Integration Of Motion Equations Research Articles

Peridynamic material model calibration based on digital image correlation experimental measurements

The non-local theory of peridynamics, based on integral equation of motion, offers a computation alternative for materials with discontinuities. In this study, the peridynamic material model parameters are calibrated using full field Digital Image Correlation measurements applied to simple tensile test experiments, circumventing thus the need for more challenging test setups/methods for the determination of the peridynamic material parameters (especially the critical stretch failure parameter). A two stage algorithm in the optimization software LS-Opt together with a MATLAB stage for performing the peridynamic simulation and an Excel stage for defining the outputs of the simulation are implemented to minimize the differences between the peridynamic model and experimental results. The material model parameters determined based on the simple test setup and the calibration procedure proposed here can then be used for modeling more challenging geometries and load cases in peridynamics.

Difference schemes for dynamics problems

To solve dynamics problems in the LIRA 10.12 software package, the difference scheme is used, known as modified central difference method. The goal of dynamic problems solving is to get a good approximation of actual dynamic response of a given structure. It's a matter of convergence conditions of the difference scheme used in numerical integration of motion equations.
 The solution of the linear dynamic problem for all possible displacements satisfies the equations
 ,
 , , – symmetric positive-definite bilinear functionals of possible work of the internal and inertial forces and motion resistance forces, they correspond to stiffness matrices, mass and damping matrices, – linear functional of possible work of the external forces.
 After approximation in spatial variables (usually the finite element method is applied) we obtain the system of ordinary differential equations
 ,
 , , – stiffness matrices, mass and damping matrices, – external forces.
 Difference scheme of modified central difference method we obtain by replacing the values of functions and derivatives with corresponding difference relations. Difference relations, which are applied in modified central difference method, approximate accelerations , velocities and displacements with an error proportional to the square of the time step.
 Approximation and stability are the convergence conditions of the difference scheme. To study the stability of difference schemes, we assume that and apply the obvious equations
 ,
 .
 Let us denote , apply the Cauchy inequality on the right-hand side and integrate the resulting inequality. Then
 ,
 where are positive constants. The boundedness follows from the Gronwall's lemma. In addition to the continual version of study of the difference schemes stability, the discrete version is also considered.
 Unconditional practical efficiency of the modified central difference method and the simplicity of its underlying principles allow it to be successfully applied to a wide range of dynamic problems.

Discretization and superintegrability all rolled into one

Abelian integrals appear in mathematical descriptions of various physical processes. According to Abel’s theorem these integrals are related to motion of a set of points along a plane curve around fixed points, which are rarely used in physical applications. We propose to interpret coordinates of the fixed points either as parameters of exact discretization or as additional first integrals for equations of motion reduced to Abelian quadratures on a symmetric product of algebraic curve.

Dynamics and Stability Analysis of Rotating Cylindrical Shells in Annular Fluid Medium

Stability and dynamics of rotating coaxial cylindrical shells conveying incompressible and inviscid fluid are investigated. The interior shell is assumed to be flexible while the exterior cylinder is rigid. Using Sander’s–Koiter theory assumptions and following Hamilton’s principle, governing equations of motion are determined in their integral form. Employing the extended Galerkin method of solution, the integral equations of motion are projected to their equivalent system of algebraic equations. Fluid equations are fundamentally based on the linearized inviscid Navier–Stokes equations. Impermeability condition on the fluid and structure interface as well as the zero radial velocity component on the exterior shell give the coupled equations of motion governing the dynamics of fluid-loaded coaxial cylindrical shells. Using the coupled fluid–structural model, stability boundaries are determined for the rotating interior shell. Various parameter studies are conducted and effects of mass ratio, gap distance between the interior and exterior shells, boundary conditions of the interior shell, length to radius ratio on the stability margins are thoroughly investigated and reported.

Twisted Cohomotopy Implies M-Theory Anomaly Cancellation on 8-Manifolds

We consider the hypothesis that the C-field 4-flux and 7-flux forms in M-theory are in the image of the non-abelian Chern character map from the non-abelian generalized cohomology theory called J-twisted Cohomotopy theory. We prove for M2-brane backgrounds in M-theory on 8-manifolds that such charge quantization of the C-field in Cohomotopy theory implies a list of expected anomaly cancellation conditions, including: shifted C-field flux quantization and C-field tadpole cancellation, but also the DMW anomaly cancellation and the C-field's integral equation of motion.

ДО АНАЛІЗУ КОНСТРУКЦІЙ КОЛІСНИХ ТРАНСПОРТНИХ ЗАСОБІВ ДЛЯ МІСЬКИХ ПЕРЕВЕЗЕНЬ ПАСАЖИРІВ

The article discusses the development of public transport as an important means of safe population mobility, especially in urban areas suffering from increasing traffic congestion. Bus or Transport Rapid Bus (BRT) or metrobus - this is the name of a sophisticated bus system. The BRT project envisages the movement of buses on dedicated and often fenced lanes. The main advantage of a metro is its complete isolation on the road from other modes of transport. Three-lane buses and trolleybuses can be used to increase passenger capacity in the BRT system. Along with the undeniable advantages of three-section articulated buses and trolleybuses, they also have disadvantages - worse maneuverability and stability of movement compared to two-section ones. In addition, the efficiency of such machines is closely linked to the passenger traffic, which may vary several times during the day. Therefore, it may be promising for trains to be made up of two (or three) buses or trolleybuses, which operate in a hitch, whose passenger capacity is similar to articulated buses and trolleybuses. During the peak hours, a bus train operates, and in the inter-peak period - each bus separately (parking of one bus on a dedicated site is possible). A comparison of these two bus trains - three-link articulated and three-link trailed according to maneuverability. The comparison is based on the differential equations of motion of these trains. The integration of motion equations is based on the original data. obtained both by the technical characteristics of the buses and by calculating the torque resistance, inertia, resistance coefficients, etc. The original equation system was solved using Maple software. The coordinates of the center of mass of the bus, which determined the overall lane (GSR) of the bus, were obtained. The analysis of the obtained data showed that the three-link trailer trains with guided axles of trailers meet the requirements of normative documents on maneuverability. The GSR of such a high-speed train is 7,193 m. For a semi-trailer high-speed train, a high-speed train with two unmanaged trailer links of the GSR is 7.39 m, which is somewhat more permissible. In this case, both controlled and unmanaged high-speed trains have insufficient turn, which can serve as a guarantee of stability.

MATHEMATICAL MODELING OF DYNAMIC PROCESSES IN THE HULL OF LIGHTLY ARMORED VEHICLE UNDER IMPULSIVE ACTION

The paper describes the questions of modeling of dynamic processes in lightly armored vehicles. The studied object is associated with a parameter set that adequately describes battle module, armored hull and suspension as a dynamic system. The obtained set of variables completely identifies this dynamic system. It is proposed to carry out numerical modeling based on the integration of motion equations by the finite element method in order to determine the rational parameters. On the basis of analysis of researches results, parameters are determined that satisfy the relevant criteria and constraints. In particular, dangerous resonance modes are detected and the criteria for disabling from these regimes are formulated.

Dynamic analysis of strength and human-comfort level of the football stadium structures at coordinated movements of audience

In the design of stadiums and other sports facilities it is necessary to take into account the dynamic performance caused by coordinated movement of groups of people. National standard technical documents do not contain guidance on the setting of dynamic loads caused by people, as well as admissible dynamic parameters of grandstands’ structures in the stadiums caused by such actions that should be taken in consideration in the course of design. Therefore, it is necessary to develop appropriate recommendations. Based on the developed recommendations, analysis of stadium structures for dynamic loads caused by the audience movement was performed in the time domain by direct integration of motion equation using certified software package Lira 10.4. It was obtained that the dynamic human-comfort level in the grandstands and in the under-grandstand facilities was ensured. Recommendations were made based on the analysis of international and Russian standard technical documents, generalization of available experience in design, construction, and operation of such objects, including similar unique large span structures. Obtained accelerations of under-grandstand facilities were analyzed in accordance with the construction standards SN 2.2.4/2.1.8.566-96. It was revealed that the limit values for some structure points have been exceeded.

A research on Performance Efficiency of Rubber Metal Support Structures

The paper scrutinizes structural behavior of lead rubber bearings by a Chinese manufacturer subjected to a single-component seismic action. Several problems were solved using specialized software complexes, which conducted forth integration of motion equations through the explicit method or response spectrum method. Depending on the calculation method, the diagram of the bearing performance was assumed to be either an actual diagram approximated by an idealized non-linear diagram or an idealized linear diagram with a specific stiffness. The computational model was assumed to be a single-mass oscillator with a lumped mass. The effort undertaken facilitated the investigation of the patterns of horizontal displacement of the bearing top relative to bottom caused by earthquakes modeled as accelerograms with different spectral compositions. The behavior of the support structure was benchmarked against similar supports by another manufacturer. The paper presents the outcomes of the research effort and draws conclusions about the efficiency of using the bearings of this particular type and model.

Stochastic Response Characteristic and Equivalent Damping of Weak Nonlinear Energy Dissipation System under Biaxial Earthquake Action

The random response characteristic of weak nonlinear structure under biaxial earthquake excitation is investigated. The structure has a SDOF (single degree of freedom) with supporting braces and viscoelastic dampers. First, it adopts integral constitutive relation and establishes a differential and integral equations of motion. Then, according to the principle of energy balance, the equation is linearized. Finally, based on the stochastic averaging method, the general analytical solution of the variance of the displacement and velocity response and the equivalent damping is deduced and derived. At the same time, the joint probability density function of the amplitude and phase and displacement and velocity of the energy dissipation structure are also given. The dynamic characteristics of a structure with viscoelastic dampers are determined as a solution to the variance of displacement response, so the equivalent damping is taken into consideration as a solution to replace the original nonlinear damping. It means it has established a unified analytical solution of stochastic response analysis and equivalent damping of a SDOF nonlinear dissipation structure with the brace under biaxial earthquake action in this paper.

Research of Influence of Soil Strength Failure on the Initial Seismic Action Transformation

The article examines the base - foundation slab system joint work with the nonlinear nature of the work and potential loss of base soil strength taking into account. As a source of the seismic action a three component accelerogram of an earthquake, rated to 9 points, was adopted. Base soil was modeled by means of a non-linear Mohr-Coulomb model, where failure conditions with specified limit values of shear strain and stress limitations with not working in tension soil have been entered. The calculation was made in LS-DYNA software package by a direct dynamic method with integration of motion equations using an explicit scheme. The article presents a comparison of results obtained with and without strength soil failure, as well as a research of original seismic impact transformation during weight loading facilities of the construction site was carried out.

Perturbed motion at small eccentricities

In the study of the motion of planets and moons, it is often necessary to have a simple approximate analytical motion model, which takes into account major perturbations and preserves almost the same accuracy at long time intervals. A precessing ellipse model is used for this purpose. In this paper, it is shown that for small eccentricities this model of the perturbed orbit does not correspond to body motion characteristics. There is perturbed circular motion with a constant zero mean anomaly. The corresponding solution satisfies the Lagrange equations with respect to Keplerian orbital elements. There are two families of solutions with libration and circulation changes in the mean anomaly close to this particular solution. The paper shows how the eccentricity and mean anomaly change in these solutions. Simple analytical models of the motion of the four closest moons of Jupiter consistent with available ephemerides are proposed, which in turn are obtained by the numerical integration of motion equations and are refined by observations.

The rolling of a body with a rotor on a moving supporting sphere

The rolling without slipping of a body with a rotor on a mobile supporting sphere in a uniform gravitational field is considered. The boundary of the body in the contact area with the support is part of a spherical surface. The central ellipsoid of inertia of the system (body+rotor) is an ellipsoid of revolution, the axis of which passes through the geometrical centre of the sphere, not, generally speaking, coinciding with the system mass centre. The supporting sphere is displaced translationally in an arbitrary way and is rotated around a vertical axis. The complete system of equations of motion of the supporting body and the rotor is obtained. Two integral equations of motion are obtained in the case of a solid of revolution. In the case when the body is a homogeneous sphere, four integral equations of motion are obtained, where the coordinates of the point of contact of the sphere and the supporting sphere are determined by quadratures, and all possible trajectories of the point of contact of the sphere and the body are indicated.

Integration of Navier-Stokes equations

The author considers the integration of the motion equations of a Newtonian fluid (Navier-Stokes equations) in vector form, taking into consideration a separation of vector Navier-Stokes equation on the two equations containing separately linear and quadratic terms. On this basis, the paper demonstrates the possibility of integration of separated motion equations of an incompressible viscous fluid, which is determined in a greater extent by the characteristics of flow: boundary conditions, axisymmetric, nonaxisymmetric flow and others.

Особенности расчета сейсмоизолированного здания по перемещениям

The article regards calculation of 16-storied building with seismic isolation in the form of elastomeric bearings on two-component accelerograms with different dominant frequencies. The problem was solved in software package LS-DYNA by forth integration of motion equations according to explicit scheme. The research showed the dependence of horizontal displacements of bearing top in relation to bottom at earthquakes given by accelerograms with different spectral structure. The article analyzes the results of the work.

Simulation of vibroshock in two-dimensional systems with inherited properties

Strongly nonlinear standing waves in two-dimensional lattice systems—formed by mutually perpendicular strings and a system of absolutely hard point bodies for connecting the strings, say—are considered. Path limiters around the bodies in the lattice system generate vibrational shock. Operator and integral equations of motion are presented. Some types of periodic standing waves are constructed. In examples, the corresponding periodic motion is described.

Complete list of first integrals for dynamic equations of motion of a solid body in a resisting medium with consideration of linear damping

A new case of integrability in the spatial problem of motion for a solid body with consideration of the nonconservative moment of forces is discussed. A nonconservative force field of action of the medium on the body is constructed. Contrary to some previous author’s works, a linear dependence of this field on the angular velocity is taken into account, although the introducing of this dependence into the components of such a field is not obvious in advance.

Multi-objective analysis of dam release flows in rivers downstream from hydropower reservoirs

This paper assesses dam releases from hydropower reservoirs in order to optimize power production and fish habitat protection. A multi-objective programming model includes output from 2-D hydraulic simulation for habitat assessment to optimize power production and fish habitat suitability as a Pareto set. To identify the optimal Pareto set two different approaches are used and compared: ε-constraint methods and non-dominant-sorting genetic algorithm (NSGA II). To formulate the ecological objective the river habitat quality is quantified by the weighted usable area (WUA). The relation between the WUA and the river flow-rate is obtained by using a 2D hydraulic model in which the hydraulic characteristics of river current – depth and velocity – are calculated by a finite difference numerical integration of two-dimensional shallow water equations on a boundary fitted non orthogonal curvilinear mesh. This approach allows the integration of motion equations on geometrically complex domains as those representing the morphology of natural watercourses. The performance of the proposed methodology is analyzed in a case study of a stretch of the Piave river downstream of the dam of the Pieve di Cadore reservoir (Belluno, Italy).

Thermal performance of a naturally ventilated cavity wall

Cavity walls are often proposed in the building envelope design as a solution for improving the thermal comfort of the occupants and reducing the adverse condensation effects on the building fabric. Although the behaviour of a non-ventilated cavity wall is well-known, more studies are required when cavity ventilation is allowed. In order to consistently predict the thermal behaviour of a naturally ventilated cavity wall, a convective model based on the integral equations of motion and enthalpy was developed and applied in the present study. The model is presented as a combination of two limiting cases of a steady laminar flow into the channel gap: fully developed flow and boundary layer flow. Conduction effects across the system are also included through a proper limiting case and then combined with the convective model. In addition a numerical CFD model was developed that provides solution for free convective flow configurations between two parallel conducting vertical walls. For comparison purposes, some test cases were simulated with the two models and a general good agreement was found between results. Finally, the integral model was applied to assess the thermal performance of a ventilated cavity wall for winter and summer conditions. Copyright © 2009 John Wiley & Sons, Ltd.

Elastic Wave Scattering on a Thin‐Walled Inclusion

AbstractThe dynamic three‐dimensional problem of interaction between an elastic wave and a thin‐walled elastic inclusion is considered. The incident wave length is comparable to the large linear size of the inclusion. Integral equations of motion are obtained by means of the asymptotic approach proposed. The deflections of a circular plate and a fragment of a spherical shell embedded in the medium are calculated in the case of axial symmetry.