As with rolling bearings, certain calculation principles must also be mastered in linear motion in order not to expose the linear guides to excessive loads and to prevent failures, a shortened service life and as well the oversizing of linear guides. It is therefore essential, for example, to calculate the service life of linear guides. Other calculations relate to the static safety factor or stiffness.
The nominal service life L10
The servie life L refers to the mileage that a component can cover before the first signs of material fatigue, which are usually visible on the raceways or the rolling elements, appear.
You may already be familiar with the nominal service life L10 from the service life calculation of rolling bearings. It is based on a statistical calculation. L10 describes the calculated service life; it refers to a single linear guide system or a group of identical linear guide systems that run under the same operating conditions and can achieve the calculated service life with a 90 % probability of failure. If you want to reduce the failure probability of 10 %, you can achieve this goal through different dimensioning. However, this can easily lead to overdimensioning of the linear guide systems. To avoid such overdimensioning and design errors, it is advisable to consult the manufacturer in the event of uncertainties.
The nominal service life L10 of linear guides is specified in kilometres. The service life of ball guides is calculated slightly differently to that of roller guides; depending on the type of linear guide, one of the two formulas must always be used.
Formula 1
for ball guides:
![\[L = \left( \frac{f_h \times f_c \times f_t}{f_w} \times \frac{C_{50}}{F_m} \right)^3 \times 5 \times 10^4\]](https://linearwizard.com/wp-content/ql-cache/quicklatex.com-a6b5f62b87a5958733cd1da0b6d3aa42_l3.png "Rendered by QuickLaTeX.com")
or
![\[L = \left( \frac{f_h \times f_c \times f_t}{f_w} \times \frac{C_{100}}{F_m} \right)^3 \times 10^5\]](https://linearwizard.com/wp-content/ql-cache/quicklatex.com-de0cf020377969e54c5f6ad95597326a_l3.png "Rendered by QuickLaTeX.com")
![\[C_{100} = 1,26 \times C_{50}\]](https://linearwizard.com/wp-content/ql-cache/quicklatex.com-7fb0e12af0cefe06b48957ee804b8f96_l3.png "Rendered by QuickLaTeX.com")
for roller guides:
![\[L = \left( \frac{f_h \times f_c \times f_t}{f_w} \times \frac{C_{100}}{F_m} \right)^{10/3} \times 10^5\]](https://linearwizard.com/wp-content/ql-cache/quicklatex.com-7aee85bd5b0949ed480004d0d65600f7_l3.png "Rendered by QuickLaTeX.com")
| L | Nominal service life (m) |
| C50 | Dynamic load rating based on 50 km (kN) |
| C100 | Dynamic load rating based on 100 km (kN) |
| fh | Hardness factor |
| fc | Contact factor |
| ft | Temperature factor |
| fw | Load factor |
| Fm | Average equivalent load |
When calculating the service life, different formulae must be used depending on the type of linear guide.
As indicated above, the operating conditions are a factor that should not be neglected when calculating the service life. The intensity of vibrations and shocks is of particular importance, and a distinction is made between five levels.
| Operating conditions | Speed (m/s) | Load factor fw |
| No or very low vibrations and shocks | ≤0.25 | 1.0…1.2 |
| Low 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 |
As in the rotative area, oscillations and vibrations can have negative effects on the contact surface between the rolling element and the raceway.
Depending on the requirements, the nominal service life L10 can also be specified in units other than kilometres. It is possible to convert the unit into hours Lh or cycles L# .
Formula 2
| | ||
| Lh | Nominal service life (h) | L# | Nominal service life (cycles) |
| s | Stroke (m) | s | Stroke (m) |
| n | Number of Strokes (min-1) | ||
![\[L_h = \frac{L \times 10^3}{2 \times s \times n \times 60}\]](https://linearwizard.com/wp-content/ql-cache/quicklatex.com-af0ef397ecc928c01f9c408c45ce6481_l3.png "Rendered by QuickLaTeX.com")
![\[L_\# = \frac{L}{2 \times s}\]](https://linearwizard.com/wp-content/ql-cache/quicklatex.com-7c45b3b6e15082a8369ae905d5218081_l3.png "Rendered by QuickLaTeX.com")
In contrast to L# , the number of double strokes or cycles (min-1) must be considered when converting to Lh .
The dynamic load rating C
According to DIN ISO 14728-1, the dynamic load rating C describes a radial load that is not variable in size and direction and that a linear guide can theoretically absorb for a nominal service life of 5 x 1010 m travelled distance. The following applies: If a nominal service life of 105 is assumed, the dynamic load rating for a nominal service life of 5 x 1010 m is multiplied by a conversion factor 1,26. This conversion factor is also used to compare the load ratings. The formula for calculating the dynamic load rating is based on DIN ISO 14728-2.
The static load rating C0
The static load rating C0 is the static radial load in the centre of the most heavily loaded contact surface between the rolling element and the raceway. The static load rating C0 corresponds to a calculated Hertz-type compression. According to DIN ISO 14728-1, this Hertz-type compression for linear guides is between 4 200 MPa and 4 600 MPa. This load results in a permanent overall deformation of the raceway. This deformation corresponds to approximately 0.0001 times the diameter of the rolling element (therefore the static safety factor fs has to be always > 1). The formula for calculating the static load rating C0 is also defined in accordance with DIN ISO.
The static safety factor fs
Another factor that needs to be calculated is the static safety factor fs . This must be considered, as unexpected or unforeseen loads and/or torques can act on the linear guide system when designing linear guides. These can be due to various causes, primarily vibrations, shocks, short start-stop cycles (strokes) or overhanging loads.
The static safety factor fs is the ratio of the static load rating C0 to the maximum occurring load Fomax . This refers to the highest amplitude; even very short-term amplitudes are taken into account. The function of the static safety factor fs is to prevent impermissible plastic deformation of the raceways and rolling elements. This factor also prevents cracks and fractures in the raceways.
Formula 3
|
![\[f_s = \frac{f_H \times f_T \times f_C \times C_0}{F_{\text{max}}}\]](https://linearwizard.com/wp-content/ql-cache/quicklatex.com-90e4fea57fd14556f6ba4e432eb719ec_l3.png "Rendered by QuickLaTeX.com")
The static safety factor fs must not be less than 1 but in practice is usually greater than 2 under normal operating conditions. The formula for the static safety factor fs is also used for ball bushings.
Finally, a brief explanation of the three influencing factors, the contact factor, hardness factor and temperature factor, follows. The contact factor fc considers the fact that the full load-capacity cannot be utilised if the carriages are arranged very close together or on block due to tolerances. From a distance between the carriages of two carriage lengths, the contact factor no longer has any influence and is 1.
The hardness factor fH is always 1 at NTN. However, if a guide is not made from carbon steel like 100Cr6, but from a material with a lower hardness, a different value would have to be calculated.
The fact that the rail hardness is reduced in a temperature range of over 100 °C is considered with the temperature factor fT . For applications with the corresponding temperature ranges, the temperature factor must therefore always be included in the calculation of the static safety factor fs .
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