Angle of twist is zero on the shaft, Mechanical Engineering

Angle of twist is zero on the shaft:

A shaft 10 m long and 100 mm diameter during its length is fixed at the ends and is subjected to two opposite torques of 8 kN-m and 10 kN-m at a distance of 3 m and 8 m from one end. Discover the fixing torques at the ends. What is the maximum angle of twist and the maximum shear stress in the shaft? At what section the angle of twist is zero on the shaft?

Take E = 80 GPa.

Solution

Σ T = 0

⇒  TA  - 8 + 10 + TD  = 0

∴          TA  + TD  = - 2

G = 80 GPa = 0.8 × 1011 N/m2              -------------- (1)

d = 100 mm = 0.1 m

 J =  (π /32)d 4 = (π/32) × (0.1)4  = 9.8 × 10- 6 m4

R = d /2= 100/2 = 0.05 m ,   ZP = 1.96 × 10- 4 m3

1451_Angle of twist is zero on the shaft.png

Figure (a)

2003_Angle of twist is zero on the shaft1.png

Figure (b) : Line Diagram

 

Portion AB

θ 1 = (TA . l )/GJ = (TA × 3 )/GJ                         ---------- (2)

Portion BC

θ2 = ((TA  - 8) × 5 )/     GJ               -------- (3)

Portion CD

θ3 = ((TA - 8 + 10) × 2)/ GJ = ((TA + 2) × 2)/ GJ                      ---------- (4)

Angle of twist at D co.r.t A = 0

(TA × 3 )/ G J + ((TA  - 8) × 5 )/ GJ + ((TA  + 2) × 2)/ GJ = 0

∴          TA  = 3.6 kN-m                                       ----------- (5)

From Eqs. (1) and (5),

3.6 + TD = - 2

TD = - 5.6 kN-m = 5.6 kN-m    (  )

Angle of Twist

θ1 = ((3.6 × 103 ) × 3) / ((0.8 × 1011 ) (9.8 × 10- 6 ))

= 1.4 × 10- 2 radians

θ2 =      ((3.6 - 8) × 103 × 5 )/(0.8 × 1011 ) (9.8 × 10- 6 )= - 2.8 × 10- 2 radians

θ3 =      (3.6 + 2) × 103  × 2 /((0.8 × 1011 ) (9.8 × 10- 6 ))= 1.4 × 10- 2 radians

θAB  = θ1 = 1.4 × 10- 2 radians

θ AC  = θ1 + θ2  = 1.4 × 10- 2 - 2.8 × 10- 2 = - 1.4 × 10- 2 radians

θAD  = θAC  + θ3 = - 1.4 × 10- 2

Shear Stress

 + 1.4 × 10- 2  = 0

τ max     = (T/ J) . R =  T/ Z P

Portion AB

τmax = 3.6 × 103 / (1.96 × 10- 4) = 18.4 × 106 N/m2  = 18.4 N/mm2

Portion BC

τ max = (3.6 - 8) × 103/ (1.96 × 10- 4)

= - 22.4 × 106 N/m2  = - 22.4 N/mm2

Portion CD

τ max      = (3.6 + 2) × 103/1.96 × 10- 6 = 28.6 × 106 N/m2  = 28.6 N/mm2

Zero Angle of Twist

It occurs between B and C

θ1 = 1.4 × 10- 2 radians

243_Angle of twist is zero on the shaft2.png

1682_Angle of twist is zero on the shaft3.png

∴ x = 2.5 m from B.

Posted Date: 1/21/2013 5:14:58 AM | Location : United States







Related Discussions:- Angle of twist is zero on the shaft, Assignment Help, Ask Question on Angle of twist is zero on the shaft, Get Answer, Expert's Help, Angle of twist is zero on the shaft Discussions

Write discussion on Angle of twist is zero on the shaft
Your posts are moderated
Related Questions
what is cone friction?


Find out the centroid of an area of a semicircle: Find out the centroid of an area of a semicircle illustrated in Figure. Solution You shall note that it is not p

Q.  A wooden beam of the rectangular cross section is subjected to bending moment of 5KNm. If the depth of section is to be double the breadth and stress in wood is not to exceed 6

what are the different method for worm gear optimization

IGES application protocol: In the year 1977, the three authors of The Initial Graphics Exchange Specification were identified collectively for their contributions to the growt

Free Body Diagram: while a body or link is isolated, the forces (active and reactive) are also illustrated on it. It is called free body diagram of the body. The essential condit


High Carbon Tool Steels Tools are implements such are employed to shape, cut or deform other materials. They are largely made in steel, although other alloys have also been i

Clutch free play: Check for free play of the clutch lever, it should be within 10-20 mm.