damping ratio ξ. This relationship is easily tested by running some trial values.
For instance, if the damping ratio ξ=0.1, then the amplitude ratio is equal to 5.0.
Similarly, if the damping ratio ξ=0.5, then the amplitude ratio is equal to 1.0.
These values are consistent with the plots presented on both Figs. 2-18 and 2-19.
The amplitude ratio computed with equation (2-91) is a useful quantity that is often referred to as the Amplification Factor for the mechanical system.
This amplitude ratio, or amplification factor is also called the Q for the reso-nance. This dimensionless quantity provides a way to describe the severity of a particular resonance, or the magnitude of the damping ratio at a resonance. In all cases, a high Q is indicative of a system with minimal damping that exhibits large amplitudes at the peak of the resonance. Systems with low available damp-ing may be easily excited, and may be susceptible to stability problems due to a lack of available system damping.
Conversely, systems with a small value for Q must be well damped by defi-nition, and this type of system will exhibit low vibration amplitudes at the reso-nant frequency. Systems with higher damping will be more difficult to excite, and will be less susceptible to a variety of stability problems.
The amplification factor for a rotating machine passing through a specific resonance (critical speed) may be evaluated analytically from the damped critical speed calculations discussed in chapter 5. Based upon the real and imaginary portions of the complex Eigenvalue, the log decrement may be computed. The amplification factor for the resonance is determined by dividing π by the calcu-lated log decrement.
From a measurement standpoint, various methods are used to determine the amplification factor based upon the vibration response data of a machine passing through the rotor resonance. A comparison of the three traditional meth-ods are reviewed in chapter 3.
The theoretical model of mechanical system behavior is closely matched by the motion of actual rotating machines. For example, Fig. 2-21 depicts a Bode plot of a high speed compressor rotor mounted between bearings. In this dia-gram, the rotational speed in RPM is plotted on the horizontal axis, with syn-chronous 1X amplitude and phase lag presented on the dual vertical axes.
This field vibration data was measured with a shaft sensing proximity probe mounted close to the coupling end journal bearing. From this plot it is apparent that a resonance occurs at a speed of 6,100 RPM. This response is the first critical speed of the rotor. Amplitude response through this resonance is moderate, and the overall phase roll through the critical is approximately 110°.
This data indicates that the mechanical system is underdamped, with a damping ratio in the vicinity of ξ=0.2. This type of transient speed behavior is normal and customary for many types of machines within the process industries.
Amplitude Ratioat Resonance
1 2 ×ξ
---=
Forced Vibration 53
The same synchronous 1X vector vibration data may be plotted in the polar coordinate format of Fig. 2-22. This data is identical to the Bode plot, but the polar presentation provides improved resolution of phase changes. As viewed from the driver end of the train, this machine rotates in a counterclockwise direction. The angular reference system begins with 0° at the probe, and the phase angles increase in a direction that is counter to shaft rotation. This is the correct angular convention, and the specific logic for this phase convention will be discussed in succeeding chapters. Note that the origin of the polar plot is coin-cident with low speed, and the plot evolves in a clockwise direction as speed increases. Again, this is normal behavior for a rotor resonance where the phase continues to increasingly lag as the unit passes through the critical speed (bal-ance reson(bal-ance) region.
Generally, both Bode and the polar plots are required to accurately define resonances. This is applicable to rotor, structural, and secondary resonances.
Although a Bode plot will allow accurate frequency identification of the critical, the type of resonance is often identified by the polar plots along the length of the rotor. A proper understanding of these transient speed plots is vital to a full com-prehension of the transient vibration behavior of the machinery.
It should also be mentioned that this data is sensitive to any type of vector offset. This is particularly true for shaft measurements made with proximity probes. These displacement transducers are susceptible to shaft surface condi-tions such as scratches, surface imperfeccondi-tions, metallurgical variacondi-tions, magne-tized segments, and eccentricity of the observed shaft surface. These types of conditions produce erroneous signals that often appear as a substantial 1X
vec-Fig. 2–21 Measured Bode Plot Of Actual Industrial Centrifugal Compressor
Fig. 2–22 Measured Polar Plot Of Actual Industrial Centrifugal Compressor
tor at low speeds. For demonstration purposes, the transient data previously shown in Fig. 2-21 is replotted in Fig. 2-23 with the inclusion of a constant 1X runout vector of 0.83 Mils,p-p at 168°.
The solid lines on this Bode plot are identical to the data in Fig. 2-21, and the dotted lines for amplitude and phase show the influence of the 0.83 Mil,p-p runout vector. Clearly this slow roll vector influences the vibration signal throughout the entire speed domain. The uncompensated critical speed peak appears at 5,900 RPM instead of the actual resonance speed of 6,100 RPM.
Vibration amplitudes at the operating speed of 9,500 RPM appear as 0.4 Mils,p-p with runout, versus the true magnitude of 0.95 Mils,p-p. Finally, the phase shift through the resonance with runout included is about 50°, whereas the properly compensated vibration signal displays a more realistic 110° phase change through the resonance region.
It is apparent that the inclusion of a slow speed runout vector can result in serious data interpretation problems. Due to the potential implications of shaft runout, the origin and various corrective measures for shaft runout will be dis-cussed in greater detail in subsequent chapters.
Overall, the relationship between the physical parameters of mass, stiff-ness, damping and the motion of a body including the displacement, velocity, acceleration, and frequency have been established in this chapter. When these fundamentals are clearly understood, complex mechanical vibration problems may be addressed, and successfully solved.
The previously discussed resonant response is quite typical for a piece of rotating machinery. In this common behavior, a rotor resonance is excited by a synchronous unbalance force during transient startup, and coastdown condi-tions. The coincidence of the excitation frequency (rotor speed) with the natural
Fig. 2–23 Measured Bode Plot Of Actual Industrial Centrifugal Compressor With Shaft Runout
Forced Vibration 55
frequency (critical speed) results in a blossoming or amplification of the vibration amplitudes. It is also apparent that during normal operating conditions, the rotor resonance is normally dormant (inactive) due to the lack of any appreciable exciting force within the bandwidth of the resonant frequency.
In actuality, a variety of rotor and structural resonances exist on every machinery train. In most instances, the various major and minor resonances are inactive due to the lack of an appropriate stimulation. However, when a machine emits a discrete excitation that is coincident with a natural frequency, that par-ticular natural frequency will become active. Similarly, when a process machine produces, or is subjected to a wide-band excitation, the resonant frequencies within the excitation bandwidth may likewise become active.
Case History 2: Steam Turbine End Cover Resonance
As an example of an excitation of a normally inactive resonance, consider a high pressure steam turbine driving a series of tandem centrifugal compressors.
The operating speed for this train normally varied between 3,500 and 3,750 RPM. The eleven stage turbine is rated at 49,000 HP, and it had a history of low vibration amplitudes with minimal evidence of any abnormalities.
Prior to a maintenance turnaround, plant rates were increased to maximize production going to storage. Under this operating condition, the machine speed was temporarily increased to 3,800 RPM. Since the rated speed for this train was 3,930 RPM, the moderate speed increase was considered to be well within the performance envelope for this unit. Unfortunately, continuous operation at this higher speed resulted in an objectionable governor housing vibration. This axial casing vibration occurred at a frequency of 15,200 CPM.
With field measurements, it was readily determined that the high vibration amplitudes were confined to the bolted end cover of the governor housing. The maximum response was measured at the middle of the cover, and minimal amplitudes were evident within the bolt circle of the cover. It was clear that the measured frequency occurred at four times machine speed (4X), and it was dis-covered that a slight speed reduction resulted in the virtual elimination of the end cover vibratory motion. Naturally, the operations personnel were against any corrections that might reduce the plant production rates.
Hence, a solution other than slowing down the turbine was required. In an effort to determine the mode shape of the cover plate, a series of additional mea-surements were obtained. A grid pattern was established on the cover plate face with approximately 5 inches between grid lines. Vibration readings filtered at four times running speed were obtained at each grid intersection. In addition, a reference accelerometer was located at the middle of the cover plate, and the tim-ing relationship to each intersection measurement point was visually deter-mined on a digital oscilloscope.
In all cases, the readings were directly in phase, and the cover was moving back and forth in unison. The accelerometer field measurements were manually converted to casing displacement at the average frequency of 15,200 CPM. Plot-ting the amplitude at each grid intersection allows the construction of the
three-dimensional plate mode shape shown in Fig. 2-24.
This diagram suggests a simple drum mode, with maximum displacement at the center of the plate, and minimum motion at the plate perimeter. Since this appears to be a resonant plate, and since it behaves like a structural resonance;
it would be desirable to confirm this hypothesis based upon the physical charac-teristics of the plate. An affirmation of this resonance may be obtained by per-forming a Finite Element Analysis (FEA). Alternatively, since this cover plate is a simple geometric structure, a set of manual calculations may be performed.
For instance, Roark’s Formulas for Stress & Strain10, includes Table 36 of Natural Frequencies of Vibration for Continuous Members. Case number 15, on page 717, within this table describes the equation for a rectangular flat plate with uniform thickness, and all four edges fixed. The natural frequency for this geometric configuration is presented as:
(2-92)
where: F = Plate Natural Frequency (Cycles / Second) a = Plate Width and Length (Inches)
K = Constant based upon ratio of Length over Width. For a square plate, K = 36.
G = Acceleration of Gravity (= 386.1 Inches / Second2) W = Plate Unit Weight (Pounds / Inch2)
D = Plate Flexural Rigidity (Pounds-Inch)
The turbine end cover was physically measured to be 30 inches square, with a thickness of 0.625 inches. The cover was attached to the turbine casing by a series of bolts that were located on a centerline of approximately 28.5 inches
Fig. 2–24 Three-Dimen-sional Mode Shape Of Vertical Steam Turbine End Cover at 15,200 CPM
10 Warren C. Young, Roark’s Formulas for Stress & Strain, 6th edition, (New York: McGraw-Hill Book Co., 1989), pp. 714-717.
25 30
Forced Vibration 57
square. Since the bolt pattern represented the point of attachment, and the zero motion perimeter, the cover dimension of 28.5 inches was used for the plate cal-culations. The plate unit weight W is computed by multiplying the plate density ρ times the plate thickness t as follows:
(2-93)
where: ρ = Plate Density = 0.283 Pounds / Inch3 t = Plate thickness = 0.625 Inches
The plate unit weight may now be computed from equation (2-93) as:
From Roark, page 714, the plate flexural rigidity D is determined from the modulus of elasticity E, Poisson’s ratio ν, and the plate thickness t as presented in equation (2-94):
(2-94)
where: E = Modulus of Elasticity (= 29.5 x 106 Pounds / Inch2)
ν = Poisson’s Ratio = 0.28 (Dimensionless)
Based on the plate thickness t, and the material constants (E and ν), the Flexural Rigidity D may now be determined with equation (2-94) as:
From these computations of the unit weight W, and the flexural rigidity D, it is now possible to compute the natural frequency of the turbine cover plate. As previously noted, the constant K in the natural frequency equation is equal to 36 due to the equal length of the sides. Combining the various values, the natural frequency is computed from equation (2-92) in the following manner:
This calculated value of 15,950 CPM is higher than the measured frequency of 15,200. However, the 750 CPM difference represents only a 5% deviation. Con-sidering that the material physical properties were estimated, and the plate
W = ρ×t
dimensions were not precise, the 5% variation in frequencies is understandable.
Thus, the measured and calculated data point towards a plate resonance as the culprit. This was verified during operation by bolting a beam with a jack-screw across the plate. Application of the jack-screw to the center of the cover plate immediately suppressed the plate resonant response. During the next scheduled turnaround, the temporary beam and jackscrew were removed, and two angle beams were welded across the inside of the cover plate. Subsequent operation at speeds up to 3,950 RPM revealed no reoccurrence of this problem.
In retrospect, this was not a serious mechanical problem, but it was an irri-tant to the Operations personnel since the vibratory response was audible on the compressor deck. Hence, sometimes problems must be corrected to satisfy the mechanics as well as the politics of the situation.