Lecture #24 SPUR GEAR TEETH CHARACTERISTICS Course Name : DESIGN OF MACHINE ELEMENTS

1. Lecture #24 SPUR GEAR TEETH CHARACTERISTICS Course Name : DESIGN OF MACHINE ELEMENTS Course Number: MET 214

2. In order to identify the shape commonly used with the teeth of a spur gear, consider the formation of a curve that is referred to as an involute as shown below. The cross section of the cylinder from which the string is unwrapped is referred to as the base circle. Usually the end of the string is shown as being unwrapped from the base circle and as a consequence it is the end of the string that traces out the involute as shown in the diagram contained on the next slide.

3. As identified in the figure below, the string is keep taut when being unwound, and as a consequence, the string always forms a tangent with the base circle regardless of how much string has been unwrapped. The line formed by any point on the involute and the corresponding point of tangency existing with the base circle is referred to as the generating line. As is evident from the figure, the length of the generating line segment existing between any point on the involute and the corresponding point of tangency increases as more of the string is unwrapped from the base circle.

4. As identified in the figure below, the base circle of a gear can be used to create a gear tooth with an involute shape. Although the next slide is from a different source than the illustration provided above, and uses a different set of symbols to characterize an involute, several equations fundamental to describing an involute are provided in the figure and accordingly, the figure should compared with the previous illustrations with the objective of being able to relate all illustrations to one anotherin a comprehensible manner.

6. As noted in the equations provided on the previous slide, the length of the string unwrapped from the base circle, defined as is related to the roll angle, defined as by the radius of the base circle, defined as . Consequently, the length of is proportional to the roll angle . Accordingly, if an involute is formed from a base circle and attached to the base circle, and the base circle rotated at a constant rate, the involute can be made to sweep across a tangent to the base circle at a constant velocity. This is illustrated in the figure provided below. Since the radius of the base circle is a, for one revolution of the base circle, the rise would be

7. Accordingly, the involute satisfies one of the criteria required for a gear in order for the gear to have the same speed ratio as the friction wheel implementation that was developed in a previous lecture. The point of contact existing between gear teeth point must travel with a constant velocity on the tangent line existing between the base circles in order for a pair of gears to produce the same speed ratio as the friction wheel implementation. The properties of an involute can be used by the teeth on both gears to enable a point of contact existing between the gear teeth to move along the line of action at a constant rate as shown in the diagram below. The line of action is shown in a vertical position to facilitate understanding of how the rotation of the base circles effects velocity along the line of action.

8. As illustrated by the drawing provided below, a tangent to the involute at any point will be perpendicular to the generating line for that point. This relationship can be proven to be true for every point on an involute. The generating line is always normal to the involute curve.

9. Accordingly, returning to the figure showing the action of two involutes in contact, repeated below for convenience, the following point should be noted. Since the tangent of each involute is perpendicular to it’s generating line, the tangents of two involutes in contact coincide only when the generating line of one involute is a continuation of the generating line of the second involute. Accordingly, the locus of the points of contact between two involutes is along the common tangent to the two base circles as shown in the figure.

10. With the following considerations in mind, the Fundamental Law of toothed Gearing will now be presented. The derivation presented below is taken from the text titled Design of Machine elements 5th edition by Spotts.

14. Key features of gear teeth are illustrated in the figures below.

15. Pitch Diameter: Diameter of pitch circle. The pitch circles or the circles that exists in the friction wheel implementation of the meshing gears. Pitch: 3 types of pitch are in common use. a) Circular pitch b) Diametral pitch c) Metric module • Circular pitch: The distance from a point on a tooth of a gear at the pitch circle to a corresponding point on the next adjacent tooth, measured along the pitch circle. The following relationship is a consequence of a gear having an integral number of teeth spaced around the periphery of the gear. where circular pitch number of teeth on gear diameter of pitch circle The following observation is evident from the requirement to meshing gears to form a mechanical system of interlocking. “The pitch of two gears in mesh must be identical”

16. As shown in the table below, standards have been established for gears. The following key features are illustrated in the figures above. 1) addendum (a): The radial distance from the pitch circle to the outside of a tooth (top land) Figures 8-11 2) dedendum (b): The radial distance from the pitch circle to the bottom of the tooth space. 3) Clearance (c) : The radial distance from the top of a tooth to the bottom of the tooth space of the mating gear when the tooth is fully engaged. c=b-a 4) Outside diameter (D0): The diameter of the circle that encloses the outside of the gear teeth. Note that D0 = D+ 2a where D= diameter of pitch circle Using the definition for Pd and standards concerning gear teeth the following relationships may be established.

17. 5) Root diameter: (DR) The diameter of the circle that contains the bottom of the tooth space. This circle is called the root circle. DR = D – 2b 6) Whole depth (ht) : The radial distance from the top of a tooth to the bottom of the tooth space. Note that ht = a + b 7) Working depth (hk): The radial distance that a gear tooth projects into the tooth space of a mating gear. Note that hk = a + a = 2a and ht = hk+ c 8) Tooth thickness (t): The arc length, measured on the pitch circle from one side of a tooth to the other side. This is sometimes called the circular thickness and has the theoretical value of one-half of the circular pitch. recall

18. 9) Tooth space: The arc length, measured on the pitch circle, from the right side of one tooth to the left side of the next tooth. Theoretically, the tooth space equals the tooth thickness. But for practical reasons, the tooth space is made large. The difference between tooth space and tooth thickness is referred to as backlash. Backlash provides space for lubrication, and relieves the demand for precision on gear teeth dimensions. 10) Face width (F): The width of the tooth measured parallel to the axis of the gear. 11) Fillet: The arc joining the involute tooth profile to the root of the tooth space. 12) Face: The surface of a gear tooth from the pitch circle to the outside circle of the gear. 13) Flank: The surface of a gear tooth from the pitch circle to the root of the tooth space, including the fillet. 14) Center distance (c): The distance from the center of the prism to the center of the gear; the sum of the pitch radii of two gears in mesh. The center distance can also be expressed in terms of the diametrical pitch.

19. 15 Pressure angle: The pressure angle is the angle between the tangent to the pitch circles and the line drawn normal (perpendicular) to the surface of the gear tooth see figure below. As established previously, where diameter of base circle pressure angle

20. Standard values of the pressure angle are established by gear manufacturers and the pressure angle of two gears in mesh must be the same. Current standard pressure angles are 200 and 250. Pressure angles of 141/20 are presently used only as replacements. Involutetooth forms for different pressure angles are shown below