That the amount of power transmitted by gearing, whether compounded or not, is equal throughout every wheel in the train, may be shown as follows:—
Referring again to [Fig. 10], it has been shown that with a 50 lb. weight suspended from a 4 inch shaft e, there would be 3033⁄100 lbs. at the perimeter of a. Now suppose a rotation be made, then the 50 lb. weight would fall a distance equal to the circumference of the shaft, which is (3.1416 × 4 = 1256⁄100) 1256⁄100 inches. Now the circumference of the wheel is (60 dia. × 3.1416 = 18849⁄100 cir.) 18849⁄100 inches, which is the distance through which the 333⁄100 lbs. would move during one rotation of a. Now 3.33 lbs. moving through 188.49 inches represents the same amount of power as does 50 lbs. moving through a distance of 12.56 inches, as may be found by converting the two into inch lbs. (that is to say, into the number of inches moved by 1 lb.), bearing in mind that there will be a slight discrepancy due to the fact that the fractions .33 in the one case, and .56 in the other are not quite correct. Thus:
| 188.49 | inches | × | 3.33 | lbs. | = | 627 | .67 | inch | lbs., | and |
| 12.56 | „ | × | 50 | „ | = | 628 | „ | „ |
Taking the next wheels in [Fig. 12], it has been shown that the 3.33 lbs. delivered from a to the perimeter of b, becomes 2.49 lbs. at the perimeter of c, and it has also been shown that c makes two revolutions to one of a, and its diameter being 40 inches, the distance this 2.49 lbs. will move through in one revolution of a will therefore be equal to twice its circumference, which is (40 dia. × 3.1416 = 125.666 cir., and 125.666 × 2 = 251.332) 251.332 inches. Now 2.49 lbs. moving through 251.332 gives when brought to inch lbs. 627.67 inch lbs., thus 251.332 × 2.49 = 627.67. Hence the amount of power remains constant, but is altered in form, merely being converted from a heavy weight moving a short distance, into a lighter one moving a distance exactly as much greater as the weight or force is lessened or lighter.
Gear-wheels therefore form a convenient method of either simply transmitting motion or power, as when the wheels are all of equal diameter, or of transmitting it and simultaneously varying its velocity of motion, as when the wheels are compounded either to reduce or increase the speed or velocity in feet per second of the prime mover or first driver of the train or pair, as the case may be.
Fig. 15.
In considering the action of gear-teeth, however, it sometimes is more convenient to denote their motion by the number of degrees of angle they move through during a certain portion of a revolution, and to refer to their relative velocities in terms of the ratio or proportion existing between their velocities. The first of these is termed the angular velocity, or the number of degrees of angle the wheel moves through during a given period, while the second is termed the velocity ratio of the pair of wheels. Let it be supposed that two wheels of equal diameter have contact at their perimeters so that one drives the other by friction without any slip, then the velocity of a point on the perimeter of one will equal that of a point on the other. Thus in [Fig. 15] let a and b represent the pitch circles of two wheels, and c an imaginary line joining the axes of the two wheels and termed the line of centres. Now the point of contact of the two wheels will be on the line of centres as at d, and if a point or dot be marked at d and motion be imparted from a to b, then when each wheel has made a quarter revolution the dot on a will have arrived at e while that on b will have arrived at f. As each wheel has moved through one quarter revolution, it has moved through 90° of angle, because in the whole circle there is 360°, one quarter of which is 90°, hence instead of saying that the wheels have each moved through one quarter of a revolution we may say they have moved through an angle of 90°, or, in other words, their angular velocity has, during this period, been 90°. And as both wheels have moved through an equal number of degrees of angle their velocity ratio or proportion of velocity has been equal.
Obviously then the angular velocity of a wheel represents a portion of a revolution irrespective of the diameter of the wheel, while the velocity ratio represents the diameter of one in proportion to that of the other irrespective of the actual diameter of either of them.