Machine Design
Gear Drive
Involute Teeth
Introduction:
An involute of a circle is a plane curve generated by a point on a tangent, which rolls on the circle without slipping or by a point on a taut string which is unwrapped from a reel.
Construction of involute teeth
Construction of involute teeth
Let A be the starting point of the involute.
The base circle is divided into equal number of parts e.g. AP1, P1 P2, P2 P3 etc.The tangents at P1, P2, P3 etc., are drawn and the lenghts P1A1, P2A2, P3A3 equal to the arcs AP1, AP2 and AP3 are set off.
Joining the points A, A1, A2, A3 etc., we obtain the involute curve AR. At any instant A3, the tangent A3T to the involute is perpendicular to P3A3 and P3A3 is the normal to the involute.
In other words, normal at any point of an involute is a tangent to the circle. Now, let O1 and O2 be the fixed centres of the two base circles as shown in Fig.(b).
Let the corresponding involutes AB and A'B' be in contact at point Q. MQ and NQ are normals to the involute at Q and are tangents to base circles.
Since the normal for an involute at a given point is the tangent drawn from that point to the base circle, therefore the common normal MN at Q is also the common tangent to the two base circles.
The common normal MN intersects the line of centres O1O2 at the fixed point P (called pitch point). Therefore the involute teeth satisfy the fundamental condition of constant velocity ratio.
From similar triangles O2 NP and O1 MP,
which determines the ratio of the radii of the two base circles. The radii of the base circles is given by
O1M = O1 P cos φ, and O2N = O2 P cos φ
where φ is the pressure angle or the angle of obliquity.
Also the centre distance between the base circles
If the centre distance is changed, then the radii of pitch circles also changes. But their ratio remains unchanged, because it is equal to the ratio of the two radii of the base circles. The common normal, at the point of contact, still passes through the pitch point. As a result of this, the wheel continues to work correctly. However, the pressure angle increases with the increase in centre distance.
The base circle is divided into equal number of parts e.g. AP1, P1 P2, P2 P3 etc.The tangents at P1, P2, P3 etc., are drawn and the lenghts P1A1, P2A2, P3A3 equal to the arcs AP1, AP2 and AP3 are set off.
Joining the points A, A1, A2, A3 etc., we obtain the involute curve AR. At any instant A3, the tangent A3T to the involute is perpendicular to P3A3 and P3A3 is the normal to the involute.
In other words, normal at any point of an involute is a tangent to the circle. Now, let O1 and O2 be the fixed centres of the two base circles as shown in Fig.(b).
Let the corresponding involutes AB and A'B' be in contact at point Q. MQ and NQ are normals to the involute at Q and are tangents to base circles.
Since the normal for an involute at a given point is the tangent drawn from that point to the base circle, therefore the common normal MN at Q is also the common tangent to the two base circles.
The common normal MN intersects the line of centres O1O2 at the fixed point P (called pitch point). Therefore the involute teeth satisfy the fundamental condition of constant velocity ratio.
From similar triangles O2 NP and O1 MP,
which determines the ratio of the radii of the two base circles. The radii of the base circles is given by
O1M = O1 P cos φ, and O2N = O2 P cos φ
where φ is the pressure angle or the angle of obliquity.
Also the centre distance between the base circles
If the centre distance is changed, then the radii of pitch circles also changes. But their ratio remains unchanged, because it is equal to the ratio of the two radii of the base circles. The common normal, at the point of contact, still passes through the pitch point. As a result of this, the wheel continues to work correctly. However, the pressure angle increases with the increase in centre distance.
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