34.1 Introduction

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Vibrations are an inherent part of all rotating machinery. Residual mass imbalance and dynamic

interaction forces between the stationary and rotating components, which are practically impossible to

eliminate, cause these vibrations. The challenge is to identify the source of vibration and control it to

within reasonable limits. Because of economic advantages, the trend in industry has been to move

towards high speed, high power, lighter and more compact machinery. This has resulted in machines

operating above their first critical speeds, which was unheard of in the past. The new operating

parameters have required concurrent development of vibration technology without which it is not

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possible to safely and reliably operate such machinery. Industry has also come to realize that vibration is

an essential phenomenon, which could be used to assess the performance, durability, and reliability of

rotating machinery.

Engineers at different levels approach the subject of vibration in rotating machinery differently. The

machinery designer has to recognize the potential sources of vibration and control them to within

acceptable levels. In the past few decades, owing to the advancement in computers and modeling

techniques, better understanding of the dynamics of rotating machinery, including the identification of

potential sources of vibration, has been realized. This has enabled designers to accurately predict the

rotordynamic behavior of machinery, allowing it to reach higher operating speeds and larger energy

capacities safely and reliably.

Approaching vibration from a different perspective, the maintenance engineer uses vibration standards

and guidelines to monitor the health of equipment for their timely repair and refurbishment. Reliable

vibration monitoring and diagnostics techniques have moved industry into predictive rather than

preventive maintenance practices, which considerably reduce plant downtimes that rely on key rotating

machinery. Premature replacement of machinery components has also been minimized. The resulting

financial and economic benefits provide an added incentive for the study and understanding of vibration

in rotating machinery.

The vibration specialist or troubleshooter has to use his knowledge of rotordynamics and his

diagnostic capabilities to solve vibration problems in rotating machinery. In most cases, it is also

important to have an understanding of the interfacial dynamics of the rotating machinery with the

surrounding system in order to solve a vibration problem.

From a safety and reliability standpoint, the public must be concerned with vibration in rotating

machinery. Their concerns are addressed through vibration standards and guidelines. These procedures

have been developed for rotating machinery by numerous organizations, both at the national and

international levels. Some of these standards are industry specific and some are equipment type specific,

while a number of them try to cover a wide range of rotating machinery. The objective of most of these

standards is to establish and control quality, safety, durability and reliable performance of rotating

machinery for the benefit of those who use or operate it.

34.1.1 History of Vibration in Rotating Machinery

Although various types of rotating machinery have been in use for many centuries, understanding of

their rotordynamic behavior did not begin until 1869 (Rankine, 1869). Since that time, there has been

steady growth in the development and understanding of the vibration behavior of rotating

machinery. A tabulation of major historical events that have contributed to this growth is presented

in Table 34.1.

* All rotating machinery vibrates to some degree. For public safety and machine reliability,

the vibrations have to be controlled to within acceptable limits.

* Modern trends towards more sophisticated, higher speed compact rotating machinery

have contributed to the rapid development in vibration technology through a better

understanding of their rotordynamics.

* Vibration technology is integrated into the areas of design, maintenance, and troubleshooting

of rotating machinery.

* From a safety and reliability standpoint, the public is protected by the implementation of

vibration standards and guidelines.

* The first publicly reported rotordynamic study was made in 1869.

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TABLE 34.1 A Chronological Listing of Major Contributions that Have Led to the Development and

Understanding of Vibration in Rotating Machinery

Year Contributor Description

1869 Rankine, W.J.M. He examined the equilibrium of a frictionless, uniform shaft

disturbed from its initial position. The resulting recorded article is

recognized to be the first on the subject of rotor dynamics. He

proposed that motion is stable below the first critical speed, is

neutral or indifferent at the critical speed, and unstable above the

critical speed

He also developed numerical formulae for critical speeds for the cases

of a shaft resting freely on a bearing at each end and for an overhanging

shaft fixed in direction at one end

1883 Greenhill, A.G. He studied the effect of end thrust and torque on the stability of

a long shaft and concluded that they were both unimportant. He

also obtained formulae for the cases of an unloaded shaft resting

on bearings at each end and fixed in direction at each end

Circa 1890 Reynolds, O. He extended the theory developed by Rankine and Greenhill for the

case of a shaft loaded with pulleys

1893 Dunkerley, S. He developed formulae for critical speeds for loaded shafts in terms

of the diameter of the shaft, weights of pulleys, the manner in

which the shaft is supported, and so on, and verified them by

experiment

He postulated that any degree of unbalance will excite the shaft

at the critical speed to very high amplitudes and that it is possible

to operate above the first critical speed. The dependence of critical

speed on the moment of inertia of the rotating pulley was

identified

1894 Rayleigh, J.W.S. He developed an approximate method to calculate the natural frequency

of a continuous beam with distributed mass and flexibility using the

energy method

1895 DeLaval, G. He was responsible for the first experimental demonstration that a

steam turbine is capable of sustained operation above the first

critical speed

1916 Timoshenko, T. He discovered the effects of transverse shear deflection on the natural

frequency of a continuous beam and applied the principle to the case

of the rotating shaft

1919 Jeffcott, H.H. He examined the effect of unbalance on the whirl amplitudes and the

forces transmitted to the bearings. The case of a light uniform shaft

supported freely on bearings at its ends and carrying a thin pulley

of mass m at the center of the span was studied. He assumed the

moment of inertia of the pulley to be negligible. Using this model,

later known as the Jeffcott model, a comprehensive theory was

developed to explain the behavior of the rotor as it passed through

the critical speed

The effect of damping on the whirl amplitude, a phase change of

angle p as it passes through the critical speed, and the concept

of synchronous rotor whirling (precession) were introduced and

explained. He also recognized that with a separation margin of 10%

on either side of a critical speed, the amplitude of vibration would

not be excessive. He demonstrated that it is better from the

vibration point of view to design the shaft with its critical speed

below the working speed rather than to have a critical speed the

same proportion above the working speed. Accordingly, he

explained the behavior of the De Laval steam turbine and the

economic advantages of operation above the critical speed

1921 Southwell, R.V.

and Gough, B.S.

They found that a torque and an end thrust of constant magnitude

lowers the critical speed of a rotating shaft, disproving Greenhill’s

earlier (1883) conclusions

(continued on next page)

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TABLE 34.1 (continued)

Year Contributor Description

1921 Holzer, H. He developed a numerical method to calculate torsional critical speeds

and mode shapes for a multidisk rotor system

1924 Newkirk, B.L. He observed that a rotor operating at a speed above the first critical

speed can enter into high, violent whirling and the center of the rotor

will precess in the forward direction at a rate equal to that of the

critical speed. Unlike in the case of synchronous whirling, if the

speed is increased beyond the initial whirl speed, the whirl amplitude

will continue to increase, eventually leading to failure. This was the

first time that it was realized that nonsynchronous unstable motion

can exist in a high-speed rotor

Based on experiments, he made the following key observations on

nonsynchronous whirling. The amplitude and the onset speed of

whirling are independent of the rotor balance. Whirling always occurs

at speeds above the critical speed, and the whirl speed is always

constant at the critical speed, regardless of the rotor speed. The

whirl threshold speed can vary even for machines of similar

construction. Whirling occurs only in built up rotors, and not in

single piece constructions. Increasing the foundation flexibility,

distortion or misalignment of the bearing housings, or introducing

damping to the foundation or increasing the axial thrust bearing

load, increased the threshold speed of whirling

1924 Kimball, A.T. Suggested that internal friction or viscous action due to bending

may cause a shaft to whirl when rotating at any speed above

the first critical speed. He postulated that the nonsynchronous

whirling observed by Newkirk was due to this phenomenon

1924 Newkirk, B.L. Based on Mr Kimball’s theory, he concluded that similar frictional

forces are generated at the mating face between the shrunk on disk

and the shaft of a built-up rotor, and the nonsynchronous whirling

observed by him was due to this effect. However, he was unable to

explain some of his experimental findings, in particular, the effects

of bearing or foundation flexibility, damping, and misalignment

1925 Newkirk, B.L. He experienced another form of nonsynchronous whirling, similar but

different to that caused by the frictional effects of a shrink-fit disk.

It occurred at rotor speeds just exceeding twice the first critical speed

on shafts mounted on journal bearings. He recognized that the oil in

the journal bearing was responsible for the violent motion and called

it oil whip. The whirl speed and direction of whirling were the same

as that for friction induced whirling, that is, the first critical speed in

the forward direction. A theory to explain how the oil film can

produce the whirling motion of a journal and to account for why

it took the same direction as rotation of the shaft was proposed.

However, the theory does not explain why whirling does not

commence until the rotor speed reaches twice the critical speed

value. The influence of foundation flexibility on the rotor stability

was also found to be confusing to Newkirk. In the case of

friction-induced whirl, he was able to totally eliminate the rotor

instability by means of a flexibly mounted bearing. When this

was tried with the journal bearings, the whirl amplitudes

magnified. External damping at the bearing was found to have a

favorable influence on whirl amplitudes

1925 Stodola, A. He developed an iterative procedure to calculate the fundamental

frequency of a vibrating system based on an assumed mode shape

1927 Stodola, A. He provided an explanation and formulae for the gyroscopic moment

effect on the critical speed of a rotor. He also introduced the notion

of synchronous and nonsynchronous reverse precession of a rotor

under specific conditions

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TABLE 34.1 (continued)

Year Contributor Description

1933 Robertson, D. In order to understand oil whip, he studied the stability of the ideal

3608 infinitely long journal bearing, and erroneously concluded that

the rotor will be unstable at all speeds and not only at speeds above

twice the critical speed value

1933 Smith, D.M. He studied the case of unsymmetrical rotors on unsymmetrical supports

and obtained four different critical speed values in comparison to the

single value for a symmetrical system. He also discussed the presence

of additional critical speeds due to gyroscopic effects of large disks

1944 Myklestad, N. A lumped parameter transfer matrix method to calculate natural

frequencies for airplane wings was developed by him

1945 Prohl, M. He developed a lumped parameter transfer matrix method for

calculating critical speeds of flexible rotors

1953 Poritsky, H. Using the small displacement theory, he derived a radial stiffness

coefficient for the journal bearings and analyzed the rotor behavior

under oil whirl conditions. He concluded that the rotor was stable

below twice the critical speed and indicated that increasing the

rotor or bearing flexibility will reduce the threshold speed of

instability. He also proposed a stability criterion for a rotor based

on the bearing and rotor stiffness

1953 Miller, D.F. He introduced a solution to the steady-state forced vibration problem,

for a beam or rotating shaft on damped, flexible end supports. The

response of the rotor to an unbalance force and the damped

resonance frequencies are calculated by this method

1955 Pinkus, O. He investigated oil whirl in various journal bearing types and made the

following major conclusions. The unbalance of the rotor has minimal

effect on stability. The threshold of instability occurs at approximately

twice the first critical speed of the rotor. In the unstable region, the

whirl frequency remained constant at the first critical speed,

irrespective of the shaft rotating speed. At speeds nearly equal to

three times the first critical speed, whipping motion stops with a

heavy shaft rotor, whereas with a light shaft rotor it does not cease.

High loads, high viscosity, flexible mountings, and bearing asymmetry

favor stability

1958 Lomakin, A. The influence of the dynamic characteristics of seals on the critical

speeds and stability of pump rotors were introduced by him

1958 Thomas, H. He proposed that an eccentric turbine rotor would generate a destabilizing

force due to the circumferential variation in clearance

1966 Gunter, E.J. Jr. He combined the different theories on whirling developed by the rotor

dynamist and the bearing specialist, and elegantly explained some of

the conflicting experimental evidence gathered thus far. He emphasized

the importance of considering the combined effects of rotor

parameters and the bearing and foundation characteristics on rotor

stability, and developed more comprehensive criteria for self-excited

whirl instability

1969 Black, H.F. He provided a comprehensive analysis of annular pressure seals on the

vibrations of pump rotors

1970 Ruhl, R. He introduced finite element models for flexible rotors for calculating

rotor critical speeds and mode shapes. These models did not take

into account gyroscopic effects and axial loading

1974 Lund, J. A transfer matrix method to calculate damped critical speeds of a rotor

taking into account the cross coupling terms as well were introduced

by him

1976 Nelson, H.

and McVaugh, J.

They extended the finite element model of a rotor to account for rotary

inertia, gyroscopic effect, and axial loads

(continued on next page)

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