## Rotor Dynamics

Lateral critical speeds are somewhat misnamed, since they are now thought of as a damped response to some form of rotor excitation. Classically, they got named when compressor (or other rotating machine) speed, whether starting or operating, happened to coincide with a responsive lateral resonance (damped natural frequency). Since the rotor of the compressor included inherent unbalance, it would excite the rotor-bearing support system. Because this occurred long before the "speed squared" effect of unbalance should have been felt, it received the name critical speed, and was tagged as a speed to avoid. Figure 9-7 shows a typical rotor response plot.

The undamped critical speed is proportional to the static deflection of a simple shaft as seen by the following equation for a mass concentrated at a single point [6].

OPERATING SPEEDS

Nc- - Rotor 1st critical, center frequency.

Cycles per minute Ncn = Critical «peed nth

Nmc Nt N

Nc- - Rotor 1st critical, center frequency.

Cycles per minute Ncn = Critical «peed nth

AF = Amplification Factor

Nt = Trip speed

A/m<: - Maximum continuous speed. 105 percent

N> = Initial (lesser) speed at .707 < peak amplitude (critical) Ni = Final (greater) speed at .707 « peak amplitude (critical) Ni-N> = Peak width at the 'half-power' point

SM = Separation Margin

CRE = Critical Response Envelope

Figure 9-7. Typical rotor response plot. (Courtesy of the American Petroleum Institute where

Nc = critical speed, rpm = static shaft deflection g = gravitational constant

All the early calculations were based on simple beam analysis. The method was improved when the summation of moments was introduced by Myklestead [5] and Prohl [6].

The interesting fact is that the two men just mentioned were initially working independently of each other. The initial analysis used infinitely stiff bearings and, while the method improved the results, it did not match the compressor critical speed test results consistently. Later work recognized the existence of a bearing oil film with elastic and damping properties.

As a journal rotates in a lubricated bearing, the viscous nature of the lubricant causes the shaft to trap oil between the journal and the bearing surface. A wedge of oil or fluid film develops and produces a hydrody-namic pressure sufficient to carry the journal load and keep the two surfaces separated. A fully developed oil film exhibits stiffness and damping characteristics which will vary in magnitude with journal rotational speed. The bearing stiffness represents the spring rate of the oil film, while damping indicates the film's ability to dissipate vibrational energy-These properties are calculated in the form of linear, non-dimensional coefficients for use in the damped unbalanced response calculation. Stiffness alone is used in the undamped calculation (see Figure 9-8).

lOflLO

lOflLO

1000

Figure 9-8. Principal non-dimensional linear stiffness and clamping coefficients for short bearing. (Courtesy of Turbocare, A Division of Demag Délavai Turbomachinery Corp., Houston facility

1000

Figure 9-8. Principal non-dimensional linear stiffness and clamping coefficients for short bearing. (Courtesy of Turbocare, A Division of Demag Délavai Turbomachinery Corp., Houston facility

When rotor geometric data, shaft diameter and lengths, element sizes and weights are put together, the mass elastic rotor system can be modeled. To complete the model, the support spring rate (oil film) is included. When the information is available, a computer analysis would include support parameters, such as pedestal and/or foundation stiffness. The results of the solution for frequency yield the undamped critical spectrum. These results are plotted in two forms. One is a mode shape diagram (deflection vs. rotor length) (see Figure 9-9). The plots are useful to identify the various modes and to check for node points in the bearing. Damping will have little or no effect if there is no movement in the bearing. To state in positive terms, damping is a dynamic property and is a function of velocity which implies movement. The second plot is a map of the response frequencies plotted against bearing stiffness (see Figure 9-10). The response frequencies are given in speed units, rpm. The operating speed is drawn in with a straight line bounded by the API 617 margin requirement. The bearing stiffness versus speed map is cross-plotted. It was mentioned previously that bearing stiffness is a function of speed. At one time, this was the extent to which rotor analysis was pursued. The figure cited would have had all interested persons happy because the points where the bearing maps cross the three response plots do not coincide with operating speed nor do they encroach into the margin specified.

BEARING STIFFNESS (Ibl/in)

Figure 9-10. Map of response frequencies plotted against bearing stiffness.

BEARING STIFFNESS (Ibl/in)

Figure 9-10. Map of response frequencies plotted against bearing stiffness.

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