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The root cause analysis of vibration problems related to critical turbomachinery is an engineering science based on what is often referred to as the "First Law of Machinery Diagnostics". The 1st Law states that displacement (vibration) is equal to the summation of the dynamic forces acting on the rotor divided by the complex dynamic support stiffness of the rotor bearing system.
As we look at the 1st Law of Machinery Diagnostics it becomes readily apparent that displacement (vibration) can change due to:
Vibration (filtered at 1X running speed for example) is not a scalar quantity. It is a VECTOR quantity and has both a magnitude (amplitude) and direction (phase angle). It is the VECTOR change in the vibration displacement that is important not simply the change in amplitude.
Consider the following example.
However, when one properly treats the vibration values as vectors, the actual vector change was from 2.0 mil pp @ 0° to 2.0 mil pp @ 180°, a vector change of 4.0 mil pp @ 180°
Determining root cause for machinery vibration problems involves applying the 1st Law of Machinery Diagnostics in conjunction with the utilization of the correct technology(ies) in order to evaluate whether or not it was the dynamic forces acting on the rotor that changed; or whether or not there has been a change in the complex dynamic support stiffness; or both. In order to make this determination, the correct data / information must be acquired. In and of itself, data is of limited value. Value is added, knowledge is obtained, and root cause is determined, when information is converted into knowledge, which leads directly to root cause determination.
In future entries in this section we will discuss the application of engineering principles and the 1st Law of Machinery Diagnostics. We will evaluate and utilize multiple data plot presentations in order to perform root cause analysis.
However, to set the stage, let us conclude this first entry with a short historical review.
The industrialization of North America from 1750-1940 is recognized as a period of rapid and impressive development of machinery technology. The use of steam power and the subsequent development of the railways from 1830 to 1870 led to the early development of the region’s industrial might.
By the 1870’s and 1880’s the westward expansion in the United States had triggered a time of great railroad expansion and growth as well. Durable and efficient railcar wheel bearings were increasingly sought after. The conventional wisdom of the day assumed that, for a plain journal (sleeve) bearing, the load caused direct contact between surfaces of the rotating shaft and the stationary bearing and that the role of the lubricant was simply to decrease friction and remove heat.
In 1883, a committee of the Institution of Mechanical Engineers hired Beauchamp Tower to find the best way to lubricate the bearings in railroad rolling stock wheel and axle assemblies. Tower was a brilliant scientist and investigator. He quickly set about the task at hand.
Tower’s experiments modeled railroad wheel bearings and included a rotating shaft supported by a partial bronze bearing. His tests were conducted with three types of oil lubrication to the partial bearing; a siphon lubricator, an oil pad, a
n oil bath.
The partial bronze bearings used in Tower’s tests were 4 inches in diameter and 6 inches long. Not surprisingly, his results showed that the bearings lubricated via the oil bath had remarkably low coefficients of friction.
In a later test, where the oil to the bearing was to be supplied via a siphon lubricator, Tower prepared the bronze partial bearing by drilling a ½ inch diameter hole in the center of the top half of the bronze bearing. When the test rig was started, and the shaft began to rotate, Tower was surprised to discover that oil flowed up and out of the hole. He tried to block this flow of oil by driving a wooden stopper into the hole. However, when the test rig was re-started, the wooden plug was slowly forced out of the hole. Tower then connected a pressure gauge to the hole and discovered that an internal (hydrodynamic) pressure was being generated within the bearing due to the relative motion of the rotating shaft and the stationary bearing.
Later experiments that Tower conducted measured the pressure at various angular and longitudinal locations around and along the bearing surface. Tower observed that the true oil film pressure was as much as twice the average pressure based on the projected area of the bearing.
Further experiments were made in order to quantify the pressures and pressure profiles throughout the bearing. Tower discovered that the summation of local hydrodynamic pressure in the bearing, multiplied by the projected bearing area was equal to the load supported by the bearing. This was the first time anyone realized that it was actually a pressure profile in the lubricant itself that was directly supporting the radial load in the wheel bearing and not the metal of the bearing itself.
Osborne Reynolds carefully studied the results of Tower’s experiments and soon realized that the observed bearing lubrication was a hydrodynamic action that was dependent on the viscosity of the lubricant. Reynolds also deduced that as the lubricant adheres to both the stationary and moving surfaces of the bearing, relative velocity between the rotor and the bearing resulted in the lubricant being dragged into a rapidly narrowing clearance space, converging in the direction of motion, where it would build up a fluid “pressure wedge” or “pressure profile” sufficient to carry the load.
Reynolds' observations and conclusions regarding the results of Tower’s experiments formed the basis of early understandings of the “wedge” shaped lubricant pressure profile. Because this pressure wedge is built up from an internal source, bearings of this type are properly classified as Internally Pressurized or Hydrodynamic Bearings.
In 1886, Reynolds further explained Tower’s experimental results, when he published the "Reynolds Equation”, which provided the mathematical model for the oil pressure distribution within the bearing. The generalized form of the Reynolds Equation is a three dimensional, partial differential equation. Determining an exact solution, for a given set of parameters, is difficult at best. Traditionally, designers have used various approximate methods to force the Reynolds Equation to yield results.
In 1904, Arnold Sommerfeld, the famous mathematician and physicist, derived approximate solutions under certain conditions for an infinitely long bearing. These are known as half and full Sommerfeld conditions. In 1952, an approximate solution for an infinitely short bearing was derived by Alastair Michell. In 1958, F.W. Ocvirk and G. B. Dubois further investigated the relationship between load capability and the minimum bearing fluid film thickness.
All of these analyses are based on certain assumptions which are: that laminar flow conditions prevail; that liquids are consistent with Newton’s definition of viscosity, e.g. the fluid viscosity is considered to be constant across the fluid film; fluid inertia forces resulting from acceleration of the liquid are small as compared to the viscous shear forces and may be neglected; and that the fluid is incompressible, e.g. the specific volume of the fluid is constant and independent of pressure. 
Problems in the design and operation of rotating machinery have been investigated separately by various researchers. The earliest recorded article on this subject was introduced by Rankine  in 1869. He examined the neutral or indifferent equilibrium conditions of a shaft operating through its first critical speed. Without taking into consideration support damping and Coriolis force, he concluded that the rotor system would be unstable if operated beyond its first critical speed. His conclusion led engineers in the next several decades to believe that rotor systems were not capable of operating beyond their first critical speed. In 1894, Dunkerly  reported his study on rotor dynamic behavior near its critical speed region. He showed that by neglecting unbalance and damping, the rotor whirling frequencies could be replaced by the lateral vibration frequencies of a simply supported system. As a result of his investigations, rotor manufacturers tried to design rotors with rigid shafts such that their critical speeds were raised beyond the operating speed range. Also in 1894, Rayleigh  presented an approximate energy method for determining the natural frequencies of continuous beams.
Timoshenko  applied Rayleigh’s method to rotating shafts and, in 1916, discovered the effects of transverse shear deflection on the natural frequency of a continuous beam. In 1919, Jeffcott  presented the concepts of shaft whirling as they are understood today; i.e., the equation for the whirl path must be expressed in terms of orthogonal force components, and damping, unbalance, elasticity, and mass must be considered.”
In the 1920’s turbine and compressor manufacturers began to design rotors with less stiffness such that the first critical speed of the system was lower than the operating speed range. Later, Newkirk  studied the instability of rotor-bearing systems and concluded that rotor dynamic behavior could not be always attributed to a critical speed resonance and the reduction of unbalance had no effect upon rotor whirl amplitudes. During the same time, a group of investigators concentrated on the study of the hydrodynamic behavior of fluid film bearings.
Using the Sommerfeld treatment, they found that the hydrodynamic profile predicts a negative fluid film pressure of the same order as the positive pressure. This negative pressure causes the fluid film to cavitate and results in radial bearing forces in the system. In 1952, Poritsky  reported that the cavitation of the fluid film can excite the “half-whirl” instability of the system.
Critical speeds of unsymmetrical rotors on unsymmetrical supports were studied in 1933 by Smith , who obtained four critical speeds, as opposed to one for a symmetrical system. Smith also discussed additional critical speeds due to gyroscopic effects of large disks. Foote, Poritsky, and Slade  applied combined unsymmetrical effects to rotating and nonrotating stiffness of a shafting system, with special emphasis on two-pole turbogenerators. The Holzer numerical solution was extended to beam and shaft vibration problems by Myklestad  and Prohl  in the mid 1940’s. The combination of the numerical methods developed by Myklestad and Prohl with another advancement; the electronic digital computer, opened the way for the analysis of complex shafting systems. Green  analyzed the effect of gyroscopic moments on the critical speeds of lumped shafting systems in 1948. In 1954, Yamamoto  presented his study on rotor forward and backward whirl due to gyroscopic effects. His report gave researchers deeper insight in the dynamic behavior of simple rotor systems.
With further development of digital computers, the simulation of complex rotor-bearing systems was made possible. The matrix transfer method of Myklestad and Prohl was fully extended into the analysis of critical speeds, stability, and forced response of a complex turborotors by Lund . Other methods such as the finite difference and finite element approach were also used in the study of rotor dynamics. The rotor system is simulated by the combination of a large number of small elastic shaft elements. Rotor weights and inertia moment effects are lumped at the mass stations, which are the interconnecting node points of the shaft elements. If a large number of node points (or shaft elements) are used, this discrete element approach usually can give a very close approximation of the real system. [16, 17]” 
(Richard Thomas is Principal Engineer, Metrix SETPOINT Contition Monitoring Group; Minden, NV)
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