API Standard ParagraphsRotordynamic Tutorial: Lateral CriticalSpeeds, Unbalance Response,Stability, Train Torsionals, and RotorBalancing - FIRST EDITION

Abstract

This document is intended to describe, discuss, and clarify the API Standard Paragraphs (SP) Section 6.8 which outlines the complete lateral and torsional rotordynamics and rotor balancing acceptance program designed by API to ensure equipment mechanical reliability.

Background material on the fundamentals of these subjects (including terminology) along with rotor modeling utilized in this analysis is presented for those unfamiliar with the subject.

The standard paragraphs are introduced with references to the appropriate background material to enhance the understanding.

This information is intended to be a primary source of information for this complex subject and is offered as an introduction to the major aspects of rotating equipment vibrations that are addressed during a typical lateral dynamics analysis.

It is not intended to be a comprehensive guideline on the execution of rotordynamics analyses but is intended to:

a) provide guidance on the requirements for analysis;

b) aid in the interpretation of rotordynamics reports;

c) provide guidance in judging the acceptability of results presented.

Organization

The document is divided into six sections:

1.

Overview

2.

Lateral Dynamic Analysis

3.

Stability Analysis

4.

Torsional Analysis

5.

Balancing of Machinery

6.

Standard Paragraphs

The individual sections have been prepared in a stand alone manner.

As a result, necessary material may be repeated in a succeeding section to provide sufficient clarity to the discussion.

Sections two through four have a parallel organization:

— Modeling criteria

— Analysis techniques and results

— Machine specific considerations

— Testing

— Applications and examples

In order to aid turbomachinery purchasers, the American Petroleum Institute’s Subcommittee on Mechanical Equipment has produced a series of specifications that define mechanical acceptance criteria for new rotating equipment.

Experience accumulated by turbomachinery purchasers over the past ten years indicates that if the API standards are properly applied, the user can be reasonably assured that the installed unit is fundamentally reliable and will, barring problems with the installation and operator misuse, provide acceptable service over its design life.

An integral component of these individual equipment specifications is contained in the API Standard Paragraphs, those specifications that are generally applicable to all types of rotating equipment.

The criteria associated with lateral and torsional rotordynamics and balancing have been categorized as standard paragraphs.

In rotating equipment specifications published by API (for example, API Standard 617—Axial and Centrifugal Compressors and Expandercompressors for Petroleum, Chemical and Gas Industry Services) there is a section on rotordynamics and balancing.

The backbone of those sections is the standard paragraphs augmented by additional information that is applicable only to the type of unit considered in the standard.

The Dynamics Standard Paragraphs originated in the Centrifugal Compressor Standard, API 617.

A timeline for the first through eighth editions with major highlights is [1]:

— 1st Edition June 1958—dynamics section added.

A critical speed separation margin defined and a balance/ vibration criterion established.

Vibration limit in operating speed range for mechanical test is a step function.

— 2nd Edition April 1963—torsional analysis added as an option.

— 3rd Edition October 1973—balance criteria and separation margin for laterals below operating speed range adjusted, rotor response required when specified, torsional analysis required on motor driven and geared units, vibration limit in operating speed range changes from step function to (12,000/MCOS)0.5 or 2.0 mils whichever is less and vibration limit added at trip speed.

— 4th Edition November 1979—AF (amplification factor) limited when passing through lateral critical speeds, lateral analysis required including a rotor response and transient torsional analysis for all synchronous motor-driven units.

— 5th Edition April 1988—removed limits on AF while passing through lateral critical speeds, acceptance defined basis rotor response analysis, lateral analysis report requirements spelled out in detail, operating seal clearance criteria added, separation margin for laterals tied to amplification factor, amplification factor less than 2.5 acceptable in any operating case, a dynamics shop test included, and balance criteria modified.

— 6th Edition February 1995—lateral paragraphs reorganized and criteria set for comparison with shop test evaluation and transient torsional required with variable speed motors, and vibration limit in operating speed range changes to (12,000/maximum continuous speed)0.5 or 1.0 mils whichever is less (had been 2.0 mils).

— 7th Edition July 2002—formula for allowable separation margins for lateral criticals changes, unbalance response test criteria clarified, stability analysis requirement included, and entire section updated.

— Proposed 8th Edition—active magnetic bearing specifications expanded including stability criteria, inclusion of damper seals into lateral analysis, high-speed balance specifications revamped, requirements for lateral and torsional reports added and influence of VFD, generators and motor faults and excitations on torsional analysis.

The complete text of the Dynamic Section of the Standard Paragraphs is included at the end of the document.

Definitions and References

Definitions are incorporated into each section of the document.

Due to very large number of references employed, they are identified at the end of each relevant section.

Fundamental Concepts of Rotating Equipment Vibrations

In order to understand the results of a rotordynamics design analysis, it is necessary to first gain an appreciation for the physical behavior of vibratory systems.

Begin by noting that all real physical systems/structures (such as buildings, bridges, and trusses) possess natural frequencies.

Just as a tuning fork has a specific frequency at which it will vibrate most violently when struck, a rotor has specific frequencies at which it will tend to vibrate during operation.

Each resonance is essentially comprised of two associated quantities: the frequency of the resonance and the associated deflections of the structure during vibration at the resonance frequency.

Resonances are often called “modes of vibration” or “modes of motion,” and the structural deformation associated with a resonance is termed “mode shape.”

The modes of vibration are important only if there is a source of energy to excite them, like a blow to a tuning fork.

The natural frequencies of rotating systems are particularly important because all rotating elements possess finite amounts of unbalance that excite the rotor at the shaft rotation frequency (synchronous frequency) and its multiples.

When the synchronous rotor frequency equals the frequency of a rotor natural frequency, the system operates in a state of resonance, and the rotor’s response is amplified if the resonance is not critically damped.

The unbalance forces in a rotating system can also excite the natural frequencies of nonrotating elements, including bearing housings, supports, foundations, piping, and the like.

Although unbalance is the excitation mechanism of greatest concern in a rotordynamics analysis, unbalance is only one of many possible lateral loading mechanisms.

Lateral forces can be applied to rotors by the following sources: impeller aerodynamic loadings, misaligned couplings and bearings, rubs between rotating and stationary components, and so on.

A more detailed list of rotor excitation mechanisms of particular interest is found in the API Standard Paragraphs, 6.8.1.1.

This subject is discussed in detail in 3.5, as well as scattered in the appropriate sections of this document.

The vibration behavior of a rotor can be described with the aid of a simple physical model.

Assume that a rotor-bearing system is analogous to the simple mass-spring-damper system presented in Figure 1-1.

NOTE 1 Multiply the quantities listed above (in US Customary Units shown) by the US to SI conversion factors to obtain the quantity in the SI units listed in the table.

NOTE 2 NA = not applicable.

In this example, the displacement response of the block to the applied force is counteracted by the block’s mass and the support’s stiffness and damping characteristics.

The undamped natural frequency of this system is calculated by determining the eigenvalue of the second order homogeneous ordinary differential equation (F = 0) for the case where the damping term is neglected (c = 0) as seen in Equation 1-3.

Since real, physical systems include damping, this needs to be included in the analysis.

The damped natural frequency of the homogeneous system (F = 0) is defined in Equation 1-4.

If the system was excited (hit by a hammer), this damped natural frequency is the frequency of vibration that would be seen as the system responds.

Note that the damped natural frequency of the system, ωd, is equal to the undamped natural frequency of the system only when system damping is negligible.

This observation underscores the fact that an undamped critical speed analysis should, in general, not be used to define the critical speeds of a rotating machine.

In the case of a rotor with unbalance, the unbalance force is defined in Equation 1-7:

This result is called “forced response analysis” and is analogous to the unbalance response analysis performed in rotordynamics studies.

The amplitude ratio depends upon frequency of the excitation and the damping in the system.

Figure 1-2 shows the amplitude ratio versus the excitation frequency.

Maximum amplitude ratio is seen where the excitation frequency equals the natural frequency of the system.

Amplitude ratio also increases as damping decreases with the amplitude becoming infinite at zero damping (a situation that is not physically practical).

There is a phase difference between the excitation and response.

This phase difference is a function of damping and reaches 90 degrees at the natural frequency.

Figure 1-3 shows the phase angle versus excitation frequency.

If a transient rather than a sinusoidal excitation excites the system, the actual response normally looks like that shown in Figure 1-4.

In this case, the response is at the natural frequency.

It decays with time based on the amount of damping.

This response is called “stable.”

In Figure 1-5, the response grows with time.

This response is called “unstable.”

While the simple, single degree of freedom system described above is useful for examining the general concepts of vibration theory, this system is clearly not representative of a turbomachine.

Recognize that the early development of rotor dynamics analysis took place in the late 19th to early 20th Centuries and calculations could only be performed with pencil and paper.

Thus, it was required to utilize sound, simplifying assumptions to permit solutions to be performed.

In 1869, Lord Rankine published an analysis that concluded operation of a rotating shaft above its first resonant speed was impossible.

This reflected the common feeling of many of the learned mechanics experts of the day.

However, in 1889, Gustav Laval built a steam turbine which he operation above the first resonance.

It was apparent that a more accurate representation of a rotating assembly was required.

Henry Jeffcott (under charter of the British Royal Society) performed an analysis with the results published in 1919.

In that analytical work, he proposed the model as shown in Figure 1-6.

It is a two degree of freedom model with all mass concentrated in a central disk.

The disk is mounted on a massless flexible shaft and It is connected to ground by rigid supports.

This model may be analyzed and solved to indicate the presence of the first undamped critical speed.

This form of model is known as the Laval (Europe) or Jeffcott (outside Europe) model.

The Laval-Jeffcott Model is used extensively to gain insight into the potential rotor dynamic performance of rotors.

While it is able to provide simple, qualitative results, more complete and comprehensive models were required.

The next level of complexity added was to replace the rigid shaft supports with springs as shown in Figure 1-7.

Then, damping was added Figure 1-8 and Figure 1-9.

As computing capabilities began to be available, more comprehensive, accurate models were developed.

These are described in considerably detail in the rest of this document.

In order to consider the effect of the combined shaft and bearing stiffness, we need to review the effective combined stiffness as shown in Figure 1-7.

As shown, the effective stiffness, Keq, is the result of the individual stiffnesses added.

The equation that describes this is:

This equation indicates that the stiffness of the combined shaft-bearing system will be less than the stiffness of the single most flexible element.

In this example, the shaft is the single most flexible element (Kshaft = 8756.5 N/mm or 50,000 lbf/in.).

According to Equation 1-8, the effective stiffness of the combined shaft-bearing spring system is only 7297.7 N/mm (41,670 lbf/in.).

To carry this analysis one step higher in complexity, consider the sketch of a rotordynamic system displayed in Figure 1-8.

Figure 1-9 shows the model with masses, springs, and dampers.

Note that this system is identical to the system just discussed except that viscous damping elements have been added to the bearing model.

All oil film bearings generate significant viscous damping forces.

Figure 1-10 displays the calculated response of the disk to a harmonic load acting at the disk for various values of shaft stiffness.

Note that as the shaft stiffness decreases, the peak response frequency decreases while the amplitude of the peak response and the sharpness of the peak both increase.

These observations are understood by noting that the decrease in shaft stiffness decreases the relative deflection of the shaft in the bearings and diminishes the magnitude of the damping forces provided by the bearings.

Thus, one may conclude that the effect of damping provided by the bearings is maximized when the shaft stiffness is large relative to the bearing stiffness.

These general concepts of vibrations will be demonstrated in considerably more detail in the sections that follow.