Relationshit Between Angle of Twist and Radius Gear
Power manual shafts — on motors and gearboxes, for case — are subjected to torque loads that result in torsion, or twisting of the shaft about its axis. Similar to structures nether tension or compression, 2 important mechanical properties of shafts under torque loads are shear stress and shear strain.
Stress is a material'due south resistance to an applied force, and strain is the deformation that results from stress. Shear stress and shear strain (which are caused past torsional loads) occur when a force is applied parallel or tangent to an area. Normal stress and normal strain (which are acquired past tension and pinch) occur when a forcefulness is applied normal (perpendicular) to an area.
The torsion, or twist, induced when torque is applied to a shaft causes a distribution of stress over the shaft's cross-sectional area. (Note that this is different from tensile and compressive loads, which produce a compatible stress over the object's cantankerous-section.)
Torque vs. Moment:
Torque is force applied at a distance that causes a modify in angular momentum. A moment is also a forcefulness applied at a distance, but it does not crusade a alter in angular momentum. In other words, torque causes a body to rotate almost an axis, whereas a moment load does non cause rotation.
Shear stress depends on the practical torque, the altitude along the radius of the shaft, and the polar moment of inertia. (Note that polar moment of inertia is a part of geometry and does not depend on the shaft material.)
τ = shear stress (N/m2, Pa)
T = applied torque (Nm)
r = distance along radius of shaft (1000)
J = polar moment of inertia (m4)
When shear stress is beingness measured at the outer edge of the shaft, the letter "c" is sometimes used in place of "r" to betoken that the radius is at its maximum.
The polar moment of inertia (aka second polar moment of expanse) for a solid cylinder is given as:
The corporeality of shear strain is adamant by the angle of twist, the distance along the radius of the shaft, and the length of the shaft. The equation for shear strain is valid in both the elastic and plastic ranges of the material. It's important to annotation that shear strain and shaft length are inversely proportional: the longer the shaft, the lower the shear strain.
γ = shear strain (radians)
r = distance along radius of shaft (m)
θ = angle of twist (radians)
L = length of shaft (grand)
As well note that at the center of the shaft (r = 0), there is no shear strain (γ = 0). Conversely, shear strain is at its maximum value (γ = γmax) at the outer surface of the shaft (r = rmax).
Like to the modulus of elasticity (E) for a body nether tension, a shaft in torsion has a belongings known as the shear modulus (too referred to as the modulus of elasticity in shear, or the modulus of rigidity). The shear modulus (Thousand) is the ratio of shear stress to shear strain. Like the modulus of elasticity, the shear modulus is governed by Hooke's Law: the relationship between shear stress and shear strain is proportional upwardly to the proportional limit of the fabric.
OR 
G = shear modulus (Pa)
Note that the procedure of yielding for a shaft in torsion is not as straightforward every bit the procedure of yielding for a structure in tension. This is because bodies subjected to tension experience a constant stress across their entire cantankerous-department. Therefore, yielding occurs simultaneously beyond the unabridged body.
Equally described above, for a shaft in torsion, the shear stress varies from nada at the center of the shaft (the centrality) to a maximum at the surface of the shaft. When the surface reaches the elastic limit and begins to yield, the interior will withal exhibit elastic beliefs for some additional corporeality of torque. At some betoken, the practical torque causes the shaft to enter its plastic region, where the strain increases while torque is constant. Only when the torque causes fully plastic behavior does the entire cross-section yield.
Feature paradigm credit: R+W America
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