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Simple machine
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ny of several elementary machines, one or more of which is found in practically every machine. The group of simple machines usually includes only the lever, wheel and axle, pulley (or block and tackle), inclined plane, wedge, and screw. However, the gear drive and hydraulic press may also be considered as simple machines. The principles of operation and typical applications of simple machines depend on several closely related concepts. See also: Block and tackle; Hydraulic press; Inclined plane; Lever; Screw; Wedge; Wheel and axle
Two conditions for static equilibrium are used in analyzing the action of a simple machine. The first condition is that the sum of forces in any direction through their common point of action is zero. The second condition is that the summation of torques about a common axis of rotation is zero. Corresponding to these two conditions are two ways of measuring work. In machines with translation, work is the product of force and distance. In machines with rotation, work is the product of torque and angle of rotation.
Work is the product of a force and the distance through which it moves. For example, the work done in raising a 10-lb object 15 ft is 150 ft-lb. In this example the work done on the weight goes into increasing the potential energy of the object. Work and energy, both potential and kinetic, have the same units, and in general the purpose of a machine is to convert energy into work. See also: Efficiency; Mechanical advantage; Work
For rotating machines, it is more convenient to consider torque and angular displacements than force and distance. Work is then expressed as the product of the torque and the angle (in radians) through which the object rotates while acted on by the torque. Torque, in turn, is the force exerted at a given radius from an axis of rotation. Thus, a 10-lb force at the end of a 15-ft crank exerts a torque of 150 lb-ft. See also: Torque
Power is the rate of doing work. For example, one horsepower is arbitrarily defined as 550 ft-lb per second, or 33,000 ft-lb per minute. See also: Machine; Power
Richard M. Phelan
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Engineering & Materials: Mechanical Engineering:Mechanical engineering
Lathe
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Select an article section Top Of Article Engine lathe Turret lathe Automatic screw machines BIBLIOGRAPHY


machine for the removal of metal from a workpiece by gripping it securely in a holding device and rotating it under power against a suitable cutting tool. The tool may be moved radially or longitudinally in respect to the turning axis of the workpiece either manually or by attached power. Forms such as cones, spheres, and related concentric-shaped workpieces as well as true cylinders can be turned on a lathe. Machining operations such as facing, boring, and threading, which are variations of the turning process, can also be performed on a lathe.
Turning equipment may be classed as being either of the horizontal or vertical type referring to the turning axis of the workpiece in the machine. The basic engine lathe is primarily a manually operated machine. Filling the gap between the engine lathe and fully automatic turning equipment is the turret lathe. Fastest and best suited to high-quantity production is the automatic screw machine.
Engine lathe
The versatile engine lathe ranges in size and design from small bench and speed lathes to large floor types. The workpiece may be held between tapered centers and rotated with the power spindle by means of a clamping device, or it may be held in a chuck, or even fixed to a rotating plate. When the work is swung between centers, the tailstock, which holds the stationary tapered center, is clamped firmly in place. Long work pieces may be supported in the middle by either a steady rest or a mechanically driven follow rest, which moves with the cutting tool.
The single-point tool is held in a tool post or block, which is supported on a cross slide and carriage. The tool may be moved radially or longitudinally in relation to the workpiece.
Angular cuts are possible, and longitudinal tapers may be cut by offsetting the tailstock, or by adjustment of a taper attachment on the rear of the lathe which actuates the cross slide.
Turret lathe
The tailstock of the engine lathe can be replaced with a multisided, indexing tool holder or turret designed to hold several tools, the machine becoming a turret lathe. The single-tool post and compound is usually replaced by a four position indexing tool post. Power cuts may be taken from both of these tool mountings either individually or simultaneously. Turret lathes are constructed in vertical models and also in horizontal models.
Horizontal turret lathes are classed as either bar or chucking machines referring to the manner in which the workpiece is held. The headstock of the bar machine is constructed so that bar stock may be slid through it and the collet on line with the turning axis of the lathe. The collet closes, holding the piece firmly. Chucking-type machines grasp a unit size workpiece in a chuck or jawed device. Chucking devices permit work of relatively large diameters to be machined.
Horizontal machines may be further classed as being either of the ram or saddle type, a designation referring to the manner in which the turret is mounted on the machine. On the ram machine, the turret is mounted on a ram or slide which rides on a saddle. When the turret is indexed for successive operations, the saddle acts as a guide for the ram in its strokes to and from the work. On the saddle-type machine, designed without a ram, the turret mounts directly on the saddle, which slides on the bedways of the lathe. Ram-type lathes with their short turret travel are generally of lighter construction than the saddle type. Excessive overhang of the ram reduces tool rigidity. Fast in operation, they perform best on small-diameter work and light chucking jobs. The rigid construction and the longer stroke of the saddle-type lathe enable it to handle both longer and heavier bar and chuck work than the ram-type machine.
The vertical turret lathe, similar in principle to the horizontal machine, is capable of handling heavier and bulkier workpieces. The vertical machine is constructed with a rotary, horizontal worktable whose diameter normally designates the capacity of the machine. Machine tables range from 30 to 74 in. (76 to 190 cm) in diameter. A crossrail mounted above the table carries a turret, which indexes in a vertical plane with tools that may be fed either across or downward. The crossrail may also carry a vertical swiveling ram with a nonindexing tool holder which feeds in a manner similar to the turret. Below the crossrail, a side head with an indexing tool holder is sometimes provided. This tool may be fed in horizontally or moved vertically. Tools may be operated simultaneously either manually or by power.
Automatic screw machines
When a high production rate of relatively small turned parts is required, automatic screw machines are used. The screw machine was originally developed from the lathe for the purpose of more economical manufacture of screws and bolts. The name has persisted even though numerous partially or completely finished products are produced on them. Generally these machines are classified as single-spindle or multiple-spindle automatics. The usual machine is of the horizontal type.
The single-spindle machine is constructed to feed bar stock through the hollow machine spindle and collet similar to a turret lathe. When the bar meets a stop, the collet closes on the piece. The manner in which the cutting tools are held may vary. Usually a small five- or six-position turret or drum indexes and feeds tools longitudinally against the end of the rotating workpiece. Turret tools may include drills, reamers, hollow mills, and counterboring tools; at times a single-threading die mounted on line with the spindle is used.
Usually two independent cam-actuated cross slides, front and rear, are provided to hold forming, grooving, or cutoff tools. Turret indexing, actuation of the collet, feeding of stock, and spindle clutch operations are automatic. Machining operations commonly performed include facing, drilling, reaming, forming, and knurling. Special attachments permit such operations as milling, index drilling, or thread chasing to be performed.
Multiple-spindle automatics employ several machine spindles arranged in a circular pattern. Each spindle is equipped to hold stock in a manner similar to a single-spindle machine. In some instances automatically operating chucks capable of holding irregularly shaped workpieces replace the collet-type chucks. Standard machines may have as many as eight rotating spindles. These in turn index in a carrier about a nonrotating turret, which holds a variety of cutting tools. As each spindle indexes and progresses around the carrier, successive machining operations are performed by the turret tools. Simultaneously, tools located in successive positions are performing their respective operations on the various workpieces. Cross slides mounted at right angles to the spindles carry the forming, grooving, or cutoff tools.
The time for the longest cut plus the allowance for such programming actions as indexing and tool traversing sets the time required to produce one finished piece. By dividing long cuts over two or more operations, the cycling time may be reduced. Machining operations performed on these machines are generally the same as those done on single-spindle automatics.
Vertical multiple-spindle machines of the semiautomatic chucking type are constructed with as many as 16 spindles. A horizontal, circular table holding the rotating chucks indexes under the vertical spindles with a different operation being performed at each station. This machine is constructed to handle much larger workpieces than horizontal automatics; on many models the speed, feed, and direction of rotation for each spindle combination may be varied from one position to the next. This permits selection of the correct feed and speed for the operation being performed at each station.
Alan H. Tuttle
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Topic Page: Engineering & Materials: Electrical & Electronics Engineering:Electrical engineering
Topic Page: Engineering & Materials: Mechanical Engineering:Mechanical engineering
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gears

Engineering & Materials: Mechanical Engineering:Mechanical engineering
Gear
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Select an article section Top Of Article Principal features Backlash Gear action Involute gear teeth Spur gears -Helical gears -Crossed helical gears Bevel gears Straight bevel gears -Spiral bevel gears -Zero bevel gears -Hypoid gears Worm gears BIBLIOGRAPHY ILLUSTRATIONS ANIMATIONS


machine element used to transmit motion between rotating shafts when the center distance of the shafts is not too large. Toothed gears provide a positive drive, maintaining exact velocity ratios between driving and driven shafts, a factor that may be lacking in the case of friction gearing which is subject to slippage. While the motion transmitted to mating gears is kinematically equivalent to that of rolling surfaces identical with the gear pitch surfaces, the action of one gear tooth on another is generally a combination of rolling and sliding motion. When the distance between shafts is large, other methods of transmission are used. See also: Belt drive; Chain drive; Rolling contact
The application of gears for power transmission between shafts falls into three general categories: those with parallel shafts, those for shafts with intersecting axes, and those whose shafts are neither parallel nor intersecting but skew (see table).
Chief types of application of gears for power transmission
Shaft relationship
Type of gearing
Parallel
Spur

Helical

Herringbone

Double helical
Intersecting
Bevel
Skew
Helical

Worm

Hypoid
Principal features
Figures 1 and 2 illustrate the principal features of toothed gears. Such terms as pitch circle, addendum circle, and root circle, being geometrical, are defined by the diagrams.
Fig. 1 Principal features of gear teeth.

Fig. 2 Action of involute gearing. (a) Gear and pinion mounted at normal center distance. (b) Gear and pinion mounted at greater than normal center distance. (After J. E. Shigley, Theory of Machines, McGraw-Hill, 1961)

Definitions for other commonly used terms for describing gears are addendum, the radial distance between the pitch circle and the addendum circle; dedendum, the radial distance between the pitch circle and the root circle; face, the tooth surface outside the pitch circle; flank, the tooth surface between the pitch circle and the root circle; clearance, the amount by which the dedendum exceeds the addendum of the mating gear; whole depth, addendum plus dedendum; and working depth, whole depth minus clearance; circular pitch, the distance from a point on one tooth to the corresponding point on the next, measured along the pitch circle; and base pitch, the similar measurement along the base circle.
Terms descriptive of a pair of spur gears in mesh are shown in Fig. 2. Where the pitch circles are tangent is pitch point P. Each pair of teeth always has its point of contact on the pressure line. The angle between this pressure line and the common tangent to the pitch circles through P is the pressure angle. As Fig. 2 illustrates, the pressure angle at which two gears actually operate depends on the distance between centers at which they are mounted. Initial contact between each pair of teeth occurs at point C, where the addendum circle of the driven gear crosses the pressure line; and final contact is at D, where the driver's addendum circle crosses the pressure line. Distance CD is the length of the path of contact. It must exceed the value of the base pitch if a pair of teeth is to come into contact before the previous pair has gone out of contact. The ratio of the length of the path of contact to the base pitch is called the contact ratio, which can be viewed as approximately the average number of pairs of teeth in contact. It is usual to design for a contact ratio of 1.4 or above to ensure smooth, continuous tooth action.
Gear tooth sizes are designated by diametral pitch, which is the number of teeth per inch of diameter of the pitch circle. Pitch circle is, in turn, the circle whose periphery is the pitch surface, or surface of an imaginary cylinder that would transmit by rolling contact the same motion as the toothed gear. A gear with a 20-in. (51-cm) pitch diameter and teeth having a diametral pitch of 2 has 40 teeth. Diametral pitch P times circular pitch p (Fig.1) equals . The diametral pitches most widely used are: 1, 11/2, 2, 21/2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 72, 96, 120, 160.
The smaller of two gears in mesh is the pinion. It has the fewer teeth and is the driving gear in a speed reduction unit. The minimum number of teeth an involute pinion can have and still run without interference between its flanks and the tips of the mating gear teeth is fixed by the tooth system. Smaller pinions are possible only if the pinion's flanks are undercut.
Backlash
The amount of which the tooth space of a gear exceeds the tooth thickness of the mating gear at the pitch circle is the backlash. It can be determined in the plane of rotation or, for helical gears, in the plane normal to the tooth face.
If mating gears have zero backlash, gears and mountings need to be dimensionally perfect. To retain zero backlash with varying operation conditions, all parts need exactly the same thermal expansion characteristic. Because of the difficulty of meeting these requirements and for lubrication, freedom-backlash is provided between gear teeth. The usual practice is to reduce the tooth thickness of each gear by an amount equal to half the desired backlash. However, in the case of a gear and a small pinion it is customary to reduce the tooth thickness on the gear only, leaving the pinion with standard tooth thickness. Backlash can also be adjusted by slight changes in the center distance between gears. Except for a small change in the pressure angle, the action of involute gear teeth is not affected by backlash or center distance adjustment.
In the case of precision gearing for control systems and similar applications, backlash results in a nonlinear relation between input and output. Several methods of reducing backlash are in use. One method is to place two identical spur gears on the same shaft, one fixed to the shaft, one free. The loose gear is attached to the fixed one by springs, which keep the composite gear in positive contact with the pinion at all times. A second method is to use tapered-tooth gearing (beveloid) with adjustment along the shaft to eliminate excessive backlash.
Gear action
A principal function of gears is to change the speed of rotation. This action is described by the velocity ratio of the gears in mesh. Ratio VR is the number of revolutions N1 of the driving gear divided by the number of revolutions N2 of the driven gear in the same time interval. For gears with teeth T1 and T2, respectively, VR is expressed as in Eq. (1).

(1) When two curved surfaces, such as the mating surface of two gear teeth, are in driving contact, there is a definite velocity ratio between the bodies. The angular velocities of the two bodies are inversely proportional to the segments into which their line of centers is divided by a line passing through their point of contact and normal to their surfaces at this point. Thus a constant angular velocity ratio between bodies in driving contact demands that the common normal to the profiles at the point of contact cut the center line at a fixed point, the pitch point. This latter statement is frequently referred to as the fundamental law of gear tooth action. Pure rolling contact between gear teeth occurs only when they are in contact at the pitch point. At all other positions the teeth have some sliding with the maximum sliding at the first and last instants of contact. Although there are a number of tooth forms that will satisfy the fundamental law, only two of them have been used to any great extent. The cycloidal tooth predominated until the late 1800s but has been replaced to a great extent by the involute gear tooth. Cycloidal teeth are still found in instruments, watches, clocks, and, occasionally, in cast and cut gears.
Gears are said to be interchangeable when any gear of the set will run with any other gear of the same set. Actually there is little need today for gears that are interchangeable. In the manufacture of the overwhelming proportion of machines pairs or groups of gears are designed to mesh with each other and no others. Standardization in manufacture is common, however. Each of the gears is made from one of a set of standardized cutters. Thus it is the standardization of the production tools rather than of the gears themselves that is of most importance.
Involute gear teeth
An involute tooth is laid out along an involute (Fig. 3), which is the curve generated by a point on a taut wire as it unwinds from a cylinder. The generating circle is called the base circle of the involute. Proportions of the tooth are fixed by the gear tooth system and the diametral pitch. The involute curve establishes the tooth profile outward from the base circle. From the base circle inward, the tooth flank ordinarily follows a radial line and is faired into the bottom land with a fillet. The basic rack form of the involute tooth has straight sides, an important property from the manufacturing standpoint.
Fig. 3 Method of generation for face of involute tooth.

Gear teeth may interfere with each other, especially where pinions have a relatively small number of teeth. In Fig. 2, point of initial contact C occurs on the pressure line to the right of (after) the point of tangency of the pressure line and the pinion's base circle. Here there is no interference. But if point C were to the left of the tangency point, this would indicate premature contact between the teeth, a contact occurring on the noninvolute portion of the pinion tooth flank below the base circle. The tip of the gear tooth, in such a case, digs into the flank of the pinion tooth.
There are several ways to eliminate interference: (1) Reduce addendum of the gear. (2) Increase pressure angle. (3) Increase backlash by increasing center distance between gears. (4) Undercut flank of the pin ion. (5) Relieve or modify face of the gear tooth.
Spur gears
In the truest sense, spur gears are only those that transmit power between parallel shafts and have straight teeth parallel to the gear axis (Fig.4 a). It is common practice, however, to group helical gears that have parallel shafts under the heading of spur gears (Fig.4 b). The pitch surfaces of gears with parallel axes are rolling cylinders, and the motion these gears transmit is kinematically equivalent to that of the rolling pitch cylinders. Spur gears are classified as external, internal, and rack and pinion. External spur gears, the most common, have teeth which point outward from the center of the gear. Internal or annular gears have teeth pointing inward toward the gear axis (Fig.4 c). A rack may be considered as a gear having an infinite pitch circle radius. Thus its pitch surface is a plane. A rack and pinion running together transform rectilinear motion into rotary motion, or vice versa.
To standardize manufacture of gears, the American Gear Manufacturers Association has adopted three basic types of spur-gear tooth, with pressure angles of 20° and 25° for coarse-pitch gears (with a diametral pitch under 20) and 20° for fine-pitch gears (diametral pitch of 20 and above).
An internal gear has the positions of the addendum and dedendum reversed from those of an external gear. This results in a different tooth action and less slippage than with an equivalent external spur gear. For a given tooth ratio, the arc of action of an internal gear is slightly greater than that of an external gear of the same size and the tooth is stronger. The nature of an internal gear makes it especially suited to closer center distances than could be used with an external gear of the same size. When it is necessary to maintain the same sense of rotation for two parallel shafts, the internal gear is especially desirable because it eliminates the need for an idler gear. These conditions make the internal gear highly adaptable to epicyclic and planetary gear trains.
Noncircular gears are used to obtain velocity ratios that vary in a precise manner. Elliptical gears are an example of noncircular gears. They provide a convenient method of obtaining a quick return for machines that do most of their work during only a portion of the drive shaft revolution. Noncircular gears are used in computing mechanisms and other devices where a prescribed varying output function is to be obtained using a linear input.
Helical gears
Gears running on parallel axes and with teeth twisted oblique to the gear axis are essentially spur gears. Because of the twist, contact is progressive across the tooth surface, starting at one edge and proceeding across the face of the tooth. The action results in reduced impact and quieter operation, particularly at high speed. Herringbone gears are equivalent to two helical gears of opposite hand placed side by side. They are especially suited for high-speed operation and eliminate the axial thrust produced by single helical gears. Double helical gears have a central groove for tool runout, making it possible to finish the teeth by a shaving operation. They can be run at even higher speeds than herringbone gears. Dimensions of a helical gear are determined on both the plane of rotation and on the plane perpendicular to the helix angle of the tooth. Thus, a helical gear has a circular pitch measured on the plane of rotation and a normal circular pitch measured on the normal plane.
Crossed helical gears
Where shafts cross obliquely, motion is transmitted by crossed helical gears (Fig. 5). The teeth are helical but differ from the teeth of worm gears in that no one tooth (thread) makes a complete turn on the pitch circle. Pitch surfaces of crossed helical gears are cylindrical as with spur gears. However, with crossed shafts, the oblique teeth have point contact rather than the line contact that occurs with parallel shafts. Analysis of the gears is based on equal components of the pitchpoint velocity on each mating gear along common normal N-N. Sliding occurs in direction T-T of the tooth elements. As with spur gears, the revolutions per unit time are inversely proportional to the numbers of teeth. Velocity ratio VR may also be expressed as in Eq. (2),

(2) where D is pitch diameter and helix angle. Helix angle is between the shaft axis and a line tangent to the tooth through the pitch point.
Fig. 5 Action of crossed helical gears.

Helical gears are referred to as right- or left-hand in the same manner as screw threads, a right-hand gear being one on which the teeth twist clockwise as they recede from an observer looking along the axis.
Bevel gears
Where shafts intersect, bevel gears transmit the motion. Such gears may be used only to change the shaft axis direction or to change speed as well as direction. Two bevel gears with equal numbers of teeth and running together with their shaft axes intersecting at 90° are called miter gears. Several forms of bevel gears are in use, including straight-tooth, spiral, and skewed bevel gears (Fig.6 ).
External bevel gears have pitch angles less than 90° (Fig. 7a). Internal bevel gears have pitch angles greater than 90°, hence their pitch cones are inverted (Fig. 7b). A crown gear is one having a pitch angle of 90° (Fig. 7c). Thus its pitch surface is a plane, and the crown gear corresponds in this respect to a rack in spur gearing.
Fig. 7 Bevel gears. (a) External. (b) Internal. (c) Crown. (After G. L. Guillet and A. H. Church, Kinematics of Machines, 5th ed., Wiley, 1950)

Straight bevel gears
The simplest form of bevel gear has straight teeth which, if extended inward, would come together at the intersection of the shaft axes. This point is also the apex of the rolling cone, which forms the pitch surface of the gear. Much of the terminology applied to bevel gears is the same as that used for spur gears. Additional terms used in reference to bevel gears are given in Fig. 8. Diametral pitch of a bevel gear is not constant across the full width of the tooth. The diametral pitch at the pitch diameter is used in fixing tooth proportions. The formative number of teeth in a bevel gear is the number of teeth that would be a spur gear whose pitch radius equaled the back cone distance of the bevel gear.
Fig. 8 Characteristics of bevel gears. (After C. W. Ham, E. J. Crane, and W. L. Rogers, Mechanics of Machinery, 4th ed., McGraw-Hill, 1948)

Speeds of the shafts of bevel gears (velocity ratio) are inversely proportional to the numbers of teeth on the gears or to the sines of the pitch angles, but not to the formative number of teeth. Use of straight bevel gears is limited to low-speed operations, ordinarily below 1000 surface feet per minute or, in the case of small gears, 1000 rpm.
Because each point on a straight tooth bevel gear remains a fixed distance from the pitch cone apex, there is no sliding along the tooth as it engages. Contact across the full tooth face occurs instantaneouslyas with spur gearsas the teeth come into mesh.
Spiral bevel gears
To provide a gradual engagement, as contrasted to the full line engagement of straight bevel gears, the teeth of spiral bevel gears are curved and oblique. Theoretically the curve is a spiral but, to facilitate manufacture, the curve is actually a circular arc which, within the tooth face width, closely approximates a spiral. This tooth inclination brings more teeth in contact at any one time than with an equivalent straight-tooth bevel gear. The result is smoother and quieter operation, particularly at high speeds, and greater load-carrying ability than with straight bevel gears of the same size.
Spiral bevel gears are used in sewing machines, motion picture equipment, machine tools, and other applications where quiet, smooth operation is essential. They should, in general, be mounted on antifriction bearings because of the axial thrust due to the oblique teeth. In the past they have been used extensively in the rear axle drives of automobiles, but are being replaced by hypoid gears.
Zero bevel gears
A special form of bevel gear has curved teeth with a zero-degree spiral angle. Thus the teeth are not oblique as is the case with spiral bevel gears. Rather, the teeth lie in the same general direction as those of an equivalent straight-tooth bevel gear, and so the gears are usually used in the same types of drives as the straight-tooth gear. As with straight bevels, they produce no axial thrust and, therefore, may be used without thrust bearings. The face that they may be produced on the same equipment as spiral, bevel, and hypoid gears makes them economically desirable.
Tooth proportions for bevel gears follow the standards established by the Gleason Works and adopted as the recommended standard of the American Gear Manufacturers Association. Tooth proportions are a function of the velocity ratio. Thus, bevel gears are not as interchangeable as spur gears.
Hypoid gears
To connect nonparallel, nonintersecting shafts, usually at right angles, hypoid gears are used. They are similar to spiral bevel gears in their general appearance. The axis of the hypoid pinion may be offset above or below the axis of the gear. The shape of the tooth is similar to that of the spiral bevel gear and gives progressive contact across the tooth. In operations these gears run even more smoothly and quietly than spiral bevel gears. To maintain line contact of the teeth, with the offset shaft, the pitch surface of the hypoid gear is a hyperboloid of revolution rather than a cone as in bevel gears.
One of the first uses of hypoid gears was in the rear-axle drive of Packard automobiles. The operating smoothness of hypoid gears, along with the lower body lines made possible by the offset pinion shaft, has made them extremely popular for automotive use. Industrial applications of the hypoid gear also take advantage of the pinion offset, which allows the mounting of any number of pinions on a single continuous shaft, a feat not possible with bevel gears. The shaft arrangement of the hypoid gear and pinion enables bearings to be placed on both sides of the gear and of the pinion. The offset axis results in a larger and, consequently, stronger hypoid pinion tooth than on an equivalent straight-tooth or spiral bevel gear. Expressed in terms of pitch diameter, a hypoid pinion has fewer teeth than a spiral bevel pinion of the same pitch diameter. It is possible to use hypoid pinions having 7, 8, or 9 teeth in contrast to a minimum of 12, 13, or 14 teethdepending on the velocity ratioon a spiral bevel pinion. Lubricants must withstand the higher loading and the sliding that occurs along the teeth of hypoid gears.
Hypoid gears are suitable for large velocity reductions; reduction ratios of 60:1 and higher are entirely feasible. In general, shaft offset should not exceed 40% of the equivalent bevel gear back cone distance and, when the loading is heavy, as in truck and tractor drives, the offset should be nearer 20% of this distance. Direction of the offset, above or below center, must be specified for any given installation. The gears should, in general, be mounted on antifriction bearings in an oil-tight case. Thrust bearings must be provided. Because of the sliding tooth action, the efficiency of hypoid gears is somewhat less than that of equivalent bevel gears.
Worm gears
A chief way to connect nonparallel, nonintersecting shafts that are at right angles is through a worm gear. The worm, ordinarily the driver, is similar to a crossed helical gear except that it has at least one complete tooth (thread) around the pitch surface. The mating gear is the worm wheel or worm gear. Worm gearing is generally used to obtain large velocity reductions with the worm as the driver and the worm wheel as the driven gear, although occasional applications, for example cream separators, have the worm wheel as the driver. The pitch surfaces of straight worms are cylinders and the involute teeth have point contact. Because the appearance of the worm is similar to that of a screw, the teeth are called threads.
The pitch of the worm is the axial distance from any point on one tooth to the corresponding point on the next tooth (Fig. 9). This must equal the circular pitch of the mating worm wheel. Lead is the axial distance the worm helix advances in one complete revolution around the pitch surface. A single thread worm has the pitch and lead equal; one revolution of such a worm will, for shafts at right angles, advance the worm wheel 1/N revolutions if N is the number of teeth on the worm wheel. A double-threaded worm has the lead equal to twice the pitch and will then advance the worm wheel 2/N revolutions per turn of the worm. Thus, worm gears follow the general rule of angular velocity ratio inversely proportional to the ratio of the numbers of teeth. Worms are right- and left-hand in the same sense as helical gears. Changing hand of the worm reverses the relative rotation of the worm wheel.
Fig. 9 Nomenclature of a single-enveloping worm gearset. Teeth of worm gear are called threads. (After J. E. Shigley, Theory of Machines, McGraw-Hill, 1961)&minusvsp22;

Improved load-carrying capacity and wear characteristics are made possible by increasing the contact between the worm and wheel (Fig. 10). Line contact is obtained by making the worm wheel surface concave to conform to the tooth profile of the worm. Still greater contact is obtained by using a concave worm as well. Known as a cone-drive or Hindley worm, this design permits greater contact surface and allows more teeth to be in contact at one time. See also: Gear train; Mechanism; Planetary gear train
Fig. 10 Nature of contact for worm gears. (a) Non-throated. (b) Single-throated. (c) Double-throated on cone. (d) Point contact. (e) Line contact. (f) Area contact. (Michigan Tool Co.)

John R. Zimmerman
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John R. Zimmerman, "Gear", in AccessScience@McGraw-Hill, http://www.accessscience.com, DOI 10.1036/1097-8542.283100, last modified: August 21, 2002.
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Engineering & Materials: Other Engineering Disciplines:Design engineering
Engineering & Materials: Mechanical Engineering:Mechanical engineering
Gear train
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Select an article section Top Of Article Classification Ratios Direction of rotation Ordinary train Transmissions Epicyclic train BIBLIOGRAPHY ILLUSTRATIONS ANIMATIONS


combination of two or more gears used to transmit motion between two rotating shafts or between a shaft and a slide. In theory two gears can provide any speed ratio in connecting shafts at any center distance, but it is often not practical to use only two gears. If the ratio is large or if the center distance is relatively great, the larger of the two gears may be excessively large. Moreover, an additional gear may be necessary simply to give the proper direction to the output gear. Belt, rope, and chain drives are frequently used in conjunction with gear trains. See also: Belt drive; Chain drive; Planetary gear train
Classification
Gear train classifications include simple, compound, reverted, epicyclic (planetary), and various combinations. The most important distinction is that between ordinary and epicyclic gear trains. In ordinary trains (Fig. 1a), all axes remain stationary relative to the frame. But in epicyclic trains (Fig. 1b), at least one axis moves relative to the frame. In Fig. 1b gear B, whose axis is in motion, is called a planet. The gears A and C are sun gears.
Fig. 1 Gear trains. (a) Ordinary. (b) Epicyclic.

An ordinary gear train is a single degree of freedom mechanism: A single input, such as an input to gear A of the train in Fig. 1a, suffices to control the motions of the other moving members. But an epicyclic gear train (Fig. 1b) has two degrees of freedom: Two inputs are necessary. In the epicyclic train of Fig. 1b, the two input members are gear A and the planet carrier, link D. Only if both these members are controlled by external agencies can the motions of gears Band Cbe predicted. Frequently one gear of an epicyclic train is fixed. This then is one of the input members with a velocity of zero revolutions per unit time.
A simple gear train is one in which each gear is fastened to a separate shaft (Fig. 1a). If at least one shaft has two or more gears fastened to it (Fig. 2), the train is compound. The train of Fig. 2 is also a reverted gear train, because the input and output shafts are in line. If the input shaft is not in line with the output shaft, the train is nonreverted.
Fig. 2 Compound reverted gear train.

The 1000-hp (750,000-W) mill drive and pinion stand in Fig. 3 are an example of a large industrial ordinary gear train. The first speed reduction is with the opposed single helical gears, and the second is through the herringbone gears. This train is compound and nonreverted.
Fig. 3 Helical gear speed reducer. (Farrel-Birmingham Co.)

Ratios
For an ordinary gear train, the ratio of the angular velocity of the last driven gear to that of the driving gear is known as the train value. The ratio of the driver's velocity to that of the last driven gear is the velocity ratio. By these definitions train value and velocity ratio are reciprocal quantities. The train value for the ordinary train of Fig. 1a is -0.25. The output gear Cturns at one-fourth the speed of the input gear Aand in the opposite direction.
For epicyclic trains the situation can be a little more complicated. Since in general the output velocity is dependent on the two input velocities, the term train value is nonspecific. However, if one gear of an epicyclic train is fixed, the output velocity is some multiple of the velocity of the nonfixed input member; the term train value then applies.
Direction of rotation
The direction of rotation of any gear in an ordinary gear train can be determined by inspection. Mating external gears have opposite directions of rotation, while an internal gear has the same direction as its mating gear.
Ordinary train
Train value is by definition

(1) Eq. (1) when the angular velocities are measured with respect to the frame supporting the gears. Referring to the ordinary train of Fig. 2, the train value is (ignoring sign) Eq. (2),

(2) where n is the speed in revolutions per unit time. This can be expanded as an identity for the entire train to Eq. (3).

(3) Each of the ratios on the right side of Eq. (3) is the train value for a pair of meshing gears. Since for any two gears in mesh the speeds vary inversely as the numbers of teeth, the train value for the whole train must be (again ignoring sign) Eq. (4), where N is the number of teeth. In Eq. (4)

(4) the number of teeth appearing in the numerator of the expression on the right are those for driving gears, while those appearing in the denominator are the numbers of teeth on the driven gears. Thus, the general expression for the magnitude of the train value for an ordinary gear train is Eq. (5).

(5) Because any two gears in mesh must have the same diametral pitch, expressions for the train value can also be written in terms of pitch diameters instead of numbers of teeth.
In the expression for the train value of the train of Fig. 2 the number of teeth on gear B cancels out. Gear B is an idler; it is both a driver and a driven gear. Its size has no effect on the train value's magnitude. It does affect, however, the sign of the train value. Idlers also are useful where a relatively large center distance must be spanned.
Transmissions
If a machine such as a machine tool or motor vehicle must be operated at any one of several output speeds, a multiple-speed gearbox, or transmission, may be used as a component part. The speed of the output shaft of a transmission can be varied by sliding gears in and out of contact or by connecting gears in continual mesh to shafts by means of clutches. A compact nine-speed transmission utilizing 10 gears and three shafts is shown in Fig. 4. Gears A, B, and C slide as a unit on the constant-speed shaft, providing three ways in which the constant speed and the intermediate shafts can be connected (A and D, or B and E, or C and G). Similarly gears H, I, and J slide as a unit on the variable-speed shaft, providing three ways in which it can be connected to the intermediate shaft (D and H, or Eand I, or F and J). Consequently nine possible combinations are available. For example, the lowest output speed is attained when A and D mesh and E and I mesh. The train value is then as in Eq. (6).

(6)
Fig. 4 Compact 9-speed transmission utilizing 10 gears and 3 shafts. (After J. R. Zimmerman, Elementary Kinematics of Mechanisms, Wiley, 1962)

The highest speed occurs when C and G mesh and D and H mesh. The train value for this combination is shown in Eq. (7).

(7) See also: Automotive transmission
Epicyclic train
An epicyclic train is named for the path described by any point on the pitch circle of a planet gear as it rolls on a sun gear. The term epicyclic refers only to the motion and is not related to the gear tooth form, which may be involute or any other form satisfying the law of gearing. An epicyclic train can be used to obtain a considerably greater velocity ratio than would be possible with an ordinary gear train of the same size.
In the compound and reverted epicyclic gear train of Fig. 5, gear A is fixed to the frame, but not to shaft 1. The planet carrier, arm or link E, is fastened to shaft 1 and carries planet gears B and C, which are fastened to a common shaft. Planet B rolls on sun gear A, while planet C rolls on sun gear D. Gear A is one of the input members; it has a velocity of 0 rpm. The other input could be either the carrier E or gear D. In the example to follow it will be assumed that the arm carrier E is the input member.
Fig. 5 Reverted epicyclic gear train.

To determine the train value of an epicyclic train with a fixed gear, the following special but very convenient technique can be used. The net motion is divided into two parts. In the first, the train is locked with no relative motion between any of its components. For this step gear A is released from the frame and locked to gear B. The locked train is rotated one revolution in an arbitrary direction. Counterclockwise rotation is assumed positive, clockwise negative. In the second step, the assembly is viewed as an ordinary gear train with the planet carrier regarded as fixed. While the fixed gear is returned to its original position by one revolution in the sense opposite that used in the first step, the consequent rotation of the other gears is noted. For any one gear, the algebraic sum of the revolutions during the two parts is the net motion of that gear for one revolution of the planet carrier. To illustrate, consider the table for the train of Fig. 5, with the number of teeth as marked for each gear. The table shows that, for each revolution of the carrier, gear D will turn 199/10,000 of a revolution in the same direction.
For epicyclic trains with no fixed gears, a modification of the two-step method is needed. In the first step, the locked train is given the same rotation as the known motion of the planet carrier. In the second, with the carrier stationary, the gear whose velocity is known is rotated sufficient turns in the proper direction to make its net motion equal to the known value.
Probably the most common epicyclic train with no fixed gear is the automobile differential. When the automobile is moving along a straight path, there is no relative motion between the bevel differential gears fastened individually to the right and left axles. But as the car makes a turn, the differential gears move relative to one another in the manner of an epicyclic bevel gear train with no fixed gears. See also: Differential; Gear; Planetary gear train
John R. Zimmerman
Donald L. Anglin
How to cite this article
Suggested citation for this article:
John R. Zimmerman, Donald L. Anglin, "Gear train", in AccessScience@McGraw-Hill, http://www.accessscience.com, DOI 10.1036/1097-8542.283500, last modified: September 11, 2002.
For Further Study
Topic Page: Engineering & Materials: Other Engineering Disciplines:Design engineering
Topic Page: Engineering & Materials: Mechanical Engineering:Mechanical engineering
DOI 10.1036/1097-8542.283500
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