The nonlinear two-degree-of-freedom system under consideration consists of the linear oscillator with a relatively big mass, which is an approximation of some continuous elastic system, and of the vibro-impact oscillator with a relatively small mass, which is an absorber of the linear system vibrations. Analysis of nonlinear normal vibration modes shows that a stable localized vibration mode, which provides the vibration regime appropriate for the elastic vibration absorption, exists in a large region of the system parameters. In this regime, amplitudes of vibrations of the linear system are small, simultaneously vibrations of the absorber are significant.

Yuri V. Mikhlin

S. N. Reshetnikova

Dynamical interaction of an elastic system and a vibro-impact absorber

José Manoel Balthazar

Nonlinear dynamics and their applications to engineering sciences

Vibrations of a nonlinear oscillator with an attached pendulum,excited by movement of its point of suspension, have been analysedin the paper. The derived differential equations of motion showthat the system is strongly nonlinear and the motions of bothsubsystems, the pendulum and the oscillator, are strongly coupledby inertial terms, leading to the so-called autoparametricvibrations. It has been found that the motion of the oscillator,forced by an external harmonic force, has been dynamicallyeliminated by the pendulum oscillations. Influence of a nonlinearspring on the vibration absorption near the mainparametric resonance region has been carried out analytically,whereas the transition from regular to chaotic vibrations has beenpresented by using numerical methods. A transmission force on thefoundation for regular and chaotic vibrations is presented aswell.

J.  Warminski

K.  Kecik

Autoparametric vibrations of a nonlinear system with pendulum

This short contribution considers the essentials of nonlinear waveproperties in typical mechanical systems such as an infinitestraight bar, a circular ring, and a flat plate. It is found thatnonlinear resonance is experienced in all the systems exhibitingcontinuous and discrete spectra, respectively. Multiwaveinteractions and the stability of coupled modes with respect tosmall perturbations are discussed. The emphasis is placed onmechanical phenomena, for example, stress amplification, althoughsome analogies with some nonlinear optical systems are alsoobvious. The nonlinear resonance coupling in a plate within theKirchhoff-Love approximation is selected as a two-dimensionalexample exhibiting a rich range of resonant wave phenomena. Thisis originally examined by use of Whitham&#39;s averaged Lagrangianmethod. In particular, the existence of three basic resonanttriads between longitudinal, shear, and bending modes is shown.Some of these necessarily enter cascade wave processes related tothe instability of some mode components of the triad under smallperturbations.

D. A. Kovriguine

G. A. Maugin

A. I. Potapov

Multiwave nonlinear couplings in elastic structures

This paper deals with effects of large amplitude on the free andforced flexural vibrations of elastic orthotropic plates ofarbitrary shape. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>-function method (RFM) is applied to obtainthe basis functions need for expansion of sought solution intoFourier series. The initial nonlinear system of differentialequations with partial derivatives is reduced to system ofordinary nonlinear differential equations by Galerkin procedure.The solving-obtained system is carried out by Runge-Kutta orGalerkin methods. The numerical results for the plate of complexform and also rectangular form and different boundary conditionshave been presented and compared with other known results.

L. V. Kurpa

T. V. Shmatko

O. G. Onufrienko

Research of nonlinear vibrations of orthotropic plates with a complex form

We considered the problem on transversal oscillations
of two-layer straight bar, which is under the action of the
lengthwise random forces. It is assumed that the layers of the bar
were made of nonhomogenous continuously creeping material and the
corresponding modulus of elasticity and creeping fractional order
derivative of constitutive relation of each layer are continuous
functions of the length coordinate and thickness coordinates.
Partial fractional differential equation and particular solutions
for the case of natural vibrations of the beam of creeping
material of a fractional derivative order constitutive relation in
the case of the influence of rotation inertia are derived. For the
case of natural creeping vibrations, eigenfunction and time
function, for different examples of boundary conditions, are
determined. By using the derived partial fractional differential
equation of the beam vibrations, the almost sure stochastic
stability of the beam dynamic shapes, corresponding to the <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>th shape of the beam elastic form, forced by a bounded axially noise
excitation, is investigated. By the use of S. T. Ariaratnam's
idea, as well as of the averaging method, the top Lyapunov exponent
is evaluated asymptotically when the intensity of excitation
process is small.

Katica (Stevanovic) Hedrih

The transversal creeping vibrations of a fractional derivative order constitutive relation of nonhomogeneous beam

In the present work, hysteresis is simulated by means of internal
variables. Analytical models of different types of hysteresis
loops enable reproduction of major and minor loops and provide
good agreement with experimental data. Using an effective
algorithm based on the analysis of wandering trajectories, an
evolution of chaotic behavior regions of oscillators with
hysteresis is presented in various parametric planes. A
substantial influence of a hysteretic dissipation value on the
form and location of these regions and also restraining and
generating effects of the hysteretic dissipation on chaos
occurrence are shown. It was demonstrated that for fixed
parameters, which govern the shape of a hysteresis loop, the
extent of pinch of this loop could be controlled by means of the
amplitude and frequency of external periodic excitation.

J. Awrejcewicz

L. Dzyubak

Modeling, chaotic behavior, and control of dissipation properties of hysteretic systems

Recent progress in the area of nonlinear modal analysis forstructural systems is reported. Systematic methods are developedfor generating minimally sized reduced-order models thataccurately describe the vibrations of large-scale nonlinearengineering structures. The general approach makes use ofnonlinear normal modes that are defined in terms of invariantmanifolds in the phase space of the system model. An efficientGalerkin projection method is developed, which allows for theconstruction of nonlinear modes that are accurate out to largeamplitudes of vibration. This approach is successfully extended tothe generation of nonlinear modes for systems that are internallyresonant and for systems subject to external excitation. Theeffectiveness of the Galerkin-based construction of the nonlinearnormal modes is also demonstrated for a realistic model of arotating beam.

Christophe  Pierre

Dongying  Jiang

Steven  Shaw

Nonlinear normal modes and their application in structural dynamics

The methods of both analysis and modeling of contact bush-shaft
systems exhibiting heat generation and wear due to friction are
presented [3–5]. From the mathematical point of view, the
considered problem is reduced to the analysis of ordinary
differential equations governing the change of velocities of the
contacting bodies, and to the integral Volterra-type equation
governing contact pressure behavior. In the case where
tribological processes are neglected, thresholds of chaos are
detected using bifurcation diagrams and Lyapunov exponents
identification tools. In addition, analytical Mel'nikov's method
is applied to predict chaos. It is shown, among the others, that
tribological processes play a stabilizing role. The following
theoretical background has been used in the analysis: perturbation
methods, Mel'nikov's techniques [7,8], Laplace
transformations, the theory of integral equations, and various
variants of numerical analysis.

Jan Awrejcewicz

Yuriy Pyryev

Regular and chaotic motion of a bush-shaft system with tribological processes

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Chief Editor, Professor Guangming Xie, is currently a full professor of dynamics and control with the College of Engineering, Peking University. His research interests include complex system dynamics and control and intelligent and biomimetic robots.

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