The transverse vibrations of a pipe containing flowing fluid: Methods of integral equations

Abstract

Abstract.Methods are developed to study the problem described in the title. Improvable lower bounds for the first eigenvalue are obtained for the low velocity-thin pipe wall case. It is shown that the eigenvalue changes from real to imaginary as the fluid velocity increases through a "critical" velocity. It is the methods which we wish to emphasize in that while we discuss them only for the present problem they are very general and especially powerful when applied to differential equations with constant coefficients.

Flow-Induced Vibration of Circular Cylindrical Structures

“…Other techniques have also been used to analyze the problem; see Li and DiMaggio (1964), Jones and Goodwin (1971), Mote (1971), Kornecki (1971).…”

Untitled

“…Goodwin [3] converted the reduced problem (2) into a Fredholm integral equation having a symmetric kernel which depends on the eigenvalue parameter; however, Goodwin's paper does not contain numerical results. The primary purpose of this paper is to describe an extension of the integral equation method used previously in [2]; to show, by means of some simple counterexamples, some advantages of these methods over those described earlier by Goodwin; and to use the new methods to compute improvable lower bounds for the eigenvalues of the "reduced" problem (2). A secondary purpose is to compare the computational aspects of the integral equation method with those of other, less general methods such as invariant embedding [10].…”

“…However, meromorphic parts of the Green's functions for higher-order problems such as (1) and the fourth-order boundary value problem describing the transverse vibration of a pipe containing flowing fluid consist of two or more terms [2], and poor results will be obtained unless the multiplicity problem is treated appropriately.…”

“…Example B shows that incorrect results can be obtained even with the Volterra formulation if the reciprocals of the pole are counted more than once; this difficulty can be avoided by using the trace formulae developed by LaGinestra [9]. It should be noted that improved lower bounds for the first eigenvalue of a pipe containing flowing fluid could be by the use of LaGinestra's formulae in [2], Example A: Consider the boundary-value problem …”

“…It should be noted that the problem is self-adjoint and has previously been studied by variational methods [1,4]. The present paper is primarily concerned with illustrating the use of an integral equation method previously described in [2] to obtain lower bounds for the eigenvalues (i2 and with comparing the computational aspects of this method with those of other, less general methods, which can also be used to determine the eigenvalues of problems with eigenvalue dependence in the boundary conditions. The historical background of the problem is given in Sec.…”