SCIENCE AND TECHNOLOGY ENGLIAH FOR MECHANICAL ENGINEERING LESSON FIVE FOUNDATIONS OF MACHINE DESIGN (PART 2) Action Of a Pair Of Mating Involute Teeth Let a and b be the base circles of a pair of mating involute gears. Line CD is a common tangent to the base circles. while ab is the line of centers. Assume that CD is part of a string being unwound from A and wound upon b, while a and b rotate together in such a manner that string CD remains taut at all times. When A and B start to rotate, point C on the string will leave circle A and move towards circle B, thus describing an involute with respect to circle A
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING LESSON FIVE FOUNDATIONS OF MACHINE DESIGN (PART 2) Action Of a Pair Of Mating Involute Teeth Let A and B be the base circles of a pair of mating involute gears. Line CD is a common tangent to the base circles, while AB is the line of centers. Assume that CD is part of a string being unwound from A and wound upon B, while A and B rotate together in such a manner that string CD remains taut at all times. When A and B start to rotate, point C on the string will leave circle A and move towards circle B, thus describing an involute with respect to circle A
SCIENCE AND TECHNOLOGY ENGLIAH FOR MECHANICAL ENGINEERING At the same time, however, one can imagine that, with respect to circle B, point C traces an involute back to its origin B. The involutes form the actual tooth outline in the ideal case. It can be proven that the basic requirement for proper gear action, namely, that no changes in speed ratio occur during the passage of any tooth, is fulfilled when the normal to the mating tooth curves at the point of contact always passes through the pitch point
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING At the same time, however, one can imagine that, with respect to circle B, point C traces an involute back to its origin B. The involutes form the actual tooth outline in the ideal case. It can be proven that the basic requirement for proper gear action , namely , that no changes in speed ratio occur during the passage of any tooth, is fulfilled when the normal to the mating tooth curves at the point of contact always passes through the pitch point
SCIENCE AND TECHNOLOGY ENGLIAH FOR MECHANICAL ENGINEERING Let us investigate whether this condition is satisfied b the involute. It is clear that since the string is taut at all times the path of the point of contact between the two involutes is a straight line. This line intersects the line of centers(AB)at P. Also, the involute is by definition normal to its generating line (i.e, the string) at all times, since the involute is a circular arc with everincreasing radius, and a radius is al ways perpendicular to its circular arc. Therefore, if we can prove that point P is the pitch point, we have satisfied all the above mentioned requirements Triangles ACP and BPD are similar since their corresponding angles are equal. Like the friction drive at the beginning of this chapter, A and b have the same circumferential velocity at points C and D. We may therefore state that
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING Let us investigate whether this condition is satisfied by the involute. It is clear that since the string is taut at all times, the path of the point of contact between the two involutes is a straight line. This line intersects the line of centers (AB) at P. Also, the involute is by definition normal to its generating line (i.e., the string) at all times, since the involute is a circular arc with everincreasing radius, and a radius is always perpendicular to its circular arc. Therefore, if we can prove that point P is the pitch point, we have satisfied all the above mentioned requirements. Triangles ACP and BPD are similar since their corresponding angles are equal. Like the friction drive at the beginning of this chapter, A and B have the same circumferential velocity at points C and D. We may therefore state that
SCIENCE AND TECHNOLOGY ENGLIAH FOR MECHANICAL ENGINEERING BD/AC=nA/nB 6.17 but BD/AC= BP/AP 6.18 and na/nR is the speed ratio. Therefore, AP and BP must be pitch radii and point P must be the pitch point Path Of Contact And Contact Ratio The path of contact is the line described by the point of contact between two mating teeth during rotation. In involute gearing, the path of contact coincides with the line of action. It begins where the addendum circle of the driven sear intersects the line of action, and ends where the addendum circle of the driving gear intersects the line of action. This definition ignores possible inter ference conditions with pinions of small tooth numbers, which are outside the scope of this text
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING BD/AC = nA / nB 6.17 but BD/AC = BP/AP 6.18 and nA /nB is the speed ratio. Therefore, AP and BP must be pitch radii and point P must be the pitch point. Path Of Contact And Contact Ratio The path of contact is the line described by the point of contact between two mating teeth during rotation. In involute gearing, the path of contact coincides with the line of action. It begins where the addendum circle of the driven sear intersects the line of action, and ends where the addendum circle of the driving gear intersects the line of action. This definition ignores possible interference conditions with pinions of small tooth numbers, which are outside the scope of this text
SCIENCE AND TECHNOLOGY ENGLIAH FOR MECHANICAL ENGINEERING The contact ratio (m) is a number indicating the average number of teeth in contact for a given pair of mating gears, and is found by dividing the length of the path of contact Z (theaction)by the base pitch, or m-length of action /base pitch Z/P 6.19 Cams cam 凸轮 A cam and its follower together form follower随动件 mechanism that converts rotary motion(often at constant speed) or oscillating motion, into a cyclical (repetitive) linear or angular motion
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING The contact ratio (mc ) is a number indicating the average number of teeth in contact for a given pair of mating gears, and is found by dividing the length of the path of contact Z (the “action”) by the base pitch, or mc = length of action / base pitch = Z / Pb 6. 19 Cams A cam and its follower together form a mechanism that converts rotary motion (often at constant speed) or oscillating motion, into a cyclical (repetitive) linear or angular motion. cam 凸轮 follower 随动件