Non-resonant Case Research Articles

Methodology of investigation the dynamics of longitudinally moving systems under the action of impulse perturbations

A methodology for researching oscillations of one-dimensional elastic systems under the influence of impulse disturbances has been developed. The mathematical model of the considered process is boundary value problems for hyperbolic equations of the second order with an irregular right-hand side and a mixed derivative with respect to linear and time variables. As for the irregularity of the right part, it takes into account the impulse effect on the research object and has the most important theoretical and practical periodic character; and with the help of the mixed derivative in the Euler variables, the longitudinal movement of the studied systems is partially taken into account. Using the basic ideas of asymptotic integration of equations with partial derivatives, the basic propositions of the wave theory of motion adapted to the considered class of problems, the principle of single-frequency oscillations in nonlinear systems with concentrated masses and distributed parameters, the method of regularization of impulse disturbances, equations in the standard form were obtained that describe the determining parameters of nonlinear oscillations for resonant and non-resonant cases. It is shown: that the longitudinal movement of the systems has a significant effect on the natural frequency of oscillations, and therefore on the period of impulse disturbances during which the resonance process takes place; the amplitude of the passage of resonance depends significantly on the speed of movement. The results obtained in the work can serve as a basis for choosing the operating modes of some classes of systems characterized by longitudinal movement in order to avoid resonant processes in them.

Leaky-Wave Analysis of TM-, TE-, and Hybrid-Polarized Aperture-Fed Bessel-Beam Launchers for Wireless-Power-Transfer Links

Bessel beams (BBs) can be generated at microwave frequencies by means of compact, planar devices based on Fabry–Perot cavity leaky-wave resonators. Most designs are based on the excitation of a single transverse magnetic/transverse electric (TM/TE) leaky mode by means of a vertical electric/magnetic dipole source. In this work, we analyze the important case of a horizontal magnetic dipole, which excites both TM and TE modes, developing an original theoretical framework suitable for resonant radiators. Analytical expressions are provided for the dominant leaky-wave aperture field. The near-field distribution is then evaluated through an accurate numerical integration of the radiating currents and validated through full-wave simulations for the relevant case of a BB launcher when excited by a waveguide-fed slot. Different designs are presented and analyzed in order to obtain TM, TE, and hybrid polarizations, finding in all cases an excellent agreement between simulations and theoretical results. Finally, we evaluate and compare the wireless-power-transfer efficiency of two coupled TM-, TE-, or hybrid-polarized BB launchers when a wireless link is established in the radiative near-field region. Interestingly, full-wave results demonstrate the superior performance of resonant BB launchers in TM or TE polarization with respect to the nonresonant hybrid case.

Parametric resonance and bifurcation analysis of thin-walled asymmetric gyroscopic composite shafts: An asymptotic study

Nonlinear geometries and cross-sectional asymmetry can be among the most critical contributing factors affecting the performance and stability of composite shafts. The influence of these factors, which should be devoted to particular attention in the design of these systems and have not been investigated yet, are evaluated analytically in this study. The shaft is simply supported, made of orthotropic multi-layers, and spinning at a constant speed. To express the nonlinear system’s behavior, which is due to the large amplitude of the vibrations, it is assumed that the shaft is under the stretching assumption. Moreover, a rectangular cross-section is used to model the asymmetry that results in parametric excitation in the system. To accurately investigate the behavior of composite material, an optimal lay-up is employed. The gyroscopic coupling is included because of Rayleigh beam theory, and Euler’s angles are employed to achieve the angular velocities. The analytical study of the parametrically excited system, obtained by the method of multiple scales, is performed in two categories of resonant and nonresonant cases. In the nonresonant case, the analytical investigation suggests that the asymmetric shaft behaves like a symmetric one therefore, the parametric excitations do not have a significant impact. This claim is confirmed by numerical results. Also, the presence of the gyroscopic coupling and hollowness of the shaft causes the beating phenomenon in the system. However, in the resonant case, the presence of parametric excitation plays a pivotal role. The results also show that under certain conditions and despite the presence of damping, asymmetric balanced rotors can have a nontrivial stable amplitude. This response exists as long as the parametric excitation effects dominate the damping effects. Although damping reduces the vibrations’ amplitude, it can improve the stability of the system and eliminates unstable responses. Furthermore, the time response and frequency response curve of the system is carefully evaluated for various geometric design parameters and operation speed. Depending on the operating speed, the system can experience supercritical or subcritical pitchfork bifurcation. In addition, a detailed description of the system’s Campbell’s diagram, damping effect, and bifurcation is provided. Furthermore, it is proved that internal resonance cannot occur in the system. The accuracy of the analytical responses of the resonant case is compared with numerical ones. Stability of trivial and nontrivial responses is discussed in the time response and phase portrait of the system simultaneously. Finally, in the undamped system, multi-frequency responses are appeared as homoclinic and heteroclinic closed orbits.

NONLINEAR EHD STABILITY OF ENCLOSED COAXIAL JETS WITH MASS AND HEAT TRANSMISSION IN THE EXISTENCE OF AN OSCILLATORY GAS VELOCITY PROFILE

In this paper, the present work investigates a nonlinear temporal instability of coaxial cylinders in porous media in the existence of an oscillating gaseous velocity. Heat and mass transfer (MHT), as well as a uniform axial electric field were all taken into account. The viscous potential flow (VPF) was employed to calculate liquid and gas velocities to make the mathematical manipulations easier. As a result of the analysis of the boundary-values problem, the cylindrical interface displacement provided a nonlinear characteristic equation. To achieve the stability inspection, a novel approach was created. The article designates both the oscillatory and uniform streaming gas. The nonlinear analysis was completed using the homotopy perturbation method (HPM), which resulted in a Klein-Gordan equation, to arrive the stability maps. Additionally, the resonance and non-resonance cases were accomplished. In the previous situations, the stability standards were theoretically derived and numerically proven using regular diagrams. It was found that the unchanging flowing had a twofold effect. The linear MHT parameter was identified to perform a twofold character in the stability setup. On the other hand, nonlinear parameters have opposing effects. The outcomes of the homogeneous gas velocities were substantially conforming.

Nonlinear Orbital Stability of Periodic Motions in the Planar Restricted Four-Body Problem

In this work we study the nonlinear orbital stability problem for periodic motions emanating from the stable relative equilibrium. To describe motions of the small body in a neighborhood of its periodic orbit, we introduce the so-called local variables. Then we reduce the orbital stability problem to the stability problem of a stationary point of symplectic mapping generated by the system phase flow on the energy level corresponding to the unperturbed periodic motion. This allows rigorous conclusions to be drawn on orbital stability for both the nonresonant and the resonant cases. We apply this method to investigate orbital stability in the case of third- and fourth-order resonances as well as in the nonresonant case. The results of the study are presented in the form of a stability diagram.

Resonant and non-resonant optimizations by multi-constraint quantum control theory in molecular rotational states

It is a promising research for optimization of quantum gate in the field of quantum computation. We investigate the feasibility of implementing the single-qubit gate (Hadamard) in molecular rotational system. By applying the Multi-constraint quantum optimal control method, the excepted final states can be achieved based on the molecular rotational states both in resonant and non-resonant cases with the control pulses. The permanent electric dipole moment is ignored in non-resonance. Besides, the zero-pulse area constraint and the constant fluence constraint are employed to optimize shapes of control pulses. Finally, we show that the Hadamard gate can be realized with the high fidelity (0.9999) and also examine the dependence of the fidelity on pulse fluence as well as the control pulse.

Numerical analysis of fully non-linear sloshing waves in an arbitrary shape tank by meshless method

This paper is devoted to develop a truly meshless numerical procedure for the fully non-linear analysis of sloshing phenomenon in an arbitrary-shape tank. For this purpose, the potential theory is considered as a liquid sloshing model, and the Lagrangian form of kinematic and dynamic boundary conditions in a moving coordinate system fixed to the tank is used to accurately capture the free surface. The multi-quadric radial basis function (MQ-RBF) augmented with the polynomial terms is employed to determine RBF-FD weights on neighboring nodes surrounding the center point for the spatial derivatives. These weight coefficients are used to solve both boundary value problems for velocity and acceleration potential. Compared with mash-based methods, the present method has the advantage of being easy to construct and flexible in dealing with the moving boundary problems. The free surface elevations, hydrodynamic pressure, and first-order natural sloshing frequencies have been reported for various geometries tanks under horizontal excitations at resonance and non-resonance cases. Fairly satisfactory agreement is observed in the numerical results and exiting results, especially for free surface elevation and first-order natural sloshing frequency.

Operator mixing in massless QCD-like theories and Poincarè–Dulac theorem

Recently, a differential-geometric approach to operator mixing in massless QCD-like theories – that involves canonical forms, obtained by means of gauge transformations, based on the Poincarè–Dulac theorem for the system of linear differential equations that defines the renormalized mixing matrix in the coordinate representation Z(x,mu ) – has been proposed in [1]. Specifically, it has been determined under which conditions a renormalization scheme exists where Z(x,mu ) may be set in a diagonal canonical form that is one-loop exact to all perturbative orders – the nonresonant diagonalizable frac{gamma _{0}}{beta _{0}} case (I) –. Moreover, the remaining cases, (II), (III) and (IV), of operator mixing, where such diagonalization is not possible, have also been classified in [1]. Accordingly, if frac{gamma _{0}}{beta _{0}}, with gamma (g)=gamma _{0} g^{2}+cdots the matrix of the anomalous dimensions and beta (g)=-beta _{0} g^{3} + cdots the beta function, either is diagonalizable but a resonant condition for its eigenvalues and the system holds (II) or is nondiagonalizable and nonresonant (III), or is nondiagonalizable and resonant (IV), Z(x,mu ) is nondiagonalizable. In the cases (II), (III) and (IV), we demonstrate that its canonical form may be factorized into the exponential of a linear combination of upper triangular nilpotent constant matrices with coefficients that asymptotically in the UV are logs of the running coupling, i.e., asymptotically loglogs of the coordinates, and a diagonal matrix as in the nonresonant diagonalizable case (I). Hence, its ultraviolet asymptotics differs intrinsically from the case (I) and, for asymptotically free theories, this is the closest analog of logCFTs. We also work out actual physics realizations of the cases (I) and (II), while we argue that the cases (III) and (IV) are ruled out by a unitarity argument in the gauge-invariant sector.

Advancing asymptotic approaches to studying the longitudinal and torsional oscillations of a moving beam

This paper analyzes the influence of kinetic and physical-mechanical parameters of systems on the characteristics of dynamic processes in moving one-dimensional nonlinear-elastic systems. Improved convenient calculation formulas have been derived that describe the laws of changing the amplitude-frequency characteristics of systems for both a non-resonant case and a resonant one. An important issue of studying the influence of the speed of movement of elements of mechanisms on the oscillations of one-dimensional nonlinear-elastic systems has not been considered in detail until now in the scientific literature. This issue relates to the vibrations of shafts in gears, pipe strings when drilling oil and gas wells, the oscillations of turbine blades and rotating turbine discs, the longitudinal vibrations of the beam as an element of structures. The main reason for this in the analytical study of dynamic processes were the shortcomings of the mathematical apparatus for solving the corresponding nonlinear differential equations that describe the laws of motion of those systems. It was found that in the case of longitudinal oscillations in the moving beam with an increase in the longitudinal speed of the medium to 10 m/s, the amplitude of the oscillation also increases by 13.5 %. However, when the longitudinal velocity of the beam is 5 m/s, the amplitude will increase by only 3 %. It is established that with the growth of the amplitude, the frequency of longitudinal oscillations decreases sharply, and if the system moves at a higher speed, for example, 20 m/s, it reduces the frequency of oscillation by about 13 %. The results reported here make it possible to assess the effect of kinetic and physical-mechanical parameters on the frequency and amplitude of oscillations. The research that involved the asymptotic method makes it possible to predict resonant phenomena and obtain engineering solutions to improve the efficiency of technological equipment.

Multipartite stationary entanglement generation in the presence of dipole-dipole interaction in an optical cavity

Engineering a stationary entanglement between atoms or ions placed at small distances is a challenging problem in quantum information science. In this paper, the stationary and dynamics of entanglement in a system of dipole-coupled qubits interacting with a single-mode optical cavity in the strong coupling regime are theoretically investigated. We find that the entanglement in the steady state can be induced and tuned in nonresonance cases by considering the dipole-dipole interaction (DDI) between the qubits. We also point out that the novel measure used in this study can quantify the net multiqubit entanglement in the system. By increasing the DDI intensity, the behavior of the system depends on the number of qubits. Interestingly, with only two qubits, the amounts of steady-state entanglement enhances as the DDI intensity increases. For a system with more than two qubits, we find that DDI in the weak range of intensity plays a constructive role in the entanglement between qubits. However, it could destroy the stationary entanglement as the interaction between qubits intensifies. The entanglement of a system consisting of more than two qubits tends to disappear for an ensemble of qubits with smaller interatomic distances that lead to stronger dipolar interaction between nonadjacent qubits. Finally, we study the atomic population transfer as well as the transmitted spectrum.

A deformed model for N-type four-level atom and a single mode field system in the presence of the Kerr medium

The aim of this paper is to study the interaction between a single mode field and four-level atom in N - configuration under nonlinear medium effect. The non-resonance case and the deformation forms in the coupling interaction between the field and the atom are included. The wave function of the proposed system is obtained when the atom is prepared initially in its excited state while the field is prepared in a coherent state. The effect of the deformation and nonlinear medium on the temporal behavior of collapse-revival, field entropy and geometric phase of the system are examined. The results show that the presence of the intensity of the coupling interaction and the non-linear medium have an important influence on the properties of these phenomena.

Possible solution to the triple alpha fine-tuning problem: Spallation reactions during planet formation

Carbon is produced during the helium burning phase of sufficiently massive stars through the triple alpha process. The 0+ energy level of the carbon nucleus allows for resonant nuclear reactions, which act to greatly increase the carbon yields compared to the non-resonant case. Many authors have argued that small changes to the energy level of this resonance would lead to a significantly lower carbon abundance in the universe, and this sensitivity is often considered an example of fine-tuning. By considering spallation reactions occurring during the process of planet formation, this paper presents a partial solution to this triple alpha fine-tuning problem. Young stellar objects generate substantial luminosities of particle radiation (cosmic rays) that can drive nuclear reactions through spallation. If the standard triple alpha process is inoperative, stars tend to synthesize oxygen (and other alpha elements) rather than carbon. Cosmic rays can interact with oxygen nuclei to produce carbon while planets are forming. The resulting carbon abundances are significant, but much smaller than those observed in our universe. However, for a range of conditions – as delineated herein – spallation reactions can result in carbon-to-oxygen ratios roughly comparable to those found on Earth and thereby obviate the triple alpha fine-tuning problem.

Codimension–one singularities of Constrained systems on [formula omitted

This paper is concerned with the dynamics near an impasse singularity of Constrained systems A(x)x˙=F(x), where A(x) is a 3×3 smooth matrix-valued function and F(x) is a smooth vector field defined on R3. Impasse singularity can be either a tangential impasse point or an impasse equilibrium point. We present all the topological types and their respective normal forms of the codimension-one impasse singularities for a large class of Constrained systems on the three-dimensional space. We split the study of impasse equilibrium points into nonresonant and resonant cases. In the first case, the constrained system presents cubic impasse bifurcation, while for the other one occurs focus–node, saddle-node, or Hopf impasse bifurcations. The tangential impasse points present three types: Lips, Bec-to-bec, and Dove's tail singularities.

Effects of linear and nonlinear dashpots forces on the dynamics of plants vibrating in a vegetation cover situation

This paper considers the nonlinear dynamics of the oscillations of an individual plant forced by the fluctuating aerodynamic drag and significantly influenced by collisions with its close neighbors. In this study, a row of identical and flexible oscillators is considered. The flexible beams theory is used to model the equation of motion of an individual plant and then plant collisions were modelled as a simple spring, linear dashpot and nonlinear dashpot interactions between adjacent plants. The multiple scales method is used to obtain the amplitude of the plant relative to wind forcing in the resonant and nonresonant oscillations cases. Numerically, chaotic oscillations, hysteresis and coexistence of attractors have been found using the bifurcation diagram, the Lyapunov exponents and the phase portraits. The influences of both type of dashpot collisions on the dynamics behaviors of the plant have been analyzed.

Wave processes in the constructions of mechanisms and buildings

In the article a methodology for researching wave processes in weakly nonlinear homogeneous one-dimensional systems – elements of machine and building structures is proposed. It is based on D'Alembert's method of constructing solutions of wave equations and the asymptotic method of nonlinear mechanics. Resonant and non-resonant cases are considered. In the non-resonant case, the influence of viscoelastic forces and small periodic disturbances on the oscillatory process is taken into account. In the resonant case, the influence of the disturbing force and the phase difference of natural oscillations is considered. Numerical methods for linear differential equations were used to analyze the obtained differential equations.

Asymptotic method and wave theory of motion in studying the effect of periodic impulse forces on systems characterized by longitudinal motion velocity

A methodology for researching dynamic processes of one-dimensional systems with distributed parameters that are characterized by longitudinal component of motion velocity and are under the effect of periodic impulse forces has been developed. The boundary problem for the generalized non-linear differential Klein–Gordon equation is the mathematical model of dynamics of the systems under study in Euler variables. Its specific feature is that the unexcited analogue does not allow applying the known classical Fourier and D'Alembert methods for building a solution. Non-regularity of the right part for the excited non-linear analogue is another problem. This study shows that the dynamic process of the respective unexcited motion can be treated as overlapping of the direct and reflected waves of different lengths but equal frequencies. Analytical dependencies have been obtained for describing the aforesaid parameters of the waves. They show that the dynamic process in such mechanical systems depends not only on their main physical and mechanical parameters and boundary conditions, but also on the longitudinal motion velocity (relative momentum). As relative momentum increases, the frequency of the process decreases. To describe excited motion, we use the principle of single frequency of oscillations in non-linear systems with concentrated masses and distributed parameters as well as regularization of periodic impulse excitations. The main idea of asymptotic integration of systems with small non-linearity into the class of dynamic systems under study has been generalized. A standard equation for the resonance and non-resonance cases has been obtained. It has been established that for the first approximation, in the non-resonance case, impulse excitation affects only the partial change of the form of oscillations. Resonance processes are possible at a specific relation between the impulse excitation period, the motion velocity of the medium, and physical-mechanical features of the body. The amplitude of transition through resonance becomes higher when impulse actions are applied closer to the middle of the body. As the longitudinal motion velocity increases, it initially increases and then decreases.

Rotation of a Planet in a Three-Body System: a Non-Resonant Case

We investigate the temporal evolution of the rotation axis of a planet in a system comprised of the planet (which we call an exo-Earth), a star (an exo-Sun) and a satellite (an exo-Moon). The planet is assumed to be rigid and almost spherical, the difference between the largest and the smallest principal moments of inertia being a small parameter of the problem. The orbit of the planet around the star is a Keplerian ellipse. The orbit of the satellite is a Keplerian ellipse with a constant inclination to the ecliptic, involved in two types of slow precessional motion, nodal and apsidal. Applying time averaging over the fast variables associated with the frequencies of the motion of exo-Earth and exo-Moon, we obtain Hamilton’s equations for the evolution of the angular momentum axis of the exo-Earth. Using a canonical change of variables, we show that the equations are integrable. Assuming that the exo-Earth is axially symmetric and its symmetry and rotation axes coincide, we identify possible types of motions of the vector of angular momentum on the celestial sphere. Also, we calculate the range of the nutation angle as a function of the initial conditions. (By the range of the nutation angle we mean the difference between its maximal and minimal values.)

Damped Mathieu Equation with a Modulation Property of the Homotopy Perturbation Method

In this article, the main objective is to employ the homotopy perturbation method (HPM) as an alternative to classical perturbation methods for solving nonlinear equations having periodic coefficients. As a simple example, the nonlinear damping Mathieu equation has been investigated. In this investigation, two nonlinear solvability conditions are imposed. One of them was imposed in the first-order homotopy perturbation and used to study the stability behavior at resonance and non-resonance cases. The next level of the perturbation approaches another solvability condition and is applied to obtain the unknowns become clear in the solution for the first-order solvability condition. The approach assumed here is so significant for solving many parametric nonlinear equations that arise within the engineering and nonlinear science.

Numerical Orbital Stability Analysis of Nonresonant Periodic Motions in the Planar Restricted Four-Body Problem

We address the planar restricted four-body problem with a small body of negligible mass moving in the Newtonian gravitational field of three primary bodies with nonnegligible masses. We assume that two of the primaries have equal masses and that all primary bodies move in circular orbits forming a Lagrangian equilateral triangular configuration. This configuration admits relative equilibria for the small body analogous to the libration points in the three-body problem. We consider the equilibrium points located on the perpendicular bisector of the Lagrangian triangle in which case the bodies constitute the so-called central configurations. Using the method of normal forms, we analytically obtain families of periodic motions emanating from the stable relative equilibria in a nonresonant case and continue them numerically to the borders of their existence domains. Using a numerical method, we investigate the orbital stability of the aforementioned periodic motions and represent the conclusions as stability diagrams in the problem’s parameter space.

Solvability for a fractional $ p $-Laplacian equation in a bounded domain

<abstract><p>In this paper we use the topological degree and the fountain theorem to study the existence of weak solutions for a fractional $ p $-Laplacian equation in a bounded domain. For the nonlinearity $ f $, we consider two situations: (1) the non-resonance case where $ f $ is $ (p-1) $-asymptotically linear at infinity; (2) the resonance case where $ f $ satisfies the Landesman-Lazer type condition.</p></abstract>