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

In my previous article, covered the various API standard section references for rotor dynamic analysis of different centrifugal machines (pumps , compressors , steam and gas turbines) and different rotating elements connecting the driver and driven equipment (Coupling and gears). In this article further cover the fundamentals of the how to perform the rotor dynamic analysis (Lateral Analysis) of rotating machine drive as per the API RP 684.

API RP 684 forms the basis of rotor dynamic analysis (lateral, stability and Torsional). The purpose of API RP 684 is to give the person who is performing and reviewing the calculations, 1) Provide guidance on the requirements for analysis , 2) Aid in interpretation of rotor dynamic reports and 3) Provide guidance in judging the acceptability of results presented.

The API RP 684 divided mainly into five different parts Introduction and overview , Lateral Rotor dynamic analysis , Stability analysis, Torsional analysis and Balancing of Machinery. Three steps are important to achieve the accurate results about the excitation frequency and amplitude values and unbalance location during the analysis and design review of machine. They are 1) Modeling Criteria , 2) Analysis techniques and Results and 3) Machine specific considerations.

Introduction and Over View

The first part of Introduction and Overview gives an idea about the rotating shaft model and how the analysis is performed. Basically the rotating shaft is considered as the spring mass system supported at it is end. The mass is basically a disc or impellers of the compressors , turbine or pump mounted on the rotor and connected by the spring elements like seals and diaphragms in between and supported by the bearing at the ends.

The above figure shows the simple spring mass system. When the force is applied on the unbalance mass it excites the system causing displacement . When the excitation frequency matches with natural frequency of simple spring mass system the maximum displacement occurs. This state is called resonance. From the governing mechanics equations of the above system is as shown below. From the governing equations the natural frequency can be derived.

From the above equations both undamped and damped critical frequency is depend the stiffness and mass , damping co-efficient in case of damped system. The stiffness and damping co-efficients will be discussed in details in the jeffcott rotor model. The simple rotor model is not sufficient for the actual rotor models. Because the actual rotor is not a single degree of freedom system as shown in the simple spring mass system. The stiffness characteristics varies widely in longitudinal and transverse directions. And Jeffcott rotor model brings lot of additional characteristic values and understanding about the Rotor models with multiple degree of freedom. The jeffcott rotor model is shown in the figure.

The stiffness used to calculate the natural frequency is combination of shaft stiffness , bearing stiffness and bearing housing support stiffness , foundation stiffness. So change in any one of the stiffness has an influence on the excitation frequency. The shaft stiffness can be calculated by using the solid mechanics formulas depending on the support. Mainly it depends on whether the disc mass is overhung or simply supported between the bearings .

The another important factor is damping coefficient. In simple terms damping force is the resistance offered by the system against the excitation force and shaft displacement due to excitation force and frequency. The damping is offered by the bearings and seals, diaphragms and other components in the rotor system. The damping co-efficient is one of the most important factor which decides the amplification factor of the induced excitation. The amplification factor is the ratio between the damping co-efficient of the system to the critical damping co-efficient of the system. The various useful relations are given below.

From the above formula it is evident that the critical damping co-efficient also depends on the stiffness of the rotor. Damping properties of the system can be modified by changing the bearing characteristics and clearances between seals, diaphragms in the rotor systems. The above paragraphs are basic to understand the rotor system. Now in the next paragraphs will see how the lateral critical speeds are calculated and analyzed and typical outputs from lateral critical speed analysis and its inferences.

Lateral Critical Speed Analysis:

There are many mechanisms involved in the lateral vibration of rotor. The main cause of loading is the unbalance in the rotor system. However the impeller aerodynamic loading, misaligned couplings, bearings and rubbing between the stationary and rotating part is also the other possible reasons for the lateral vibration of the machine.

Undamped Critical Speed Analysis:

Lateral critical analysis first starts with the Undamped critical speed analysis. It is an preliminary analysis to understand the general dynamic behavior of the machine. The analysis is performed without considering damping and unbalances masses in the system. The output of this analysis is undamped critical speed map. The undamped critical speed map is the plot between the Critical speed in rpm or cpm versus the stiffness of support. The typical undamped critical speed is shown below.

The excitation of the machine occurs at the point in which the intersection of stiffness line with critical speed line. Means that the vibration amplitude is maximum at this point. The vibration behavior is depends on the stiffness of the rotor. Whether it is an stiff rotor or flexible rotor. The stiff rotor is the rotor system in which the shaft stiffness in very very high compared to the bearing, bearing support stiffness. In the flexible rotor system it is the inverse of stiff rotor, the bearing , bearing support system is very very high compared to the shaft stiffness.

The advantages of the unbalanced response analysis is 1) It is planar and two dimensional , 2) It provides an preliminary location of excitation points and where the unbalance mass need to be placed during the further analysis to understand the machine behavior.

Damped Unbalance Response Analysis:

The damped response analysis is performed to understand the actual vibration amplitudes expected from the machine considering the damping effects. The objective of this analysis to determine whether the machine meets the various API standard requirements indicated in the respective standards.For different requirements refer to the link below (previous article).

Typically the following are the minimum points covered.

a) Adequate separation margin between the critical speeds and operating speeds

b) The probe vibration limit is not exceed with in the specified operating speed range even with twice the maximum allowable residual unbalance present

c) No rubbing will occur even if the rotor's balance state degrades to the probe vibration limit

The unbalance mass location for this analysis is selected based on the output mode shape of Undamped critical speed analysis. The distribution of this unbalance becomes as important as its amount. This is due to the fact that to excite a natural frequency of any system, the forces must not be at the node points of the natural frequency's mode shape. All modern day damped unbalance response analysis are performed by using the algorithm developed by Lund and Orcutt.

The Output of the Damped unbalance response analysis is Bode Plot. Bode plot is the plot between the shaft displacement from it is Centre line vs speed of the machine. The pecks in the bode plot's are the critical speeds of the machine.The sample bode plot of charge gas compressor from API RP684 is shown below.

In general the vibration amplitude shall not exceed the 75% of clerance limit between the rotor and stationary part as established in API Standard paragraphs.

Stability Analysis:

The stability analysis is performed to understand the behavior of shaft displacement with respect to time when it is excited. It requested to understand whether the vibration is stable or unstable. Means amplitude of vibration decreases with time it is stable vibration, if it increase with time then it is unstable vibration. I will try to cover the stability analysis more in detail in the next article.

Machine Specific Considerations:

1) Steam Turbine: For Steam turbine bearing support flexibility and partial steam admission force are specifically to be considered and impact on the outcome of the lateral critical speed analysis.

2) Electric Motors: In electric motors defining the winding mass and diameter in the rotor dynamic model is difficult. This needs special consideration to get accurate results.

3) Gear Box: In gear boxes the gear loading pattern and tooth mesh frequency has a significant roles in the outcome of the analysis.

4) Power Turbines and FCC Power recovery expanders: The inlet temperature are close to 1200 deg F. The stiffness and damping co-efficient's shall be considered at this temperature. Mostly the rotors of these machines are stiff and their first undamped critical speeds are well above the operating speed.

5) Axial Compressors: The modelling techniques depends on the type of shaft construction. Typically the rotors are constructed in four different ways. They are disc-on-shaft, shrink fit, stacked disk with tie bolts, drum rotors with studs or tie-bolts and solid rotors. Generally rotor stability is not a problem for these machines.

6) Centrifugal Compressors:

a) Multistage Compressor: The damping of fluid films of gas seals, bearing oil seals and bearing damping has important influence on the critical speed , stability of machine.

b) Overhung Compressor: The gyroscopic effects also known as "Morton's effect" to be considered in the analysis of overhung compressor rotors.

c) Integrally Geared Compressor: It is like overhung machine analysis but the bull gear with additional loading to be considered.

API Testing and Results:

The testing is performed as required by various API machine standards to compare the actual vibration levels with the estimated levels from the lateral critical analysis. Some points related remembered while performing and comparing the unbalance verification testing.

a) Residual unbalances and forces still exist in the rotor.

b) Only few choices exist for the addition of trial weight in the rotor at shop test bench. The one possible location is coupling. If the coupling space and size is small cannot add more trial weight. At this condition it is difficult to excite the first mode.

c) Applied trial weight is normally limited in size.

d) There are some risks which should be avoided during the testing. The elevated vibration levels in testing may be due to in accurate testing or calculations. In order to prevent the machine damage test should be stopped immediately.

e) The limitation in measurment capabilities at test bench , rotors balanced in high speed balancing facilities adding weight at mid span, quarter span and other combination not possible in the test bench.

In Section 2.10 of the API RP 684 titled Standard paragraphs is used in most of API machine standards. This section provide the formula's for calculating the Unbalance mass, Separation Margin and correction factor to be applied against the calculated peak to peak amplitude.

There are numerous Journal references given in API RP684 under each section. Please refer to them on case by case basis to discover further about the individual topics and deepen the

I will cover the Stability and Torsional analysis basics and methodology in next article.

References:

API RP 684 Second Edition