discussion
3 @Pðf, ZÞ h~ ðfÞ @Z
Z¼0
df ¼
yL 6
~ ð0Þ dc i Ei dZ i¼1 1 X
ð12dÞ
Numerical results for the pressure field, friction factor, dimensionless load carrying capacity and side leakage flow were
Fig. 5. Comparison of the potentials as a function of the eccentricity for l ¼1: (a) friction factor; (b) dimensionless side leakage flow; (c) dimensionless load carrying capacity; (d) attitude angle.
E.N. Santos et al. / Tribology International 52 (2012) 161–169
obtained from a code developed in the FORTRAN 95/2003 programming language. The code was implemented on a computational platform with INTELs CORETM i7 M620 2.67 GHz processor, and the system given by Eqs. (9) and (11) was simultaneously handled through the DBVPFD and DZREAL subroutines, both from the IMSL Library [37]. A relative error target of 10 5 in the subroutine DBVPFD was employed throughout the computations, for varying the values of the dimensionless eccentricity, e, and of the aspect ratio, l ¼D/L. A typical CPU running time for the computations such as those shown in Table 5 is about 350 s. First, an analysis of convergence behavior of the following potentials is made, namely, angle that characterizes the lubricant film length, yL, attitude angle, j, dimensionless load carrying capa~ , and dimensionless Pmax at the middle plane of the radial city, W bearing. For this purpose, the influence of different truncation orders (NT) in the solution of the pressure equation was investigated. A comparison with those results obtained from a simplified solution (valid for e small) is also made. Table 1 brings the convergence behavior of such potentials for e ¼ 10 5 and l ¼10 5. The convergence is established with 30 ~ and Pmax terms in the summations for yL and j, while W experience lower convergence rates, reaching such convergences with NT between 50 and 70 terms and NT between 110 and 130 terms, respectively. There is also an excellent agreement of the results obtained in the present analysis with those of an analytical solution by considering the hypothesis that e is very small (see Appendix), within four significant digits for yL and j, whereas ~ and Pmax; this good two significant digits are obtained for W agreement ensures verification of the computer code developed. A similar analysis is shown in Table 2 for the case of e ¼10 5 and l ¼0.5. Furthermore, excellent convergence rates are clearly observable, with full convergence to four digits being achieved for yL and j, even for lower truncation orders (approximately NT¼30). The ~ and Pmax reach three converged digits with a truncapotentials W tion order of around NTE30 to 50 terms. The excellent agreement of the results with those of an analytical solution shown in the appendix once again ensures the verification of such results. Tables 3 and 4 address the convergence behavior of same potentials for the cases of e ¼0.5 and l ¼10 5 and of e ¼0.5 and l ¼0.1, respectively. For such situations, higher convergence rates are obtained for all potentials analyzed, with four converged digits for yL and j (also approximately NT ¼30) and a full ~ and Pmax (NT E90 to 110 terms convergence to three digits for W for the case of e ¼0.5 and l ¼10 5 and NT E50 to 70 terms for the case of e ¼0.5 and l ¼0.1). Therefore, after this convergence analysis, the subsequent computations were performed using up to seventy terms (NT r70) in the eigenfunction expansions. Table 5 illustrates the results for the dimensionless load carrying capacity obtained in the present work and for those available in the literature, for l ¼D/L ¼1 and 4, and different eccentricities, i.e., e ¼0.1, 0.4, 0.5, 0.6 and 0.8. In general, a reasonable agreement is observed among the present results with those from the literature, in particular for l ¼ 4, while for l ¼1 the literature results lose adherence with the present converged results as the eccentricity increases. Fig. 3 shows a comparison of the present GITT results against those obtained with an analytical solution for very small eccentricities. In this analysis, the dimensionless pressure field in the circumferential direction is shown for e ¼10 5 and l ¼10 5 and e ¼ 10 5 and l ¼0.5 at the journal bearing medium plan. The excellent agreement again warrants the verification of the results produced with the present GITT approach. Fig. 4 gives the present GITT results for the dimensionless pressure field in the circumferential direction for l ¼ 1 and e ¼ 0.4 at the journal bearing medium plan, which are compared with those results obtained by Williams and Symmons [6], Sharma
167
Fig. 6. Comparison of the maximum pressure as a function of the axial position for e ¼ 0.4 and l ¼ 1.
et al. [17] and Mokhiamer et al. [20]. Fig. 5 shows a comparison between the present results with those of Mokhiamer et al. [20] for the potentials of friction factor, dimensionless side leakage flow, dimensionless load carrying capacity and attitude angle for l ¼1. Also, in Fig. 6, the maximum pressure as a function of the axial position for e ¼0.4 and l ¼1 is compared with the results of Mokhiamer et al. [20]. In all cases analyzed in Figs. 4 to 6, an excellent agreement among the set of results is noted, once again confirming the present GITT results. Finally, Fig. 7 show pressure field patterns in the radial journal bearing for different values of the governing parameters: e ¼10 5 and l ¼10 5 (Fig. 7(a)); e ¼0.5 and l ¼10 5 (Fig. 7(b)); e ¼10 5 and l ¼1 (Fig. 7(c)) and e ¼0.5 and l ¼1 (Fig. 7(d)). In Fig. 7(a) and (b) can be observed that for long journal bearings (l ¼ D/L-0), the circumferential pressure field does not depend on the axial position, Z. This is explained by the fact that the pressure gradient in the Z direction to be negligible in relation to the gradient in the y direction (qP/qy 4 4qP/qZ), i.e., for long journal bearings, the flow in the y direction is more important that in the Z direction. It is also verified that the pressure in the journal bearing is proportional to e (see Figs. 4 to 6). The variation in the pressure field is due to e for regulating the wedge effect. In other words, this effect increases for increasing values of the eccentricity and, as a consequence, the lubricant shear increases, proportional to hydrodynamic pressure, as can be observed in Eqs. (12). On the other hand, from Fig. 7(c) and (d), it is observed that the pressure field varies in the Z direction due to the fact of the journal bearing not being long (l ¼D/L¼1). Similarly to Fig. 7(a) and (b), the pressure is proportional to the eccentricity. Therefore, by comparing Fig. 7(a) and (b) to (c) and (d), it is verified that the pressure field is inversely proportional to l ¼D/L, thus demonstrating that there is a dependence of this field in relation to this parameter, which determines if a journal bearing is long or short.
5. Conclusions A hybrid numerical–analytical solution for a problem of hydrodynamic lubrication related to radial journal bearings was
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E.N. Santos et al. / Tribology International 52 (2012) 161–169
Fig. 7. Patterns of pressure field for different values of the governing parameters: (a) e ¼ 10 5 and l ¼ 10 5; (b) e ¼ 0.5 and l ¼10 5; (c) e ¼ 10 5 and l ¼ 1; (d) e ¼ 0.5 and l ¼ 1.
advanced by extending the ideas in the Generalized Integral Transform Technique (GITT). Convergence behaviors were extensively undertaken in the eigenfunction expansions, for the pressure field and for the journal bearing potentials, which have demonstrated the adequacy and robustness of the proposed hybrid solution approach. The present analysis has also considered different values of the eccentricity, e, and of the aspect ratio, l, in order to present physical aspects concerning this journal bearing problem. The GITT approach, due to its hybrid numerical– analytical nature based on eigenfunction expansions, is well suited for benchmarking purposes, and its successful employment in the present journal bearing analysis with relative lower computational costs and computational efficiency permits a natural extension of its application to other journal bearing problems such those by taking into account dynamic cavitation (mass-conserving models), bearing deformations and heat exchange (thermohydrodynamic (THD) and thermal elastohydrodynamic (TEHD) analyses).
Appendix. Analytical solution for very small eccentricity An approximation considered in hydrodynamic lubrication is to assume that a journal bearing has a very small eccentricity (e-0). 3 Therefore, it can be considered that the term h~ is approximately 3 ~ equal to unity in Eq. (1) (h 1). As a consequence, the Reynolds
equation together with the boundary conditions are given by:
2
yL l 2
@2 PðZ, fÞ @2 PðZ, fÞ þ ¼ f ðfÞ 2 @Z2 @f
Pð0, fÞ ¼ 0;
Pð1, fÞ ¼ 0
PðZ, 0Þ ¼ 0;
@PðZ, 1Þ ¼ PðZ, 1Þ ¼ 0 @f
ðA:1aÞ
ðA:1b; cÞ ðA:1d2fÞ
where 0 2 f ðfÞ ¼ 6yL h~ ðfÞ ¼ 6eyL sinðfyL Þ
ðA:1gÞ
The problem presented by Eq. (A.1) can be solved through the Classical Integral Transform Technique (CITT) [38]. For this purpose, the same integral transform operations can be performed over Eq. (A.1), i.e., the partial differential Eq. (A.1a) and the boundary conditions given by Eqs. (A.1d) and (A.1e) are ~ ðZÞ, integrated over the domain [0,1] in the Z multiplied by c i direction, so that one obtains the following transformed ordinary differential system for the transformed potentials, P i ðfÞ, with their respective boundary conditions: 2
d Pi ðfÞ df
2
2
bi Pi ðfÞ ¼ hi ðfÞ
Pi ð0Þ ¼ 0;
dP i ð1Þ ¼0 df
ðA:2aÞ
ðA:2b; cÞ
E.N. Santos et al. / Tribology International 52 (2012) 161–169
where yl 2 2 b2i ¼ L mi ; hi ðfÞ ¼ 6ey2L sinðfyL ÞLi ; 2 Z 1 ½1cosðmi Þ pffiffiffiffiffi c~ i ðZÞdZ ¼ Li ¼ mi N i 0
ðA:2d2fÞ
Here, the eigenquantities (eigenvalues, eigenfunctions and norms) and the integral transform pair are the same as those given in the Section 3 by Eqs. (5a) to (7b). The ordinary differential equation given by Eq. (A.2a) together with boundary conditions (A.2b,c) allow one to obtain an analytical solution for the transformed potentials, P i ðfÞ, in the form: 2 6ey L yL cosðyL Þ sinhðbi fÞ ðA:3Þ P i ðfÞ ¼ 2 L 2i sinðfyL Þ coshðbi Þ bi ðb þ y Þ i
L
Next, by introducing Eq. (A.3) into the inverse formula given by Eq. (7b), the following analytical equation for the pressure distribution is the result: 1 X Li sinðmi ZÞ yL cosðyL Þ sinhðbi fÞ 2 pffiffiffiffiffi PðZ, fÞ ¼ 6eyL sinðfyL Þ 2 2 coshðbi Þ bi Ni i ¼ 1 ðbi þ yL Þ ðA:4Þ Therefore, the angle yL can be calculated from the introduction of Eq. (A.4) into additional boundary condition (A.1f), to yield 1 X Li sinðmi ZÞ yL cosðyL Þ sinhðbi Þ pffiffiffiffiffi ¼0 ðA:5Þ sinðyL Þ 2 2 coshðbi Þ bi Ni i ¼ 1 ðb þ y Þ i
L
Eq. (A.5) provides a transcendental equation for determining the angle yL. For computational purposes, such a summation must be truncated into a number of NT terms. After that, the subroutine DZREAL from the IMSL Library [37] with a tolerance of 10 9 is ~ , j, Cf also employed for this determination. The parameters W and Q~ S are also calculated from their usual definitions. References [1] Cameron A. The Principles of Lubrication. London: Longmans Green; 1966. [2] Cameron A. Basic Lubrication Theory. New Delhi: Wiley Eastern Ltd; 1987. [3] Sivak B, Sivak M. The numerical solution of the Reynolds equation by a modified Ritz method. Wear 1981;72:371–6. [4] Tayal SP, Sinhasan R, Singh DV. Analysis of hydrodynamic journal bearings having non-Newtonian lubricants. Tribology International 1982;15:17–21. [5] Chandrawat HN, Sinhasan R. A comparison between two numerical techniques for hydrodynamic journal bearing problems. Wear 1987;119:77–87. [6] Williams PD, Symmons GR. Analysis of hydrodynamic journal bearings lubricated with non-Newtonian fluids. Tribology International 1987;20: 119–224. [7] Sinhasan R, Chandrawat HN. An elastohydrodynamic study on two-axialgroove journal bearings. Tribology International 1988;21:341–51. [8] Sinhasan R, Chandrawat HN. Analysis of a two-axial-groove journal bearing including thermoelastohydrodynamic effects. Tribology International 1989;22:347–53. [9] El-Gamal HA. Analysis of the steady state performance of a wedge-shaped hydrodynamic journal bearing. Wear 1995;184:111–7. [10] Banwait SS, Chandrawat HN. Study of thermal boundary conditions for a plain journal bearing. Tribology International 1998;31:289–96.
169
[11] Blanco CJC, Prata A.T. Optimization and simulation of thrust circular bearings with planes sector-shaped. Industrial Lubrication and Tribology 2012; in Press. [12] Stefani F, Rebora A. Steadily loaded journal bearings: Quasi-3D mass-energyconserving analysis. Tribology International 2009;42:448–60. [13] Warner PC. Static and dynamic properties of partial journal bearings. ASME Journal of Basic Engineering 1963;85:247–57. [14] Ritchie GS. The prediction of journal loci in dynamically loaded internal combustion engine bearings. Wear 1975;35:291–7. [15] Reason BR, Narang IP. Rapid design and performance evaluation of steady state journal bearings — a technique amenable to programmable hand calculators. ASLE Transactions 1982;25:429–44. [16] Chadan S. Lubrication theory for couple stress fluids and its application to short bearings. Wear 1982;80:281–90. [17] Sharma D, Athre K, Biswas S. Iyenger SRK. Solution of Reynolds’ equation for a non-Newtonian lubricant in a journal bearing implementing the moving boundary conditions. Tribology International 1991;24:85–9. [18] Capone G, Agostino V, Guida D. A finite length plain journal bearing theory. ASME Journal of Tribology 1994;116:648–53. [19] Hirani H, Rao TVVLN, Athre K, Biswas S. Rapid performance evaluation of journal bearings. Tribology International 1997;30:825–34. [20] Mokhiamer UM, Crosby WA, El-Gamal HA. A study of a journal bearing lubricated by fluids with couple stress considering the elasticity of the liner. Wear 1999;224:194–201. [21] Vignolo GG, Barila´ DO, Quinzani LM. Approximate analytical solution to Reynolds equation for finite length journal bearings. Tribology International 2011;44:1089–99. [22] Cotta RM. Hybrid numerical–analytical approach to nonlinear diffusion problems. Numerical Heat Transfer Part B 1990;127:217–26. [23] Cotta RM. Integral Transforms in Computational Heat and Fluid Flow. Boca Raton: CRC Press; 1993. [24] Cotta RM. Benchmark results in computational heat and fluid flow: — the integral transform method. International Journal of Heat and Mass Transfer (invited paper) 1994;37:381–94. [25] Cotta RM, Mikhailov MD. Heat Conduction: Lumped Analysis, Integral Transforms, Symbolic Computation. Chichester: Wiley-Interscience; 1997. [26] Cotta RM. The Integral Transform Method in thermal and Fluids Sciences and Engineering. New York: Begell House; 1998. [27] Santos CAC, Quaresma JNN, Lima JA. Benchmark Results for Convective Heat Transfer in Ducts: — The Integral Transform Approach. Rio de Janeiro: E–Papers; 2001. [28] Cotta RM, Mikhailov MD. Hybrid methods and symbolic computations. In: Minkowycz WJ, Sparrow EM, Murthy JY, editors. Handbook of Numerical Heat Transfer. 2nd edn. New York: Wiley; 2006. p. 493–522. [29] Sphaier LA, Cotta RM, Naveira-Cotta CP, Quaresma JNN. The UNIT algorithm for solving one-dimensional convection-diffusion problems via integral transforms. International Communications in Heat and Mass Transfer 2011;38:565–71. [30] Pe´rez Guerrero JS, Cotta RM. Integral transform solution of developing laminar duct flow in Navier–Stokes formulation. International Journal for Numerical Methods in Fluids 1995;20:1203–13. [31] Pe´rez Guerrero JS, Cotta RM. Benchmark integral transform results for flow over a backward-facing step. Computers & Fluids 1996;25:527–40. [32] Pe´rez Guerrero JS, Quaresma JNN, Cotta RM. Simulation of laminar flow inside ducts of irregular geometry using integral transforms. Computational Mechanics 2000;25:413–20. ~ [33] Castelloes FV, Quaresma JNN, Cotta RM. Convective heat transfer enhancement in low Reynolds number flows with wavy walls. International Journal of Heat and Mass Transfer 2010;53:2022–34. [34] Monteiro ER, Macˆedo EN, Quaresma JNN, Cotta RM. Laminar flow and convective heat transfer of non-Newtonian fluids in doubly connected ducts. International Journal of Heat and Mass Transfer 2010;53:2434–48. [35] Silva RL, Quaresma JNN, Santos CAC, Cotta RM. Integral transforms solution for flow development in wavy wall ducts. International Journal of Numerical Methods for Heat and Fluid Flow 2011;21:219–43. [36] Liu S. On boundary conditions in lubrication with one dimensional analytical solutions. Tribology International 2012;48:182–90. [37] IMSL Library, 1991. ‘‘Math/Lib’’, Houston, Texas. ¨ zisik MN. Unified Analysis and Solutions of Heat and Mass [38] Mikhailov MD, O Diffusion. New York: John Wiley; 1984.
Contents lists available at SciVerse ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Integral transform solutions for the analysis of hydrodynamic lubrication of journal bearings E.N. Santos a, C.J.C. Blanco b, E.N. Macˆedo c, C.E.A. Maneschy a, J.N.N. Quaresma c,n a
´, UFPA, Campus Universita ´rio do Guama ´, Rua Augusto Corrˆea, 01, 66075–110 Bele´m, PA, Brazil School of Mechanical Engineering, Universidade Federal do Para ´, UFPA, Campus Universita ´rio do Guama ´, Rua Augusto Corrˆea, 01, 66075–110 Bele´m, PA, Brazil School of Environmental and Sanitary Engineering, Universidade Federal do Para c ´, UFPA, Campus Universita ´rio do Guama ´, Rua Augusto Corrˆea, 01, 66075–110 Bele´m, PA, Brazil School of Chemical Engineering, Universidade Federal do Para b
a r t i c l e i n f o
abstract
Article history: Received 2 December 2011 Received in revised form 6 March 2012 Accepted 22 March 2012 Available online 6 April 2012
This work deals with analysis of hydrodynamic lubrication of radial journal bearings. The Reynolds equation was treated in order to obtain a hybrid numerical–analytical solution through the Generalized Integral Transform Technique (GITT) for the problem. A parametric analysis is done to investigate the influence of typical governing parameters for such a physical situation. Numerical results for engineering parameters such as pressure field, friction coefficient, axial flow rate and dimensionless load capacity were thus produced as functions of such parameters. Comparisons with results presented in the literature were also performed in order to verify the present results, as well as to demonstrate the consistency of the final results and the capacity of the GITT approach in handling journal bearing problems. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Hydrodynamic lubrication Journal bearings Lubrication theory Integral transforms
1. Introduction The field for application of radial journal bearings is enormous, and includes shipbuilding, industrial machinery and equipment, transportation industry, among others. Research and simulation of physical phenomena involved in the operation of these bearings are very important technologies because they allow evaluation of the phenomena involved with their performance, such as the workload and coefficient of friction. The analysis of journal bearings is probably the most important part of the classical hydrodynamic lubrication theory and, it is also most difficult and complex due to integration of the Reynolds equation. The journal bearings are used mainly for decreasing the friction existing between solid parts of rotating machines and weakening the load variations supported by them. The journal bearing must support the load carried with minimal lost energy and low wear [1,2]. Hydrodynamic lubrication has been studied by many researchers with different numerical techniques for solving the Reynolds equation either for isothermal or non-isothermal flows. Sivak and Sivak [3] obtained a numerical solution of the Reynolds equation by modified Ritz method. Tayal et al. [4] investigated the effect of nonlinearity on the performance of journal bearings with finite width by using the finite element method (FEM). Chandrawat and Sinhasan [5] presented
n
Corresponding author. Tel.: þ55 91 32017837; fax: þ55 9132017848. E-mail address: [email protected] (J.N.N. Quaresma).
0301-679X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.triboint.2012.03.016
a comparison between the Gauss-Seidel iterative method and the linear complementarity approach for determining the pressure field in the analysis of plain and two-axial groove journal bearings in laminar flow operation. Williams and Symmons [6] analyzed a procedure, based on the finite difference method together with a technique known as SIMPLE, to solve the Navier–Stokes equations for the steady three-dimensional flow of a non-Newtonian fluid into the journal bearing with finite-breadth. Sinhasan and Chandrawat [7] presented an elastohydrodynamic study of two-axial-groove journal bearings. Similarly, Sinhasan and Chandrawat [8] have included thermoelastohydrodynamic effects in journal bearings and the FEM has been used to solve the governing equations. Steady performance of a wedge-shaped hydrodynamic journal bearing was analyzed by El-Gamal [9], in which a method of perturbation was used to solve the Reynolds-like equation governing the pressure inside the bearing. Banwait and Chandrawat [10] analyzed non-isothermal plain journal bearing problems by using the linear complementarity and FEM approaches to solve the related governing equations. Blanco and Prata [11] have used the finite volume method (FVM) to simulate and optimize the thrust bearings. Stefani and Rebora [12] applied a finiteelement approach to thermoelastohydrodynamic lubrication analysis applied to the problem of steadily loaded journal bearings. Their results proved be consistent with those from experimental and numerical works in the literature. Various techniques for performance analysis of journal bearings also are presented in the literature. Among them, interesting approximations consider infinitely long bearing for simplifying the solution of the Reynolds equation. Warner [13] used a side
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Nomenclature Ai Bi c Cf Ci D Di e Ei f F~ gi h h~ hi L Li NT Ni p P Pmax Pi Q~ s R u U u~ v w ~ w ~ W ~ 1 W
Coefficient defined in Eq. (13a) Coefficient defined in Eq. (13b) Radial clearance Friction factor Coefficient defined in Eq. (13c) Journal diameter Coefficient defined in Eq. (13d) Eccentricity Coefficient defined in Eq. (13e) Coefficient defined in Eq. (A.1g) Dimensionless friction force Coefficient defined in Eq. (8d) Film thickness Dimensionless film thickness Coefficient defined in Eq. (A.2e) Bearing length Coefficient defined in Eq. (A.2f) Truncation order for the pressure expansion Normalization integral Pressure field Dimensionless pressure field Dimensionless maximum pressure Transformed potentials Dimensionless side leakage flow Journal radius Circumferential velocity component Journal velocity Dimensionless circumferential velocity component Radial velocity component Axial velocity component Dimensionless axial velocity component Dimensionless load carrying capacity Dimensionless load component along the line of centers
flow leakage factor to improve the solution accuracy of long bearing approximation. In similar fashion, Ritchie [14] introduced the short bearing solution by the Galerkin method to improve the accuracy of short bearing approximation at high eccentricity. A simple and precise solution for the infinitely long and infinitely narrow bearings is presented by Reason and Narang [15]. This technique shows good results when compared to those from the FEM approach. Chandan [16] derived a generalized Reynolds equation to include couple stress effects in the analysis of short journal bearings. Sharma et al. [17] solved the Reynolds equation for a non-Newtonian lubricant for a finite width journal bearing through a finite difference scheme and showed that the lubricant non-Newtonian behavior has an improved beneficial effect for the case of relatively short bearings. An analytical solution for a second order model was developed by Capone et al. [18]; this model reduces infinitely long and infinitely narrow bearing theory in limit cases characterized for a parameter pair (L/D, e). Hirani et al. [19] have modified the analysis of Reason and Narang [15] to find a rapid method for evaluating the significant design parameters of journal bearings. Mokhiamer et al. [20] have analyzed the performance of finite journal bearings lubricated with a fluid with couple stresses taking into account the elastic deformation of the liner and concluded that the couple stress influence is significant. Recently, Vignolo et al. [21] obtained an approximate analytical solution to the Reynolds equation for finite length journal bearings by using a regular perturbation method.
~ 2 W x y Yi z
Dimensionless load component perpendicular to the line of centers Circumferential bearing coordinate Radial bearing coordinate Vector solution for the transformed potentials Axial bearing coordinate
Greek letters
a bi
e Z y yL l
m mi x f
j ci c~ i
Cavitation angle Parameter defined in Eq. (A.2d) Dimensionless eccentricity Dimensionless axial bearing coordinate Dimensionless circumferential bearing coordinate Angle that characterizes the lubricant film length Aspect ratio Viscosity Eigenvalues defined in Eq. (6a) Dimensionless radial bearing coordinate Normalized dimensionless circumferential bearing coordinate Attitude angle Eigenfunctions defined in Eq. (6b) Normalized eigenfunctions
Subscripts and superscripts i L max s 1, 2 –
Expansion index Related to angle that characterizes the lubricant film length Maximum value Related to side leakage flow Related to components of the load carrying capacity Integral transformed quantities Related either to dimensionless quantities or to normalized eigenfunctions
Regarding the solution methodology, the so-called Generalized Integral Transform Technique (GITT) [22–29] has been successfully employed in the solution related the mathematical modeling of several problems in the field of heat and fluid flow. However, specifically for applications involving moderate and lower Reynolds number flows, one may cite the works of Pe´rez Guerrero ~ et al. [33], and Cotta [30,31], Pe´rez Guerrero et al. [32], Castelloes Monteiro et al. [34] and Silva et al. [35]. In this context, the present work aims at applying the ideas in the GITT solution methodology to solve a general formulation of the Reynolds equation. Such a hybrid numerical–analytical approach is an eigenfunction expansion methodology for solving linear or nonlinear in multiphysics problems, especially those not a priori transformable by the classical approach. An extensive parametric analysis is done in order to investigate the influence of typical governing parameters for such physical situation. Comparisons with results for typical situations are performed to demonstrate the consistency of the final results and to show the capacity of the GITT approach for handling journal bearing problems.
2. Mathematical formulation Fig. 1 shows a schematic representation and nomenclature used for the mathematical formulation of a radial journal bearing problem. In it, c represents the radial clearance, which is the
E.N. Santos et al. / Tribology International 52 (2012) 161–169
163
Fig. 1. Schematic representation for the problem of radial journal bearing: (a) geometric configuration; (b) coordinates system; (c) load components.
difference between the bearing and journal radii. The center of the journal and bearing are located at points O and O0 , respectively. Also, e is the eccentricity, the distance between these two centers, while ho is the minimum film thickness and occurs at the centerline. The film thickness at any point is represented by h. Therefore, the formulation of radial journal bearings problems is done through the classical Reynolds equation, which is developed by considering mass and momentum balances. It is also considered steady and incompressible laminar flow of Newtonian fluid with constant viscosity. Forces due to lubricant inertia are neglected and it is not taken account pressure variations in axial and in the clearance directions. Within the simplifying hypotheses the classical Reynolds equation in dimensionless form is written as: 0 yL l 2 @ ~ 3 @PðZ, fÞ @ ~3 @PðZ, fÞ h ðfÞ h ðfÞ ð1Þ þ ¼ 6yL h~ ðfÞ @Z @Z @f @f 2
3. Solution methodology
subjected to the following boundary conditions:
3.1. Eigenvalue problem
Pð0, fÞ ¼ 0;
Pð1, fÞ ¼ 0 and PðZ, 0Þ ¼ 0;
@PðZ, 1Þ ¼ PðZ, 1Þ ¼ 0 @f ð2a2eÞ
– Choose an appropriate auxiliary eigenvalue problem; – Develop the integral transform pair for performing the integral transformation and inversion operations; – Integral transformation of the original partial differential equation; – Numerically solve the resulting coupled ordinary differential equations system for the transformed potentials; – Recall the analytical inverse formula to construct the hybrid solution of the desired original potential.
In accordance with the steps stated above, the following choice for the eigenvalue problem is made: 2
where, ~ fÞ ¼ 1 þ ecosðfy Þ; hð L
In the application of the Generalized Integral Transform Technique (GITT) [22–35] for solving the problem given by Eqs. (1) to (3), the following basic steps are considered:
0 h~ ðfÞ ¼ eyL sinðfyL Þ
ð3a; bÞ
The following dimensionless groups were used to obtain Eqs. (1) to (3):
y ¼ x=R; x ¼ y=c; Z ¼ z=L; f ¼ y=yL ; h~ ¼ h=c; e ¼ e=c; ~ ¼ w=U; P ¼ pc2 =ðmURÞ l ¼ 2R=L; u~ ¼ u=U; w
ð4a2jÞ
In Eq. (4d) yL is the angle that characterizes the lubricant film length, which depends on the cavitation boundary, and is taken into account in the Reynolds equation by considering Reynolds or Swift–Steiber boundary conditions, such as evidenced in references by Cameron [1,2], Sharma et al. [17] and Liu [36]. In the present work, such boundary conditions are given by Eqs. (2), where in this case, yL ¼ p þ a, with a being the cavitation angle.
d ci ðZÞ þ m2i ci ðZÞ ¼ 0 dZ2
ci ð0Þ ¼ 0;
ð5aÞ
ci ð1Þ ¼ 0
ð5b; cÞ
Problem (5) is readily solved analytically, to yield the eigenvalues, mi, and the eigenfunctions, ci(Z), as
mi ¼ ip, i ¼ 1,2,. . .;
ci ðZÞ ¼ sinðmi ZÞ
ð6a; bÞ
It can be shown that the eigenfunctions, ci(Z), enjoy the following orthogonality property with their corresponding normalization integral, Ni: ( Z 1 Z 1 0, for i aj 1 ; Ni ¼ ci ðZÞcj ðZÞdZ ¼ c2i ðZÞdZ ¼ ; Ni , for i ¼ j 2 0 0 ð6c; dÞ
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Table 1 ~ and Pmax for the case of e ¼ 10 5 Convergence behavior of the potentials yL, j, W and l ¼ 10 5. NT
yL (degrees)
j (degrees)
~ 103 W
Pmax 104
10 30 50 70 90 110 130
257.5 257.5 257.5 257.5 257.5 257.5 257.5 257.5
70.91 70.91 70.91 70.91 70.91 70.91 70.91 70.91
0.130 0.134 0.135 0.135 0.135 0.135 0.135 0.136
0.871 0.836 0.829 0.826 0.825 0.824 0.823 0.819
a
a Results obtained from an approximate solution for e-0 given in the Appendix.
~ ðZÞ, integrated over the domain [0,1] in Eq. (1), is multiplied by c i the Z direction, and the inverse formula, Eq. (7b), is employed in place of the pressure field, P(Z,f), as well as the orthogonality property when applicable. The same operations can be performed over the boundary conditions given by Eqs. (2c) and (2d), to furnish the following system: " 0 # 2 d Pi ðfÞ 3h~ ðfÞ dP i ðfÞ yL l 2 2 þ mi Pi ðfÞ ¼ g i ðfÞ ð8aÞ 2 ~ fÞ df 2 hð df Pi ð0Þ ¼ 0;
dP i ð1Þ ¼0 df
ð8b; cÞ
The coefficient that appears in Eq. (8a) is defined as g i ðfÞ ¼
0 0 Z 6yL h~ ðfÞ 1 ~ 6yL h~ ðfÞ ½1cosðmi Þ pffiffiffiffiffi c ð Z Þd Z ¼ i 3 3 mi Ni h~ ðfÞ 0 h~ ðfÞ
ð8dÞ
The ODE system defined by Eqs. (8) depends on yL itself, which is calculated with the additional boundary condition given by Eq. (2e). Therefore, using the inverse formula, Eq. (7b), in such additional condition, one obtains the following transcendental equation for the yL calculation: 1 X
c~ i ðZÞPi ð1Þ ¼ 0
ð9Þ
i¼1
Fig. 2. Algorithm of the solution procedure.
3.2. Integral transform pair The above orthogonality property allows the definition of the following integral transform pair: Z 1 P i ðfÞ ¼ c~ i ðZÞPðZ, fÞdZ, transf orm ð7aÞ 0
PðZ, fÞ ¼
1 X
c~ i ðZÞP i ðfÞ, inverse
ð7bÞ
i¼1
pffiffiffiffiffi ~ ðZÞ ¼ c ðZÞ= N are the normalized eigenfunctions. where c i i i 3.3. Integral transformation of the original partial differential equation To obtain the transformed ordinary differential system for the transformed potentials, P i ðfÞ, the partial differential equation,
Eq. (9) is solved through the DZREAL subroutine from the IMSL Library [37] with a prescribed tolerance of 10 9. This subroutine ¨ finds the real zeros of a real function using Muller’s method. In this computation, it is necessary to truncate the infinite expansion in a sufficiently large number of terms (e.g., NT), so as to achieve the user-prescribed relative error target for obtaining the angle that characterizes the lubricant film length. One can see from Eq. (9) that the calculation of the yL depends on the variable Z; however, the present approach has shown that the same values are obtained for yL even for different values of Z. This can be explained by the fact that the boundary conditions given by Eqs. (2a) to (2e) also must be satisfied in this computational process. Once the yL is determined, the transformed potentials, Pi ðfÞ, can be obtained. For this purpose, in order to solve the transformed ODE system, efficient numerical algorithms for boundary value problems with stiff characteristics are to be employed, such as the subroutine DBVPFD also from the IMSL Library [37], which offers an automatic adaptive scheme for local error control of the numerical results for the transformed potentials. The features on this subroutine are focused on solving a (parameterized) system of differential equations with boundary conditions at two points, using a variable order, variable step size finite difference method with deferred corrections. It is then necessary to rewrite the transformed ODE system as a first order
E.N. Santos et al. / Tribology International 52 (2012) 161–169
one, by introducing the following dependent variables: dY i dPi ðfÞ ¼ Y i þ NT ¼ ; df df
Y i ¼ Pi ðfÞ;
Therefore, by making use of Eqs. (10), the transformed system can be rewritten as: " 0 # dY i dY i þ NT 3h~ ðfÞ yL l 2 2 ¼ Y i þ NT ; ¼ mi Y i þg i ðfÞ, Y i þ NT þ ~ fÞ df df 2 hð
2
dY i þ NT d P i ðfÞ ¼ , 2 df df
i ¼ 1,2,3,. . .,NT
ð10a2cÞ
i ¼ 1,2,3,. . .,NT Y i ð0Þ ¼ 0;
Table 2 ~ and Pmax for the case of e ¼ 10 Convergence behavior of the potentials yL, j, W and l ¼0.5.
ð11a; bÞ
Y i þ NT ð1Þ ¼ 0
ð11c; dÞ
5
NT
yL (degrees)
j (degrees)
~ 104 W
Pmax 104
10 30 50 70
233.0 233.0 233.0 233.0 233.0
80.57 80.60 80.60 80.60 80.60
0.560 0.560 0.560 0.560 0.560
0.490 0.489 0.489 0.489 0.489
a
165
1x10-4 Lines - GITT solution Symbols - Approximate analytical solution
9x10-5
=10-5;
=10-5
8x10-5 7x10-5 6x10-5
Results obtained from an approximate solution for e-0 given in the Appendix.
P
a
=10-5 ; =0.5
5x10-5 4x10-5
Table 3 ~ and Pmax for the case of e ¼ 0.5 and Convergence behavior of the potentials yL, j, W l ¼10 5. NT
yL (degrees)
j (degrees)
~ W
Pmax
10 30 50 70 90 110
219.7 219.7 219.7 219.7 219.7 219.7
58.30 58.30 58.30 58.30 58.30 58.30
6.19 6.37 6.40 6.42 6.43 6.43
4.76 4.57 4.53 4.52 4.51 4.51
3x10-5 2x10-5 1x10-5 0 0
20 40 60 80 100 120 140 160 180 200 220 240 260
(degrees)
Table 4 ~ and Pmax for the case of e ¼ 0.5 and Convergence behavior of the potentials yL, j, W l ¼0.1. NT
yL (degrees)
j (degrees)
~ W
Pmax
10 30 50 70 90
219.5 219.7 219.7 219.7 219.7
59.63 60.22 60.26 60.26 60.27
5.75 5.75 5.75 5.75 5.75
4.59 4.48 4.48 4.47 4.47
Fig. 3. Comparison of the GITT and approximate analytical results for the dimensionless pressure field in the circumferential direction for e ¼10 5 and l ¼ 10 5 and e ¼ 10 5 and l ¼0.5.
Table 5 Comparison of the dimensionless load carrying capacity with the literature results for different eccentricities e and aspect ratios l. Aspect ratio (l )
1
4
Authors
Reason and Narang [15] FEM [15] Sharma et al. [17] Hirani et al. [19] GITT – present work Reason and Narang [15] FEM [15] Sharma et al. [17] Hirani et al. [19] GITT – present work
Eccentricity (e) 0.1
0.4
0.5
0.6
0.8
0.228
–
1.722
–
6.924
0.228 – 0.228 0.239
– 1.2386 – 1.22
1.584 – 1.62 1.77
– 2.7206 – 3.91
5.964 7.5788 6.204 8.08
0.0192 –
0.1782 –
1.2216
0.0198 – 0.0192 0.0196
0.1740 – 0.1770 0.287
1.1304 1.1855 1.1616 1.24
– 0.1123 – 0.196
– 0.2967 – 0.429
Fig. 4. Comparison of the dimensionless pressure field in the circumferential direction for e ¼0.4 and l ¼1.
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E.N. Santos et al. / Tribology International 52 (2012) 161–169
At this point, since the transformed potentials, P i ðfÞ, are obtained, the inverse formula given by Eq. (7b) is recalled to calculate the bearing and flow parameters. Therefore, by introducing Eq. (7b) in the usual definitions of the dimensionless load components, the friction force and the side leakage flow, one obtains the expressions for computation of such parameters, respectively, as Z 1Z 1 1 X ~ 1 ¼ yL W PðZ, fÞcosðfyL ÞdfdZ ¼ yL Ai C i ð12aÞ 0
~ 2 ¼ yL W
Z
0
1Z 0
F~ ¼ yL
1
PðZ, fÞsinðfyL ÞdfdZ ¼ yL 0
0
2i¼1
yL Q~ s ¼ 6
1 0
1 1X
Z
Bi C i
"
ð12bÞ
# Z 2p=yL ~ fÞ @P hð 1 þ df dZdf ¼ yL ~ fÞ ~ fÞ 2yL @f 0 hð hð 1
C i Di
1 0
1 X i¼1
Z 2p=yL Z
þ
i¼1
where the various coefficients that appear in Eqs. (12) are calculated from: Z 1 Z 1 Ai ¼ Pi ðfÞcosðfyL Þdf; Bi ¼ P i ðfÞsinðfyL Þdf; 0
Ci ¼
Z
0
Di ¼
0
1
c~ i ðZÞdZ;
Z 2p=yL 0
~ fÞ dP i ðfÞ df; hð df
Ei ¼
Z
1 0
3 h~ ðfÞP i ðfÞdf
ð13a2eÞ
Fig. 2 shows the algorithm of the scheme for the solution procedure. Now, from Eq. (12) above, the dimensionless load ~ , the attitude angle, j, and the friction factor carrying capacity, W Cf are also calculated from their usual definitions, respectively, as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ 2 þW ~ ¼ W ~ 2 ; j ¼ tan1 ðW ~ 2 =W ~ 1 Þ; C f ¼ F~ =W ~ W ð14a2cÞ 1 2
ð12cÞ 4. Results and discussion
3 @Pðf, ZÞ h~ ðfÞ @Z
Z¼0
df ¼
yL 6
~ ð0Þ dc i Ei dZ i¼1 1 X
ð12dÞ
Numerical results for the pressure field, friction factor, dimensionless load carrying capacity and side leakage flow were
Fig. 5. Comparison of the potentials as a function of the eccentricity for l ¼1: (a) friction factor; (b) dimensionless side leakage flow; (c) dimensionless load carrying capacity; (d) attitude angle.
E.N. Santos et al. / Tribology International 52 (2012) 161–169
obtained from a code developed in the FORTRAN 95/2003 programming language. The code was implemented on a computational platform with INTELs CORETM i7 M620 2.67 GHz processor, and the system given by Eqs. (9) and (11) was simultaneously handled through the DBVPFD and DZREAL subroutines, both from the IMSL Library [37]. A relative error target of 10 5 in the subroutine DBVPFD was employed throughout the computations, for varying the values of the dimensionless eccentricity, e, and of the aspect ratio, l ¼D/L. A typical CPU running time for the computations such as those shown in Table 5 is about 350 s. First, an analysis of convergence behavior of the following potentials is made, namely, angle that characterizes the lubricant film length, yL, attitude angle, j, dimensionless load carrying capa~ , and dimensionless Pmax at the middle plane of the radial city, W bearing. For this purpose, the influence of different truncation orders (NT) in the solution of the pressure equation was investigated. A comparison with those results obtained from a simplified solution (valid for e small) is also made. Table 1 brings the convergence behavior of such potentials for e ¼ 10 5 and l ¼10 5. The convergence is established with 30 ~ and Pmax terms in the summations for yL and j, while W experience lower convergence rates, reaching such convergences with NT between 50 and 70 terms and NT between 110 and 130 terms, respectively. There is also an excellent agreement of the results obtained in the present analysis with those of an analytical solution by considering the hypothesis that e is very small (see Appendix), within four significant digits for yL and j, whereas ~ and Pmax; this good two significant digits are obtained for W agreement ensures verification of the computer code developed. A similar analysis is shown in Table 2 for the case of e ¼10 5 and l ¼0.5. Furthermore, excellent convergence rates are clearly observable, with full convergence to four digits being achieved for yL and j, even for lower truncation orders (approximately NT¼30). The ~ and Pmax reach three converged digits with a truncapotentials W tion order of around NTE30 to 50 terms. The excellent agreement of the results with those of an analytical solution shown in the appendix once again ensures the verification of such results. Tables 3 and 4 address the convergence behavior of same potentials for the cases of e ¼0.5 and l ¼10 5 and of e ¼0.5 and l ¼0.1, respectively. For such situations, higher convergence rates are obtained for all potentials analyzed, with four converged digits for yL and j (also approximately NT ¼30) and a full ~ and Pmax (NT E90 to 110 terms convergence to three digits for W for the case of e ¼0.5 and l ¼10 5 and NT E50 to 70 terms for the case of e ¼0.5 and l ¼0.1). Therefore, after this convergence analysis, the subsequent computations were performed using up to seventy terms (NT r70) in the eigenfunction expansions. Table 5 illustrates the results for the dimensionless load carrying capacity obtained in the present work and for those available in the literature, for l ¼D/L ¼1 and 4, and different eccentricities, i.e., e ¼0.1, 0.4, 0.5, 0.6 and 0.8. In general, a reasonable agreement is observed among the present results with those from the literature, in particular for l ¼ 4, while for l ¼1 the literature results lose adherence with the present converged results as the eccentricity increases. Fig. 3 shows a comparison of the present GITT results against those obtained with an analytical solution for very small eccentricities. In this analysis, the dimensionless pressure field in the circumferential direction is shown for e ¼10 5 and l ¼10 5 and e ¼ 10 5 and l ¼0.5 at the journal bearing medium plan. The excellent agreement again warrants the verification of the results produced with the present GITT approach. Fig. 4 gives the present GITT results for the dimensionless pressure field in the circumferential direction for l ¼ 1 and e ¼ 0.4 at the journal bearing medium plan, which are compared with those results obtained by Williams and Symmons [6], Sharma
167
Fig. 6. Comparison of the maximum pressure as a function of the axial position for e ¼ 0.4 and l ¼ 1.
et al. [17] and Mokhiamer et al. [20]. Fig. 5 shows a comparison between the present results with those of Mokhiamer et al. [20] for the potentials of friction factor, dimensionless side leakage flow, dimensionless load carrying capacity and attitude angle for l ¼1. Also, in Fig. 6, the maximum pressure as a function of the axial position for e ¼0.4 and l ¼1 is compared with the results of Mokhiamer et al. [20]. In all cases analyzed in Figs. 4 to 6, an excellent agreement among the set of results is noted, once again confirming the present GITT results. Finally, Fig. 7 show pressure field patterns in the radial journal bearing for different values of the governing parameters: e ¼10 5 and l ¼10 5 (Fig. 7(a)); e ¼0.5 and l ¼10 5 (Fig. 7(b)); e ¼10 5 and l ¼1 (Fig. 7(c)) and e ¼0.5 and l ¼1 (Fig. 7(d)). In Fig. 7(a) and (b) can be observed that for long journal bearings (l ¼ D/L-0), the circumferential pressure field does not depend on the axial position, Z. This is explained by the fact that the pressure gradient in the Z direction to be negligible in relation to the gradient in the y direction (qP/qy 4 4qP/qZ), i.e., for long journal bearings, the flow in the y direction is more important that in the Z direction. It is also verified that the pressure in the journal bearing is proportional to e (see Figs. 4 to 6). The variation in the pressure field is due to e for regulating the wedge effect. In other words, this effect increases for increasing values of the eccentricity and, as a consequence, the lubricant shear increases, proportional to hydrodynamic pressure, as can be observed in Eqs. (12). On the other hand, from Fig. 7(c) and (d), it is observed that the pressure field varies in the Z direction due to the fact of the journal bearing not being long (l ¼D/L¼1). Similarly to Fig. 7(a) and (b), the pressure is proportional to the eccentricity. Therefore, by comparing Fig. 7(a) and (b) to (c) and (d), it is verified that the pressure field is inversely proportional to l ¼D/L, thus demonstrating that there is a dependence of this field in relation to this parameter, which determines if a journal bearing is long or short.
5. Conclusions A hybrid numerical–analytical solution for a problem of hydrodynamic lubrication related to radial journal bearings was
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E.N. Santos et al. / Tribology International 52 (2012) 161–169
Fig. 7. Patterns of pressure field for different values of the governing parameters: (a) e ¼ 10 5 and l ¼ 10 5; (b) e ¼ 0.5 and l ¼10 5; (c) e ¼ 10 5 and l ¼ 1; (d) e ¼ 0.5 and l ¼ 1.
advanced by extending the ideas in the Generalized Integral Transform Technique (GITT). Convergence behaviors were extensively undertaken in the eigenfunction expansions, for the pressure field and for the journal bearing potentials, which have demonstrated the adequacy and robustness of the proposed hybrid solution approach. The present analysis has also considered different values of the eccentricity, e, and of the aspect ratio, l, in order to present physical aspects concerning this journal bearing problem. The GITT approach, due to its hybrid numerical– analytical nature based on eigenfunction expansions, is well suited for benchmarking purposes, and its successful employment in the present journal bearing analysis with relative lower computational costs and computational efficiency permits a natural extension of its application to other journal bearing problems such those by taking into account dynamic cavitation (mass-conserving models), bearing deformations and heat exchange (thermohydrodynamic (THD) and thermal elastohydrodynamic (TEHD) analyses).
Appendix. Analytical solution for very small eccentricity An approximation considered in hydrodynamic lubrication is to assume that a journal bearing has a very small eccentricity (e-0). 3 Therefore, it can be considered that the term h~ is approximately 3 ~ equal to unity in Eq. (1) (h 1). As a consequence, the Reynolds
equation together with the boundary conditions are given by:
2
yL l 2
@2 PðZ, fÞ @2 PðZ, fÞ þ ¼ f ðfÞ 2 @Z2 @f
Pð0, fÞ ¼ 0;
Pð1, fÞ ¼ 0
PðZ, 0Þ ¼ 0;
@PðZ, 1Þ ¼ PðZ, 1Þ ¼ 0 @f
ðA:1aÞ
ðA:1b; cÞ ðA:1d2fÞ
where 0 2 f ðfÞ ¼ 6yL h~ ðfÞ ¼ 6eyL sinðfyL Þ
ðA:1gÞ
The problem presented by Eq. (A.1) can be solved through the Classical Integral Transform Technique (CITT) [38]. For this purpose, the same integral transform operations can be performed over Eq. (A.1), i.e., the partial differential Eq. (A.1a) and the boundary conditions given by Eqs. (A.1d) and (A.1e) are ~ ðZÞ, integrated over the domain [0,1] in the Z multiplied by c i direction, so that one obtains the following transformed ordinary differential system for the transformed potentials, P i ðfÞ, with their respective boundary conditions: 2
d Pi ðfÞ df
2
2
bi Pi ðfÞ ¼ hi ðfÞ
Pi ð0Þ ¼ 0;
dP i ð1Þ ¼0 df
ðA:2aÞ
ðA:2b; cÞ
E.N. Santos et al. / Tribology International 52 (2012) 161–169
where yl 2 2 b2i ¼ L mi ; hi ðfÞ ¼ 6ey2L sinðfyL ÞLi ; 2 Z 1 ½1cosðmi Þ pffiffiffiffiffi c~ i ðZÞdZ ¼ Li ¼ mi N i 0
ðA:2d2fÞ
Here, the eigenquantities (eigenvalues, eigenfunctions and norms) and the integral transform pair are the same as those given in the Section 3 by Eqs. (5a) to (7b). The ordinary differential equation given by Eq. (A.2a) together with boundary conditions (A.2b,c) allow one to obtain an analytical solution for the transformed potentials, P i ðfÞ, in the form: 2 6ey L yL cosðyL Þ sinhðbi fÞ ðA:3Þ P i ðfÞ ¼ 2 L 2i sinðfyL Þ coshðbi Þ bi ðb þ y Þ i
L
Next, by introducing Eq. (A.3) into the inverse formula given by Eq. (7b), the following analytical equation for the pressure distribution is the result: 1 X Li sinðmi ZÞ yL cosðyL Þ sinhðbi fÞ 2 pffiffiffiffiffi PðZ, fÞ ¼ 6eyL sinðfyL Þ 2 2 coshðbi Þ bi Ni i ¼ 1 ðbi þ yL Þ ðA:4Þ Therefore, the angle yL can be calculated from the introduction of Eq. (A.4) into additional boundary condition (A.1f), to yield 1 X Li sinðmi ZÞ yL cosðyL Þ sinhðbi Þ pffiffiffiffiffi ¼0 ðA:5Þ sinðyL Þ 2 2 coshðbi Þ bi Ni i ¼ 1 ðb þ y Þ i
L
Eq. (A.5) provides a transcendental equation for determining the angle yL. For computational purposes, such a summation must be truncated into a number of NT terms. After that, the subroutine DZREAL from the IMSL Library [37] with a tolerance of 10 9 is ~ , j, Cf also employed for this determination. The parameters W and Q~ S are also calculated from their usual definitions. References [1] Cameron A. The Principles of Lubrication. London: Longmans Green; 1966. [2] Cameron A. Basic Lubrication Theory. New Delhi: Wiley Eastern Ltd; 1987. [3] Sivak B, Sivak M. The numerical solution of the Reynolds equation by a modified Ritz method. Wear 1981;72:371–6. [4] Tayal SP, Sinhasan R, Singh DV. Analysis of hydrodynamic journal bearings having non-Newtonian lubricants. Tribology International 1982;15:17–21. [5] Chandrawat HN, Sinhasan R. A comparison between two numerical techniques for hydrodynamic journal bearing problems. Wear 1987;119:77–87. [6] Williams PD, Symmons GR. Analysis of hydrodynamic journal bearings lubricated with non-Newtonian fluids. Tribology International 1987;20: 119–224. [7] Sinhasan R, Chandrawat HN. An elastohydrodynamic study on two-axialgroove journal bearings. Tribology International 1988;21:341–51. [8] Sinhasan R, Chandrawat HN. Analysis of a two-axial-groove journal bearing including thermoelastohydrodynamic effects. Tribology International 1989;22:347–53. [9] El-Gamal HA. Analysis of the steady state performance of a wedge-shaped hydrodynamic journal bearing. Wear 1995;184:111–7. [10] Banwait SS, Chandrawat HN. Study of thermal boundary conditions for a plain journal bearing. Tribology International 1998;31:289–96.
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