On the existence of periodic solution for equation of motion of thick beams having arbitrary cross section with tip mass
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On the existence of periodic solution for equation of motion of thick beams having arbitrary cross section with tip mass under harmonic support motion Abbas Rastgoo Æ Farzad Ebrahimi Æ Ali Feyz Dizaji
Received: 19 October 2006 / Accepted: 13 November 2006 / Published online: 22 December 2006 Springer Science+Business Media B.V. 2006
Abstract A cantilever beam having arbitrary cross section with a lumped mass attached to its free end while being excited harmonically at the base is fully investigated. The derived equation of vibrating motion is found to be a non-linear parametric ordinary differential equation, having no closed form solution for it. We have, therefore, established the sufficient conditions for the existence of periodic oscillatory behavior of the beam using Green’s function and employing Schauder’s fixed point theorem.The derived equation of vibration motion is found to be a non-linear parametric ordinary differential equation, having no closed form solution for it. To formulate a simple, physically correct dynamic model for stability and periodicity analysis, the general governing equations are truncated to only the first mode of vibration. Using Green’s function and Schauder’s fixed point theorem, the necessary and sufficient conditions for periodic oscillatory behavior of the beam are established. Consequently, the phase domain of periodicity and stability for various values of physical characteristics of the beam-mass system and harmonic base excitation are presented.
A. Rastgoo (&) F. Ebrahimi A. F. Dizaji Department of Mechanical Engineering, Tehran University, North Kargar Street, Tehran, Iran e-mail: [email protected]
Keywords Fixed point theorem Non-linear vibration Timoshenko beam Green’s function Nomenclature A(x) Cross-sectional area E Young’s modulus of elasticity G Modulus of rigidity I(x) Cross-sectional moment of inertia K Shear deformation coefficient L Beam length m Tip mass Q Amplitude of harmonic excitation t Time T Kinetic energy u Longitudinal elastic deflections U Base excitation displacement v Transverse elastic deflections V Potential displacement x Reference variable along beam x–y Cartesian coordinate Y Maximum of a function in a bounded region of phase space q(x) Mass per unit length fi (t) Generalized coordinates for the beam element’s elastic orientation s Dimensionless period of vibration ni (t) Generalized coordinates for the beam element’s elastic deflection wi (x) Rotational eigenfunctions (modal shapes) of a clamped-mass Timoshenko beam
123
30
Fi (x) Transverse eigenfunctions (modal shapes) of a clamped-mass Timoshenko beam W Frequency of excitation
> > p 0txs < 2 ffiffiaffi sinðpffiffiaffis=2Þ ð27Þ G¼ pffiffiffi > s=2Þ > > cospffiffiaffi ðx ptffiffiþ ffi 0 x t s: : 2 a sinð as=2Þ
It follows from Schauder’s fixed point theorem that there exists a fixed point of = for } such that the integral equation (26) has a solution in the confined set given by Eq. 29 (Elfimov et al. 1985; Hale 1970). In order to follow the procedure, we first search for Y given by Eq. 29. Since
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