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Fundamentals of Stress and Vibration
[A Practical guide for aspiring Designers / Analysts] 1. Mathematics for Structural mechanics
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[Fig 1.62: Involute]
Let the involute originate at (O) and (α) be the initial lean angle of the line joining (O) to the center
of the base circle. It should be noted that, (O’O), which is the arc length (R θ), is equal to the length
b
of the tangent (O’O’’), that is, O O = R θ ≅ O O
′
′′
′
b
Let (x,y) be a point on the involute generated by a tangent drawn to the base circle at (O’).
Let (O O’’) subtend an angle (ϕ) at the center of the base circle.

The radius at (O’) makes an angle (θ + α) with the horizontal.

Let us start by computing the coordinates (x , y ). From the [Fig 1.62], we have:
1
1
x = R cos θ + α and y = R sin θ + α
1
b
b
1
Similarly (x, y) is given as:
x = R cos θ + α + R θ sin θ + α and y = R sin θ + α − R θ cos θ + α
b
b
b
b

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,
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