Paul E. Sandin - Robot Mechanisms and Mechanical Devices Illustrated (779750), страница 22
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An additional gear pair isemployed as shown in Figure 2-18.Output speedn2 n4 = 12 n1 + ROutput torqueT4 = 2T3 = 2RT2Output hp Rn1 + n2 hp = T2 63, 025 hp increase Rn1 ⌬hp = T2 63, 025 Speed variationn4 max − n4 min =12R(n2 max − n2 min)9394Chapter 2Figure 2-18Figure 2-19Indirect Power Transfer DevicesChapter 2Indirect Power Transfer Devices95Figure 2-20 A variable-speedtransmission consists of two setsof worm gears feeding a differential mechanism. The output shaftspeed depends on the differencein rpm between the two inputworms. When the worm speedsare equal, output is zero. Eachworm shaft carries a cone-shapedpulley.
These pulley are mountedso that their tapers are in opposite directions. Shifting the position of the drive belt on thesepulleys has a compound effect ontheir output speed.Speed range increase differential (Figure 2-19). This arrangementachieves a wide range of speed with the low limit at zero or in the reversedirection.TWIN-MOTOR PLANETARY GEARS PROVIDESAFETY PLUS DUAL-SPEEDMany operators and owners of hoists and cranes fear the possible catastrophic damage that can occur if the driving motor of a unit should failfor any reason.
One solution to this problem is to feed the power of twomotors of equal rating into a planetary gear drive.Power supply. Each of the motors is selected to supply half therequired output power to the hoisting gear (see Figure 2-21). One motordrives the ring gear, which has both external and internal teeth. The second motor drives the sun gear directly.Both the ring gear and sun gear rotate in the same direction. If bothgears rotate at the same speed, the planetary cage, which is coupled to96Chapter 2Indirect Power Transfer DevicesFigure 2-21 Power flow fromtwo motors combine in a planetary that drives the cable drum.the output, will also revolve at the same speed (and in the same direction). It is as if the entire inner works of the planetary were fusedtogether.
There would be no relative motion. Then, if one motor fails, thecage will revolve at half its original speed, and the other motor can stilllift with undiminished capacity. The same principle holds true when thering gear rotates more slowly than the sun gear.No need to shift gears. Another advantage is that two working speedsare available as a result of a simple switching arrangement. This makes isunnecessary to shift gears to obtain either speed.The diagram shows an installation for a steel mill crane.HARMONIC-DRIVE SPEED REDUCERSThe harmonic-drive speed reducer was invented in the 1950s at theHarmonic Drive Division of the United Shoe Machinery Corporation,Beverly, Massachusetts. These drives have been specified in many highperformance motion-control applications.
Although the Harmonic DriveDivision no longer exists, the manufacturing rights to the drive have beensold to several Japanese manufacturers, so they are still made and sold.Most recently, the drives have been installed in industrial robots, semiconductor manufacturing equipment, and motion controllers in militaryand aerospace equipment.The history of speed-reducing drives dates back more than 2000years. The first record of reducing gears appeared in the writings of theRoman engineer Vitruvius in the first century B.C. He described wooden-Chapter 2Indirect Power Transfer Devices97Figure 2-22 Exploded view of atypical harmonic drive showingits principal parts.
The flexsplinehas a smaller outside diameterthan the inside diameter of thecircular spline, so the ellipticalwave generator distorts the flexspline so that its teeth, 180º apart,mesh.tooth gears that coupled the power of water wheel to millstones forgrinding corn. Those gears offered about a 5 to 1 reduction. In about 300B.C., Aristotle, the Greek philosopher and mathematician, wrote abouttoothed gears made from bronze.In 1556, the Saxon physician, Agricola, described geared, horsedrawn windlasses for hauling heavy loads out of mines in Bohemia.Heavy-duty cast-iron gear wheels were first introduced in the mideighteenth century, but before that time gears made from brass and othermetals were included in small machines, clocks, and military equipment.The harmonic drive is based on a principle called strain-wave gearing, a name derived from the operation of its primary torque-transmittingelement, the flexspline.
Figure 2-22 shows the three basic elements ofthe harmonic drive: the rigid circular spline, the fliexible flexspline, andthe ellipse-shaped wave generator.The circular spline is a nonrotating, thick-walled, solid ring withinternal teeth. By contrast, a flexspline is a thin-walled, flexible metalcup with external teeth. Smaller in external diameter than the insidediameter of the circular spline, the flexspline must be deformed by thewave generator if its external teeth are to engage the internal teeth of thecircular spline.When the elliptical cam wave generator is inserted into the bore of theflexspline, it is formed into an elliptical shape.
Because the major axis ofthe wave generator is nearly equal to the inside diameter of the circular98Chapter 2Indirect Power Transfer DevicesFigure 2-23 Schematic of a typical harmonic drive showing the mechanical relationship between the two splines and the wave generator.spline, external teeth of the flexspline that are 180° apart willengage the internal circular-spline teeth.Modern wave generators are enclosed in a ball-bearingassembly that functions as the rotating input element. Whenthe wave generator transfers its elliptical shape to the flexspline and the external circular spline teeth have engaged theinternal circular spline teeth at two opposing locations, a positive gear mesh occurs at those engagement points. The shaftattached to the flexspline is the rotating output element.Figure 2-23 is a schematic presentation of harmonic gearing in a section view.
The flexspline typically has two fewerexternal teeth than the number of internal teeth on the circularspline. The keyway of the input shaft is at its zero-degree or12 o’clock position. The small circles around the shaft are theball bearings of the wave generator.Figure 2-24 is a schematic view of a harmonic drive inthree operating positions. In Figure 2-24A, the inside and outside arrows are aligned. The inside arrow indicates that thewave generator is in its 12 o’clock position with respect to thecircular spline, prior to its clockwise rotation.Figure 2-24 Three positions of the wave generator: (A) the 12 o’clock orzero degree position; (B) the 3 o’clock or 90° position; and (C) the 360°position showing a two-tooth displacement.Chapter 2Indirect Power Transfer DevicesBecause of the elliptical shape of the wave generator, full toothengagement occurs only at the two areas directly in line with the majoraxis of the ellipse (the vertical axis of the diagram).
The teeth in line withthe minor axis are completely disengaged.As the wave generator rotates 90° clockwise, as shown in Figure 2-24B,the inside arrow is still pointing at the same flexspline tooth, which hasbegun its counterclockwise rotation. Without full tooth disengagement atthe areas of the minor axis, this rotation would not be possible.At the position shown in Figure 2-24C, the wave generator has madeone complete revolution and is back at its 12 o’clock position. The insidearrow of the flexspline indicates a two-tooth per revolution displacementcounterclockwise. From this one revolution motion the reduction ratioequation can be written as:GR =FSCS − FSwhere:GR =gear ratioFS = number of teeth on the flexsplineCS = number of teeth on the circular splineExample:FS = 200 teethCS = 202 teethGR =200202 − 200= 100 : 1 reductionAs the wave generator rotates and flexes the thin-walled spline, the teethmove in and out of engagement in a rotating wave motion.
As might beexpected, any mechanical component that is flexed, such as the flexspline, is subject to stress and strain.Advantages and DisadvantagesThe harmonic drive was accepted as a high-performance speed reducerbecause of its ability to position moving elements precisely. Moreover,there is no backlash in a harmonic drive reducer. Therefore, when positioning inertial loads, repeatability and resolution are excellent (one arcminute or less).Because the harmonic drive has a concentric shaft arrangement, theinput and output shafts have the same centerline.
This geometry contributes to its compact form factor. The ability of the drive to providehigh reduction ratios in a single pass with high torque capacity recommends it for many machine designs. The benefits of high mechanical99100Chapter 2Indirect Power Transfer Devicesefficiency are high torque capacity per pound and unit of volume, bothattractive performance features.One disadvantage of the harmonic drive reducer has been its wind-up ortorsional spring rate. The design of the drive’s tooth form necessary for theproper meshing of the flexspline and the circular spline permits only onetooth to be completely engaged at each end of the major elliptical axis ofthe generator.
This design condition is met only when there is no torsionalload. However, as torsional load increases, the teeth bend slightly and theflexspline also distorts slightly, permitting adjacent teeth to engage.Paradoxically, what could be a disadvantage is turned into an advantage because more teeth share the load. Consequently, with many moreteeth engaged, torque capacity is higher, and there is still no backlash.However, this bending and flexing causes torsional wind-up, the majorcontributor to positional error in harmonic-drive reducers.At least one manufacturer claims to have overcome this problem withredesigned gear teeth.
In a new design, one company replaced the original involute teeth on the flexspline and circular spline with noninvoluteteeth. The new design is said to reduce stress concentration, double thefatigue limit, and increase the permissible torque rating.The new tooth design is a composite of convex and concave arcs thatmatch the loci of engagement points. The new tooth width is less than thewidth of the tooth space and, as a result of these dimensions and proportions, the root fillet radius is larger.FLEXIBLE FACE-GEARS MAKE EFFICIENTHIGH-REDUCTION DRIVESA system of flexible face-gearing provides designers with a means forobtaining high-ratio speed reductions in compact trains with concentricinput and output shafts.With this approach, reduction ratios range from 10:1 to 200:1 for single-stage reducers, whereas ratios of millions to one are possible formulti-stage trains.
Patents on the flexible face-gear reducers were heldby Clarence Slaughter of Grand Rapids, Michigan.Building blocks. Single-stage gear reducers consist of three basicparts: a flexible face-gear (Figure 2-25) made of plastic or thin metal; asolid, non-flexing face-gear; and a wave former with one or more slidersand rollers to force the flexible gear into mesh with the solid gear atpoints where the teeth are in phase.The high-speed input to the system usually drives the wave former.Low-speed output can be derived from either the flexible or the solidface gear; the gear not connected to the output is fixed to the housing.Chapter 2Indirect Power Transfer Devices101Figure 2-25 A flexible face-gearis flexed by a rotating wave former into contact with a solid gearat point of mesh.
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