Angle of Twist For a homogeneous shaft of constant area and G over a length L, under a torque T, the angular displacement of one end relative to the other is 1.5.40 For a shaft consistin
Trang 11-84 Section 1
where J = the polar moment of inertia of the cross-sectional area; for a solid circle, J = πr4/2; for a tube,
Power Transmission
The power P transmitted by a shaft under torque T and rotating at angular velocity ω is
where ω = 2πf; f = frequency of rotation or number of revolutions per second.
Angle of Twist
For a homogeneous shaft of constant area and G over a length L, under a torque T, the angular
displacement of one end relative to the other is
(1.5.40)
For a shaft consisting of segments with various material and/or geometric properties, under several different torques in each, the net angular displacement is calculated from the vector sum of the individual twists,
(1.5.41)
The right-hand rule is used for a sign convention for torques and angles: both T and φ are positive, with the thumb pointing outward from a shaft and the fingers curling in the direction of torque and/or rotation,
as in Figure 1.5.26 Note that regardless of the number of torques applied to a shaft at various places along its length, there is only one torque at a given cross section, and this torque is a constant in that segment of the shaft (until another external torque is encountered, requiring a different free-body diagram)
Inelastic Torsion
A shaft may plastically deform under an increasing torque, yielding first in its outer layers and ultimately throughout the cross section Such a shaft is analyzed by assuming that the shear strains are still linearly
FIGURE 1.5.25 Shear stress distributions in a shaft.
FIGURE 1.5.26 Right-hand rule for positive torque and angle.
J=( / )( )π 2 r04 −r i4
φ = TL
JG
φ =∑T L
J G
i i
i i