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Document Title
Fundamentals of Stress and Vibration Chapter Title
[A Practical guide for aspiring Designers / Analysts] 2. Engineering Mechanics

Putting (b = 1) in the exponents of length, we get: (a + d = 4)

Subtracting (a + c = 2) from (a + d = 4), we get: (d – c = 2), by where, (d = 2 + c) and also (a = 2 – c).

Rewriting equation (2.4) in terms of exponent ‘c’, we get:

2
c
c
T = v (2−c) ρ N r (2+c) = T = v 2−c ρ Nr r

(Nr) is equivalent to the tangential velocity of the blade (ωr). Therefore, combining tangential
velocity (Nr) with velocity (v), assuming that air hugs the aero foil perfectly (no slip), we get:
T = v 2−c ρ Nr r = T = v 2−c ρ v r = T = v ρ r
2
c
2
2
c
2
Recasting the above expression in terms of RPM (N), we get:
Or T = (Nr) 2−c ρ Nr r = T = N r ρ
c
2
2 4
Further, for the power generated, we get:
2
3
Power = Thrust ∗ Velocity = P = T ∗ v = P = v ρ r
The above expression for power (P) can also be written as: P = N ρ r
5
3
Where, [N (RPM) * r (radius) = v (velocity)]

This clearly establishes that, the typical fan power is equal to cube of the RPM.

Example 3: let us relate power to torque and angular velocity.

Mathematically, power is given by: P = T ω - - - - (2.6)
a
b

Where, (a and b) are exponents of their respective terms.

Parameters Units Dimensions

J 2 −3
Power (P) Joules/second newton meter/second ( T )
s

Torque (T) newton meter(Nm) (ML T )
2 −2
1 −1
Angular velocity (ω) radians/second T
t

Page 6 QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,