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Unformatted text preview: the total energy values for static and dynamic conditions are identical. If the velocity is increased, the impact values are considerably reduced. For further information, see Ref. 10. 10.6.6 Steady and Impulsive Vibratory Stresses For steady vibratory stresses of a weight, W, supported by a beam or rod, the deflection of the bar, or beam, will be increased by the dynamic magnification factor. The relation is given by dynamic = Static x dynamic magnification factor An example of the calculating procedure for the case of no damping losses is - S ^ X i _ ( ^ ) 2 (10.65) where a) is the frequency of oscillation of the load and O) n is the natural frequency of oscillation of a weight on the bar. For the same beam excited by a single sine pulse of magnitude A in./sec 2 and a sec duration, then for t < a a good approximation is S static (A/g) T 1 / a}\ "I <*-. = t J ^ / y [sin - - ^ ( - ) sin ^ J (10.66) \47TOjJ where A/g is the number of g's and o> is TT/a. 10.7 SHAFTS, BENDING, AND TORSION 10.7.1 Definitions TORSIONAL STRESS. A bar is under torsional stress when it is held fast at one end, and a force acts at the other end to twist the bar. In a round bar (Fig. 10.23) with a constant force acting, the straight line ab becomes the helix ad, and a radial line in the cross section, ob, moves to the position od. The angle bad remains constant while the angle bod increases with the length of the bar. Each cross section of the bar tends to shear off the one adjacent to it, and in any cross section the shearing stress at any point is normal to a radial line drawn through the point. Within the shearing proportional limit, a radial line of the cross section remains straight after the twisting force has been applied, and the unit shearing stress at any point is proportional to its distance from the axis. TWISTING MOMENT, T, is equal to the product of the resultant, F, of the twisting forces, multiplied by its distance from the axis, p. RESISTING MOMENT, T r , in torsion, is equal to the sum of the moments of the unit shearing stresses acting along a cross section with respect to the axis of the bar. If dA is an elementary area of the section at a distance of z units from the axis of a circular shaft (Fig. 10.23), and c is the distance from the axis to the outside of the cross section where the unit shearing stress is r, then the unit shearing stress acting on dA is (TZ/C) dA, its moment with respect to the axis is (TZ 2 Ic) dA, an the sum of all the moments of the unit shearing stresses on the cross section is J (rz 2 /c) dA. In Fig. 10.23 Round bar subject to torsional stress. this expression the factor J z 2 dA is the polar moment of inertia of the section with respect to the axis. Denoting this by 7, the resisting moment may be written rJIc....
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