Tutorial on the API Standard Paragraphs Covering Rotor Dynamics and Balancing An Introduction to Lateral Critical and Train Torsional Analysis and Rotor Balancing API PUBLICATION 684 FIRST EDITION, FE[.]
Scope
This document is intended to describe, discuss, and clarify
Section 2.8 of the API Standard Paragraphs(SP) (Revision
Section 2.8 details the rotor dynamics acceptance program established by API to ensure the mechanical reliability of equipment Prior to delving into the standard paragraphs, foundational information on rotor dynamics, including key terminology and rotor modeling, is provided for readers who may be new to the topic This introductory material serves to highlight the essential aspects of rotating equipment vibrations that are typically examined in a lateral dynamics analysis, rather than acting as a comprehensive resource on this intricate subject.
Introduction to Rotor Dynamics
The mechanical reliability of rotating equipment is significantly influenced by the decisions made by both the purchaser and vendor before manufacturing Equipment designed with advanced computer-aided engineering methods tends to have fewer issues compared to those without such analysis Even if mechanical acceptance tests are conducted before delivery, discovering design-related problems during these tests can disrupt the unit's cost and delivery schedule Therefore, it is crucial for purchasers to require a design analysis and review prior to construction, as relying solely on mechanical acceptance tests may lead to accepting equipment that could cause problems post-installation.
In order to aid turbomachine purchasers, the American
Petroleum Institute’s Subcommittee on Mechanical Equip- ment has produced a series of specifications that define me- chanical acceptance criteria for new rotating equipment.
Over the past decade, turbomachine purchasers have found that proper application of API standards ensures a fundamentally reliable installed unit As long as there are no installation issues or operator misuse, users can expect acceptable service throughout the unit's design life The API standards serve as the foundation for these equipment specifications.
Standard Paragraphs, those specifications that are generally applicable to all types of rotating equipment The rotating
Tutorial on the API Standard Paragraphs Covering Rotor Dynamics and Balancing: An Introduction to Lateral Critical
And Train Torsional Analysis and Rotor Balancing
SECTION 1—ROTOR DYNAMICS: LATERAL CRITICAL ANALYSIS
The IHS-reproduced guidelines outline a comprehensive design and machine evaluation program aimed at ensuring that vibrations from equipment remain within acceptable limits during normal operation This three-phase program includes: a) modeling and analysis of the proposed or pre-manufactured design, emphasizing the need for significant features in the computer model to accurately reflect the dynamic behavior of the assembled unit; b) evaluation of the proposed design with established acceptance and rejection criteria; and c) shop testing and evaluation of the assembled machine to verify performance.
Acceptance/rejection criteria are offered for the completed machine, based on rotor vibrations measured during the me- chanical acceptance test.
Implementing the program according to the relevant API standards allows turbomachine vendors to address most design issues before manufacturing With the adoption of the updated API acceptance criteria for rotor-bearing system designs, there has been a significant reduction in the occurrence of design-related vibration problems during unit testing.
The responsibility of the purchaser is to ensure the proper implementation of the plan by the equipment manufacturer as outlined by the specific API standards.
The introduction consists of three key components: a concise glossary of essential terminology with definitions and discussions, a tutorial covering the basic principles of vibrations in rotating equipment, and an overview of the standard procedures involved in a generic rotor-dynamics design analysis.
The amplification factor (AF) quantifies a rotor-bearing system's sensitivity to vibration caused by unbalance near its lateral critical speeds An AF greater than 10 suggests significant rotor vibration, potentially leading to contact between critical components like labyrinth seals and bearings with stationary parts during high vibration periods Conversely, an AF below 5 indicates low sensitivity to unbalance at critical speeds, suggesting a more stable operation The impact of the amplification factor on rotor response is illustrated in Figure 1-1, which also details the calculation methods for AF based on damped response or vibration measurements.
A Bodé plot visually represents a rotor's synchronous vibration amplitude and phase angle relative to shaft rotation speed It is commonly produced from a rotor damped unbalance response analysis Recent revisions of the API Standard Paragraphs indicate that a rotor system's critical speeds are identified using the response amplitude data shown in the Bodé plot An example of a Bodé plot can be found in Figure 1-2.
A Campbell diagram visually represents the natural frequencies of equipment trains or individual units alongside potential harmonic excitation frequencies This graph effectively highlights the acceptable operating speeds, ensuring that vibration excitation mechanisms, such as unbalance, do not disrupt critical resonance frequencies, including lateral critical speeds, blade resonance frequencies, and torsional natural frequencies Refer to Figures 1-3 and 1-4 for illustrative examples.
Critical speed refers to the rotational speed of a shaft that aligns with the resonance frequency of a non-critically damped rotor system (AF > 2.5) As outlined in API Standard Paragraph 2.8.1.3, the critical speed frequency is identified as the peak vibration response frequency depicted in the Bodé plot, which is derived from damped unbalance response analysis and shop test data.
1.2.2.5 Critical speed of concern is any critical speed to which the acceptance criteria of this standard is applicable.
Critical speeds of concern include (a) any speed below the unit's operating range, (b) any speed within the operating range, and (c) the first critical speed above the maximum continuous operating speed (MCOS) Additionally, in specific situations, other critical speeds may also be significant, such as those that are integer multiples of the electric line frequency or other excitation mechanisms For further details on potential critical speed excitation mechanisms, refer to API Standard Paragraphs 2.8.1.5.
An undamped critical speed map illustrates a rotor's undamped critical speeds based on the combined bearing and support stiffness This graph typically displays only the first three undamped critical speeds of the rotating element To enhance the understanding of the rotor dynamic characteristics of the rotor-bearing system, calculated bearing stiffness is often cross-plotted on the critical speed map However, it is important to note that this map is not used for calculating the rotor's critical speeds due to the exclusion of factors such as bearing and seal damping, as well as aerodynamic cross-coupling.
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Vibration amplitude (mm p-p) Vibration amplitude (mils p-p)
= Rotor first critical, center frequency, cycles per minute.
= Initial (lesser) speed at 0.707 × peak amplitude (critical).
= Final (greater) speed at 0.707 × peak amplitude (critical).
= Peak width at the half-power point.
Figure 1-1—Evaluating Amplification Factors (AFs) from Speed-Amplitude Plots
Reproduced by IHS under license with API
5 shoe tilting pad bearing load between pads orientation Length (pad) =
Journal Diameter = Clearance (dia) = χ (Pad Arclength) = δ (Preload) = α (Offset) =
Figure 1-2—A Sample Bode Plot: Calculated Damped Unbalance and Phase Responses of an Eight-Stage 12 MW(16,000 HP) Steam Turbine
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1st train torsional natural frequency
Range of unacceptable torsional natural frequencies
Range of acceptable torsional natural frequencies
90% speed Motor operating speed 110% speed
Figure 1-3—Sample Train Campbell Diagram for a Typical Motor-Gear-Compressor Train
Reproduced by IHS under license with API
1st stage blade natural frequency
5th train torsional natural frequency
4th train torsional natural frequency 3rd train torsional natural frequency
1st compressor lateral critical 2nd train torsional natural frequency
1st turbine lateral critical 1st train torsional natural frequency
Range of Acceptable Torsional Natural Frequencies
Natural frequency (CPM) 90% speed operating speed 110% speed
Figure 1-4—Sample Train Campbell Diagram for a Typical Turbine-Compressor Train
This tutorial focuses on the API standards related to rotor dynamics and balancing It highlights that the energy dissipation mechanisms in rotating systems, including material damping and friction, are typically minimal compared to the damping effects provided by bearings and oil film seals.
Frequency refers to the number of cycles of a repetitive motion occurring within a specific time frame It is commonly calculated as the reciprocal of the period of the motion and is typically expressed in cycles per second (cps or Hertz) or cycles per minute (cpm) The use of cpm allows for easy comparison between the measured vibration and the shaft's rotating frequency.
The logarithmic decrement (log dec) is defined as the natural logarithm of the ratio of two successive amplitude peaks in free harmonic vibration It is mathematically derived from the real part of the damped system eigenvalues obtained during rotor dynamic stability analysis The log dec serves as an important measure of rotor system stability and is considered in undamped critical speed analysis.
1-5 displays a sample critical speed map.
1.2.2.7 A damped unbalanced response analysis is a calcu- lation of the rotor’s response to a set of applied unbalances.
The applied unbalance synchronously excites the rotor, causing its response to occur at the shaft's rotation speed Analyzing the damped unbalance response requires considering all steady-state linearized forces, including bearing and seal stiffness, viscous damping, and support effects, to predict a machine's critical speed characteristics Typically, the results of this analysis are illustrated in Bodé plots.
Rotor Bearing System Modeling
Modeling is the crucial process for conducting engineering analyses of physical systems To ensure the accuracy of the model, it is essential to integrate several verification checks into the modeling procedure.
This tutorial provides an overview of the API standard paragraphs related to rotor dynamics and balancing It is important to note that the undamped critical speed analysis is conducted by the equipment manufacturer solely upon customer request.
Shaft flexibility can significantly reduce the lateral critical speeds of a rotor-bearing system and negatively impact the rotor dynamics of rotating equipment However, it is not the mere presence of shaft flexibility that causes degradation, but rather excessive flexibility in comparison to the support bearings When bearing stiffness is low relative to the rotor's bending stiffness, the fundamental critical speed is primarily influenced by the bearings' stiffness and rotor mass In contrast, if the bearings are much stiffer than the shaft's bending stiffness, the undamped critical speeds are mainly determined by the rotor's mass and bending stiffness The ideal scenario is a stiff rotor with soft bearings, while a flexible rotor with stiff bearings is undesirable The undamped critical speed map serves as a valuable tool to assess when shaft flexibility may exceed acceptable limits, potentially jeopardizing the safe and reliable operation of a turbomachine.
The undamped critical speed map, illustrated in Figure 1-5, features two distinct regions On the left side, known as the stiff rotor area, the critical speed lines exhibit a positive and nearly constant slope due to the rotor's bending stiffness being significantly greater than the bearing stiffness Conversely, the right side represents the stiff bearing region, where critical speeds remain unchanged despite increasing bearing stiffness, as the bearing stiffness far exceeds the rotor's bending stiffness The critical speed lines' asymptotic values in this region are commonly referred to as the rigid bearing critical speeds.
Cross-plotting calculated bearing coefficients on undamped critical speed maps helps to determine the damped unbalance response characteristics of rotor-bearing systems When bearing stiffness intersects a critical speed line in the stiff rotor region, the amplification factor is typically low (less than 5.0), indicating a well-damped response to unbalance near critical speeds In contrast, if the stiffness intersects on the flat part of the critical speed line, the bearings are significantly stiffer than the rotor, resulting in a high amplification factor (greater than 12.0) and a highly amplified response to unbalance during operation near critical speeds.
Examining the undamped mode shapes related to the first three undamped critical speeds for both soft and stiff bearing cases enhances the understanding of the relationship between the undamped critical speed map and other lateral dynamics analyses, such as damped unbalance response analysis In soft bearing scenarios, shaft deflections are minimal compared to bearing deflections, allowing the damping from the bearings to effectively reduce rotor vibrations caused by unbalance Conversely, in stiff bearing cases, shaft deflections are significant relative to bearing deflections, resulting in small damping forces even with high bearing damping coefficients Consequently, rotor vibrations from unbalance and other forces can be significantly amplified at critical speeds when bearings are much stiffer than the rotor bending stiffness.
Mode shapes related to undamped lateral natural frequencies can be derived from undamped critical speed analysis, typically using bearing principal stiffnesses at the unit's normal operating speed However, these calculated frequencies may not align perfectly with the unit's critical speeds, which are determined by the intersection of critical speed lines and bearing stiffnesses The undamped mode shapes, which are two-dimensional and independent of unbalance distribution, reflect the mass-elastic model and exhibit consistent characteristics for beam rotors For instance, the fundamental critical speed mode shape displays both rigid body motion and some shaft bending, while the third critical speed mode shape shows only shaft bending These distinct mode shapes enable verification of critical speed calculations, and the three lowest frequency undamped lateral natural frequencies can serve as a benchmark for assessing damped lateral response and rotor stability.
Reproduced by IHS under license with API
A ci = Response amplitude at i th critical.
The applied balance magnitude, denoted as U, is crucial in understanding dynamic rotor mode shapes These mode shapes, or modal diagrams, illustrate the amplitude of response at key locations, including critical clearances, coupling engagement planes, bearings, and seals.
1.4.4 ROTOR DYNAMIC (DAMPED EIGENVALUE) STABILITY
Prior to constructing centrifugal compressors, it is crucial for purchasers to assess the rotor dynamic stability through a stability or damped eigenvalue analysis as part of the design audit This analysis aims to identify the potential for large amplitude subsynchronous vibrations during normal operation If the design lacks sufficient stability, modifications to the bearing or shaft design must be made before construction Addressing rotor stability issues post-construction can be challenging, as significant improvements are often unattainable without compromising unit speed and flow capacity Engineers typically manage unstable rotor vibrations by adjusting lubricant supply temperatures and reducing inlet pressure, but these measures may only provide temporary relief, necessitating a redesign for a long-term solution.
The analysis of stability in high-pressure centrifugal compressors is complex due to unpredictable factors such as aerodynamic interactions, labyrinth seal effects, and friction from components Consequently, stability assessments often rely on experiential calibration based on unit service and design characteristics Notably, significant subsynchronous vibrations have been observed in compressors despite calculations suggesting sound designs Key influences on rotor stability include the relative stiffness of the shaft and bearings, the design of the rotor and seals, and destabilizing forces from process conditions.
Current technology assesses rotor stability primarily through damped eigenvalues, which are calculated at the rotor's operating speed A damped eigenvalue is represented as a complex number in the form \( s = p \pm i \omega_d \), where \( p \) denotes the damping exponent and \( \omega_d \) signifies the oscillation frequency.
The impact of the damping exponent on rotor motion is illustrated in Figures 1-12 and 1-13 A negative damping exponent indicates stable rotor vibrations, where the vibration envelope diminishes over time Additionally, the undamped mode shapes effectively represent the relative displacements of the shaft when the rotor operates near its critical speed.
Thus, given a vibration amplitude at a probe location during operation near a critical speed, one may estimate the vibra- tion amplitude at other locations on the rotating element.
The damped unbalance response analysis is essential for assessing lateral rotor dynamics, including critical speeds and amplification factors, as outlined by API Figure 1-1 illustrates a sample rotor response plot from the API Standard Paragraphs, highlighting the definitions of critical speeds, amplification factors, and critical speed separation margins based on calculated or measured rotor response data.
The accuracy of calculated results relies heavily on the detail included in the rotor system model This model must account for various effects, which will be elaborated on in subsequent sections After constructing the unit model, the vendor assesses the lateral response to specific amounts of unbalance applied at designated locations on the rotor, as outlined by API standards to ensure that critical points of concern are effectively excited.
The damped unbalance response analysis, as mandated by the API Standard Paragraphs, will present key findings including Bodé response plots, which illustrate the amplitude and phase of vibration caused by applied unbalance in relation to the rotor's operating speed.
Modeling Methods and Considerations
Accurate computer simulations of physical systems rely on precise models to yield reliable results This section outlines effective methods used by the authors over many years to model key components of solid shaft rotor-bearing systems While tie-bolt or built-up rotors can be modeled accurately with these techniques, it is important to note that they may present greater modeling challenges compared to solid shaft designs For instance, the bending stiffness of the rotor can be influenced by tie-bolt stretch, and non-linear axial face friction forces between rotor segments may become significant if there is relative movement during operation Additionally, small high-speed built-up rotors may not be adequately represented by simply applying cylindrical beam elements.
In such cases, sophisticated finite element analysis of the rotat- ing element may be necessary to build an equivalent beam el- ement model that permits accurate prediction of results.
An accurate model of a rotor system is a model that per- mits accurate calculation of the actual rotor system’s dy-
Eigenvalue of a viscously damped system: s = p + i ωd p = damping exponent ωd = frequency of oscillation
-ept ept x xo = Real (est) x x o envelope (p < 0) displacement
Figure 1-12—Motion of a Stable System Undergoing Free Oscillations
Minimizing the number of elements in rotor modeling, as reproduced by IHS under license with the API model, can lead to numerical issues Additionally, this approach offers the advantage of reducing the time required for engineers to generate data files and for computers to execute calculations.
1.5.2.1 Division of Rotor into Discrete Sections
The modeling process involves the analyst dividing the rotor into elements that correspond to changes in the outside diameter (OD) or inside diameter (ID) After this initial division, further refinement of the model is typically necessary It is recommended that the length to diameter ratio of any section should not exceed 1.0, with a preferred ratio of 0.5, and should not be less than 0.10.
To ensure accurate calculation of the first three critical speeds, the model must have sufficient resolution It is also essential to avoid large length changes in adjacent shaft elements, as this can lead to numerical calculation issues The dynamic characteristics of the rotor are effectively represented when its mass-elastic properties are accurately modeled For a basic rotor dynamics design audit, a complete model of the rotating assembly can be created by combining lumped inertia shaft and disk elements While shaft elements provide both inertia and stiffness to the global model, disk elements contribute inertia only Although more complex element types can be utilized, they may complicate the model Generally, most turbomachinery in petroleum plants can be adequately modeled using the lumped inertia shaft and disk elements discussed in this tutorial.
After determining the elements for analysis, the engineer must create a detailed geometric description of the rotor using an adequate number of selected elements Figure 1-14 illustrates the rotor schematics alongside its lumped parameter model It is essential to impose certain constraints on the use of lumped inertia shaft elements to ensure the accuracy of the rotor models Using too few elements can lead to a model that lacks the necessary resolution to effectively represent the intricate mass-elastic properties.
Eigenvalue of a viscously damped system: s = p + i ωd ω d = frequency of oscillation p = damping exponent
Figure 1-13—Motion of an Unstable System Undergoing Free Oscillations x x o (Dimensionless)
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External lumped inertia for coupling
External lumped inertia input, Station No 7 Ext W7 = impeller weight.
Ext I p7 = impeller polar moment of inertia.
Ext l T7 = impeller transverse moment of inertia.
Parameter Length Diameter Weight Moment of Inertia Bending stiffness Damping
US Units in. in. lbf lbf • in. lbf/in. lbf • sec/in.
Figure 1-14—Schematic of a Lumped Parameter Rotor Model
I Ti = i th station lumped transverse moment of inertia.
I pi = i th station lumped polar moment of inertia.
K si = i th shaft stiffness section bending stiffness.
The assembly process of the model can lead to the accumulation of numerical round-off errors due to the creation of a consulting shaft stiffness This issue arises when these stiffness values are combined.
When uncertain about modeling a feature in a rotating element, analysts can assess the sensitivity of results to different modeling approaches For instance, strict adherence to certain guidelines may prevent the modeling of short circumferential grooves on compressor shafts, which are essential for positioning split rings and securing thrust collars Typically, these design features are overlooked if reducing the diameter of the element does not impact critical results or modeshapes However, if a geometric feature significantly influences the results, it may indicate a fundamental flaw in the rotor's design.
When analysts face challenges in modeling a rotating assembly due to complex rotor geometries, they can create an equivalent model through advanced analysis techniques For instance, the bending characteristics of a stub shaft connected to the second stage impeller of an overhung gas pipeline compressor can be accurately assessed using finite element analysis, as illustrated in Figure 1-15 The presence of large counter-bored bolt holes significantly reduces the lateral bending stiffness of the stub shaft After completing the static bending analysis, a corresponding lumped parameter beam-type model can be developed, which maintains the same bending stiffness at the lumped inertia locations as used in rotor dynamics analysis.
1.5.2.2 Addition of External Masses and Inertial
Not all components of a rotating assembly, such as impellers and sleeves, significantly influence the bending stiffness of a rotor Typically, these components are assumed to have minimal impact, leading to an underestimation of the fundamental critical speed, though the discrepancy is usually under 10 percent The difference between calculated and observed critical speeds is influenced by the flexibility of the rotor; more flexible shafts experience a greater impact from shrunk-on components This assumption is valid for standard rotor dynamics analysis, as highly flexible rotors typically operate well above their first critical speed Additionally, rotor stability and the location of the second critical speed are less affected by shaft sleeves, often yielding conservative predictions when their stiffening effects are ignored If critical speeds and amplification factors vary by more than 5 percent, the predictive model may need refinement based on mechanical testing data.
The inertial characteristics of a rotating assembly are influenced by components that are shrunk onto the shaft, necessitating their inclusion in the model Typically, this is achieved by adding lumped inertias at the mass centers of these components In certain cases, such as with motor cores, it may be essential to create detailed inertia distributions for the shrunk-on parts Common additional masses found in most rotors include impellers, disks, couplings, sleeves, balance pistons, and thrust collars.
Particular machines will have specific masses that must be added, including the following: a Armature windings in electric motors. b Shrunk-on gear meshes. c Wet impeller mass and inertia in pumps.
It is imperative that the rotor model properly account for these masses and any additional rotating masses that may be peculiar to a particular system.
1.5.2.3 Addition of Stiffening Due to Shrink Fits and Irregular Sections
Most rotating assemblies feature non-integral components like collars, sleeves, and impellers that are shrunk onto the shaft during assembly While these components typically do not enhance the lateral stiffness of the shaft, larger and longer shrink fits may necessitate modeling them as contributing to shaft stiffness Vendors must assess the significance of shrink fits on a case-by-case basis, often relying on experience with similar units Although modal testing of vertically hung rotors can provide insights into the stiffening effects of shrunk-on components, such tests may overstate these effects due to centrification at normal operating speeds.
Non-circular rotor cross-sections are common in the midspan areas of electric motors and generators These elec- trical machines frequently possess integral or welded-on
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Finite element analysis (FEA) is employed to determine the effective bending stiffness of complex geometry rotating components This calculated bending stiffness is subsequently transformed into an equivalent cylindrical section, which can be utilized in lateral rotor dynamics analysis software.
3D Finite Element Model of Stub Shaft
Cross-Section of Rotating Element with Complex Geometry Component
Figure 1-15—3D Finite Element Model of a Complex Geometry Rotating Component
Tilting pad bearings, favored in the process industry over the past two decades, feature multiple pads that rotate on pivot points, enabling them to adapt to changes in journal position during operation These bearings exhibit various designs, reflecting the unique requirements and philosophies of different manufacturers An exaggerated schematic of an operating journal bearing illustrates that the journal's position is typically below the bearing centerline, resulting in a cavitated oil film in the upper half and a converging film thickness in the lower half, which creates a pressurized oil film wedge to support the rotating journal For the journal to maintain static equilibrium, the oil film forces must balance in both horizontal and vertical directions, leading to displacement not only downward but also sideways.
The displacement of the journal from the center of the bearing is a nonlinear function of the applied load, leading to linearized bearing stiffnesses (k xx, k xy, k yx, and k yy) that also vary nonlinearly with the journal’s equilibrium position Similarly, the linearized damping coefficients (c xx, c xy, c yx, and c yy) generated by the bearing are influenced by both the bearing geometry and the applied load Table 1-3 provides ranges of stiffness and damping coefficients for different bearing designs, highlighting how variations in bearing type and load can significantly impact the lateral dynamic characteristics of a machine.
To understand the impact of bearings on the dynamic characteristics of a rotor system, it is essential to calculate the linearized bearing dynamic coefficients Various commercial and academic software tools are available for engineers to compute these coefficients, typically utilizing finite element or finite difference methods to solve Reynolds' equation, which governs thin hydrodynamic films These tools can incorporate multiple factors, such as heat generation from fluid shearing, fluid turbulence, and variations in oil supply temperatures Ultimately, the analysis yields a set of eight linearized hydrodynamic bearing coefficients, which play a crucial role in enhancing the stiffness of the rotor midspan.