A prominent member of the power-transmission linear-actuator family is the ball screw assembly, a device that converts torque to thrust, in which rolling motion of the balls provides energy transmission efficiency over 90%. That’s twice the efficiency of sliding-type lead screws. Nearly all the power (torque times angular velocity) can be used as linear power (force times linear velocity), moving the ball nut and any attached member. The advantages: lower energy consumption, less wear, long-lasting preload and precision, and predictable life. But those advantages can be lost because of poor choice of end-fixity conditions.

You can draw free-body diagrams for the screw, nut, and bearing end-mounts, Figure 1. Every action is opposed by an equal and opposite reaction. If the screw rotates, then the nut must translate for power transmission to occur. To react the force, one component must be fixed with respect to linear motion.

Because most screws are electric-motor- driven, the screw is usually constrained with respect to linear motion. Typically, it is installed on bearings that are mounted on the same base as the motor. The forces of load and acceleration created on the ball nut are thereby reacted by an axial thrust at the bearing end (or ends) of the screw. This is why the method used to mount the screw has a major impact on the system’s load capacity, stiffness, responsiveness, and speed.

The methods of fixing ball-screw ends and their effects on performance are many.

### Supported vs. fixed

Mounting of ball-screw ends is described as either *supported* or *fixed*. A supported end offers one focal point of the ball screw and does not react bending moments; instead, the angle between the axis of the screw and the rotary bearing increases, Figure 2. A fixed end can react moment loads because it is based on two rotary bearings sufficiently spaced apart so the ball screw remains perpendicular to the planes of the rotary bearings. Effective centers 1.5 times nominal screw diameter are usually enough. Figure 3 shows details of the fixed end of a screw, including hardware.

Two advantages of a screw with a fixed end are greater column strength and higher critical speed. More about these later. The advantages of a supported end are compactness and lower cost. A fixed end is generally harder to align and install than a supported end, so installation cost is a factor.

It is the function of linear motion bearings, which are typically an integral part of a linear motion system, to support side loads (radial loads). Apart from considerations of end fixity, side-loading could introduce misalignments and strains in the ball nut and reduce its expected travel life. Therefore, you might ask, “Why must end fixity counteract bending forces?” The answer has to do with column strength and critical speed.

From simplest to more difficult mounting arrangements, a ball screw can be:

• Fixed at one end and free at the other. With one end free, the ball screw is easily bent because the free end can move like a cantilever beam.

• Supported at both ends.

• Fixed at one end and supported at the other.

• Fixed at both ends.

Figure 4 shows these cases. The relative lengths correspond to lengths that would result in the same critical speed for equal-diameter ball screws.

### Column strength

The ability of a ball screw to avoid buckling under a compressive load is called its *column strength*. The screw must carry an axial load that is equal and opposite the load generated on the ball nut by the motor’s torque. In general, column strength is the controlling design parameter because, for long columns, it is much lower than the material’s strength in compression.

Because the length-to-diameter ratio is important in column buckling, it is no surprise that the compressive load strength of ball screws is a function of length. A length-to-diameter ratio (*L/D*) exceeding 100:1 requires special design consideration; consult the manufacturer regardless of anticipated loads or fixity.

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To support the same amount of load without buckling, a ball screw with both ends fixed can be 1½ times as long as a ball screw with both ends merely supported, and 2½ times as long as a ball screw with a free end. Because of these large differences, manufacturer’s charts typically provide data for each of the four types of end fixity, Figure 4.

### Stiffness

The axial rigidity or deflection due to load of a ball-screw assembly is its *stiffness*. Total system stiffness includes the stiffness of the screw, balls, ball nut, support bearings, and mounts.

Because of its long length with respect to other components, the ball screw is typically the greatest source of deflection. The axial deflection, *e*, of a ball screw is directly proportional to load, *P*, and length, *L*, and it is inversely proportional to the cross-sectional area, *A*, at the root diameter, and Young’s modulus (the modulus of elasticity), *E*, thus: *e = PL/AE*

For example, if the root diameter is 1 in. and the load is 100 lb, with *E* = 293106 psi for steel, then

*e = (4/29π) × 10 ^{-4}L*

which reduces to

*e = 0.000044L*

If *L* is 100 in., for example, the deflection can be substantial, particularly in precision applications. Such deflections introduce dynamic, systemic errors, which can be reduced through careful design. Fixed-fixed bearing arrangements offer the highest stiffness for the system as a whole. Deflections can be reduced by allowing some of the load to be carried as tensile load in the portion of the ball screw toward the direction of motion, Figure 5.

Stiffness is mentioned here as one additional factor to consider in ball-screw system design. Its importance depends on the application. The ball nut assembly and the bearings and blocks also contribute deflection to the system. Like the ball screw, each is characterized by a spring rate,

*K = P/e*

with units of lb/in. Spring rates add in analogy with the summation of electrical resistances in a parallel circuit:

1/*K _{total}* = 1/

*K*+ 1/

_{screw}*K*+ 1/

_{nut}*K*+ 1/

_{balls}*K*

_{bearings}### Critical speed

Trends in machinery design demand improved productivity through quicker cycle times and hence, faster actuation. Because fast cycle times are typically accompanied by the need for high duty cycles and longer travel life, selecting the right ball screw can help.

Surprisingly, limiting speed usually results from the fundamental frequency of vibration (natural frequency) of the ball screw. Just as a guitar string vibrates at a frequency determined by its length, tension, and mass per unit length, a ball screw also has a characteristic frequency. A screw behaves like a string on a bass guitar. For example, a 100-in. long, 1-in. root diameter ball screw with both ends fixed vibrates at about 18 Hz.

End fixity is analogous to string tension; in the example just given, the same screw would vibrate at about 8 Hz with both ends supported rather than fixed.

This critical speed is a direct consequence of the characteristic frequency of vibration of the ball screw. If the rotational frequency of the screw matches the screw’s characteristic frequencies, slight imbalances in the screw can resonate, causing the screw to absorb energy and “whip.” Excessive bending and bowing then would keep the screw from working properly.

Critical speed can be expressed as

*N = 4,800,000D _{R}C/L^{2}*

where

*N* = Critical speed, rpm

*C* = End fixity constant, with values as given in Figure 5 for the four types of end fixity

*D _{R}* = Root diameter, in.

*L* = Length, in.

If you multiply *N* in Table 1 by the lead of the screw and allow a safety margin (say, multiply by 0.8), then you get an approximation of the recommended maximum operating speed of the ball screw shaft in ipm.

Probably the most significant trend in use of ball screws is the demand for fast linear speeds using high lead advances with small-diameter screws. Rotational speeds greater than 2,400 to 3,600 rpm and linear speeds of 30 to 50 ips are becoming more common. Moreover, these specifications are being met using long ball screws, and still providing long life and high reliability.

**James A. Babinski** is Senior Product Engineer at Thomson Saginaw Ball Screw Co. Inc., Saginaw, Mich. He has a mechanical engineering degree from Saginaw Valley State University, and 13 years experience engineering ball screws and designing automation equipment. Mr. Babinski is a consulting member of the American Society of Mechanical Engineers Technical Committee 43 on Ball Screws, Subcommittee of the American National Standards Committee B5 — Machine Tools.