Take off-center forces into account when estimating the life of linear-motion systems.

**Chris Blaszczyk Manager of Product Development Automation Components Div. Misumi USA Schaumburg, Ill. **

**Edited by Jessica Shapiro**

The life of components in linear-motion systems (LMS) was discussed previously (Machine Design, Dec. 13, 2007, pp. 84-87). The systems represented in that article were assumed to be centrally loaded. While on-axis loading assumptions can be used to estimate the life of systems with off-axis loading, the equations in this article will further refine the accuracy of component-life predictions for systems with noncentral loads.

**Horizontal Axis**

When the load acting on a horizontal LMS is offset from the axis of motion and from the center of the system, the load doesnâ€™t transmit equally to all four loading points. The points closer to the acting load take a larger portion of the load; the points further from the acting load are unloaded accordingly.

The loads on points 1 through 4 are then given by:

*P _{1}* =

*W*/4 + (

*x*)/(2

_{0}W*X*) + (

*y*)/2

_{0}W*Y*

*P _{2}* =

*W*/4 - (

*x*)/(2

_{0}W*X*) + (

*y*)/2

_{0}W*Y*

*P _{3}* =

*W*/4 + (

*x*)/(2

_{0}W*X*) - (

*y*)/2

_{0}W*Y*

*P _{4}* =

*W*/4 - (

*x*)/(2

_{0}W*X*) - (

*y*)/2

_{0}W*Y*

Where *P _{i}*= the load on each corner of the LMS,

*W*= applied load,

*X*= total length of the system in the direction of motion,

*Y*= total length of the system perpendicular to the direction of motion, and

*x*and

_{0}*y*are the offsets of the acting load from the center point of the system.

_{0}If* c _{1}* is the dynamic load rating of the individual rollers on a two-shafted system like the one depicted here, the dynamic load rating is:

*C _{s}* = 2.88

*c*

_{1}The life of each roller can then be calculated by:

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

*f*)]

_{w}P_{i}^{10/3}

where *f _{t}* = the temperature coefficient for temperatures over 100Â°C and

*f*= the load coefficient based on the type of application. Common values of the load coefficient can be found in the previous article.

_{w}**Vertical axis**

In the case of vertical shafts where a downward force is counteracted by an upward thrust, the rollers see forces in the two directions perpendicular to the axis of motion. The loads on the points are as follows:

*P _{1}* =

*P*=

_{2}*P*=

_{3}*P*= (

_{4}*l*)/(2

_{1}W*X*)

*P _{1s}* =

*P*=

_{2s}*P*=

_{3s}*P*= (

_{4s}*y*)/(2

_{0}W*X*)

where* l _{1}* = the distance between the point of application of the thrust and the point of application of the downward force out of the shaftsâ€™ plane, and

*y*is the distance between the midpoint between the two shafts and the location of the downward force. The thrust is assumed to be applied at the center point of the system although it may be out of the shaftsâ€™ plane. After calculating the load on each support point, you can calculate the life span as above.

_{0}**Right angles to horizontal**

When the system is oriented such that the shaft axis is horizontal and the load is downward, perpendicular to the axis of motion, there are three sets of loads to be considered. The out-of-plane loads are the same as in the vertical axis case above. The in-plane loads for the points closer to the load-application point are:

*P _{1s}* =

*P*=

_{3s}*W*/4 + (

*x*)/(2

_{0}W*X*)

The in-plane loads for the points further from the point of load application are:

*P _{2s}* =

*P*=

_{4s}*W*/4 - (

*x*)/(2

_{0}W*X*)

Once again, these loads can be converted into lifespan estimates as above.

**Acceleration and Deceleration**

Even if the applied load acts on the center point of an LMS, the load becomes unevenly distributed on the four rolling points during acceleration and deceleration. In a horizontal double-shaft system in which a downward force is applied at the center point, the force on each rolling point is *W*/4 at rest and at constant speed. During acceleration, the force on the lagging rollers is:

*P _{1}* =

*P*=

_{3}*W*/4 [1 + (2

*V*)/(

_{1}l_{1}*gt*)]

_{1}XThe force on the leading rollers is:

*P _{2}* =

*P*=

_{4}*W*/4 [1 - (2

*V*)/(

_{1}l_{1}*gt*)]

_{1}XIn these equations, *V* is the speed at the end of the acceleration in mm/sec, *l _{1}* is the vertical distance between the application of the load and the application of the thrust,

*g*is the gravitational constant, 9.8 x 10

^{3}mm/sec

^{2},

*t*is the time in seconds over which the system accelerated, and

*X*is the linear system span in the axial direction.

When decelerating the equations are the same, except that the leading rollers take the higher than average load. For off-axis loading systems, combine the acceleration equations with those above that fit the system. The loads obtained can be used to calculate an average load on each roller over a lifetime of starting and stopping before plugging the average into the lifeprediction calculations.

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