How this instrument works
Root-mean-square velocity takes three separate speed readings — not three components of one motion, but three independent measurements of the same directionless speed — and combines them by squaring each one, averaging the squares, then taking the square root of that average. The squaring step is the entire point: it happens before the averaging, not after, so a single large reading pulls the result upward more than a plain arithmetic mean ever would. Feed the formula 10, 20, and 30 m/s and it returns roughly 21.6 m/s, not the 20 m/s a straight average gives.
The reason engineers reach for this particular average rather than a plain one is that kinetic energy, dynamic pressure, and vibration power all scale with the square of speed, not speed itself. A vibration analyst running a diagnostic on a pump bearing, logging three velocity readings across a test cycle, cares about RMS velocity precisely because ISO 20816-1 grades machine health against RMS velocity limits in millimetres per second — a metric built to reflect the destructive potential of the motion, which a simple mean would understate whenever the readings are uneven rather than steady.
It is easy to confuse this calculation with finding the magnitude of a three-dimensional velocity vector from its x, y, and z components, since both involve squaring three numbers and taking a square root. They are not the same operation: vector magnitude skips the division entirely, √(v_x² + v_y² + v_z²), because it combines three perpendicular pieces of one motion into a single speed. This formula divides by 3 because it is instead averaging three separate measurements of one non-directional speed — mixing the two up gives a physically wrong answer, not just an imprecise one.
- Enter your first measured speed in the Speed, sample 1 field, in metres per second.
- Enter the second reading in Speed, sample 2 — order does not matter, since squaring removes it.
- Enter the third reading in Speed, sample 3 to complete the set of three.
- Read the result in Root-mean-square velocity, and compare it to the plain average of your three numbers to see how much the spread matters.
Worked example — readings of 10, 20, and 30 m/s
Three speed readings taken during a test run come back as 10, 20, and 30 m/s. Squaring each gives 100, 400, and 900; summing gives 1400; dividing by 3 gives 466.667; and the square root of that is 21.6024689947 m/s — about 21.6 m/s, the exact figure this instrument returns for these three inputs.
That 21.6 m/s sits above the plain average of the same three numbers, which is exactly 20 m/s — the identical average a steady run of 20, 20, and 20 m/s would also produce. Only the RMS figure tells the two situations apart: the steady set returns an RMS velocity of exactly 20 m/s, while the uneven set returns 21.6 m/s, a real difference that matters wherever the quantity driving wear, drag, or vibration damage rises with the square of speed rather than with speed itself.
Questions
Why does dividing by 3 make sense here?
Because the formula is averaging three independent measurements of one directionless speed, not combining three perpendicular components of a single motion. A true velocity-vector magnitude, √(v_x² + v_y² + v_z²), never divides by the count; this one does, because 3 is the sample size being averaged, not a count of spatial dimensions. Confusing the two gives a different — and wrong — number.
Why is the RMS result higher than the plain average?
Squaring happens before averaging, and squaring always exaggerates whichever readings are largest. For 10, 20, and 30 m/s the plain average is 20 m/s, but the RMS velocity is about 21.6 m/s, because the 30 m/s reading contributes 900 to the sum of squares versus only 100 from the 10 m/s reading — nine times as much weight for a value only three times as large.
Can two sets of readings share the same average speed but different RMS velocity?
Yes, and it happens constantly. Three readings of 20, 20, and 20 m/s and three readings of 10, 20, and 30 m/s both average to exactly 20 m/s, yet their RMS velocities differ — 20 m/s for the steady set against 21.6 m/s for the uneven one. Whenever the underlying spread matters, as it does for anything driven by v², RMS is the figure that actually distinguishes them.
What kind of work actually uses RMS velocity like this?
Vibration analysis is the clearest case: ISO 20816-1 evaluates rotating-machine health against RMS velocity limits in millimetres per second, measured across several readings during a diagnostic run, because vibration damage tracks the square of velocity rather than its plain average. The same logic applies anywhere destructive or dissipated power scales with v² — gust loading on a structure, dynamic pressure on a surface, or fatigue from an unevenly loaded cycle.
Does the order I enter the three speeds in matter?
No. Addition is commutative, so v₁² + v₂² + v₃² comes out identical no matter which reading goes in which field — entering 10, 20, 30 or 30, 10, 20 returns the same 21.6024689947 m/s either way. Only the three values themselves, not their order, determine the result.
What happens if all three speeds are identical?
The RMS velocity equals that same value exactly, with no inflation at all. Three readings of 20 m/s each give an RMS velocity of precisely 20 m/s, because squaring, averaging, and rooting a set with zero spread just returns the original number — a useful sanity check that confirms the formula is behaving correctly before you trust it on real, uneven data.