General Equations Of Motion Research Articles

Reflection and refraction of a plane sound wave in an elastic plate with an inhomogeneous anisotropic coating

In paper the problem of the reflection and refraction of a plane harmonic sound wave by a homogeneous isotropic elastic plate with an continuously inhomogeneous anisotropic elastic coating is considered. It is assumed that the plate is adjoin to ideal fluids. The propagation of small perturbations in a ideal fluid in the case of steady state oscillations is described by Helmholtz’s equation. The propagation of elastic waves in a uniform isotropic elastic plate is described by scalar and vector Helmholtz’s equations for longitudinal and transverse waves. Oscillations of an inhomogeneous anisotropic elastic coating are described by general motion equations of the continuous medium. The boundary-value problem for the system of ordinary second order differential equations is constructed for determination of the displacement field in inhomogeneous anisotropic coating. An analytical description of the reflected and transmitted through the plate acoustic fields is obtained. Particular cases when the material of an inhomogeneous coating is transversely isotropic and isotropic are considered. The results of numerical calculations of dependence of reflection coefficient of a homogeneous isotropic plate with a transversely isotropic coating from the incidence angle of plane wave are presented.

Free vibration of FGM conical–spherical shells

Natural frequencies of a conical–spherical functionally graded material (FGM) shell are obtained in this study. It is assumed that the conical and spherical shell components have identical thickness. The system of joined shell is made from FGMs, where properties of the shell are graded through the thickness direction. The first order shear deformation theory of shells is used to investigate the effects of shear strains and rotary inertia. The Donnel type of kinematic assumptions are adopted to establish the general equations of motion and the associated boundary and continuity conditions with the aid of Hamilton’s principle. The resulting system of equations are discretized using the semi-analytical generalized differential quadrature (GDQ) method. Considering various types of boundary conditions for the shell ends and intersection continuity conditions, an eigenvalue problem is established to examine the vibration frequencies. After proving the efficiency and validity of the present method for the case of thin isotropic homogeneous joined shells with the data of conventional finite element software, parametric studies are carried out for the system of combined moderately thick conical–spherical joined shells made of FGMs and various types of end supports.

Data-driven selection of coarse-grained models of coupled oscillators

Systematic discovery of reduced-order closure models for multi-scale processes remains an important open problem in complex dynamical systems. Even when an effective lower-dimensional representation exists, reduced models are difficult to obtain using solely analytical methods. Rigorous methodologies for finding such coarse-grained representations of multi-scale phenomena would enable accelerated computational simulations and provide fundamental insights into the complex dynamics of interest. We focus on a heterogeneous population of oscillators of Kuramoto type as a canonical model of complex dynamics, and develop a data-driven approach for inferring its coarse-grained description. Our method is based on a numerical optimization of the coefficients in a general equation of motion informed by analytical derivations in the thermodynamic limit. We show that certain assumptions are required to obtain an autonomous coarse-grained equation of motion. However, optimizing coefficient values enables coarse-grained models with conceptually disparate functional forms, yet comparable quality of representation, to provide accurate reduced-order descriptions of the underlying system.

Ordinary Differential Equation with Left and Right Fractional Derivatives and Modeling of Oscillatory Systems

We consider the principle of least action in the context of fractional calculus. Namely, we derive the fractional Euler–Lagrange equation and the general equation of motion with the composition of the left and right fractional derivatives defined on infinite intervals. In addition, we construct an explicit representation of solutions to a model fractional oscillator equation containing the left and right Gerasimov–Caputo fractional derivatives with origins at plus and minus infinity. We derive a representation for the composition of the left and right derivatives with origins at plus and minus infinity in terms of the Riesz potential, and introduce special functions with which we give solutions to the model fractional oscillator equation with a complex coefficient. This approach can be useful for describing dissipative dynamical systems with the property of heredity.

Ricci-inverse gravity: A novel alternative gravity, its flaws, and how to cure them

We introduce a novel theory of gravity based on the inverse of the Ricci tensor, that we call the anticurvature tensor. We derive the general equations of motion for any Lagrangian function of the curvature and anticurvature scalars. We then demonstrate a no-go theorem: no Lagrangian that contains terms linear in any positive or negative power of the anticurvature scalar can drive an evolution from deceleration to acceleration, as required by observations. This effectively rules out many realizations of this theory, as we illustrate in detail in a particular case. Finally, we speculate on how to circumvent the no-go theorem.

Lorentz Violation at the Level of Undergraduate Classical Mechanics

In this paper, we use the classical limit of the Standard-Model Extension to explore some generic features of Lorentz violation. This classical limit is formulated at the level of undergraduate physics. We first discuss the general equations of motion and then concentrate on three specific systems. First, we consider the theoretical aspects of pendulum motion in the presence of Lorentz violation, followed by some sample experimental results. The experimental bounds we achieve, in the range of 10−3, are not competitive with the current bounds from atomic clocks; rather, our experiment illustrates some common ideas and methods that appear in Lorentz-violation studies. We then discuss how Newton’s 2nd Law must be treated with caution in our model. Finally, we introduce a computational simulation of a binary star system that is perturbed by Lorentz-violating effects. This simulation shows some interesting behavior that could be the subject of future analytical studies.

Influence of the shape of the pressureless trapezoidal channel and roughness on the pressure loss of the machine channels of the pumping stations

In this work, we consider the general equation of fluid motion in pressureless channels and the functional dependences of the coefficient of hydraulic friction on the Reynolds number, relative roughness, and the shape of the live section of the channel, as well as the resistance formulas for the simplest channels with respect to the cross-sectional shape (round and infinitely wide rectangular), and then for channels with a more complex cross-sectional shape. To take into account the effect (on the pressure loss) of the channel cross-sectional shape and the presence of a flow with a free surface in it, additional correction factors are introduced (using the hydraulic radius concept). In conclusion, it is concluded that the values of the mentioned correction factors can and should be refined only as a result of the relevant experiments. The work also examined the patterns of hydraulic resistance in engine channels of the correct form of a live section with uniform turbulent fluid motion. Formulas of hydraulic resistance in the machine channels of a simple and complex outline of a live section are given.

Free vibration of joined cylindrical–hemispherical FGM shells

Free vibration response of a joined shell system including cylindrical and spherical shells is analyzed in this research. It is assumed that the system of joined shell is made from a functionally graded material (FGM). Properties of the shells are assumed to be graded through the thickness. Both shells are unified in thickness. To capture the effects of through-the-thickness shear deformations and rotary inertias, first-order shear deformation theory of shells is used. The Donnell type of kinematic assumptions is adopted to establish the general equations of motion and the associated boundary and continuity conditions with the aid of Hamilton’s principle. The resulting system of equations is discretized using the semi-analytical generalized differential quadrature method. Considering the clamped and free boundary conditions for the end of the cylindrical shell and intersection continuity conditions, an eigenvalue problem is established to examine the vibration frequencies of the joined shell. After proving the efficiency and validity of the present method for the case of thin isotropic homogeneous joined shells, some parametric studies are carried out for the system of combined moderately thick cylindrical–spherical shell system. Novel results are provided for the case of FGM joined shells to explore the influence of power-law index and geometric properties.

The effects of morphometric elements of the channel on hydraulic resistance of machine channels of pumping stations

Presents the results of hydraulic studies in pressureless and pressure water conduits, the results of which showed that the dependences obtained for calculating the hydraulic resistance in round pressure pipes cannot be distributed without the corresponding adjustments of pressureless channels (provided that the diameter of the pipe is replaced in the corresponding calculations with a value where hydraulic radius). In particular, it follows (and this is confirmed by the data of the corresponding experimental studies published in the literature) that in the case of pressureless channels, the pressure loss coefficient, and therefore the pressure loss, depends not only on the relative roughness and Reynolds number, but also on the shape of the cross section the channel. Moreover, if for hydraulically smooth pressureless channels this dependence can be considered to a first approximation clarified, then for channels whose wetted surface is characterized by the presence of a certain roughness, the question of its influence, as well as the shape of the channel cross section on the pressure loss, is far from roughly approximate solution. The paper also considers the general equation of fluid motion in pressureless channels and the functional dependences of the coefficient of hydraulic friction on the Reynolds number, relative roughness, and on the shape of the living section of the channel; resistance formulas are first for the simplest with respect to the cross-sectional shape of the channels (round and infinitely wide rectangular), and then for channels with a more complex cross-sectional shape. To take into account the effect (on the pressure loss) of the channel cross-sectional shape and the presence of a flow with a free surface in it, additional correction factors are introduced (using the hydraulic radius concept) as well as new dependencies and formulas for determining the hydraulic friction coefficient are given, s taking into account the influence of morphometric elements of the channel on the hydraulic resistance of the machine channels of pumping stations.

Study of the Variation of the β Factor and the Radius Through the Hollow Cathode Z-Pinch by Using Snowplow Model Simulation

The Hollow cathode z-pinch is designed and operated to investigate the ejected plasma through the hollow cathode. The hollow cathode is simulated using the snow plow model. The model uses three phases: the radial inward phase, the reflected shock phase and expansion phase. The differential equation of plasma motion and its electric circuit equation are derived and solved numerically using the code. The snow plow model suggested that the plasma radius decreases to zero, so this problem was solved adding the effect of the internal kinetic pressure and the shock wave into the general equation of motion using β factor. The effect of β factor (ratio of the kinetic pressure of plasma to magnetic pressure) is studied on plasma temperature, plasma radii, plasma velocity, pinching time and plasma elongation distance. The simulated results illustrate that the β factor varies from 0 to 0.79. The data illustrate that the elongation velocity of plasma is increased by decreasing the β factor, while the elongation displacement of the plasma is increased by increasing of the ring radii of the hollow cathode. The code illustrates that a multi-pinch of the CS could be appeared at high energy.

3D control stretched length of lambda-phage WLC DNA molecule by nonlinear optical tweezers

In this paper, the general Langevin equations of motion for the polystyrene bead linked to the lambda-phage worm-like chain DNA molecule embedded in the fluid under the nonlinear optical tweezers is derived in 3D space. Using the finite difference method, the dynamical properties of the bead trapped by the nonlinear optical tweezers using a thin layer of Acid Blue 29 are numerically studied. Results in, the stretched length of the lambda-phage worm-like chain DNA molecule can be controlled in 3D space by finely tuning of the laser power.

Formation Control of Multi Unmanned Aerial Vehicle Systems Based on DDS Middleware

In this paper formation control of Unmanned Aerial Vehicle System (UAVs) is introduced. L 1 controller with potential field technique and Data Distribution Service (DDS) middleware used for the navigation of the agents, which is multi-UAVs. L 1 adaptive controller used to stabilizing the general equations of motion of each UAVs and the potential field technique used to formalize the followers around the leader. The exchanging data between the leader and the followers done through publisher/subscriber DDS middleware. This L 1 controller has a high performance coming from a robust adaptation of it. Robustness of the L 1 controller verified using Matlab Simulation. The Lyapunov method provided the analysis and stability of the framework of UAVs.

Exact solution of free vibration of adjacent buildings interconnected by visco-elastic dampers

ABSTRACT With consideration of a high-rise coupled building system, a flexible beams-based analytical model is setup to characterize the dynamic behavior of the system. The general motion equation for the two beams interconnected by multiple viscous/visco-elastic dampers is rewritten into a non-dimensional form to identify the minimal set of parameters governing the dynamic characteristics. The corresponding exact solution suitable for arbitrary boundary conditions is presented. Furthermore, the methodology for computing the coefficients of the modal shape function is proposed. As an example, the explicit expression of the modal shape function is derived, provided only one damper is adopted to connect the adjacent buildings. Finally, to validate the proposed methodologies, three case studies are performed, in which the existence of the overdamping and the optimal damping coefficient are revealed. In the case of using one damper in connecting two similar buildings, the estimating equations for the first modal damping ratio are formulated.

Unified picture of electromagnetic and gravitational forces in two-spinor language

The well known Geodesic Equation of General Relativity is newly formulated in Weyl two-spinor language in a way susceptible of being generalized via a set of two-spinor equations, equivalent to the Lorentz Force of Electrodynamics, obtained in previous studies. General spinor equations of motion for a test charged spin 1/2 particle in arbitrary electromagnetic and gravitational fields are obtained describing not only spacetime trajectories but also spin precession. For the case of the Schwarzschild metric and radial trajectories, spinor equations unifying gravitational and electric forces are obtained and solved for a weak gravitational field (or long distance from the source).

Unloading Wave in a Cylindrical Net

Based on the general case equations of motion of a net, equations of motion of a cylindrical net are constructed. Variants of wave propagation in the case of the elastic fibers-based net are determined. The problem of the propagation of unloading waves in a pre-stretched net is solved. The problem is solved by the method of characteristics. The problem is illustrated by calculations.

Прохождение звука через упругую пластину с неоднородным покрытием, граничащую с вязкими жидкостями

В статье рассматривается задача об отражении и прохождении плоской звуковой волнычерез однородную упругую пластину с непрерывно-неоднородным упругим покрытием,граничащую с вязкими жидкостями. Полагается, что законы неоднородности покрытияпластины описываются дифференцируемыми функциями.Распространение малых возмущений в вязкой жидкости в случае установившихся коле-баний описывается скалярным и векторным уравнениями Гельмгольца. Распространениеупругих волн в однородной изотропной упругой пластине описывается двумя волновымиуравнениями для продольных и поперечных волн. Колебания неоднородного изотропногоупругого покрытия описываются общими уравнениями движения сплошной среды.Для нахождения поля смещений в неоднородном покрытии построена краевая задачадля системы обыкновенных дифференциальных уравнений второго порядка.Получено аналитическое описание отраженного и прошедшего через пластину акусти-ческих полей.Представлены результаты численных расчетов зависимостей коэффициентов отраже-ния и прохождения продольных волн от угла падения плоской волны.

Chiral active matter: microscopic ‘torque dipoles’ have more than one hydrodynamic description

Many biological systems, such as bacterial suspensions and actomyosin networks, form polar liquid crystals. These systems are ‘active’ or far-from-equilibrium, due to local forcing of the solvent by the constituent particles. In many cases the source of activity is chiral; since forcing is internally generated, some sort of ‘torque dipole’ is then present locally. But it is not obvious how ‘torque dipoles’ should be encoded in the hydrodynamic equations that describe the system at the continuum level: different authors have arrived at contradictory conclusions on this issue. In this work, we resolve the paradox by presenting a careful derivation, from linear irreversible thermodynamics, of the general equations of motion of a single-component chiral active fluid with spin degrees of freedom. We find that there is no unique hydrodynamic description for such a fluid in the presence of torque dipoles of a given strength. Instead, at least three different hydrodynamic descriptions emerge, depending on whether we decompose each torque dipole as two point torques, two force pairs, or one point torque and one force pair—where point torques create internal angular momenta of the chiral bodies (spin), whereas force pairs impart centre of mass motion that contributes to fluid velocity. By considering a general expansion of the Onsager coefficients, we also derive a new shear-elongation parameter and cross-coupling viscosity, which can lead to unpredicted phenomena even in passive polar liquid crystals. Finally, elimination of the angular variables gives an effective polar hydrodynamics with renormalized active stresses, viscosities and kinetic coefficients. Remarkably, this can include a direct contribution of chiral activity to the equation of motion for the polar order parameter, which survives even in ‘dry’ active systems where the fluid velocity is set to zero.

Analytical investigation on nonlinear vibration behavior of an unbalanced asymmetric rotor using the method of multiple scales

In this paper, the vibration behavior of a rotor with an asymmetric shaft subjected to unbalanced forces is analyzed theoretically. The model is a rotor composed of a rigid disk and a flexible shaft. The shaft is considered to be a beam with a rectangular cross section. The general equations of motion were first derived by considering the effect of high order large deformation in bending. In this process, a continuous shaft, gyroscopic effects, and rotor mass unbalance are taken into account to study the rotor’s nonlinear vibratory behavior near the main resonances. The equations are discretized using the Rayleigh–Ritz method. The obtained equations are nonlinear coupled differential equations which are solved using the multiple-scales method. It can be concluded from the results that nonlinearity due to an asymmetric shaft severely affects the resonance behavior of the rotor.

Effective field theory of hairy black holes and their flat and de Sitter limits

Effective theory of fluctuations based on underlying symmetry plays a very important role in understanding the low energy phenomena. Using this powerful technique we study the fluctuation dynamics, keeping in mind the following central question: does the effective theory of black holes provide any information about the possible existence of hair? Assuming the symmetry of the hair is that of the underlying black hole space-time, we start by writing down the most general action for the background and the fluctuations in the effective field theory framework. Considering the Schwarzschild and Schwarzschild--de Sitter black hole background with spherically symmetric hair, we derive the most general equation of motion for the fluctuations. For a particular choice of theory parameters, quasinormal modes corresponding to those fluctuations appear to have distinct features compared to those of the usual black hole quasinormal modes. The background equations from the effective theory Lagrangian, on the other hand, seem to suggest that the underlying theory of the hair under consideration should be a higher derivative in nature. Therefore, as a concrete example, we construct a class of higher derivative scalar field theory which gives rise to spherically symmetric hair through the background cosmological constant. We also calculate the quasinormal modes whose behavior turns out to be similar to the one discussed from the effective theory.

Рассеяние звуковых волн цилиндром с радиально-неоднородным упругим покрытием в плоском волноводе

В статье рассматривается задача рассеяния звуковых волн абсолютно жестким цилиндром с радиально-неоднородным изотропным упругим покрытием в плоском волноводе. Полагается, что волновод заполнен однородной идеальной жидкостью, одна его граница является абсолютно жесткой, а другая - акустически мягкой, законы неоднородности материала покрытия цилиндра описываются дифференцируемыми функциями, гармоническая звуковая волна возбуждается заданным распределением источников на сечении волновода. В случае установившихся колебаний распространение малых возмущений в идеальной жидкости описывается уравнением Гельмгольца. Колебания неоднородного изотропного упругого цилиндрического слоя описываются общими уравнениями движения сплошной среды. Для нахождения поля смещений в неоднородном покрытии построена краевая задача для системы обыкновенных дифференциальных уравнений второго порядка. Первичное поле возмущений представлено совокупностью собственных волн волновода. Давление рассеянного цилиндрическим телом поля ищется в виде потенциала простого слоя. Построена функция Грина для уравнения Гельмгольца, удовлетворяющая заданным граничным условиям на стенках волновода и условиям излучения на бесконечности. Функция плотности распределения источников ищется в виде разложения в ряд Фурье. Для нахождения коэффициентов этого разложения получена бесконечная линейная система уравнений. Проведено усечение бесконечной системы и ее решение найдено методом обратной матрицы. Получены аналитические выражения для рассеянного акустического поля в разных областях волновода.