Glossary term

Shaft Bow

Permanent or temporary curvature of a rotor or shaft, diagnosed with slow-roll runout, 1x phase evidence, centerline movement, clearance review and thermal history.

Branch: Mechanical Engineering
Glossary type: phenomenon
Content: Glossary term
Updated: Jun 25, 2026
Revision: v1.0.0 · reviewed

Definition

phenomenon

Shaft bow is curvature of a shaft or rotor that makes the shaft centerline deviate from the intended straight rotation axis.

Shaft bow may be permanent, such as a bent shaft or rotor sag that remains after cooldown, or temporary, such as thermal bow caused by uneven heating, rub heating, hot shutdown, asymmetric cooling or process temperature gradients. In rotating machinery it can create slow-roll runout, synchronous 1x vibration, phase behavior that follows the bowed high spot, reduced clearance and higher rub risk.

Shaft bow is curvature of a shaft or rotor. The shaft centerline is no longer straight relative to the intended rotation axis, so one angular position becomes a high spot. In a rotating machine this can appear as slow-roll runout, synchronous 1x vibration, changing phase, reduced clearance, seal contact or a rub that grows as temperature changes.

The bow may be permanent or temporary. Permanent bow can come from plastic bending, handling damage, long-term sag, manufacturing error, residual stress relief or a severe rub. Temporary bow can come from uneven heating, hot shutdown, asymmetric cooling, steam admission, process temperature gradients, oil starvation or local friction heat during a rub.

Diagnostic Role

Shaft bow matters because it can imitate or amplify other faults. A bowed rotor may look like unbalance because it creates a repeatable 1x displacement pattern. It may look like runout because it is visible during slow roll. It may also cause rotor rub because the bowed high spot consumes clearance before dynamic orbit amplitude is added.

A practical review asks four questions:

is the once-per-revolution high spot present during slow roll;
does the phase stay tied to the same shaft angular location;
does the bow change with temperature, soak time or turning-gear operation;
does the combined bow, centerline shift and orbit amplitude leave enough clearance?

Shaft bow should not be diagnosed from a single spectrum. It needs angular reference, slow-roll evidence, shaft-relative displacement, thermal history and physical plausibility.

Slow-Roll Evidence

At slow roll, dynamic forces are usually small enough that a proximity-probe signal can be treated as a measurement of repeatable shaft-related effects:

x_{sr}(\theta)\approx x_{bow}(\theta)+x_{geo}(\theta)+x_{elec}(\theta)+e(\theta)

where $x_{bow}$ is the bowed-shaft contribution, $x_{geo}$ is geometric runout, $x_{elec}$ is electrical runout and $e$ is residual measurement error. The angular coordinate $\theta$ comes from a once-per-revolution reference.

For a simple bowed high spot, the radial contribution at one probe plane may be approximated as:

x_{bow}(\theta)=e_b\cos(\theta-\theta_b)

where $e_b$ is the bow eccentricity at that axial station and $\theta_b$ is the angular location of the high spot.

This model is a screen, not a proof. A real rotor can have different bow at different axial stations, probe angles, material effects, residual magnetism and bearing support motion. The value of the slow-roll test is that it preserves amplitude and phase before operating-speed forces dominate the data.

1x Force Implication

If shaft bow moves mass eccentricity away from the rotation axis, it can create a synchronous force similar to residual unbalance. A simplified force estimate is:

F_b=m e_b \Omega^2

where $m$ is the effective rotating mass associated with the bowed section, $e_b$ is eccentricity and $\Omega$ is angular speed.

Example: a rotor section has effective mass:

m=22\ \text{kg}

Slow-roll review estimates bow eccentricity:

e_b=35\ \mu\text{m}=35\times10^{-6}\ \text{m}

At:

n=3600\ \text{rpm}

the angular speed is:

\displaystyle \Omega=\frac{2\pi n}{60}=377\ \text{rad/s}

The simplified synchronous force estimate is:

F_b=22(35\times10^{-6})(377)^2=109\ \text{N}

This does not prove the measured vibration will equal a particular amplitude. Bearings, supports, damping, mode shape and speed relative to critical speed control the response. It does show why a small bow can become important at operating speed.

Thermal Bow Screen

For a crude thermal-bow screen, a linear temperature difference across a shaft diameter can be converted into approximate curvature:

\displaystyle \kappa\approx\frac{\alpha\Delta T}{D}

where $\alpha$ is coefficient of thermal expansion, $\Delta T$ is the temperature difference across diameter $D$ , and $\kappa$ is curvature. If the bowed span is approximated as an arc between supports, the midspan offset is roughly:

\displaystyle e_{mid}\approx\frac{\kappa L^2}{8}

Example: a steel rotor span has:

\alpha=12\times10^{-6}\ /\text{K}

\Delta T=8\ \text{K}

D=0.16\ \text{m}

L=1.2\ \text{m}

The curvature screen gives:

\displaystyle \kappa=\frac{12\times10^{-6}(8)}{0.16}=6.0\times10^{-4}\ \text{m}^{-1}

and the approximate midspan bow is:

\displaystyle e_{mid}=\frac{(6.0\times10^{-4})(1.2)^2}{8}=108\ \mu\text{m}

This simplified number is large enough to demand caution. The next engineering step is not to accept the estimate as exact; it is to check temperature evidence, turning-gear history, slow-roll phase and clearance margin.

Clearance and Rub Risk

Bow is dangerous when it consumes clearance together with mean shaft position and dynamic orbit amplitude:

c_{rem}=c-r_c-a_{orbit}-e_b

where $c$ is radial running clearance, $r_c$ is mean shaft-centerline offset, $a_{orbit}$ is the relevant dynamic orbit radius and $e_b$ is the bowed high-spot contribution at the clearance location.

Suppose:

c=180\ \mu\text{m}

r_c=65\ \mu\text{m}

a_{orbit}=58\ \mu\text{m}

e_b=35\ \mu\text{m}

The remaining clearance screen is:

c_{rem}=180-65-58-35=22\ \mu\text{m}

That is a small margin. If thermal bow grows by another $25\ \mu\text{m}$ during a hot restart, the simplified margin becomes negative and rotor rub becomes plausible.

Shaft bow is not electrical runout. Electrical runout is an apparent probe signal caused by target material or electromagnetic effects. Shaft bow is physical curvature, although both can appear as repeatable slow-roll 1x content.

Shaft bow is not ordinary runout. Runout is a measurement result relative to a datum and can include geometry, setup, electrical effects and bow. Shaft bow is one physical cause that may contribute to runout.

Shaft bow is not residual unbalance. Unbalance is mass eccentricity. Bow can create mass eccentricity or clearance problems, but a balance correction may hide symptoms without removing the bent or thermally bowed condition.

Shaft bow is not rotor rub. A rub is contact. Bow can cause or result from a rub, especially when local heating bends the rotor further, but the diagnosis needs contact evidence before calling the fault a rub.

Validation and Release

A defensible shaft-bow review states the probe locations, probe calibration, slow-roll speed, angular reference, runout compensation method, phase convention, shaft temperature history, turning-gear status, bearing clearance, shaft centerline position, orbit amplitude, operating speed and inspection evidence.

Useful release evidence includes repeatable slow-roll amplitude and phase, stable phase at operating speed, no worsening temperature trend, adequate clearance margin, acceptable probe linear range, inspection for rub marks, and a repeat run after thermal soak or controlled cooldown when thermal bow is suspected.

Release should be withheld when the bow grows with temperature, phase walks during run-up, clearance margin is small, rub evidence appears, slow-roll compensation is undocumented or the machine needs a balance correction that changes strongly between runs.

Common Mistakes

Do not balance away a bowed-rotor symptom without checking slow-roll evidence. A balance shot may reduce 1x at one speed while the rotor still has a clearance or rub problem.

Do not treat all slow-roll 1x content as electrical runout. Mechanical bow, geometric runout and probe-target effects must be separated as far as the evidence allows.

Do not remove the DC shaft position before checking clearance. A bowed high spot may be acceptable when the shaft is centered and unsafe when the mean centerline has moved toward a bearing or seal boundary.

Limits

The force, thermal-bow and clearance equations are simplified engineering screens. Real rotors have distributed mass, multiple supports, overhung sections, anisotropic bearings, mode shapes, thermal gradients along the shaft, residual stress, oil-film effects, casing motion and uncertain probe geometry.

The practical goal is decision quality: identify whether bow is plausible, whether it is changing, whether it threatens clearance or vibration limits, and what evidence is needed before running, balancing, inspecting or releasing the machine.

REF

Disciplines

Shaft Bow

Definition

Diagnostic Role

Slow-Roll Evidence

1x Force Implication

Thermal Bow Screen

Clearance and Rub Risk

Validation and Release

Common Mistakes

Limits

See also

Diagnostic Role

Slow-Roll Evidence

1x Force Implication

Thermal Bow Screen

Clearance and Rub Risk

Distinction from Related Terms

Validation and Release

Common Mistakes

Limits