All about rigidity in linear actuation

The default method for increasing ballscrew rigidity is to increase the preload of the ball nut. However, the ball nut is one of the most rigid components in the ballscrew assembly … so increasing ball nut rigidity has little effect on the overall system rigidity (except when upgrading from a ball nut with no preload to one that is preloaded). Also, higher preload causes higher frictional torque, which means more heat is generated and more torque is required to drive the screw. A double ball nut is somewhat more rigid than a single nut but again, the effect is minimal due to the nut’s already high rigidity relative to the other components.

Ballscrew rigidity is calculated:

Where: R tot = Rigidity of the screw system (N/μm) R S = Rigidity of the screw shaft (N/μm) R N = Rigidity of the ball nut (N/μm) R B = Rigidity of the support bearings (N/μm) R H = Rigidity of the ball nut and bearing housings (N/μm)

The limiting factor for rigidity in a ballscrew assembly is typically the screw shaft. The rigidity of the screw shaft depends on its modulus of elasticity, diameter (which determines its crosssectional area), and unsupported length. The screw’s modulus of elasticity is dependent on the material (typically steel) and its diameter is primarily determined by the required thrust force and speed. The unsupported length of the screw shaft is determined by the stroke and the end bearing arrangement.

While the screw’s material and diameter are generally set by the application, the end bearing arrangement (and hence, its influence on unsupported length) is chosen based on factors such as speed, buckling load, and rigidity.

There are four common end bearing combinations for ballscrew assemblies. From most to least rigid, they are fixed-fixed, fixed-floating, floating-floating, and fixed-free. A fixed end uses an angular contact thrust bearing, for support against both radial and axial loads, while a floating end uses a simple radial bearing, with no support for axial loads. A free end has no bearing support.

Using the stiffest bearing arrangement – fixed-fixed – increases the rigidity of the screw shaft by four times compared to the least rigid arrangement of fixed-free. This is due to two design advantages. First, using angular thrust bearings on both ends allows forces to be transmitted through the screw on both sides of the ball nut. In addition, with thrust bearings on both ends, the maximum distance between the ball nut and either one of the fixed bearings occurs when the nut is at the middle of the stroke, which means the unsupported length is one-half the length of the screw.

Another effective method to increase ballscrew rigidity is to change the way the assembly operates: hold the screw stationary and rotate the ball nut. In a rotating nut design, because the screw itself does not rotate, there is no need for support bearings on the ends of the screw shaft. Instead, the ball nut is supported by a large angular contact thrust bearing.

The screw shaft is rigidly mounted at both ends, which allows torsional moments to be transmitted to the mounts at each end of the screw. While a rotating nut assembly is very rigid, its construction can pose dimensional problems and interference issues for applications that were designed with standard rotating screw assemblies in mind.

RIGIDITY IN LINEAR ACTUATORS AND LINEAR STAGES

Although there are no industry standards that define linear actuators and linear stages, generally accepted terminology indicates that:

A linear actuator is typically constructed with an aluminum extrusion or base

A linear stage is typically built on a flat, machined steel or granite base.

This distinction implies that linear actuators can provide longer strokes and use a variety of drive mechanisms (belt, screw, rack and pinion) while stages generally have higher rigidity and use highprecision linear guides and drive mechanisms (typically a ballscrew or linear motor) for excellent travel and positioning accuracies.

But one actuator design — the U-shaped linear actuator — defies these specifications ... using an extruded steel base to provide rigidity and travel accuracy specifications that rival some linear stages.

The use of a steel (rather than aluminum) profile makes the U-shaped design extremely rigid and allows manufacturers to offer a linear actuator with the high travel and positioning accuracies typically found in more precise (and more expensive) linear stages.

The steel base can also be machined to provide a reference edge for precise alignment with other machine components ... or with other actuators in a multi-axis system.

With their very high rigidity, U-shaped linear actuators can be better suited than other designs to applications where the actuator is supported only on one end.

These include two and three-axis Cartesian systems, for example. In the U-shaped actuator design, the linear guide system is integrated — there is no guide rail. Instead, the raceways that would normally be found on the guide rail are ground into the inside of the base. The carriage or table is analogous to a linear bearing block turned inside-out, with the balls riding on the outside.

This leaves the center portion of the carriage available to accommodate the ballscrew nut. This construction principle makes the entire actuator extremely compact, with a width-to-height ratio of approximately 2:1. For example, a U-shaped actuator with a width of 60 mm is only 33 mm high. The most common cross-sections (width x height) are 40 x 20 mm, 50 x 26 mm, 60 x 33 mm, and 86 x 46 mm … although other sizes are offered as well.

Some manufacturers offer U-shaped linear actuators made from extruded aluminum profiles, with steel inserts for the linear guide raceways. Aluminum versions lack the rigidity of steel designs, but they offer a very compact profile. In addition, they are often dimensionally interchangeable with steel versions where an application might benefit from a lower-cost option. Note that the use of a steel (rather than aluminum) profile makes the U-shaped design extremely rigid and allows manufacturers to offer a linear actuator with the high travel and positioning accuracies typically found in more precise (and more expensive) linear stages. While steel versions of U-shaped linear actuators use ballscrew drives almost exclusively, aluminum designs are more likely to be offered with both ballscrew and leadscrew drive options.

Originally developed for high-precision applications such as semiconductor wafer handling and medical diagnostic dispensing — for which space constraints don’t allow a typical linear stage — U-shaped linear actuators are now used in a wide variety of industries and applications. These include plasma welding, automated assembly, and optical inspection.

One of the driving factors behind the widespread adoption of U-shaped actuators is that they are the only linear actuator design with dimensional interchangeability between manufacturers.

However, it’s important to note that due to differing guideway and ballscrew designs, technical specifications (such as load capacity, speed, or rigidity) can vary between manufacturers and product lines — even for products with the same cross-sectional size and mounting dimensions.

MORE MECHANICAL PROPERTIES: STIFFNESS AND DEFLECTION

Note that in most linear-motion contexts, stiffness is the rigidity of an assembly and the extent to which it resists deformation under loading. That’s in contrast with rigidity that we've just explained — the inability of a material to be to deformed by some force.

Recall from the classic stress-strain curve how various aspects of material strength (including tensile strength, yield strength, and fracture strength) relate. While we often think of materials and structures in terms of strength, technically strength is a measure of how much force a material can withstand before permanent deformation or failure occurs. For proper running of linear guides, actuators, and other motion components, it’s typically more important to know how much deflection the object will experience under a given load — in other words, the more important property is the object’s stiffness.

A material’s stiffness indicates its ability to return to its original shape or form after an applied load is removed.

When a material is subjected to a load — including its own unsupported weight, an external applied load, or both — it experiences stress and strain. Stress σ is an internal force on the material caused by the load, and strain ε is the deformation of the material that results from this stress. The ratio of stress (force per unit area) to strain (deformation per unit length) is referred to as the modulus of elasticity denoted E. This ratio of stress to strain is also referred to as a material’s elastic modulus, tensile modulus, or Young’s modulus.

According to Hooke’s Law, the modulus of elasticity is the slope of the linear portion of the stress-strain curve, up to the proportional limit … also called the elastic limit.

A material that is strong can withstand high loads without permanent deformation. A material that is stiff can withstand high loads without elastic deformation. Another material property sometimes confused with strength or stiffness is hardness. Hardness defines a material’s ability to resist localized (surface) deformation often due to friction or abrasion.

Unlike strength, a material’s stiffness or modulus of elasticity is an inherent property of the material … and external factors such as temperature or material processing have very little effect on its value.

But in practical applications, the stiffness of a structure depends on both the material’s modulus of elasticity and the structure’s geometry in terms of planar moment of inertia (also referred to as second moment of area). Planar moment of inertia I expresses how the material’s area is distributed around the axis of motion.

The product of modulus of elasticity and planar moment of inertia is sometimes referred to as the material’s flexural rigidity EI.

In equations for deflection, both stiffness factors — the modulus of elasticity E and the planar moment of inertia I — appear in the denominator. This makes sense because deflection is inversely related to stiffness. So the higher the material’s modulus of elasticity (and the higher the object’s planar moment of inertia) the less the structure will deflect under a given load.

Tubular linear motors see increased use in automation

Matt Prellwitz | Motion control product manager ● Beckhoff Automation

Ironless tubular motors are suitable for anarray of existing and new linear motion scenarios.They usually replace pneumatic and hydraulicoptions for linear actuation. That’s in part becauseironless tubular motors offer a compact design,easy integration, and high precision and dynamics… features that are useful in packaging, machinetool, paper, textile, and food industries.

In combination with the Beckhoff AX8000 or AX5000 servo drives, AA2518 ironless tubular motors is suitable for linear motion … especially in confined spaces. The AA2518 leverages water cooling to deliver a peak force of 1,050 N and acceleration to 8 m/sec.

The magnetic rods are 35 mm in diameter and available in lengths of 700 mm and 1,000 mm. The absence of additional mechanical parts minimizes wear, resulting in lower maintenance costs than systems that use a spindle.

For more information, visit www.beckhoff.com/drivetechnology.