Unbalance Response Analysis

What Unbalance Response Analysis Computes

Unbalance response analysis computes the steady-state vibration of the rotor-bearing system under the synchronous (1×) unbalance force, as a function of the rotational speed. The output is the vibration amplitude and phase at each speed, at each location on the rotor. The analysis is performed across the operating speed range — from zero to the maximum speed — and it shows the response through each critical speed. The amplitude shows a peak at each critical speed (resonance); the phase shows a 180-degree shift through each critical speed. The response is the primary tool for assessing whether the rotor can operate safely: the amplitude at the critical speed determines the bearing load, the shaft stress and the clearance margin, and the amplitude in the operating range determines the steady-state vibration level.

Unbalance response analysis computes the rotor vibration amplitude and phase as functions of speed. It shows the response through each critical speed — the peaks and the phase shifts — and it is the primary tool for assessing whether the rotor can operate safely across its speed range.

The Synchronous Response

The unbalance force is synchronous — it rotates with the shaft at the 1× frequency. The response is therefore also synchronous — the rotor orbits at the 1× frequency, with the amplitude and phase determined by the proximity to the critical speed. Below the first critical speed, the response amplitude is small — the excitation frequency is well below the natural frequency, and the rotor moves quasi-statically with the unbalance force. As the speed approaches the critical speed, the amplitude rises — the excitation approaches the natural frequency, and the amplification factor increases. At the critical speed, the amplitude peaks — the excitation is at the natural frequency, and the response is limited only by the damping. Above the critical speed, the amplitude falls again — the excitation is above the natural frequency, and the rotor response decreases. For a supercritical machine, the amplitude after passing through the critical may be lower than before — the rotor "centres" itself as the speed-dependent stiffening dominates.

• Below critical speed: small amplitude — quasi-static response, rotor follows the unbalance
• Approaching critical: amplitude rises — excitation approaching natural frequency, amplification increasing
• At critical speed: amplitude peaks — excitation at natural frequency, response limited by damping only
• Above critical speed: amplitude falls — excitation above natural frequency, response decreasing
• Supercritical: amplitude may be lower than subcritical — rotor "centres" itself

AMPLITUDE AND PHASE THROUGH A CRITICAL SPEED

The Bode plot — amplitude and phase vs speed — is the standard presentation of the unbalance response. The amplitude shows a peak at the critical speed; the phase shows a 180-degree shift. The shape of the peak — its height and its width — is determined by the damping: a lightly damped rotor has a tall, narrow peak; a heavily damped rotor has a low, broad peak. The phase shift is centred at the critical speed: below the critical, the response is approximately in phase with the force (0 degrees); at the critical, the phase is 90 degrees; above the critical, the phase is approximately 180 degrees (out of phase). The phase shift is the signature of resonance — it is the most reliable indicator of a critical speed, because it is less sensitive to the unbalance magnitude than the amplitude.

bode-plot-critical-speed

The Bode plot — amplitude and phase vs speed — is the signature of the unbalance response. The amplitude peaks at the critical speed; the phase shifts 180° through it. The peak height is determined by the damping; the phase shift is the most reliable indicator of the critical speed.

The Influence of Damping on the Peak

The peak amplitude at the critical speed is approximately F/(2·ζ·k), where F is the unbalance force at the critical speed, ζ is the damping ratio and k is the modal stiffness. The peak is inversely proportional to the damping — doubling the damping halves the peak. This is why bearing damping is so important for rotordynamic design: the bearing is usually the primary source of damping, and increasing the bearing damping is the most effective way to reduce the resonant response. For a lightly damped rotor (ζ = 0.02), the peak amplification factor is about 25 — the resonant response is 25 times the static deflection under the same force. For a heavily damped rotor (ζ = 0.10), the amplification factor is about 5 — the peak is five times lower. The damping also affects the width of the peak: a lightly damped rotor has a narrow, tall peak (the response is high only in a narrow speed band around the critical); a heavily damped rotor has a broad, low peak (the response is moderate over a wider speed band).

  Peak amplitude at critical speed:

  x_peak ≈ F / (2 · ζ · k)

  where  F = unbalance force at critical speed = m_e · e · ω²_critical
         ζ = damping ratio
         k = modal stiffness

  Amplification factor:  H ≈ 1 / (2·ζ)

  ζ = 0.02:  H ≈ 25  (light damping — tall narrow peak)
  ζ = 0.05:  H ≈ 10
  ζ = 0.10:  H ≈ 5   (heavy damping — low broad peak)

  →  Doubling damping halves the peak amplitude
  →  Bearing damping is the primary control on resonant response

Response at Bearings and Shaft Locations

The unbalance response is evaluated at several locations on the rotor, each serving a different assessment purpose. At the bearings, the response determines the dynamic bearing load — the cyclic force that the bearing must carry in addition to the static load. The dynamic bearing load affects the bearing fatigue life and the housing stress. At the shaft mid-span (or at the location of maximum mode-shape amplitude), the response determines the maximum shaft deflection — the cyclic bending that the shaft must endure. The shaft deflection affects the shaft fatigue stress and the clearance margin. At locations where clearance is critical — blade tips, seals, labyrinth teeth — the response determines the rub risk: if the vibration amplitude exceeds the clearance, the rotor contacts the stator, producing a potentially destructive event (Article 15). The unbalance response analysis must report the amplitude at all these locations, and the engineer must assess each against its acceptance criterion — bearing load, shaft stress, clearance margin.

• Bearing locations: dynamic bearing load — affects bearing fatigue life and housing stress
• Shaft mid-span: maximum shaft deflection — affects shaft fatigue stress and clearance
• Clearance-critical locations (blade tips, seals): rub risk — if amplitude exceeds clearance, contact occurs
• Report amplitude at all locations; assess each against its acceptance criterion

Cross-Link: Critical Speeds and Campbell Diagram

The unbalance response analysis is the quantitative complement to the Campbell diagram (Article 07) and the critical-speed analysis (Article 04). The Campbell diagram identifies where the crossings are — the critical speeds. The unbalance response analysis computes what happens at each crossing — the amplitude and the phase. The two are used together: the Campbell diagram for screening and identification, the response analysis for evaluation and acceptance. A critical speed on the Campbell diagram that shows a low response amplitude in the unbalance response analysis may be acceptable; one that shows a high response requires mitigation — more damping, better balancing, or a design change to move the critical speed. The engineer who has both the Campbell diagram and the unbalance response has a complete picture of the rotordynamic behaviour across the operating range.

The Campbell diagram identifies where the critical speeds are. The unbalance response analysis computes what happens at each one. The two are used together — the diagram for identification, the response for evaluation. A complete rotordynamic assessment requires both.

Key takeaways

• Unbalance response analysis computes the rotor vibration amplitude and phase as functions of speed, driven by the synchronous (1×) unbalance force. It is the primary tool for assessing whether the rotor's balance state is adequate for the operating speed range.
• The response amplitude peaks at the critical speed, where the excitation frequency coincides with the natural frequency. The peak amplitude is limited by the damping — more damping means a lower peak.
• The phase shifts by 180 degrees through the critical speed — below resonance, the response is in phase with the force; above, it is 180 degrees out of phase. The phase shift is the signature of resonance.
• The response is evaluated at the bearings (for load assessment), at the shaft mid-span (for maximum deflection) and at any location where clearance is critical (for rub risk).