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### Ball Screw Design Equations and Selection Criteria

Machine Design Resources

Gear Design and Engineering

A ball screw transforms rotational motion into transitional motion. As a result, the shaft is subject to loads:

Ball Screw Design Equations and Selection Criteria:

- Applied load
- Required Torque
- Maximum Rotational Speed
- Resonance (bending) of threaded shaft
- DN force ball screw
- Ball screw life, basic dynamic rating
- Stiffness
- Accuracy

**Driving Torque to Obtain Thrust**

Where:

T = Driving torque (Nm)

F_{a} = Axial thrust force on ball screw (N)

l = Screw lead (m)

η = Efficiency

**Thrust Required**

Thrust is equal to the sum of all forces acting in the axial direction

F_{a} = F_{M} + F_{f} + F_{i} + F_{g}

F_{M} = Machining force (N)

F_{f} = Frictional force (N)

F_{i} = Inertial force (N)

F_{g} = Gravitational force (N)

**Stresses from Applied Loads**

Where:

F_{a} = Applied axial force (N)

r_{tr} = root radius = root diameter / 2 (m)

T = Applied Torque (N-m)

π = 3.14159

**Equivalent (Von Mises) Stress **

**Compressive Load **

P_{1} = Buckling load [N]

l_{b} = Distance between mounting positions [m]

E = Elastic modulus [Pa]

I = Second moment of inertia [m^{4}]

λ = Support factor

Fixed – free: λ = 0.25

Fixed – supported: λ = 2.0

Fixed – fixed: λ = 4.0, Fixed -supported = 2.0, Fixed-free = 0.25

Fixed - Free Configuration

Simple - Simple Configuration

Fixed - Simple Configuration

Fixed - Fixed Configuration

**Basic Static Loading **

- When ball screws are subjected to excessive loads in static condition (non rotating shaft), local permanent deformations are caused between the track surface and the steel balls.
- When the amount of this permanent deformation exceeds a certain degree, smooth movement will be impaired.

Where:

C_{oa} = Basic static load rating [N, kgf, lbf]

F_{s} = Static safety factor

F_{a} = Load on shaft in axial direction [N, kgf, lbf]

Use |
F_{s} (Lower Limit) |
---|---|

Normal | 1.0 - 2.0 |

With Impacts and Vibrations |
3.2 - 3.0 |

**Permissible Ball Screw Speed**

When the speed of a ball screw increases, the ball screw will approach its natural frequency, causing a resonance and the operation will become impossible.

Where:

n_{c} = Critical speed [min-1]

*l*_{b} = Distance between supports [m]

E: Elastic modulus [Pa]

I = Second moment of inertia [m4]

ρ = Density [kg/m3]

A = Root cross sectional area [m2]

λ = Support factor

Fixed – free: λ = 1.875

Supported – supported: λ = 3.142

Fixed – supported: λ = 3.927

Fixed – fixed: λ = 4.730

**Spindle Speed and DN Drive**

n = V_{a} / *l*

Where:

n = Revolutions per second ( s^{-1} )

V_{a} = Axial Speed (m/s)

*l* = Lead (m)

DN unless otherwise specified

DN ≤ 70000

D = Ball circle diameter (mm)

N = Revolutions per minute (min^{-1})

**Dynamic Load Rating C _{a} and Life **

The basic load rating C_{a} is the load in the shaft direction with 90% of a group of the same ball screws operating individually will reach a life of 10^{6} (1 million) revolutions.

Where:

L = Rotation life [rev]

C_{a} = Basic dynamic load rating [N, kgf, lbf]

*f _{w}* = Load factor

F

_{a}= Load in shaft direction [N, kgf, lbf]

Use | f_{w} |
---|---|

Smooth operation no impacts | 1.0 - 1.2 |

Normal operation | 1.2 - 1.5 |

Operation with impacts and vibrations | 1.5 - 2.5 |

Running Lengths depending on Usage

Running distance during acceleration:

Running distance during deceleration:

References:

THK Co. Ltd

Nook Industries

University of Utah, Mechanical Engineering ME EN 7960