How to calculate the effect of preload on ball screw axial deflection

In a recent post, we looked at the methods for inducing preload in a ball screw assembly and the effects it has on performance. One of those effects is that preload reduces axial deflection (displacement) and increases rigidity. But preload also increases the applied load on the screw assembly and reduces life.

To determine the ideal preload amount for reducing ball screw axial deflection without significantly reducing life, let’s look at the relationship between force and deflection for a preloaded ball nut.

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There are several ways to induce preload in a ball screw assembly, with the most common methods being: 1) use two separate ball nuts with a spring or spacer between them, or 2) use a single nut with a lead shift, which creates the effect of two opposing ball nuts. Regardless of which method is used, the ball nut can be treated as two separate pieces: Nut 1 and Nut 2.

The graph below shows the effects of preload on each nut, in terms of the axial force that each ball nut experiences and the axial displacement of each. The axial displacement values and axial forces are defined as follows:

δ_ao = axial displacement due to preload

δ₁ = total axial displacement of Nut 1

δ₂ = total axial displacement of Nut 2

δ_a = axial displacement due to applied axial force

F_a0 = axial force due to preload

F₁ = total axial force on Nut 1

F₂ = total axial force on Nut 2

F_a = applied axial force

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Preload creates an axial force on the nut (F_a0), which is partially taken by Nut 1 and partially taken by Nut 2. As a result, each nut experiences an axial displacement, δ_ao, due to the preload axial force.

When an external axial load (F_a) is applied to the screw, the axial force on Nut 1 increases from F_ao to F₁. On the graph above, if we follow the point where F_a0 intersects the curve for Nut 1, to the point where F₁ intersects the curve, we see that the axial displacement of Nut 1 is increased by an amount equal to δ_a. The displacement of Nut 1 is now equal to δ₁ (δ_a0 + δ_a).

The relationship between the external axial load and the loads on N1 and Nut 2 must remain in equilibrium: F₁ – F₂ = F_a. Therefore, the effect of the external load on Nut 2 is to reduce its axial force from F_a0 to F₂. This, in turn, reduces the displacement of Nut 2 to δ₂ (δ_a0 – δ_a).

The relationship between axial elastic deformation and axial load, according to Hertz law of point contact (see page 9-15 in this guide from MIT), is given as:

δ_a ∝ F_a^2/3

From the graph above, we can form the equations:

δ_a0 = k*F_a0^2/3

and

2*δ_a0 = k*F_t^2/3

(Where k is a proportionality constant.)

Setting these two equations equal to one another and simplifying gives us:

2*k*F_a0^2/3 = k*F_t^2/3

2 = (F_t/F_ao)^2/3

F_t = 2.83*F_a0

This means that the maximum axial load is approximately three times the preload force. At this load, the amount of axial elastic deformation on the preloaded ball screw is half as much as the elastic deformation of a non-preloaded version.

Given this relationship, the theoretically ideal preload amount would be approximately 1/3 of the maximum axial load. However, even though the “ideal” preload for reducing ball screw axial deflection is relatively high, preload increases heat and wear and reduces the life of the ball screw. Therefore, ball screw preload is often limited to an amount between 5 and 13 percent of the basic dynamic load capacity, C.