CHAPTER I.

=THE TEETH OF GEAR-WHEELS.=

PAGE =Gear-Wheels.= Spur-wheels, bevel-wheels, mitre-wheels, crown-wheels, annular or internal wheels 1 Trundle-wheels, rack and pinion-wheel and tangent screw, or worm and worm-wheel 1 The diameter of the pitch circle of 1 =Gear-Wheel Teeth.= The face, the flank, the depth or height 1 The space, the pitch line, the point, the arc pitch, the chord pitch, the line of centres 2 Rules for finding the chord pitch from the arc pitch; table of natural sines; diametral pitch; finding the arc from the diametral pitch; table of arc and diametral pitches 3 =Gear-Wheels.= The driver and follower, a train of gears 3 Intermediate gears 3 The velocity of compounded wheels 4 Finding the diameters of the pitch circles of 4 Considered as revolving levers 5 Calculating the revolutions of, and power transmitted by 5 The angular velocity of 6 =Gear-Wheels.= Hunting tooth in, stop motion of 7 =Gear-Wheel Teeth.= The requirements and nature of the teeth curves 7 Cycloidal curves for the faces of; epicycloidal and involute curves; the hypocycloidal curve; method of forming or generating the epicycloidal and hypocycloidal curves for the faces and flanks of gear teeth 8 Applications of the epicycloidal and hypocycloidal curves in the formation of gear teeth 9 The diameter of the circle for generating the epicycloidal and hypocycloidal curves; graphical demonstration that the flank curves are correctly formed to work with the face curves of the other wheel 10 Graphical demonstration that the curves are correct independent of either the respective sizes of the wheels, or of the curve generating circles 11 =Gear-Wheels.= Hand applications of the rolling or generating circle to mark the tooth curves for a pair of wheels 12 =Gear-Wheel Teeth.= The variation of curve due to different diameters of wheels or of rolling circles 12 Tracing the path of contact of tooth upon tooth in a pair of gear-wheels; definition of the "arc of approach;" definition of the "arc of recess;" demonstration that the flanks of the teeth on the driver or driving-wheel have contact with the faces of the driven wheel during the arc of approach, and with the flanks of the driven wheel during the arc of recess 13 Confining the action of the teeth to one side only of the line of centres, when motion rather than power is to be conveyed 13 Demonstration that the appearance or symmetry of a tooth has no significance with regard to its action 14 Finding how many teeth will be in constant action, the diameter of the wheels, the pitch of the teeth, and the diameter of the rolling circle being given 15 Example of the variation of tooth form due to variation of wheel diameter 15 =Gear Teeth.= Variation of shape from using different diameters of rolling circles 16 Thrust on the wheel shafts caused by different shapes of teeth 16 =Gear-Wheels.= Willis' system of one size of rolling circle for trains of interchangeable gearing 16 Conditions necessary to obtain a uniform velocity of 16 =Gear Teeth.= The amount of rolling and of sliding motion of 16 The path of the point of contact of 16 The arcs of approaching and of receding contact 16 Lengths of the arcs of approach and of recess 16 The influence of the sizes of the wheels upon the arcs of contact 17 Influence of the size of the rolling circle upon the amount of flank contact 18 Demonstration that incorrectly formed teeth cannot correct themselves by wear 18 The smaller the diameter of the rolling circle, the less the sliding motion 18 Influence of the size of the rolling upon the number of teeth in contact in a given pair of wheels 19 Demonstration that the degrees of angle the teeth move through exceed those of the path of contact, unless the tooth faces meet in a point 19 Influence of the height of the teeth upon the number of teeth in contact 20 Increasing the arc of recess without increasing the arc of approach 20 Wheels for transmitting motion rather than power 21 Clock wheels 21 Forms of teeth having generating or rolling circles, as large or nearly as large as the diameters of the wheels 21 =Gear-Wheels.= Bevel 21 The principles governing the formation of the teeth of bevel- wheels 22 Demonstration that the faces of the wheels must be in line with the point of intersection of the axis of the two shafts 22 =Gear Teeth.= Method of finding the curves of, for bevel gear 22 =Gear-Wheels.= Internal or annular 23 to 27 Demonstration that the teeth of annular wheels correspond to the spaces of spur-wheels 23 =Gear-Wheels Internal.= Increase in the length of the path of contact on spur-wheels of the same diameter, and having the same diameter of generating or rolling circle 23 Demonstration that the teeth of internal wheels may interfere when spur-wheels would not do so 23 Methods of avoiding the above interference 23 Comparison of, with spur-wheels 23 The teeth of: demonstration that it is practicable to so form the teeth faces that they will have contact together as well as with the flanks of the other wheel 24 Intermediate rolling circle for accomplishing the above result 24 The application of two rolling circles for accomplishing the above result 24 Demonstration that the result reached by the employment of two rolling circles of proper diameter is theoretically and practically perfect 24 Limits of the diameters of the two rolling circles 25 Increase in the arc of contact obtained by using two rolling circles 25 Demonstration that the above increase is on the arc of recess or receding contact, and therefore gives a smooth action 25 Demonstration that by using two rolling circles each tooth has for a certain period two points of contact 25 The laws governing the diameters of the two rolling circles 25 Practical application of two rolling circles 26 Demonstration that by using two rolling circles the pinion may contain but one tooth less than the wheel 26 The sliding and rolling motion of the teeth of 27