The calculation bases for screw drives are extensive and range from the calculation of the static safety factor to the critical speed, the buckling load and the nominal service life. All of this awaits you below. If you are looking for further calculation bases, you will find them in our catalogue.
Definition of screw drives
There are three main values to consider for the nut: the outer diameter of the nut body D1 , the flange diameter D2 and the pitch circle diameter of the flange D3 . Further parameters are the nominal ball diameter Dw , the static axial load rating C0a and the dynamic axial load rating Ca . The nut length must also be taken into account.
Ball screw drive ∅d1 = Outer spindle diameter ∅d2 = Spindle core diameter ∅Dpw = Ball centre-to-centre diameter P = Pitch ß = Pitch angle | Mother ∅D1 = Outer diameter of the nut body ∅D2 = Flange diameter ∅D3 = Pitch circle diameter of the flange Ball ∅Dw = Nominal diameter of the ball C0a = Static axial load rating Ca = Dynamic axial load rating |
In this overview you will find all the mathematical definition criteria for ball screws, nuts and balls at a glance.
Calculation of the drive torque
The efficiency is the percentage of the applied energy that is converted into active power. The formulas and parameters required for this are as follows.
Formula 4: Calculation of the drive torque T | Formula 5: Calculation of the axial force Fa | ||
| T | Drive torque | m | Mass |
| Fa | Axial force | a | Acceleration |
| P | Pitch | µ | Coefficient of friction of the drive system |
η | Efficiency from conversion from rotation into linear movement | ||
![\[T = \frac{F_a \times P}{2 \times \pi \times \eta}\]](https://linearwizard.com/wp-content/ql-cache/quicklatex.com-398a0910f34c632f7ddce21c3a40ccea_l3.png "Rendered by QuickLaTeX.com")
![\[F_a = m \times a \times \mu\]](https://linearwizard.com/wp-content/ql-cache/quicklatex.com-038025f9351cd9b8416fb7e80688581d_l3.png "Rendered by QuickLaTeX.com")
Both the efficiency η and the axial force Fa are important for determining the drive torque T.
Load ratings: the axial static load rating C0a and axial dynamic load rating Ca
The axial static load rating C0a describes the constant axial load that generates a total plastic deformation of 0.00001 times the ball diameter. According to DIN, the axial load ratings – both axial static and axial dynamic – based on tolerance class 5. The axial static load rating C0a is calculated on the basis of DIN 69051-4.
The calculation of the axial static load rating C0a is also relevant in addition to the calculation of the axial dynamic load rating Ca . The axial dynamic load rating Ca is defined as the axial load that is unchanging in size and direction and under which a ball screw drive theoretically achieves a service life of 106 revolutions. The axial dynamic load rating Ca for ball screws is also determined in accordance with DIN 69051-4.
With regard to the axial load ratings, it is important to note that the relevant information in the NTN catalogue refers to an optimum load distribution on all loaded balls for ball screws of tolerance class 5. For tolerance classes deviating from tolerance class 5, a correction factor fac must be taken into account, which can be read from the table.
The greater the pitch error of a screw, the less guarantee there is that all balls will bear equally. Consequently, if not all balls loaded evenly, the axial load rating decreases. It is therefore necessary to correct the axial load rating by taking the tolerance class into account.
| Tolerance class | |||
| 0, 1, 3, 5 | 7 | 10 | |
| Correction factor fac | 1.0 | 0.9 | 0.7 |
Correction factors for the different tolerance classes according to DIN ISO 3408-5.
The static safety factor fs
The static safety factor fs must also be taken into account. This takes into account the fact that ball screw drives can be exposed to unforeseen loads. These can have various causes, such as vibrations, shocks or short start-stop cycles. The static safety factor fs is used to prevent impermissible, permanent plastic deformation of both raceways and rolling elements of the ball screw. It is essential that this factor is appropriately observed to ensure the stability and functionality of the respective application with ball screws and to guarantee reliable operation under various load conditions. The static safety factor fs must always be ≥ 1. If the assumed loads fluctuate greatly and tend to be unpredictable, a higher safety factor fs should be used.
Although the calculation of the static safety factor for ball screws is not the same, it is similar to that for linear guides. For ball screws, the two influence factors (the hardness factor fH and the temperature factor fT) are multiplied by the basic static load rating C0a and divided by the maximum axial load Fmax .
Formula 6
|
![\[f_s = \frac{f_H \times f_T \times C_{0a}}{F_{\text{max}}}\]](https://linearwizard.com/wp-content/ql-cache/quicklatex.com-11f74d0cadfc8207b4c7ca6720417a55_l3.png "Rendered by QuickLaTeX.com")
As with linear guides checking the static safety factor is also important for ball screw drives.
Even if there is no clear rule, there are certain guide values or recommendations as to how high this factor should be as a minimum. The factor depends on the movement speeds acting on the ball screw drive and how heavy the loads and how intense the vibrations and shocks are.
| Operating conditions | Static safety factor fs |
• Slow movements • Low loads • No vibrations and shocks | 1.0 … 1.3 |
• Slow movements • Low loads • Light vibrations and shocks | 1.2 … 1.7 |
• Slow movements • Medium loads • Vibrations and shocks | 1.5 … 2.5 |
• Fast movements • High loads • Vibrations and shocks | 2.0 …4.0 |
• Fast movements • High loads • Strong vibrations and shocks | 3.0 … 8.0 |
Ideally, you need a low static safety factor fs , but in extreme conditions or highly loaded applications this can (far) exceed 3.0.
The nominal service life L10
The nominal service life L10 is the calculated service life that can be achieved with a 90 % probability of survival for ball screws under normal operating conditions. Several influencing factors are required to calculate the nominal service life of ball screws. These include the load factor fw , the hardness factor fH and the temperature factor fT . Furthermore, the axial dynamic load rating Ca and the average axial load Fm are required.
Formula 7 | |
| L | Nominal service life [min-1] |
| Ca | Axial dynamic load rating [kN] |
| fw | Load factor |
| fH | Hardness factor |
| fT | Temperature factor |
| Fm | Average axial load [kN] |
![\[L = \left( \frac{f_T \times f_H}{f_W} \times \frac{C_a}{F_m} \right)^3 \times 10^6\]](https://linearwizard.com/wp-content/ql-cache/quicklatex.com-a2148320286ad9732f5af5500de4107f_l3.png "Rendered by QuickLaTeX.com")
On linearwizard.com you will also find information on calculating the service life of linear guides and ball bushings.
With regard to the load factor fw , there are recommendations that depend on the operating conditions, more precisely on the strength of the vibrations and shocks that act on the screw drives in the individual application. If the conditions cannot be precisely predicted in advance, the load factor should be taken into account with a certain degree of certainty or based on experience with comparable applications.
| Operating conditions | Speed [m/s] | Load factor fw |
| No or very low vibrations and shocks | ≤0.25 | 1.0 … 1.2 |
| Light vibrations and shocks | 0.25…≤1.0 | 1.2 … 1.5 |
| Medium vibrations and shocks | 1.0…≤2.0 | 1.5 … 2.0 |
| Strong vibrations and shocks | >2.0 | 2.0 … 3.5 |
| Short-stroke applications | 3.5 … 5.0 |
The load factor results from the operating conditions.
The nominal service life L10 can also be specified in units other than revolutions. Depending on the requirement, it can be specified in kilometers Ls, in hours Lh or in cycles L# .
Formula 8
Formel 12
| | | |||
| L | nominelle Lebensdauer [min-1] | L | nominelle Lebensdauer [min-1] | L | nominelle Lebensdauer [min-1] |
| Ls | nominelle Lebensdauer [km] | Lh | nominelle Lebensdauer [h] | L# | nominelle Lebensdauer [Zyklen] |
| P | Spindelsteigung [mm] | nm | Mittlere Betriebsdrehzahl [min-1] | P | Spindelsteigung [mm] |
| ED | Einschaltdauer [%] | s | Verfahrweg | ||
![\[L_\# = \frac{L \times P}{2 \times s}\]](https://linearwizard.com/wp-content/ql-cache/quicklatex.com-1ca116a7a39daff3e054a9d7d83a453c_l3.png "Rendered by QuickLaTeX.com")
![\[L_s = \frac{L \times P}{10^6}\]](https://linearwizard.com/wp-content/ql-cache/quicklatex.com-e8400eefda862a7d685b4e31b34dc7da_l3.png "Rendered by QuickLaTeX.com")
L
Nominal service life [min-1]
Ls
Nominal service life [km]
P
Pitch [mm]
![\[L_h = \frac{L}{n_m \times 60 \times ED}\]](https://linearwizard.com/wp-content/ql-cache/quicklatex.com-14401fa644a3faa8108411aaa7e2e775_l3.png "Rendered by QuickLaTeX.com")
L
Nominal service life [min-1]
Lh
Nominal service life [h]
nm
Average operating speed [min-1]
ED
Duty cycle [%]
Here you will find the formulas for converting the service life into kilometers, hours or cycles.
Further calculation principles can be found in NTN catalogue for ball screws. Information on mounting screw drives can be found in the corresponding article.
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