Rotational vs. linear motion with gearing

A gear rack, when mated with a pinion, is the only gear pair that can transform rotary motion into linear motion.

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Spur gear pairs, helical gear pairs, bevel gear pairs, and worm gear pairs all transfer rotary motion from one shaft to another. In the case of spur gears, the drive shaft and the driven shaft are parallel; however, the rotation of the output shaft is opposite to that of the input shaft. For helical gears, the drive shaft and the driven shaft are parallel when the helix angle is identical for both gears, but the hand of the gears are opposite. If the helix angles are different, then the shafts will not be parallel. In the case of the helix angle for both gears being 45 degrees and the hands being the same, helical gears will operate at 90 degrees to one another. Regardless of the shaft orientation, the rotational direction of the output shaft will be opposite to that of the input shaft. For both bevel gears and worm gears, the preferred positioning between the drive shaft and the driven shaft is 90 degrees. For bevel gears, it is possible to produce sets with shaft angles other than 90 degrees; however, doing so greatly reduces the permissible size of the tooth flank. For bevel gears, the gear shaft will always rotate in the opposite direction of the pinion shaft. For worm pairs, the worm gear will always rotate in the opposite direction of the worm shaft.

All of the gear types above will transmit rotary motion from one shaft to another. In doing so, they will change the rotational direction, they will change the speed of rotation, and they will change the transmitted torque. The one gear system that differs from all of these is the gear rack. A gear rack, when mated with a pinion, is the only gear pair that is able to transform rotary motion into linear motion. In its simplest form, a gear rack is a spur gear with an infinite pitch radius. This allows the gear rack to mesh with a simple spur gear or, if the gear rack is cut with helical teeth, then it will mate with a helical pinion.

For spur gears and straight tooth bevel gears, the line of action is perpendicular to the shafts. As such, there are no axial thrust loads. For helical gears, spiral bevel gears, and worm gear pairs, the helix or spiral angles induce forces in the axial direction. For these reasons, thrust bearings must be properly sized and used in applications using these gears. Spur gears can readily use radial bearings.

For a rack-and-pinion application, the determination of bearings for the drive shaft will be determined by the type of drive pinion. A spur pinion can use simple radial bearings, but a helical pinion will require axial thrust bearings.

Rack and pinion motion is limited to a simple oscillating motion. When the pinion rotates in one direction, it translates along the rack. When the rotation is reversed, the direction of travel along the rack will reverse. The length of travel is limited only by the length of the rack and the number of rotations of the pinion. From a manufacturing perspective, racks can only be produced in continuous lengths of six feet. This is due to a variety of factors: The first is the availability of material in extended lengths. The second is the ability to transport racks of greater lengths. The most important is the amount of deformation that will occur both during manufacture and during shipment as the length increases.

One solution to this issue is to produce gear racks with finished ends. This is a process in which the final tooth on each end of the gear rack is ground off slightly so that, when two racks are assembled end to end, the result is a rack assembly that functions as one continuous piece (Figure 1).   

Figure 1

Another consideration for gear racks is the total height of the rack. If the height from the base of the rack to the pitch line is too small, the rack will not be able to resist the upward forces and will bend at a tooth root resulting in failure due to bending. It is best practice to set the minimum height of the gear rack, as measured from the base of the rack to the pitch line, at least 1.15 times the face width of the pinion. For best wear, the face width of the rack should be equal to the face width of the pinion. (Figure 2)

Figure 2

The center distance will change based on the pitch radius of the pinion. It is recommended that the pinion have a minimum of 20 teeth; however, the maximum number of teeth is based on the design envelope.

When the rack face width is too small, the rack will twist. When the rack is too short, the rack will deflect. Both conditions should not only be avoided, but they could also make the designed rack impossible to manufacture. 

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is general manager of KHK USA Inc, a subsidiary of Kohara Gear Industry with a 24-year history of working in the industrial automation industry. He is skilled in assisting engineers with the selection of power-transmission components for use in industrial equipment and automation. Dengel is a member of PTDA and designated as an intern engineer by the state of New York. He is a graduate of Hofstra University with a Bachelor’s of Science in Structural Engineering.