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Ball Screw Design Equations and Selection Criteria

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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

Driving Torque to Obtain Thrust

Where:

T = Driving torque (Nm)
Fa = 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

Fa = FM + Ff + Fi + Fg

FM = Machining force (N)
Ff = Frictional force (N)
Fi = Inertial force (N)
Fg = Gravitational force (N)

Stresses from Applied Loads

Axial Load

Torsional load

Where:

Fa = Applied axial force (N)
rtr = root radius = root diameter / 2 (m)
T = Applied Torque (N-m)
π = 3.14159

Equivalent (Von Mises) Stress

Von Mises stress

Stress equation

Compressive Load

Compressive load

P1 = Buckling load [N]
lb = Distance between mounting positions [m]
E = Elastic modulus [Pa]
I = Second moment of inertia [m4]
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

Fixed - Free Configuration

Simple - Simple Configuration

Simple - Simple Configuration

Fixed - Simple Configuration

Fixed - Simple Configuration

Fixed - Fixed 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.

Basic Static Loading

Where:

Coa = Basic static load rating [N, kgf, lbf]
Fs = Static safety factor
Fa = Load on shaft in axial direction [N, kgf, lbf]

Use Fs (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.

Permissible Ball Screw Speed

Where:

nc = Critical speed [min-1]
lb = 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 = Va / l

Where:

n = Revolutions per second ( s-1 )
Va = 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 Ca and Life

The basic load rating Ca is the load in the shaft direction with 90% of a group of the same ball screws operating individually will reach a life of 106 (1 million) revolutions.

Dynamic Load Rating

Where:

L = Rotation life [rev]
Ca = Basic dynamic load rating [N, kgf, lbf]
fw = Load factor
Fa = Load in shaft direction [N, kgf, lbf]

Use fw
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 acceleration:

Running distance during deceleration:

Running distance during deceleration

References:

THK Co. Ltd
Nook Industries
University of Utah, Mechanical Engineering ME EN 7960

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