caa a22 4500 467913692 CHVBK 20180406152856.0 cr unu---uuuuu 170328e20060301xx s 000 0 eng 10.1007/s10999-006-9011-1 doi (NATIONALLICENCE)springer-10.1007/s10999-006-9011-1 On the existence of periodic solution for equation of motion of thick beams having arbitrary cross section with tip mass under harmonic support motion [Elektronische Daten] [Abbas Rastgoo, Farzad Ebrahimi, Ali Dizaji] 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. Springer Science+Business Media, Inc., 2006 Fixed point theorem nationallicence Non-linear vibration nationallicence Timoshenko beam nationallicence Green's function nationallicence A ( x ) : Cross-sectional area nationallicence E : Young's modulus of elasticity nationallicence G : Modulus of rigidity nationallicence I ( x ) : Cross-sectional moment of inertia nationallicence K : Shear deformation coefficient nationallicence L : Beam length nationallicence m : Tip mass nationallicence Q : Amplitude of harmonic excitation nationallicence t : Time nationallicence T : Kinetic energy nationallicence u : Longitudinal elastic deflections nationallicence U : Base excitation displacement nationallicence v : Transverse elastic deflections nationallicence V : Potential displacement nationallicence x : Reference variable along beam nationallicence x - y : Cartesian coordinate nationallicence Y : Maximum of a function in a bounded region of phase space nationallicence ρ( x ) : Mass per unit length nationallicence ζ i ( t ) : Generalized coordinates for the beam element's elastic orientation nationallicence τ : Dimensionless period of vibration nationallicence ξ i ( t ) : Generalized coordinates for the beam element's elastic deflection nationallicence ψ i ( x ) : Rotational eigenfunctions (modal shapes) of a clamped-mass Timoshenko beam nationallicence Φ i ( x ) : Transverse eigenfunctions (modal shapes) of a clamped-mass Timoshenko beam nationallicence Ω : Frequency of excitation nationallicence $${\Re^2}$$ : Two-dimensional metric space nationallicence $${\Im}$$ : Banach space nationallicence · : Partial derivative with respect to the time variable (t) nationallicence ′ : Partial derivative with respect to the position variable (x) nationallicence Rastgoo Abbas Department of Mechanical Engineering, Tehran University, North Kargar Street, Tehran, Iran aut Ebrahimi Farzad Department of Mechanical Engineering, Tehran University, North Kargar Street, Tehran, Iran aut Dizaji Ali Department of Mechanical Engineering, Tehran University, North Kargar Street, Tehran, Iran aut International Journal of Mechanics and Materials in Design Springer Netherlands 3/1(2006-03-01), 29-38 1569-1713 3:1<29 2006 3 10999 https://doi.org/10.1007/s10999-006-9011-1 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10999-006-9011-1 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Rastgoo Abbas Department of Mechanical Engineering, Tehran University, North Kargar Street, Tehran, Iran aut NATIONALLICENCE 700 1- Ebrahimi Farzad Department of Mechanical Engineering, Tehran University, North Kargar Street, Tehran, Iran aut NATIONALLICENCE 700 1- Dizaji Ali Department of Mechanical Engineering, Tehran University, North Kargar Street, Tehran, Iran aut NATIONALLICENCE 773 0- International Journal of Mechanics and Materials in Design Springer Netherlands 3/1(2006-03-01), 29-38 1569-1713 3:1<29 2006 3 10999 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer