Ball Screw Selection and Calculations - University of Utah

[Pages:14]Ball Screw Selection and Calculations

ME EN 7960 ? Precision Machine Design Topic 4

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

4-1

Ball Screw Selection Criteria

Applied Load Thrust force Required torque

Maximum Rotational Speed

Resonance (bending) of threaded shaft DN value

Ball Screw Life Basic dynamic load rating

Ball Screw Selection

Accuracy

Stiffness

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

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1

Based on Load

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

? Thrust force (the sum of all external forces such as machining load, gravity, friction, inertial forces, etc.).

? Torque required to generate the thrust force.

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

4-3

Driving Torque to Obtain Thrust

T = Fal 2

T: driving torque [Nm]

Fa: thrust force [N] l: screw lead [m] : efficiency

Source: THK Co., Ltd.

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

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2

Required Thrust

? The thrust is 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]

Source: THK Co., Ltd.

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

4-5

Stresses from Applied Loads

axial

=

Fa rtr2

torsional

=

2T rtr3

The equivalent (Von Mises) stress:

eq =

+ 3 2

axial

2 torsional

eq

=

4Fa dt2r

1

+

12l 2 2dt2r

2

max: Permissible stress [147 MPa]

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

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3

Von Mises Stress [Pa]

Graphic Solution

3 .108 2. 5 .108

2 .108 1. 5 .108

1 .108 5 .107

Von Mises Stress in Shaft (Fa = 2344N)

1mm lead 2mm lead 4mm lead 5mm lead

max = 147 MPa

0 0. 00 1

0. 00 2

0. 00 3

0.004 0.005 0.006 0.007

Root Diameter [m]

0. 00 8

0. 00 9

1mm lead -> dtr > 4.6mm 2mm lead -> dtr > 4.8mm 4mm lead -> dtr > 5.2mm 5mm lead -> dtr > 5.5mm

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

4-7

Permissible Compressive Load

? Buckling Load

P1

=

2EI lb2

P1: Buckling load [N]

lb:

Distance between mounting positions [m]

E:

Elastic modulus [Pa]

I:

Second moment of inertia [m4]

:

Support factor

Fixed ? free:

= 0.25

Fixed ? supported:

= 2.0

Fixed ? fixed:

= 4.0

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

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4

Fixed-Free Mount

Source: THK Co., Ltd.

Inexpensive but only applicable for short ball screws and/or slow speeds.

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

4-9

Fixed-Supported Mount

Most commonly used mounting setup.

Source: THK Co., Ltd.

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

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5

Fixed-Fixed Mount

Source: THK Co., Ltd.

Overconstrained mounting setup for applications where high stiffness, accuracy, and high shaft speed is required. Ball screw needs to be prestretched to avoid buckling in the case of thermal expansion.

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

4-11

Basic Static Load Rating Coa

? 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.

Coa fs Fa

Coa:

Basic static load rating [N, kgf, lbf]

fs:

Static safety factor

Fa:

Load on shaft in axial direction [N, kgf, lbf]

Use conditions Normal operation

fs (lower limit) 1.0 ? 2.0

Operation with impacts and vibrations 2.0 ? 3.0

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

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6

Permissible 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.

nc

=

602 2lb 2

EI A

= 152dtr E 2lb2

nc:

Critical speed [min-1]

lb:

Distance between supports [m]

E:

Elastic modulus [Pa]

I:

Second moment of inertia [m4]

:

Density [kg/m3]

A:

Root cross sectional area [m2]

:

Support factor

Fixed ? free:

= 1.875

Supported ? supported:

= 3.142

Fixed ? supported:

= 3.927

Fixed ? fixed:

= 4.730

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

4-13

Spindle Speed and DN Value

? Shaft speed

n = va l

n:

Revolutions per second [s-1]

va:

axial speed [m/s]

l:

lead [m]

? DN Value. Unless specified otherwise:

DN 70000

D: N:

Ball circle diameter [mm] Revolutions per minute [min-1]

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

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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.

L

=

Ca f w Fa

3 ?106

L:

Rotation life [rev]

Ca:

Basic dynamic load rating [N, kgf, lbf]

fw:

Load factor

Fa:

Load in shaft direction [N, kgf, lbf]

Use conditions Smooth movement without impacts Normal movements Movement with impacts and vibrations

fw 1.0 ? 1.2 1.2 ? 1.5 1.5 ? 2.5

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

4-15

Example

? Mass of axis: 350kg ? Maximum velocity: 20m/min ? Acceleration time: 0.05s ? Bearing friction factor: 0.003 ? Machining force: 500N ? Length of work piece: 500mm ? Length of travel at maximum speed: 100mm ? Orientation of axis: horizontal

ME EN 7960 ? Precision Machine Design ? Ball Screw Calculations

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