Nonlinear dynamics of a reduced multimodal Timoshenko beam subjected to thermal and mechanical loadings

Meccanica, Aug 2014

Large amplitude vibrations of a Timoshenko beam under an influence of temperature are analysed in this paper. In the considered model the temperature increases instantly and the heat is uniformly distributed along the beams length and cross-section. The mathematical model, represented by partial differential equations takes into account thermal and mechanical loadings. Next, the problem is reduced by means of the Galerkin method, considering the first three natural vibration modes of a simply supported beam in the both ends. The influence of the temperature on amplitudes and localisation of the resonance zones and stability of the solutions is studied numerically and analytically by the multiple time scale method. The bifurcation points, existence of unstable lobes and transition from regular to chaotic oscillations are shown.

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Nonlinear dynamics of a reduced multimodal Timoshenko beam subjected to thermal and mechanical loadings

Anna Warminska 0 1 2 Emil Manoach 0 1 2 Jerzy Warminski 0 1 2 0 J. Warminski (&) Department of Applied Mechanics, Lublin University of Technology , Nadbystrzycka 36, 20-618 Lublin, Poland 1 E. Manoach Institute of Mechanics, Bulgarian Academy of Sciences , Acad. G. Bonchev 4, 1113 Sofia, Bulgaria 2 A. Warminska Department of Thermodynamics , Fluid Mechanics and Aviation Propulsion Systems, Lublin University of Technology , Nadbystrzycka 36, 20-618 Lublin, Poland Large amplitude vibrations of a Timoshenko beam under an influence of temperature are analysed in this paper. In the considered model the temperature increases instantly and the heat is uniformly distributed along the beams length and crosssection. The mathematical model, represented by partial differential equations takes into account thermal and mechanical loadings. Next, the problem is reduced by means of the Galerkin method, considering the first three natural vibration modes of a simply supported beam in the both ends. The influence of the temperature on amplitudes and localisation of the resonance zones and stability of the solutions is studied numerically and analytically by the multiple time scale method. The bifurcation points, existence of unstable lobes and transition from regular to chaotic oscillations are shown. 1 Introduction The beams are fundamental structural elements with application in many branches of the industry. Frequently, these structures are subjected to dynamic loading leading to large amplitude vibrations. Large vibrations introduce a geometrical type of non-linearity that influences the dynamic behaviour of a structure. In this case the structures stiffness, and consequently the resonance frequencies and mode shapes, are amplitude dependent. Linear and nonlinear vibrations of beams have been deeply investigated for many years and were reviewed, for example, in the books of Nayfeh and Mook [1] and of Nayfeh and Pai [2]. Nonlinear vibrations of the EulerBernoulli or shear deformable beam models have been studied there and the influence of the nonlinear terms on the bifurcation scenario and possible resonances have been discussed in details. The advanced composite beam theory has been presented in [3]. The beam models considered various configurations of lamina with reinforced fibers orientation, closed or open cross-section shapes. The beams are also used to model rotating blades dynamics, for example blades of a helicopter rotor [4, 5] or blades of turbines. Many authors use the classical FEM [6, 7] and semi-analytical methods [8, 9] to study this problem. In most of the analysis the environmental conditions are neglected. One of very important factors which has to be considered is temperature, which may vary in high ranges in real mechanical or aerospace applications. Temperature variations can and do affect substantially the vibration response of a structure. Thermal loads introduce stresses due to thermal expansion, which lead to changes in the modal properties. The basic problems of the thermoelastic vibrations can be found in the books of Boley and Weiner [10], Nowacki [11] and Thorton [12]. Although the temperature and elastic behaviours are in fact coupled [12, 13] for thin structures it is often reasonable to assume that the temperature distribution is independent of the deformation or that the structure gets the elevated temperature instantly. This approach is widely used to model the thermoelastic behaviour of structures. The geometrically nonlinear vibrations of structures at the elevated temperature are studied by many authors as [1416], etc. In [17] and [18] thermomechanical, geometrically nonlinear vibrations of plates and beams, correspondingly, are studied. The authors found a very reach nonlinear dynamic behaviour of the system including, periodic, quasi-periodic and chaotic oscillations. A thermomechanical model of the vibration of a Timoshenko beam after its one mode reduction is studied by multiple time scale method in [19]. In the present work the study is extended by using three mode reduction of the beams model for thermoelastic vibration. The goal of this paper is to show specific dynamic phenomena of geometrically nonlinear vibrations of a Timoshenko beam subjected to thermal and mechanical loadings. The phenomena, such as bifurcations, non-periodic or chaotic oscillations which arise due to varying temperature are taken into account in the study. 2 A model of a Timoshenko beam under thermo-mechanical loadings The considered structure is a beam made of elastic composite material subjected to thermal and mechanical loadings. The beam orientation together with coordinates and indicated length l, thickness h, and width b, is presented in Fig. 1. The mathematical model of the Timoshenko beam presented in Fig. 1 has been derived in papers [13, 16]. The dimensionless equations of motion have the form: Fig. 1 Schematic beam model with indicated coordinates and dimensions where u, w, w are di (...truncated)


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Anna Warminska, Emil Manoach, Jerzy Warminski. Nonlinear dynamics of a reduced multimodal Timoshenko beam subjected to thermal and mechanical loadings, Meccanica, 2014, pp. 1775-1793, Volume 49, Issue 8, DOI: 10.1007/s11012-014-9891-3