Chris Blaszczyk

Manager of Product Development

Automation Components Div.

Misumi USA

Schaumburg, Ill.

A fundamental question engineers face when designing a motion system is determining how long it will last. Movement in linear-motion systems (LMS) creates stresses between rolling elements and their corresponding contact surfaces. Over time, these stresses can lead to subsurface metal-fatigue cracks that eventually propagate to the surface and cause metal flaking, or spalling.

A common measure of operational life for motion systems is the total travel distance needed to initiate flaking. A number of factors come into play when calculating life span, but the key to accurate life span data is the common, real-life problems LMS encounter — and their effects on mechanical components.

The type of materials that make up system components will affect these calculations to some degree. But the equations and schematics in this article are generally applicable and will serve as guides to calculating system lifespan.

**System basics **

The first step is to define basic parameters.

**Basic dynamic load rating,** *C*, is the constant load applied in a constant direction that lets LMS of the same design, operating under the same conditions, travel 50,000 m, without 90% of the material surface flaking due to rolling-contact fatigue. LMS manufacturers generally supply this data.

**Basic static load rating**, *C _{o}*, is the static load exerted on contacting parts under maximum stress, defined as the point where total permanent deformat ion in the rolling element and rolling contact surface equals 0.0001 times the rolling-element diameter.

**Allowable static moment**, *M _{s}*, if applicable, is a function of the roll, pitch, and yaw moments (

*M*,

_{1}*M*, and

_{2}*M*) and the critical static-moment load that acts on the system. It is set according to the permanent deformation defined for the basic static-load rating,

_{3}*C*.

_{o}**Rated life span** is the total travel distance that 90% of LMS of the same design can endure under the same rated conditions, without rolling and sliding elements suffering excessive wear, rolling fatigue, or flaking.

**Predicting life **

For ideal systems, calculate rated life span, *L*, from the basic dynamic load rating and various loads on the system, using the following equations. For ball bearings,

*L* = 50(*C*/*P*)^{3}

where *P* = the load acting on the system.

For roller bearings,

*L* = 50(*C*/*P*)^{10/3}.

Realistically, engineers must consider loads caused by vibration and impact, as well as load distribution as the system moves back and forth. Further, operating temperatures and component makeup can significantly affect life span. Engineers should, thus, consider these additional real-world factors.

**Surface hardness**. In any linear-motion system, the shaft or rail surface must be hard enough to withstand contact with the bearings. Otherwise, the allowable loads and life span are typically lower. Use the graph to calculate the hardness coefficient, *f _{h}*.

**Temperature**. Temperatures that exceed 100ºC in LMS can degrade the hardness of components and reduce allowable loads, when compared to systems used at room temperature. It reduces life span as well. Use the graph to calculate the temperature coefficient, *f _{t}*.

**Bearing contact. **In general, the number of linear bearings mounted on a shaft, rail, or raceway varies with the application. Loading may not be uniform, so allowable loads depend on the number of bearings on the same shaft. Contact coefficient, *f _{c}*, is defined in the table.

**Loading.** In a perfect world, engineers would have the exact specs regarding material weights, inertia forces, load moments, and other factors when calculating the overall load acting on LMS. Realistically, unpredictable vibration, impact, and other external factors can make precise data difficult to come by. Therefore, simplify life-span calculations using load coefficient, *f _{w}*, as shown in the accompanying table.

When all the parameters are considered, the ideal life equations change as follows:

For ball bearings,

*L* = 50((*f _{h}f_{t}f_{c}*/

*f*)(

_{w}*C*/

*P*))

^{3}.

For roller bearings,

*L* = 50((*f _{h}f_{t}f_{c}*/

*f*)(

_{w}*C*/

*P*))

^{10/3}.

For cases with constant stroke length and cycle times, calculate lifespan in hours using:

*L _{h}* = (1,000

*L*)/((2

*l*)60).

_{s}n_{1}These equations hold for linear bushings, ball bushings, linear guides, and similar components.

Rated loads for slideways are based on the number of rolling elements and their orientation. For slideways with one shaft and offset loading, as shown in the accompanying graphic, dynamic load rating, *C _{s}*, is defined as:

*C _{s}* = (

*Z*/2)

^{3/4}

*c*

_{1}where *Z* = the number of rolling elements and *c _{1}* is the basic dynamic load rating per roller.

Static load rating, *C _{os}*, is:

*C _{os}* = (

*Z*/2)

*c*

_{0}where *c _{0}* = the basic static load rating per roller.

For vertically loaded shafts and two shafts loaded in parallel, dynamic load rating is:

*C _{s}* = (

*Z*/2)

^{3/4}

*c*(2)

_{1}^{7/9}

and static load rating is:

*C _{os}* =

*Zc*.

_{0}Calculate life span for slideways using:

*L* = 50(*f _{t}C_{s}*/

*f*)

_{w}P^{10/3}.

**Speed loads **

Often, the load, *P*, acting on a linear-motion system is not constant. For instance, loads obviously change when a system has start/stop reciprocating motion or transfers objects in one direction and returns empty. In such cases, we need to determine and apply the “speed load” to life-span calculations.

Here are several common cases, as shown in the Fluctuating Loads graphic.

1. Load changes in steps over a travel distance. First, specify travel distance *l*_{1} subject to load *P*_{1}, travel distance *l*_{2} subject to load *P*_{2}, and so on. Mean load *P _{m}* can be found from:

P=_{m}^{3}√ | 1 | (P_{1}^{3}l_{1}+P_{2}^{3}l_{2} | … | +P_{n}^{3}l_{n)} |

l |

where *l* = the total travel distance.

2. Load changes almost linearly. Here, approximate mean load using:

*P _{m}* ≈ (1/3)(

*P*+ 2

_{min}*P*)

_{max}where *P _{min}* and

*P*= the minimum and maximum fluctuating loads, respectively.

_{max}3. Load change resembles a sinusoidal curve. In such cases, approximate mean load from:

*P _{m}* ≈ 0.65

*P*.

_{max}For partial-sinusoidal curves, approximate mean load from:

*P _{m}* ≈ 0.75

*P*.

_{max}Finally, due to unpredictable forces and vibrations that can act on LMS — both at rest and when moving — it is always advisable to apply a safety factor when calculating allowable loads. Divide the basic static load rating by a static safety factor to determine allowable loads. We recommend a lower limit on the safety factor of 1 to 2 for normal operating conditions; 2 to 4 when the application requires smooth travel; and 3 to 5 when there are vibrations or impact loading.