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c -2.69 X 104

-1.11 X 105 _

The negative signs imply a positive stiffness because the restoring load is opposite the applied load. Note that for this bearing configuration there is very strong cross-coupling, evidenced by the magnitude of the off-diagonal terms.

Critical Mass The cross-coupled spring and damping coefficients provide a convenient way of representing a bearing in a stability analysis. They reduce the fluid film bearing to a spring-mass system (see Figure 20), and consequently stability and dynamics problems are simplified considerably.

Consider a journal of mass M operating in a bearing. The journal can be considered to have two degrees of freedom, x and y. The governing equations are

TABLE 3 Two-Groove Cylindrical Bearing Geometry and Operating Conditions

160° (10° grooves on either side) 0.0025 in (0.064 mm) 5000 rpm

Load direction is vertical downward

FIGURE 20 Point mass representation of a bearing supported on cross-coupled springs and dampers.

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FIGURE 20 Point mass representation of a bearing supported on cross-coupled springs and dampers.

FIGURE 21 Interpretation of the growth factor a and orbital frequency v.

FIGURE 21 Interpretation of the growth factor a and orbital frequency v.

MX + Dxxx + Dxyy + K„x + Kxyy = 0 My + Dyyy + Dyjc + Kyyy + Kyxx = 0 Assume a sinusoidal response to the form y = ye where ß is a complex variable:

By the Euler expansion of ebt, another way to write Equations 43 and 44 is x = x0eai(cos o)t + i sin «i ) y = yoeat(cos «t + i sin «t)

An interpretation of a and « is shown in Figure 21. The real part of b = a is called the growth or attenuation factor. The imaginary part is the frequency of vibration. A positive real part means that the response to a disturbance grows in time. The growth factor is similar to the logarithmic mean decrement, which is common in vibration theory:

n where t = period of vibration xn+1 = amplitude at time n + 1 xn = amplitude at time n

Thus, the growth factor a is a measure of the growth or decay of the journal to a small disturbance. A positive growth factor implies an instability.

The solutions to Equations 41 and 42 are obtained by substituting Equations 43 and 44, which produces the following:

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