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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

P₁ = Buckling load [N]
l_b = 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

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	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.

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 10⁶ (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	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 deceleration:

References:

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