June 30 2011 -
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<p style="\"margin-left:" 160px\"="">Richard Thomas, P.E.

The following statement is often referred to as the *“1 ^{st} Law of Machinery Diagnostics”*:

**Vibration Displacement (Vector Quantity) = **

**Summation of Forces Acting on the Rotor System (Vector Quantity) divided by the System Dynamic Stiffness (Vector Quantity)**

The 1^{st} Law reminds us that vibration is not a scalar quantity, it is a vector quantity in that it has both a magnitude and a direction. Additionally, the 1^{st} Law acknowledges that if the vibration displacement changes in amplitude, either increasing or decreasing, it is due to either a change in the applied forces to the rotor bearing system, a change in the dynamic stiffness of the rotor bearing system, or both.

Although beyond the scope of this discussion, it can readily be shown that the system dynamic stiffness for a simple Jeffcott rotor can be approximated by:

K* _{Dynamic Stiffness}*= (K-MΩ

In Equation 1, K is the spring stiffness, M is the rotor modal mass, D is the rotor/bearing viscous modal damping, and ω is a term that represents the frequency of the vibration precession (or orbiting), which may be different from the rotation speed, Ω. λ is the Fluid Circumferential Average Velocity Ratio of the fluid in the bearing or seal. The first two terms to the left of the equal sign, contained in the first set of parentheses of Equation 1, are the spring stiffness K (shaft, bearing, support pedestal, etc.) and the mass-inertial stiffness, –MΩ^{2}. The spring and mass-inertia stiffnesses form the *Direct Dynamic Stiffness*. It is called Direct because the *Direct Dynamic Stiffness* acts directly along the line of action of the summation of forces that are applied to the rotor system. Also, the Mass-Inertia Stiffness, –MΩ^{2}, is a speed squared dependent term. When the magnitude of the Mass-Inertia Stiffness, –MΩ^{2}, is equal to the Spring Stiffness K, the *Direct Stiffness* becomes zero; i.e. the undamped natural frequency or critical speed of the rotor / bearing system.

The next two terms, contained in the second set of parentheses, of Equation 1, are the *Quadrature Dynamic Stiffness Terms*, i.e. the Damping Stiffness, +jDω, and Tangential Stiffnesses, –jDλΩ[1]. It is called *Quadrature* because, as indicated by the +j (i.e., square root of -1), acts at 90° to the line of action of the summation of the applied forces.

When a fluid, either liquid or gas, is constrained within the space between two concentric cylinders, one rotating and one stationary, the fluid in the clearance between the two cylinders will be set into circumferential motion. This can happen in rotating machinery within the fluid-lubricated bearings, seals, around pump impellers, or in any fluid filled gap between the rotor and the stator.

In a hydrodynamic, fluid film bearing, the fluid velocity at the surface of the bearing is zero, while the fluid velocity at the surface of the rotor is equal to the rotor surface speed. The fluid near the rotor surface moves at a slightly slower velocity that continues to decrease with the distance from the rotor and reaches zero at the bearing surface. It is easy to see that the fluid must have some overall average velocity which is less than the rotor speed, and that the faster the rotor turns, the faster the average fluid velocity must be. In fact, previous work has shown that the fluid circumferential average angular velocity in the bearing can be expressed as λΩ, where λ is the Fluid Circumferential Average Velocity Ratio, and Ω is the rotor rotative speed (radians/sec), with the value of λ normally less than ½X; i.e., 0.43X to 0.48X.

If a radial load is applied to a rotor operating in the center of a fluid-lubricated bearing, the rotor will be displaced from the center of the bearing. The reduced clearance on one side will restrict the flow of fluid around the journal bearing clearance. Because of this restriction, the fluid has to slow down as the available flow area gets smaller. As the fluid slows down, the pressure in this region increases. The pressure exerts a force on the rotor. This force can be separated into a radial part, which points back toward the center of the bearing, and a tangential part, which acts at 90° to the radial force, in the direction of fluid flow; i.e. shaft rotation. Both the radial and tangential forces are proportional to the displacement of the rotor from the center of the bearing. Thus, the lubricating fluid acts like a spring with a complex stiffness. The radial part of the fluid wedge stiffness is referred to as the *Direct Dynamic Stiffness* and the tangential part of the fluid wedge stiffness is referred to as the *Quadrature Dynamic Stiffness* as defined by Equation 1 above.

It is this complicated spring effect that provides the primary support for the rotor. The rotor and the fluid pressure "wedge" move until the fluid spring forces in the bearing exactly balance the summation of radial loads applied to the rotor and the rotor settles into an equilibrium position.

If the rotor is disturbed from the equilibrium position, the direct spring stiffness acts to return the rotor to the equilibrium position. However, the net force produced by the damping stiffness acting in the direction opposite of rotation and the tangential stiffness acting in the direction of rotation prevents that from happening and the rotor orbits (vibrates) around the equilibrium position at some displacement amplitude.[2]

If the forces that perturb the rotor could be removed, the orbit would spiral back to the equilibrium position. It is the complicated effect of these forces that maintains rotor stability.

The overall system *Dynamic Stiffness* is comprised of a series of springs from the rotor to ground with the weakest (smallest) spring being the fluid film stiffness of the bearing itself. This fact is logical in that it is the weakest spring that will control vibration. Furthermore the resulting *Dynamic Stiffness* resulting from these springs in series will always be less than the stiffest spring in the series.

In the case of the bearing fluid film, vibration energy is absorbed and dissipated via the direct damping term,Dω , which is the rotor pushing on the fluid – the shock absorber effect.

**However, for adequate damping to occur within the bearing, the bearing must be rigidly mounted to the foundation spring and the foundation spring rigidly connected to ground**.

__Any looseness between the bearing and the housing, the housing to the foundation, the foundation to ground will result in unwanted seismic relative motion which will adversely affect damping within the bearing. Any looseness between the bearing and its housing, the housing and the foundation, or the foundation and ground will reduce overall damping which can lead to elevated vibration amplitudes at one or more frequencies.__

It is always important during maintenance to not only verify the internal clearance of a radial bearing but also to verify the "crush" or "interference" fit between a radial bearing and the bearing housing.

Most bearing designs call for a 0 clearance ("line to line") or a sligh crush ("interference") fit of 25.4 to 38.1 μm (1 to 1.5 mil). That being said, spherically seated radial bearings often will have specifications that allow for a clearance "over the top" of the bearing. As long as the rotor vibration levels are low a radial bearing with a clearance fit between it and its bearing housing will often provide satisfactory results. However, if rotor vibration amplitudes become abnormal or excessive, the loose radial bearing will have lower than anticipated direct damping due to the loss of relative motion between the bearing and the rotor as the rotor vibrates and the bearing tends to move in phase with the rotor because it is "loose" in the housing.

From a purely rotordynamic perspective, radial bearings should at all times be firmly secured within the bearing housing so that under all operating conditons, there is no possibility for the bearing to move. Relative motion should always occur within the radial bearing on the fluid film, not between the radial bearing and its housing.

[1] Cross coupled stiffness

[2] *A Short Course in the Practical Application of Rotordynamics as a Tool for Machinery Diagnostics*; Thomas, G. Richard, Paper presented at the Canadian Machinery Vibration Association Annual Meeting / Seminar; 29 October 2009

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