# Roller Screw Calculation and Selection

## High Static and Dynamic Capacity

Transmission of the load from the nut to the roller screw shaft is provided through the planetary rollers' engagement. The number of contact points is larger, therefore the load-carrying capacity of roller screws is much higher than that of ball screws.

Roller screws are also available with a wide variety of Lead × Pitch combinations, with leads as small as 2mm.

Roller screws can be used with high acceleration and deceleration rates as well as with a high rotational speed. Additionally, there is no problem with losing bearing balls - the nut can be easily removed from the screw shaft frequently.

## Basic Dynamic Load Ratings C and L10 Life

Dynamic load rating is used to calculate the fatigue life of a NRS planetary roller screw. The dynamic load rating is defined as a load, constant in magnitude and direction under which 90% of a statistically significant number of apparently identical planetary roller screws reach an operating life of 106 revolutions (L10).

## Static Load Ratings (C0) and Safety Factors (S0)

Static load rating C0 is a load that would cause a permanent deformation at the most heavily loaded contact equal to 0.0001 of the curvature diameter of the rolling element. In order to prevent deformations that could impair the proper function and the operating noise of the planetary roller screw, a safety factor S0 should be used when selecting a roller screw on the basis of its static load rating.

The S0 factor should not be less than 3. For operations with quasi-static load applications (i.e. presses) where the load occurs primarily on the same portion of the stroke, it is recommended to use higher S0. If size constraints prevent the use of larger screws and the operation of the device is such that the S0 approaches 1, please contact our engineering department.

## Theoretical Life

Theoretical life L10 or Lh is the operating time reached by 90% of a group of apparently identical planetary roller screws operating under the same conditions. The theoretical life is calculated as follows:

 L10 = ( C )3 P

If operation reliability higher than 90% is required, then the theoretical life must be corrected by using a reliability factor (an) according to the table. Ln = L10 × an

 Reliability (%) -a0 90 1 95 0.62 96 0.53 97 0.44 98 0.33 99 0.21

Operating loads can be defined by physical characteristics (i.e. masses, inertia, etc.) that operate on the screw. For systems with varying conditions, such as changes of load magnitude and duration as well as speed, the simple calculation cannot be employed and an equivalent load should be assessed.

The equivalent load is a calculated mean operating load used for determining life and is dependent upon load pattern.

The equivalent load can be computed using the following formula:

 P = ∛ q1 × n1× Fax13 + q2 × n2 × Fax23 + ... + qn × nn × Faxn3 q1 × n1 + q2 × n1 + q2 × n2 +... + qn × nn

The equivalent speed can be computed as follows:

 neq = (q1 × n1 + q2 × n2 +...+ qn × nn) 100

Where

Fax(1,2,n) = applied load in the individual time step
n(1,2,n) = screw rotational speed (RPM) in the individual load steps
q(1,2,n) = time step in (%)

Preloaded nuts are used to eliminate axial lash and to increase system rigidity. Preload is detrimental to the operating life and should be selected carefully. The preload magnitude should be accounted for in the equivalent load calculation so its impact on the system life can be determined.

Preload magnitude should be selected as a function of the operating conditions. In case the varying steps cannot be easily identified, the preload magnitude can be assessed as follows:

 Fp = Fmax 2.83

P = Fp - 0.65 × Fax (for Fax < 2.83 × Fp) (N) P = 0 (for Fax 2.83 × Fp) (N)

## Relieved nut (or half-nut)

P = Fp - 0.35 × Fax (for Fax < 2.83 × Fp) P = 0 (for Fax 2.83 × Fp)

Where

P = resulting equivalent load (N)

## Rigidity of a roller screw

The rigidity of a roller screw assembly is a function of several parameters, such as: nut rigidity, bearing support rigidity, screw shaft rigidity, mounting housing rigidity as well as the mounting arrangement. If known, all of the parameters can be assembled in a formula as follows:

 Cδt = ( 1 + 1 + 1 + 1 )-1 Cδs Cδn Cδb Cδh

Where

Cδt = total system rigidity (N/μm)
Cδs = screw shaft rigidity (N/μm)
Cδn = screw nut rigidity (N/μm)
Cδb = support bearing rigidity (N/μm)
Cδh = housing rigidity (N/μm)

The screw rigidity can be calculated as follows:

Cδs = 165 × d02 × fe

Where

fe = factor dependent on end-support configuration. (See Image 1 in Figure 1 Below).

The nut rigidity can be calculated as follows:

Cδn = fn × (Fax)

The factor fn can be supplied upon request. The customer must determine the rigidity of the bearings and housing.

## Column Strength

If the screw is subjected to compressive loads, then a verification of its suitability to the loading conditions can be performed as follows:

 Fax allowed = fsc × d04 × 104 L²

Where

fsc = factor dependent on end-support configuration (See Image 2 in Figure 1 Below).
d0 = screw nominal diameter (mm)
L = free-length (mm)

## Critical Speed

The maximum achievable rotational velocity of planetary roller screws is affected by the following parameters:

• Rotational speed capability of the nut (and planetary train)
• Diameter and free length of the screw (for rotating screw shafts)
• End support configuration (for rotating screws)
• Rotation member (nut or screw)

While the rotational capability of the nut can be easily assessed since it depends upon the maximum rotational factor DMn (mean diameter of the planetary train × rotational velocity n), the critical speed of the screw shaft must be calculated for each application. This value is normally considered the threshold at which the screw will start to resonate (1st order). The nut DMn factor equals 140,000.

The critical speed is calculated as follows:

 nmax = fsn × d0 × 107 (RPM) L2

Where

nmax = allowable screw rotational velocity (RPM)
fsn = factor dependent upon the end-support configuration (See Image 3 in Figure 1 Below).
d0 = screw nominal diameter (mm)
L = screw free-length (mm)

## Efficiency and Driving Torque

Efficiency of the NRS planetary roller screw is dependent upon its operating parameters. The friction of the system is dependent upon varying factors that cannot be easily summarized here. To simplify the selection of the screw size, the following formulae can be used.

 η1 = 1 1+ (ff × d0/ pho)

(for transforming rotary motion into axial motion)

 η1 = 1 1− (ff × d0/ pho)

(for transforming axial motion into rotary motion)

Where

ff = friction factor

## Torque Required

To move an axial load at constant speed, the screw will require a motor torque and its magnitude can be calculated as follows:

 Mt = Fax × pho × 10-3 2 × π × η1
Mt = drive torque (N · m)

By contrast, to restrain an axial load, the screw must be equipped with a brake and the restraining torque is calculated as follows:

 Mb = Fax × pho × η2× 10-3 2 × π
Mb = brake torque (N · m)

Note: The start-up torque required will be greater than the calculated Mt above.

## Lubrication & Maintenance

NRS planetary roller screws, like all rolling element systems, must be lubricated in order to operate properly.

The screws can be lubricated with oil or grease. The application demands will dictate which media is more suited for the task.

### Grease Lubrication

Typically NLGI Grade 2 greases are used for roller screws. The grease used must not contain solid additives in any form. Greases suitable for lubricating screws must contain EP additives as well as anti-wear additives.

The lubricant characteristics, the amount to be used and its replenishment interval are a function of the application. Factors such as load, stroke length, operating temperature, environment cleanliness, and operating speed will impact the lubricant suitability and durability.

Nook engineers will gladly provide guidance on the selection of suitable grease as well as the maintenance interval.

### Oil Lubrication

Nook E-900L is available in a 32 oz. bottle for applications that require oil lubrication (see page 86 of our (Precision Screw Assemblies Catalog). Applications that operate with high speeds and continuous motion may operate only with oil lubrication. The basic oil viscosity, the presence of additives and the lubricant flow should be assessed during the design phase.

Nook engineers will gladly provide guidance on the selection of a suitable oil, as well as the proper flow, to insure the system operates as intended.

Figure 1