By I.M. Rapoport

The motions of drinks in relocating packing containers represent a vast classification of difficulties of significant useful significance in lots of technical fields. The impression of the dynamics of the liquid at the motions of the box itself is a finest and intricate point of the overall topic, even if one considers purely the rigid-body motions of the box or its elastic motions to boot. it's so much becoming consequently that this translation of Professor Rapoport's booklet has been undertaken so swiftly following its unique book, which will make available this really designated account of the mathematical foundations underlying the therapy of such prob lems. on the grounds that so much of this sizeable physique of research has been constructed over the last decade by way of scientists within the USSR, and has accordingly been largerly unavailable to these not able to learn Russian, this quantity will absolutely be of significant price to many folks. H.

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1 N J ee = SSS (~2+'I')2)Qd'V+ ~ Q. =1 N 8 jV +a. SS Je"l=-SSS~TlQdV+~Q. =1 8/v +a. =1 8/v +a. J£e=-SSS~CQd'V+ ±Q. "e. 4) cos (x, ~) cos(z, ~)+cos (x, 1']) cos (z, 1'])+ cos (x, q cos (z, C)=O; cos (y, ~) cos (z, ~)+cos (y, 1']) cos (z, 1'])+ cos (y, C) cos (z, C)= 0 According to Eq. 5) We find from Eqs. 6) (conPd) TJ) X X X [COS(y, ~)COS(Z, TJ)+COS(Z, ~)COS(y, TJ)]+ +~C[COS(y, ~)COS(Z, q+COS(Z, ~)COS(y, C)] + +TJC[COS(y, TJ)COS(Z, C)+COS (Z, TJ) COS (y, C)]. 7) 'f'y='f'acos(y, ~)+'f'~cos(y, TJ)+'f',cos(y, C); 'f'z='f'I;COS(Z, ~)+'f'~cos(z, YJ)+'f', cos (z, q According to the above equations, we will have iJ'f'x iJ'f'l; iJ'f' iJ'f' -iJ-=-cos(x, ~)+-~ COS (x, TJ)+ - ' cos (x, q; n iJn iJn iJn iJ'f'y iJ'f'a iJ'f' - i J - = - cos(y, ~)+-~ n iJn iJn iJ'f' _z = iJn iJ'f' cos(y, TJ) + - ' cos(y, C); iJ'f'l; iJ'f' iJ'f' -cos(z, ~)+ -~ cos (z, TJ)+ - ' cos (z, q iJn iJn iJn From the above equation and Eq.

10) according to which, the Neumann problem stated by Eq. 7) has the particular solution Q" -.. 11) where C, is an arbitrary constant. Substituting the above equations into Eq. )-~Fo· from which it is found that SS-rds=O ". 12) (we recall that s. , while ;:. denotes the radius vector of its center of gravity). Substituting the above equation into Eq. 11), for Po (x, y. 13) We now consider the boundary-value problem of Eqs. 8).

Z rotates slowly. Disregarding the Coriolis acceleration and the acceleration of transport;x (~xr), we will set in Eq. 2) r +w=wo+-X dt iJt2 in the same manner as in deriving the force and moment equations in Chap. 1. 1) in this case will take the form - Q= -(I -- r+ --) (wo-g+-X - -+ 39 d", dt .. 3) Dynamics of Elastic Containers 40 In accordance with the limitations imposed by us on the ex- ternal surface forces in Chap. 5) -+ while 8q (x, y, z, t) is a vector of surface forces which reduces to the minor principal vector F (t) and the minor principal moment M(t), ss s -+ 8qds=F, sss -+ ....