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Hydrodynamic Lubrication 2009 Part 4

Tham khảo tài liệu 'hydrodynamic lubrication 2009 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 3.4 Finite Length Bearings 45 L dp -j j 3x 3x Ax 3 Pi j 1 - Pi j 3 p j - Pi j-1 d h dp - hi j 1 2 Az hi j-1 2 Az dz 3z Az dh _ hi 1 2 j - i-1 2 j dx Ax 3.69 3.70 3.71 Substituting these into Eq. 3.68 and solving for Pi j we obtain Pi j in the following form Pi j - a0 a1 Pi 1 j a2Pi-1j 3Pij 1 a4Pij-1 3.72 i 1 2 m - 1 j 1 2 n - 1 where Pi j is the pressure at the nodal point i j and a0 a1 a2 a3 and a4 are constants given at the respective nodal point. If the number of nodal points at which the pressure is to be calculated is N then we have N equations of the form of Eq. 3.72. The boundary conditions are given as the pressure at the points on the boundary. By solving these simultaneously the pressures at respective nodal points are obtained. One of the methods to solve these simultaneous equations is elimination typically Gauss elimination method. Pressure at all the nodal points can thereby be found in a finite number of operations. Further if we sweep the lubricating domain two dimensionally with the calculation of Eq. 3.72 at each nodal point in consecutive order starting from a suitable nodal point and if this is repeated a sufficient number of times then it is expected that the pressure obtained at each nodal point gradually approaches the true value of the pressure. This is called the iterative method successive approximation method . In this case the calculation will be repeated until the following relation is satisfied J l Pi j k - Pi j k-1l i j _ --------------- e 3.73 2jl Pi j kl i j where Pụ k are pressures obtained in the kth calculation Pi j k-1 are those in the previous calculation and e is a sufficiently small allowable error. The pressure Pit j k obtained in the kth calculation will be a solution Pitj. For a high accuracy of calculation a value of 10-6 -10-12 for example is used for e. Further the convergence in the region of rapid pressure rise can be improved by introducing the following new variable P into Reynolds equation Eq 3.68 2 . h and P .