The motions of liquid-filled pipe reaches in which long wavelength assumptions are valid can be described by Poisson-coupled axial stress waves in the pipe and in the liquid column, and in the piping structure, by torsional and flexural waves. Based on linearized assumptions, a simultaneous solution of the wave equations is presented. Eigenvalues and mode shapes are derived for the variables fluid pressure and displacements, and pipe forces and displacements. The results are assembled into a transfer matrix, which represents the motion of a single pipe section. Combined with point matrices that describe specified boundary conditions, an overall transfer matrix for a piping system can be assembled. Corresponding state vectors can then be evaluated to predict the piping and liquid motion, and the accompanying forces. The results from two experimental piping systems are compared with the ones obtained by the modal analysis method.

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