Terms In Equations Of Motion Research Articles

Entanglement entropy in the σ-model with the de Sitter target space

We derive the formula of the entanglement entropy between the left and right oscillating modes of the σ-model with the de Sitter target space. To this end, we study the theory in the cosmological gauge in which the non-vanishing components of the metric on the two-dimensional base space are functions of the expansion parameter of the de Sitter space. The model is embedded in the causal north pole diamond of the Penrose diagram. We argue that the cosmological gauge is natural to the σ-model as it is compatible with the canonical quantization relations. In this gauge, we obtain a new general solution to the equations of motion in terms of time-independent oscillating modes. The constraint structure is adequate for quantization in the Gupta–Bleuler formalism. We construct the space of states as a one-parameter family of Hilbert spaces and give the Bargmann–Fock and Jordan–Schwinger representations of it. Also, we give a simple description of the physical subspace as an infinite product of D12+ in the positive discreet series irreducible representations of the SU(1,1) group. We construct the map generated by the Hamiltonian between states at two different values of time and show how it produces the entanglement of left and right excitations. Next, we derive the formula of the entanglement entropy of the reduced density matrix for the ground state acted upon by the Hamiltonian map. Finally, we determine the asymptotic form of the entanglement entropy of a single mode bi-oscillator in the limit of large values of time.

Eigenvectors phase correction in inverse modal problem

The solution of the inverse modal problem for the spatial parameters of mechanical and structural systems is heavily dependent on the quality of the modal parameters obtained from the experiments. While experimental and environmental noises will always exist during modal testing, the resulting modal parameters are expected to be corrupted with different levels of noise. A novel methodology is presented in this work to mitigate the errors in the eigenvectors when solving the inverse modal problem for the spatial parameters. The phases of the eigenvector component were utilized as design variables within an optimization problem that minimizes the difference between the calculated and experimental transfer functions. The equation of motion in terms of the modal and spatial parameters was used as a constraint in the optimization problem. Constraints that reserve the positive and semi-positive definiteness and the inter-connectivity of the spatial matrices were implemented using semi-definite programming. Numerical examples utilizing noisy eigenvectors with augmented Gaussian white noise of 1%, 5%, and 10% were used to demonstrate the efficacy of the proposed method. The results showed that the proposed method is superior when compared with a known method in the literature.

On the optimal shape parameters of distinct versions of RBF meshless methods for the bending analysis of plates

This study intends to computationally analyze single-layer plates. Reissner–Mindlin plate deformation theory was combined with the radial basis function (RBF) meshless method. Considering the first order shear deformation theory, the RBF meshless method formulation for isotropic plates analysis is presented. According to the motion equations in terms of displacement for isotropic homogeneous laminates, non-zero laminate stiffness is determined. Transverse displacement results are evaluated for various thickness values where the plate is simply supported in all edges. In this work, it is also shown that Wendland's RBF formulation with optimal values of the shape parameter formulation using several mesh densities produces deflections of single-layer laminate plates agreeing well with analytical solutions. The obtained RBF solution proves that the supporting numerical methodology is efficient and robust.

Free Vibration of Thin Shallow Elliptical Shells

This research presents a study of the free vibration of thin, shallow elliptical shells. The equations of motion for the elliptical shell, which are developed from Love's equations, are coupled and nonlinear. In this research, a new approach is introduced to uncouple the transverse motion of the shallow elliptical shell from the surface coordinates. Through the substitution of the strain-compatibility equation into the differential equations of motion in terms of strain, an explicit relationship between the curvilinear surface strains and transverse strain is determined. This latter relationship is then utilized to uncouple the spatial differential equation for transverse motion from that of the surface coordinates. The approach introduced provides a more explicit relationship between the surface and transverse coordinates than could be obtained through use of the Airy stress function. Angular and radial Mathieu equations are used to obtain solutions to the spatial differential equation of motion. Since the recursive relationships that are derived from the Mathieu equations lead to an infinite number of roots, not all of which are physically meaningful, the solution to the eigenvalue problem is used to determine the mode shapes and eigenfrequencies of the shallow elliptical shell. The results of examples demonstrate that the eigenfrequencies of the thin shallow elliptical shell are directly proportional to the curvature of the shell and inversely proportional to the shell's eccentricity.

Millicharged fermion vacuum polarization in a cosmic magnetic field and generation of CMB elliptic polarization

The contribution of one loop milli-charged fermion vacuum polarization in cosmic magnetic field to the cosmic microwave background (CMB) polarization is considered. Exact and perturbative solution of the density matrix equations of motion in terms of the Stokes parameters are presented. For linearly polarized CMB at decoupling time, it is shown that propagation of CMB photons in cosmic magnetic field would generate elliptic polarization (circular and linear) of the CMB due to milli-charged fermion vacuum polarization. Analytic expressions for the degree of circular polarization and rotation angle of polarization plane of the CMB are presented. Depending on the ratio of milli-charged fermion relative charge to mass, $\epsilon/m_\epsilon$, and CMB observation frequency, it is shown that the acquired CMB degree of circular polarization could be of the order of magnitude $P_C(T_0)\sim 10^{-10}- 10^{-6}$ in the best scenario. The effect studied generates CMB polarization even in the case when the CMB is initially in thermal equilibrium. Limits on the magnetic field amplitude due to prior decoupling CMB polarization are presented.

On the motion of a gyro in the presence of a Newtonian force field and applied moments

This work focuses on the motion of a dynamical model that consists of a symmetric rigid body (gyro) that rotates about a fixed point similar to Lagrange’s gyroscope. This body is acted upon by external forces represented by a Newtonian force field, gyro torques about the principal axes of inertia of the gyro and perturbing moments acting on the same axes. Assuming that, the gyro initially has a high angular velocity about the dynamic axis of symmetry. The averaging technique is used to obtain a more appropriate averaging system for the governing system of equations of motion in terms of a small parameter. Therefore, the analytical solutions of this system for two applications, depending on different forms of perturbing moments, are presented. These solutions are represented graphically to clarify the effectiveness of the different parameters of the body on the motion.

Application of the Nonlinear Tschauner-Hempel Equations to Satellite Relative Position Estimation and Control

In this paper we develop the nonlinear motion equations in terms of the true anomaly varying Tschauner–Hempel equations relative to a notional orbiting particle in a Keplerian orbit, relatively close to an orbiting primary satellite to estimate the position of a spacecraft. A second orbiting body in Earth orbit relatively close to the first is similarly modelled. The dynamic relative motion models of the satellite and the second orbiting body, both of which are modelled in terms of independent relative motion equations, include several perturbing effects, such as the asymmetry of the Earth gravitational field resulting in the Earth's oblateness effect and the third body accelerations due to the Moon and the Sun. Linear control laws are synthesised for the primary satellite using the transition matrix so it can rendezvous with the second orbiting body. The control laws are then implemented using the state estimates obtained earlier to validate the feedback controller. Thus, we demonstrate the application of a Linear Quadratic Nonlinear Gaussian (LQNG) design methodology to the satellite rendezvous control design problem and validate it.

Two-point correlators revisited: fast and slow scales in multifield models of inflation

We study the structure of two-point correlators of the inflationary field fluctuations in order to improve the accuracy and efficiency of the existing methods to calculate primordial spectra. We present a description motivated by the separation of the fast and slow evolving components of the spectrum which is based on Cholesky decomposition of the field correlator matrix. Our purpose is to rewrite all the relevant equations of motion in terms of slowly varying quantities. This is important in order to consider the contribution from high-frequency modes to the spectrum without affecting computational performance. The slow-roll approximation is not required to reproduce the main distinctive features in the power spectrum for each specific model of inflation.

A simple analytical model for free vibration of orthotropic and functionally graded rectangular plates

With the widespread application of composite structures in engineering, the static and dynamic analyses of these structures have become important in the past years. Composite and functionally graded plates have been frequently being used and are important parts of many engineering structures. Numerous plate theories and solution procedures have been presented to predict the linear dynamic response of composite and functionally graded plates. This study aims to develop a simple analytical model to predict the free vibration response of orthotropic and functionally graded rectangular plates with clamped and simply-supported boundary conditions. To this end, Mindlin’s shear deformation plate theory has been used to obtain the equations of motion in terms of transverse displacement and rotations of mid-plane of the plate. After expressing the rotation parameters in terms of the transverse displacement, they have been substituted into equation of transverse motion of the plate. Then, for each boundary condition, an appropriate function has been assumed for the transverse displacement of the plate. Using the orthogonality of these functions, the natural frequencies of the plate have been determined. Some examples have been provided to validate the proposed model.

Quantum Kramers model: Corrections to the linear response theory for continuous bath spectrum.

Decay of the metastable state is analyzed within the quantum Kramers model in the weak-to-intermediate dissipation regime. The decay kinetics in this regime is determined by energy exchange between the unstable mode and the stable modes of thermal bath. In our previous paper [Phys. Rev. A 42, 4427 (1990)PLRAAN1050-294710.1103/PhysRevA.42.4427], Grabert's perturbative approach to well dynamics in the case of the discrete bath [Phys. Rev. Lett. 61, 1683 (1988)PRLTAO0031-900710.1103/PhysRevLett.61.1683] has been extended to account for the second order terms in the classical equations of motion (EOM) for the stable modes. Account of the secular terms reduces EOM for the stable modes to those of the forced oscillator with the time-dependent frequency (TDF oscillator). Analytic expression for the characteristic function of energy loss of the unstable mode has been derived in terms of the generating function of the transition probabilities for the quantum forced TDF oscillator. In this paper, the approach is further developed and applied to the case of the continuous frequency spectrum of the bath. The spectral density functions of the bath of stable modes are expressed in terms of the dissipative properties (the friction function) of the original bath. They simplify considerably for the one-dimensional systems, when the density of phonon states is constant. Explicit expressions for the fourth order corrections to the linear response theory result for the characteristic function of the energy loss and its cumulants are obtained for the particular case of the cubic potential with Ohmic (Markovian) dissipation. The range of validity of the perturbative approach in this case is determined (γ/ω_{b}<0.26), which includes the turnover region. The dominant correction to the linear response theory result is associated with the "work function" and leads to reduction of the average energy loss and its dispersion. This reduction increases with the increasing dissipation strength (up to ∼10%) within the range of validity of the approach. We have also calculated corrections to the depopulation factor and the escape rate for the quantum and for the classical Kramers models. Results for the classical escape rate are in very good agreement with the numerical simulations for high barriers. The results can serve as an additional proof of the robustness and accuracy of the linear response theory.

An Optimized RBF Analysis of an Isotropic Mindlin Plate in Bending

This study focuses on the numerical analysis of an isotropic plate under a uniform distributed load. The problem is solved considering the Reissner-Mindlin plate deformation theory combined with a Radial Basis Function (RBF) meshless method. Thus, in this work, the RBF meshless method formulation for the analysis of thick isotropic plates assuming a first order shear deformation theory is shown with detail. Additionally, the essential boundary conditions, material properties and geometrical characteristics of the model are fully presented. In this work, the nonzero laminate stiffnesses are determined through equations of motion in terms of displacements for homogeneous laminates.The transverse displacement results are evaluated for various thickness values where the plate is simply supported in all edges. Additionally, the shape parameter governing the RBF formulation is optimized using several mesh densities. The obtained numerical results are compared to the exact analytical solution available in the literature. Accurate RBF solutions are obtained, supporting the efficiency of the numerical methodology.

Forward dynamics of variable topology mechanisms—The case of constraint activation

Many mechanical systems exhibit changes in their kinematic topology altering the mobility. Ideal contact is the best known cause, but also stiction and controlled locking of parts of a mechanism lead to topology changes. The latter is becoming an important issue in human–machine interaction. Anticipating the dynamic behavior of variable topology mechanisms requires solving a nonsmooth dynamic problem. The core challenge is a physically meaningful transition condition at the topology switching events. Such a condition is presented in this article. Two versions are reported, one using projected motion equations in terms of redundant coordinates, and another one using the Voronets equations in terms of minimal coordinates. Their computational properties are discussed. Results are shown for joint locking of a planar 3R mechanisms and a 6DOF industrial manipulator.

Nonlocal transient electrothermomechanical vibration and bending analysis of a functionally graded piezoelectric single-layered nanosheet rest on visco-Pasternak foundation

ABSTRACTThis study develops the transient thermoelectromechanical vibration and bending analysis of a functionally graded piezoelectric nanosheet rest on visco-Pasternak’s foundation. Nonlocal elasticity theory as well as classical plate theory is used to implement basic equations of the nanosheet. The plate is resting on visco-Pasternak’s foundation and subject to mechanical, thermal, and electrical loadings. Hamilton’s principle is used for derivation equations of motion in terms of displacement components. As a first case study and for validation of the responses of the system, nonlocal vibration analysis of nanosheet is studied. The effects of nonlocal parameter and nonhomogeneous index of nanosheet are studied on the fundamental frequencies of the system. As the main objective of this study, the electrothermal bending results of the nanosheet are studied. The effects of some important parameters such as nonlocal parameter, nonhomogeneous index, thickness, distribution of electric potential, and damping are calculated on the maximum deflection of the sheet under various thermal and electrical loadings.

Free vibration, wave propagation and tension analyses of a sandwich micro/nano rod subjected to electric potential using strain gradient theory

Strain gradient theory is used to study free vibration, wave propagation and tension analyses of a sandwich micro/nano rod made of piezoelectric materials under electric potential. The structure is resting on a Pasternak’s foundation medium. Love’s rod model is used for derivation of displacement field. The piezoelectric face sheets are subjected to two-dimensional electric potential including an applied voltage at top of plate and a cosine term along the thickness direction. Hamilton’s principle is used to derive governing equations of motion in terms of axial displacement and electric potential. Three distinct behaviors of the present problem including free vibration, wave propagation and tension analyses are performed. Some important numerical results are presented in detail to capture the effect of materials length scales and applied voltage on the different behaviors of microrod.

Employing sinusoidal shear deformation plate theory for transient analysis of three layers sandwich nanoplate integrated with piezo-magnetic face-sheets

In this paper, based on the sinusoidal shear deformation plate theory, equations of motion for a sandwich nanoplate containing a nano core and two integrated piezo-magnetic face-sheets are derived. The piezo-magnetic face-sheets are subjected to three dimensional electric and magnetic potentials. Nonlocal piezo-magneto-elastic relations are derived in a thermal environment. Hamilton’s principle is used to derive seven equations of motion in terms of three deformation components of mid-surface, two shear components and electric and magnetic potentials. Natural frequencies of the sandwich nanoplate are derived in terms of nonlocal parameter. After finding solutions to the governing equations of motion, the effect of important parameters of the nanoplate are investigated on the mechanical, electrical and magnetic components of the nanoplate. Based on the present study, with increasing applied electric potential, dimensionless deflection is decreased and maximum electric and magnetic potentials are increased. Furthermore, with increasing applied magnetic potential, deflection is increased and maximum electric and magnetic potentials are decreased significantly. The numerical results of this problem indicate that one can control deformation or stress in the nano structure by changing the applied electric and magnetic potentials.

Influence of weak and strong gyroscopic effects on light aircraft dynamics

Purpose The purpose of this study is to investigate the influence of gyroscopic effects on the dynamic stability and the response of light aircraft to manoeuvres following either a rapid deflection of the control surfaces or wind gust. Design/methodology/approach The analyses were conducted for several different mathematical models of aircraft motion, which allowed for the investigation of the relationship between introduced simplifying assumptions and the aircraft response, including non-linear terms in equations of motion expressing the influence of inertial coupling. The analytical and experimental methods (measurements in the wind tunnel for the scaled model and during flight tests of I-31T prototype aircraft) were used. Findings It was found that gyroscopic moments are induced mainly by the propeller, and their influence on dynamic stability of a light aircraft is negligible. However, these phenomena in manoeuvring flight investigation should not be excluded, although for general aviation (GA) aircraft, they are not strong. Hence, two types of gyroscopic effects depending on the level of steady flight disturbances were distinguished. The authors differentiated weak gyroscopic effects, corresponding to classical dynamic stability, and strong gyroscopic effects, corresponding to rapid manoeuvres. Practical implications Conclusions include some findings on the nature of gyroscopic effects (i.e. sensitivity of flight stability versus turboprop power unit parameters) and practical recommendations for aircraft designers dealing with new configurations of GA aircraft. Originality/value The analysis focuses on the assessment of the flight dynamics of light aircraft with a novel, compact, lightweight, fast-rotating turbopropeller engine and strong/weak gyroscopic effects.

Dynamics of Flexible Body Negotiating a Curve

When a rigid body negotiates a curve, the centrifugal force takes a simple form which is function of the body mass, forward velocity, and the radius of curvature of the curve. In this simple case of rigid body dynamics, curve negotiation does not lead to Coriolis forces. In the case of a flexible body negotiating a curve, on the other hand, the inertia of the body becomes function of the deformation, curve negotiations lead to Coriolis forces, and the expression for the deformation-dependent centrifugal forces becomes more complex. In this paper, the nonlinear constrained dynamic equations of motion of a flexible body negotiating a circular curve are used to develop a systematic procedure for the calculation of the centrifugal forces during curve negotiations. The floating frame of reference (FFR) formulation is used to describe the body deformation and define the nonlinear centrifugal and Coriolis forces. The algebraic constraint equations which define the motion trajectory along the curve are formulated in terms of the body reference and elastic coordinates. It is shown in this paper how these algebraic motion trajectory constraint equations can be used to define the constraint forces that lead to a systematic definition of the resultant centrifugal force in the case of curve negotiations. The embedding technique is used to eliminate the dependent variables and define the equations of motion in terms of the system degrees of freedom. As demonstrated in this paper, the motion trajectory constraints lead to constant generalized forces associated with the elastic coordinates, and as a consequence, the elastic velocities and accelerations approach zero in the steady state. It is also shown that if the motion trajectory constraints are imposed on the coordinates of a flexible body reference that satisfies the mean-axis conditions, the centrifugal forces take the same form as in the case of rigid body dynamics. The resulting flexible body dynamic equations can be solved numerically in order to obtain the body coordinates and evaluate numerically the constraint and centrifugal forces. The results obtained for a flexible body negotiating a circular curve are compared with the results obtained for the rigid body in order to have a better understanding of the effect of the deformation on the centrifugal forces and the overall dynamics of the body.

Modelling of the joint motion of nonisothermal liquid film and gas flow in a microchannel: numerical simulation of full Navier-Stokes equations

New full-statement 3D mathematical model of joint motion of thin liquid film and gas in a microchannel at local heating developed by taking into account the heat transfer by flows, evaporation and condensation, as well as the heat transfer at the gas-liquid interface is derived. The model is based on the full system of the Navier-Stokes equations, taking into account the convective terms of motion equations in the phases. Comparison of the numerical results obtained using the model based on the full Navier-Stokes equations and using the simplified model developed in the framework of the thin layer approximation has been performed. The comparison shows that at low Reynolds numbers, simplified model well describes all the main characteristics of the gas and liquid motion. With the gas Reynolds numbers significant increase difference between numerical results starts to grow.

WKB Approxiation of the Dirac Equation with a Supersymmetric Extension

A general scheme of the self-consistent construction of a semiclassical approximation for the Dirac equation in an external gauge field in which the standard Dirac operator is replaced by the Dirac operator with a supersymmetric extension is presented. It is shown that in contrast to the usual WKB method, here the expansion must be carried out over half-integer powers of the Planck constant ħ. The first four terms of the semiclassical expansion of the wave function are obtained in explicit form. It is shown that generalization of the initial Dirac operator leads to the appearance of new additional terms in the semiclassical equation of motion for the spin of a particle in an external field, which thus requires a modification of the Lagrangian of the spinning particle. The result so obtained is used to construct mappings between two Lagrangian descriptions of a classical color-charged spinning particle, one of which possesses local supersymmetry, and the other not. It is shown that in order for the mappings to be one-to-one it is necessary to add new additional terms to the Lagrangian without supersymmetry, obtained within the framework of the semiclassical approximation of the Dirac operator with supersymmetry.

Closed form Newton–Euler dynamic model of flexible manipulators

SUMMARYIn this paper, a closed-form dynamic model of flexible manipulators is developed, based on the Newton–Euler formulation of motion equations of flexible links and on the adoption of the spatial vector notation. The proposed model accounts for two main innovations with respect to the state of the art: it is obtained in closed form with respect to the joints and modal coordinates (including the quadratic velocity terms) and motion equations of the whole manipulator can be computed for any arbitrary shape of the links and any possible link cardinality starting from the output of several commercial (finite element analysis) FEA codes. The Newton–Euler formulation of motion equations in terms of the joint and elastic variables greatly improves the simulation performances and makes the model suitable for real-time control and active vibration damping. The model has been compared with literature benchmarks obtained by the classical multibody approach and further validated by comparison with experiments collected on an experimental manipulator.