Campbell Diagram
A frequency-vs-speed map that reveals critical speeds, gyroscopic splitting, and resonance hazard zones in rotating machinery — from micro-turbines to multi-megawatt compressor trains.
Definition
A Campbell diagram (also called a whirl speed map or interference diagram) is a graph that plots the natural frequencies of a rotor-bearing system on the vertical axis against rotational speed on the horizontal axis. Diagonal excitation-order lines (1×, 2×, 3×…) are superimposed; wherever an excitation line crosses a natural-frequency curve, a critical speed exists. The diagram is the primary tool for determining whether a machine's operating range is safely separated from resonance conditions.
In a sentence: the Campbell diagram answers one question — "At which speeds will this rotor resonate, and how close are those speeds to where I plan to operate?"
Historical Background
Wilfred Campbell published the concept in 1924 while studying circumferential waves in steam-turbine discs at General Electric. His original chart plotted disc vibration modes against rotational speed to predict where destructive resonances would appear during operation.
The approach filled a gap that had troubled engineers since the 1890s. W. J. M. Rankine's 1869 shaft-whirling analysis had incorrectly predicted that supercritical operation was impossible. Gustaf de Laval proved otherwise by running a steam turbine above its first critical speed in 1889. Henry Jeffcott's landmark 1919 paper finally explained why supercritical operation is stable, but Campbell's diagram gave engineers the visual tool to predict exactly where those dangerous speeds lie — and how to design around them.
Over the following decades, the concept expanded from disc vibrations to full lateral rotor analysis, torsional analysis, and even acoustics. Today, every major API, ISO, and IEC standard for rotating machinery either requires or recommends Campbell-diagram analysis.
Anatomy of the Diagram
A Campbell diagram carries four families of information on a single plot. Understanding each layer is necessary before you can read intersections correctly.
Axes
The horizontal axis is rotational speed, typically in RPM or Hz. The vertical axis is frequency, in Hz or CPM. When both axes use the same unit, the 1× excitation line runs at exactly 45° — a useful visual check that the scale is correct.
Natural-Frequency Curves
Each curve represents one vibration mode of the rotor-bearing-support system. In the simplest case (rigid bearings, no gyroscopic effects), these curves are horizontal lines because the natural frequencies do not change with speed. In reality, gyroscopic moments and speed-dependent bearing stiffness cause the curves to slope, split, or both.
Modes are labeled by deflection shape: first bending (one antinode), second bending (two antinodes with one node), third bending, and so on. Torsional and axial modes may also be plotted if relevant.
Forward and Backward Whirl
When gyroscopic effects are significant, each non-spinning natural frequency splits into two curves as speed increases:
- Forward whirl (FW): the mode precesses in the same direction as the shaft rotation. Gyroscopic stiffening pushes its frequency up.
- Backward whirl (BW): the mode precesses opposite to rotation. Gyroscopic softening pushes its frequency down.
Forward whirl modes are the primary concern for unbalance-driven resonance because unbalance excites synchronous forward precession.
Excitation-Order Lines
These are straight diagonal lines radiating from the origin. Each line represents an excitation whose frequency is a fixed multiple of rotational speed:
| Line | Relationship | Typical Source |
|---|---|---|
| 1× | f = 1 × RPM/60 | Mass unbalance, shaft bow |
| 2× | f = 2 × RPM/60 | Misalignment, cracked shaft, ovality |
| 3×, 4×… | f = n × RPM/60 | Gear mesh, vane/blade pass, coupling defects |
| 0.43–0.48× | f ≈ 0.45 × RPM/60 | Oil whirl in fluid-film bearings |
| Blade-pass | f = Z × RPM/60 | Number of blades Z × running speed |
Intersection Points = Critical Speeds
Each intersection between an excitation line and a natural-frequency curve marks a potential resonance. The RPM value at that intersection is a critical speed for that particular mode-excitation combination. If the operating range includes or is close to that RPM, the machine risks high vibration amplitudes.
Interactive Campbell Diagram
The SVG below shows a typical Campbell diagram for a two-bearing, flexible-shaft rotor. Hover over elements to identify modes, excitation lines, and critical speed intersections.
Fig. 1 — Campbell diagram for a flexible two-bearing rotor. Gold circles mark critical speeds (CS₁, CS₂). The amber band shows the operating-speed range 9,000–12,000 RPM.
How to Read and Interpret a Campbell Diagram
Step-by-Step Reading Procedure
Identify the Operating Speed Range
Locate the vertical band or tick marks indicating minimum and maximum continuous operating speeds. In Fig. 1, this is 9,000–12,000 RPM.
Trace the 1× Line First
The 1× synchronous line is the most critical because unbalance — present in every rotor — excites at 1× running speed. Find every point where it crosses a forward-whirl curve.
Read Horizontal Coordinates at Intersections
Each intersection's x-coordinate is a critical speed. Record each one along with the mode number it involves.
Check 2× and Higher-Order Intersections
Repeat for 2×, 3×, blade-pass, and sub-synchronous lines. These intersections are secondary critical speeds — lower energy than 1× but still capable of causing vibration problems, especially if the excitation source is strong.
Calculate Separation Margins
For each critical speed, compute the percentage distance to the nearest edge of the operating range. Compare against applicable standards (API 617, API 612, ISO, OEM spec).
Evaluate Curve Slopes
Steep upward-sloping FW curves indicate strong gyroscopic effects — common in overhung rotors. Nearly flat curves suggest the system is bearing-stiffness dominated.
Identify Hazard Zones
If two critical speeds bracket the operating range with insufficient margins, the design must be modified: bearing stiffness, shaft diameter, support stiffness, or operating speed must change.
⚠️ A common misunderstanding: backward-whirl modes rarely respond to unbalance excitation because unbalance produces only forward precession. Intersections with BW curves are usually not true operational critical speeds — they are included on the diagram for completeness and for cases where other excitation sources exist (e.g., reverse-rotating flow in seals).
Understanding Separation Margins
Safe operation demands that the operating speed range lies far enough from every critical speed so that the resonance amplification is tolerable. The required margin depends on the sharpness of the resonance peak, quantified by the amplification factor (AF).
- A low AF (< 2.5) means heavy damping — the rotor can operate close to or even at the critical speed without excessive vibration.
- A high AF (> 8) means a sharp peak — even a few percent deviation from the critical speed causes dangerous amplitude growth.
Typical industrial practice calls for 15–30% separation, but the exact requirement depends on the governing standard and the AF value.
Gyroscopic Effects and Frequency Splitting
When a spinning disc precesses (wobbles), gyroscopic moments arise that couple the motion in two perpendicular planes. This coupling splits what would be a single natural frequency at zero speed into two distinct frequencies at any non-zero speed.
The Physics
The equation of motion for a rotor with gyroscopic effects takes the form:
where M is the mass matrix, C the damping matrix, G the skew-symmetric gyroscopic matrix (proportional to spin speed Ω), and K the stiffness matrix. Because G is speed-dependent, the eigenvalues — and therefore the natural frequencies — change with Ω.
What Determines the Splitting Magnitude?
The ratio of polar moment of inertia (Ip) to diametral moment of inertia (Id) controls how strongly the gyroscopic effect acts. Disc-like components (Ip/Id > 1) produce strong splitting. Long, slender shaft sections (Ip/Id ≈ 0) produce negligible splitting.
Overhung rotors (single-stage pump impellers, turbocharger wheels, cantilevered grinding wheels) exhibit the most pronounced gyroscopic splitting. In these designs, the forward-whirl first critical speed can be 20–40% higher than the zero-speed natural frequency, meaning the Campbell diagram differs dramatically from a simple "flat-line" model. Running a flat-line analysis for an overhung rotor will underpredict the first FW critical and overpredict the first BW critical, potentially leading to incorrect operating-speed decisions.
How Bearing Type Shapes the Campbell Diagram
Bearings connect the rotor to the stator and define the boundary conditions that determine natural frequencies. Different bearing technologies produce fundamentally different diagram shapes.
| Bearing Type | Stiffness Behavior | Effect on Campbell Curves | Additional Concerns |
|---|---|---|---|
| Rolling Element (ball, roller) | Nearly constant with speed | Natural frequency curves are approximately flat (horizontal) unless gyroscopic effects dominate | Defect frequencies (BPFO, BPFI, BSF) add excitation lines at non-integer orders |
| Fluid-Film (Journal) | Stiffness and damping increase with speed (Sommerfeld number changes) | Curves slope upward more steeply than gyroscopic effect alone would produce | Cross-coupled stiffness can cause instability (oil whirl/whip); add 0.43–0.48× sub-synchronous line |
| Tilting-Pad Journal | Stiffness increases with speed; minimal cross-coupling | Similar slope to plain journal but with better stability | Preferred for high-speed compressors per API 617 |
| Active Magnetic | Programmable via control algorithm; can be constant, increasing, or adaptive | Curves can be intentionally shaped to move critical speeds away from operating range | Control-loop bandwidth limits maximum achievable stiffness at high frequencies |
| Gas (Foil/Aerostatic) | Stiffness increases sharply with speed; very low damping | Steeply rising curves; high-Q resonances | Low damping makes separation margins even more critical |
Anisotropic Supports
When the bearing support pedestal or foundation has different stiffness in the horizontal and vertical directions, each mode further splits into horizontal and vertical variants. The Campbell diagram then shows even more curves — a horizontal FW, a vertical FW, a horizontal BW, and a vertical BW for each mode. This is typical in horizontal machines with flexible foundations.
API 617 and Separation-Margin Requirements
For centrifugal and axial compressors in petroleum, chemical, and gas service, API Standard 617 (8th Ed., 2014; 9th Ed., 2022) mandates a rigorous Campbell-diagram analysis as part of the lateral rotordynamic study.
The API 617 Separation-Margin Formula
where SM is the required separation margin (%) and AF is the amplification factor from the unbalance-response (Bode) plot at that critical speed.
| AF Value | SM per Formula | Interpretation |
|---|---|---|
| < 2.5 | No SM required | Critically damped; may operate at the critical speed |
| 3.5 | 8.5% | Moderate damping; small margin sufficient |
| 5.0 | 12.1% | Typical for tilting-pad bearings |
| 8.0 | 14.4% | Sharp peak; larger margin needed |
| 12.0 | 15.4% | Very sharp; approaching 16% cap |
| > ~11 | ≤ 16% (capped) | API caps SM at 16% for CS below min speed |
Applying This to the Campbell Diagram
During design review, the engineer reads each critical speed from the Campbell diagram, then checks the corresponding AF from the Bode plot. If SMactual ≥ SMrequired, the design passes. If not, the engineer must modify bearings, shaft geometry, or operating range until all margins are met.
Other standards with similar requirements: API 612 (steam turbines), API 613 (gear units), API 672 (packaged air compressors), ISO 10814 (tolerance of critical-speed proximity), ISO 22266 (mechanical vibration of non-reciprocating machines). Each uses slightly different formulas or fixed-percentage thresholds, but all rely on the Campbell diagram as the source data.
Creating a Campbell Diagram: Analytical vs. Experimental
Analytical (FEA / Transfer Matrix) Approach
Build the Rotor Model
Discretize the shaft, discs, impellers, couplings, and sleeves into beam elements (Timoshenko or Euler-Bernoulli) or 3D solid/shell elements. Include mass, stiffness, and gyroscopic terms.
Define Bearing Properties
Input speed-dependent stiffness and damping coefficients (8 coefficients for each fluid-film bearing: Kxx, Kxy, Kyx, Kyy, Cxx, Cxy, Cyx, Cyy). For rolling-element bearings, use constant stiffness values.
Set Speed Range and Increments
Define a speed sweep from 0 to at least 115% of maximum continuous speed (per API 617 trip-speed requirement), with fine enough RPM increments (typically 100–500 RPM steps) to capture curve shapes accurately.
Solve the Complex Eigenvalue Problem
At each speed step, solve det(K + iΩG − ω²M) = 0 to find natural frequencies ωn (imaginary parts) and damping (real parts). The imaginary parts become the y-coordinates on the Campbell diagram.
Plot and Overlay Excitation Lines
Plot all modes vs. speed, add 1×, 2×, and other relevant excitation lines, and mark intersections.
Experimental Approach (From Field Data)
When a machine already exists, a Campbell diagram can be extracted from vibration measurements during a run-up or coastdown:
- Mount accelerometers or proximity probes at bearing locations.
- Record vibration continuously during a slow startup (or coastdown after trip).
- Generate a waterfall (cascade) plot: a stack of FFT spectra taken at successive RPM values.
- Identify frequency peaks at each RPM slice — these are the natural frequencies excited by whichever order dominates.
- Plot the peak frequencies vs. RPM to produce an experimental Campbell diagram.
Coastdown tests often produce cleaner data than startups because the machine decelerates smoothly without the torque fluctuations of a motor starting. Run the coastdown from trip speed to rest with continuous high-resolution data acquisition (≥ 4,096 lines, 0.5-second averaging). If the machine uses a VFD, program a linear ramp at 50–100 RPM/second for best spectral resolution.
Applications by Machine Type
| Machine | Typical Speed Range | Key Campbell-Diagram Concerns | Governing Standard |
|---|---|---|---|
| Centrifugal Compressor | 3,000–60,000 RPM | Multiple critical speeds; fluid-film bearing instability; seal cross-coupling; typically 2–4 modes below trip speed | API 617 |
| Steam Turbine | 3,000–15,000 RPM | Blade-pass excitation; thermal bow shifting modes during warmup; disc modes at high orders | API 612 |
| Gas Turbine | 3,600–30,000 RPM | Dual-spool designs require separate Campbell diagrams for each spool; squeeze-film damper effects | API 616 / OEM |
| Electric Motor / Generator | 750–36,000 RPM | Electromagnetic excitation at 2× line frequency; VFD-driven motors require sweep through resonances | API 541 / IEC 60034 |
| Pump | 1,000–12,000 RPM | Overhung impeller with strong gyroscopic effects; vane-pass excitation; wear-ring stiffness changes over time | API 610 |
| Machine-Tool Spindle | 5,000–60,000+ RPM | Preloaded angular-contact bearings; speed-dependent preload loss softens frequencies at high speed | ISO 15641 / OEM |
| Turbocharger | 30,000–300,000 RPM | Floating-ring bearings with complex inner/outer film dynamics; sub-synchronous whirl common | OEM / SAE |
| Wind Turbine Gearbox | 10–20 RPM (rotor); up to 1,800 RPM (HSS) | Torsional Campbell diagram for gear-mesh resonances; multiple speed ratios | IEC 61400 / AGMA |
Design-Phase Uses
During design, the Campbell diagram guides decisions about shaft diameter, bearing placement, bearing type, and impeller/disc geometry. Moving a critical speed by just 10% may require changing bearing span by 50 mm or shaft diameter by 5 mm — the diagram shows engineers exactly how much shift is needed.
Troubleshooting Uses
If a machine develops high 1× vibration at a specific speed, the Campbell diagram quickly shows whether that speed coincides with a predicted critical. If it does, the solution is either to change the operating speed, add damping (e.g., squeeze-film damper), or improve balancing quality. If it does not, the high vibration likely has a different root cause such as mechanical looseness or bearing defect.
Operating Guidance
The Campbell diagram defines forbidden speed ranges — RPM bands where sustained operation is not allowed because a critical speed falls within the band. Variable-speed machines (VFD-driven compressors, turbine-generator sets with load-following) must have their Campbell diagrams reviewed to ensure no continuous-duty operating point sits in a forbidden band. Transient passage through a critical speed during startup or shutdown is acceptable if the acceleration rate is high enough to prevent amplitude buildup.
Measure What the Diagram Predicts
The Balanset-1A portable analyzer records vibration data you need for experimental Campbell diagrams — spectrum vs. RPM during run-up and coastdown. Two-plane balancing in the field. From €1,975.
Related Diagrams and Plots
The Campbell diagram is one of several interrelated visualizations in rotordynamic analysis. Each serves a distinct purpose.
Campbell Diagram
Axes: natural frequency vs. rotational speed.
Shows: where critical speeds will occur (predictive). Based on eigenvalue analysis or extracted from waterfall data.
Bode Plot
Axes: vibration amplitude & phase vs. rotational speed.
Shows: measured response during actual run-up/coastdown. Confirms critical-speed locations and provides amplification factors for margin calculations.
Waterfall (Cascade) Plot
Axes: frequency spectrum vs. rotational speed (3D).
Shows: full spectral content at each RPM step. Source data for extracting experimental Campbell diagrams. Reveals all excitation orders simultaneously.
Undamped Critical-Speed Map
Axes: natural frequency vs. bearing stiffness (not speed).
Shows: how critical speeds shift as support stiffness changes. Used in early design to bracket the bearing stiffness range before generating the full Campbell diagram.
Orbit Plot
Axes: X-displacement vs. Y-displacement at a single speed.
Shows: the shape of shaft motion at a specific RPM. Forward whirl produces a circular orbit; backward whirl produces a retrograde ellipse.
Stability Map
Axes: logarithmic decrement (or real eigenvalue) vs. speed.
Shows: where the system is stable (positive damping) vs. unstable (negative damping). A Campbell diagram extended by one dimension.
Practical Example: High-Speed Compressor
Consider a centrifugal compressor designed for 15,000 RPM continuous operation (250 Hz), with trip speed at 17,250 RPM (115%).
Campbell Diagram Results
- 1st FW Critical (1×): 5,200 RPM (86.7 Hz) — safely below operating range.
- 2nd FW Critical (1×): 19,800 RPM (330 Hz) — above trip speed.
- 1st FW × 2×: 2,600 RPM — only relevant during startup; passed through quickly.
Margin Check
Minimum operating speed: 12,000 RPM. Separation from 1st FW critical at 5,200 RPM:
The AF at this critical from the Bode plot is 4.2, yielding a required SM of 10.7% per the API 617 formula. Actual SM of 56.7% far exceeds the requirement — no issue.
Separation from 2nd FW critical at 19,800 RPM to trip speed 17,250 RPM:
The AF at this critical is 6.5, yielding a required SM of 13.6%. Actual SM of 14.8% passes, but marginally. The engineer flags this in the report and recommends verifying the exact AF during shop mechanical running tests.
If fouling increases impeller mass by 3%, the 2nd FW critical drops from 19,800 to approximately 19,200 RPM, reducing the separation margin to 11.3% — below the required 13.6%. This scenario must be captured in the sensitivity analysis submitted with the API datasheet.
Software Tools for Campbell Diagrams
Campbell diagrams are produced by both general-purpose FEA platforms and dedicated rotordynamics packages.
| Tool | Type | Notes |
|---|---|---|
| ANSYS Mechanical (Rotordynamics) | General FEA | Full 3D solid + beam models; built-in Campbell chart post-processor; requires damped modal analysis with RGYRO |
| Siemens Simcenter 3D | General FEA | Superelement reduction for multi-rotor systems; integrated orbit and stability plots |
| DyRoBeS | Dedicated rotordynamics | Beam-element based; fast; widely used in compressor and turbine OEMs per API 684 tutorial |
| XLTRC² (Texas A&M) | Dedicated rotordynamics | Spreadsheet-based workflow; strong bearing coefficient library; popular in pump and compressor analysis |
| MADYN 2000 | Dedicated rotordynamics | German-developed; FE + transfer-matrix hybrid; excellent for torsional + lateral coupled analyses |
| COMSOL Multiphysics | General FEA | Rotordynamics module for custom models; programmable post-processing |
| Bently Nevada System 1 / ADRE | Condition monitoring | Extracts experimental Campbell diagrams from field vibration data; real-time tracking |
Common Mistakes When Using Campbell Diagrams
1. Ignoring Gyroscopic Effects
Running an undamped, zero-speed modal analysis and assuming those frequencies are the critical speeds. This produces flat lines that miss forward/backward splitting entirely. Always solve the speed-dependent eigenvalue problem.
2. Using Too Coarse a Speed Increment
If the RPM step is 2,000 RPM in a machine running at 10,000, you might miss a narrow crossing entirely. Use increments of 100–500 RPM for reliable curve definition.
3. Confusing Campbell and Bode
The Campbell diagram predicts where criticals are; the Bode plot shows how severe they are. Both are required for a complete rotordynamic evaluation per API 617.
4. Neglecting Foundation and Support Flexibility
A rotor model with rigid supports will produce different critical speeds than the same rotor on a real flexible foundation. Include pedestal and foundation compliance in the model.
5. Forgetting Temperature and Load Effects
Bearing clearances change with temperature, altering stiffness coefficients. Process-gas density affects seal cross-coupling. The Campbell diagram should be run at both minimum and maximum clearance / density conditions.
6. Treating All Intersections as Equally Dangerous
A 1× intersection with the first forward mode is far more dangerous than a 4× intersection with a high backward mode. Prioritize by excitation energy and mode type.
Need On-Site Vibration Data?
Balanset-1A captures vibration spectra during run-up/coastdown for waterfall plots and experimental Campbell diagrams. Two-channel, two-plane, ISO 1940 compliant. Ships worldwide via DHL Express.
Frequently Asked Questions
What is the difference between a Campbell diagram and a Bode plot?
A Campbell diagram plots the system's natural frequencies against rotational speed — it predicts at which speeds critical conditions exist. A Bode plot plots actual measured (or calculated) vibration amplitude and phase against rotational speed — it shows how much the rotor vibrates at those critical speeds. Engineers use the Campbell diagram for design and the Bode plot for verification. Both are required by API 617 for compressor certification.
What separation margin does API 617 require from critical speeds?
API 617 uses the formula SM = 17 × {1 − [1/(AF − 1.5)]}, where AF is the amplification factor at that critical speed. If AF < 2.5, no margin is required because the resonance is overdamped. For typical tilting-pad bearings (AF = 4–8), required margins range from 10% to 15%. The maximum required SM is capped at 16% for critical speeds below minimum operating speed. For critical speeds above maximum continuous speed, the same formula applies but the margin is calculated as a percentage of the maximum continuous speed.
Why do natural frequencies split into forward and backward whirl on the Campbell diagram?
Gyroscopic moments from spinning discs couple the rotor's motion in two perpendicular planes. This coupling creates two distinct precession patterns: forward whirl (precession in the same direction as shaft rotation, stiffened by the gyroscopic effect) and backward whirl (precession opposite to rotation, softened by the effect). The higher the disc's polar-to-diametral inertia ratio, the stronger the splitting. At zero speed, there is no gyroscopic moment, so both modes coalesce to a single frequency.
Can you create a Campbell diagram from field measurements?
Yes. Record vibration during a continuous startup (or coastdown) using accelerometers or proximity probes at bearing housings. Process the time-domain data into a waterfall (cascade) plot — a series of FFT spectra at each RPM increment. Extract the peak frequencies at each RPM step, then plot those peaks against RPM. The result is an experimental Campbell diagram. Coastdowns tend to give cleaner data because there are no motor-starting torque transients. Aim for a deceleration rate of 50–100 RPM/s and use at least 4,096 FFT lines for good frequency resolution.
What excitation orders should be included on a Campbell diagram?
At minimum, always include the 1× line (unbalance — the single most common excitation source in all rotating machinery). Add 2× for misalignment, shaft ovality, or cracked shafts. For turbomachinery, include blade-pass frequency (number of blades × 1×) and vane-pass frequency. For geared systems, include gear-mesh frequency. For machines with fluid-film bearings, add a 0.43–0.48× line for oil whirl. If the machine has a known defect pattern (e.g., coupling with 6 jaws), include that order (6×).
How does bearing type affect the shape of a Campbell diagram?
Rolling-element bearings have nearly constant stiffness across the speed range, so natural-frequency curves remain almost flat (horizontal) — the only slope comes from gyroscopic effects. Fluid-film (journal) bearings increase in stiffness with speed as the oil film thins and becomes stiffer, causing natural-frequency curves to rise more steeply. Tilting-pad journal bearings behave similarly but produce less cross-coupling, improving rotor stability. Active magnetic bearings can be programmed to shift stiffness in real time, allowing engineers to reshape the Campbell diagram dynamically to avoid resonances.
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