From Operating-Speed Range to Defensible Rotordynamic Substantiation

The Complete Chain

This article walks through the complete chain of rotordynamic substantiation, from the operating-speed envelope to the engineering conclusion. Each step has been the subject of an earlier article in this category. Here, the steps are assembled into the sequence that a real rotordynamic assessment follows — not as a rigid prescription, but as a checklist that ensures every link is visited before the assessment is declared complete.

Assess the rotor across its operating speed range — not at one static configuration. The complete chain runs from the speed envelope through the Campbell diagram, the response analysis, the test correlation and the sensitivity, to the engineering conclusion.

THE CHAIN

The chain is presented as an ordered list. Each step feeds the next, and each step feeds back to the previous ones. The chain is walked more than once — the first pass is the screening assessment, the second is the detailed analysis, the third is the test-correlated assessment, and the fourth is the substantiation.

1. Operating-speed envelope — what is the speed range, the steady-state operating speeds, the transient conditions (run-up, run-down), the dwell times and the over-speed conditions?
2. Rotor configuration — what is the shaft geometry, the disk locations and masses, the bearing locations, the coupling and the support arrangement?
3. Mass and inertia — what is the mass distribution, the centre of gravity, the transverse inertia and the polar inertia at each disk?
4. Bearing and support properties — what are the direct stiffness, cross-coupled stiffness, damping and speed-dependent properties of each bearing? What is the support (housing) flexibility?
5. Speed-dependent modal analysis — compute the natural frequencies and mode shapes at a series of speeds from zero to maximum, including gyroscopic, centrifugal and bearing-speed effects.
6. Campbell diagram — plot the mode branches and the excitation orders; identify the crossings within and near the operating-speed envelope.
7. Forcing and order definition — identify the excitation sources: unbalance (1×), misalignment (2×), blade passing (V×), engine order, harmonic torque. Define the forcing magnitudes.
8. Response analysis — compute the unbalance response (amplitude and phase vs speed) at the bearings, the shaft mid-span and the clearance-critical locations.
9. Clearance, load and fatigue assessment — check the vibration amplitude against the bearing load capacity, the shaft fatigue strength and the clearance margin at each location.
10. Stability analysis — compute the eigenvalues at each speed; identify the onset speed of instability if any; verify that the operating range is stable.
11. Test correlation — perform the run-up/run-down test; compare the measured critical speeds, modes, amplitudes and phases with the model; update the model to match.
12. Sensitivity analysis — assess the effect of variation in bearing stiffness, damping, unbalance and support flexibility on the critical speeds and the response.
13. Engineering conclusion — is the rotor safe across its operating-speed range? Are the critical speeds separated from the operating range or managed by damping? Is the response acceptable? Is the system stable? Is the assessment defensible?

Operating-Speed Envelope

The chain begins with the operating-speed envelope — the complete definition of the speeds at which the machine operates. This includes: the steady-state operating speeds (the normal operating speed and any alternative speeds), the transient conditions (run-up from rest to operating speed, run-down from operating speed to rest, overspeed testing), the dwell times (how long the machine stays at each speed — a machine that dwells near a critical speed is more at risk than one that passes through quickly), and the over-speed conditions (the maximum speed the machine can experience, including fault conditions and overspeed trip). The operating-speed envelope is the x-axis range of the Campbell diagram and the speed range of the response analysis. A complete definition of the envelope is the foundation of a defensible assessment — an incomplete definition ("it runs at 3,000 rpm") misses the transient and dwell conditions that may be the most critical.

• Steady-state operating speeds — normal and alternative
• Transient conditions — run-up, run-down, overspeed test
• Dwell times — how long at each speed; dwelling near a critical is high-risk
• Over-speed conditions — maximum speed including fault and trip
• The envelope is the x-axis of the Campbell diagram and the range of the response analysis

Rotor Configuration and Properties

The rotor configuration and properties define the model. The shaft geometry (length, diameter, material, steps and changes of section), the disk locations and their properties (mass, transverse inertia, polar inertia), the bearing locations and their properties (stiffness, damping, cross-coupling, speed dependence), the coupling (type, stiffness, inertia), and the support flexibility (housing stiffness) together determine the system's dynamic behaviour. Each property must be known accurately — the model is only as good as its inputs. The bearing properties are usually the most uncertain — they come from the bearing manufacturer's data or from the bearing analysis, and they may need to be updated during the test correlation. The mass and inertia properties are usually well known from the CAD model. The support flexibility is often uncertain — it depends on the housing geometry, the bolted joints and the foundation, and it may need to be measured or updated.

• Shaft geometry: length, diameter, material, section changes
• Disk properties: mass, transverse inertia, polar inertia at each location
• Bearing properties: stiffness, damping, cross-coupling, speed dependence — most uncertain
• Coupling: type, stiffness, inertia
• Support flexibility: housing stiffness — often uncertain, may need measurement or updating

Speed-Dependent Modal Analysis and Campbell Diagram

The speed-dependent modal analysis (Articles 07, 09) computes the natural frequencies and mode shapes at a series of speeds, including the gyroscopic, centrifugal and bearing-speed effects. The Campbell diagram (Article 07) plots the mode branches and the excitation orders, and the crossings within or near the operating envelope are the potential critical speeds. The Campbell diagram is the screening tool — it identifies where the problems may be, and it drives the subsequent response analysis. A crossing on the Campbell diagram is not automatically a problem (Article 07) — the response analysis determines whether the amplitude is acceptable. But a crossing that is not identified is a problem that is missed — the Campbell diagram must be complete, covering all modes and all orders in the operating range.

• Speed-dependent modal analysis: natural frequencies and mode shapes at a series of speeds
• Includes gyroscopic, centrifugal and bearing-speed effects
• Campbell diagram: mode branches + excitation orders → crossings = potential critical speeds
• The Campbell diagram is the screening tool — it identifies where to focus the response analysis
• A crossing not identified is a problem missed — the diagram must be complete

Response, Stability and Assessment

The response analysis (Article 10) computes the unbalance response — the vibration amplitude and phase at each speed, at each location. The stability analysis (Article 16) computes the eigenvalues — the system's net damping at each speed, and the onset speed of instability if any. The clearance and fatigue assessment checks the response amplitude against the bearing load capacity, the shaft fatigue strength and the clearance margin. Each of these is a quantitative check against a physical limit, and each must be reported and assessed. The response may be acceptable at the critical speed (if the damping is adequate), the stability may be acceptable (if the onset speed is above the operating range), and the clearance may be adequate (if the amplitude is below the gap). If any check fails, the design must be modified — more damping, better balancing, a design change to move the critical speed, or a bearing change to improve stability.

• Response analysis: amplitude and phase vs speed at each location
• Stability analysis: eigenvalues, onset speed of instability
• Clearance and fatigue assessment: amplitude vs bearing load, shaft fatigue, clearance gap
• Each check is quantitative — against a physical limit
• If any check fails: modify the design (damping, balancing, critical speed, bearing type)

Test Correlation and Sensitivity

The test correlation (Article 17) verifies the model against the real machine — the critical speeds, modes, amplitudes and phases must agree. The sensitivity analysis assesses the robustness — how much do the critical speeds and the response change if the bearing stiffness, the damping, the unbalance or the support flexibility varies within their uncertainty range? A design that is sensitive — a critical speed that moves into the operating range if the bearing stiffness changes by 10% — is a fragile design. A design that is robust — the critical speed stays clear of the operating range even with realistic variation — is a defensible design. The sensitivity analysis identifies the parameters that most affect the result, and it quantifies the margin against variation. The combination of test correlation and sensitivity analysis is the basis for engineering confidence: the model matches the machine (test), and the result is robust to realistic variation (sensitivity).

Test correlation verifies the model. Sensitivity analysis assesses robustness. A design that matches the test and is robust to parameter variation is a defensible design. A design that matches the test but is sensitive to variation is a fragile design — the margin is real, but it is not robust.

The Engineering Conclusion

The engineering conclusion is the final step: is the rotor safe across its operating-speed range? The conclusion is based on the complete evidence chain — the speed envelope, the rotor configuration, the bearing properties, the Campbell diagram, the response analysis, the stability analysis, the clearance and fatigue assessment, the test correlation and the sensitivity. The conclusion includes: the critical speeds and their separation from the operating range, the response amplitudes and their margin against the limits, the stability margin and the onset speed, the test correlation quality, and the sensitivity of the result to parameter variation. A defensible conclusion is one that can be explained: the engineer can trace the chain from the speed envelope to the conclusion, identify the critical speeds, quantify the margins, describe the stability, present the test correlation, and explain why the rotor is safe. The engineer who can only say "the vibration is within limits" has not made a defensible assessment — the rotor is a system, and the conclusion must address the system across its entire speed range.

A defensible rotordynamic substantiation is one whose dynamics can be explained — the speed envelope, the modes, the critical speeds, the crossings, the response, the stability, the test correlation and the sensitivity are all documented and consistent. The engineer who can trace the chain from the speed envelope to the conclusion has made a defensible assessment.

ASSESS THE ROTOR ACROSS ITS OPERATING SPEED RANGE

The overarching principle is that the rotor must be assessed across its operating speed range — not at one static configuration. The modes change with speed, the critical speeds are speed-dependent, the excitation frequencies move with speed, and the response and stability are functions of speed. A single-speed analysis — even at the operating speed — misses the transient conditions, the critical-speed crossings during run-up and run-down, and the speed-dependent stability. The complete assessment covers the full speed range, from zero to the maximum overspeed, and it addresses every link in the chain: modes, critical speeds, crossings, response, stability, clearance, fatigue, test and sensitivity. This is what it means to substantiate a rotor: to demonstrate, with analysis and test, that it is safe across its entire operating-speed range.

Assess the rotor across its operating speed range — not at one static configuration. The modes, the critical speeds, the excitation, the response and the stability all change with speed. A single-speed analysis misses the behaviour that matters. The complete assessment covers the full speed range, from zero to the maximum overspeed.

Key takeaways

• Assess the rotor across its operating speed range — not at one static configuration. The modes, the critical speeds and the response all change with speed, and a single-speed analysis misses the speed-dependent behaviour.
• The complete chain runs from the operating-speed envelope through rotor configuration, mass and inertia, bearing properties, speed-dependent modal analysis, Campbell diagram, forcing definition, response analysis, clearance and fatigue assessment, test correlation, sensitivity and engineering conclusion.
• A defensible rotordynamic substantiation is one whose dynamics can be explained — the speed envelope, the modes, the critical speeds, the crossings, the response, the stability and the test correlation are all documented and consistent.
• The chain is iterative. The Campbell diagram identifies the crossings; the response analysis evaluates them; the test correlation verifies the model; the sensitivity analysis assesses the robustness. Each step feeds back to the previous.