Vladimir Stojanović

Moment Lyapunov Exponents and Stochastic Stability of a Three-Dimensional System on Elastic Foundation Using a Perturbation Approach

In this paper, the stochastic stability of the three elastically connected Euler beams on elastic foundation is studied. The model is given as three coupled oscillators. Stochastic stability conditions are expressed by the Lyapunov exponent and moment Lyapunov exponents. It is determined that the new set of transformation for getting Ito∧ differential equations can be applied for any system of three coupled oscillators. The method of regular perturbation is used to determine the asymptotic expressions for these exponents in the presence of small intensity noises. Analytical results are presented for the almost sure and moment stability of a stochastic dynamical system. The results are applied to study the moment stability of the complex structure with influence of the white noise excitation due to the axial compressive stochastic load.

Effects of Axial Compression Forces, Rotary Inertia and Shear on Forced Vibrations of the System of Two Elastically Connected Beams

This chapter covers 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 forces. Scientific contribution is presented through the analytical solutions in forms of three cases of forced vibrations - Harmonic arbitrarily continuous excitation, the continuous uniform harmonic excitation and harmonic concentrated excitation. Analytical solutions were obtained by using the modal analysis method. Based on the results derived in this chapter, it can be made a conclusion that the differences in the approximations of the solutions depending of the used model gave a good solutions just in cases of Timoshenko’s and Reddy-Bickford’s theory for thick beams in higher modes also in forced vibrations regime and must be taken into account.

Free Vibrations and Stability of an Elastically Connected Double-Beam System

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 on the natural frequencies and amplitudes. Verification of obtained results is shown by comparison with results of the existed classical models. New scientific contribution in this chapter is formulation 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 (Rayleigh’s model, Timoshenko’s model, Reddy - Bickford’s model). It is formulized the static stability condition of the two elastically connected beams of different types with analytical expressions for the various 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 are increasing with the increasing the mode of vibration.

The Effects of Rotary Inertia and Transverse Shear on Vibrations and Stability of the System of Elastically Connected Reddy-Bickford Beams on Elastic Foundation

Chapters 6 analyzed free vibration of the multiple elastically connected beam system of Reddy-Bickford’s type on an elastic foundation under the influence of axial forces with the comparison of the frequency and stability theoretical research for all four types of the beam’s theory. Analytical solutions for the natural frequencies and the critical buckling forces are determined by the trigonometric method and verified numerically as in case in the previous chapter. In the case of the Reddy-Bickford’s model, it is shown that the natural frequency provides the best solution approximation.

Geometrically Non-linear Vibrations of Timoshenko Damaged Beams Using the New p-Version of Finite Element Method

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 p-version of finite element method and provides better approximations of solutions with a small number of degrees of freedom used in numerical analysis.

The Effects of Rotary Inertia and Transverse Shear on the Vibration and Stability of the Elastically Connected Timoshenko Beam-System on Elastic Foundation

Chapter 5 analyzed free vibration of the multiple elastically connected beams of Timoshenko’s type on an elastic foundation under the influence of axial forces. Analytical solutions for the natural frequencies and the critical buckling forces are determined by the trigonometric method and verified numerically. It is shown that the fundamental natural frequency in the first mode of the multiple beam system tends to the value of the natural frequency of the system with one beam resting on an elastic foundation with the tendency of increasing the number of connected beams with the same stiffness of the layers between.

Static and Stochastic Stability of an Elastically Connected Beam System on an Elastic Foundation

Chapter 4 considers the static and stochastic stability of the elastically connected three beams on an 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 system and it is presented comparison study of the static stability between mechanical systems with one, two and three beams on an elastic foundation. 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 the one beam on elastic foundation.