Rotor Stability Analysis; A Review on Tools and Methods

To start from the end and according to Mr. Bently [1];

Fact #1: Because the Logarithmic Decrement presents the ratio of the two eigenvalue components, it loses information. The Root Locus Plot is superior to the logarithmic decrement because it displays both components of the eigenvalue. It allows us to see how rapidly a particular eigenvalue approaches the instability threshold with changes in either rotor speed or other parameters. At the same time, it allows us to see how the natural frequencies of the system are changing.

Fact #2: The Campbell Diagram presents only frequency information, and static frequency information at that; it would be desirable to be able to see stability information on the same plot. The Root Locus Plot presents eigenvalue data that represents both stability and frequency information, and it shows how natural frequencies change with rotor speed.

From the two above-mentioned facts, following conclusions can be made;

1. There are three methods for rotor system stability analysis; Logarithmic Decrement, Campbell Diagram and Root Locus Plot;
2. The Root Locus Plot is the preferred rotor system stability analysis tool compared to the other two methods.

All the three methods are reviewed here, in conjunction with the concept of stability (of a rotor system).

Stability

In terms of social and human sciences, stability means “a situation in which things happen as they should and there are no harmful changes” or more interestingly, “a condition in which someone’s mind or emotional state is healthy” [5]. These definitions are pretty much close to stability of a rotor system.

There are other definitions of stability relating to being stationary or not moving or staying in the same state, which is not true for a rotor system. The main characteristic of a rotor is that it rotates!

For a rotor system, particularly, there are a few different definitions of stability, all leading to the same result;

• A mechanical system is unstable if, when it is disturbed, it tends to move away from the original equilibrium condition [1];
• A rotor system is unstable when the destabilizing forces exceed stabilizing (damping) forces [2].

Rotor System Model

Detailed derivation and solution of rotor system model equations are beyond the scope of this article. Only a summary is presented here with reference to [1].

Rotor system equation of motion for free vibration is given by Equation (1). In this equation, M, D and K are system mass, damping and stiffness and the term K-jDλΩ is the tangential stiffness, in which λ is fluid circumferential average velocity ratio and Ω is rotor speed.

Substituting r as a general position vector solution in the form of Equation (2) into the equation of motion leads to the characteristic equation of the rotor system as presented by Equation (3).

Roots of the characteristic equation are eigenvalues of the rotor system given by Equation (4) and equation (5). Eigenvalue components, γ₁, γ₂ and ω_d are complicated functions of M, D, K, λ and Ω. γ is the growth/decay rate and ω_d is the undamped natural frequency.

The complete vibration of the rotor system is given by Equation (6), consisting of r₁, the forward precession and r₂, the reverse precession.

The growth/decay rate for each eigenvalue can be calculated from Equation (7) and Equation (8).

Rotor Stability Criteria

If the total vibration is decaying, the rotor system is stable and if it is growing, the rotor system is unstable (Figure 1). Growing and decaying of rotor system vibration is examined by the growth/decay rate;

• γ < 0: The rotor system is stable;
• γ = 0: The rotor system is at the threshold of instability;
• γ < 0: The rotor system is unstable.

Figure 1: Vibration with damped critical speed [3]

Logarithmic Decrement

Logarithmic Decrement has a simple definition. It is the natural logarithm of the ratio of two consecutive vibration peak amplitudes:

Based on this simple definition, rotor stability criteria can be derived;

• If any vibration peak amplitude is smaller than its previous peak amplitude (A_n+1<A_n), that means the vibration is being damped or dissipated and rotor system is stable. Because the amplitudes ratio is bigger than 1, its natural logarithm is positive. Hence, rotor system stability is achieved when δ>0.
• If the vibration amplitude is growing in each time period (A_n+1>A_n), the rotor system will be unstable and with the amplitudes ratio smaller than 1, the natural logarithm will be negative. Therefore, rotor system is unstable when δ<0.

Logarithmic decrement can be expressed in terms of the ratio of rotor system eigenvalue components [1];

This expression is helpful in noticing the relationship between rotor stability criteria by logarithmic decrement and by growth/decay rate. The negative sign denotes that if γ<0, then δ>0 and rotor system is stable. Similarly, if γ>0, then δ<0 and rotor system is unstable.

Campbell Diagram

The Campbell Diagram is a representation of a rotating machinery (train) rotor system natural frequencies or resonant speeds (on the vertical axis) versus all the excitation frequencies or operating speeds of each individual rotor (on the horizontal axis).

A sample Campbell Diagram is shown in Figure 2. In this Campbell Diagram, both lateral natural frequencies (solid lines) and torsional natural frequencies (dotted lines) are included. The diagonal line denotes 1X or synchronous vibration.

Figure 2: A sample Campbell Diagram

The Campbell Diagram is to be used in the following way:

1. Find the point of intersection of a horizontal line (or a natural frequency) with the 1X line;
2. Ideally, no vertical line (or an excitation frequency) shall be in the vicinity of the point of intersection.

It is recommended that 2X and 3X diagonal lines be shown on the Campbell Diagram as well, and similar intersections be checked.

The problem with the Campbell Diagram is that the natural frequencies shown by the horizontal lines are associated with the situation that the machinery is at its operating speed. The Campbell Diagram, while useful, represents an oversimplification of rotor response [1];

• Many damped natural frequencies in a rotor system change with rotor speed;
• The Campbell Diagram presents only static frequency information;
• It would be desirable to be able to see stability information on the same plot.

Root Locus Plot

It can be seen from equation (7) and Equation (8) that the growth/decay rate is a function of rotor speed. This is true for the other eigenvalue component, damped natural frequency, too. This means that the vibration behavior of the system changes with rotor speed [1].

The Root Locus Plot is a representation of eigenvalue components, γ and ω_d on the horizontal and vertical axis of an XY diagram respectively (Figure 3). Any point on the plot is associated with a specific rotor speed. Connecting all the points, gives the root locus plot of the rotor system.

Figure 3: A sample Root Locus Plot [4]

From the stability criteria, it is evident that the left half plane (γ<0) is the stable operating region and the right half plane (γ>0) correspond to unstable rotor system behavior. On the root locus plot, it is easy to follow how the eigenvalues, and mainly the growth/decay rate, change with rotor speed and which rotor speed corresponds to the threshold of instability.

For the sample rotor system with the Root Locus Plot is shown in Figure 3, the threshold of instability is 1050 rad/s (Figure 4) which is caused by the forward precession. The further left on the plot that the root exists, the more damping exists to control the decay rate of the rotor motion. The reverse root poses no threat to the stability because it moves toward left of the plot [4].

Figure 4: The sample Root Locus Plot; forward precession [4]

At this point, the two facts at the beginning of this article are worth reading again.

References

[1] Bently, Hatch, 2002, “Fundamentals of Rotating Machinery Diagnostics,” Bently Pressurized Bearing Press.

[2] Moore, “Rotordynamics Tutorial: Theory, Practical Applications and Case Studies,” Southwest Research Institute.

[3] Gunter, 2001, “INTRODUCTION TO ROTOR DYNAMICS - Critical Speed and Unbalance Response Analysis,” RODYN Vibration Analysis, Inc.

[4] Grant, 1999, “Root locus analysis – An excellent tool for rotating machine design and analysis,” ORBIT Second/Third Quarters 1999.

[5] Macmillan Dictionary, www.macmillandictionary.com.