Nonlinear dynamics of a reduced multimodal Timoshenko beam subjected to thermal and mechanical loadings
Anna Warminska
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1
2
Emil Manoach
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1
2
Jerzy Warminski
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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)