Torsional equation , Mechanical Engineering

Torsional equation:

Derive the Torsional equation T/J = Π  /R = /L

Or

Derive an expression for the shear stress in shaft subjected to a torque.

Sol.: Assume,

T  = Maximum twisting torque or twisting moment

D = Diameter of shaft

R = Radius of shaft

J  = Polar moment of Inertia

τ= Maximum Permissible Shear stress (Fixed for given material)

G = Modulus of rigidity

θ= Angle of twist (Radians) = angle D'OD L  = Length of shaft.

?= Angle D'CD = Angle of Shear strain

 

2090_Torsional equation.png

Than Torsion equation is: T/J =   τ/R = G. θ /L

Let the shaft is subjected to a torque or twisting moment 'T'. And hence every C.S. of this shaft will be subjected to shear stress.

Now distortion at the outer surface = DD'

Shear strain at outer surface = Distortion/Unit length tan?         = DD'/CD

i.e. shear stress at the outer surface (tan? ) = DD'/or  = DD'/L           ...(i)

Now DD' = R.θ              or      ?= R .  θ /L    ...(ii)

Now G = Shar stress induced/shear strain produced

G =   τ/(R. θ /L);

or;                                             τ/R = G. θ /L                              ...(A);

This equation is called Stiffness equation.

Hear G,  θ , L are constant for a given torque 'T'. That is proportional to R

If τ r  be the intensity of shear stress at any layer at a distance 'r' from canter of the shaft, then;

1566_Torsional equation1.png

Now from equation (ii) T = ( τ/R)   J

or                                              τ/R = T/J;                                   ...(B)

This equation is called as strength equation

The combined equation A and B; we get

T/J =   τ/R = G.  τ/L

This equation is called as Torsion equation.

From the relation             T/J =   τ/R ; We have  T =   τ.J/R =  τ .ZP

For the given shaft I and R are constants and IP/R is thus constant and is called as POLAR MODULUS(ZP). of the shaft section.

Polar modulus of section is thus measure of strength of shaft in the torsion.

TORSIONAL RIGIDITY or Torsional Stiffness (K): = G.J/L = T

Posted Date: 10/23/2012 2:32:37 AM | Location : United States







Related Discussions:- Torsional equation , Assignment Help, Ask Question on Torsional equation , Get Answer, Expert's Help, Torsional equation Discussions

Write discussion on Torsional equation
Your posts are moderated
Related Questions
Differentiate between hot working and cold working process

proof of work being a path function

Determine the economic order quantity: Determine the economic order quantity and the reorder point. Given Annual demand (D) = 2,000 units Average daily demand (d) = 2,00

1.5kw/ton of refrigeration is required to maintain temp of -40c in the refrigerator.if the cycle works on carnot cycle.determine 1.cop,2.sink temp 3. heat rejejected to the sink/to

Find out the safe power transmitted by pulley: A belt drives pulley of 200mm diameter such that ratio of tensions in tight side and slack side is 1.2. If maximum tension in b

Creating 3-d solid models Two basic approaches to create solid models are :    (i)  Constructive solid geometry (CSG or c-rep), and (ii)  Boundary representation (b-rep).

Q. Nozzle Design Loads? Minimum nozzle design loads shall be in accordance. F = Axial force on centre line of nozzle (lbs); Maximum Transverse Force =1.5F F R = Resultan

Define What is Nose Radius? The nose radius is provided to increase finish and strength of the cutting tip of the tool. Small radii will produce smooth finish and are used on t

Torque Transmission : The clutch should be able to transmit the maximum torque of the engine under all conditions. It is usually designed to transmit 125 to 150 percent of the max

• For the ASTM dogbone specimens, the ¾ inch hole specimens, and the notched specimen tabulate the failure load, in pounds. • Show plots of stress and strain for all specimens w