The calculation on pages link to link is used for the preliminary selection of monorail guidance systems. They allow an approximate calculation of the equivalent static and dynamic bearing loads.

In order to achieve precise design of linear guidance elements in relation to basic rating life and static load safety factor, it is necessary to calculate the bearing load in a statically indeterminate system and the internal load distribution of the linear guidance elements (Loading of individual rolling elements, Figure 1). This requires a complex calculation process.

For this reason, INA developed the rolling bearing analysis program BEARINX^® which can be used to calculate linear and rotary bearings as a part of the complete system (e.g. machine tool, automotive gearbox, etc.) and thereby ensure reliable designs.

The linear module of BEARINX^® can be used to calculate linear guidance elements in multi-axis systems (e.g. machine tools) under any load combination down to the level of the rolling element contact. The integral analysis method can be used to investigate the influence of nearly all parameters of the complete system on relevant results.

This sophisticated calculation model takes account of all the elasticities in the system, ranging from the rigidity of the saddle plate and guideways through to the non-linear deflection behaviour of the rolling elements.

In order to determine even more precisely the pressure between the rolling elements and raceway in linear recirculating roller bearing and guideway assemblies, the end profiling of the rolling elements is also taken into consideration. The adjacent construction is assumed to be rigid in the first instance but can, if necessary, be modelled on an elastic basis by means of reduced rigidity matrices (e.g. from FE calculation).

This model gives significantly more precise results than calculation programs that only take account of elasticity in rolling contact. This means an increased level of security in the design.

BEARINX^® allows the calculation of systems with any number of: travel axes, linear guidance elements and linear drives, load situations, loads and masses.

The results provided by BEARINX^® include the static load safety factor, the basic rating life and the displacements that arise from the elasticity of the bearing arrangement.

Calculation using BEARINX^® is available as a service.

The linear calculation program BEARINX^® online assists in the calculation and design of the linear guidance system, Figure 2; for information and registration, please visit: www.schaeffler.com. A fee will be charged for usage.

The input data for the calculation program should be compiled from the design brief (with clearly dimensioned drawings or diagrams in at least two views). Here is a step-by-step guide based on a simple example to show the dimensioning process.

The relevant factors for calculation, apart from the linear guidance elements and the drive system for the table, are those components that induce loads on the linear guidance elements (the inherent mass of the components or their inertia forces), Figure 3.

The table co-ordinate system is a Cartesian, right hand co-ordinate system.

The directions in the table co-ordinate system are defined as follows, Figure 4:

• X axis: travel direction of the table
• Y axis: main load direction on the system (direction of weight)
• Z axis: derived from the right hand rule (lateral direction).

The (translational) position of the table co-ordinate system is freely selectable. It is recommended that this should be located centrally between the carriages for directions X and Y.

The translational position of the linear guidance elements is stated in relation to the table co-ordinate system. In order to determine the torsion angle of the linear guidance elements, their co-ordinate system is rotated about the X axis into the table co-ordinate system, Figure 5.

The translational position of the drives (support function in the traverse direction) is stated in relation to the table co-ordinate system as Y and Z co-ordinates, Figure 6.

The mass of the components is concentrated at a mass point at its centre.

The translational position of the centres is again stated in relation to the table co-ordinate system, Figure 7.

External loads such as machining forces on the linear table, are stated in relation to the table co-ordinate system.

The following must be stated, Figure 8:

• in which of the defined load cases the load acts on the table co‑ordinate system
• the position of its loading point
• the force and moment components.

In order to depict the working cycle of the machine, a duty cycle must be described. This is composed of the motion parameters of the machine and their loading due to external loads (e.g. machining forces).

On the basis of a speed/time diagram, the working cycle should be subdivided logically into individual load cases, Figure 9, to .

With the aid of the basic motion formulae for uniform motion (v = const.) or uniform acceleration (a = const.) as appropriate, the missing values (travel, acceleration) can then be determined.

The following simplified example describes the travel of a linear table.

The circled numbers  to describe the load cases in Figure 9.

Complex travel cases can in certain circumstances be usefully reduced by combination. Please consult the Schaeffler Group engineering service on this matter.

In t₁ (0,05 s) to v₁ (0,5 m/s), Figure 9, .

In t₂ (0,045 s) to v₂ (0,05 m/s), Figure 9, .

At v₃ (0,05 m/s) for t₃ (1,105 s);
additional effect of machining force, Figure 9, .

Position:

• x = –520 mm
• y = –270 mm
• z = –260 mm.

Value:

• M_x = 720 Nm
• F_x = 24 Nm
• M_y = 24 Nm
• F_z = 20 Nm.

In t₄ (0,0025 s) to v₄ (0 m/s), Figure 9, .

In t₅ (0,025) to v₅ (–0,5 m/s); opposing direction, Figure 9, .

At v₆ (–0,5 m/s) for t₆ (0,135 s);
opposing direction, Figure 9, .

In t₇ (0,0257 s) to v₇ (0 m/s), Figure 9, .

For t₈ (1,5 s), at v₈ (0 m/s), Figure 9, .