Axial Stiffness of Ball Screw Drives Across Nut Positions

01 / What is being evaluated

Define the load, measurement point and reference

The same feed axis can show different elastic displacement when the same thrust is applied near the axial locating end and farther away. The screw is elastic, but position also changes the loaded shaft length and the relationship between supports and joints. A nut-stiffness value alone rarely explains the response of the complete axis.

This article evaluates quasi-static, small incremental displacement along the screw axis. The reference is the common rigid structure supporting the end assemblies, and the rotational coordinate is held constant. Torsion, control-loop motion, thermal drift, guide pitch and working-point offset are excluded. Add those effects separately where the application requires them; the resulting curves are not a complete machine-position error prediction.

Stiffness k = ΔF/Δδ and compliance c = Δδ/ΔF are reciprocals. Using increments makes clear that a slope is being evaluated around a specified preload and contact state. If loading separates a contact, causes slip or enters a strongly nonlinear range, use the appropriate force–displacement relationship instead of extending one constant stiffness indefinitely.

02 / Equivalent paths

Build each series branch before combining the ends

With axial reaction at the left end only, load returns through the nut, left screw segment and left support. A right-end radial support does not automatically provide a second axial path. If both ends are designed to restrain axial displacement, the load at the nut can return through two paths to the same reference structure. They carry load at a common node displacement, so the paths act in parallel.

k_bL and k_bR represent the effective axial stiffness of each support bearing assembly and its local housing connection. Combine each shaft segment in series with its own support to obtain k_L and k_R. A is the selected equivalent axial section. For a stepped shaft, add the segment compliances using their individual lengths and areas rather than using the largest visible diameter throughout.

This network assumes stable end references and branches coupled only through the common node. If base bending moves the two supports, or a plate couples their reactions, add the appropriate base compliance or coupled model. Assigning the same physical plate deformation independently to both branches can count it twice and obscure the actual relationship.

03 / Position example

Compare displacement under the same 600 N increment

Use an independent model with axial support spacing L = 1,000 mm, equivalent solid-rod diameter 16 mm and E = 210,000 N/mm², giving A = 201.06 mm². Assign each end support an effective stiffness of 150 N/μm, the nut contact 180 N/μm and the common connection 120 N/μm. These assumed values explain the network; they are not measurements or ratings for an HZMotion model.

Apply an axial increment F = 600 N and assume the contact state remains unchanged throughout loading. Compare axial support at the left end only with axial support at both ends, keeping the other parameters the same. At x = 500 mm, each shaft segment has stiffness about 84.45 N/μm. In series with a 150 N/μm support, each end branch becomes approximately 54.03 N/μm.

In the single-end model, moving the nut away from the axial locating end lengthens the shaft segment transmitting load and increases its compliance. In the symmetric dual-end model, both ends provide a return path. At midspan neither branch is short, so total stiffness is lowest there and the curve is symmetric. Unequal end-support stiffness would change that symmetry.

At midspan, the two parallel branches give k_end ≈ 108.06 N/μm. After adding nut and common-connection compliance, the complete modeled axis has only about 43.21 N/μm. This explains why improving the supports can yield less overall improvement than a shaft-only calculation suggests: the load still crosses other compliant components.

04 / Load sharing

Support reactions follow stiffness, not an automatic half split

At x = 200 mm in the example, the shorter, stiffer left path carries about 415.2 N of the increment and the right path about 184.8 N. Only at midspan do they each carry 300 N. This division helps organize support checks, but it is not the complete internal contact load of each bearing. Bearing arrangement, preload and load direction still have to be included.

Initial tensile preload and the incremental force in this model are separate accounting items. Nut-node movement can increase tension in one segment and reduce it in the other, or produce a compression increment under some initial conditions. Combine initial state and increments with consistent signs to determine net tension/compression, maintained contact and bearing loads. Thermal changes in support spacing require an additional calculation.

Replacing the nut with a stiffer configuration does not remove a long single-end shaft path or a flexible mounting connection. Conversely, a stiff support network can still be limited by the nut bracket or carriage attachment. Locate the main displacement contributions before comparing a larger section, different support layout, improved joint or revised measurement reference.

Also distinguish slope from hysteresis. Smooth reversible displacement is consistent with elastic compliance. A direction-dependent loop, drifting zero or discontinuity can require checks for friction, clearance, slip and contact changes. Those effects should not all be hidden inside a single low-stiffness constant.

05 / Measurement

Measure the axis, then separate accessible contributions

At representative nut positions, record axial force and displacement with an agreed loading arrangement. Keep the load point, rotational restraint, temperature, measurement direction and reference consistent. The displacement reference should move as little as possible under load; a bending sensor bracket adds its own compliance to the reading.

1. Record the initial stateConfirm assembly, preload and temperature, and use the approved static-load range.
2. Cover several positionsChoose representative near-end, middle and far-end locations with the same loading arrangement.
3. Load and unload in stepsRecord actual force and settled displacement at each step, retaining loading direction.
4. Compare incremental slopesCalculate ΔF/Δδ within the same contact regime; do not fit across a clearance or slip discontinuity.
5. Return to the structureRelate measured compliance to the actual support, nut and joints before choosing a change and repeating the measurement.

HZMotion actuators and HZM worktables combine screw drive, guides and mounting structure into motion units; support units and nut brackets provide component options. Family names identify structures but do not supply every stiffness needed in this model. Start with the HZM structure description and support-unit catalog, then request model-specific force–displacement or structural information.

Specify allowable displacement at a stated load instead of asking only for “high rigidity.” Include the working-position range, force direction, initial preload, mounting base and measurement reference. Those details let component data be translated into an axis requirement.

06 / Questions

Common axial-stiffness modeling questions