Understanding Campbell Diagrams in Rotor Dynamics
Definition: What is a Campbell Diagram?
A Campbell diagram (also known as a whirl speed map or interference diagram) is a graphical representation used in rotor dynamics that plots the system’s natural frequencies against rotational speed. The diagram is an essential tool for identifying critical speeds—the operating speeds at which resonance can occur—and for assessing whether adequate separation margins exist between operating speeds and these critical conditions.
Named after Wilfred Campbell, who developed the concept in the 1920s for analyzing aircraft engine vibrations, the Campbell diagram has become indispensable for designing and analyzing all types of high-speed rotating machinery, from turbines and compressors to electric motors and machine tool spindles.
Structure and Components of a Campbell Diagram
A Campbell diagram consists of several key elements that together provide a complete picture of a rotor system’s dynamic behavior:
The Axes
- Horizontal Axis (X-axis): Rotational speed, typically expressed in RPM (revolutions per minute) or Hz (Hertz)
- Vertical Axis (Y-axis): Frequency, usually in Hz or CPM (cycles per minute), representing the natural frequencies of the system
Natural Frequency Curves
The diagram shows curved or straight lines representing how each natural frequency of the rotor system changes with rotational speed. For most systems:
- Forward Whirl Modes: Natural frequencies that increase with speed due to gyroscopic stiffening effects
- Backward Whirl Modes: Natural frequencies that decrease with speed (less common, more prevalent in certain bearing types)
- Each mode (first bending, second bending, etc.) is represented by a separate curve
Excitation Lines
Diagonal straight lines superimposed on the diagram represent potential excitation sources:
- 1X Line: Passes through the origin at 45° (when axes have same scale), representing synchronous excitation from unbalance
- 2X Line: Representing twice-per-revolution excitation (from misalignment or other sources)
- Other Multiples: 3X, 4X, etc., for higher harmonic excitations
- Sub-synchronous Lines: Fractional multiples like 0.5X for phenomena like oil whirl
Intersection Points (Critical Speeds)
Where an excitation line crosses a natural frequency curve, a critical speed exists. At this speed, the excitation frequency matches the natural frequency, causing resonance and potentially dangerous vibration amplification.
How to Read and Interpret a Campbell Diagram
Identifying Critical Speeds
The primary purpose of a Campbell diagram is identifying critical speeds:
- Find intersections between excitation lines (1X, 2X, etc.) and natural frequency curves
- The horizontal coordinate of each intersection indicates a critical speed
- The more intersections present, the more critical speeds exist in the operating range
Assessing Separation Margins
Safe operation requires adequate “separation margin” between operating speeds and critical speeds:
- Typical Requirement: ±15% to ±30% separation from critical speeds
- Operating Speed Range: Usually indicated as a vertical band on the diagram
- Acceptable Design: Operating range should not overlap with critical speed zones
Understanding Mode Shapes
Different curves on the diagram correspond to different vibration modes:
- First Mode: Usually the lowest frequency curve, representing simple bending (like a jumping rope with one hump)
- Second Mode: Higher frequency, S-curve shape with a node point
- Higher Modes: Increasingly complex deflection patterns
Creating a Campbell Diagram
Campbell diagrams are generated through computational analysis or experimental testing:
Analytical Approach
- Build Mathematical Model: Create a finite element model of the rotor-bearing-support system
- Include Speed-Dependent Effects: Account for gyroscopic moments, bearing stiffness changes, and other speed-dependent parameters
- Solve Eigenvalue Problem: Calculate natural frequencies at multiple rotational speeds
- Plot Results: Generate curves showing how natural frequencies vary with speed
- Add Excitation Lines: Overlay 1X, 2X, and other relevant excitation lines
Experimental Approach
For existing machinery, Campbell diagrams can be created from test data:
- Perform startup or coastdown tests while continuously recording vibration
- Generate a waterfall plot showing vibration spectrum vs. speed
- Extract natural frequency peaks from the data
- Plot extracted frequencies vs. speed to create an experimental Campbell diagram
Applications in Machine Design and Analysis
Design Phase Applications
- Speed Range Selection: Determine safe operating speed ranges that avoid critical speeds
- Bearing Design: Optimize bearing location, type, and stiffness to position critical speeds appropriately
- Shaft Sizing: Adjust shaft diameter and length to move critical speeds away from operating ranges
- Support Structure Design: Ensure foundation and pedestal stiffness doesn’t create undesirable critical speeds
Troubleshooting Applications
- Resonance Diagnosis: Determine if high vibration is due to operating near a critical speed
- Speed Change Evaluation: Assess the impact of proposed speed increases or decreases
- Modification Analysis: Predict effects of machine modifications (added mass, stiffness changes, bearing replacements)
Operating Guidance
- Startup/Shutdown Procedures: Identify speed ranges to pass through quickly to minimize time at critical speeds
- Variable Speed Operation: Define safe speed ranges for variable speed drives
- Speed Restrictions: Establish forbidden speed ranges where operation should be avoided
Special Considerations and Advanced Topics
Gyroscopic Effects
For flexible rotors, gyroscopic moments cause natural frequencies to split into forward and backward whirl modes. The Campbell diagram clearly shows this splitting, with forward modes typically increasing and backward modes decreasing with speed.
Bearing Effects
Different bearing types affect the Campbell diagram differently:
- Rolling Element Bearings: Relatively constant stiffness, producing nearly horizontal natural frequency lines
- Fluid-Film Bearings: Stiffness increases with speed, causing natural frequencies to rise more steeply
- Magnetic Bearings: Active control can modify natural frequencies based on control algorithms
Anisotropic Systems
When rotor systems have different stiffness in different directions (asymmetric bearings or supports), the Campbell diagram must show separate curves for horizontal and vertical vibration modes.
Campbell Diagram vs. Other Rotor Dynamic Plots
Campbell Diagram vs. Bode Plot
- Campbell Diagram: Shows natural frequencies vs. speed, predicts where critical speeds will occur
- Bode Plot: Shows measured vibration amplitude and phase vs. speed, confirms actual critical speed locations
Campbell Diagram vs. Interference Diagram
The terms are sometimes used interchangeably, though “interference diagram” typically emphasizes the intersection points (interferences) between natural frequencies and excitation orders.
Practical Example
Consider a high-speed compressor designed to operate at 15,000 RPM (250 Hz):
- Campbell Diagram Shows: First critical speed at 12,000 RPM (1X), second critical speed at 22,000 RPM (1X)
- Analysis: Operating speed of 15,000 RPM is safely between the two critical speeds with adequate margins (25% below second critical, 20% above first critical)
- Operating Guidance: During startup, accelerate quickly through 12,000 RPM to minimize time at first critical speed
- Speed Increase Study: If considering operation at 18,000 RPM, Campbell diagram shows this would reduce separation margin from second critical to unacceptable 18%—modification would require bearing or shaft redesign
Modern Software and Tools
Today, Campbell diagrams are typically generated using specialized software:
- Rotor dynamics analysis packages (MADYN, XLTRC, DyRoBeS, ANSYS, etc.)
- Built-in plotting functions in vibration analysis software
- Post-processing tools for experimental data
- Integration with condition monitoring systems for real-time tracking
These tools allow rapid what-if analysis, optimization studies, and correlation between predicted and measured behavior, making Campbell diagrams more accessible and useful than ever for engineers working with rotating machinery.
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