. Machine design. prevent the tendency of the bendingto open the joints, and the fastenings should have thesame tensile strength as the rim of the wheel. 86. Rotating Discs. The formulas derived in will only apply in the case of thin rims and cannotbe used for discs or for rims having any considerabledepth. The determination of the stresses in a rotatingdisc is a complicated and difficult problem, if the ma-terial is regarded as perfectly elastic. A rational solution of this problem may be found inStodolas Steam Turbines, pp. 157-69. For the pur- 12 178 MACHINE DESIGN. poses of this tre
. Machine design. prevent the tendency of the bendingto open the joints, and the fastenings should have thesame tensile strength as the rim of the wheel. 86. Rotating Discs. The formulas derived in will only apply in the case of thin rims and cannotbe used for discs or for rims having any considerabledepth. The determination of the stresses in a rotatingdisc is a complicated and difficult problem, if the ma-terial is regarded as perfectly elastic. A rational solution of this problem may be found inStodolas Steam Turbines, pp. 157-69. For the pur- 12 178 MACHINE DESIGN. poses of this treatise an approximate solution is pre-ferred, the elasticity of the metal being method of treatment is much simpler, and as themetals used are imperfectly elastic (especially the castmetals) the results obtained will probably be as reliableas any—for practical use. The following discussion is an abstract of one givenby Mr. A. M. Levin in the American Machinist * thenotation being changed Fig. 80. 87. Plain Discs. Let Fig. 80 representa ring of uniformthickness t, havingan external diameterD and an internaldiameter d, all ininches. Let v=external ve-locity in feet per sec-ond. Let a. angular velocity =-77- r=radius to center of gravity of half ring in of metal per cubic value of r for a half-ring is easily proved to be : _2_ Dz-d* 3tt or £>2- 1 18tt d2D3 in inches D2-d % in feet. * American Machinist, Oct. 20, 1904, ROTATING DISCS. 179 The weight of the half-ring is : W=l(D2-cl2)tw o and its centrifugal force : n Wa2r aHw^—d?) ,QQs c=~r= isr~ m Substituting for a its value in terms of v : c_Uwv\JJ-dz) /g9x Now if we assume the stress on the area at AB dueto the centrifugal force to be uniformly distributed :(and here lies the approximation) then will the tensilestress on the section be e C ±wv\D2 + Dd+d2) ,lnAv S=(D=Wt= gU * (1°0) For a solid disc : s^Jf- (101) For a thin ring : S^J^f (102) on the same as in equation (89)
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