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Fundamentals of Stress and Vibration 1. Mathematics for Structural mechanics
[A Practical guide for aspiring Designers / Analysts]
If the involute were to originate from the horizontal instead of an arbitrary angle (α), meaning

(α = 0) , then, the equations for (x) and (y) are given as:

x = R cosθ + R θ sinθ and y = R sinθ − R θ cos θ
b
b
b
b
The coordinates (x, y) go to form the expression for the involute curve.

ͳǤͳͳǤ͵ ‘‰ƒ”‹–Š‹ ’‹”ƒŽ

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ƒ••Ž‡•• •’”‹‰ †ƒ’‡” •›•–‡Ǥ

[Fig 1.63: Logarithmic spiral]

The rate of change of radius with respect to the angle (θ) is proportional to the instantaneous
radius (r).

dr dr dr
Mathematically, this is given as: ∝ r or = cr or = c dθ - - - - (1.77)
dθ dθ r
To find a relationship between (r) and (θ), let us integrate equation (1.77) on both sides.

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 61