Gear mechanisms

Gear mechanisms are used for transmitting motion and power from one shaft to another by means of the positive contact of successively engaging teeth. In about 2,600B.C., Chinese are known to have used a chariot incorporating a complex series of gears like those illustrated in Fig.2.7. Aristotle, in the fourth century B .C .wrote of gears as if they were commonplace. In the fifteenth century A.D., Leonardo da Vinci designed a multitude of devices incorporating many kinds of gears. In comparison with belt and chain drives ,gear drives are more compact ,can operate at high speeds, and can be used where precise timing is desired. The transmission efficiency of gears is as high as 98 percent. On the other hand, gears are usually more costly and require more attention to lubrication, cleanliness, shaft alignment, etc., and usually operate in a closed case with provision for proper lubrication.

Gear mechanisms can be divided into planar gear mechanisms and spatial gear mechanisms. Planar gear mechanisms are used to transmit motion and spatial gear mechanisms. Planar gear mechanisms are used to transmit motion and power between parallel shafts ,and spatial gear mechanisms between nonparallel shafts.

Types of gears

Spur gears. The spur gear has a cylindrical pitch surface and has straight teeth parallel to its axis as shown in Fig. 2.8. They are used to transmit motion and power between parallel shafts. The tooth surfaces of spur gears contact on a straight line parallel to the axes of gears. This implies that tooth profiles go into and out of contact along the whole facewidth at the same time. This will therefore result in the sudden loading and sudden unloading on teeth as profiles go into and out of contact. As aresult, vibration and noise are produced.

Helical gears. These gears have their tooth elements at an angle or helix to the axis of the gear(Fig.2.9). The tooth surfaces of two engaging helical gears inn planar gear mechanisms contact on a straight line inclined to the axes of the gears. The length of the contact line changes gradually from zero to maximum and then from maximum to zero. The loading and unloading of the teeth become gradual and smooth. Helical gears may be used to transmit motion and power between parallel shafts[Fig. 2.9(a)]or shafts at an angle to each other[Fig. 2.9(d)]. A herringbone gear [Fig. 2.9(c)] is equivalent to a right-hand and a left-hand helical gear placed side by side. Because of the angle of the tooth, helical gears create considerable side thrust on the shaft. A herringbone gear corrects this thrust by neutralizing it , allowing the use of a small thrust bearing instead of a large one and perhaps eliminating one altogether. Often a central groove is made around the gear for ease in machining.

Bevel gars. The teeth of a bevel gear are distributed on the frustum of a cone. The corresponding pitch cylinder in cylindrical gears becomes pitch cone. The dimensions of teeth on different transverse planes are different. For convenience, parameters and dimensions at the large end are taken to be standard values. Bevel gears are used to connect shafts which are not parallel to each other. Usually the shafts are 90 deg. to each other, but may be more or less than 90 deg. The two mating gears may have the same number of teeth for the purpose of changing direction of motion only, or they may have a different number of teeth for the purpose of changing both speed and direction. The tooth elements may be straight or spiral, so that we have plain and spiral bevel gears. Hypoid comes from the word hyperboloid and indicates the surface on which the tooth face lies. Hypoid gears are similar to bevel gears, but the two shafts do not intersect. The teeth are curved, and because of the nonintersection of the shafts, bearings can be placed on each side of each gear. The principal use of thid type of gear is in automobile rear ends for the purpose of lowering the drive shaft, and thus the car floor.

Worm and worm gears. Worm gear drives are used to transmit motion and ower between non-intersecting and non-parallel shafts, usually crossing at a right angle, especially where it is desired to obtain high gear reduction in a limited space. Worms are a kind of screw, usually right handed for convenience of cutting, or left handed it necessary. According to the enveloping type, worms can be divided into single and double enveloping. Worms are usually drivers to reduce the speed. If not self-locking, a worm gear can also be the driver in a so called back-driving mechanism to increase the speed. Two things characterize worm gearing (a) large velocity ratios, and (b) high sliding velocities. The latter means that heat generation and power transmission efficiency are of greater concern than with other types of gears.

Racks. A rack is a gear with an infinite radius, or a gear with its perimeter stretched out into a straight line. It is used to change reciprocating motion to rotary motion or vice versa. A lathe rack and pinion is good example of this mechanism.

Geometry of gear tooth

The basic requirement of gear-tooth geometry is the provision of angular velocity rations that are exactly constant. Of course, manufacturing inaccuracies and tooth deflections well cause slight deviations in velocity ratio; but acceptable tooth profiles are based on theoretical curves that meet this criterion.

The action of a pair of gear teeth satisfying this requirement is termed conjugate gear-tooth action, and is illustrated in Fig. 2.12. The basic law of conjugate gear-tooth action states that as the gears rotate, the common normal to the surfaces at the point of contact must always intersect the line of centers at the same point P called the pitch point.

The law of conjugate gear-tooth can be satisfied by various tooth shapes, but the only one of current importance is the involute, or, more precisely, the involute of the circle. (Its last important competitor was the cycloidal shape, used in the gears of Model T Ford transmissions.) An involute (of the circle) is the curve generated by any point on a taut thread as it unwinds from a circle, called the base circle. The generation of two involutes is shown in Fig. 2.13. The dotted lines show how these could correspond to the outer portion of the right sides of adjacent gear teeth. Correspondingly, involutes generated by unwinding a thread wrapped counterclockwise around the base circle would for the outer portions of the left sides of the teeth. Note that at every point, the involute is perpendicular to the taut thread, since the involute is a circular arc with everincreasing radius, and a radius is always perpendicular to its circular arc. It is important to note that an involute can be developed as far as desired outside the base circle, but an involute cannot exist inside its base circle.

Let us now develop a mating pair of involute gear teeth in three steps: friction drive, belt drive, and finally, involute gear-tooth drive. Figure 2.14 shows two pitch circles. Imagine that they represent two cylinders pressed together. If slippage does not occur, rotation of one cylinder (pitch circle) will cause rotation of the other at an angular velocity ratio inversely proportional to their diameters. In any pair of mating gears, the smaller of the two is called the pinion and the larger one the gear. (The term “gear” is used in a general sense to indicate either of the members, and also in a specific sense to indicate the larger of the two.) Using subscripts p and g to denote pinion and gear, respectively.

In order to transmit more torque than is possible with friction drive alone, we now add a belt drive running between pulleys representing the base circles, as in Fig 2.15. If the pinion is turned counterclockwise a few degrees, the belt will cause the gear to rotate in accordance with correct velocity ratio. In gear parlance, angle Φ is called the pressure angle. From similar triangles, the base circles have the same ratio as the pitch; thus, the velocity ratio provided by the friction and belt drives are the same.

In Fig. 2.16 the belt is cut at point c, and the two ends are used to generate involute profiles de and fg for the pinion and gear, respectively. It should now be clear why Φ is called the pressure angle: neglecting sliding friction, the force of one involute tooth pushing against the other is always at an angle equal to the pressure angle. A comparison of Fig. 2.16 and Fig.2.12 shows that the involute profiles do indeed satisfy the fundamental law of conjugate gear-tooth action. Incidentally, the involute is the only geometric profile satisfying this law that maintains a constant pressure angle as the gears rotate. Note especially that conjugate involute action can take place only outside of both base circles.

Nomenclature of spur gear

The nomenclature of spur gear (Fig .2.17) is mostly applicable to all other type of gears.

The diameter of each of the original rolling cylinders of two mating gears is called the pitch diameter, and the cylinder’s sectional outline is called the pitch circle. The pitch circles are tangent to each other at pitch point. The circle from which the involute is generated is called the base circle. The circle where the tops of the teeth lie is called the dedendum circle. Similarly, the circle where the roots of the teeth lie is called the dedendum circle. Between the addendum circle and the dedendum circle, there is an important circle which is called the reference circle. Parameters on the reference circle are standardized. The module m of a gear is introduced on the reference circle as a basic parameter, which is defined as m=p/π. Sizes of the teeth and gear are proportional to the module m.

The addendum is the radial distance from the reference circle to the addendum circle. The dedendum is the radial distance from the reference circle to the dedendum circle. Clearance is the difference between addendum and dedendum in mating gears. Clearance prevents binding caused by any possible eccentricity.

The circular pitch p is the distance between corresponding side of neighboring teeth, measured along the reference circle. The base pitch is similar to the circular pitch is measured along the base circle instead of along the reference circle. It can easily be seen that the base radius equals the reference radius times the cosine of the pressure angle. Since, for a given angle, the ratio between any subtended arc and its radius is constant, it is also true that the base pitch equals the circular pitch times the cosine of the pressure angle. The pressure angle is the angle between the normal and the circumferential velocity of the point on a specific circle. The pressure angle on the reference circle is also standardized. It is most commonly 20º(sometimes 15º).

The line of centers is a line passing through the centers of two mating gears. The center distance (measured along the line of centers) equals the sum of the pitch radii of pinion and gear.

Tooth thickness is the width of the tooth, measured along the reference circle, is also referred to as tooth thickness. Width of space is the distance between facing side of adjacent teeth, measured along the reference circle. Tooth thickness plus width of space equals the circular pitch. Backlash is the width of space minus the tooth thickness. Face width measures tooth width in an axial direction.

The face of the tooth is the active surface of the tooth outside the pitch cylinder. The flank of the tooth is the active surface inside the pitch cylinder. The fillet is the rounded corner at the base of the tooth. The working depth is the sum of the addendum of a gear and the addendum of its mating gear.

In order to mate properly, gears running together must have: (a) the same module; (b) the same pressure angle; (c) the same addendum and dedendum. The last requirement is valid for standard gears only.

Rolling-Contactbearings

The rolling-contact bearing consists of niier and outer rings sepatated by a number of rolling elements in the form of balls ,which are held in separators or retainers, and roller bearings have mainly cyinndrical, conical , or barrelcage.The needles are retainde by integral flanges on the outer race,

Bearigs with rolling contact have no skopstick effect,low statting torqeu and running friction,and unlike as in journal bearings. The coefficient of friction varies little with load or opeed.Probably the outstanding of a rolling-contant beating over a sliding bearing is its low statting friction.The srdinary sliding bearing starts from rest with practically metal to metal contact and has a high coefficient of friction as compared with that between rolling members.This teature is of particular important in the case of beatings whcch vust carry the same laode at test as when tunning,for example.less than one-thirtieth as much force is required to start a raliroad freight car equopped with roller beatings as with plain journal bearings.However.most journal bearing can only carry relatively light loads while starting and do not become heavily loaded until the speed is high enough for a hydrodynamic film to be built up.At this time the friction id that in the luvricant ,and in a properly designed journal bearing the viscous friction will be in the same order of magnitude ad that for a that for a rolling-conanct bearing.