×

zbMATH — the first resource for mathematics

On the zeros of solutions of beam equations. (English) Zbl 0683.73031
For three model equations of beam theory it is shown that oscillatory classical solutions have zeros, under reasonable hypotheses. The method of proofs consists in a (clever) adaptation of known techniques, partly developed by the author, for the study of linear and nonlinear hyperbolic equations. It would therefore seem that the main interest of the paper is on the applicative side; unfortunately, a discussion of the mechanical meaning of the main hypotheses has not been included.
Reviewer: P.Podio-Guidugli

MSC:
74H45 Vibrations in dynamical problems in solid mechanics
35L35 Initial-boundary value problems for higher-order hyperbolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cazenave, T.; Haraux, A., Propriétés oscillatoires des solutions de certaines équations des ondes semi-linéaires, C. R. Acad. Sci. Paris Sér. I Math., 298, 449-452 (1984) · Zbl 0571.35074
[2] Kahane, C., Oscillation theorems for solutions of hyperbolic equations, Proc. Amer. Math. Soc., 41, 183-188 (1973) · Zbl 0277.35005
[3] Kopáčková, M.; Vejvoda, O., Periodic vibrations of an extensible beam, Časopis Pěst. Mat., 102, 356-363 (1977) · Zbl 0384.73042
[4] Kreith, K., Sturmian theorems for characteristic initial value problems, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 47, 139-144 (1969) · Zbl 0194.41502
[5] Kusano, T.; Yoshida, N., Forced oscillations of Timoshenko beams, Quart. Appl. Math., 43, 167-177 (1985) · Zbl 0572.73062
[6] Menzala, G. P., On global classical solutions of a nonlinear wave equation, Applicable Anal., 10, 179-195 (1980) · Zbl 0441.35037
[7] Narazaki, T., On the time global solutions of perturbed beam equations, Proc. Fac. Sci. Tokai Univ., 16, 51-71 (1981) · Zbl 0474.35010
[8] Pagan, G., Oscillation theorems for characteristic initial value problems for linear hyperbolic equations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 55, 301-313 (1973) · Zbl 0298.35035
[9] Timoshenko, S.; Young, D. H.; Weaver, W. Jr., Vibration Problems in Engineering (1974), New York: John Wiley, New York
[10] Yoshida, N., Forced oscillations of extensible beams, SIAM J. Math. Anal., 16, 211-220 (1985) · Zbl 0569.73063
[11] Yoshida, N., On the zeros of solutions to nonlinear hyperbolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 106, 121-129 (1987) · Zbl 0651.35056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.