On a spectral problem in vibration mechanics  Computation of elastic tanks partially filled with liquids
Abstract
Theoretical and numerical analysis of the vibrations of a fluid in a deformable shell. Analysis is based on linear theory. The general equations of the coupled system formed by the fluid and the elastic shell are set up, and the problem is formulated in variational terms. By an appropriate choice of function spaces, an associated spectral problem is formulated. The existence of a spectrum of eigenfrequencies and eigenfunctions which defines the vibration modes is proved. The theoretical study is complemented with a numerical analysis by the finite element method. The approximate problem is studied by matrix methods. Some numerical results are presented for the first stage of the Diamant launch vehicle. The results identify the mode of vibration characterizing the Pogo effect  an instability due to coupling between the main structure (tanks and shells) and the secondary structure (pipes and pumps) and the thrust.
 Publication:

Journal of Mathematical Analysis and Applications
 Pub Date:
 August 1975
 Bibcode:
 1975JMAA...51..272B
 Keywords:

 Elastic Shells;
 Hydroelasticity;
 Liquid Filled Shells;
 Liquid Sloshing;
 Structural Vibration;
 Tanks (Containers);
 Boundary Value Problems;
 Eigenvalues;
 Shell Stability;
 Spectrum Analysis;
 Variational Principles;
 Vibration Mode;
 Fluid Mechanics and Heat Transfer