M
ACHINE
ELEMENTS
MODULE 2
Bodies in Pure Rolling Contact
Module 2
LEARNING OUTCOMES
1. Compute the speed and diameter of rolling cylinders.
2. Compute the speed and cone angle of rolling cones.
M
ACHINE
ELEMENTS
PURE ROLLING CONTACT
Consists of such a relative motion of two lines or surfaces that the
consecutive points or elements of one come successively into contact with
those of the other in their order. There is no slipping between two
surfaces which have pure rolling contact; that is, all points in contact
have the same linear speed.
M
ACHINE
ELEMENTS
CYLINDERS IN PURE ROLLING CONTACT
❑ External Contact
✓ Rotation is in opposite
direction
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝐴 (𝜔𝐴 ) = 2𝜋𝑅N
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝐵 (𝜔𝐵 ) = 2𝜋𝑅1 𝑁1
𝑆𝑝𝑒𝑒𝑑 𝑅𝑎𝑡𝑖𝑜 𝑀 =
𝑁 𝑅1
=
𝑁1
𝑅
𝐶𝑒𝑛𝑡𝑒𝑟 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐶
𝐶 = 𝑅 + 𝑅1
2𝐶 = 𝐷 + 𝐷1
M
ACHINE
ELEMENTS
CYLINDERS IN PURE ROLLING CONTACT
❑ Internal Contact
✓ Rotation is in same
direction
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝐴 (𝜔𝐴 ) = 2𝜋𝑅N
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝐵 (𝜔𝐵 ) = 2𝜋𝑅1 𝑁1
𝑆𝑝𝑒𝑒𝑑 𝑅𝑎𝑡𝑖𝑜 𝑀 =
𝑁 𝑅1
=
𝑁1
𝑅
𝐶𝑒𝑛𝑡𝑒𝑟 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐶
𝐶 = 𝑅 − 𝑅1
2𝐶 = 𝐷 − 𝐷1
M
ACHINE
ELEMENTS
SAMPLE PROBLEM 1
A cylinder 24 in. in diameter on shaft V drives by pure rolling contact another cylinder on shaft T.
Has an angular speed of 600 radians per minute. Shaft T turns 148.25 rpm in the opposite
direction.
a. Calculate the diameter of cylinder of shaft T and the distance between the axes of the two
shafts.
b. Calculate the diameter of cylinder of shaft T and the distance between the axes of the two
shafts if they turn in the same direction.
M
ACHINE
ELEMENTS
SAMPLE PROBLEM
M
ACHINE
ELEMENTS
CONES IN PURE ROLLING CONTACT
❑ External Contact
✓ Rotation is in opposite
direction
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝐴 (𝜔𝐴 ) = 2𝜋𝑅N
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝐵 (𝜔𝐵 ) = 2𝜋𝑅𝑁1
𝑆𝑝𝑒𝑒𝑑 𝑅𝑎𝑡𝑖𝑜 𝑀 =
𝑁 𝑅1
=
𝑁1
𝑅
𝑆ℎ𝑎𝑓𝑡 𝐴𝑛𝑔𝑙𝑒 𝜃 = 𝛼 + 𝛽
M
ACHINE
ELEMENTS
CONES IN PURE ROLLING CONTACT
𝑁 𝑠𝑖𝑛𝛽
𝑠𝑖𝑛𝛽
𝑠𝑖𝑛𝛽
=
=
=
𝑁1 𝑠𝑖𝑛𝛼 𝑠𝑖𝑛(𝜃 − 𝛽) 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝛽 − 𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝛽
𝑠𝑖𝑛𝛽
(
)
𝑐𝑜𝑠𝛽
𝑁
𝑡𝑎𝑛𝛽
=
=
=
𝑁1 𝑠𝑖𝑛𝜃 − 𝑐𝑜𝑠𝜃 ( 𝑠𝑖𝑛𝛽 ) 𝑠𝑖𝑛𝜃 − 𝑐𝑜𝑠𝜃 𝑡𝑎𝑛𝛽
𝑐𝑜𝑠𝛽
∴
tan 𝛽 =
sin 𝜃
(𝑁1 Τ𝑁) + cos 𝜃
∴
tan 𝛼 =
sin 𝜃
(𝑁Τ𝑁1 ) + cos 𝜃
M
ACHINE
ELEMENTS
CONES IN PURE ROLLING CONTACT
❑ Internal Contact
✓ Rotation is in the
same direction
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝐴 (𝜔𝐴 ) = 2𝜋𝑅N
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝐵 (𝜔𝐵 ) = 2𝜋𝑅𝑁1
𝑆𝑝𝑒𝑒𝑑 𝑅𝑎𝑡𝑖𝑜 𝑀 =
𝑁 𝑅1
=
𝑁1
𝑅
𝑆ℎ𝑎𝑓𝑡 𝐴𝑛𝑔𝑙𝑒 𝜃 = 𝛼 − 𝛽
tan 𝛽 =
sin 𝜃
(𝑁1 Τ𝑁) − cos 𝜃
tan 𝛼 =
sin 𝜃
𝑐𝑜𝑠𝜃 − (𝑁Τ𝑁1 )
M
ACHINE
ELEMENTS
SAMPLE PROBLEM 2
A turns 100 rpm and B 150 rpm. They are connected by rolling cones. Calculate the cone
angle of each cone. If the base of Cone on A is 3 inches from the vertex, calculate the
diameters of both cones.
M
ACHINE
ELEMENTS