Grammel’s method and parametric vibrations

Authors: Kuporosova I.S.
Published in issue: #5(34)/2019
DOI: 10.18698/2541-8009-2019-5-483
Category: Mechanics | Chapter: Mechanics of Deformable Solid Body
Keywords: Grammel’s method, transverse bending vibrations, ideally elastic vertical cantilever, parametric vibrations, Hamilton principle, parametric resonance, Mathieu equation, instability of oscillatory system
Published: 06.06.2019

This article reviews the problem of transverse bending vibrations of a rectilinear elastic beam. The bending of the beam is straight, the system vibrations are small. In reality, this system can be considered as a mast or factory straight pipe. In the theory of equations of mathematical physics, there is the concept of duality. This article outlines an approach based on the Grammel’s method. It is shown that if geometric boundary conditions are satisfied as preliminary conditions (the Bubnov – Galerkin’s method), then the force conditions will be natural and vice versa, as in this case, and especially in the Grammel’s method.

References

[1] Feodos’yev V.I. Soprotivlenie materialov [Strength of materials]. Moscow, Nauka Publ., 1967 (in Russ.).

[2] Kolesnikov K.S., ed. Kurs teoreticheskoy mekhaniki [Course of theoretical mechanics]. Moscow, Bauman MSTU Publ., 2017 (in Russ.).

[3] Timoshenko S.P., Young D.H., Weaver W. Vibration problems in engineering. Wiley, 1974. (Russ. ed.: Kolebaniya v inzhenernom dele. Moscow, Fizmatlit Publ., 1959.)

[4] Biderman V.L. Prikladnaya teoriya mekhanicheskikh kolebaniy [Applied theory of mechanical oscillations]. Moscow, Vysshaya shkola Publ., 1972. (in Russ.).

[5] Strelkov S.P. Vvedenie v teoriyu kolebaniy [Introduction into oscillation theory]. Sankt-Petersburg, Lan’ Publ., 2005 (in Russ.).

[6] Butenin N.V. Teoriya kolebaniy [Oscillation theory]. Moscow, Vysshaya shkola Publ., 1963 (in Russ.).

[7] Il’in M.M., Kolesnikov K.S., Saratov Yu.S. Teoriya kolebaniy [Oscillation theory]. Moscow, Bauman MSTU Publ., 2003 (in Russ.).

[8] Butenin N.V., Lunts Ya.L., Merkin D.R. Kurs teoreticheskoy mekhaniki. T. 2 [Course of theoretical mechanics. Vol. 2]. Moscow, Nauka Publ., 1985 (in Russ.).

[9] Zhuravlev V.F. Osnovy teoreticheskoy mekhaniki [Fundamentals of theoretical mechanics]. Moscow, Fizmatlit Publ., 1997 (in Russ.).

[10] Il’in M.M., Pozhalostin A.A., Tusheva G.M. Kolebaniya lineynoy sistemy s odnoy stepen’yu svobody [Oscillations of linear system with one degree of freedom]. Moscow, Bauman MSTU Publ., 2002 (in Russ.).