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Glossary term

Unbalance Response

Speed-dependent vibration amplitude and phase caused by residual rotating mass eccentricity in a rotor-bearing system.

Branch
Mechanical Engineering
Glossary type
model
Content
Glossary term
Updated
Revision
v1.0.1 ยท reviewed

Definition

model

Unbalance response is the speed-dependent vibration amplitude and phase produced when residual mass eccentricity creates a rotating force in a rotor-bearing system.

Unbalance response analysis predicts how a rotor responds to the synchronous 1x force generated by residual unbalance. It connects unbalance magnitude, speed, modal mass, stiffness, damping, critical speed, vibration amplitude and phase. Engineers use it to assess balancing, run-up behaviour, operating-speed margins, bearing loads, acceptance limits and whether a 1x vibration peak is plausible for an imbalance mechanism.

Unbalance response is the vibration response produced when a rotor has residual rotating mass eccentricity. The unbalance is commonly written as:

U=m_e e

where U is unbalance, m_e is the equivalent unbalanced mass and e is eccentricity radius. At angular speed \Omega, the rotating unbalance force amplitude is:

F_u=U\Omega^2

This force acts at 1x shaft speed. The measured response depends not only on U, but also on rotor mass distribution, bearing stiffness, damping, support flexibility, mode shape, speed, sensor direction and proximity to a critical speed.

Engineering Role

Unbalance response analysis helps engineers decide whether a 1x vibration peak is consistent with residual mass eccentricity, whether a machine can pass through a critical speed, whether balancing is likely to help, and whether an operating speed has enough separation from a resonant condition.

A typical unbalance response review looks at amplitude and phase versus speed. Below a critical speed, the displacement response is usually small and has limited phase lag. Near a lightly damped critical speed, amplitude can rise sharply and phase changes rapidly. Above the critical speed, amplitude may reduce, while phase approaches the post-resonance side of the response. This phase progression is one reason tachometer reference, order tracking, orbit plots and waterfall spectra are valuable during run-up and coast-down tests.

Unbalance response is not a diagnosis by itself. Misalignment, looseness, rubs, bent shafts, electrical forcing, aerodynamic excitation and measurement faults can also create strong 1x or near-1x content. A defensible conclusion needs speed trend, phase stability, sensor evidence, machine geometry and repeatability.

Evidence Boundary for Balancing

A useful unbalance-response review treats vibration as a vector, not only as a scalar amplitude. At a specified speed and measurement point, the synchronous response can be written as:

V=Ae^{j\phi}

where A is the 1x amplitude and \phi is phase relative to a once-per-revolution reference. If the phase is unstable, the speed is drifting, the tachometer reference is unreliable or the response is strongly nonlinear, a balance correction based on that vector is weak evidence.

For a trial-weight or correction-weight workflow, the measured change in vibration vector is often screened as:

\Delta V=V_{trial}-V_0

where V_0 is the original synchronous response. A coherent \Delta V supports the idea that the machine is responding to a controlled unbalance change. A small, random or phase-inconsistent \Delta V suggests that looseness, rub, process excitation, poor sensor mounting, speed change, mode interaction or measurement error may be dominating the observation.

The measurement boundary also matters. Shaft displacement from proximity probes, bearing housing velocity and casing acceleration can tell different stories because bearings, supports and frames filter the rotor motion. A high casing 1x peak can be consistent with unbalance, but the engineer should still check runout, orbit shape, support flexibility, bearing condition, critical-speed proximity and whether the response follows a repeatable amplitude-phase trend through speed.

Simplified Response Model

For a single-mode approximation with modal mass M, natural angular frequency \omega_n and damping ratio \zeta, define the speed ratio:

\displaystyle r=\frac{\Omega}{\omega_n}

The displacement amplitude caused by unbalance can be approximated by:

\displaystyle X=\frac{U}{M}\frac{r^2}{\sqrt{(1-r^2)^2+(2\zeta r)^2}}

The phase lag of displacement relative to the rotating force is:

\displaystyle \phi=\tan^{-1}\left(\frac{2\zeta r}{1-r^2}\right)

with quadrant interpreted correctly. The equation is a teaching model, not a complete rotor-dynamic analysis. Real machines can have multiple modes, anisotropic bearings, gyroscopic splitting, cross-coupled fluid forces, nonlinear supports and separate measurement directions.

Worked Example: Speed-Dependent Amplitude and Phase

A simplified fan rotor has estimated residual unbalance:

U=0.0004\ \text{kg m}

modal mass:

M=80\ \text{kg}

first critical frequency:

f_n=30\ \text{Hz}

and damping ratio:

\zeta=0.08

The natural angular frequency is:

\omega_n=2\pi f_n=2\pi(30)=188.5\ \text{rad/s}

The unbalance displacement scale is:

\displaystyle \frac{U}{M}=\frac{0.0004}{80}=5.0\times10^{-6}\ \text{m}

or:

5.0\ \mu\text{m}

Evaluate three speeds.

SpeedForcing frequencySpeed ratioDisplacement amplitudePhase lag
1200\ \text{rpm}20\ \text{Hz}0.6673.93\ \mu\text{m}10.9^\circ
1800\ \text{rpm}30\ \text{Hz}1.00031.25\ \mu\text{m}90.0^\circ
2400\ \text{rpm}40\ \text{Hz}1.33311.02\ \mu\text{m}164.7^\circ

At 1800\ \text{rpm} the forcing frequency equals the natural frequency, so the denominator is governed by damping:

\sqrt{(1-1^2)^2+(2(0.08)(1))^2}=0.16

and:

\displaystyle X=5.0\ \mu\text{m}\frac{1}{0.16}=31.25\ \mu\text{m}

The peak velocity amplitude at that speed is:

v_{pk}=\Omega X=188.5(31.25\times10^{-6})=5.89\times10^{-3}\ \text{m/s}

The RMS velocity is:

\displaystyle v_{RMS}=\frac{v_{pk}}{\sqrt{2}}=4.16\ \text{mm/s RMS}

Engineering comment: the same residual unbalance produces a much larger response near the critical speed than below it. Balancing reduces U, but changing bearing stiffness, support stiffness or damping changes the response shape. If the measured 1x phase does not follow a coherent speed-dependent trend, the engineer should not assume simple unbalance.

Unbalance response is not residual unbalance alone. Residual unbalance is the forcing source. Unbalance response is the machine motion caused by that source after stiffness, damping, mass and speed are included.

Unbalance response is not critical speed. Critical speed identifies a speed range where response can be amplified. Unbalance response predicts amplitude and phase across the speed range.

Unbalance response is not order tracking. Order tracking labels the synchronous component by shaft order. Unbalance response explains how the 1x component should change with speed for a physical unbalance model.

Unbalance response is not a field-balancing procedure. Influence-coefficient balancing uses measured vibration vectors and trial weights to choose correction mass. Unbalance response analysis is the dynamic model that explains why such vectors change with speed and phase.

Validation and Common Mistakes

A defensible unbalance response review states the unbalance convention, speed range, operating load, bearing and support model, assumed modal mass, damping, critical-speed estimate, sensor type, measurement direction, phase reference, filtering, uncertainty and acceptance criterion.

Common mistakes include:

  • diagnosing unbalance from a 1x peak without checking phase stability, speed trend and machine geometry;
  • applying a single-degree-of-freedom formula to a rotor with multiple close modes or anisotropic bearings without stating the limitation;
  • ignoring runout, bent shaft, looseness, rubs, electrical forcing or process excitation;
  • using casing vibration as if it were shaft motion without considering transfer through bearings and supports;
  • balancing through an unstable or nonlinear response where the trial weight does not produce a coherent vector change;
  • comparing run-up and coast-down response without noting ramp rate, load, thermal state and filtering;
  • treating low vibration away from resonance as proof that critical-speed behaviour is acceptable.
REF

See also

  1. Rotor DynamicsMechanical Engineeringterm
  2. Critical SpeedMechanical Engineeringquantity
  3. ResonanceMechanical Engineeringphenomenon
  4. Natural FrequencyMechanical Engineeringquantity
  5. Damping RatioAutomation and Control Engineeringquantity
  6. Frequency ResponseAutomation and Control Engineeringmodel
  7. Modal AnalysisMechanical Engineeringmethod
  8. Mode ShapeMechanical Engineeringquantity
  9. BearingMechanical Engineeringdevice
  10. Cross-Coupled StiffnessMechanical Engineeringquantity
  11. RunoutMechanical Engineeringmetric
  12. Rotor RubMechanical Engineeringphenomenon
  13. Mechanical LoosenessMechanical Engineeringphenomenon
  14. Shaft BowMechanical Engineeringphenomenon
  15. Shaft Coupling MisalignmentMechanical Engineeringphenomenon
  16. Vibration IsolationMechanical Engineeringmethod
  17. TachometerMechanical Engineeringdevice
  18. EncoderAutomation and Control Engineeringdevice
  19. Order TrackingMechanical Engineeringmethod
  20. Waterfall SpectrumMechanical Engineeringmethod
  21. Orbit PlotMechanical Engineeringmethod
  22. Campbell DiagramMechanical Engineeringmethod
  23. DisplacementNaval and Marine Engineeringquantity
  24. Fast Fourier TransformMathematical Engineeringmethod
  25. Sampling TheoremTelecommunications Engineeringtheorem
  26. AliasingComputer Engineeringphenomenon
  27. Signal-to-Noise RatioTelecommunications Engineeringmetric
  28. ValidationIndustrial and Management Engineeringmethod
  29. Uncertainty AnalysisMathematical Engineeringmethod
  30. Failure ModeIndustrial and Management Engineeringconcept
  31. ReliabilityIndustrial and Management Engineeringmetric
  32. Mechanical Vibration and Rotating Machinery ReliabilityMechanical Engineering
  33. Mechanical Vibration and Rotating Machinery Reliability Formula SheetMechanical Engineering
  34. Mechanical Vibration and Rotating Machinery Reliability ExercisesMechanical Engineering
  35. Rotating Machinery Vibration Commissioning and Reliability Baseline ProjectMechanical Engineering
  36. Rotating Machinery Imbalance Vibration Diagnosis Case StudyMechanical Engineering
  37. Critical Speed Avoidance Band Run-Up Case StudyMechanical Engineering
  38. Oil Whirl Speed Sweep Trip Decision Case StudyMechanical Engineering
  39. Seal-Induced Rotor Instability Case StudyMechanical Engineering
  40. Marine Propulsion and Ship Power SystemsNaval and Marine Engineering
  41. Ship Power Systems and Propulsion Plant Formula SheetNaval and Marine Engineering
  42. Sea Trial Speed, Power, and Fuel Consumption Validation ProjectNaval and Marine Engineering
  43. Influence CoefficientMechanical Engineeringmethod
  44. Oil WhirlMechanical Engineeringphenomenon
  45. Proximity ProbeMechanical Engineeringdevice
  46. Electrical RunoutMechanical Engineeringphenomenon
  47. Shaft Centerline PlotMechanical Engineeringmethod
  48. Once-Per-Revolution ReferenceMechanical Engineeringmethod
  49. Operating Deflection ShapeMechanical Engineeringconcept
  50. TransmissibilityMechanical Engineeringmetric
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