Page 134 - Improving Machinery Reliability

106 Improving Machinery Reliability

subsynchronous vibration which tracks approximately half-speed up to the point
where the speed is two times the first critical. As the speed increases, the subynchro-
nous vibration will remain near the first critical speed. These types of instabilities
can generally be solved by changing the bearing design to a pressure dam, elliptical,
or offset-half bearing or changing to a tilting pad bearing.
A second type of instability vibrations can occur on any rotor, including those
with tilted pad bearings. The vibrations will usually occur near the rotor first critical
speed or may track running speed at some fractional speed. These types of instability
vibrations are sometimes called self-excited vibrations since the motion of the rotor
creates the forcing mechanism that causes the instability. l8
A third type of instability is called a forced non-synchronous instability and can be
caused by stage stall in the compressor last stages or by acoustical resonances in the
~ystern.'~~~~ type of instability usually occurs at 10%-20% of running speed as
This
dictated by the acoustical response characteristics of the diffuser and passage geometry.
The predominant method used in performing a stability analysis is to calculate the
damped (complex) eigenvalues and logarithmic decrement (log dec) of rotor, bear-
ing, and seal assembly.21 A positive log dec indicates that a rotor system is stable,
whereas a negative log dec indicates an unstable system. Experience has shown that
due to uncertainties in the calculations, the calculated log dec should be greater than
0.3 to ensure stability. The damped eigenvalue and log dec are sometimes plotted in
a synchronous stability map. Damped eigenvalues generally occur near the shaft crit-
ical speeds; however, in some heavily damped rotors they can be significantly differ-
ent from the responses due to unbalance.
Rotor stability programs are available that can model the rotor stability for most of
the destabilizing mechanisms; however, some of the mechanisms that influence it are
not clearly understood.22 It has been well documented that increased horsepower,
speed, discharge pressure, molecular weight, and pressure ratio can cause a decrease
in the rotor stability. Many units that are stable at low speeds and pressure become
unstable at higher values. To predict the stability of a rotor at the design operating
conditions, the rotor shaft, bearings, and seals are modeled and the log dec is calcu-
lated as a function of aerodynamic loading. An equation, based on experience from
several instability problems, includes many of the factors that have shown to be
important in rotor stability such as horsepower, speed, diameter of impeller, density
ratio across the compressor, impeller and diffuser restrictive dimensions, and molec-
ular weight of fluid.I5 This equation can be used to predict the approximate aerody-
namic loading that the unit should be able to withstand. The aerodynamic loading is a
cross coupling term, usually applied near the center of the rotor. Conceptually, it can
be thought of as a component that detracts from the stabilizing forces in the system.
In the normal audit procedures, the stability is calculated as a function of aerody-
namic loading with a computer model of the rotor, bearings, and seals. In the evalua-
tion of the stability, it is desirable to have a log dec at zero aerodynamic loading
greater than 0.3 and still greater than 0.1 at the calculated aerodynamic loading. The
log dec should be calculated for the range of bearing and seal properties expected as
shown in Figure 3-21, which gives a plot for the lowest forward mode and indicates
the estimated aerodynamic loading.