Parametric stability of a double pendulum with variable length and with its center of mass in an elliptic orbit

<p style='text-indent:20px;'>We consider the planar double pendulum where its center of mass is attached in an elliptic orbit. We consider the case where the rods of the pendulum have variable length, varying according to the radius vector of the elliptic orbit. We make an Hamiltonian view of the problem, find four linearly stable equilibrium positions and construct the boundary curves of the stability/instability regions in the space of the parameters associated with the pendulum length and the eccentricity of the orbit.</p>

Nuclear Adiabatic Effects of Electron-Positron Pairs on the Dynamical Instability of Very-Massive Stars

The adiabatic effects of electron-positron pair-production on the dynamical instability of very-massive stars is investigated from stellar progenitors of carbon-oxygen cores within the range of 64 M < MCO < 133 M both with and without rotation. At a very high temperature and relatively low density; the production of electron-positron pairs in the centres of massive stars leads the adiabatic index to below 4/3. The adiabatic quantities are evaluated by constructing a model into a thermodynamically consistent electron-positron equation of state (EoS) table. It is observed that the adiabatic indices in the instability regions of the rotating models are fundamentally positive with central temperature and density. Similarly, the mass of the oxygen core within the instability region has accelerated the adiabatic indices in order to compress the star, while the mass loss and adiabatic index in the non-rotating model exponentially decay. In the rotating models, a small amount of heat is required to increase the central temperature for the end fate of the massive stars. The dynamic of most of the adiabatic quantities show a similar pattern for all the rotating models. The non-rotating model may not be suitable for inducing the instability. Many adiabatic quantities have shown great effects on the dynamical instability of the massive stars due to electron-positron pair-production in their centres. The results of this work would be useful for better understanding of the end fate of very-massive stars.

INVESTIGATION OF THE NORMAL VIBRATION MODES STABILITY IN SOME ESSENTIALLY NONLINEAR SYSTEMS

In the paper stability of nonlinear normal modes is analyzed by two approaches. One of them is the method of Ince algebraization, when a new independent variable associated with the unperturbed solution is introduced in the problem. In this case equations in variations are transformed to equations with singular points. The problem of determination of solutions corresponding to boundaries of the stability/ instability regions is reduced here to the problem of determination of functions that have singularity at the mentioned points. Such solutions can be obtained in the form of power series, which coefficients are satisfying a system of homogeneous linear algebraic equations. The condition ensuring the existence non-trivial solutions for such systems determines the boundaries between the stability / instability regions in the system parameter space. An advantage of the Ince algebraization is that we do not use the time-presentation of the solution when studying its stability. Other approach to investigating steady state stability is associated with the classical Lyapunov definition of stability. The analytical-numerical test proposed in the paper can be applied to a stability problem when the problem has no analytical solution. It also allows to obtain boundaries between the stability / instability regions in the system parameter space. In the present paper the first approach is used to analyze stability of normal vibration modes in the system of connected oscillators on the essentially nonlinear elastic support, and the second one is used to analyze stability of a horizontal vibration mode in the so-called stochastic absorber.

Free Vibration and Instability Analysis of Sandwich Plates with Carbon Nanotubes-Reinforced Composite Faces and Honeycomb Core

In this study, the instability regions of a honeycomb sandwich plate are investigated for different end conditions under periodic in-plane loading. The core layer of the sandwich plate is made of carbon nanotube (CNT)/glass fiber-reinforced honeycomb and the face layers of CNT/glass fiber- reinforced laminated composite. The governing equations are derived using classical laminated plate theory (CLPT) and solved numerically by using finite element formulation. The effectiveness of the developed finite element formulation is demonstrated by comparing the results in terms of natural frequencies with those available in the literature. The effects of CNT wt.% on the core material, CNT wt.% on the skin material, ply orientation and various end conditions on the variation of natural frequencies, loss factors and instability regions are studied. Finally, some inferences for the effects of CNT reinforcement on the honeycomb sandwich plate subjected to the periodic in-plane loads are discussed.

OPTIMIZATION OF ULTRASOUND MEDICAL PARAMETERS TOOLS

Для многих видов медицинских вмешательств требуется применение ультразвуковых инструментов с различными характеристиками. Используются инструменты, совершающие продольные колебания, значительно реже - инструменты с изгибами и крутильными колебаниями, либо достаточно длинные ультразвуковые медицинские инструменты, либо короткие, но тонкие. В таких инструментах часто наблюдается так называемая динамическая потеря устойчивости, когда прямолинейный инструмент, совершающий продольные колебания, внезапно начинает совершать изгибные колебания, амплитуда которых бывает настолько высока, что приводит к разрушению инструмента. Такое явление также называют параметрическим резонансом ультразвуковых инструментов. Цель статьи - анализ условий и параметров, позволяющих минимизировать травматичность применения ультразвуковых медицинских инструментов, исследование в динамике устойчивости ультразвуковых низкочастотных медицинских инструментов. Для определения оптимального набора параметров динамической устойчивости изгибных колебаний ультразвуковых низкочастотных медицинских инструментов используется уравнение Матье-Хилла. В этом аспекте решение задачи сводится к определению: 1) границ областей неустойчивости уравнения Матье; 2) границ областей неустойчивости при разных значениях коэффициента возбуждения; 3) границ областей неустойчивости с применением метода малого параметра. Для исследования динамической устойчивости уравнения колебаний прямолинейного стержня переменного сечения достаточно выполнить расчет коэффициентов уравнения Матье и использовать диаграмму Айнса-Стретта для нахождения точек попадания в область устойчивости. Результаты расчетов показали, что инструменты, изготовленные из титана, обладают высокой динамической устойчивостью, что практически исключает вероятность их разрушения при проведении медицинских операций. Полученные характеристики медицинских инструментов указывают на эффективность их применения в медицинской практике Many types of medical interventions require the use of ultrasound instruments with different characteristics. Instruments that perform longitudinal vibrations are used, much less often-instruments with bends and torsional vibrations, or rather long ultrasound medical instruments, or short, but thin. In such instruments, the so-called dynamic loss of stability is often observed, when a straight-line tool that performs longitudinal vibrations suddenly begins to make bending vibrations, the amplitude of which is so high that it leads to the destruction of the tool. This phenomenon is also called parametric resonance of ultrasonic instruments. The purpose of the article is to analyze the conditions and parameters that allow minimizing the traumaticity of the use of ultrasonic medical instruments, to study the dynamics of the stability of ultrasonic low-frequency medical instruments. The Mathieu-Hill equation is used to determine the optimal set of parameters for the dynamic stability of bending vibrations of ultrasonic low-frequency medical instruments. In this aspect, the solution of the problem is reduced to the definition of: 1) the boundaries of the instability regions of the Mathieu equation; 2) the boundaries of the instability regions at different values of the excitation coefficient; 3) the boundaries of the instability regions using the small parameter method. To study the dynamic stability of the equation of oscillations of a rectilinear rod of variable cross-section, it is sufficient to calculate the coefficients of the Mathieu equation and use the Ains-Strett diagram to find the points of falling into the stability region. The results of the calculations showed that the instruments made of titanium have a high dynamic stability, which practically eliminates the possibility of their destruction during medical operations. The obtained characteristics of medical instruments indicate the effectiveness of their use in medical practice

Hot Processing Map of an Al–4.30 Mg Alloy under High One-Pass Deformation

The high one-pass deformation behaviors of mass-produced Al–4.30Mg alloy are investigated in the temperature ranging of 350 °C–500 °C, the strain rate ranging of 0.01 s−1–1 s−1 and the reduction ranging of 50–75%. 3D processing maps are constructed by the superimposition of the instability map and the power dissipation map at the true strain of 0.69, 0.92, 1.20 and 1.38. When the true strain increases from 0.69 to 1.38, the average apparent activation energy (Q) decreases from 140.3 kJ/mol to 112.7 kJ/mol, indicating the reduction of the hot deformation energy barrier. The heating caused by a large strain of 1.38 greatly reduces the Q and improves processing efficiency. The instability regions at the strain of 0.69 appear at two domains, namely 350 °C/1.0 s−1 and 450 °C/1.0 s−1; whereas, the instability regions disappear at the strain of 1.38. The maximum efficiency of power dissipation is about 48%, which occurs at both domains of 440–480 °C/0.01 s−1/0.69 true strain and 470–500 °C/1.0 s−1/1.20 true strain. High-efficiency domains represent the optimized deformation conditions which are verified by stress-strain curves and microstructure characterization, in which the local dynamic recrystallization is observed and the power dissipates mainly by dynamic recrystallization during deformation.

Exact stability and instability regions for two-dimensional linear autonomous multi-order systems of fractional-order differential equations

Necessary and sufficient conditions are explored for the asymptotic stability and instability of linear two-dimensional autonomous systems of fractional-order differential equations with Caputo derivatives. Fractional-order-dependent and fractional-order-independent stability and instability properties are fully characterised, in terms of the main diagonal elements of the systems’ matrix, as well as its determinant.

A THEORETICAL STUDY OF OSCILLATIONS OF TWO IMMISCIBLE FLUIDS IN A LIMITED TANK

In the paper, the nonlinear oscillations of a two-layer fluid that completely fills a limited tank are theoretically studied. To determine any smooth function on the deflected interface, the Taylor series expansions are considered using the values of the function and its normal derivatives on the undisturbed interface of the fluids. Using two fundamental asymmetric harmonics, which are generated in two mutually perpendicular planes, the differential equations of nonlinear oscillations of the two-layer fluid interface are investigated. As a result, the frequency-response characteristics are presented and the instability regions of the forced oscillations of the two-layer fluid in the cylindrical tank are plotted, as well as the parametric resonance regions for different densities of the upper and lower fluids. The Bubnov-Galerkin method is used to plot instability regions for the approximate solution to nonlinear differential equations. At the final stage of the work, the nonlinear effects resulting from the interaction of fluids with a rigid tank that executes harmonic oscillations at the interface of the fluids are theoretically studied.