Page 599 - Dynamics of Mechanical Systems

0593_C17_fm Page 580 Tuesday, May 7, 2002 7:12 AM

580 Dynamics of Mechanical Systems

Let Q be the intersection point of the Y-axis and the base circle. Q is then at the origin of
the involute curve. The arc TQ then has the same length as the generating line segment
TP. Hence, we have:

TP = TQ = r β (17.4.7)
b
The angle β may be considered as the unwrapping angle. It is often helpful to express the
coordinates of P in terms of β. From Figure 17.4.5 we have:

r = OP = OT + TP = r + β = r 1+ ( β 2 ) (17.4.8)
2
2
2
2
2
2
2 2
r
b b b
or
+
r = r 1 β 2 (17.4.9)
b
Also, from Figure 17.4.5 we can obtain the coordinates x and y as follows:
=
+
OP OT TP (17.4.10)
But, OT may be expressed as:
OT = r cosγ n + r sin γ n (17.4.11)
b x b y

and as:
OT = r sinβ n + r cosβ n (17.4.12)
b x b y
Similarly, TP may be expressed as:

TP = r β ( −sin γ n + cos γ n ) (17.4.13)
b x y

and as:
TP = r β ( −cos β n + sin β n ) (17.4.14)
b x y
Then, from Eq. (17.4.10) we have:

OP = r sinβ n + r cosβ n − r cosβ n + r sinβ n
β
β
b x b y b x b y
= ( r sinββ n ) + ( r cosβ + sinβ n ) β (17.4.15)
− cosβ
b x b y
= x n + y n
x y
or

− cosβ
β
x = (sinββ ) and y = (cosβ + sinβ ) (17.4.16)
r
r
b b