## Oscilacije i stabilnost sistema elastično povezanih Timošenkovih greda

Vibration and stability of systems of elastically connected Timoshenko beams

##### Метаподаци

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The presented work consists of seven parts which are separately formed by chapters. The first
chapter relates to the introductory discussion and review of previous research in the theory of
elastic and related damaged structures. It is one of the ways to perform partial differential
equations of motion of mechanical systems and provides a basic overview of the methods used.
Chapters 2-6 are devoted to the analysis of linear elastic oscillations. The seventh chapter is
devoted to geometric nonlinear oscillations of damaged beams using the new finite element
method.
Free oscillations and static stability of two elastically connected beams are considered in Chapter
2. At various examples it is shown analytically obtained results and impacts of some mechanical
parameters of the system to the natural frequency and amplitudes. Verification of the obtained
analytical results is shown by comparison with results of the existed classical models. New
scientific contribution in this chapter is for...mulation of the new double-beam model described with
new derived equations of motion with rotational inertia effects and with inertia of rotation with
transverse shear (Timoshenko’s model, Reddy - Bickford 's model). It is formulized the static
stability conditions of the two elastically connected beams of different types with analytical
expressions for the values of critical forces. Numerical experiments confirmed the validity of the
analytical results obtained by comparing the results of the models existing in the literature. From
chapter 2 it can be concluded that the effects of rotational inertia and transverse shear must be
taken into account in the model of thick beams because errors that occur by ignoring them increase
with the mode of vibration.
Chapter 3 presents the solution for forced vibrations of two elastically connected beams of
Rayleigh’s, Timoshenko’s and Reddy – Bickford’s type under the influence of axial force. Scientific
contribution is presented analytical solutions for forms of three types of forced vibration -
Harmonic arbitrarily continuous excitation, the continuous uniform harmonic excitation and
harmonic concentrated excitation. Analytical solutions were obtained by using the modal analysis
method. The chapter also presents the analytical solutions of forced vibration for the case of
harmonic excitation effects are concentrated on one of the beams under effect of compressive axial
forces. Based on the results derived in this chapter, it can be made conclusion that the differences
in the approximations of the solutions depending of the used model gave a good solutions just in
case of Timoshenko and Reddy-Bickford theory for thick beams in higher modes. Classical theories
don’t give a quite good results.
Chapter 4 considers the static stability of the elastically connected two and three beams and single
beam on elastic foundation. It is derived a new set of partial differential equations for static
analysis of deflections and critical buckling force of the complex mechanical systems. It is
analytically determined critical buckling force for each system individually. It is concluded that
the system is the most stable in the case of one beam on elastic foundation.
Chapters 5 and 6 analyzed free vibration of more elastically connected beams of Timoshenko and
Reddy-Bickford's type on elastic foundation under the influence of axial forces. Analytical
solutions for the natural frequencies and the critical force are determined by trigonometric method
and verified numerically.
Chapter 7 presents geometrically nonlinear forced vibrations of damaged Timoshenko beams. In
the study it is developed new p-version of finite element method for damaged beams. The
advantage of the new method is compared with the traditional method and provides better
approximations of solutions with a small number of degrees of freedom used in numerical analysis.
Scientific contribution is in two topics-computational mechanics and non-linear vibrations of
beams. It is concluded that traditional method can’t give good approximations of solutions in the
case of very small width of the damage. This benefit is also shown in comparison with obtained
results in the commercial software Ansys.
A new p-version finite element is suggested to deal with geometrically non-linear vibrations of
damaged Timoshenko beams. The novelty of the p-element comes from the use of new
displacement shape functions, which are function of the damage location and, therefore, provide
for more efficient models, where accuracy is improved at lower computational cost. In numerical
tests in the linear regime, coupling between cross sectional rotation and longitudinal vibrations is
discovered, with longitudinal displacements suddenly changing direction at the damage location
and with a peculiar change in the cross section rotation at the same place. Geometrically nonlinear,
forced vibrations are then investigated in the time domain using Newmark’s method and
further couplings between displacement components are found.