The elasticity and resistance of the materials of engineering . —Equations of Motion and Equilibrium in Polar Co-ordinates. The relation, in space, existing between the polar andrectangular systems of co-ordinates is shown in Fig. i. Theangle ^ is measured in the plane ZF and from that o( XV;while ip is measured normal to ZV in a plane which containsOX. The analytical relation existing between the two systemsis, then, the following : X = r sin tpy y = r cos ip cos cp^ and s = r cos tp sin (p. The indefinitely small portion of material to be consideredleaked. It is limited by the co-ordinate pl
The elasticity and resistance of the materials of engineering . —Equations of Motion and Equilibrium in Polar Co-ordinates. The relation, in space, existing between the polar andrectangular systems of co-ordinates is shown in Fig. i. Theangle ^ is measured in the plane ZF and from that o( XV;while ip is measured normal to ZV in a plane which containsOX. The analytical relation existing between the two systemsis, then, the following : X = r sin tpy y = r cos ip cos cp^ and s = r cos tp sin (p. The indefinitely small portion of material to be consideredleaked. It is limited by the co-ordinate planes located by 28 ELASTICITY TV AMORPHOUS SOLID BODIES. [Art. 8. q) and f, and concentric spherical surfaces with radii r andr -j- dr. The directions r, (p and ^, at any point, are rectangu-lar; hence, the sums of the forces acting on the small portionof the material, taken in these directions, must be found andput equal to m ~dP m d^dt- V and in d^GD ~di^ in which expressions, p, rj and od represent the strains in thedirection of r, q) and ^ Those forces which act on the faces aJi, bd and cd will beconsidered negative, and those which act on the other facespositive. The notation will remain the same as in the preceding Ar-ticles, except that the three normal stresses will be indicatedby N^, a; and N^. Art. 8.] EQUATIONS IN POLAR CO-ORDINATES. 2g Forces acting along r. — Nr , r dtp r cos ?/- dq), -{-Nr , r^ cos tp dip dcp + (^^^r^ dr = r^ dr -\- 2r Nr dr\ cos ip dip dcp. — T^r • ^ dip dr, + T^r. r dip dr -j j^ dq) . r dip dr, dcp — T^r • ^ <^os ip dcp dr. + T^r r costp dcp dr -\- ( -^—5^^^-—^ <///? = cos ip -—tt^ ^^ — T^r sin ip dipjr dq) dr. — N^ , r dip dr . si^i aOc = — N^ . r dtp dr . cos ip dcp; on face ce, — N^ . rcos tp dcp dr . si7t aOb = — N^ . r cos ip dcp dr , dip\ on face be. Forces acting along cp.—• Tr^ . r cos tp dq) r dip. + Tr^ . T-cosip dq) dip + (^-^^^^-^ r^-^ ^r+ 2r Tr^d^ cos Ip dip dcp. 30 ELASTICITY IN
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