### Day 2

Topics of discussion:
Free play
Stiffness hysteresis
Unstable relaxation
Tooth ripple
Manufacturing variation

The first article in this series examined a pseudo-ideal gear reducer to demonstrate that free play between adjacent parts becomes lost motion. Furthermore, an increase in load proportionally increases rotational deflection due to reducer characteristic stiffness. These two characteristics are illustrated in Figure 1. Because lost motion at the output affects applications, input is locked and all torque and deflection is measured at the reducer output. This situation gives the most useful information.

Precision gear reducers typically exhibit less than 10 arc-min of backlash, a tenfold improvement over common reducers where some free play is actually desirable. Though this is extremely low, it is still consequential in applications requiring tight tolerances. Lost motion can cause errors in systems that use encoders or resolvers on the motor for positioning, especially when there are many directional changes. Zooming in on the plot of precision reducer performance shows the complexity of the reducer’s actual behavior.

### Free play

Free play deflection of a pseudo-ideal device requires no torque, so it is shown as a vertical line in Figure 1. In reality, a small torque must be applied to push a component through its free play zone.

Energy follows the path of least resistance. As an input shaft initially turns, the first components to move are those with the least resistance to travel. Requiring the least torque, Figure 2 shows that their resulting motion for a given input is highest. Successive components require increasing amounts of torque for the same resultant motion, reflected by a decreasing slope. After the plot section corresponding to the most resistive component, reducer deflection as a function of torsional stiffness follows.

### Stiffness hysteresis

Because of running friction (a factor in all moving components) it takes more torque to rotate a gear reducer’s output shaft. Once free play is taken up, running friction still resists output shaft rotation, and results in torsional deflection. Consequently, less rotational deflection occurs (than with zero friction) when torsional load increases, and more deflection occurs when torsion decreases. This difference in rotational deflection for a given load appears as hysteresis in figure 2. Frictionless rotational deflection is somewhere between the upper and lower boundaries.

### Unstable relaxation

Once free play is eliminated, reducer components become fully compressed. But the position of some components (gear teeth in particular) may not be stable; torque keeps them in place. Consider Figure 3. As torque is removed, unstable components relax into unloaded stable positions, with those requiring the most holding torque moving first; those requiring less and less torque follow (a few components may remain in unstable positions, restrained by residual friction). The roll-off seen in Figure 2 corresponds to this relaxation of unstable components.

### Tooth ripple and manufacturing variation

Backlash and stiffness could be found by working backward from a rotational deflection vs. torque plot. By sifting out frictional effects, free play would become apparent. But note the caveat: the backlash and free play obtained would be unique to the gear reducer position at the time of data acquisition.

The reason for this is twofold. First, all manufactured components exhibit surface variations. Second, gear meshing is not homogeneous; free play varies as a single gear tooth mates with different teeth at every gear revolution. The latter effect is known as tooth ripple. Interestingly, the complete rotation of a single system part doesn’t reset the cycle. For example, the input shaft on a 5:1 planetary reducer might rotate 155 times before all components return to their original position. And every alignment pattern has a different level of backlash. Because measuring lost motion for every possible tooth combination is impractical, most manufacturers take a statistically determined number of measurements to obtain a representative value.

Gear teeth entering and exiting a mesh cause a higher frequency ripple. As gears rotate, meshed teeth swing through a range of angles, taking up different amounts of the open space between opposing gear teeth. As a result, free play between them varies.

Manufacturing variation occurs when interacting features are not identical. Individual gear tooth profiles are sometimes not machined uniformly. Other times, gears are mounted eccentrically on their axes. These and other inconsistencies cause variations that change the contributed free play of a part as it rotates.

Michael Trull is senior design engineer at Design & Development Engineering, Apex, N.C. Jon Mailey is an applications engineer with Parker Hannifin’s Zenith Products Division, Sanford, N.C. Jon can be reached at (877) 959-4327 or zenithgearhead@parker.com.