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

Abstract

The 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 and numerical results were presented, with the efficiency of the method being discussed in the process of their presentation.

1. Introduction

The 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 finite element methods of analysis [1–8]. 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 field of tribology.

2. Model

The 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 𝜕ℎ𝜕𝜃³𝜕𝑝𝜕𝜃+𝑅²𝜕ℎ𝜕𝑧³𝜕𝑝𝜕𝑧=12𝜇𝑅²𝑐𝜀̇cos𝜃+𝜀𝜑̇−𝜔sin𝜃, (1) where 𝜃 and 𝑧 are the bearing coordinates, 𝜇 is the dynamic oil viscosity, ℎ is the oil film thickness at time 𝑡, and 𝑐 is the radial clearance; 𝜔=(𝜔_𝑗+𝜔_𝑏)/2−𝜓̇ is the average angular velocity of journal and bearing relative to the external load vector 𝐹(𝑡), 𝜔_𝑗 is the angular velocity of the journal, 𝜔_𝑏 is the angular velocity of the bearing, and 𝜓̇ is the angular velocity of the load; 𝜑̇=𝑑𝜑/𝑑𝑡 is the angular velocity of the attitude angle, and 𝜑 is the attitude angle, that is, an angle between the applied load direction and the center line of the journal and the bearing; 𝜀̇ is the radial velocity of the journal center, and 𝜀𝜑̇ is the tangential velocity of the journal center.

The geometry of a dynamically loaded journal bearing and its notations are shown in Figure 1, for the purpose of defining the mobility.

Figure 1 

Journal bearing geometry for dynamic loading.

The right hand side of (1) includes unknowns 𝜀̇ and 𝜑̇. These 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 𝐹^𝜀=𝐹cos𝜑=−^𝐿₀^𝜃2_𝜃1𝐹𝑝cos𝜃𝑑𝜃𝑑𝑧,_𝜑=−𝐹sin𝜑=^𝐿₀^𝜃2_𝜃1𝑝sin𝜃𝑑𝜃𝑑𝑧. (2) The positive portion of the hydrodynamic pressure distribution lies between 𝜃₁ and 𝜃₂, given by the following.

For short bearing, 𝜃₁=tan⁻¹𝜀̇𝜀𝜑̇,𝜃𝜔−₂=𝜃₁+𝜋,when𝜀𝜑̇𝜃𝜔−>0,₁−𝜋,when𝜀𝜔−𝜑̇>0. (3) For infinitely long bearing, 𝜃₁=tan⁻¹2+𝜀²𝜀̇2𝜀𝜑̇,𝜃𝜔−₂=𝜃₁+𝜋,when𝜀𝜑̇𝜃𝜔−>0,₁−𝜋,when𝜀𝜔−𝜑̇>0. (4) It is important to note that angles 𝜃₁ and 𝜃₂ change with the work conditions of journal bearing. Thus, 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.

In [3], in order to simplify the solution of Reynolds’ equation, Booker proposed to express the 𝜀̇ and 𝜑̇ as function of mobility components as ̇𝐹()𝜀=𝑐/𝑅²𝑀𝜇𝐿𝐷^𝜀,̇𝐹()𝜑=𝑐/𝑅²𝑀𝜇𝐿𝐷𝜀^𝜙+𝜔. (5) In [4], these two functions, 𝑀^𝜀 and 𝑀^𝜙, can be considered as the components of a mobility vector 𝑀^→, which is defined as a ratio of velocity to force. If 𝛽 is mobility direction angle, as it is illustrated in Figure 2, we may write 𝑀^𝜀=−𝑀^→cos𝛽,𝑀^𝜑=−𝑀^→sin𝛽. (6)𝑀^→ is a function only of the position vector ^→𝜀 and the ratio of 𝐿/𝐷. The components of the mobility vector, 𝑀^→(𝜀,𝐿/𝐷), that is, 𝑀^𝜀 and 𝑀^𝜙 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 𝜕ℎ𝜕𝜃³𝜕𝑝𝜕𝜃+𝑅²𝜕ℎ𝜕𝑧³𝜕𝑝𝜕𝑧=12𝜇𝑅²𝑐𝐹(𝑐/𝑅)²𝑀𝜇𝐿𝐷^𝜀𝐹()cos𝜃+𝜀𝑐/𝑅²𝑀𝜇𝐿𝐷𝜀^𝜙sin𝜃. (7) Equation (7) can be solved numerically and new pressure distribution is thus obtained. Bearing load capacity can be calculated to balance the applied load.

Figure 2 

Representation of mobility vector and its components.

The 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 (𝐿/𝐷) is less than or equal to 0.75 [5], a bearing can be assumed to act as a short bearing. The short bearing assumption is used to eliminate the pressure term in the circumferential direction from Reynolds equation, that is, the first term on the left hand side of (7). Because of this assumption, (𝜕/𝜕𝜃)[ℎ³(𝜕𝑝/𝜕𝜃)]≪𝑅²(𝜕/𝜕𝑧)[ℎ³(𝜕𝑝/𝜕𝑧)]. Hence, Reynolds equation can be reduced to: 𝑅²𝜕ℎ𝜕𝑧³𝜕𝑝𝜕𝑧=12𝜇𝑅²𝑐𝐹(𝑐/𝑅)²𝑀𝜇𝐿𝐷^𝜀𝐹()cos𝜃+𝜀𝑐/𝑅²𝑀𝜇𝐿𝐷𝜀^𝜙sin𝜃. (8) By integrating (8) twice, the hydrodynamic pressure along the 𝑧 axis is obtained as 𝑝=6𝜇𝑐ℎ³𝐹(𝑐/𝑅)²𝑀𝜇𝐿𝐷^𝜀𝐹()cos𝜃+𝜀𝑐/𝑅²𝑀𝜇𝐿𝐷𝜀^𝜙×𝑧sin𝜃₂+2𝑐₁𝑧+𝑐₂. (9) Using the boundary condition that 𝑝(𝜃,±𝐿/2)=0, we can evaluate the constants in (9): 𝑐₁=0, 𝑐₂=−𝐿²/4. Therefore, the hydrodynamic pressure is 𝑝=6𝜇𝑐ℎ³𝐹(𝑐/𝑅)²𝑀𝜇𝐿𝐷^𝜀𝐹()cos𝜃+𝜀𝑐/𝑅²𝑀𝜇𝐿𝐷𝜀^𝜙×𝑧sin𝜃₂−𝐿²4. (10) Substituting (10) into (2) leads to 𝐹^𝜀=−^𝐿₀^𝜃2_𝜃1𝑝()=𝜃,𝑧cos𝜃𝑑𝜃𝑑𝑧_𝜃2𝜃1𝜇𝐿³𝑅𝑐²𝐹(𝑐/𝑅)²𝑀𝜇𝐿𝐷^𝜀𝐹()cos𝜃+𝜀𝑐/𝑅²𝑀𝜇𝐿𝐷𝜀^𝜙×sin𝜃cos𝜃(1+𝜀cos𝜃)³𝐹𝑑𝜃,_𝜑=^𝐿₀^𝜃2_𝜃1𝑝()=𝜃,𝑧sin𝜃𝑑𝜃𝑑𝑧_𝜃2𝜃1𝜇𝐿³𝑅𝑐²𝐹(𝑐/𝑅)²𝑀𝜇𝐿𝐷^𝜀𝐹()cos𝜃+𝜀𝑐/𝑅²𝑀𝜇𝐿𝐷𝜀^𝜙×sin𝜃sin𝜃(1+𝜀cos𝜃)³𝑑𝜃. (11) Solving (2) and (11) simultaneously produces 𝑀^𝜀=𝐼₃²⁰cos𝜑+𝐼₃¹¹sin𝜑2(𝐿/𝐷)²𝐼₃²⁰𝐼₃⁰²−𝐼₃¹¹𝐼₃¹¹,𝑀^𝜙=𝐼₃⁰²cos𝜑+𝐼₃¹¹sin𝜑2(𝐿/𝐷)²𝐼₃²⁰𝐼₃⁰²−𝐼₃¹¹𝐼₃¹¹, (12) where 𝐼_𝑛^𝑙𝑚=∫_𝜃(sin^𝑙𝜃cos^𝑚𝜃/(1+𝜀cos𝜃)^𝑛)𝑑𝜃.

The solution of the above equations is complex and involves a good deal of mathematical manipulation. However, Booker [1] gave the numerical solution to (12), for positive force, 𝐹>0, 𝑀^𝜍=(1−𝜍)^5/2𝜋(𝐿/𝐷)²,𝑀^𝜅()=−4𝜅1−𝜍^3/2𝜋²(𝐿/𝐷)² (13) and, for negative force, 𝐹<0, 𝑀^𝜍=(1+𝜍)^5/2𝜋(𝐿/𝐷)²,𝑀^𝜅=4𝜅(1+𝜍)^3/2𝜋²(𝐿/𝐷)² (14) with 𝑀^𝜀=𝑀^𝜍cos𝜑+𝑀^𝜅𝑀sin𝜑,_𝜑=−𝑀^𝜍sin𝜑+𝑀^𝜅cos𝜑,𝜍=𝜀cos𝜑,𝜅=𝜀sin𝜑. (15) For a specified external load 𝐹(𝑡), (13)–(15) are solved for 𝜀̇ and 𝜑̇ and then integrated with respect to time, in order to obtain the journal center trajectory, that is, 𝜀(𝑡), 𝜑(𝑡).

Similarly, the above scheme can be applied to infinitely long bearings. But for a bearing of finite length, both hydrodynamic pressure terms in the Reynolds equation are significant and cannot be neglected in the calculation of the pressure distribution. However, no closed form analytical exists for (7). Therefore, 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 fit approach, see, for example, Goenka [6]. When the ratio of bearing length to bearing diameter ranges between 0.75 and 4, a bearing is assumed to be a finite length bearing.

3. Numerical Procedure

With a knowledge of the hydrodynamic pressure distribution, the load carrying capacity can be determined and (𝜀̇, 𝜑̇) can be modified 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 film profile 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.(1)Make an initial guess for 𝜀,𝜑,𝜀,̇𝜑̇.(2)Determine 𝜃₁ and 𝜃₂, then obtain the pressure distribution.(3)Calculate 𝜀̇ and 𝜑̇ using mobility data.(4)Evaluate 𝜀,𝜑 at next time step by using 𝜀^𝑡+1=𝜀^𝑡+𝜀Δ̇𝑡, 𝜑^𝑡+1=𝜑^𝑡+𝜑̇Δ𝑡.(5)Repeat steps (2)-(4) until the last time step in one engine cycle.(6)Check for the convergence of 𝜀 and 𝜑 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,𝜀,̇𝜑̇, and repeat steps (2) to (6) if the results have not converged.

4. Results and Discussion

A rigid bearing is analyzed using the scheme developed in above. The 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. The 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^∘×60/1050×360≈0.1143s. The time corresponding to one degree crank angle is 1.59×10⁻⁴s. The bearing is divided into a grid (𝑁_𝑥×𝑁_𝑧) with a grid size Δ𝑥×Δ𝑧. A grid size with 37×21 grid points is used in the program code.

Table 1 

Bearing characteristics.

Figure 3 

The bearing load.

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 influences the first instants of simulation. Thus, the cyclic behavior of the journal, independent of the assumed initial values, is reached very quickly. In Figure 4 the journal eccentricity in 𝑥 and 𝑦 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 film thickness.

Figure 4 

Journal center eccentricity.

The minimum film thickness is related to the eccentricity ratio and radial clearance as follows: ℎ_min=𝑐(1−𝜀). (16) The variation of the minimum film thickness during the dynamic simulation of journal bearing gives the value of the absolute minimum film thickness. This 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 film thickness is 2.5 μm, and, in practical, the minimum film thickness should be at least (1.0–1.5) × 2.5 μm [7]. Figure 5 shows the variation of the minimum film thickness with the angle. The dashed horizontal line in the same figure represents that safe film 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.

Figure 5 

The minimum film thickness.

Acknowledgments

The present research has been funded by National Natural Science Foundation of China (Grant no. 50979016), Program for New Century Excellent Talents in University of Ministry of Education of China, and Special Fund for Basic Research Program of Higher Education of China (Code: HEUCF100308). The authors would like to sincerely express their appreciation.