Rotating Unbalance Force
Rotating Unbalance Force
Any rotating component that is not perfectly mass-balanced about its rotation axis — a fan blade with a manufacturing tolerance, a motor rotor with a slightly off-center winding — generates a centrifugal force that rotates along with the shaft, exciting vibration in whatever supports the machine. This rotating unbalance force is one of the most common sources of machinery vibration in practice, which is why rotating equipment is routinely balanced to a specified tolerance before installation.Unbalance force grows with the square of rotational speed, so a machine that ran acceptably at a lower speed can develop a serious vibration problem if its operating speed is increased — even a small residual unbalance, tolerable at low speed, can become a significant exciting force at high speed, which is why balance tolerances tighten considerably for high-speed rotating equipment like turbines and precision spindles.
The centrifugal unbalance force is F_unb = m_u·r_u·omega^2, the unbalance mass times its eccentricity times the square of the angular rotational speed. where m_u is the unbalance mass, r_u is the eccentricity (radius) at which that mass acts, f_rot is the rotational speed in cycles per second, omega is the corresponding angular speed, and F_unb is the resulting rotating centrifugal force.
Inputs
m_u := 0.05 kg=0.05 kg
r_u := 0.1 m=100 mm
f_rot := 50 Hz=50 Hz
Converting rotational speed from cycles per second to angular speed puts it into the units the centrifugal force formula requires.
omega := 2·pi·f_rot=314.16 Hz
The centrifugal force formula shows why unbalance is so speed-sensitive: it depends on the square of angular speed, so doubling rotational speed quadruples the resulting force for the same unbalance mass and eccentricity.
F_unb := m_u·r_u·omega^2=493.48 N
Results
An unbalance force of about 493 N at this rotational speed is a substantial rotating load for such a small mass (only 50 grams) — a clear illustration of how strongly this force depends on the omega-squared term rather than the unbalance mass itself. If this machine were to double its operating speed, the same physical unbalance would produce roughly four times this force, which is why balance quality (measured in balance grade standards) becomes progressively more critical as design speed increases. This rotating force excites vibration at exactly the rotational frequency, which is one of the key diagnostic signatures used in vibration analysis to identify unbalance as opposed to other fault types like misalignment or bearing wear.
Assumptions & limitations
- Unbalance is modeled as a single discrete point mass at a fixed eccentricity, not a distributed mass imperfection.
- Steady-state rotation at constant speed — transient effects during speed changes are not included.
- Rigid rotor assumed (no flexible-rotor mode shape effects at high speed).
- The force found here is the rotating force magnitude; the resulting vibration amplitude also depends on the support stiffness and damping.