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A rotational order expresses a signal’s frequency relative to a specified reference shaft’s rotational frequency:Thus $q=1$ means one oscillation per reference revolution and $q=1/2$ means one oscillation per two reference revolutions. A rotational order is dimensionless and need not be an integer. It is especially useful when a machine changes speed: a shaft-synchronous feature can keep the same rotational order while moving through many frequencies in an ordinary frequency spectrum.^[ref]

The reference is essential. Suppose shaft A turns at $600\mathrm{rpm}=10\mathrm{Hz}$ and shaft B at $1500\mathrm{rpm}=25\mathrm{Hz}$. The same $100\mathrm{Hz}$ signal has rotational order $10$ relative to A and $4$ relative to B. Changing the reference has changed the dimensionless label, not the measured vibration. Likewise, a 24-tooth gear turning at $900\mathrm{rpm}$ has a gear mesh frequency of $24\times 15=360\mathrm{Hz}$. Its mating 8-tooth gear turns at $2700\mathrm{rpm}$, giving the same gear mesh frequency but rotational orders $24$ and $8$ relative to the two shafts.

For a component with unwrapped phase $\varphi (t)$ and reference angle $\theta (t)$, the instantaneous version iswhere the reference angular velocity is nonzero and both angles use the same units. A constant rotational order gives $\varphi (t)=q\theta (t)+{\varphi }_0$. This explains why angular resampling makes a speed-following component periodic as a function of shaft angle. A vibration fixed at $100\mathrm{Hz}$ instead changes from rotational order $10$ to $5$ when the reference shaft accelerates from $600$ to $1200\mathrm{rpm}$.

An order tracking measurement therefore pairs a vibration sensor, such as an accelerometer, with a shaft-angle or speed reference, such as a tachometer. An order peak identifies a speed relationship; it does not by itself establish which part is damaged. A resonance can greatly change the amplitude as speed changes even when the excitation’s rotational order stays fixed.

A simple gear train transmits rotation through individual intermediate gears, each intermediate gear engaging its neighbors without a separate, differently sized gear rigidly sharing that intermediate rotation. For external meshes on fixed parallel axes, consecutive signed ratios multiply and intermediate tooth counts cancel:where $m$ is the number of external meshes. An intermediate idler gear changes placement and potentially output direction, but not this ideal endpoint tooth-count ratio.^[ref]

For example, a chain with tooth counts 18, 30, 42 and 72 has three meshes andAt input speed $800$ revolutions per minute, output speed is $200$ revolutions per minute in the opposite direction. Removing one intermediate gear while maintaining a compatible two-mesh layout preserves the fourfold reduction but reverses the direction conclusion.

The cancellation concerns kinematics, not every engineering consequence. Intermediate tooth counts affect physical sizes, axis positions and angular speeds of intermediate rotating bodies; additional meshes and supports can change gear efficiency and backlash. Moreover, if two different intermediate gears are rigidly coupled, their tooth counts do not cancel by the same argument: that arrangement is a compound gear train.

A spur gear is a cylindrical gear with straight teeth parallel to its rotation axis. A conventional external pair transmits motion between parallel axes. Unlike the inclined teeth of a helical gear, the ideal straight-tooth geometry produces no axial thrust from the tooth-normal contact force; there are still tangential and separating loads that the mounting must carry.^[ref]

For compatible spur gears, reference diameter is $d=mN$, where $m$ is gear module and $N$ is tooth count. Standard reference axis separation is $(d_1+d_2)/2$. Two external spur gears have signed angular velocity relation ${\omega }_2/{\omega }_1=-N_1/N_2$: they rotate in opposite directions. Matching gear module alone is insufficient; tooth profiles and gear pressure angles must also be compatible.

For an original example, take gear module $3\mathrm{mm}$ and tooth counts 20 and 60. Reference diameters are $60\mathrm{mm}$ and $180\mathrm{mm}$, so standard axis separation is $120\mathrm{mm}$. Driving the smaller spur gear at $600$ revolutions per minute gives the larger one $200$ revolutions per minute in the opposite direction. In a steady ideal transmission, an input torque of $2\mathrm{N}\mathrm{m}$ becomes $6\mathrm{N}\mathrm{m}$ at the output. The corresponding input gear tangential force is $2/0.030\approx 66.7\mathrm{N}$; multiplying this same force by the output reference radius $0.090\mathrm{m}$ recovers the output torque.

A simple bench demonstration marks one tooth on each spur gear and counts complete turns, testing the three-to-one gear ratio. Measuring transmitted torques and rotational speeds additionally tests gear efficiency. Observing the rotation ratio alone cannot establish a tooth's allowable load or the losses caused by friction and backlash.