**2. Literature review**

6 Performance Evaluation of Bearings

[Chauhan and Sehgal: 2008].

**Figure 3.** Schematic diagram of offset-halves journal bearing

The offset-halves journal bearing has been commonly used as a lobed bearing in which two lobes are obtained by orthogonally displacing the two halves of a cylindrical bearing. Offsethalves journal bearings (Fig. 3) are frequently used in gear boxes connecting turbine and generator for the power generation industries. These also find applications where primary directions of force, constant direction of rotation are found or high bearing load capacity, long service life, high stiffness, and damping values are the main characteristics under consideration. If the equipment is operated at full power, these requirements can be met by lemon bore bearings. Lemon bore bearing is a variation on the plain bearing where the bearing clearance is reduced in one direction and this bearing has a lower load carrying capacity than the plain bearings, but is more susceptible to oil whirl at high speeds [W2]. However, equipment must often be operated at lower performance levels, particularly in the times of reduced current needs. It is precisely under these conditions that lemon bore bearings may provide unstable conditions, which may require equipment shut down to avoid damage. Offset-halves journal bearings have the durability equal to lemon bore bearings while these show stiffness and damping properties which permit light loads at high rotational speeds. It also offers the advantage of a long, minimally convergent inlet gap, resulting in high load carrying capacity. At the same time, the externally applied force and the compression resulting from the horizontal displacement of the bearing halves accurately holds the shaft in the lubricant film. This effect produces excellent hydrodynamic characteristics, such as elastic rigidity and damping by the oil film. Thus, the offset-halves journal bearings prove to be technical alternative to conventional lemon bore bearings

Upper Lobe

Lower Lobe

This section of the chapter provides details of research carried out on hydrodynamic bearings in general and, offset-halves and elliptical journal bearings in particular. There is enormous information available on the theoretical and experimental work related to the circular journal bearings. However, such work pertaining to non-circular journal bearings especially offset-halves and elliptical journal bearings are limited and hence, the theoretical and experimental works pertaining to non-circular journal bearings have been summarized:

Pinkus and Lynn [1956] have presented an analysis of elliptical journal bearings based on the numerical solution of Reynolds equation for finite bearings with the assumption that there is no heat loss to the surroundings. They have supplemented the solution of differential equation by additional work on the nature of the oil flow, power loss, and eccentricity in elliptical journal bearings. Wilcock [1961] has worked towards the possibility

of displacing the lobe centers of two-lobe journal bearings orthogonally with respect to the mid-radius of the lobe. The author shows that when the lobe displacement is in a direction opposite to the shaft surface motion, and the bearing is centrally loaded, shaft stiffness orthogonal to the load vector is substantially increased. At the same time, vertical stiffness essentially remains unchanged and minimum film thickness is decreased; particularly at low loads, while oil flow is increased. Author also carried out an analysis for a bearing having in cross-section two arcs (each subtending an angle of 1500), L/D=1/2, and with the arc centers each displaced from the geometric center by half the radial clearance. Singh et al. [1977] have reported that non-circular bearings are finding extensive use in high speed machinery as they enhance shaft stability, reduce power losses and increase oil flow (as compared to circular bearings), thus reducing bearing temperature. The authors had presented a solution to analyze the elliptical bearings, using a variational approach. Crosby [1980] has solved full journal bearing of finite length for thermohydrodynamic case in which the energy transmitted by conduction is included. The effect of temperature variation across the film thickness on bearing performance is investigated by the author. Singh and Gupta [1982] have considered the stability limits of elliptical journal bearings supporting flexible rotors. The Reynolds equation was solved numerically for several values of the eccentricity ratio, the L/D ratio and the dimensionless velocity of the journal centre. The authors observed that the operating load, ellipticity, L/D ratio and shaft flexibility significantly affect the limit of stable operation. The authors also reported that elliptical bearings are suitable for stiff and moderately flexible rotors. Tayal et al. [1982] have investigated the effect of nonlinear behavior of additive-fortified lubricants which behave as non-Newtonian fluid on the performance characteristics of the finite width elliptical journal bearing. The finite element method with Galerkin's technique was used to solve the Navier-Stokes equations in cylindrical co-ordinates that represent the flow field in the clearance space of a bearing using Newtonian fluids, and then the non-Newtonian effect was introduced by modifying the viscosity term for the model in the iterations. Booker and Chandra et al. [1983] have compared the performance of different bearing configurations namely offset-halves, lemonbore, three-lobe and four-lobe bearing at the same load capacity and speed. During the comparison, the authors have considered the steady state and stability characteristics. Govindachar [1984] have suggested that Novel 'offset' designs offer attractive possibilities in several applications for which conventional journal bearings are only marginally satisfactory. They considered one such prototypical problem in rotating machinery (the support of a rigid rotor turning at high speed under gravity). The problem has been studied by the authors through a numerical example for both dimensional and non-dimensional parametric studies. The authors show that the stability of full journal bearing system is significantly improved by moderate offset and is fairly insensitive to small departures from optimal design values. Singh and Gupta [1984] have theoretically predicted the stability of a hybrid two-lobe bearing which is obtained by displacing the lobe centers of an elliptical bearing. It has been found that an orthogonally displaced bearing is more stable than the conventional bearings. Mehta and Singh [1986] have analytically analyzed the dynamic behaviour of a cylindrical pressure dam bearing in which centers of both halves are displaced. Authors observed that the stabilities of a cylindrical pressure dam bearing can be Thermal Studies of Non-Circular Journal Bearing Profiles: Offset-Halves and Elliptical 9

increased many times by displacing the centers of two halves. It has been reported by the authors that the bearing so obtained is even superior to elliptical and half elliptical pressure dam bearings in stability. Read and Flack [1987] developed a test apparatus on which an offset-halves journal bearing of 70 mm diameter journal was tested at five vertical loads and two rotational speeds. Singh et al. [1989] have studied the effects of linear elastic deformation and lubricant viscosity with pressure and temperature for an elliptical bearing and solved 3-D equations for momentum, continuity and elasticity to obtain pressure in the lubricant flow-field and deformation in the bearing liner. Basri and Gethin [1990] have carried out a theoretical analysis of the thermal behaviour of orthogonally displaced, threelobe, and four-lobe bearing geometries. The thermal analysis illustrates the implication of the type selection with regards to the parameters of load-carrying ability, power loss, lubricant requirements, and operating temperatures. The comparisons presented by the authors, show that all profiles considered have inferior load-carrying ability and less extreme thermal effects when compared with the cylindrical geometry along with significantly larger lubricant supply requirements. Hussain et al. [1996] have predicted the temperature distribution in various non-circular journal bearings namely two-lobe, elliptical, and orthogonally displaced. The work reported by them is based on a twodimensional treatment following Mc Callion's approach in which the Reynolds and energy equations in oil film are decoupled by neglecting all pressure terms in the energy equation. Ma and Taylor [1996] experimentally investigated the thermal behaviour of a two-axialgroove circular bearing and an elliptical bearing, both 110mm in diameter. Both bearings were tested at specific loads upto 4 MPa and rotational frequencies up to 120 Hz. The authors have measured the power losses & flow rates directly, and the detailed temperature information was collected. The results presented by them show that the thermal effects are significant in both bearings. Banwait and Chandrawat [2000] have analyzed the effect of journal misalignment on the thermohydrodynamic performance characteristics of elliptical journal bearing and solved the generalized Reynolds equation for the oil-film pressure distribution. The energy and heat conduction equations are used by the authors for determining the oil-film, bush and journal temperature fields. This work has reported the important features observed in static performance characteristics of thermohydrodynamic analysis of misaligned elliptical bearing along with the isopressure curves, pressure profiles, isotherms and temperature profiles. Sehgal et al. [2000] have presented a comparative analysis of three types of hydrodynamic journal bearing configurations namely, circular, axial groove, and offset-halves. It has been reported by the authors that the offset bearing runs cooler than an equivalent circular bearing with axial grooves. A computer-aided design of hydrodynamic journal bearing is provided considering thermal effects by Singh and Majumdar [2005]. They have solved the Reynolds equation simultaneously along with the energy equation and heat conduction equations in bush and shaft to obtain the steady-state solution. A data bank is generated by the authors that consists of load, friction factor and flow rate for different L/D & eccentricity ratios. Sharma and Pandey [2007] have carried out a thermohydrodynamic lubrication analysis of infinitely wide slider bearing assuming parabolic and Legendre polynomial temperature profile across the film thickness. It was observed that the temperature approximation across the film thickness by Legendre

of displacing the lobe centers of two-lobe journal bearings orthogonally with respect to the mid-radius of the lobe. The author shows that when the lobe displacement is in a direction opposite to the shaft surface motion, and the bearing is centrally loaded, shaft stiffness orthogonal to the load vector is substantially increased. At the same time, vertical stiffness essentially remains unchanged and minimum film thickness is decreased; particularly at low loads, while oil flow is increased. Author also carried out an analysis for a bearing having in cross-section two arcs (each subtending an angle of 1500), L/D=1/2, and with the arc centers each displaced from the geometric center by half the radial clearance. Singh et al. [1977] have reported that non-circular bearings are finding extensive use in high speed machinery as they enhance shaft stability, reduce power losses and increase oil flow (as compared to circular bearings), thus reducing bearing temperature. The authors had presented a solution to analyze the elliptical bearings, using a variational approach. Crosby [1980] has solved full journal bearing of finite length for thermohydrodynamic case in which the energy transmitted by conduction is included. The effect of temperature variation across the film thickness on bearing performance is investigated by the author. Singh and Gupta [1982] have considered the stability limits of elliptical journal bearings supporting flexible rotors. The Reynolds equation was solved numerically for several values of the eccentricity ratio, the L/D ratio and the dimensionless velocity of the journal centre. The authors observed that the operating load, ellipticity, L/D ratio and shaft flexibility significantly affect the limit of stable operation. The authors also reported that elliptical bearings are suitable for stiff and moderately flexible rotors. Tayal et al. [1982] have investigated the effect of nonlinear behavior of additive-fortified lubricants which behave as non-Newtonian fluid on the performance characteristics of the finite width elliptical journal bearing. The finite element method with Galerkin's technique was used to solve the Navier-Stokes equations in cylindrical co-ordinates that represent the flow field in the clearance space of a bearing using Newtonian fluids, and then the non-Newtonian effect was introduced by modifying the viscosity term for the model in the iterations. Booker and Chandra et al. [1983] have compared the performance of different bearing configurations namely offset-halves, lemonbore, three-lobe and four-lobe bearing at the same load capacity and speed. During the comparison, the authors have considered the steady state and stability characteristics. Govindachar [1984] have suggested that Novel 'offset' designs offer attractive possibilities in several applications for which conventional journal bearings are only marginally satisfactory. They considered one such prototypical problem in rotating machinery (the support of a rigid rotor turning at high speed under gravity). The problem has been studied by the authors through a numerical example for both dimensional and non-dimensional parametric studies. The authors show that the stability of full journal bearing system is significantly improved by moderate offset and is fairly insensitive to small departures from optimal design values. Singh and Gupta [1984] have theoretically predicted the stability of a hybrid two-lobe bearing which is obtained by displacing the lobe centers of an elliptical bearing. It has been found that an orthogonally displaced bearing is more stable than the conventional bearings. Mehta and Singh [1986] have analytically analyzed the dynamic behaviour of a cylindrical pressure dam bearing in which centers of both halves are displaced. Authors observed that the stabilities of a cylindrical pressure dam bearing can be increased many times by displacing the centers of two halves. It has been reported by the authors that the bearing so obtained is even superior to elliptical and half elliptical pressure dam bearings in stability. Read and Flack [1987] developed a test apparatus on which an offset-halves journal bearing of 70 mm diameter journal was tested at five vertical loads and two rotational speeds. Singh et al. [1989] have studied the effects of linear elastic deformation and lubricant viscosity with pressure and temperature for an elliptical bearing and solved 3-D equations for momentum, continuity and elasticity to obtain pressure in the lubricant flow-field and deformation in the bearing liner. Basri and Gethin [1990] have carried out a theoretical analysis of the thermal behaviour of orthogonally displaced, threelobe, and four-lobe bearing geometries. The thermal analysis illustrates the implication of the type selection with regards to the parameters of load-carrying ability, power loss, lubricant requirements, and operating temperatures. The comparisons presented by the authors, show that all profiles considered have inferior load-carrying ability and less extreme thermal effects when compared with the cylindrical geometry along with significantly larger lubricant supply requirements. Hussain et al. [1996] have predicted the temperature distribution in various non-circular journal bearings namely two-lobe, elliptical, and orthogonally displaced. The work reported by them is based on a twodimensional treatment following Mc Callion's approach in which the Reynolds and energy equations in oil film are decoupled by neglecting all pressure terms in the energy equation. Ma and Taylor [1996] experimentally investigated the thermal behaviour of a two-axialgroove circular bearing and an elliptical bearing, both 110mm in diameter. Both bearings were tested at specific loads upto 4 MPa and rotational frequencies up to 120 Hz. The authors have measured the power losses & flow rates directly, and the detailed temperature information was collected. The results presented by them show that the thermal effects are significant in both bearings. Banwait and Chandrawat [2000] have analyzed the effect of journal misalignment on the thermohydrodynamic performance characteristics of elliptical journal bearing and solved the generalized Reynolds equation for the oil-film pressure distribution. The energy and heat conduction equations are used by the authors for determining the oil-film, bush and journal temperature fields. This work has reported the important features observed in static performance characteristics of thermohydrodynamic analysis of misaligned elliptical bearing along with the isopressure curves, pressure profiles, isotherms and temperature profiles. Sehgal et al. [2000] have presented a comparative analysis of three types of hydrodynamic journal bearing configurations namely, circular, axial groove, and offset-halves. It has been reported by the authors that the offset bearing runs cooler than an equivalent circular bearing with axial grooves. A computer-aided design of hydrodynamic journal bearing is provided considering thermal effects by Singh and Majumdar [2005]. They have solved the Reynolds equation simultaneously along with the energy equation and heat conduction equations in bush and shaft to obtain the steady-state solution. A data bank is generated by the authors that consists of load, friction factor and flow rate for different L/D & eccentricity ratios. Sharma and Pandey [2007] have carried out a thermohydrodynamic lubrication analysis of infinitely wide slider bearing assuming parabolic and Legendre polynomial temperature profile across the film thickness. It was observed that the temperature approximation across the film thickness by Legendre

Polynomial yields more accurate results in comparison to Parabolic Profile approximation. However, the former is algebraically more complex to tackle in comparison to the later; the authors also observed that the film temperatures computed by Parabolic Profile approximation are lower in comparison to Legendre Polynomial approximation. Further, it has been concluded by them that the computational time taken in solution of coupled governing equation with both temperature profile approximation have only marginal difference. Mishra et al. [2007] have considered the non-circularity in bearing bore as elliptical and made a comparison with the circular case to analyse the effect of irregularity. It was reported that, with increasing non-circularity, the pressure gets reduced and temperature rise obtained is less. Chauhan et al. [2010] have presented a thermohydrodynamic study for the analysis of elliptical journal bearing with three different grade oils. The authors have made a comparative study of rise in oil-temperatures, thermal pressures, and load carrying capacity for three commercial grade oils. Rahmatabadi et al. [2010] have considered the non-circular bearing configurations: two, three and four-lobe lubricated with micropolar fluids. The authors have modified the Reynolds equation based on the theory of micropolar fluids and solved the same by using finite element methods. It has been observed by the authors that micropolar lubricants can produce significant enhancement in the static performance characteristics and the effects are more pronounced at larger coupling numbers.

Thermal Studies of Non-Circular Journal Bearing Profiles: Offset-Halves and Elliptical 11

In present chapter, the thermal studies of non-circular journal bearings: offset-halves and

The film thickness equations for offset-halves journal bearing (Fig. 1 (a)) are given as Sehgal

1 1 cos sin

 

1 1 cos sin

 

In these equations, *h* represents film thickness for circular journal bearing, *C* represents

horizontal split axis in the direction of rotation. *Cm* denotes minimum clearance when

The film thickness equations for elliptical journal bearing (Fig. 1 (b)) are given as Hussain et

, for 0<

2 cos ;

 

sin tan

 *E E M M* 

> *EM* cos

represents eccentricity ratio from 0-180 Deg. (upper lobe) and 180-360

represents attitude angles from 0-180 Deg. (upper lobe)

   

 

 (0<

 (180<

2 2 2

 2 cos

<180) (1)

<360) (2)

denotes offset factor

<180 (3)

<360) (4)

1

*m*

; *h m M*

*C C <sup>E</sup> C* 

represents angle measured from the

 

 

 

 1 1 1 cos *hc E m M* 

> , for (180<

> > 1

2 

In eqns. (3 and 4), *h* represents film thickness for elliptical journal bearing, *EM* represents

and 180-360 Deg. (lower lobe) respectively and *Ch* represents horizontal clearance for elliptical journal bearing. Film thickness represented by eqns. 1, 3 corresponds to upper lobe

 2 2 1 cos *hc E m M* 

 

represents eccentricity ratio, and

journal centre is coincident with geometric centre of the bearing,

elliptical have been presented using thermohydrodynamic approach.

2 2 *<sup>m</sup> h c* 

2 2 *<sup>m</sup> h c* 

denotes attitude angle.

Different parameters used in eqns. (3) & (4) are given as:

 *E E M M* 

1

sin tan

*EM* cos 

> ,

whereas eqns. 2, 4 represents film thickness for lower lobe of the.

; <sup>1</sup>

2 2 2

 

2

**3.1. Film thickness equation** 

et al. [2000]:

radial clearance,

( / *C C <sup>m</sup>* ), and

al. [1996]:

1 

 ,

Deg. (lower lobe) respectively, 1 2

1

 

elliptical Ratio, 1 2
