Equations Of Motion Research Articles

A note on field equations in generalized theories of gravity

In the work [Phys. Rev. D 84, 124 041 (2011)], to obtain a simple and economic formulation of field equations for generalised theories of gravity described by the Lagrangian −gLgαβ,Rμνρσ , the key equality ∂L/∂gμνRαβκω=2PμλρσRνλρσ was derived. In this note, it is demonstrated that such an equality can be directly derived from an off-shell Noether current associated with an arbitrary vector field. As byproducts, a generalized Bianchi identity related to the divergence for the expression of field equations, together with the Noether potential, is obtained. On the basis of the above, we further propose a systematic procedure to derive the equations of motion from the Noether current, and then this procedure is extended to more general higher-order gravities endowed with the Lagrangian encompassing additional terms of the covariant derivatives of the Riemann tensor. To our knowledge, both the detailed expressions for field equations and the Noether potential associated with such theories are first given at a general level. All the results reveal that using the Noether current to determine field equations establishes a straightforward connection between the symmetry of the Lagrangian and the equations of motion and such a remedy even can avoid calculating the derivative of the Lagrangian density with respect to the metric.

A variational-based non-smooth contact dynamics approach for the seismic analysis of historical masonry structures

A variational formulation of the non-smooth contact dynamics method is proposed to address the dynamic response of historical masonry structures modeled as systems of 3D rigid blocks and subjected to ground excitation. Upon assuming a unilateral-frictional contact law between the blocks, the equations of motions are formulated in a time-discrete impulse theorem format in the unknown block velocities and contact impulses. The variational structure of the problem to be solved at each time step is proven. On that basis, the numerical method requires at each time step to perform a collision detection that identifies antagonist contact points based on the given structural configuration, to solve a second-order conic programming problem that outputs block velocities and contact impulses, and to update the structural configuration for the solution to advance in time. As a merit of the formulation, large-scale problems can be robustly and efficiently addressed thanks to the convex setting of the time-step optimization problem. Numerical results are presented to test the computational performances of the proposed approach. Benchmark problems provide numerical evidence that the formulation is consistent with event-driven solutions based on the classical Housner impact model. The dynamic response, failure domains, and fragility functions of real-size masonry structures are then explored under ground impulse or earthquake excitation. The obtained results prove the reliability of the present computational method for the dynamic analysis and seismic assessment of historical masonry constructions of engineering interest.

Horseshoes and spiral waves: capturing the 3D flow induced by a low-mass planet analytically

ABSTRACT The key difficulty faced by 2D models for planet–disc interaction is in appropriately accounting for the impact of the disc’s vertical structure on the dynamics. 3D effects are often mimicked via softening of the planet’s potential; however, the planet-induced flow and torques often depend strongly on the choice of softening length. We show that for a linear adiabatic flow perturbing a vertically isothermal disc, there is a particular vertical average of the 3D equations of motion that exactly reproduces 2D fluid equations for arbitrary adiabatic index. There is a strong connection here with the Lubow–Pringle 2D mode of the disc. Correspondingly, we find a simple, general prescription for the consistent treatment of planetary potentials embedded within ‘2D’ discs. The flow induced by a low-mass planet involves large-scale excited spiral density waves that transport angular momentum radially away from the planet and ‘horseshoe streamlines’ within the coorbital region. We derive simple linear equations governing the flow that locally capture both effects faithfully simultaneously. We present an accurate coorbital flow solution allowing for inexpensive future study of corotation torques, and predict the vertical structure of the coorbital flow and horseshoe region width for different values of adiabatic index, as well as the vertical dependence of the initial shock location. We find strong agreement with the flow computed in 3D numerical simulations, and with 3D one-sided Lindblad torque estimates, which are a factor of 2–3 lower than values from previous 2D simulations.

Re‐derivation and mathematical analysis for linear peridynamics model for arbitrary Poisson ratio's material

AbstractThis paper is concerned with the modeling and mathematical analysis of linear peridynamic model for arbitrary Poisson ratio's material. Based on the fundamental laws of dynamics, we re‐derive the bond‐based peridynamic model for anisotropic materials by relaxing certain assumptions. Through this process, we draw several significant conclusions, such as the relationship between the equivalent strain energy density hypothesis and the convergence of the peridynamic operator to the classical Navier operator. Additionally, the well‐posedness of time‐dependent peridynamic equations of motion is established. Finally, some necessary conditions for the material stability of anisotropic material are given.

Unsymmetricity effects on seismic performance of multi-story buildings

The unsymmetrical configurations in buildings lead to non-uniform distributions in their strength, mass, and stiffness, and they are consequently prone to damage during seismic hazards. In this study, the seismic performance of multi-story buildings with 5, 8, 10, and 12 stories of square, ‘L’, ‘T’, and ‘U’-shaped buildings have been investigated. The research deals with the variation of the natural time periods and how it affects the seismic performance of unsymmetrical multi-story buildings. The coupled and uncoupled equations of motion, based on the symmetricity of the buildings about both axes, were solved to obtain natural time periods that influence the spectral acceleration of the ground accelerations. Six important ground accelerations were considered. Nonlinear static analysis, such as pushover analysis, was also carried out on all the buildings. Comparisons were made on the seismic behavior of both the symmetrical and unsymmetrical structures. The results revealed that the spectral acceleration influences dynamic responses, such as base shear, base moment, base torsion, roof displacement, roof rotation, and story drifts of the buildings. Moreover, it was found that even though the ‘L’-shaped buildings are unsymmetrical about both axes, they are less vulnerable than the ‘T’ and ‘U’-shaped buildings, which are unsymmetrical about one axis.

Nonlocal buckling analysis of five-layer laminated nanocomposites on kerr foundation: A refined zigzag theory approach

In this paper, the buckling analysis of a five-layer laminated nanocomposite resting on a Kerr foundation is presented. In order to describe the non-continuous behavior of composite plate through its thickness, the displacement field is determined using the Refined Zigzag Theory (RZT). Additionally, the constitutive relations of piezo electromagnetic isotropic materials, orthotropic composites, and Functionally Graded Porous Materials (FGPMs) are presented. With respect to Bi- and Uniaxial loading in the nanoplate, the Hamilton’s principle is utilized to derive the equation of motion of this nanoplate. To study the small-scale effect in nanoplates, both Nonlocal Eringen Theory and Nonlocal Strain Gradient Theory (NSGT) are employed to account for nonlocal effects. Finally, the coupled equations of motion are solved using the Differential Quadrature Method (DQM). This paper introduces the newly used Kerr foundation and its effect on the buckling analysis. It also investigates the influence of plate dimensions, piezo electromagnetic terms, boundary conditions, and loading on the dimensionless critical buckling load.

Chaos and resonance of nonlinear FG shallow shells reinforced by oblique stiffeners

This paper presents an analysis of the nonlinear dynamics of functionally graded (FG) shallow shells that are simply supported, reinforced with oblique stiffeners, and subjected to external excitation. The properties of these oblique stiffened functionally graded (OSFG) shallow shells vary across their thickness, following a power law distribution based on the volume fractions. Employing the first-order shear deformation theory (FSDT) coupled with the von Kármán nonlinear strain-displacement relations, the derivation of the nonlinear equations of motion is accomplished through the application of Hamilton's principle. The model of FG shallow shells reinforced with oblique two-series stiffeners is developed, allowing for the angles of the two stiffeners to be similar or different from each other. In this context, the employment of the first-order shear deformation theory (FSDT) results in the enhancement of the complexity in modeling the oblique stiffeners. Galerkin's method is applied to discretize the given equations, converting them into a nonlinear dynamical system with two degrees of freedom. This system incorporates both quadratic and cubic nonlinearities and considers the effects of external excitation. The analysis of this research particularly focuses on the resonant scenario involving principal resonance and 1:1 internal resonance. Through the asymptotic perturbation method, a four-dimensional nonlinear averaged equation is obtained. For the first time, a numerical investigation of the research reveals critical insights into the behavior of the system, such as waveforms, phase diagrams, and Poincaré maps, highlighting the effects of different angles of the stiffeners and variations of material parameters on the nonlinear dynamics of these shells.

Role of Rastall gravity in constructing new spherically symmetric stellar solutions

This paper formulates new stellar solutions corresponding to a static spherical spacetime through the minimal gravitational decoupling in the context of Rastall theory. For this, we initially adopt a seed source to be the isotropic and then add a new gravitating matter source whose impact is controlled by the decoupling parameter. The addition of the later source enhances the degrees of freedom in the corresponding equations of motion. To reduce them, we apply the geometric transformation only to a radial metric potential that divides the field equations into two sets, each of them govern the impact of only one source. Afterwards, we take into account the Schwarzschild solution to make the metric potentials known. Moreover, different constraints on the components of newly added source are employed to develop some suitable solutions of the second system. We construct three different solutions to the total fluid distribution and check their graphical behavior for certain parametric values. Finally, we conclude all our resulting models to be agree with the energy conditions in this extended theory.

Discrete differential geometry-based model for nonlinear analysis of axisymmetric shells

In this paper, we propose a novel one-dimensional (1D) discrete differential geometry (DDG)-based numerical method for geometrically nonlinear mechanics analysis (e.g., buckling and snapping) of axisymmetric shell structures. Our numerical model leverages differential geometry principles to accurately capture the complex nonlinear deformation patterns exhibited by axisymmetric shells. By discretizing the axisymmetric shell into interconnected 1D elements along the meridional direction, the in-plane stretching and out-of-bending potentials are formulated based on the geometric principles of 1D nodes and edges under the Kirchhoff–Love hypothesis, and elastic force vector and associated Hessian matrix required by equations of motion are later derived based on symbolic calculation. Through extensive validation with available theoretical solutions and finite element method (FEM) simulations in literature, our model demonstrates high accuracy in predicting the nonlinear behavior of axisymmetric shells. Importantly, compared to the classical theoretical model and three-dimensional (3D) FEM simulation, our model is highly computationally efficient, making it suitable for large-scale real-time simulations of nonlinear problems of shell structures such as instability and snap-through phenomena. Moreover, our framework can easily incorporate complex loading conditions, e.g., boundary nonlinear contact and multi-physics actuation, which play an essential role in the use of engineering applications, such as soft robots and flexible devices. This study demonstrates that the simplicity and effectiveness of the 1D discrete differential geometry-based approach render it a powerful tool for engineers and researchers interested in nonlinear mechanics analysis of axisymmetric shells, with potential applications in various engineering fields.

Seebeck and Nernst effects in the mixed state of strongly fluctuating superconductors

We study Seebeck and Nernst effects in strongly fluctuating superconductors by calculating the Seebeck and Nernst coefficients in the Ginzburg–Landau Lawrence–Doniach model with thermal noise. Strongly thermal fluctuations are represented by the Langevin thermal noise. To capture the interesting fluctuation effects, we use the self-consistent Gaussian approximation to treat cubic term in the equation of motion. Our results including all Landau levels are to treat arbitrary magnetic fields. The Seebeck and Nernst coefficients are calculated for arbitrary temperature and magnetic field. The results are compared to experimental data on Fe-based superconductor Fe1+yTe0.6Se0.4 in both the superconducting state below the critical temperature Tc and the normal state above Tc. Our calculation supports the picture of phase disordering of fluctuations above Tc in Fe-based superconductors.

Periodic free vibrations of composite laminates with curvilinear fibres and CNTs

This article addresses the combined effect of using curvilinear fibres and carbon nanotubes (CNTs) reinforcements in the non-linear modes of vibration of laminated composite plates. To arrive at the material properties of the three-phase composite material, a two-step hierarchic procedure is followed. A modified version of the Halpin–Tsai model is employed to predict the Young’s modulus of the CNT enriched resin and expressions, deduced from equilibria of a unit cell where a fibre is embedded in resin, are applied to obtain the diverse elasticity moduli of the three-phase composite. Moderately large displacements are considered, with von Kármán strain–displacement relations. Although the presented model is an equivalent single layer one, it applies to thick plates, because a Third-order Shear Deformation Theory (TSDT) is followed. The set of autonomous non-linear equations of motion is reduced using static condensation and a modal basis with selected modes, chosen after a convergence analysis. The reduced set of equations of motion is solved by the shooting method. Numerical tests considering plates with diverse curvilinear fibre paths, CNT contents and thicknesses are carried out. The results obtained are thoroughly analysed.

Time Evolution and Thermal Renormalization Group Flow in Cosmology

Time-evolution of the Universe as described by the Friedmann equation can be coupled to equations of motion of matter fields. Quantum effects may be incorporated to improve these classical equations of motion by the renormalization group (RG) running of their couplings. Since temporal and thermal evolutions are linked to each other, astrophysical and cosmological treatements based on zero-temperature RG methods require the extension to finite-temperatures. We propose and explore a modification of the usual finite-temperature RG approach by relating the temperature parameter to the running RG scale as T≡kT=τk (in natural units), where kT is acting as a running cutoff for thermal fluctuations and the momentum k can be used for the quantum fluctuations. In this approach, the temperature of the expanding Universe is related to the dimensionless quantity τ (and not to kT). We show that by this choice dimensionless RG flow equations have no explicit k-dependence, as it is convenient. We also discuss how this modified thermal RG is used to handle high-energy divergences of the RG running of the cosmological constant and to “solve the triviality”of the ϕ4 model by a thermal phase transition in terms of τ in d=4 Euclidean dimensions.

Gyroscopic rotor dynamics simulation with anisotropy of elastic and damping characteristics of the support

In the work, the differential equations of motion of the gyroscopic rotor, built taking into account the anisotropy of stiffness and damping of the flexible support, are solved analytically, by the method of harmonic balance, convenient for obtaining separately amplitude-frequency and phase-frequency characteristics in the direction of oscillations. The equations of the non-stationary process are obtained by the method of changing amplitudes. It has been found that when the linear stiffness of the elastic support is different in two orthogonal directions, two critical velocities and the corresponding resonant regions arise. In the area of each critical speed, there are two amplitude-frequency curves of oscillations of the main direction and the direction perpendicular to it, respectively. The geometric nonlinearity of damping suppresses the elevations of these amplitude-frequency curves more significantly than linear damping. If only one of the two directions has a damping nonlinearity, then its effect is on the amplitude-frequency curves of the corresponding critical speed. It is more efficient to control resonant amplitudes for smooth resonant transitions by enhancing the linear damping with geometrically nonlinear damping. The results of the analytical solution of the equations of motion agree well with the results of direct modeling and experimental studies.

A modified linear drag induced deceleration using a transformation of Newton’s second equation of motion

The drag force exerted on a body moving in a fluid is empirically known to follow a linear drag law at low Reynolds numbers and a quadratic drag law at very high Reynolds numbers. The major challenge with the empirical description of the drag lies in the uncertainty of the description of the drag coefficient as a function of velocity. If the function is precisely known, it may be possible to find a general formula for the drag, which would, in principle, be applicable for all Reynolds numbers. The general law can consequently be tested using the linear and quadratic drag laws within their validity regimes to confirm its reliability and validity. The current absence of such a general law implies that the prediction of the dynamics of a body moving under drag forces in the intermediate regime of the Reynolds number can only be approximate and inaccurate since it is not guided by any valid law. In this study, we derive a general relation for the drag force, which may be applicable in all the Reynolds number regimes. We achieve this through a simple transformation of Newton’s second equation for uniformly decelerated motion. It turns out that our general relation is a modification of the linear drag law. The effect of the introduced correction term in the dynamics of a projectile is evidently promising, as can be seen in the more realistic results generated.

Studying Binary Formation under Dynamical Friction Using Hill’s Problem

Using the equations of motion from Hill’s problem, with added accelerations for different forms of dynamical friction, we provide the (to-date) broadest scale-free study of friction-driven binary formation in gaseous disks and stellar clusters. We focus mainly on binary formation between stellar-mass black holes in active galactic nuclei (AGNs), considering both gas dynamical friction (GDF) from AGN disks and stellar dynamical friction (SDF) from the nuclear star cluster. We first find simple, dimensionless friction coefficients that approximate the effects of standard models for GDF and SDF. We perform extensive simulations of Hill’s problem under such friction, and we present a picture of binary formation through encounters between single stars on nearby orbits, as a function of friction parameter, eccentricity, and inclination. Notably, we find that the local binary formation rate is a linear function of the friction coefficient so long as the friction is weak. Due to the dimensionless nature of our model problem, our findings are generalizable to binary formation at all scales (e.g., intermediate-mass black holes in a star cluster, planetesimals in a gaseous disk).

New angular momentum conservation laws for electromagnetic waves interacting with dirac fields

Abstract Global conservation laws of angular momentum are well-known in the theory of light-matter interaction. However, local conservation laws, i.e. the conservation law of angular momentum at every point in space, remain unexplored especially in the context of relativistic Dirac-Maxwell fields. Here, we use the QED Lagrangian and Noether's theorem to derive a new local conservation law of angular momentum for Dirac-Maxwell fields in the form of the continuity relation for linear momentum. We separate this local conservation law into four coupled motion equations for spin and orbital angular momentum (OAM) densities. We introduce a helicity current tensor, OAM current tensor, and spin-orbit torque in the motion equations to shed light on the local dynamics of spin-OAM interaction and angular momentum exchange between Maxwell and Dirac fields. We elucidate how our results translate to classical electrodynamics using the example of plane wave interference as well as a dual-mode optical fiber. Our results shine light on angular momentum phenomena related to the relativistic interaction of electromagnetic waves and Dirac fields.

A theoretical framework for robust implementation of in situ measurements of ocean currents and waves in dynamics of mooring systems

We develop an approximated method to solve analytically the equations of motion that describe mooring line dynamics in a one-dimensional model. For the first time, we derive integral closed-form expressions to compute dynamic properties of mooring lines subject to ocean currents and waves of arbitrary time and spatial dependence, in terms of modified Bessel functions. This is done by decomposing the mooring line in three regions where different approximations and mathematical techniques of solution are carried out. Our analytical results provide a robust framework to simulate and analyze extreme realistic oceanic events when data from in situ ocean observation systems are available, regardless of the resolution or coarseness of subsurface measurements and even for long acquisition times. In order to prove the advantages of this approach, we have processed data from two stations in the National Data Buoy Center of the National Oceanic and Atmospheric Administration. From simulations with ocean currents data, we have gained insights into the coupling of the spatial modulation of ocean currents with the characteristic wavelengths of elastic lines. From simulations with ocean waves data, we have defined a scheme to analyze wave data and identify the contribution of each subset of frequency peaks to the net fluctuation of mooring line tension. This could be useful for classification of irregular waves based on their impact on mooring line tension. The development of better tools that integrate theoretical and experimental findings is necessary for the assessment of marine structures under the environmental conditions associated with climate change.

Vibration analysis of rotating variable-angle-tow composite cylindrical structures via high-fidelity shell models

The paper focuses on rotordynamics analyses of various rotating variable-angle-tow (VAT) composite cylindrical structures using both low- and high-fidelity structural models. To model the spatially varying composite shell structures, the Carrera Unified Formulation (CUF) is employed. The CUF provides a hierarchical and automated approach for developing shell finite element models. Various shell models are created using different kinematics models based on Taylor or Lagrange expansion functions. The linearized equations of motion include the Coriolis and initial stress contributions. Several VAT composite curved panels and thin and thick cylinders have been analyzed to validate the presented approach and provide benchmark solutions. The results are discussed and compared with commercial software solutions. The results demonstrate the good accuracy and reliability of the proposed numerical methodology.

Low-velocity impact response of sandwich plates with corrugation star-shaped honeycomb hybrid core

Compared with traditional lightweight corrugation and honeycomb cores, the novel cellular structure exhibiting a negative Poisson's ratio possesses distinctive mechanical deformation features, making it suitable for modeling lightweight sandwich structures. Therefore, the concept of combining the auxetic honeycomb core with folded corrugations is proposed to construct a new type of corrugation star-shaped honeycomb (SSH) hybrid core for studying the dynamic behavior of sandwich panels subjected to low-velocity impact. Integrate Hertz elasticity theory and first-order shear deformation theory (FSDT) to develop an equivalent analytical model, and derive the equations of motion through Hamilton principle. To model contact force interactions during dynamic processes, a spring-mass model is utilized. Analytical solutions are derived for predicting transverse displacement with Duhamel's principle and Navier's method. Numerical simulations are conducted using the Abaqus commercial software, and the validity of the results is confirmed by comparing them with findings in the existing literature. Based on this, effective strategies for enhancing the sandwich panel's resistance to low-velocity impacts are proposed by examining the influence of different side length ratios, thickness ratios, and cell concave angles. In comparison to the corrugation re-entrant hexagonal honeycomb hybrid core sandwich panel structure, the corrugation SSH hybrid core sandwich panel structure reduces transverse displacement by 33.6% at the same impact velocity.

Convection induced by centrifugal and Coriolis buoyancy in a rotating Hele-Shaw reactor

The study of heat and mass transfer in a Hele-Shaw cell rotating around a perpendicular axis has various advanced technological applications. These include the design of microfluidic devices and continuous-flow chemical microreactors, to name a couple. In this setup configuration, the quasi-two-dimensional design allows for recording the density field using optical methods, and the rotation enables control of this field through spatially distributed inertial forces. As is known, in the limit of an infinitely thin layer, the Coriolis force vanishes within a standard mathematical model. However, experimental observations of fluid flow in a rotating Hele-Shaw cell indicate the opposite. In this paper, we show that the correct derivation of the equation of motion under the Hele-Shaw approximation leads to the appearance of a Boussinesq-type term for the Coriolis force. To study the effect of the Coriolis buoyancy, we consider the problem of fluid stability during the internal generation of a transfer component, which can be either the concentration of the dissolved substance or the temperature of the medium. The careful study of system dynamics involves linear stability analysis, weakly nonlinear analysis, and direct numerical simulation. The general properties of the disturbance spectrum are analyzed. The branching of solutions near the first bifurcation is studied using the technique of multiple time scales. A stationary convection is replaced by an oscillatory one under the action of the Coriolis force, as demonstrated by weakly nonlinear analysis. Finally, we investigate the nonlinear dynamics using direct numerical simulation.