Page 167 - Marks Calculation for Machine Design

P1: Rakesh
January 4, 2005
Brown˙C03
Brown.cls
3
where (ρ) is the density of the disk in (slugs/ft ) or (kg/m ) and (ω) is the angular velocity
of the disk in (rad/s). 14:16 ADVANCED LOADINGS 3 149
Notice that the thickness (t) is not in any of these equations, as no variation is allowed
perpendicular to the plane of rotation.
The tangential stress (σ t ) is a maximum at the inside radius (r i ) of the disk, given in
Eq. (3.44) as
2
3 + ν 1 − ν r
max i
σ t = σ o 1 + (3.44)
4 3 + ν r 2
o
where the radial stress (σ r ) is zero.
√
The radial stress (σ r ) is a maximum at a radius ( r i r o ), given in Eq. (3.45) as
3 + ν 2
max r i
σ r = σ o 1 + (3.45)
8 r o
where the tangential stress (σ t ) is given by Eq. (3.46) as

√ 3 + ν r 2
r i r o 1 − ν r i i
σ t = σ o 1 + 2 + (3.46)
8 r 2
3 + ν r o
o
As these two stresses are both positive, and are the principal stresses (σ 1 ) and (σ 2 ), and
as rotating disks are usually made of ductile materials, the distortion-energy theory will be
the most accurate predictor of whether the design is safe. Therefore, the factor-of-safety for
a rotating thin disk is given by Eq. (3.47) as
1/2
2 2
σ + σ − σ 1 σ 2 1
1 2
= (3.47)
S y n
where (S y ) is the yield strength of the material.
However, for a stress element at the inside radius (r i ), the principal stress (σ 1 ) will be
the maximum tangential stress (σ t max ) and the principal stress (σ 2 ) will the radial stress
(σ r ) that is zero at the inside radius, making this a uniaxial stress element, and so any of
the theories for ductile materials would apply.
Consider the following example to conclude this section.
U.S. Customary SI/Metric
Example 1. Determine whether the design of Example 1. Determine whether the design of
a thin high-speed saw blade is safe, where a thin high-speed saw blade is safe, where
ω = 1,000 rpm ω = 1,000 rpm
r o = 3ft = 36 in r o = 1m
r i = 1in r i = 2.5 cm = 0.025 m
t = 0.25 in t = 0.6 cm = 0.006 m
3
3
ρ = 15.2 slug/ft (steel) ρ = 7,850 kg/m (steel)
S y = 50 kpsi (steel) S y = 350 MPa (steel)
ν = 0.3 (steel) ν = 0.3 (steel)