The transverse vibrations of a pipe containing flowing fluid: Methods of integral equations
Authors:
Louise H. Jones and Bruce E. Goodwin
Journal:
Quart. Appl. Math. 29 (1971), 363-374
DOI:
https://doi.org/10.1090/qam/99757
MathSciNet review:
QAM99757
Full-text PDF Free Access
Abstract |
References |
Additional Information
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.
L. H. Jones, The transverse vibrations of a pipe containing flowing fluid, Master’s Thesis, University of Delaware, Newark, Delaware
E. P. Hamilton and B. E. Goodwin, The inverse problem of the calculus of variations, in Analytic methods in mathematical physics (ed. R. P. Gilbert and R. G. Newton), Gordon and Breach, New York, 1970
- Bruce E. Goodwin, On the realization of the eigenvalues of integral equations whose kernels are entire or meromorphic in the eigenvalue parameter, SIAM J. Appl. Math. 14 (1966), 65–85. MR 193455, DOI https://doi.org/10.1137/0114006
- Rudolf Iglisch, Über lineare Integralgleichungen mit vom Parameter abhängigem Kern, Math. Ann. 117 (1939), 129–139 (German). MR 1438, DOI https://doi.org/10.1007/BF01450013
G. W. Housner, Bending vibrations of a pipeline containing flowing fluid, J. Appl. Mech. 19, 205–208 (1952)
R. H. Long, Jr., Experimental and theoretical study of transverse vibrations of a pipe containing flowing fluid, J. Appl. Mech. 22, 65–68 (1955)
- Holt Ashley and George Haviland, Bending vibrations of a pipe line containing flowing fluid, J. Appl. Mech. 17 (1950), 229–232. MR 0037170
- George H. Handelman, A note on the transverse vibration of a tube containing flowing fluid, Quart. Appl. Math. 13 (1955), 326–330. MR 74256, DOI https://doi.org/10.1090/S0033-569X-1955-74256-9
- F. G. Tricomi, Integral equations, Pure and Applied Mathematics, Vol. V, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. MR 0094665
F. I. Niordson, Experimental and theoretical study of transverse vibrations of a tube containing flowing fluid, Trans. Roy. Inst. Tech., Stockholm 1953, 73 pp.
- M. R. Speigel, The summation of series involving roots of transcendental equations and related applications, J. Appl. Phys. 24 (1953), 1103–1106. MR 57356
- Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968
- William E. Boyce and Bruce E. Goodwin, Random transverse vibrations of elastic beams, J. Soc. Indust. Appl. Math. 12 (1964), 613–629. MR 175396
- William E. Boyce and Bruce E. Goodwin, Random transverse vibrations of elastic beams, J. Soc. Indust. Appl. Math. 12 (1964), 613–629. MR 175396
L. H. Jones, Improved lower bounds for the eigenvalues of a class of boundary value problems, SIAM J. Appl. Math. (Submitted)
L. H. Jones, The transverse vibrations of a pipe containing flowing fluid, Master’s Thesis, University of Delaware, Newark, Delaware
E. P. Hamilton and B. E. Goodwin, The inverse problem of the calculus of variations, in Analytic methods in mathematical physics (ed. R. P. Gilbert and R. G. Newton), Gordon and Breach, New York, 1970
B. E. Goodwin, On the realization of the eigenvalues of integral equations whose kernels are entire or meromorphic in the eigenvalue parameter, SIAM J. Appl. Math. 14, 65–85 (1966)
R. Iglisch, Über Linear Integralgleichungen mit vom Parameter abhängigem Kern, Math. Ann. 117, 129–139 (1939)
G. W. Housner, Bending vibrations of a pipeline containing flowing fluid, J. Appl. Mech. 19, 205–208 (1952)
R. H. Long, Jr., Experimental and theoretical study of transverse vibrations of a pipe containing flowing fluid, J. Appl. Mech. 22, 65–68 (1955)
H. Ashley and G. Haviland, Bending vibrations of a pipeline containing flowing fluid, J. Appl. Mech. 17, 229–232 (1950)
G. H. Handelman, A note on the transverse vibration of a tube containing flowing fluid, Quart. Appl. Math. 13, 326–330 (1956)
F. G. Tricomi, Integral equations, Pure and Appl. Math., vol. 5, Interscience, New York, 1957
F. I. Niordson, Experimental and theoretical study of transverse vibrations of a tube containing flowing fluid, Trans. Roy. Inst. Tech., Stockholm 1953, 73 pp.
M. R. Spiegel, The summation of series involving roots of transcendental equations and related applications, J. Appl. Phys. 24, 1103–1106 (1953)
W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and theorems for the special functions of mathematical physics, 3rd ed., Die Grundlehren der math. Wissenschaften, Band 52, Springer-Verlag, New York 1966
B. E. Goodwin and W. E. Boyce, The vibrations of a random elastic string: The method of integral equations, Quart. Appl. Math. 12, 261–266 (1964)
W. E. Boyce and B. E. Goodwin, Random transverse vibrations of elastic beams, J. Soc. Indust. Appl. Math. 12, 613–629 (1964)
L. H. Jones, Improved lower bounds for the eigenvalues of a class of boundary value problems, SIAM J. Appl. Math. (Submitted)
Additional Information
Article copyright:
© Copyright 1971
American Mathematical Society
