ENT345 Mechanical Components Design Dr. Haftirman Mechanical Engineering Program School of Mechatronic Engineering Universiti Malaysia Perlis UniMAP Page 1 1) LOAD AND STRESS ANALYSIS i. Principle stress ii. The maximum shear stress iii. The endurance strength of shaft. 1) Problem 3- 71 A countershaft carrying two-V belt pulleys is shown in the figure. Pulley A receives power from a motor through a belt with the belt tensions shown. The power is transmitted through the shaft and delivered to the belt on pulley B. Assume the belt tension on the loose side at B is 15 percent of the tension on the tight side. a) Determine the tensions in the belt on pulley B, assuming the shaft is running at a constant speed. b) Find the magnitudes of the bearing reaction forces, assuming the bearings act as simple supports. c) Draw shear-force and bending moment diagrams for the shaft. If needed, make one set for the horizontal plane and anther set for the vertical plane. d) At the point of maximum bending moment, determine the bending stress and the torsional shear stress. e) At the point of maximum bending moment, determine the principal stresses and the maximum shear stress. Solution Assume the belt tension on the loose side at B is 15 percent of the tension on the tight side. T 2 = 0.15 T 1
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ENT345 Mechanical Components Design
Dr. Haftirman Mechanical Engineering Program School of Mechatronic Engineering Universiti Malaysia Perlis UniMAP Page 1
1) LOAD AND STRESS ANALYSIS
i. Principle stress
ii. The maximum shear stress
iii. The endurance strength of shaft.
1) Problem 3- 71
A countershaft carrying two-V belt pulleys is shown in the figure. Pulley A
receives power from a motor through a belt with the belt tensions shown.
The power is transmitted through the shaft and delivered to the belt on
pulley B. Assume the belt tension on the loose side at B is 15 percent of
the tension on the tight side.
a) Determine the tensions in the belt on pulley B, assuming the shaft is
running at a constant speed.
b) Find the magnitudes of the bearing reaction forces, assuming the
bearings act as simple supports.
c) Draw shear-force and bending moment diagrams for the shaft. If
needed, make one set for the horizontal plane and anther set for the
vertical plane.
d) At the point of maximum bending moment, determine the bending
stress and the torsional shear stress.
e) At the point of maximum bending moment, determine the principal
stresses and the maximum shear stress.
Solution
Assume the belt tension on the loose side at B is 15 percent of the tension on the
tight side. T2 = 0.15 T1
ENT345 Mechanical Components Design
Dr. Haftirman Mechanical Engineering Program School of Mechatronic Engineering Universiti Malaysia Perlis UniMAP Page 2
a) Determine the tensions in the belt on pulley B, assuming the shaft
is running at a constant speed.
∑𝑻 = 𝟎
(𝟑𝟎𝟎 − 𝟒𝟓)𝑵(𝟏𝟐𝟓)𝒎𝒎 + (𝑻𝟐 − 𝑻𝟏)𝑵(𝟏𝟓𝟎)𝒎𝒎 = 𝟎
𝟑𝟏𝟖𝟕𝟓𝑵𝒎𝒎 + (𝟎. 𝟏𝟓𝑻𝟏 − 𝑻𝟏)𝑵(𝟏𝟓𝟎)𝒎𝒎 = 𝟎
𝟑𝟏𝟖𝟕𝟓𝑵𝒎𝒎 − 𝟏𝟐𝟕. 𝟓𝑻𝟏 = 𝟎
ENT345 Mechanical Components Design
Dr. Haftirman Mechanical Engineering Program School of Mechatronic Engineering Universiti Malaysia Perlis UniMAP Page 3
𝑻𝟏 = 𝟐𝟓𝟎𝑵𝒎𝒎 On pulley B
𝑻𝟐 = (𝟎. 𝟏𝟓)𝟐𝟓𝟎𝑵𝒎𝒎 = 𝟑𝟕. 𝟓 𝑵𝒎𝒎
𝑻 = 𝑻𝟏 + 𝑻𝟐 = 𝟐𝟓𝟎 + 𝟑𝟕. 𝟓 = 𝟐𝟖𝟕. 𝟓 𝑵𝒎𝒎
b) Find the magnitudes of the bearing reaction forces, assuming the
bearings act as simple supports.
Y 31.82 N 45 N
31.82N T2=0.15 T1
X 212.132 N 300 N
A 45˚ B
Z 212.132 N
T1
∑𝑀𝑜𝑦 = 0
𝟑𝟒𝟓𝒄𝒐𝒔𝟒𝟓°(𝟑𝟎𝟎) − 𝟐𝟖𝟕. 𝟓(𝟕𝟎𝟎) + 𝑹𝑪𝒛(𝟖𝟓𝟎) = 𝟎
𝑹𝑪𝒛 = −𝟏𝟓𝟎. 𝟕 𝑵
∑𝐹𝑧 = 0
𝑹𝑶𝒛 − 𝟑𝟒𝟓𝒄𝒐𝒔𝟒𝟓° + 𝟐𝟖𝟕. 𝟓 − 𝟏𝟓𝟎. 𝟕 = 𝟎
𝑹𝑶𝒛 = 𝟏𝟎𝟕. 𝟐 𝑵
∑𝑀𝑜𝑧 = 0
𝟑𝟒𝟓𝒔𝒊𝒏𝟒𝟓°(𝟑𝟎𝟎) + 𝑹𝑪𝒚(𝟖𝟓𝟎) = 𝟎
𝑹𝑪𝒚 = −𝟖𝟔. 𝟏𝟎 𝑵
ENT345 Mechanical Components Design
Dr. Haftirman Mechanical Engineering Program School of Mechatronic Engineering Universiti Malaysia Perlis UniMAP Page 4
∑𝐹𝑦 = 0
𝑹𝑶𝒚 + 𝟑𝟒𝟓𝒄𝒐𝒔𝟒𝟓° − 𝟖𝟔. 𝟏𝟎 = 𝟎
𝑹𝑶𝒛 = −𝟏𝟓𝟕. 𝟗 𝑵
y 243.952
O A B C X
300 mm 400 mm 150 mm
-157.9 86.1
ENT345 Mechanical Components Design
Dr. Haftirman Mechanical Engineering Program School of Mechatronic Engineering Universiti Malaysia Perlis UniMAP Page 5
Z 243.952 150.7 N
O A B C X
300 mm 400 mm 150 mm
107.2 287.5
c) Draw shear-force and bending moment diagrams for the shaft. If
needed, make one set for the horizontal plane and anther set for
the vertical plane.
y 243.952
O A B C X
300 mm 400 mm 150 mm
-157.9 86.1
V (N)
86.1
O
-157.9
ENT345 Mechanical Components Design
Dr. Haftirman Mechanical Engineering Program School of Mechatronic Engineering Universiti Malaysia Perlis UniMAP Page 6
M (Nm)
O x
-47.37
Z 243.952 150.7 N
O A B C X
300 mm 400 mm 150 mm
107.2 287.5
V (N)
136.8
O
-107.2 150.7
ENT345 Mechanical Components Design
Dr. Haftirman Mechanical Engineering Program School of Mechatronic Engineering Universiti Malaysia Perlis UniMAP Page 7
M (Nm) 22.56
O x
-32.16
d) At the point of maximum bending moment, determine the bending
stress and the torsional shear stress.
The critical location is at A where both planes have the maximum
bending moment. Combining the bending moments from the two
planes;
𝑀 = √(−47.37)2 + (−32.16)2 = 57.26 𝑁𝑚
The torque transmitted through the shaft from A to B is
𝑇 = (300 − 45)(0.125) = 31.88 𝑁𝑚
The bending stress and the torsional stress are both maximum are on
the outer surface of a stress element.
𝜎 =𝑀𝑐
𝐼=
32𝑀
𝜋𝑑3=
32(57.26)
𝜋(0.020)3= 72.9𝑥106𝑃𝑎 = 72.9𝑀𝑃𝑎
𝜏 =𝑇𝑟
𝐽=
16𝑇
𝜋𝑑3=
16(31.88)
𝜋(0.020)3= 20.3𝑥106𝑃𝑎 = 20.3𝑀𝑃𝑎
ENT345 Mechanical Components Design
Dr. Haftirman Mechanical Engineering Program School of Mechatronic Engineering Universiti Malaysia Perlis UniMAP Page 8
e) At the point of maximum bending moment, determine the principal
stresses and the maximum shear stress.
𝜎1, 𝜎2 =𝜎𝑥
2± √(
𝜎𝑥
2)2
+ (𝜏𝑥𝑦)2
=72.9
2± √(
72.9
2)2
+ (20.3)2 =
𝜎1 = 78.2𝑀𝑃𝑎
𝜎2 = −5.27𝑀𝑃𝑎
𝜏𝑚𝑎𝑥 = √(𝜎𝑥
2)2
+ (𝜏𝑥𝑦)2
= √(72.9
2)2
+ (20.3)2 = 41.7 𝑀𝑃𝑎
ENT345 Mechanical Components Design
Dr. Haftirman Mechanical Engineering Program School of Mechatronic Engineering Universiti Malaysia Perlis UniMAP Page 9
2) Problem 3-73
A gear reduction unit uses the countershaft shown in the figure. Gear A
receives power from another gear with the transmitted force FA applied at
the 20˚ pressure angle as shown. The power is transmitted through the shaft
and delivered through gear B through a transmitted force FB at the pressure
angle shown.
a) Determine the force FB, assuming the shaft is running at a constant
speed.
b) Find the magnitudes of the bearing reaction forces, assuming the
bearings act as simple supports.
c) Draw shear-force and bending-moment diagrams for the shaft. If
needed, make one set for the horizontal plane and another set for the
vertical plane.
d) At the point of maximum bending moment, determine the bending
stress and the torsional shear stress.
e) At the point of maximum bending moment, determine the principal
stresses and the maximum shear stress.
ENT345 Mechanical Components Design
Dr. Haftirman Mechanical Engineering Program School of Mechatronic Engineering Universiti Malaysia Perlis UniMAP Page 10
FB
Y FA
20˚ 25˚
Z
a) Determine the force FB, assuming the shaft is running at a constant
speed.
∑𝑻 = 𝟎
−𝟏𝟏𝟎𝟎𝟎(𝒄𝒐𝒔𝟐𝟎°)(𝟑𝟎𝟎) − 𝑭𝑩(𝒄𝒐𝒔𝟐𝟓°)(𝟏𝟓𝟎) = 𝟎
𝑭𝑩 = 𝟐𝟐𝟖𝟏𝟎 𝑵
b) Find the magnitudes of the bearing reaction forces, assuming the