Introduction to Rotor Dynamics Analysis as API Standard related to Centrifugal Machines -Part III-Overview of API RP684 - Stability Analysis

In this article will present an overview about the rotor stability analysis. It is a continuation to lateral critical analysis. The stability analysis is also called as Damped eigen value or damped natural frequency analysis. This is performed to check the effectiveness of the the damping present in rotor system in order to limit the excitation of rotor system due to various forces.

The solution for differential equation of vibration model of single degree freedom system with shaft stiffness (K) and damping co-efficient (C) will result in the complex equation for the displacement. The complex equation contains two parts real and imaginary. In damped eigen value or damped natural frequency analysis real part of the eigen value is studied. The rate of decay of oscillation is the amount of damping present in the system. The stability analysis is conducted in the two level's Level 1 and Level 2. Before going into details first it is important to know some basic terminologies like log decrement, cross coupling forces. The formula and the lograthamic decrement is shown below. For stable systems with positive rate of decay the log decrement values will be in positive. I.e The Vibration amplitude will decrease with respect to time. For unstable systems with negative rate of decay the log decrement values will be in negative. i.e The vibration amplitude will increase with respect to time.

The stability analysis becomes an important part of rotor dynamic analysis and design due to major studies on failure cases of compressor in 1960's and 1970's. The papers presented by the KayBob, Smith [1], Ekofisk, Booth [2] received the attention of the industry about the importance of performing the stability analysis. The paper presented by the Ruhl and Booker [6] published in 1972 gave the detailed algorithm for performing the stability analysis using both transfer matrix, Finite element analysis techniques and called it as Muller's method.

The various factors are that are causing the stability problems include aerodynamic cross coupling, Internal friction, Rotating stall, Entrapped fluids , synchronous thermal instability also known as "Morton's effect". Many development and research activities are going on to define all these effect in accurate computational modeling and simulations during the rotor dynamic studies supplier experience to be taken into consideration with respect to similar kind of equipment design conditions and field operating experience while defining and agreeing upon the log decrement values , safety margin between the Purchaser and Supplier.

The aerodynamic cross coupling is one of the major factor causing the sub-synchronous instability. The aerodynamic cross coupling is defined as the interaction of fluid forces with structural surfaces (both rotating and stationary). It is mainly created due to the tip leakage flows , secondary flow in an impeller shroud or stator cavity flows. The aerodynamic cross coupling varies depending on the type of seals.

The Alford developed an equation to calculate the aerodynamic cross coupling for the axial stages.

The Wachel developed an equation to calculate the aerodynamic cross coupling for the centrifugal stages. It is also known as the Modified Alford's equation.

Due to the limitation on the computation ability in the 1970's and 1980's the experience plot's were developed. The databases available at that time were used to develop the experience plot's which acted as an primary screening criteria to define the level of stability analysis required for centrifugal machines and acceptance criteria.There were four different experience charts were developed , first one from Sood ,after that from Fulton , Kirk and Shemeld including addition details and improvement of Sood experience plot.

Sood's was the first rotor stability criteria , but the problem in using it is it does not contain any scale. The next one with Sood / Fulton included scale. i.e - it is little bit modified from original sood's plot with more scale values on both axis based on the data available at that time.

The Kirk further developed an experience plot based on the Critical speed ratio and Pressure parameter.

After Kirk the Shemeld developed an experience map further using the Fulton criteria with additional details.

Standard Paragraphs:

The standard paragraph sections of the API RP 684 is replicated in the API 617 paragraphs for the Level 1 and Level 2 stability analysis. The level 1 stability analysis is performed to demonstrate in more conservative way that the Level 2 analysis is not required and adequate safety margin's are exist to contain the sub synchronous stability.

Some important points about the Level 1 Analysis:

• If the First critical speed of the machine is above the maximum continuous speed of the machine the stability analysis is not required. For other centrifugal compressor, steam turbines , axial rotors stability analysis shall be performed by the supplier.
• When the tilt pad bearings are used the analysis shall be performed with synchronous tilt pad co-efficients.
• The cross coupling forces shall be calculated using the Modified alford equation for centrifugal compressor and alford equation for Steam turbine , Axial flow rotor's.
• The applied cross coupling for the Level 1 stability analysis is from Zero to Minimum of

1) A level equal to ten times the anticipated cross coupling forces Qa (or)

2) The amount of cross coupling required to produce Zero log decrement

Level 1: Screening Criteria

The Level 2 analysis shall be performed considering the detailed cross coupling forces and individual elements in the rotor system if the follow criteria's are meet / established from Level 1 stability analysis.

1) For Centrifugal Compressors;

• if Qo/Qa < 2.0, Then Level II analysis is required.
• if Log decrement is < 0.1, Then Level II analysis is required
• if 2.0 < Qo/Qa < 10 and CSR in Region B otherwise no Level II analysis is required.

2) For Steam Turbine and Axial flow rotors;

• if Log decrement is < 0.1 , Then Level II analysis is required

Level II Stability Analysis:

In Level II stability analysis all the sources which will cause the synchronous and sub-synchronous instability in the rotor system including labyrinth seals , balance piston, impeller /blade flow, shrink fits and shaft material hysteresis. The frequency and log decrement of 1st forward damped mode shall be calculated for the following condition.

a) Rotor and support system only

b) For the addition of each group of de-stabilizing effects utilized in the analysis

c) Complete model including all destabilizing forces (Final log decrement).

The Acceptance criteria for the Level II stability analysis:

• If final Log decrement value is greater than 0.1, Then rotor stability is acceptable.
• If not then the vendors experience on similar kind of equipment, its field operating experience shall be considered by Purchaser and final acceptable log decrement to be agreed between the Supplier and Purchaser to demonstrate that the rotor system has adequate safety margin exist in rotor system.

Based on the studies and analysis the stability problems in the rotor can be solved by Decreasing/ Eliminating the excitation forces or Increasing the effective damping of system. The ways to increase the effectiveness of damping is described in table 3.9.3 of the API RP 684 and decreasing/ eliminating the excitation forces is described in table 3.9.1.This can be used as an guideline / starting point.

The stability of testing of the Machine can be performed at machine test bench by conducting the Full pressure , Load and Speed test such as PTC Type I. Some machine specific consideration and sub-synchronous stability frequency's are given below.

• The sub synchronous in-stability caused at exactly 50% of synchronous speed is due to dry friction rub. It is also called as "Rub Induced Instability".
• The in stability of rotor system observed at 80 to 95% of the synchronous speed is due to entrapped fluids in the rotor system.
• The in stability in the rotor system at 5 to 20% f the shaft speed near the surge line is due to Rotating stall.
• The in stability in the rotor system at 1x speed is due to thermally induced shaft bending . Also known as "Morton's effect".
• The vibration instability induced at 30% to 50% of synchronous speed is due to fluid induced instabilities [ Oil Whirl and Whip].

Note: The Figures and Formula's in the Article are from API RP 684 . It is simplified an overview and guideline based on API RP 684 stability analysis chapters. For more information/inferences reader's are requested to read the Individual sections and Reference bibliography given under each section of stability analysis.