cyclogramThe critical speed
A spinning screw is a shaft between bearings with a natural frequency of bending. Turned near it, it whips. The screw catalogues give the speed as
n_c = f · d_r / L² · 10⁷ rpm, with d_r and L in mm allowed = 0.8 · n_cwhere d_r is the root diameter, L the length between the bearings, and f a factor for how the ends are held. For a Ø16 mm screw — root 13.6 mm — 500 mm between bearings:
| Ends | f | n_c |
|---|---|---|
| fixed – free | 3.4 | 1850 rpm |
| supported – supported | 9.7 | 5277 rpm |
| fixed – supported | 15.1 | 8214 rpm |
| fixed – fixed | 21.9 | 11 914 rpm |
The root diameter is the screw's core under the thread; the page takes 0.85 times the nominal diameter unless you type the catalogue's figure.
The longest screw for a speed
L_max = √( f · d_r · 10⁷ · 0.8 / n )At 3600 rpm the Ø16 mm screw may be 676 mm between bearings with one fixed and one supported end, or 814 mm with both ends fixed. At 800 mm, fixed – supported, its critical speed falls to 3209 rpm, and the page flags the move.
D·N — the nut's own limit
D·N = d · n_screw ≤ about 70 000 for a standard screwThe balls circulating in the nut have a speed limit of their own. 16 mm at 3600 rpm is 57 600. Stretch the same move into a triangle profile and the screw reaches 4800 rpm and a D·N of 76 800 — over. High-speed screw series allow more; take the figure from the catalogue.
When the screw is too fast
- Shorten the span between the bearings, or stiffen the ends — fixed – fixed raises the critical speed by 2.3 times over supported – supported.
- Take a larger diameter: the critical speed rises with it, though so do the D·N and the screw's inertia.
- Take a longer lead: the screw turns slower for the same table speed.
- For long, fast strokes, a belt drive has no critical speed at all.
Questions
Why only 80 % of the critical speed?
Real supports are never as stiff as the formula assumes, and the screw must not run close to its resonance. 80 % is the common catalogue allowance.
What root diameter should I enter?
The catalogue's root (core) diameter of the thread. Without it, the page takes 0.85 times the nominal diameter, a typical value.
Which end supports does my screw have?
Fixed means a pair of angular-contact bearings that hold the shaft's angle; supported means a single bearing that lets it tilt; free means no bearing at that end.
Which length do I enter?
The length between the bearings. For a fixed – free screw, catalogues measure from the bearing to the nut at its farthest position.
Sources
- Ball-screw catalogues of NSK, THK and Bosch Rexroth — the critical-speed formula, the support factors and the D·N limit.
Related guides
- Axis sizing calculatorThe whole guided flow: motion, mechanism, motor, regeneration, and the move as a bar.
- Ball screw motor sizingSpeed, reflected inertia, torque per phase, RMS, critical speed and D·N for a ball screw axis.
- Move time and motion profilePeak speed, acceleration and jerk from a distance and a move time; trapezoid, triangle and S-curve.
- Belt drive motor sizingPulley and motor speed, reflected inertia, torque, belt force — and the gearbox a belt axis needs.
- Servo motor sizingPeak torque at speed, RMS over the cycle, maximum speed and inertia ratio, each with its margin.
Every term on this page, defined: the axis and motor sizing glossary.