Mobility Method Applied to Calculate the Lubrication Properties of Bearing under Dynamic Loads

e analytical mobility method for dynamically loaded journal bearings was presented, with the intent to include it in a general computational program, such as the dynamic analysis program, that has been developed for the dynamic analysis of general mechanical systems. An illustrative example andnumerical results were presented, with the efficiency of themethod being discussed in the process of their presentation.


Introduction
e mobility method is widely used for analyzing rigid journal bearings due to its computational efficiency when compared to other approaches such as the multigrid techniques based on the Elrod algorithm and the �nite element methods of analysis [1][2][3][4][5][6][7][8].It is still a mainstay of engine bearing performance calculation because of its rapid prediction ability even though many other comprehensive methods have been developed for complicated bearing analysis.In fact, the mobility method is sufficiently accurate and easy to understand and apply by designers and engineers that do not need to be expert in the �eld of tribology.

Model
e mobility method introduced by Booker [1] is a technique for direct evaluation of the instantaneous velocity of the journal center in terms of the applied load and the bearing geometry.By determining the velocity of the bearing center, the orbits of the journal center can be obtained.
From [2], the Reynolds equation for a dynamically loaded journal bearing with an isothermal lubrication in the polar coordinate system can be written as where  and  are the bearing coordinates,  is the dynamic oil viscosity, ℎ is the oil �lm thickness at time , and  is the radial clearance;  =   +   )/2  ψ is the average angular velocity of journal and bearing relative to the external load vector ),   is the angular velocity of the journal,   is the angular velocity of the bearing, and ψ is the angular velocity of the load; φ = / is the angular velocity of the attitude angle, and  is the attitude angle, that is, an angle between the applied load direction and the center line of the journal and the bearing; ε is the radial velocity of the journal center, and  φ is the tangential velocity of the journal center.e geometry of a dynamically loaded journal bearing and its notations are shown in Figure 1, for the purpose of de�ning the mobility.e right hand side of (1) includes unknowns ε and φ.ese two unknowns can be determined by integrating the trajectory of the journal center.If the pressure distribution is known, the load carrying capacity components can be obtained to balance the applied load, and the balance can be written as e positive portion of the hydrodynamic pressure distribution lies between  1 and  2 , given by the following.
For short bearing, For in�nitely long bearing, It is important to note that angles  1 and  2 change with the work conditions of journal bearing.us, for each instant of simulation, the pressure is function of the journal bearing geometric properties, its kinematic characteristics, and varies in both circumferential and axial directions.
F 2: Representation of mobility vector and its components.
In [3], in order to simplify the solution of Reynolds' equation, Booker proposed to express the ε and φ as function of mobility components as In [4], these two functions,   and   , can be considered as the components of a mobility vector  , which is de�ned as a ratio of velocity to force.If  is mobility direction angle, as it is illustrated in Figure 2, we may write is a function only of the position vector ⃗  and the ratio of .e components of the mobility vector,   ( ), that is,   and   can be computed numerically.If mobility data for a particular bearing is available, it is relatively simple to solve (1).Substituting ( 5) into (1), we can obtain Equation ( 7) can be solved numerically and new pressure distribution is thus obtained.Bearing load capacity can be calculated to balance the applied load.e above discussion can lead to a procedure for the use of the mobility method to analyze dynamically loaded journal bearings.When the ratio of bearing length to bearing diameter () is less than or equal to 0.75 [5], a bearing can be assumed to act as a short bearing.e short bearing assumption is used to eliminate the pressure term in the circumferential direction from Reynolds equation, that is, the �rst term on the le� hand side of (7).Because of this assumption, ( 3 (   2 ( 3  e solution of the above equations is complex and involves a good deal of mathematical manipulation.However, Booker [1]  ( For a speci�ed external load (, (13)-( 15) are solved for ε and φ and then integrated with respect to time, in order to obtain the journal center trajectory, that is, (, (. Similarly, the above scheme can be applied to in�nitely long bearings.But for a bearing of �nite length, both hydrodynamic pressure terms in the Reynolds equation are signi�cant and cannot be neglected in the calculation of the pressure distribution.However, no closed form analytical exists for (7).erefore, numerical methods are extensively used to solve (7).From a vast number of solutions obtained bearing hydrodynamic lubrication problems, the mobility components can be expressed in a closed formation using a curve �t approach, see, for example, �oenka [6].When the ratio of bearing length to bearing diameter ranges between 0.75 and 4, a bearing is assumed to be a �nite length bearing.

Numerical Procedure
With a knowledge of the hydrodynamic pressure distribution, the load carrying capacity can be determined and ( ε, φ) can be modi�ed to ensure the applied load is balanced.When the balance is achieved, with the knowledge of the journal center velocity, journal movement over a time step can be determined.A new �lm pro�le may thus be obtained.A new pressure distribution for the new instantaneous operating conditions is then required, and the process is repeated until convergence is obtained.A computational scheme for using the mobility method in the analysis of a bearing is as follows.

Results and Discussion
A rigid bearing is analyzed using the scheme developed in above.e related bearing parameters are listed in Table 1, and Figure 3 shows the diagram of the dynamic load applied to the bearing during one engine cycle.For a four-stroke engine, there are 720 degree crank angle in one engine cycle.e time step corresponding to the analysis is one degree crank angle; thus, there are 720 time-steps in one engine cycle.When the engine speed is 1050 rpm, the total time corresponding to one engine cycle is 720 ∘ × 60/00 × 360 ≈ 0.43 s. e time corresponding to one degree crank angle is .9 × 0 4 s. e bearing is divided into a grid (  ×   ) with a grid size  × .A grid size with 37 × 2 grid points is used in the program code.
It should be noted that, with this approach of mobility method, the choice of the initial conditions does not affect the relative journal bearing motion and only in�uences the �rst instants of simulation.us, the cyclic behavior of the journal, independent of the assumed initial values, is reached very quickly.In Figure 4 the journal eccentricity in  and  direction is shown as function of the crank angle.In dynamically loaded journal bearings both the eccentricity and attitude angle vary with the cycle of applied load, and the correct design should ensure that the combination of load and speed rotation does not lead a dangerous small minimum �lm thickness.
e minimum �lm thickness is related to the eccentricity ratio and radial clearance as follows: the absolute minimum �lm thickness.is value should be greater than the average asperity height of the journal bearing surfaces in order to prevent the metal-to-metal contact and, consequently, to avoid the friction and wear, besides keeping the hydrodynamic approach valid.From the journal bearing design view point, the safe value for �lm thickness is 2.5 m, and, in practical, the minimum �lm thickness should be at least (1.0-1.5)× 2.5 m [7]. Figure 5 shows the variation of the minimum �lm thickness with the angle.e dashed horizontal line in the same �gure represents that safe �lm thickness.By observing Figure 5, it is clear that the effective hydrodynamic lubrication is performed on the journal bearing, meaning that the journal and bearing surfaces are completely separated.

F 1 :
Journal bearing geometry for dynamic loading.

1 F 4 :
ℎ min =  (  ).(16)e variation of the minimum �lm thickness during the dynamic simulation of journal bearing gives the value of Journal center eccentricity.
(. Hence, Reynolds equation can be reduced to: (sin  cos  (1   cos   . gave the numerical solution to (12), for positive force,   , (7)ck for the convergence of  and  between the values of the initial guess and the last time step.(7)Set the values of the last time step as new guess of   ε φ, and repeat steps (2) to (6) if the results have not converged.