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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