An investigation into natural vibrations of fluid–structure interaction systems subject to Sommerfeld radiation condition

A fluid–structure interaction system subject to Sommerfeld’s condition is defined as a Sommerfeld system which is divided into three categories: Fluid Sommerfeld (FS) System, Solid Sommerfeld (SS) System and Fluid Solid Sommerfeld (FSS) System of which Sommerfeld conditions are imposed on a fluid boundary only, a solid boundary only and both fluid and solid boundaries, respectively. This paper follows the previous initial results claimed by simple examples to further mathematically investigate the natural vibrations of generalized Sommerfeld systems. A new parameter representing the speed of radiation wave for generalized 3-D problems with more complicated boundary conditions is introduced into the Sommerfeld condition which allows investigation of the natural vibrations of a Sommerfeld system involving both free surface and compressible waves. The mathematical demonstrations and selected examples confirm and reveal the natural behaviour of generalized Sommerfeld systems defined above. These generalized conclusions can be used in theoretical or engineering analysis of the vibrations of various Sommerfeld systems in engineering.

[1]  W. G. Price,et al.  A power–flow analysis based on continuum dynamics , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[2]  Roger Ohayon,et al.  Fluid-Structure Interaction: Applied Numerical Methods , 1995 .

[3]  W. Thomson.  Theory of vibration with applications , 1965 .

[4]  W. G. Price,et al.  A mixed finite element method for the dynamic analysis of coupled fluid–solid interaction problems , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[5]  W. G. Price,et al.  Variational principles of nonlinear dynamical fluid–solid interaction systems , 1997, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[6]  W. G. Price,et al.  Mixed finite element substructure—subdomain methods for the dynamical analysis of coupled fluid-solid interaction problems , 1996, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[7]  Jing T. Xing.  The natural vibration of fluid-structure interaction systems subject to the sommerfeld radiation condition , 2006 .

[8]  Transmission and Radiation of Sound by Thin Plates , 1999 .

[9]  R. Eatock Taylor.  A review of hydrodynamic load analysis for submerged structures excited by earthquakes , 1981 .

[10]  W. G. Price,et al.  NATURAL VIBRATION OF A BEAM—WATER INTERACTION SYSTEM , 1997 .

[11]  R. Geroch.  Partial Differential Equations of Physics , 1996, gr-qc/9602055.

[12]  D. Brill,et al.  Acoustic spectrogram and complex‐frequency poles of a resonantly excited elastic tube , 1984 .

[13]  K. Bathe.  Finite Element Procedures , 1995 .

[14]  Andrew M. Stuart,et al.  A First Course in Continuum Mechanics: Bibliography , 2008 .

[15]  J. N. Newman.  The Theory of Ship Motions , 1979 .

[16]  Dr. M. G. Worster.  Methods of Mathematical Physics , 1947, Nature.

[17]  Guo Chang-qing.  FLUID-STRUCTURE INTERACTED VIBRATION OF AN ELEMENT MODEL OF PARALLEL-PLATE FUEL ASSEMBLY , 2007 .

[18]  F. Rellich,et al.  Über das asymptotische Verhalten der Lösungen von ...u + ...u = 0 in unendlichen Gebieten. , 1943 .

[19]  Jing Tang Xing,et al.  Natural vibration of two-dimensional slender structure–water interaction systems subject to Sommerfeld radiation condition , 2007 .

[20]  Eugene P. Wigner,et al.  Formulas and Theorems for the Special Functions of Mathematical Physics , 1966 .

[21]  A. Sommerfeld.  Partial Differential Equations in Physics , 1949 .

[22]  W. G. Price,et al.  Hydroelasticity of Ships , 1980 .