Frequency formula for a class of fractal vibration system

Four fractal nonlinear oscillators (The fractal Duffing oscillator, fractal attachment oscillator, fractal Toda oscillator, and a fractal nonlinear oscillator) are successfully established by He’s fractal derivative in a fractal space, and their variational principles are obtained by semi-inverse transform method. The approximate frequency of the four fractal oscillators are found by a simple frequency formula. The results show the frequency formula is a powerful and simple tool to a class of fractal oscillators.

Implicit implementation of the nonlocal operator method: an open source code

In this paper, we present an open-source code for the first-order and higher-order nonlocal operator method (NOM) including a detailed description of the implementation. The NOM is based on so-called support, dual-support, nonlocal operators, and an operate energy functional ensuring stability. The nonlocal operator is a generalization of the conventional differential operators. Combined with the method of weighed residuals and variational principles, NOM establishes the residual and tangent stiffness matrix of operate energy functional through some simple matrix without the need of shape functions as in other classical computational methods such as FEM. NOM only requires the definition of the energy drastically simplifying its implementation. The implementation in this paper is focused on linear elastic solids for sake of conciseness through the NOM can handle more complex nonlinear problems. The NOM can be very flexible and efficient to solve partial differential equations (PDEs), it’s also quite easy for readers to use the NOM and extend it to solve other complicated physical phenomena described by one or a set of PDEs. Finally, we present some classical benchmark problems including the classical cantilever beam and plate-with-a-hole problem, and we also make an extension of this method to solve complicated problems including phase-field fracture modeling and gradient elasticity material.

THEORETICAL BASIСS OF PROVIDING THE TECHNICAL CHARACTERISTICS OF MILITARY AND CIVIL VEHICLES BY JUSTIFICATION OF THE FORM AND PROPERTIES OF THE MATERIALS OF CONTACTING ELEMENTS

Elements of constructions of modern military and civil vehicles usually work in conditions of high contact loads. Аt the stage of their creation, strength studies are carried out using traditional models of contact of bodies of nominal shape. Нowever, the real structural elements have deviations from such models, which are due to design and technological factors: macrodeviation of the shape, surface roughness, strengthening etc. Such perturbations of nominal parameters have a significant effect on the distribution of contact pressure between the elements of military and civil vehicles, however, traditional methods for studying the stress-strain state of contacting bodies do not make it possible to take such factors into account fully, collectively and exhaustively. To eliminate the existing contradiction, a semi-analytical method is proposed, which is based on the development of variational principles and boundary-element sampling. The created models make it possible to take into account the regularities of the influence of shape perturbations and properties of the surface layers of contacting bodies on the stress-strain state. As a result, it becomes possible to justify favorable perturbations by strength criteria. Such models and methods are offered to the work, and on their basis it’s proposed the implementation of research elements of military and civil vehicles for appointment to ensure world class the technical and tactically technical characteristics. Ключові слова: military and civilian vehicles; design and technological factor; stress-strain state; contact interaction; strength

Nonpotentiality of a diffusion system and the construction of a semi-bounded functional

The wide prevalence and the systematic variational principles are used in mathematics and applications due to a series of remarkable consequences among which the possibility to establish the existence of the solutions of the initial equations, and the determination of stable approximations of the solutions of the considered equations by the so-called variational methods. In this connection, it is natural for a given system of equations to investigate the problem of the existence of its variational formulations. It can be considered as the inverse problem of the calculus of variations. The main goal of this work is to study this problem for a diffusion system of partial differential equations. A key object is the criterion of potentiality. On its ground, the nonpotentiality of the operator of the given boundary value problem with respect to the classical bilinear form is proved. This system does not admit a matrix variational multiplier of the given form. Thus, the diffusion system cannot be deduced from the classical Hamilton’s principle. We posed the question that whether there exists a functional semi-bounded on solutions to the boundary value problem. We have done the algorithm of the constructive determination of such a functional. The main value of constructed functional action will be in applications of direct variational methods.

SUBSTANTIATION OF DESIGN DECISIONS OF ELEMENTS OF MILITARY OBJECTS IN CONDITIONS OF CONTACT AND PLASTIC DEFORMATION

New design solutions, technologies and materials are required to improve tactical and technical characteristics of military equipment. Often this implies operation in such conditions as contact interaction and elasto-plastic deformations of materials. New models and research methods are developed for better utilization of modern materials and improved performance of military equipment. They account directly for complex physical and structural nonlinearities. The properties of conventional and novel materials are determined both in bulk and on surfaces at microstructural level. This will enable physically adequate and mathematically correct analysis of stress-strain state. The new advanced design solutions will emerge through the objective-driven search by means of parametric modeling. The project will extend traditional local problem statements with newly developed variational principles that account for structural and physical nonlinearity and are suitable for parameterization. This will create the basis for fundamental analysis of torsion bar suspensions, hydrovolumetric and gear drives and other crucial components of combat vehicles, engineering solutions for domestic manufacturers of military equipment that will bring their tactical and technical characteristics to highest modern standards. Keywords: contact interaction; stress-strain state; intermediate layer; contact pressure; contact area; plastic deformation

A NOVEL VARIATIONAL PERSPECTIVE TO FRACTAL WAVE EQUATIONS WITH VARIABLE COEFFICIENTS

This paper aims at establishing two different types of wave models with unsmooth boundaries by the fractal calculus, and their fractal variational principles are successfully designed by employing the fractal semi-inverse transform method. A new approximate technology is proposed to solve the two fractal models based on the variational principle and fractal two-scale transform method. Finally, two numerical examples show that the proposed method is efficient and accurate, which can be extended to solve different types of fractal models.