Ball screws

The Ball screws module rates a ball screw and nut over a load spectrum. It gives the rating life after ISO 3408-5, the static safety, the shaft's critical speed and buckling load, the d·n value, the efficiency and the drive torque. With the ball diameter entered, it also rates the nut from its internal geometry, computes the axial rigidity after ISO 3408-4 and solves the nut ball by ball. The module sits in the Motion and drives group of the navigation.

Ball screws stand on their own: they are not linked to a shaft or a guide. The Ball screw and linear guides template holds a guide and its ball screw drive together, with a preloaded nut. The Ball screw lift (single nut, no preload) template is a vertical axis on a nut with backlash; Preloaded and non-preloaded nuts compares the two.

Inputs

Choose + Ball screw at the top of the list. The inspector shows six groups.

Ball screw shaft and nut

Mounting

• Unsupported length (mm): between the bearings, or from the fixed bearing to the nut.
• End fixity: Fixed – fixed, Fixed – supported (default), Supported – supported or Fixed – free. It sets the critical speed and buckling factors.
• Compresses the shaft: Either direction (default), Positive forces or Negative forces. Choose the force direction that compresses the shaft between the nut and the axial (fixed) bearing; the other direction puts it in tension. Keep Either direction when the bearing position is not known or both ends hold the shaft axially.
• d·n limit: 70 000 by default. High-speed screws allow more; take the value from the maker.

Rating (ISO 3408-5)

This group is collapsed. Reliability (90 % by default, up to 99 %) applies the reliability factor of Table 3. Standard tolerance grade (0, 1, 3, 5, 7 or 10), Ball track hardness (HV10; 654 and above gives no reduction) and Steel melting (air melted fm 1.0 up to vacuum remelted fm 1.71) give the modified ratings.

Load spectrum

Axial force, speed and time share: one row per operating step with Force (N), Speed (rpm) and Share (%). The shares are normalised. A negative force is load direction 2. Choose + Row to add a step.

Balls (internal geometry)

Leave Ball diameter Dw (solves ball by ball) empty for a catalogue rating only. With a ball diameter the group shows Contact angle (45°), Screw shaft conformity r/Dw and Nut conformity r/Dw (0.55), Loaded turns per circuit (2.5), Circuits (2), Balls per turn (as many as fit), Nut outer diameter (1.75 × d₀), Unloaded balls in the return per turn, Screw shaft bore, Groove depth / Dw (0.3), Screw shaft takes the load out (At the nut's far end or At the flange end), and an optional Radial load on the nut and Tilting moment on the nut.

The radial load and tilting moment act on the ball model only, as the ISO ratings assume a purely axial load.

Requirements

Collapsed: Min life (20 000 h by default) and Min static safety (2).

How the life is found

The rating follows ISO 3408-5:

1. The equivalent speed and the cubic mean load are taken per load direction, weighted by revolutions (clause 7.1, eqs. 23–27).
2. A preloaded double nut shares each step between its two nuts with the operational preload 0.6·Fpr (clause 7.1.3, eqs. 28–34). Above $2^{1.5}\times 0.6\times F_{pr}\approx 1.70\times F_{pr}$ one nut lifts off and the loaded nut carries the force alone.
3. Each direction (or nut) gets $L={(C_{am}/F_m)}^3\cdot {10}^6$ revolutions. The two lives combine to ${(L_1^{-10/9}+L_2^{-10/9})}^{-9/10}$ (clause 7.2.3, eq. 39).
4. The modified rating $C_{am}=C_a\cdot f_h\cdot f_{ac}\cdot f_m$ (clause 6.2, Tables 1 and 2) and the reliability factor of Table 3 give the life that is checked (clause 7.3). The nominal life uses the plain Ca at 90 %.

The life exponent 3 and the 10/9 combination are the same ideas as in ISO 281 rating life.

The critical speed and buckling load are not part of ISO 3408. The program takes the shaft as a beam of the root diameter: $n_{cr}=\frac{60}{2\pi }\cdot \frac{{\lambda }^2}{l^2}\cdot \sqrt{\frac{EI}{\rho A}}$ and the Euler load ${\pi }^{2}EI/{(K\cdot l)}^2$, with λ and K from the end fixity, E 206 000 MPa and ρ 7.85 t/m³. The checks allow 80 % of n_cr and 50 % of the buckling load.

Results

The results column shows the checks and the key results Rating life (with the equivalent load and speed, and the nominal life) and Critical speed (with the running speed and the 80 % limit). With the ball model it adds Life from the balls and Ball contact pressure.

The Details card lists Equivalent loads 1 / 2, Modified ratings Cam / C0am, the ISO 3408-5 ratings Ca / C0a and Axial rigidity (ISO 3408-4) from the geometry, Static safety, Buckling load, d·n, Efficiency / drive torque, Preload torque, Travel over the life and Linear speed.

With the ball model the centre shows a 3D view of the nut cut open. The view shows the contact pressure as Stress shape, Painted or Balls, coloured by Load or Stress. Below it are the charts Load on each ball along the nut and Contact pressure, screw and nut.

Checks

The template

The Ball screw and linear guides template's ball screw is Ø32 × 10 mm (d₂ 27.5 mm, Ca 30 000 N, C₀a 60 000 N, preload 600 N), 1100 mm long, fixed–supported, with three steps: 3500 N at 300 rpm for 15 %, 900 N at 1500 rpm for 55 % and 300 N at 2000 rpm for 30 %. It gives:

All checks pass. The d·n check is the closest, at 91 %.

With 6.35 mm balls the template also reports ISO 3408-5 ratings from the geometry of Ca 21 335 N and C₀a 45 380 N (15 balls a turn), an axial rigidity of 77.59 N/µm (shaft 113.5, nut 245.3 N/µm) and, from the ball model, a largest contact pressure of 1786 MPa, a ball static safety of 13.01 and a life from the balls of 121 164 h scaled to Ca. The checked life uses the catalogue Ca you enter. The ratings and life from the geometry are a comparison: if they are much lower than the catalogue values, check the geometry inputs with the maker.

Fixing a failing check

1. Life: a larger screw or nut (higher Ca), more circuits, a lower preload, or a better steel (fm). Check the load spectrum for unrealistic peaks.
2. Critical speed: a shorter unsupported length, Fixed – fixed ends, a larger root diameter, or a larger lead for the same linear speed.
3. Buckling: the same levers as the critical speed. Better, place the axial bearing so that the working load puts the shaft in tension, and set Compresses the shaft to match.
4. d·n: a smaller diameter or a larger lead, or a high-speed screw with a higher limit from the maker.
5. Static safety: a larger C₀a, or limit the peak force.

Preloaded and non-preloaded nuts

A single nut without preload (Nut preload 0) has a small axial play, the backlash, between the balls and the grooves. Under a force one flank of each groove carries all the balls; when the force changes sign the nut shifts across the backlash and the other flank takes over. The program rates each load direction on its own flank (Equivalent loads 1 / 2) and combines the two lives (eq. 39). Nothing lifts off, because there is no second nut to lift.

A preloaded nut is rated as a double nut: two halves pressed against each other by the preload, each carrying half of the circuits. There is no backlash, and the nut is stiff in both directions. Under a force one half takes more and the other is relieved, until it lifts off (above about 1.70 × the preload in ISO 3408-5 with the operational preload, 2.83 × the full preload in the ball model). Above lift-off the loaded half carries the force alone with half the balls.

A preload pays where the force changes sign: a machine axis that pushes and pulls, or must position without play. Where the force keeps one sign, such as the weight on a vertical axis, the balls stay on one flank anyway and the backlash never shows. There a single nut carries the load on all its balls, has no preload torque and gives the longer life.

The lift template

The Ball screw lift (single nut, no preload) template is a vertical lift with the motor and the fixed bearing at the bottom. A Ø40 × 10 mm screw (d₂ 34.1 mm, Ca 38 000 N, C₀a 80 000 N, 6.35 mm balls in 2 circuits × 2.5 turns, nut Ø67 mm) is 1100 mm long between its bearings, fixed–supported. It lifts a 180 kg carriage with a 320 kg payload, holds it at the top and lowers the empty carriage:

All three forces are positive: the weight pushes the nut down in every step, so Compresses the shaft is Positive forces. The holding step adds time but no revolutions; it counts for the static safety, not for the life.

The same template with a preload of 2000 N (about 5 % of Ca) shows what the preload changes:

Without preload the 96 balls share the load almost evenly and the ball model's life agrees with the ISO life within 7 %. With the preload the lifting force nearly lifts one half off: the other half's 48 balls carry it, at twice the ball load. The ISO life drops only a little, because ISO 3408-5 counts the operational preload of 0.6 × 2000 N, which lifts off below the lifting force.

The other results of the template:

All checks pass; d·n is the closest at 86 %. The ball screw lift tutorial builds the template step by step.

Standards and scope

• Life after ISO 3408-5:2006 and rigidity after ISO 3408-4:2006. The 1989 drafts DIN 69051-4 and DIN 69051-6 differ in several constants; the program follows the ISO parts.
• The lift-off of the preloaded double nut is taken where eq. 29 reaches the force $(2^{1.5}\cdot f_{op}\cdot F_{pr})$, not at the $2^{1.5}\cdot F_{pr}$ that eq. 28 prints, so the loaded nut never carries less than the force.
• axm verify checks the rigidity against the worked example of ISO 3408-4 Annex A, the preloaded nut loads, eq. 39 and Table 3.
• Not covered: thermal growth of the screw, lubrication and wear, the support bearings, the motor sizing, and fatigue of the screw shaft or nut body.

From the command line

axm new axis.axm.json --template linear-axis
axm ball-screw update axis.axm.json BS1 --patch '{"fixity":"fixedFixed"}'
axm solve axis.axm.json

axm new lift.axm.json --template ball-screw-lift
axm ball-screw update lift.axm.json BS1 --patch '{"preload":2000}'
axm solve lift.axm.json

axm guide guides-screws lists every field.

Related

• Open the Ball screw and linear guides example
• Open the Ball screw lift example
• Tutorial: ball screw lift
• Linear guides
• ISO 281 rating life
• Hertz contact and subsurface stress
• Checks and safeties
• Units