5. Rotordynamics and Machine Stability

Introduction and Context

Rotating machinery converts, transmits, or controls energy through shafts, rotors, bearings, couplings, gears, seals, and supporting structures. Turbines, compressors, pumps, generators, electric motors, machine-tool spindles, turbochargers, and vehicle drivetrains all contain rotating elements whose dynamic behaviour can determine performance, reliability, and safety. A rotor that is statically well balanced may nevertheless experience large vibration because of resonance, gyroscopic action, bearing flexibility, aerodynamic forces, or torsional excitation.

Rotordynamics is the study of the motion, vibration, stability, and energy exchange of rotating systems. It combines rigid-body dynamics, vibration theory, elasticity, fluid-film lubrication, numerical modelling, and control engineering. The central engineering problem is not merely to reduce vibration; it is to ensure that the rotor remains dynamically stable throughout its operating range, including acceleration, steady operation, shutdown, transient loading, and fault conditions.

The distinction between static and dynamic behaviour is important. Static balance concerns the location of the mass centre, whereas dynamic balance also accounts for the orientation of the principal inertia axis. Similarly, a shaft may be adequately designed for static strength but fail through fatigue caused by repeated bending or torsional vibration. Rotordynamic analysis therefore links mechanical design with operational decision-making.

Learning Objectives

After studying this chapter, the learner should be able to:

1. Explain critical speed, shaft whirl, gyroscopic effects, and rotor–bearing interaction.

2. Formulate simple lateral and torsional rotor models and interpret their natural frequencies.

3. Distinguish synchronous, subsynchronous, and supersynchronous vibration.

4. Evaluate the effects of unbalance, bearing properties, damping, and support flexibility.

5. Explain balancing methods and the conditions leading to rotor instability.

6. Interpret Campbell diagrams, stability maps, orbit plots, and vibration spectra.

7. Select appropriate vibration-control and condition-monitoring strategies.

8. Apply rotordynamic reasoning to an industrial failure or design case.

The Rotordynamic System

A practical rotor is a distributed elastic body supported by bearings and connected to other machines. For analysis it is commonly idealised as a rigid disk on a flexible shaft, a Jeffcott rotor, a lumped-parameter rotor, or a finite-element beam model. The choice depends on the required accuracy. A preliminary design may use a single-degree-of-freedom model, while a high-speed compressor may require a three-dimensional finite-element model with anisotropic bearing coefficients and seal forces.

The principal motions are lateral (or bending) vibration, torsional vibration, and axial vibration. Lateral motion involves displacement of the shaft centreline perpendicular to its axis. Torsional vibration involves angular oscillation about the shaft axis. Axial vibration occurs along the axis and is often associated with thrust bearings, gears, combustion forces, or hydraulic excitation. These motions may be coupled; for example, a misaligned coupling can produce both lateral and axial forces, while a geared rotor may couple torsional and lateral modes.

Important terminology includes:

• Rotor: A rotating assembly including shaft, disks, impellers, sleeves, and attached components.
• Support: The bearings, housing, foundation, and structural elements that constrain rotor motion.
• Unbalance: A noncoincident relationship between the mass centre and the axis of rotation.
• Whirl: Orbital motion of the rotor centreline or journal around a bearing or geometric centre.
• Spin: Rotation of the rotor about its own shaft axis.
• Critical speed: A rotational speed at which excitation coincides with a lateral natural frequency.
• Stability: The tendency of perturbations to decay or grow with time.

Spin and whirl need not have the same frequency or direction. Forward whirl is in the same direction as rotor spin; backward whirl is opposite. This distinction is essential in diagnosing bearing-induced instability and rotor–stator contact.

Lateral Vibration and Critical Speeds

For a simple undamped rotor with mass \(m\), lateral stiffness \(k\), and displacement \(x\),

\[

m\ddot{x}+kx=F(t).

\]

Its natural frequency is

\[

\omega_n=\sqrt{k/m}.

\]

If the rotor rotates at angular speed \(\Omega\) and has eccentricity \(e n), the synchronous unbalance force is approximately

\[

F_u=m e \Omega^2.

\]

For a damped single-degree-of-freedom model,

\[

m\ddot{x}+c\dot{x}+kx=m e\Omega^2\sin(\Omega t).

\]

The steady-state amplitude is

\[

X=\frac{m e\Omega^2}{\sqrt{(k-m\Omega^2)^2+(c\Omega)^2}}.

\]

Near \(\Omega=\omega_n\), the response becomes large, although damping limits the peak. This speed is called a critical speed or lateral resonance speed. In a distributed rotor, several bending modes exist and therefore several critical speeds may occur.

The term rigid-body critical speed refers to a low-frequency mode governed mainly by support stiffness, whereas flexible critical speeds are associated with shaft bending. A rotor may pass through one or more critical speeds during start-up. Crossing speed is not necessarily dangerous if the passage is rapid, the damping is adequate, the response remains within clearance limits, and no destabilising force is present. Continuous operation close to a critical speed, however, is usually unacceptable unless specifically designed for it.

The ratio of excitation frequency to natural frequency is the frequency ratio \(r=\Omega/\omega_n\). The phase of vibration changes markedly through resonance. Below resonance, displacement is approximately in phase with the unbalance force; near resonance, it is about ninety degrees out of phase; above resonance, it approaches opposition. Phase information is therefore as useful as amplitude in field diagnosis.

Shaft Whirl and Gyroscopic Effects

A rotating disk possesses polar and diametral mass moments of inertia. When its axis tilts, the angular momentum changes direction and produces a gyroscopic moment. For a disk with polar moment \(I_p\), spinning speed \(\Omega\), and precession rate \(\dot{\theta}\), the gyroscopic contribution is proportional to

\[

I_p\Omega\dot{\theta}.

\]

This term makes the equations of motion speed-dependent. In a simplified two-coordinate model with lateral displacements \(x\) and \(y\), the gyroscopic matrix introduces cross-coupling between these coordinates. Consequently, forward and backward whirl frequencies separate as rotational speed increases. The forward mode generally increases in frequency, while the backward mode decreases, although the exact trend depends on rotor configuration and support conditions.

A Campbell diagram plots natural frequencies against rotational speed and superimposes excitation orders such as 1X, 2X, or blade-pass frequency. Intersections indicate potential resonances. The diagram is a primary design tool because it shows not only where critical speeds occur but also how gyroscopic effects shift them.

Whirl may also be forced or self-excited. Unbalance produces mainly synchronous, or 1X, forced response. Fluid-film bearings, seals, aerodynamic cross-forces, and internal friction can generate forces dependent on displacement and velocity. If the net energy supplied by these forces exceeds damping losses, a small perturbation grows and the rotor becomes unstable. A self-excited whirl often appears at a subsynchronous frequency, commonly below running speed.

Rotor–Bearing Dynamics

Bearings do more than support static load. They determine the rotor’s stiffness, damping, cross-coupled forces, heat generation, and stability margin. Rolling-element bearings are often represented by stiffness and damping coefficients that may vary with load, preload, clearance, and contact geometry. Fluid-film bearings are described by pressure fields generated in the lubricant film. Linearisation about an operating equilibrium yields direct and cross-coupled coefficients:

\[

\begin{bmatrix}F_x\\F_y\end{bmatrix}

=-\begin{bmatrix}K_{xx}&K_{xy}\\K_{yx}&K_{yy}\end{bmatrix}

\begin{bmatrix}x\\y\end{bmatrix}

-\begin{bmatrix}C_{xx}&C_{xy}\\C_{yx}&C_{yy}\end{bmatrix}

\begin{bmatrix}\dot{x}\\\dot{y}\end{bmatrix}.

\]

The direct terms resist motion in the same direction, while cross-coupled terms produce forces perpendicular to the displacement or velocity. Cross-coupled stiffness can promote forward whirl and reduce stability. Bearing selection must therefore consider dynamic coefficients rather than only static load capacity.

A rotor-bearing system can be represented in matrix form as

\[

M\ddot{q}+(C+G)\dot{q}+Kq=F(t),

\]

where \(M\) is the mass matrix, \(C\) the damping matrix, \(G\) the gyroscopic matrix, and \(K\) the stiffness matrix. In a speed-dependent model, bearing and seal coefficients are functions of \(\Omega\). Complex eigenvalue analysis determines modal frequencies and decay rates. A mode with a positive real growth rate is unstable; a negative growth rate indicates decay.

Support flexibility must not be ignored. A stiff rotor on flexible pedestals can have a lower rigid-body critical speed than expected. Foundation resonance, looseness, cracked grout, and casing flexibility may create additional modes. In industrial practice, rotor, bearing, housing, foundation, and coupling should be considered as one dynamic system.

Torsional Vibration

Torsional vibration is angular oscillation caused by fluctuating torque. A two-inertia model with inertias \(J_1\) and \(J_2\), connected by torsional stiffness \(k_t\) and damping \(c_t\), has a relative torsional natural frequency approximately given by

\[

\omega_t=\sqrt{k_t\left(\frac{1}{J_1}+\frac{1}{J_2}\right)}.

\]

Excitation may arise from reciprocating engines, electromagnetic torque pulsation, gear-mesh variation, propeller loading, or cyclic compressor torque. Resonance occurs when an excitation order approaches a torsional natural frequency. Unlike lateral vibration, torsional motion may be poorly detected by casing accelerometers because the shaft can twist significantly while housing vibration remains modest.

Torsional stresses are related to torque by

\[

\tau=\frac{Tr}{J_p},

\]

where \(T\) is torque, \(r\) is radial position, and \(J_p\) is the polar second moment of area. Repeated torsional stress can cause fatigue, fretting at couplings, gear damage, and shaft fracture. Solutions include changing shaft stiffness or inertia distribution, adding a tuned torsional damper, modifying excitation orders, and controlling drive torque.

Balancing and Alignment

Balancing reduces the rotating force and moment produced by mass eccentricity. Static balancing corrects the resultant centrifugal force in a single plane. Couple balancing corrects the resultant moment by placing correction masses in different axial planes. Most long rotors require two-plane dynamic balancing.

The influence-coefficient method measures vibration response at selected speeds, applies a trial weight, determines the change in amplitude and phase, and calculates correction weights. Field balancing is valuable when the assembled rotor differs from its manufactured condition because of coupling, impeller, deposits, thermal distortion, or assembly errors.

Balancing cannot correct every vibration problem. Misalignment commonly produces strong 1X and sometimes 2X components, but the exact pattern depends on coupling type and machine structure. Mechanical looseness, bent shafts, rubs, bearing defects, electrical asymmetry, and aerodynamic excitation can produce similar symptoms. Correct diagnosis requires phase, orbit, spectrum, operating condition, and inspection evidence.

Alignment concerns the relative position and angular orientation of connected shafts. Thermal growth must be included in alignment targets. Excessive offset or angular misalignment creates forces that can overload bearings and couplings even when the rotor is well balanced.

Stability, Rubs, and Rotor–Stator Interaction

A stable rotor returns toward its equilibrium after a small disturbance. Instability may result from cross-coupled fluid forces, internal damping, seal forces, aerodynamic loading, bearing preload, or operating outside the design envelope. Oil whirl is a fluid-film phenomenon in which the journal centre orbits at a subsynchronous frequency. Oil whip occurs when this whirl locks onto a rotor natural frequency and can produce a severe, persistent vibration.

Rotor–stator rubs occur when vibration or thermal growth closes the clearance between rotating and stationary parts. A light rub may generate harmonics, fractional harmonics, changes in orbit shape, and temperature rise. A severe rub can transfer energy to the rotor, distort the shaft, damage seals, and trigger a thermal bow. Rubs are especially dangerous because the original vibration mechanism may be amplified by the contact itself.

Stability margins should be assessed over the full speed and load range. A design that is stable at rated speed may become unstable during a process upset, a change in lubricant viscosity, seal replacement, or operation at a different pressure ratio. Acceptance criteria should therefore include transient response, not only steady-state amplitude.

Measurement, Modelling, and Digital Analysis

A useful instrumentation system may include proximity probes for shaft displacement, accelerometers for housing vibration, keyphasors or once-per-revolution sensors for phase reference, torque meters for torsional response, temperature sensors, and pressure measurements. Proximity probes are particularly valuable for fluid-film bearing machines because they reveal shaft orbit relative to the bearing.

Common analytical displays include:

• Spectrum: vibration amplitude versus frequency; useful for identifying 1X, harmonics, subsynchronous components, and bearing frequencies.
• Waterfall plot: spectra displayed over changing speed or time.
• Bode plot: amplitude and phase versus speed during run-up or coast-down.
• Orbit plot: horizontal displacement plotted against vertical displacement, revealing whirl direction, ellipticity, and rub behaviour.
• Campbell diagram: modal or measured frequencies plotted against rotational speed.
• Polar plot: amplitude and phase represented during a speed sweep.

Finite-element rotor models divide the shaft into beam elements and include disks, bearings, seals, couplings, and support structures. Model credibility depends on correct mass properties, boundary conditions, bearing coefficients, damping assumptions, and validation against measured data. Digital twins can combine a calibrated model with live sensor data to estimate operating deflection shape, remaining stability margin, or the probability of a developing fault. Machine-learning methods can support anomaly detection, but they should not replace physical interpretation, sensor validation, or engineering judgment.

Design and Control Strategies

A robust rotordynamic design begins with an accurate definition of speed range, operating loads, transient events, allowable vibration, clearances, and maintenance conditions. The following strategies are commonly combined:

1. Mass and geometry control: reduce eccentricity, improve concentricity, avoid abrupt shaft changes, and control rotor runout.

2. Critical-speed separation: place critical speeds sufficiently away from continuous operating speeds, allowing for manufacturing and temperature uncertainty.

3. Damping enhancement: use suitable bearings, squeeze-film dampers, tuned absorbers, or structural damping treatments.

4. Bearing and seal optimisation: reduce harmful cross-coupled coefficients and ensure adequate lubrication.

5. Support modification: increase pedestal or foundation stiffness where flexible supports create unacceptable modes.

6. Active control: use magnetic bearings, active dampers, or variable-speed control to modify dynamic response.

7. Operational control: accelerate through resonance appropriately, avoid unstable speed bands, and use protective trips based on validated limits.

Magnetic bearings permit non-contact support and active force control. They can provide high-speed operation and online adaptation but require sensors, power electronics, backup bearings, and fail-safe control. Variable-speed drives can avoid resonance or reduce excitation, but speed changes may also pass through critical regions and alter electromagnetic torque pulsation.

Illustrative Case Study: Subsynchronous Vibration in a Centrifugal Compressor

Background

An illustrative gas-processing plant operates a high-speed centrifugal compressor supported by tilting-pad journal bearings. The compressor normally runs at 9,000 rpm and has a rated capacity near its design point. Shaft displacement probes are installed at both bearings.

Situation and Problem

After a seal replacement, vibration increased gradually. The dominant component was approximately 0.45X rather than 1X. Overall casing vibration remained moderate, but shaft orbit amplitude increased during operation at high discharge pressure. The machine occasionally approached its trip limit during process disturbances.

Analysis

A spectrum and waterfall plot showed a subsynchronous component whose frequency increased with speed until it approached a rotor-bearing natural frequency. The orbit was predominantly forward and became larger as load increased. Balance correction was considered but rejected as the first action because unbalance would primarily produce synchronous 1X vibration. Inspection confirmed that the replacement seal had higher cross-coupled stiffness than the original design. Process data also showed operation near a region of high aerodynamic cross-force.

A rotor-bearing model was updated using the revised seal coefficients. Complex eigenvalue analysis predicted a reduced stability margin and a threshold speed close to the observed operating condition. The result demonstrated the importance of including seals and operating pressure in rotordynamic analysis.

Approach and Solution

The plant temporarily avoided the unstable speed and pressure combination. Engineers then selected a seal design with lower cross-coupled force, verified bearing preload, and established an operating limit based on stability analysis. The protection system was reviewed so that shaft displacement, rate of change, and process variables were evaluated together rather than relying only on casing acceleration.

Outcome

The subsynchronous response decreased substantially after the seal modification, and the machine operated below the revised vibration limit throughout the permitted envelope. The root cause was not inadequate balancing but a change in rotor–seal dynamics.

Learning Points

The case illustrates that vibration frequency is diagnostic evidence. A large subsynchronous component suggests instability, fluid-film effects, or self-excited whirl rather than simple unbalance. It also demonstrates that component replacement must preserve dynamic characteristics, not merely dimensional compatibility.

Challenges, Limitations, and Professional Responsibilities

Rotordynamic predictions are sensitive to uncertain parameters. Bearing coefficients vary with temperature, viscosity, load, clearance, and speed. Seal forces may be difficult to estimate, and damping is often poorly known. Simplified models can therefore provide misleading precision. Sensitivity studies and measured-data validation are essential.

Common mistakes include treating every 1X vibration as unbalance, ignoring phase, analysing the rotor without its supports, using nominal rather than operating temperature, neglecting transient passage through critical speed, and accepting a replacement seal or coupling without reviewing its dynamic effect. Another mistake is relying exclusively on overall vibration severity; a moderate but rapidly growing subsynchronous component may be more dangerous than a larger stable synchronous component.

Professional practice requires traceable assumptions, calibrated instruments, documented alarm limits, competent review of protection systems, and conservative action when evidence indicates instability. Safety-critical rotating equipment should have independent trips, appropriate overspeed protection, and maintenance procedures that prevent unsafe operation. Data collected for predictive maintenance should also be governed so that alarm thresholds, model updates, and automated recommendations are auditable.

Current Developments and Best Practices

Modern rotordynamics increasingly combines high-fidelity finite-element models, online monitoring, operational modal analysis, and physics-informed analytics. Digital twins can update model parameters from measured speed, temperature, pressure, and vibration. Cloud or plant-edge systems can compare current behaviour with validated operating envelopes, while automated order tracking separates speed-related components from structural resonances.

Best practice is to:

• define the complete operating envelope before design approval;
• model rotor, bearings, seals, couplings, supports, and foundation together;
• use both amplitude and phase-based diagnostics;
• validate predicted critical speeds and damping with run-up or coast-down data;
• distinguish forced response from self-excited instability;
• monitor shaft displacement as well as casing vibration where appropriate;
• review dynamic consequences of every major component change;
• maintain clear alarm, trip, and emergency procedures; and
• combine automated analytics with experienced engineering review.

Chapter Summary

Rotordynamics evaluates the vibration and stability of rotating systems whose behaviour depends on mass distribution, shaft flexibility, gyroscopic effects, bearings, seals, supports, and operating conditions. Critical speeds occur when excitation approaches lateral natural frequencies, while gyroscopic effects split forward and backward whirl modes. Rotor–bearing models use mass, damping, stiffness, and gyroscopic matrices, with cross-coupled coefficients playing a major role in instability.

Unbalance produces mainly synchronous response and is addressed through static or dynamic balancing. Torsional vibration results from fluctuating torque and requires separate modelling because it may not be visible in casing vibration. Subsynchronous vibration, growing orbit amplitude, and positive modal growth rates can indicate self-excited instability. Effective engineering practice combines modelling, run-up testing, spectrum and orbit analysis, appropriate bearing and seal design, damping, active control, operational limits, and validated protection systems.

Key Terms

Critical speed: Rotational speed at which an excitation frequency coincides with a system natural frequency.

Campbell diagram: Plot of modal or measured frequencies against rotational speed.

Forward whirl: Orbital motion in the same direction as rotor spin.

Backward whirl: Orbital motion opposite to rotor spin.

Gyroscopic effect: Speed-dependent moment caused by changes in the direction of angular momentum.

Unbalance: Eccentricity or angular misalignment between the mass distribution and rotation axis.

Dynamic balancing: Correction of both resultant centrifugal force and rotating couple.

Synchronous vibration: Vibration occurring at the rotational frequency, commonly called 1X.

Subsynchronous vibration: Vibration with frequency below the rotational frequency.

Cross-coupled stiffness: Force-producing stiffness component acting in a direction transverse to displacement.

Rotor-bearing system: Coupled dynamic system consisting of rotor, bearings, supports, and often seals.

Oil whirl: Subsynchronous fluid-film bearing instability involving journal orbiting.

Oil whip: Severe whirl condition in which subsynchronous motion locks onto a rotor natural frequency.

Torsional vibration: Oscillatory angular motion about the shaft axis.

Bode plot: Amplitude and phase plotted against rotational speed.

Orbit plot: Plot of shaft displacement in two orthogonal directions.

Squeeze-film damper: Fluid damper that dissipates vibration energy through viscous film motion.

Stability margin: Measure of the separation between operating conditions and onset of dynamic instability.

Rotor rub: Contact between rotating and stationary components.

Operating deflection shape: Measured vibration shape of a machine under operating excitation.

Review Questions

Short Answer Questions

1. Define critical speed and explain why damping affects its vibration peak.

2. Distinguish spin from whirl.

3. What is the role of the gyroscopic matrix in rotor analysis?

4. Why is torsional vibration sometimes difficult to detect using casing accelerometers?

5. State two diagnostic differences between unbalance and self-excited instability.

Long Answer Questions

1. Derive and interpret the steady-state response of a damped unbalanced rotor.

2. Explain how bearing direct and cross-coupled coefficients influence rotor stability.

3. Discuss the purpose and interpretation of a Campbell diagram, Bode plot, and orbit plot.

4. Explain static, couple, and two-plane dynamic balancing, including their limitations.

5. Describe a systematic design strategy for avoiding dangerous critical speeds.

Analytical / Application Questions

1. A rotor exhibits rapidly increasing forward whirl at 0.45X while 1X vibration remains moderate. Diagnose the likely mechanism and propose a measurement and corrective-action plan.

2. A compressor seal is replaced by one with greater cross-coupled stiffness. Predict the likely effect on stability and identify the model parameters that should be reassessed.

3. A drive train has two large inertias connected by a flexible coupling and shows periodic torque oscillation near a known excitation order. Explain how you would determine whether torsional resonance is present and recommend two remedies.

22. Multiple Choice Questions with Answers

Question 1

A rotor with mass m, stiffness k, and negligible damping reaches its primary lateral resonance when rotational speed is approximately:

A. k/m

B. sqrt(k/m)

C. m/k

D. sqrt(mk)

Correct Answer: B. sqrt(k/m)

Question 2

Which vibration component most strongly suggests a self-excited rotor instability rather than ordinary unbalance?

A. A stable 1X component

B. A subsynchronous component with growing amplitude

C. A small 2X component during startup

D. A constant broadband noise floor

Correct Answer: B. A subsynchronous component with growing amplitude

Question 3

The principal effect of gyroscopic action in a rotating rotor is to:

A. Eliminate all critical speeds

B. Separate forward and backward whirl frequencies

C. Remove torsional vibration

D. Make bearing stiffness independent of speed

Correct Answer: B. Separate forward and backward whirl frequencies

Question 4

Why can a seal replacement alter rotor stability even when its dimensions are unchanged?

A. It always changes rotor mass substantially

B. It can change cross-coupled fluid forces

C. It eliminates gyroscopic moments

D. It prevents all shaft bending

Correct Answer: B. It can change cross-coupled fluid forces

Question 5

A large 1X vibration component is measured after assembly, and its phase changes consistently with trial weights. The most appropriate initial technique is:

A. Two-plane influence-coefficient balancing

B. Immediate replacement of all bearings

C. Increasing process pressure

D. Ignoring phase information

Correct Answer: A. Two-plane influence-coefficient balancing

Question 6

In a Campbell diagram, a resonance is most likely indicated by:

A. An intersection between an excitation order and a modal-frequency curve

B. A horizontal line at zero frequency

C. A decrease in rotor mass only

D. The absence of any speed dependence

Correct Answer: A. An intersection between an excitation order and a modal-frequency curve

Question 7

Which measurement is particularly useful for identifying shaft orbit and whirl direction in a fluid-film bearing machine?

A. A single casing temperature sensor

B. Orthogonal proximity probes with a phase reference

C. An isolated microphone

D. A static pressure gauge only

Correct Answer: B. Orthogonal proximity probes with a phase reference

Question 8

The approximate torsional natural frequency of two inertias connected by torsional stiffness depends on:

A. Only the shaft radius

B. The stiffness and the reciprocals of both inertias

C. Only the bearing clearance

D. Casing acceleration amplitude alone

Correct Answer: B. The stiffness and the reciprocals of both inertias

Question 9

Oil whip differs from ordinary oil whirl because oil whip generally:

A. Occurs only at zero speed

B. Locks subsynchronous motion onto a rotor natural frequency

C. Is exactly equal to 1X unbalance

D. Can occur only in rolling-element bearings

Correct Answer: B. Locks subsynchronous motion onto a rotor natural frequency

Question 10

Which design action directly increases the separation between a continuous operating speed and a critical speed?

A. Operating continuously at resonance

B. Changing shaft or support stiffness after analysing the mode

C. Removing all phase sensors

D. Increasing cross-coupled stiffness

Correct Answer: B. Changing shaft or support stiffness after analysing the mode