The following values were used: v = 6000 lb (2721 kg), SAE 20 oil, L = 3 in (76 mm), D = 3 in (76 mm), N = 1800 rpm, t,= 120°F (48.9°C), c = 0.008 in (0.076 mm).

mechanism for self-excited rotor whirl. Whirl is manifest as an orbiting of the journal at a subsynchronous frequency, usually close to one-half the rotating speed. Whirl is usually destructive and must be avoided.

It is the purpose of this section to provide some insight into bearing dynamics, present some background on analytical methods and representations, and discuss some particular bearings and factors that can influence dynamic characteristics. Dynamic performance data and sample problems are presented for several bearing types.

The Concept of Cross Coupling As mentioned in the opening paragraphs of this subsection, a journal bearing derives load capacity from viscous pumping of the lubricant through a small clearance region. To generate pressure, the resistance to pumping must increase in the direction of the fluid flow. This is accomplished by a movement of the journal such that the clearance distribution takes on the form of a tapered wedge in the direction of rotation, as shown in Figure 1.

The attitude angle g in Figure 1 is the angle between the load direction and the line of centers. Thus, the displacement of the journal is not along a line that is coincident with the load vector, and a load in one direction causes not only displacements in that direction, but orthogonal displacements as well.

Similarly, a displacement of the journal in the bearing will cause a load opposing the displacement and a load orthogonal to it. Thus, strong cross-coupling influences are introduced by the mechanism by which a bearing operates. The concept of cross-coupling is significant in dynamic characteristics.

It is the cross-coupling characteristics of a journal bearing that can promote self-excited instabilities in the form of bearing whirl. Motion in one direction produces orthogonal forces that in turn cause orthogonal motion. The process continues, and an orbital motion of the journal results. This orbital motion is generally in the same direction as shaft rotation and subsynchronous in frequency. Half-frequency whirl is a self-excited phenomenon and does not require external forces to promote it.

Cross-Coupled Spring and Damping Coefficients For dynamic considerations, a convenient representation of bearing characteristics is a cross-coupled spring and damping coefficients. These are obtained as follows (refer to Figure 1):

1. The equilibrium position to support the given load is established by computer solution of Reynolds' equation.

2. A small displacement to the journal is applied in the y direction. A new solution of Reynolds' equation is obtained, and the resulting forces in the x and y directions are produced. The spring coefficients are as follows:

where Kxy = the stiffness in the x direction due to y displacement

AFx = the difference in x forces between displaced and equilibrium positions Ay = the displacement from the equilibrium position in the y direction Kyy = the stiffness in the y direction due to y displacement AFy = the difference in y forces between displaced and equilibrium positions

3. The journal is returned to its equilibrium position and an x displacement is applied. Similar reasoning produces Kxx and Kyx.

The cross-coupled damping coefficients are produced in a similiar manner, except, instead of displacements in the x and y direction, velocities in these directions are consecutively applied with the journal in the equilibrium position. The mechanism for increasing the load capacity is squeeze film in which the last term on the right-hand side of

Equation 14 is actuated. Thus, for most fixed bearing configurations, eight coefficients exist: four spring and four damping. The total force on the journal is

where Ft = force in the ith direction.

Repeated subscripts imply the following summation:

It should be realized that the cross-coupled spring and damping coefficients represent a linearization of bearing characteristics. When they are used, the equilibrium position should be accurately determined, as the coefficients are valid for only a small displacement region encompassing the equilibrium position of the journal. This is true because the spring and damping coefficients remain constant for only a small region of the equilibrium position.

Consider the two-groove cylindrical bearing shown in Figure 1, with the geometric and operating conditions indicated in Table 3. The computer solution (also the performance curves in Figure 13) produces the following results:

Minimum film thickness hM = 0.00125 in (0.032 mm)

The spring and damping coefficients are

Spring coefficients, lb/in (kg/mm):

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