What is a Campbell diagram?

A Campbell diagram plots a rotor's natural frequencies as a function of running speed and shows critical speeds as the intersections with engine-order lines (k times rotational speed). Because the gyroscopic effect splits each vibration mode into a forward whirl and a backward whirl branch as the speed rises, the diagram is a family of curves rather than a single line. Crossings with the excitation lines mark the speeds at which resonance occurs, letting an engineer see at a glance how much margin separates the operating speed from each critical speed.

What is the difference between forward and backward whirl?

Forward whirl (FW) is the mode in which the rotor whirls in the same direction as its rotation; the gyroscopic effect raises its natural frequency as the speed grows. Backward whirl (BW) whirls against the rotation, so its natural frequency falls instead. In real machines the FW branch is usually the troublesome one, because it can resonate with synchronous (1x) excitation from unbalance and misalignment. The FW x 1x crossing is therefore the first critical speed evaluated in design.

What do the engine-order lines (kx) represent?

Engine-order lines are straight lines whose slope equals the order k, representing excitation frequencies that are proportional to running speed. The 1x line corresponds to synchronous excitations such as unbalance and misalignment; 2x to coupling effects and shaft eccentricity; 3x and higher to gear meshing and blade-passing frequencies. Each crossing of a natural-frequency branch with one of these lines is a separate critical speed, so the diagram shows whether multiple resonances fall inside the operating range.

How much safety margin is needed from a critical speed?

Industry standards for rotating machinery, such as API 612 (steam turbines) and API 684, recommend a separation margin of roughly 15 to 20 percent or more between the operating speed and the nearest critical speed. When the simulator's nearest-critical-speed margin drops below 10 percent the design enters a zone where stiffness, span or shaft diameter must be re-tuned to move the natural frequency, or where additional damping must be added. Sub-critical and super-critical operation also use different margin philosophies, so the side on which the nearest critical speed sits matters as well.

Campbell Diagram Simulator — Rotor Critical Speeds

Parameters

Mode 1 natural frequency f_n1

Mode 2 natural frequency f_n2

Gyroscopic coupling g

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Operating speed

RPM

"Sweep" automatically drives the operating speed from 300 to 6000 RPM so you can watch it pass through each critical speed.

Results

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1x mode 1 critical speed

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1x mode 2 critical speed

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Nearest critical speed

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Margin (relative to operating)

Campbell Diagram (Frequency vs Speed)

Cyan = mode 1 (FW solid, BW dashed) / orange = mode 2 / white = engine-order lines (1x solid, 2x dashed, 3x dotted) / red dot = critical speed / yellow vertical line = operating speed

Critical Speed Map (Bar Chart)

Each bar = critical speed in RPM / yellow horizontal line = operating speed / the red bar marks the critical speed nearest the operating speed

Theory & Key Formulas

When a rotor spins, the gyroscopic effect splits each vibration mode into a forward whirl (FW) and a backward whirl (BW). With rotational speed $\Omega$ in rev/s, rest natural frequency $f_n$ in Hz, and gyroscopic coupling $g$:

Forward-whirl frequency (whirling with the rotation):

$$f_\text{FW}(\Omega) = f_n + g\,\Omega$$

Backward-whirl frequency (whirling against the rotation):

$$f_\text{BW}(\Omega) = f_n - g\,\Omega$$

k-th order engine-order excitation line:

$$f_\text{exc} = k\,\Omega$$

Critical speed where the kx line crosses the FW branch of mode m:

$$\Omega_\text{crit} = \frac{f_{n,m}}{k - g} \quad,\quad N_\text{crit}\,[\text{RPM}] = 60\,\Omega_\text{crit}$$

Operating close to any of these critical speeds turns even small unbalance into large vibration, so designs typically keep at least a 15 to 20 percent separation margin from the nearest critical speed.

What is the Campbell Diagram Simulator

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Whenever I open a turbine or compressor design report, "Campbell diagram" appears straight away. What exactly are we supposed to read off it?

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In one picture it tells you "at which running speed the rotor will resonate". The horizontal axis is rotational speed (RPM), the vertical axis is the natural frequency (Hz). Critical speeds are where the excitation lines (1x, 2x, 3x...) that grow with speed cross the natural-frequency branches. In the simulator above, the yellow vertical line is the operating speed and the red dots are the critical speeds.

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Wait, I thought the natural frequency was a fixed number. Why are the lines curved?

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That is the interesting part of rotor dynamics — gyroscopic effects come into play. As the rotor spins, each mode splits into a "forward whirl (FW)" branch that whirls with the rotation and rises in frequency, and a "backward whirl (BW)" branch that whirls against the rotation and falls. Push the "gyroscopic coupling g" slider up to 0.3 and you can see the cyan and orange lines fan apart visibly.

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I see! What about those slanted white 1x, 2x, 3x lines — what do they represent?

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Those are the synchronous excitation frequencies. 1x is unbalance and misalignment, 2x is coupling effects and shaft eccentricity, 3x and above is gear meshing or blade passing. As a formula it is $f = k\,\Omega$, so the kth-order line has slope k. In the simulator 1x is solid, 2x is dashed, and 3x is dotted.

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At 1500 RPM operating speed it shows the nearest critical at 1655 RPM with only a 10.3 percent margin. Is that safe?

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It is borderline. Standards such as API 612 and 684 ask for at least 15 to 20 percent separation, so 10.3 percent is in the redesign zone. Either move the natural frequency by changing the shaft diameter or the bearing support stiffness, or add damping to suppress the resonant peak. Try sliding f_n2 to 90 Hz in the simulator and watch the nearest critical speed shift.

Frequently Asked Questions

For teaching purposes the model is reduced to the two lowest representative modes of a rotor (typically the first and second bending modes, or the translational and conical modes). Real machines have dozens of densely packed modes, but only the two or three lowest ones usually fall inside the operating range, so the essence of a critical-speed map is captured by two modes times three excitation orders. Enter f_n1 and f_n2 as the natural frequencies at rest (Omega = 0).

g is a dimensionless gyroscopic coupling. In real machines it depends on the ratio of the polar to transverse moment of inertia of the disk, the bearing layout, and the rotor's length-to-diameter ratio. Long generator rotors give roughly 0.05 to 0.10, while thicker rotating bodies such as flywheels and turbine disks give about 0.15 to 0.25. The slider here covers 0.00 to 0.30; setting it to zero degenerates to the classical model in which natural frequencies do not depend on speed.

Synchronous excitation (1x) rotates with the shaft and so cannot directly excite the BW branch. Non-synchronous excitations such as fluid forces inside seals, rubbing, or internal friction, as well as external disturbances such as earthquakes, can excite BW, so it cannot be ignored entirely in design. This tool always shows BW as a dashed line but does not place red dots at its crossings: the FW x kx crossings (red dots) are the most important critical speeds in practice.

The operating speed sits exactly on a resonance, so even a tiny unbalance grows into large vibration amplitude. This is the classical "passing through a critical speed" problem in rotor-dynamics textbooks. A brief passage during start-up or shutdown is fine because damping limits the amplitude, but staying on a critical speed in steady operation is forbidden. In the simulator's bar chart this corresponds to the yellow operating line touching the tip of one of the bars. The cure is either to move the operating speed so the margin is again 15 to 20 percent, or to push the critical speed out of the operating range by tuning stiffness and mass.

Real-World Applications

Steam and gas turbine design: In thermal, nuclear and geothermal steam turbines, in jet engines and in industrial gas turbines, several critical speeds line up above and below the operating range (typically 3000 to 3600 RPM for utility turbines, tens of thousands of RPM for jet engines). Drawing a Campbell diagram early in the design and securing a 15 percent or larger separation margin (per API 612) is standard practice. Blade-passing frequencies (NPF) are added as additional excitation lines so blade-vibration interference is also evaluated.

Turbomachinery (compressors and pumps): In centrifugal and axial compressors, and in large boiler feed-water pumps, multiple impeller stages on a long shaft push natural frequencies close together. Because the operating speed has to pass through several critical speeds, the start-up and shutdown sequence is planned on the Campbell diagram so each critical speed is traversed quickly. Squeeze-film dampers and magnetic bearings are also chosen here when extra damping is needed.

Automotive engines and drivetrains: Torsional vibration of crankshafts and propeller shafts uses the same framework. The horizontal axis becomes engine speed and the vertical axis is the torsional natural frequency; for a 4-cylinder engine the dominant excitation is 2x (firing order), for a V8 it is 4x. Torsional vibration dampers — the rubber inside flywheels or dual-mass arrangements — are sized from this diagram to suppress amplitudes within the critical range.

Wind turbines and condition monitoring: In a large wind-turbine rotor and nacelle, the blade edgewise and flapwise modes, the tower bending mode, and the generator-shaft torsional mode all interact. Because the rotor speed varies, the Campbell diagram is used to identify "permanent resonance bands" and the variable-speed control is programmed to avoid sitting at dangerous speeds. Vibration data from condition-monitoring systems (CMS) is overlaid on the Campbell plot to detect new excitations or natural-frequency shifts early.

Common Misconceptions and Cautions

The most common misconception is to assume "there is one critical speed". Textbook examples of the Jeffcott rotor really do produce only one critical speed, but only because they consider one mode times one excitation. Real rotors have several modes, and orders other than 1x — such as 2x, 3x, and blade-passing frequencies — are often dominant, so several critical speeds always appear above and below the operating range. The six red dots produced by this simulator (two modes times three orders) reproduce that situation. The correct workflow is to check every critical speed near the operating range, not only the single nearest one.

The next pitfall is to design while ignoring the gyroscopic effect. Setting g = 0 in the simulator turns the natural-frequency curves into horizontal lines and reduces the critical speed to $N_\text{crit}=60 f_n / k$. Real disks and impellers always have a gyroscopic effect, so the correct expression is $N_\text{crit}=60 f_n/(k-g)$. For example, with f_n = 80 Hz, k = 1, and g = 0.1, ignoring the gyroscope predicts 4800 RPM but the real value is 5333 RPM — a 533 RPM gap. This is one of the most common reasons predictions disagree with measurements, so the gyroscopic effect must always be included in the analysis.

Finally, beware of the oversimplification "if the operating speed is far from any critical speed, all is well". The Campbell diagram is a linear, viscously damped model. In real machines, non-linear and non-conservative forces such as seal forces, fluid forces, and internal friction can excite backward whirl or sub-critical modes. The critical speeds shown by this tool correspond to "synchronous excitation of the forward whirl", which alone is not enough to declare a design safe. Detailed designs require a full eigenvalue analysis (complex modal analysis and stability analysis), and this simulator should be used as a tool for concept-stage understanding and initial scoping.

Campbell Diagram Simulator — Rotor Critical Speeds

What is the Campbell Diagram Simulator

Frequently Asked Questions

Real-World Applications

Common Misconceptions and Cautions

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