Dynamic Analysis of Engine Bearings *

This paper presents a simple methodology to evaluate the stiffness and damping coefficients of an engine bearing over a load cycle. A rapid technique is used to determine the shaft ‘limit cycle’ under engine dynamic loads. The proposed theoretical model is based on short and long bearing approximations. The results obtained by present approximation are compared with those obtained by numerical method. The influence of thermal effects on the stiffness and damping coefficients is predicted by using a simplified thermal analysis. In order to illustrate the application of the proposed scheme, one engine main bearing and a connecting rod bearing are analysed.


INTRODUCTION
The most convenient way to analyse a complete system is to divide it into separate sub-systems or sub-structures (Parszewski, 1989), analyse each sub-system individually with less time-consuming methods, and then assemble into the whole system.In the present paper the journal bearing is treated as a sub-system of internal-combustion engine, and a semi-analytical method is proposed to evaluate dynamic characteristics of such a journal bearing subjected to dynamic load.
The principal bearings in internal combustion engines support the large dynamic loads.The gas force due to combustion/compression pressure in the engine cylinder and inertia forces due to reciprocating and unbalanced rotating masses contribute to the engine bearing loads.These loads vary markedly during a cycle of the crankshaft rotation.Under such dynamic loads, the behaviour ofthe rotor-bearing system becomes non-linear and to study rotor-bearing dynamics a complete non- linear transient simulation is used.Such an analysis involves the simultaneous solutions of dynamic Reynolds equation and equations of motion, for locating the journal centre at each time step.and equations of motion involve many iterations and are expensive in skill and time.The other analyses (Booker, 1971; Ritchie, 1975) make use of some closed form approximate solution to the dynamic Reynolds equation for reducing computa- tion time.The authors have reported recently a rapid method for analysis of dynamically loaded journal bearings (Hirani et al., 1998), wherein analytical pressure expression was proposed to eliminate the need of time-consuming and tedious iterations for pressure calculations.A two-dimen- sional non-linear root finding Newton-Raphson method with globally converging strategy was applied to evaluate the velocity of rotor-centre at each time step.The journal displacement at next time step was determined by using Runga-Kutta method.The results obtained match well with those using FEM (Goenka, 1984) at much less computation cost.
The knowledge of hydrodynamic damping and elastic properties of the journal bearing is desirable for accurate determination of the critical speeds of crankshaft and for the anticipation of its behaviour in the neighbourhood of such speeds.Moreover, it is very important to examine how dynamic coeffi- cients vary in a crankshaft cycle.If this could be observed, it would be useful for analysis of engine vibrations.Unfortunately, the constant values of stiffness and damping coefficients have received more attention, and variation in these coefficients is neglected in vibration analysis of crankshaft.Obviously, one reason for this is the numerical methodology, which takes hours of execution time, and no reliable closed-form solutions are available in literature.This paper provides a conceptually simple and rapid means for calculating the stiffness and damping coefficients at each time step.
The stiffness and damping coefficients of oil films do not explicitly appear in the governing equation of non-linear system analysis.The dynamic coef- ficients can be determined by using Reynolds equation with a first order pressure perturbation (Lund and Thomson, 1978).This gives five equa- tions, one for pressure and other four for partial derivatives of pressure with respect to rotor displacement and velocity components.The jour- nal velocities and displacements can be deter- mined by solving first equation (Reynolds equation without perturbation) with equations of motion by using authors' (Hirani et al., 1998), or any other (Booker, 1971;Ritchie, 1975) technique.In the present study remaining four equations (Reynolds equation for pressure with perturbation of displace- ment and velocity components) are solved for short and long bearing approximations.The pressure derivatives for finite bearing are determined by taking harmonic average of short and long bearing pressure derivatives.The integration of these pressure derivatives along the axial direction is carried out analytically and Weddle's numerical inte- gration formula is employed in circumferential direction.
Most of the engine bearing analyses are based on the isothermal simplification.However, thermal considerations do play an important role in the engine bearing dynamics.For a given load the thermal effects modify the value of relative eccen- tricity that leads to a large change in dimensional stiffness and damping coefficients.A complete thermal analysis requires simultaneous solution of generalised Reynolds equation, energy and heat transfer equations with proper boundary condi-  tions (Paranjpe, 1996), which is referred as thermo- hydrodynamic (THD) analysis.The THD analysis gives probable realistic solution but is computa- tionally intensive and requires a significant amount of time and effort in development.In the present study a simplified thermal analysis is used which is rapid and provides reasonable predictions of the performance.

THEORY
The co-ordinate system for a complete (360) journal bearing operating under extremely general dynamic condition is defined in Fig. 1.For a journal bearing the following assumptions are normally made: body forces are negligible, pressure varia- tion across the film thickness is ignored, the bearing surface curvature is large compared to film thickness, no slip condition at fluid/bearing surfaces, the lubricant viscosity and density are constant.The Reynolds equation with these assumptions is given as: R900 N + -0--72z2 6# coN+ 2-0-7 (1) In a dynamically loaded journal bearing, the bearing is subjected to variable loads and speeds.In such a situation, a forced motion takes place, consequently generated pressure profile in oil film varies considerably with load fluctuation, and Eq. ( 1) is required to be solved in a series of time steps.Under such conditions, determination of dynamic coefficients at a static equilibrium position gives unrealistic predictions.The realistic results can be obtained by repeated calculations for dynamic coefficients at each time interval along the locus of the journal centre.This can be achieved by assuming the first order pressure perturbation at each time step, namely: P P' + PxAx + PyAy + PkAk + P:oA. (2)  Similarly, the film thickness" h h' + Ax cos 0 + Ay sin 0, h' C(1 + e cos 0').
The journal velocities and displacements at a particular instant (where external force can be assumed constant) can be evaluated by solving Eq. ( 4) along with equations of motion by using technique mentioned in (Hirani et al., 1998).
For known rotor centre displacement and velocity components, the stiffness and damping coefficients can be obtained by solving Eqs. ( 5)-( 8).
In this study a new approach is presented for evaluation of dynamic coefficients.This technique requires solutions for short and long bearing approximations.The short bearing approximation, which is commonly used (Hattori, 1993) in practice, neglects the oil film pressure gradients in the circumferential (0) direction, and the partial deri- vatives of pressure with respect to displacement and velocity components can be derived for short bearing as: Similarly, the partial derivatives (Px, P,, P, and P2) for long bearing approximation can be derived from Eqs. ( 5)-( 8) by ignoring the terms representing variation in axial direction: (2 @ 2)

THERMAL ANALYSIS
The mechanical friction losses in an engine journal bearing are due to shearing of the oil film and generation of pressure due to hydrodynamic and squeezing actions.These losses, appear as heat, raise the temperature of the lubricant within the clearance space, lower its operating viscosity, increase relative eccentricity and therefore affect the dynamic coefficients.
A major portion of generated heat is dissipated through the lubricant (convection) and transmitted by the lubricated surfaces (conduction).For simpli- city either the heat conducted is neglected (Barwell  and Lingard, 1980), or convection as a known portion of total generated heat is considered in energy balance equation.In the second case, which is more realistic, energy balance is given as: Oil Mass Flow rate x Heat Capacity of oil Temperature rise--Power loss or, (8) where, cr is the fraction of the total heat generated, and carried away by the lubricant.A number of related theoretical works have indicated that for lightly loaded bearing this fraction is about 0.5, for moderately loaded it should beto 0.8, and recently Paranjpe (1996) used a value of c equal to 0.9 for heavily loaded bearings.The authors have suggested a value of o-equal to eccentricity ratio (Hirani, 1997) and validated this assumption for steadily loaded bearing by obtaining a more realistic prediction by this assumption (Hirani  et al., 1997).
The oil flow Q, from an engine bearing involves the hydrodynamic flow QH caused by the shaft rotation and the film pressure gradient, together with the feed pressure flow Qp, which is the direct result of oil being forced through the bearing by supply pressure.Martin (1993) provided suitable combinations of QH and Qp for various feeding arrangements.In the present analysis the axial flow Q is determined by using same formulations as in (Martin, 1993).
The effective temperature rise can be deter- mined by using energy balance, Eq. ( 18).The effective viscosity can be calculated by using temperature-viscosity relation.Here Walther's temperature-viscosity relation is used.which indicates matching in trend and close agreement in the magnitude of the coefficients.
In order to illustrate the application of proposed methodology to dynamically loaded journal Eccentricity ratio FIGURE 3 Comparison of damping coefficients (L/D=0.5).

RESULTS AND DISCUSSION
To validate the proposed approximate model for stiffness and damping coefficients, results obtained by using the present study for steadily loaded bearings (-0, -0) with LID ratio equal to 0.5 and 1.0, are compared with the results obtained by numerical results (Lund and Thomson, 1978).In the present analysis, unbalance rotor mass is not considered because of unavailability of data.
Normally force due to unbalance mass is negligible compared to large engine bearing load, so results will not alter much.The unbalanced mass force can Example 1 To study the variation in stiffness and damping coefficients, the Ruston and Hornsby 6 VEB-X Mk III big end connecting rod is investigated.The rectangular components of bearing load are shown in Fig. 6.The engine and bearing data are as follows: connecting rod length 0.782 m, crank length 0.184 m, C 82.55 lam, D 0.2032 m, L 0.057 m, # 0.015 Pa.s, coj -207r rad/s.
The predicted journal locus for this bearing by using a rapid method (Hirani et al., 1998) is given in Fig. 7.The stiffness and damping coefficients are evaluated by proposed scheme, mentioned in Section 2. Figures 8 and 9 show variation in the stiffness and damping coefficients over a crankshaft cycle.value.Accordingly it can be concluded that the mean value of these coefficients gives unrealistic results and it is suggested that the variation of these coefficients should be considered in engine vibration studies.The present analysis contains analytical pressure gradients, which reduce time- consuming iterations, and make determination of dynamic coefficients easy.be however included easily.The rotor acceleration can be approximated by Newton's backward difference method, and inertia force can be included in the equations of motion.A typical automotive crankshaft main bearing is analysed to illustrate the application of suggested thermal analysis.The load data are given in   F. (N)   Angle (Deg) F (N) F, (N)   Angle (Deg) F. (N)   /, (N)    In the present analysis, the properties of typical SAE 30 oil are used for thermal analysis.The Walther's viscosity-temperature relation is used, to represent the viscosity variation of this oil with temperature as follows: //; poll0 10(8s83-34389)tg(r) 0.6].
The journal orbit for journal of the main bearing obtained by using isothermal (Hirani et al., 1998), and thermal analysis is given in Fig. 10.The simplified thermal analysis follows same trend as isothermal case, but gives better prediction of eccentricity ratio.The temperature effects increase eccentricity ratio, which is seen in Fig. 10.
Figures 11-14 show variation of stiffness and damping coefficients for thermal and isothermal (constant viscosity)case.Five to ten percent variations in dynamic coefficients between iso- thermal and thermal analysis can be visualised from these figures.The required computing time for presented simple thermal analysis is comparable to that taken by isothermal analysis, and it .givesrealistic results compared to isothermal analysis.Accordingly, one can recommend the thermal analysis for simulation of stiffness and dynamic coefficients.Tribology Transactions, 39, 636-644.Parszewski, Z.A. and Krynicki, K., 1989, Rotor-bearing system stability: composition approach with bearing shape function presentation, Tribology Transactions, 32, 517-523.Ritchie, G.S., 1975, The prediction of journal loci in dynami- cally loaded internal combustion engine bearings, Wear, 135, 291-297.

FIGURE 7 FIGURE 8
FIGURE 7 Journal orbit in clearance circle (VEB bearing).
FIGURE 9 Variation of damping coefficients.

FIGURE 13
FIGURE 11 Stiffness coefficients (Kxx and Kvy) for thermal and isothermal analysis.
student for allowing us to use his computer program based on Lund's finite-difference tech- nique(Lund and Thomson, 1978) for comparison (Figs.2-5) with the present study.NOMENCLATURE Bxx, Bxy, Byx, Byy damping coefficients, N soil viscosity, Pa.s co-ordinate in circumferential direction, tad density of lubricant, kg/m angular speed of journal, rad/s angular speed of rotation, rad/s Paranjpe, R.S., 1996, A study of dynamically loaded engine bearings using a transient thermohydrodynamic analysis,

E
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Table II .
The bearing and lubricant data are as