Equations Of Motion Research Articles

Spin-lattice relaxation with non-linear couplings: Comparison between Fermi's golden rule and extended dissipaton equation of motion.

Fermi's golden rule (FGR) offers an empirical framework for understanding the dynamics of spin-lattice relaxation in magnetic molecules, encompassing mechanisms like direct (one-phonon) and Raman (two-phonon) processes. These principles effectively model experimental longitudinal relaxation rates, denoted as T1-1. However, under scenarios of increased coupling strength and nonlinear spin-lattice interactions, FGR's applicability may diminish. This paper numerically evaluates the exact spin-lattice relaxation rate kernels, employing the extended dissipaton equation of motion formalism. Our calculations reveal that when quadratic spin-lattice coupling is considered, the rate kernels exhibit a free induction decay-like feature, and the damping rates depend on the interaction strength. We observe that the temperature dependence predicted by FGR significantly deviates from the exact results since FGR ignores the higher order effects and the non-Markovian nature of spin-lattice relaxation. Our methods can be easily extended to study other systems with nonlinear spin-lattice interactions and provide valuable insights into the temperature dependence of T1 in molecular qubits when the coupling is strong.

Shear-free inhomogeneous energy density in 4D Einstein-Gauss-Bonnet spherical systems

Abstract We explore the inhomogeneity factors for the initially regular relativistic spheres in 4D-Einstein-
Gauss-Bonnet (EGB) theory. The corresponding equations of motion are derived once the generic
expressions for the kinematical variables are obtained for spherically symmetric self-gravitating
system. By using the non-zero divergence of the stress-energy tensor, the independent components
of Bianchi identities are constructed. To enable a thorough explanation of the inhomogeneity of
the particular shear free matter distribution, we computed two distinct components of evolution
equations employing the Weyl tensor. We then investigate the requisite variables for the irregularity
by looking at particular scenarios in both the adiabatic and non-adiabatic domains. These instances
demonstrate how, in addition to other factors, the Gauss-Bonnet terms contribute to the regularity
requirements of the collapsing fluid.

Nonlocal strain gradient-based nonlinear vibration analysis of nonlinear FG three-phase HJCNT/MWCNT/epoxy composite microbeam resonators using a modified micromechanical model

This paper aims to study nonlinear dynamic behavior of functionally graded (FG) three-phase composite microbeam resonators made of an epoxy matrix and two reinforcements namely multi-walled carbon nanotubes (MWCNTs) and hetero-junction carbon nanotubes (HJCNTs). The effective mechanical properties of the composite microbeam are obtained using the modified Halpin-Tsai micromechanical model. The microbeam surrounding medium is simulated using a two-parameter elastic foundation. The von-Karman’s geometric nonlinearity relations are incorporated and the equations of motion are derived based on the nonlocal strain gradient Euler–Bernoulli beam model. A new closed-form analytical solution is obtained using the homotopy perturbation method. The effects of vibration amplitude, nanofiber volume fraction, nanofiber distribution pattern, small-scale parameters and the foundation parameters on the nonlinear frequency and deflection of the FG three-phase composite microbeams are studied in detail. The findings of the paper are valuable for researchers in the field of microbeam resonators.

D-terms in generalised complex geometry

Generalised Complex Geometry provides a natural interpretation of the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supersymmetry conditions for warped solutions of type II supergravity as differential equations on polyforms on the internal manifold. Written in this language the supersymmetry conditions correspond to calibration conditions for probe D-branes: D-string, domain-wall or space-filling branes, depending on the directions they span in the non-compact four-dimensional space. The BPS condition corresponding to the calibration of space-filling D-branes has been reformulated by Tomasiello, eliminating the explicit dependence on the metric. We generalise this derivation to the case of non-supersymmetric backgrounds violating the domain-wall and D-string calibration conditions. We use this reformulation to derive constraints that the ten-dimensional solutions with BPS space-filling sources must respect in order to dimensionally reduce to solutions of four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supergravity with non-vanishing F-terms and potentially non-vanishing D-terms. We give the equations of motion for the class of type II vacua satisfying these constraints in the language of pure spinors. We investigate how restrictive these constraints are for the class of type IIB SU(3) backgrounds with BPS space-filling O5-planes.

An Innovative Approach for Precise Identification of the Lowest Unoccupied Molecular Orbital Using the Parametric Equation of Motion.

The lowest unoccupied molecular orbital (LUMO) plays a crucial role in quantum chemistry, but current quantum chemistry calculations fail to provide useful virtual orbitals, making it challenging to explore various processes such as photochemical reactions, electron attachment, reduction, or excitation processes. The LUMO obtained from the self-consistent field (SCF) solution can not be relied upon and needs to be identified as they are often present among the continuum states having almost similar energies. The nuclear charge stabilization method has been proven useful in identifying LUMO. Herein, we have proposed the application of parametric equations of motion (PEM) in conjunction with nuclear charge stabilization method to identify the LUMO obtained from the SCF solution exhibiting stability with different basis sets including diffuse functions.

Primary Resonances in Inclined Cantilever Beam with Tip-Mass: A Parameter-Splitting Multiple-Scales Approach

The transverse static load, resultant from self-weight, significantly impacts both the response and the nonlinear dynamic behavior of a flexible inclined cantilever with a large lumping tip mass under harmonic base motion. However, the transverse static load was not taken into account and the mode shape of a pure cantilever beam was directly adopted during Galerkin’s procedure in the formulations of a cantilever carrying a large lumping tip mass in many previous studies. In this paper, (1) the static load effect caused by the self-weight in an inclined cantilever is considered by using a coordinate transformation, (2) the efficacy of adopting a pure cantilever beam mode shape to analyze a cantilever beam carrying a large lumping mass is studied and (3) a recently proposed semi-analytical method named parameter-splitting multiple-scales method is extended to analyze the cantilever studied to examine its effectiveness. First, the extended Hamilton’s principle is utilized to formulate the equation of motion of the cantilever studied. After that, the mode shape of a pure cantilever and the exact mode shape of a cantilever carrying a lumping mass are separately adopted in Galerkin’s method to transform the partial differential equation into ordinary differential equations. The frequency-response curves of the discretized nonlinear differential equations obtained by the numerical continuation method and the parameter-splitting multiple-scales method are compared to address ① the transverse static load effect on the frequency-response and nonlinear dynamical behavior of the beam, ② the discrepancies between the resultant frequency-response curves obtained by using two different mode shapes and ③ the efficacy of the parameter-splitting multiple-scales method on solving strongly nonlinear practical problem.

Classical Kerr-Schild double copy in bigravity for maximally symmetric spacetimes

A generalized Kerr-Schild ansatz for bigravity, already considered in the literature, which leads to linear interactions between the metrics is used to study the bigravity equations in the context of the double copy. By contracting the resulting spin-2 field bigravity equations of motion using Killing vector fields, as is usually carried out in general relativity, we arrive to the single and zeroth copy equations for the mentioned ansatz. For the case of stationary solutions, we obtain two Maxwell and two conformally coupled scalar field equations for the single and zeroth copies respectively, and the linear interactions are absent. In the time-dependent case we obtain equations for the fields which are coupled. By decoupling these equations and at the zeroth copy level, we recover a massless and a massive field whose mass is proportional to the Fierz-Pauli mass and depends on the coefficients of the interaction potential between the metrics. This has been also previously documented in the literature and is now reinterpreted within the context of the double copy proposal.

Flowing between string vacua for the critical non-Abelian vortex with a deformation of N=2 Liouville theory

It has been shown that non-Abelian solitonic vortex string supported in four-dimensional (4D) N=2 supersymmetric QCD (SQCD) with the U(2) gauge group and Nf=4 quark flavors becomes a critical superstring. This string propagates in the ten-dimensional space formed by a product of the flat 4D space and an internal space given by a Calabi-Yau noncompact threefold, namely, the conifold. The spectrum of low-lying closed string states was found and interpreted as a spectrum of hadrons in 4D N=2 SQCD. In particular, the lowest string state appears to be a massless Bogomol’nyi-Prasad-Sommerfield (BPS) baryon associated with the deformation of the complex structure modulus b of the conifold. It was recently shown that the Coulomb branch of the associated string sigma model which opens up at strong coupling can be described by N=2 Liouville theory. Building on these results we switch on quark masses in 4D N=2 SQCD and study the interpolation of the initial U(2) SQCD with Nf=4 quarks to the final SQCD with the U(4) gauge group and Nf=8 quarks. To find the true string vacuum which arises due to the mass deformation we solve the effective supergravity equations of motion associated with the deformed world sheet Liouville theory. We show that the massless BPS baryon b survives the deformation and that finding of the spectrum of low-lying massive hadrons in the final SQCD is linked to the Calogero problem. Published by the American Physical Society 2024

Transient Dynamic Response of Generally Shaped Arches under Interval Uncertainties

This paper endeavors to investigate the characteristics of the transient dynamic response of a generally shaped arch when influenced by uncertain parameters while being subjected to specific external excitation. The equations of motion of the generally shaped arches are derived by the differential quadrature (DQ) method, and the deterministic dynamic responses are calculated using the Newmark-β method. By employing the Chebyshev inclusive function, an interval method based on a non-intrusive polynomial surrogate model is developed, and the uncertain dynamic responses are reckoned by enabling numerical simulations. The results of the proposed interval method are compared with those obtained from the scanning method for validation. The effects of various shapes and rise span ratios on the dynamic responses are investigated through a parametric study. The results suggest that the degree of fluctuation in the uncertain dynamic behavior is influenced by the type of parameter. Additionally, the responses of each shaped arch decrease with the increase in the rise span ratios, and with the same rise span ratio, the deterministic responses and corresponding uncertain responses are also affected by the shape of the arch, and they are considered to be at a minimum when the arch shape is parabolic. This study will enhance understanding of the dynamic properties of arches with uncertainties and provide some basis for the assessment and health monitoring of arch structures.

When the Anomalistic, Draconitic and Sidereal Orbital Periods Do Not Coincide: The Impact of Post-Keplerian Perturbing Accelerations

In a purely Keplerian picture, the anomalistic, draconitic and sidereal orbital periods of a test particle orbiting a massive body coincide with each other. Such degeneracy is removed when post-Keplerian perturbing acceleration enters the equations of motion, yielding generally different corrections to the Keplerian period for the three aforementioned characteristic orbital timescales. They are analytically worked out in the case of the accelerations induced by the general relativistic post-Newtonian gravitoelectromagnetic fields and, to the Newtonian level, by the oblateness of the central body. The resulting expressions hold for completely general orbital configurations and spatial orientations of the spin axis of the primary. Astronomical systems characterized by extremely accurate measurements of orbital periods like transiting exoplanets and binary pulsars may offer potentially viable scenarios for measuring such post-Keplerian features of motion, at least in principle. As an example, the sidereal period of the brown dwarf WD1032 + 011 b is currently known with an uncertainty as small as ≃10−5s, while its predicted post-Newtonian gravitoelectric correction amounts to 0.07s; however, the accuracy with which the Keplerian period can be calculated is just 572 s. For double pulsar PSR J0737–3039, the largest relativistic correction to the anomalistic period amounts to a few tenths of a second, given a measurement error of such a characteristic orbital timescale as small as ≃10−6s. On the other hand, the Keplerian term can be currently calculated just to a ≃9 s accuracy. In principle, measuring at least two of the three characteristic orbital periods for the same system independently would cancel out their common Keplerian component, provided that their difference is taken into account.

Stochastic Analysis for the Embedded Single-Walled Carbon Nanotube Under Random Vibrations

It is inevitable that carbon nanotubes (CNTs) are subjected to random vibrations due to environmental noise, which may impair their mechanical properties. The nonlinear dynamics associated with single-walled carbon nanotubes (SWCNTs) under random noise has been investigated in this work. First, based on the nonlocal theory and the Euler–Bernoulli beam model with fixed supports at both ends, the governed equations and boundary conditions are established according to Hamilton’s principle. Second, the Galerkin method is used to approximate the differential equations of motion and the stochastic averaging method (SAM) is applied to predict the approximate response of the original system. Subsequently, a detailed parametric study is conducted to analyze the nonlinear random vibration response of the SWCNT with nonlocal effects, and the effectiveness of the proposed method is verified by comparing the analytical results with those from the Runge–Kutta method. Finally, by solving the backward Kolmogorov (BK) equation and the generalized Pontryagin (GP) equation simultaneously, the conditional reliability function (CRF) and mean first-passage time (MFPT) for determining the reliability of SWCNT systems are obtained.

Variational Gaussian approximation for the magnetic Schrödinger equation *

In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of particles under the influence of a magnetic field. The solution of the time-dependent Schrödinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac–Frenkel variational principle. For the approximation we derive ordinary differential equations of motion for the parameters of the variational solution. Moreover, we prove L 2-error bounds and observable error bounds for the approximating Gaussian wave packet.

Numerical Simulation of Folding Tail Aeroelasticity Based on the CFD/CSD Coupling Method

This paper presents a CFD/CSD coupling method for aeroelastic simulation of folding tail morphing aircraft. The unsteady aerodynamic analysis is based on an in-house computational fluid dynamics (CFD) solver for the Euler equations, and emphasis is made on developing an efficient dynamic mesh method for the tail’s hybrid fold motion/elastic vibration deformation. The structural dynamic analysis is based on the computational structural dynamics (CSD) technique for solving the structural equation of motion in modal space. The aeroelastic coupling was achieved through successive iterations of CFD and CSD computations in the time domain. An adaptive multi-functional morphing aircraft allowing tail fold motion was selected to be studied. By using the developed method, aeroelastic simulation and mechanism analysis for fixed configurations at different folding angles and for variable configurations during the folding process were performed. The influence of folding rate on tail aeroelasticity and its influence mechanism were also analyzed.

Quasi-Analytical Solution of Kepler’s Equation as an Explicit Function of Time

Although Kepler’s laws can be empirically proven by applying Newton’s laws to the dynamics of two particles attracted by gravitational interaction, an explicit formula for the motion as a function of time remains undefined. This paper proposes a quasi-analytical solution to address this challenge. It approximates the real dynamics of celestial bodies with a satisfactory degree of accuracy and minimal computational cost. This problem is closely related to Kepler’s equation, as solving the equations of motion as a function of time also provides a solution to Kepler’s equation. The results are presented for each planet of the solar system, including Pluto, and the solution is compared against real orbits.

Fast Calculation for Vehicle–Road Coupled Response Based on Cyclic Road Modeling Method

In order to improve the computational efficiency of the vehicle–road coupled response in long-distance driving, this paper proposes a cyclic road modeling method to equivalent an infinitely long road. The road is built using a closed model, and the vehicle–road coupled response is calculated assuming that the vehicle numerically cyclic driving. First of all, the concept of the cyclic road model is introduced, and the modal analysis is performed. Secondly, the principle for determining the optimal length of the cyclic road model is proposed based on the attenuation properties of the road response in time and space. Next, the vibration analysis of the vehicle–road coupled system is conducted, and its equation of motion is constructed using the mode superposition method (MSM). Further, the optimal length of the cyclic road model is derived regarding to road parameters via numerical analysis. At last, the application of this cyclic road model on computing the vehicle–road coupled response in long-distance driving is discussed via numerical simulation. The results indicate that this model has a consistent performance with the widely used models, while significantly improves computational efficiency. The proposed method also provides a new idea for the fast modeling of roads.

Analytic Solutions for Drag and Magnus forces

Abstract When a spinning object moves in air, it is affected by three forces: the gravitational force, the drag force, and the Magnus force. The equations of motion for such an object are nonlinear, making it difficult to find analytic solutions. Hence, numerical solutions using computers are often preferred. This paper provides two examples of analytic solutions of the equations of motion with the drag and Magnus forces: nearly vertical motion and zero-gravity motion. In these two cases, we find analytic solutions for the velocity and the position of the object, avoiding mathematical difficulties. It is suitable for university students studying physics.

Equation of motion for taut-line buzzers

Equation of motion for taut-line buzzers

Analysis of a Mechanism Used to Operate an Oscillating Separator

This article presents a comparative study of two different kinds of processes that produce oscillatory motion on a work surface during the mechanical separation process. The investigation began with determining the trajectory produced by the oscillating separator’s active component of the classical drive mechanism. Based on this, a second mechanism—the six-bar mechanism—was created using the WATT program, and a mathematical analysis was conducted. The comparative examination of the two mechanisms was carried out using OriginPro, Mathcad, and Roberts software. This study’s findings all point to the same conclusion: the newly developed mechanism produces the same trajectory as the classical mechanism when viewed through the lens of the reference element, or the element that causes the oscillatory movement. However, when looking at the operating parameters, there was a noticeable difference in the movement’s speed and the angle of the crank when producing its maximum speed. Theoretically, this new mechanism increases the speed at which solid particles move across a work surface. However, this difference cannot be characterized as positive or negative because further research is required to determine how the nature of solid particles and the work surface’s inclination affect this process, in addition to this mechanism. The identification of the mathematical equations of motion for the constituent parts of the mechanism under study is the novelty produced of this paper.

Prestress Force and General Excitation Identification for Plate-like Concrete Bridges

Prestress force dominates the carrying capacity of concrete girders, and is vital for bridge health monitoring. Many vibrational-based methods have been proposed to determine prestress using bridge responses induced by known excitations, which means that they can barely use the normal traffic of in-service bridges as excitation to achieve long-term monitoring. Moreover, most studies are based on beam theory, which may not be precise for plate-like bridges. Hence, this paper establishes a motion equation for a prestressed slab via Kirchhoff’s plate theory and proposes a two-step procedure to assess the prestress and general excitation simultaneously through only bridge responses. The excitation is determined in the first step via the Load Shape function method and used as input for the prestress identification via state-space formulation in the second step. A numerical study on a prestressed plate subjected to a moving load is conducted. Considering different levels of measurement noise and load speed, the proposed method can determine the prestress and moving load with 7.33% and 10.18% error, respectively. A laboratory test on a prestressed box girder subjected to a fixed cyclic load is performed, the prestress and cyclic load are both determined to have good stability, and the errors are under 11.32%.

Optical pulse-induced ultrafast antiferrodistortive transition in SrTiO3

The ultrafast dynamics of the antiferrodistortive phase transition in perovskite SrTiO3 is monitored via time-domain Brillouin scattering. Using femtosecond optical pulses, we initiate a thermally driven tetragonal-to-cubic structural transformation and detect the crystal phase through changes in the frequency of Brillouin oscillations (BO) induced by propagating acoustic phonons. Coupling the measured BO frequency with a spatiotemporal heat diffusion model, we demonstrate that, for a sample kept in the tetragonal phase, deposition of sufficient thermal energy induces a rapid transformation of the heat-affected region to the cubic phase. The initial phase change is followed by a slower reverse cubic-to-tetragonal phase transformation occurring on a timescale of hundreds of picoseconds. We attribute this ultrafast phase transformation in the perovskite to a structural resemblance between atomic displacements of the R-point soft optic mode of the cubic phase and the tetragonal phase, both characterized by anti-phase rotation of oxygen octahedra. The structural relaxation time exhibits a strong temperature dependence consistent with the prediction of the equation of motion describing collective oxygen octahedra rotation based on the energy landscape of the phenomenological Landau theory of phase transitions. Evidence of such a fast structural transition in perovskites can open up new avenues in information processing and energy storage sectors.