# Ball Screw Column Strength and Critical Speed

## End Fixity

End fixity refers to the method by which the ends of the screw are supported. The degree of end fixity is related to the amount of restraint of the ends of the screw. It is important to determine the proper end fixity in order to determine the appropriate screw size for a give application.

When determining if an end is fixed, it is not sufficient to just restrain the bearing axially. In order to be truly fixed a dual bearing, separated by 1.5 the journal diameter, or a multiple bearing set is needed.

## Column Strength

When a screw is loaded in compression, its limit of elastic stability can be exceeded and the screw will fail through bending or buckling. Use the following formula in determining the column strength of a given screw.

WHERE:

• Fc = Permissible axial load to buckling (N)
• E = Modulus of longitudinal elasticity (2.05 x 105 N/mm2)
• I = Minimum secondary moment of screw shaft cross section (mm4)
• dr = Screw shaft root diameter (mm)
• C = Factor determined by supporting method of ball screws
• One end fixed other end free n=0.25
• Both ends simply supported n=1
• One end fixed other end simply supported n=2
• Both ends fixed n=4

NOTE: No safety factor incorporated into above equation. Nook recommends using a safety factor of at least Fc × 0.5.

## Critical Speed

The speed that excites the natural frequency of the screw is referred to as the critical speed. Resonance at the natural frequency of the screw will occur regardless of the screw orientation (vertical, horizontal etc.) or if the system is designed such that the nut rotates about the screw.

The critical speed will vary with the diameter, unsupported length, end fixity and rpm. Since critical speed can also be affected by shaft straightness and assembly alignment, it is recommended that the maximum speed be limited to 80% of the calculated critical speed. The theoretical formula to calculate critical speed in rpm is:

WHERE:

• ncr = Permissible operating speed for critical speed (rpm)
• L = Distance between supports (mm)
• E = Modulus of longitudinal elasticity (2.05 Ã— 105 N/mm2)
• I = Minimum second area moment of Inertia of Screw shaft cross section (mm4)
• dr = Screw shaft root diameter (mm)
• g = Acceleration of gravity (9.81 Ã— 103 mm/sec2)
• γ = Specific weight (7.71 Ã— 10-5 N/mm3)
• A = Minimum Cross sectional Area of Screw Shaft (mm2)
• λ = Factor determined by supporting method of ball screws
• One end fixed other end free λ = 0.59 π
• Both ends simply supported λ = π
• One end fixed other end simply supported λ = 1.25 π
• Both ends fixed λ = 1.49 π

Assuming Earth gravity at Sea Level, a simplified formula for calculating critical speed is available:

Where d and L are both measured in millimeters and Cs is from the following table:

• One end fixed other end free Cs = 0.36
• Both ends simply supported Cs = 1.00
• One end fixed other end simply supported Cs = 1.56
• Both ends fixed Cs = 2.23

NOTE: No safety factor incorporated into above equations. Nook recommends using a safety factor of at least ncr × 0.8.

## D × N value

The critical speed is also limited by the ball circle diameter × rpm value. This value is not to exceed 70,000.

WHERE:

• D: Ball circle diameter - BCD (mm)
• N : Number of revolutions per minute (rpm)
Figure 1