# Ball Screw Selection

The selection of the correct ball screw and nut for a particular application involves five interrelated factors. Before attempting to determine the ball screw and nut combination, the following values must be known:

• Load measured in pounds or newtons
• Speed measured in inches or millimeters per minute
• Length between bearings measured in inches or millimeters
• Life expectancy
• End fixity type

The loads that need to be considered are the static loads, dynamic loads, reaction forces and any external forces affecting the screw. See Load definitions section on page 74 of our Precision Screw Assemblies Catalog for details.

## Speed

The travel rate (linear speed) is the rpm at which the screw or nut is rotating multiplied by the lead of the screw.

## Length

Unsupported length of the screw.

## Life Expectancy

The dynamic load ratings shown on the product specification pages indicate the load that can be carried for 1,000,000 inches of travel for inch screws and 1,000,000 revolutions for metric screws.

The charts on pages 82-83 relate life to load. In applications where the load is relatively constant over the entire stroke, use the highest load to select the ball screw to provide a factor of extra life. For applications where the loads vary significantly, an equivalent load can be calculated using the following formula:

 Lm= ∛ %1(L1)3 + %2(L2)3 + %3 (L3)3 + ... + %n (Ln)3 100

## Where

Ln = each increment of load
%n = percent of stroke at load Ln

## For Example

L1 = 150#
L2 = 225#
L3 = 725#
%1 = 30%
%1 = 45%
%1 = 25%
 Lm = ∛ 30(150)3 + 45(225)3 + 25(725)3 100
Lm = 466 lbs.

The life required is determined by multiplying the total stroke in inches by the total number of strokes required for the designed life of the equipment. To calculate the travel life for a ball nut other than at rated load use the following formula:

 Tx = ( Fr )3 × Tr Fx

Where

Tx = Travel other than rated load. Life is given in inches or meters.
Fr = Rated Load in pounds or kilonewtons.
Fx = Actual or Equivalent load in pounds or kilonewtons.
Tr = Rated Travel Life. For inch screws this is equal to 1,000,000 inches. For Metric Screws this is equal to the ball nut lead in meters times one million revolutions.

## 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. three basic types of end fixity are described in (See Image 1 in Figure 1 Below)

"Simple" end fixity can be provided through a single bearing support. Multiple or spaced pairs of bearings are more rigid than a "simple" support, but, because of their inherent compliance are not truly "fixed". A screw can be supported with different combinations of end fixity. (See Image 2 in Figure 1 Below)

## 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 so 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 the maximum speed be limited to 80% of the calculated critical speed. The theoretical formula to calculate critical speed in rpm is:

 N = Cs × 4.76 × 106 × d L²

## Where

N = Critical Speed (rpm)
d = Root Diameter of Screw (inch)
L = Length Between Bearing Supports (inch)
Cs = 0.36 for one end fixed, one end free
1.00 for both ends simple
1.47 for one end fixed, one end simple
2.23 for both ends fixed

The critical speed chart on page 85 or 159 is provided to quickly determine the minimum screw size applicable for Nook EZZE-MOUNT™ designs. Maximum travel rate is also limited by ball velocity. The ball velocity is a function of the ball circle diameter and rotational speed. Ball velocity is limited by a maximum DN (ball circle diameter × rpm). The charts show the maximum speed based on the DN value for each screw in parentheses.

If the selected ball screw does not meet the speed criteria, consider the following options:

1. Increase screw lead and reduce rpm
2. Change end fixity (e.g. simple to fixed)
3. Increase screw diameter

The final consideration should be to recheck the selected screw against all three of the design criteria; life, column strength and critical speed.

## Column Strength

When a screw is loaded in compression (see compression load definition on page 74), its limit of elastic stability can be exceeded and the screw will fail through bending or buckling.

The theoretical formula to calculate the column strength in pounds is:

 Pcr= 14.03 × 106 ×Fc ×d4 L²

Where

Fc = End Fixity Factor
0.25 for one end fixed, one end free
1.00 for both ends supported
2.00 for one end fixed, one end simple
4.00 for both ends rigid
d = Root Diameter of Screw (inch)
L = Distance between nut and load carrying bearing (inch)

The column strength chart on page 16 may be used to verify that the screw can carry the required load without buckling.

The charts show the theoretical limitations of each screw on a separate line. The lines are limited horizontally by the slenderness ratio and vertically by the maximum static capacity of the bronze nut. Actual load is limited by the maximum nut capacity.

If the selected screw does not meet compression load criteria, consider the following options:

1. Change end fixity (e.g. simple to fixed)
2. Design to use screw in tension
3. Increase screw diameter
Figure 1