Mechanics of Solids 2011 Part 7 pot

1-84 Section 1 where J = the polar moment of inertia of the cross-sectional area; for a solid circle, J = πr 4 /2; for a tube, . Power Transmission The power P transmitted by a shaft under torque T and rotating at angular velocity ω is P = Tω (1.5.39) 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 1.5.25Shear stress distributions in a shaft. FIGURE 1.5.26Right-hand rule for positive torque and angle. Jrr i =−(/)( )π2 0 44 φ= TL JG φ= ∑ TL JG ii ii . 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. 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. 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