Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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TORSION OF SHAFTS

TORSION OF CIR CULAR  SHAFTS
Theory of Pure Torsion

Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

Torsional Moment of Resistance:

Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

ts= Shear intensity at the surface of the shaft
R = Radius of shaft
G = Modulus of rigidity of shaft material
l = Length of shaft
q = Angular movement due to strain in length of the shaft
T = total moment of resistance offered by the cross-section of the shaft
I|p= Polar moment of Inertia of the section of the shaft

Assumptions in the theory of pure torsion: 

  • The material of the shaft is uniform throughout. 
  • Twist along the shaft is uniform. 
  • Shaft is of uniform circular section throughout, which may be hollow or solid. 
  • Cross section of the shaft, which are plane before  twist remain plane after twist. 
  • All radii which are straight  before twist remain straight after twist:

Polar modulus:

Polar modulus = 

Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

The greatest twisting moment which a given shaft section can resist = Max. permissible shear stress × Polar Modulus

T = ts Zp 

  • For solid shaft,

Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
for hollow shaft,  
 Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Where, do= outer diameter
di = inner diameter
Torsional rigidity:

Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering 
Where, G = rigidity modulus
Ip = Polar moment of Inertia
The quantity GIp is called torsional rigidity. It is the torque required to produce a twist of 1 radian per unit length of the shaft.

Power Transmitted by a shaft:
 Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
= Torque × angle turned per second
Where, P = Power transmitted (kW)
N = rotation per minute (rpm)
T = mean torque (kNm)

SHAFTS IN SERIES AND SHAFTS IN PARALLEL
(a) shafts in series:
• Torque T will be same for both the shafts.
• The twists 1q and 2q will be different for both the shafts.
Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical EngineeringTorsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Where, T = Torque
G1, G2 = Modulus of rigidity for shafts 1& 2
l1, l2 = length of shaft 1&2

Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
if l1 = l2 G1 = G2 them
Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Where, q1,q2 = angleof twi,st
Ip1, Ip2 = polar moments of inertia

(b) Shafts in parallel:

  •  In this case applied torque T is distributed to two shafts.

Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  • T = T1 + T2
  • The angle of twist will be same for each shaft,

q1 = q2 = q

  • T = T1 + T2 =

Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
COMPARISON BETWEEN SOLID AND HOLLOW SHAFTS
Let hollow shaft and solid shafts have same material and length.
D0 = external diameter of hollow shaft
Di = nD= Internal diameter of hollow shaft
D = Diameter of the solid shaft

Case (i): When the hollow and solid shafts have the same torsional strength.

  • In this case polar modulus section of two shafts would be equal.
  • Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
  • USE: %Saving in weight can be calculated for same torsional strength.

Case (ii): When the hollow and solid shafts are of equal weights.

  • In this case torsional strength is compared.

Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  • USE: ratio of strength for same weight can be calculated.

Case (iii) : When the diameter of solid shaft is equal to the external diameter of the hollow shaft.

  • Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

SHEAR AND TORSIONAL RESILIENCE
Shear resilience:
Let t = shear stress intensity at faces of a square block

  • Strain energy stored per unit volume

Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering= (uniform through the section)
Where G = rigidity modulus.

Torsional resilience:

  • In this case shear stress due to torsion varies uniformly form zero at the axis to the maximum value ts at the surface.
  • Strain energy stored, per unit volume

Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  • for hollow shaft,

Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Where, D = outer diameter of hollow shaft
d = internal diameter of hollow shaft
 

The document Torsion of Shafts - Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering is a part of the Mechanical Engineering Course Mechanical Engineering SSC JE (Technical).
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