Page 120 - Improving Machinery Reliability

92 Improving Machinery Reliability

In the design stage, it is not possible to know the exact installed configuration
with regard to bearings (clearance, preload) and balance (location of unbalance).
Usually a mechanical test will be limited to one configuration (clearance, preload,
unbalance), which may not show any problem. Changes introduced later by spare
parts during turnarounds may change sensitive dimensions, which may result in a
higher response. For this reason, some satisfactorily operating machines change
vibration characteristics after an overhaul.

Undamped Natural Frequencies. If the principal bearing stiffnesses are plotted
on the critical speed map (Figure 3-2), the location of the undamped natural frequen-
cies (critical speeds) are identified. By calculating the bearing stiffness and damping
coefficients over the expected range of bearing clearances, preload, and viscosity
variation, the anticipated range of criticals can be estimated. For rotor systems with
tilting-pad bearings, which do not have cross-coupling stiffnesses, the measured crit-
icals will be near these undamped intersections. For bearings with significant cross-
coupling stiffness and damping values, the damped critical speeds are usually higher
than the undamped critical speeds.
The vertical and horizontal bearing stiffnesses calculated for minimum and maxi-
mum clearances are plotted on the critical speed map in Figure 3-2. In this example
the vertical stiffnesses did not change significantly with the change in clearance;
however, the horizontal stiffness changed by a factor of 4 to 1. This change illus-
trates the importance of considering the various combinations of clearances in the
calculations. The intersection of these stiffness curves defines the undamped hori-
zontal and vertical critical speeds.

Undamped Mode Shapes. An undamped mode shape is associated with each
undamped natural frequency (critical speed) and can be used to describe the rotor
vibration characteristics. For a vertical stiffness of 464,000 Ibs/in., the mode shapes for
the first and second undamped natural frequencies are shown in Figures 3-6 and 3-7.
The plotted mode shapes were calculated assuming no damping. The actual vibra-
tion mode shapes and response frequencies during operation can vary depending upon
the unbalance distribution and damping. The shaft vibrations at any shaft running
speed can be calculated for different unbalances using a rotor response program.

Evaluation of Critical Speed Map Calculations. To summarize, in the evaluation
of the adequacy of the rotor from the critical speed map and the mode shapes, the
following items should be examined.

1. The proximity of the critical speed to running speed or speed range. The
undamped lateral speeds should not coincide with the running speed. In order to
determine if the actual critical speed will cause excessive vibrations, it is neces-
sary to perform a rotor response to unbalance analysis.
2. The location of the critical speed relative to the support stiffness. If the critical
speed is near the rigid bearing criticals (flexible shaft region), increasing the
bearing stiffness will not increase the critical speed. Also vibration amplitudes
will be low at the bearings and therefore low damping will be available. This