By I.M. Rapoport
The motions of beverages in relocating packing containers represent a vast category of difficulties of serious useful value in lots of technical fields. The impact of the dynamics of the liquid at the motions of the box itself is a finest and complicated element of the overall topic, even if one considers in basic terms the rigid-body motions of the box or its elastic motions besides. it really is such a lot becoming for this reason that this translation of Professor Rapoport's publication has been undertaken so briskly following its unique book, as a way to make available this particularly special account of the mathematical foundations underlying the therapy of such prob lems. for the reason that so much of this significant physique of study has been constructed over the last decade by means of scientists within the USSR, and has accordingly been largerly unavailable to these not able to learn Russian, this quantity will certainly be of serious worth to many folks. H.
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Additional info for Dynamics of Elastic Containers: Partially Filled with Liquid
1 SS -; :; :: ds= S Iv +'. 25) 28 Dynamics of Elastic Containers N lXJ(= SSS (y2+ z2)Qdv + ~Q. SS Cfx o;nx ds; V SW+~ =1 N lXY= - SSS xyQdv +~Q. =1 + • SjV N lxz=- J'SS xZQdv+ ~Q. UXYQdV+ ~Q. =1 SjV N +'. cpy 0;: ds; a;~< ds; lyy=SSS(X +z )QdV+ ~Q. =1 N lyz= - S5S yZQdv + ~Q. =1 SjV N +". lzx= - SSS xZQdv +~Q. =1 SjV +'. N lzy= - SSS YZQdv +~Q. =1 +'. N lzz= SSS (x2+y2)Qdv + ~Q. SS cpz 0;: ds v =1 SIV+n Substituting Eq. 25) into Eq. 24), we obtain -+ d2a m(wo-g)=Fo+F ( t ) -dt2 -; --Jo .... 27) dwy +1zz dWz)-+_ +(1zx dwx dt +1 zYdt dt ez- The first of these equations will subsequently be called the force equation.
I-QdV+ ~Q. =1 _02j + SS a. cP at; ds SIV ) According to Eq. ds= \ r;unds+ a. S IV + U7- f yds-7-. dS l Thus, the first of Eqs. 19) can be replaced by the equivalent formula JS N (; (t)= S ;Qdv+ ~QY ax -;unds+ U7-f. 21) According to Eqs. 21), Eqs. 18) can be transformed into - d2a m(g-wo) - -+Fo+F(t)=O; dt2 ~ -+ a X -+ -+- (g-wo) - SSS rX -+ ..... v N -~ Q. y=1 -+ ~ - )Qdv(dixr dOl .... -+ dt an dt2 SS -cp doo . ii'f d2~ ~ - ds- -+M(t)=O SIV +a. 23) is the total mass of the body and the fluids that it contains.
Thus, the moment equation can be simplified by selecting such directions for Fluid Pressure on the Wetted Surface oj the Cavity 31 these coordinate axes for which the centrifugal moments of inertia lXII' IxZt and lllz become zero.  MOMENTS OF INERTIA OF A SOLID BODY CONTAINING FLUID MASSES Together with the x, y, z coordinate system, we now consider any other orthogonal system, 6, '1'), ~. which is rigidly tied to the undeformed elastic body and which has a common origin with the former coordinate system.