Vibrations of a pendulum consisting of a bob suspended from a wire: the method of integral equations

Author:
Hyun J. Ahn

Journal:
Quart. Appl. Math. **39** (1981), 109-117

MSC:
Primary 70J05; Secondary 45J05

DOI:
https://doi.org/10.1090/qam/613954

MathSciNet review:
613954

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The vibrations of a vertical pendulum consisting of a bob suspended from a wire are studied by the method of integral equations and the composite method, respectively. The composite method combines the minimum principles and the method of integral equations. This problem consists of the fourth-order differential equation and the boundary conditions dependent on the eigenvalue parameter. Lower bounds are established for the lowest natural frequencies by both methods. Numerical results are presented. Both theoretical and computational efficiencies are illustrated and the method of integral equations is stressed.

**[1]**H. J. Ahn,*Operator-valued functions of the eigenvalue*, Doctoral dissertation, University of Delaware, June, 1975 MR**2625205****[2]**W. E. Boyce, R. C. DiPrima, and G. H. Handelman,*Vibrations of rotating beams of constant section*, Proc. Second U.S. Natl. Congress of Appl. Mech. 122-152 (1954) MR**2938649****[3]**R. Courant and D. Hilbert,*Methods of mathematical physics*, vol. 1, Interscience, New York, 1953 MR**0065391****[4]**B. E. Goodwin,*On the realization of the eigenvalues of integral equations whose kernels are entire or meromorphic in the eigenvalue parameter*, J. SIAM Appl. Math.**14**, 65-85 (1966) MR**0193455****[5]**G. H. Handelman, W. E. Boyce, and H. Cohen,*Vibrations of a uniform, rotating beam with tip mass*, Proc. Third U.S. Natl. Congress of Appl. Mech. 175-180 (1958) MR**0105898****[6]**G. H. Handelman and J. B. Keller,*Small vibrations of a slightly stiff pendulum*, Proc. Fourth U.S. Natl. Congress of Appl. Mech., 195-202 (1962) MR**0153150****[7]**F. B. Hildebrand,*Method of applied mathematics*, Prentice-Hall New Jersey, 1961 MR**0175336****[8]**R. Iglisch,*Uber lineare Integralgleichungen mit vom Parameter abhangigem Kern*, Math. Ann.,**117**, 129-139 (1939) MR**0001438****[9]**E. L. Ince,*Ordinary differential equations*, Dover, New York, 1956 MR**0010757****[10]**H. Jeffreys and B. Jeffreys,*Methods of mathematical physics*, Cambridge University Press, New York, 1972**[11]**A. V. Laginestra and W. E. Boyce,*Convergence and evaluation of sums of reciprocal powers of eigenvalues of boundary value problems nonlinear in the eigenvalue parameter*, SIAM J. Math. Anal.**5**, 64-90 (1974) MR**0336251****[12]**R. V. Southwell, Aero. Res. Comm. (U.K.), Reports and memoranda 486; H. Lamb and R. V. Southwell,*The vibrations of a spinning disc*, Proc. Roy. Soc. (A),**99**, 272-280 (1921)**[13]**M. R. Spiegel,*The summation of series involving roots of transcendental equations and related applications*, J. Appl. Phys.**24**, 1103-1106 (1953) MR**0057356****[14]**K. Yosida,*Lectures on differential and integral equations*, Interscience, New York, 1960 MR**0118869**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
70J05,
45J05

Retrieve articles in all journals with MSC: 70J05, 45J05

Additional Information

DOI:
https://doi.org/10.1090/qam/613954

Article copyright:
© Copyright 1981
American Mathematical Society