Page 72 - Analysis and Design of Machine Elements

Analysis and Design of Machine Elements
50
Solution:
Steps Computation Results Units

1. The number From Eq. (2.35) N = 0.01 × 10 7
1
of cycles ( ) m
7
to failure N i N = N 0 −1 N = 0.0751 × 10
2
i
at each stress i 7
level i N = 0.2497 × 10
1
We have
( ) m
( 300 ) 9
7
N = N 0 −1 = 10 × = 0.01 × 10 7
1
1 500
( ) m
( 300 ) 9
7
N = N −1 = 10 × = 0.0751 × 10 7
2 0
400
2
( ) m
( 300 ) 9
7
N = N 0 −1 = 10 × = 0.2497 × 10 7
3
3 350
2. The number From Miner’sruleinEq. (2.32) n = 4.21 × 10 4
3
of remaining n n n
cycles N 1 + N 2 + N 3 = 1
for 350 MPa 1 2 3
( )
10 4 10 5 n 3
4 × + + = 1
0.01 × 10 7 0.0751 × 10 7 0.2497 × 10 7
Therefore
7
n = 0.2497 × 10 ×
3 ( )
1 10 4 10 5 4
− − = 4.21 × 10
4 0.01 × 10 7 0.0751 × 10 7
2.4 Contact Strength
Previous discussions focus on strength analyses within an element to prevent body fail-
ure, such as yielding and fatigue fracture. This section deals with surface strength or
contact strength in localized regions, with an aim to prevent surface failure.
2.4.1 Hertzian Contact Stresses
Contact is one of the most common methods of transmitting forces in a machine. When
elements make contact with each other, a pair of equal and opposite forces generate
according to the action-reaction law. Typical examples are the force transmission
between a pair of meshing gears or rolling contact bearings. Theoretically, the contact
between curved surfaces of elements is a point or a line. When curved elements are
loaded, contact areas deviate elastically from the basic surface curvatures, high contact
stresses are correspondingly developed within small contact areas.
Contact stress (also called Hertzian contact stress) refers to the localized stress that
develops as two curved surfaces come in contact and deform slightly under imposed