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In many industries’ applications, one deals with pipes conveying fluid. Besides, the implementation of advanced materials in manufacturing of new-brand structures is developing. Consequently, in this paper, one of the most important dynamics analyzes that an advanced pipe conveying fluid may meet it is addressed. For the first time, linear and nonlinear dynamics of a pipe reinforced with graphene nanoplatelets (GPL pipe) that conveys pulsating flow is presented. The Euler–Bernoulli beam model follows the plug flow model in order to formulate the problem. The method of multiple scales (MMS) applied to the discretized couple nonlinear gyroscopic governing equations with the aim of specifying the instability region and the steady-state response of the GPL pipe subjected to the principal parametric resonance of one of its modes, as a consequence of pulsating flow. Respectively, the Floquet theory and the Runge–Kutta fourth-order (RK4) method are applied to the linear governing equations and nonlinear governing equations for the sake of confirming the current MMS results. It is deduced that a small amount usage of GPL reinforcement phase improves substantially the resistance against static instability, and the critical fluid excitation amplitude fraction, meanwhile declines the instability region bandwidth, and the steady-state response of the pipe. In contrary, it is confirmed that a heavier fluid makes reverse the preceding statements with respect to a lighter fluid. The aforementioned findings shed light on the essential design keys that outstandingly affect the mechanics of the GPL pipe conveying pulsating fluid that lead and conduct forthcoming studies and open new horizons for designers in industry implementations.

The spinning circular solar sail is a promising spacecraft for long-duration missions. This work reveals its structural dynamic and stability behavior under the periodically time-varying solar radiation pressure and gravitational force. The geometric stiffness generated by the centrifugal force due to spinning and the coupling effect between the deformation and solar radiation pressure are taken into account. The von Kármán plate theory is adopted by neglecting the high-frequency in-plane vibrations and considering the effect of the in-plane internal force on the transverse vibration. The partial differential equation of the spinning solar sail is derived and further spatially discretized into periodically time-varying equations of motion. Effects of Poisson ratio and radius ratio on natural frequencies and mode shapes are analyzed, and curve veering phenomena are then observed. Steady-state periodic responses of the solar sail under the solar radiation pressure with different orbit distances, incident angles, and spinning angular velocities are analyzed. The stability analysis is rigorously performed by the Floquet theory rather than the commonly used approach of conducting the eigenvalue analysis at a series of specific discrete time nodes. Moreover, the stability boundary associated with transverse vibrations is determined, which contributes to the parameter design of the spinning solar sail.

We study the evolution of a one-dimensional model atom with δ-function binding potential, subjected to a dipole radiation field E(t)x with E(t) a 2π/ω-periodic real-valued function. We prove that when E(t) is a trigonometric polynomial, complete ionization occurs, i.e. the probability of finding the electron in any fixed region goes to zero as t → ∞.

For ψ(x, t = 0) compactly supported and general periodic fields, we decompose ψ(x, t) into uniquely defined resonance terms and a remainder. Each resonance is 2π/ω periodic in time and behaves like the exponentially growing Green's function near x = ±∞. The remainder is given by an asymptotic power series in t^-1/2 with coefficients varying with x.

We review Floquet formalism of quantum electron pumps. In the Floquet formalism the quantum pump is regarded as a time dependent scattering system, which allows us to go beyond the adiabatic limit. It can be shown that the well-known adiabatic formula given by Brouwer can be derived from the adiabatic limit of Floquet formalism. We compare various physical properties of the quantum pump both in the adiabatic and in the non-adiabatic regime using the Floquet theory.

The derivation of a time-dependent Schrödinger equation (TDSE) from a time-independent Schrödinger equation (TISE) in the coherent state representation is considered for the special case of a simple coupled atom-field system described by the soluble Jaynes-Cummings model. The derivation shows why, from the outset, a linear combination of energy eigenstates, instead of a single state, must be used in order to obtain a TDSE for general states. Moreover, this study leads to a method of solving a TDSE by simply solving a TISE.

We present a new method for the numerical computation of Lyapunov exponents of periodic trajectories which is crucial in the investigation of dynamics. The computational time of this method is merely a period of the periodic trajectory considered, when Lyapunov exponents can be approximated as precise as one wants. Our method stems from a combination between Floquet theory on periodic linear differential systems and Lie group methods in structure preserving algorithms on manifolds. The Lyapunov exponents of a periodic trajectory of the Lorenz system and a periodic solitary wave of the nonlinear Schrödinger equation with periodic coefficients are investigated by using the method.

Some patterns of synchrony/asynchrony in the dynamics of coupled cell systems can be predicted by symmetry. However, in the system without symmetry, the different patterns of periodic solutions may exist as well. We consider a general model including three cells with multiple time delays that connect in any possible manner. Our approach is based on the analytic construction by using a perturbation procedure together with the Fredholm alternative theory. Then we employ the Poincaré–Lindstedt series expansion to compute the Floquet exponents which determine the stability. Finally, we resort to numerical computation to get some insights about the leading term in the Floquet exponents, and the numerical simulations are given to confirm the theoretical results.

The harmonic balance method truncates the Fourier series in a finite number of terms. In this paper we show that the truncated residues may be important to determine the stability of the approximated solution and that the truncated residues in the stability analysis can fully be considered without increasing the number of equations in the original solution. Therefore, the high order superharmonic and subharmonic responses and the cascade of bifurcations to irregular attractor can be accurately approximated by just the first few terms of the Fourier series so that analytical prediction is possible. A harmonically driven oscillator with quadratic nonlinearity is taken as examples. The explicitly analytical solutions are obtained for the steady state solutions and for the high order superharmonic approximation. The stabilities of the solutions are determined by the Floquet theory. It is shown that the predicted stability of the solution can be qualitatively different with and without the consideration of the feed forward residues. The second-, fourth- and eighth-order subharmonic analytical bifurcation solutions are calculated to obtain the cascades of bifurcations to irregular attractor. The improved analytical harmonic approximations are compared with other results and with numerical solutions. It is proved that a two superharmonic expansion with appropriate subharmonic is sufficient for determining the characteristics of the solutions of a harmonically driven oscillator with quadratic nonlinearity.

Recently, nonlinearities have been shown to play an important role in increasing the extracted energy of vibration-based energy harvesting systems. In this paper, we study the dynamical behavior of a piecewise linear (PWL) spring-mass-damper system for vibration-based energy harvesting applications. First, we present a continuous time single degree of freedom PWL dynamical model of the system. Different configurations of the PWL model and their corresponding state-space regions are derived. Then, from this PWL model, extensive numerical simulations are carried out by computing time-domain waveforms, state-space trajectories and frequency responses under a deterministic harmonic excitation for different sets of system parameter values. Stability analysis is performed using Floquet theory combined with Filippov method, Poincaré map modeling and finite difference method (FDM). The Floquet multipliers are calculated using these three approaches and a good concordance is obtained among them. The performance of the system in terms of the harvested energy is studied by considering both purely harmonic excitation and a noisy vibrational source. A frequency-domain analysis shows that the harvested energy could be larger at low frequencies as compared to an equivalent linear system, in particular, for relatively low excitation intensities. This could be an advantage for potential use of this system in low frequency ambient vibrational-based energy harvesting applications.

Response, stability, and bifurcation of roll oscillations of a biased ship under regular sea waves are investigated. The primary and subharmonic response branches are traced in the frequency domain employing the Incremental Harmonic Balance (IHB) method with a pseudo-arc-length continuation approach. The stability of periodic responses and bifurcation points are determined by monitoring the eigenvalues of the Floquet transition matrix. The primary and higher-order subharmonic responses experience a cascade of period-doubling bifurcations, eventually culminating in chaotic responses detected by numerical integration (NI) of the equation of motion. Bifurcation diagrams are obtained through the period-doubling route to chaos. Solutions are aided with phase portrait, Poincaré map, time history and Fourier spectrum for better clarity as and when required. Finally, the same ship model is investigated under variable excitation moments that may result from different wave heights in regular seas. The biased ship roll model exhibits primary and subharmonic responses, jump phenomena, coexistence of multiple responses, and chaotically modulated motion. The stable, periodic, and steady-state roll responses obtained by the IHB method are validated by the NI method. Results obtained by both methods are found to agree very well.

We study electromagnetic wave propagation in a periodic and frequency dependent material characterized by a space- and frequency-dependent complex-valued permittivity. The spectral parameter relates to the time-frequency, leading to spectral analysis of a holomorphic operator-valued function. We apply the Floquet transform and show for a fixed quasi-momentum that the resulting family of spectral problems has a spectrum consisting of at most countably many isolated eigenvalues of finite multiplicity. These eigenvalues depend continuously on the quasi-momentum and no nonzero real eigenvalue exists when the material is absorptive. Moreover, we reformulate the special case of a rational operator-valued function in terms of a polynomial operator pencil and study two-component dispersive and absorptive crystals in detail.

We consider a two-patch SI model of integrated pest management with dispersal of both susceptible and infective pests between patches. A biological control, consisting of the periodic release of infective pests and a chemical control, consisting of periodic and impulsive pesticide spraying, are employed in order to maintain the size of the pest population below an economically acceptable level. It is assumed that the spread of the disease which is inflicted on the pest population through the use of the biological control is characterized by a nonlinear force of infection expressed in an abstract form. A sufficient condition for the local stability of the susceptible pest-eradication periodic solution is found using Floquet theory for periodic systems of ordinary differential equations, an analysis of the influence of dispersal between patches being performed for several particular cases. Our numerical simulations indicate that an increase in the amount but not in the frequency of pesticide use may not result in control. We also show that patches which are stable in isolation can be destabilized by dispersal between patches.

Prüfer transformation is more effective and flexible in studying the spectral analysis of boundary value problem than using the classical methods in operator theory. The goal of this paper is to study Prüfer approach to spectral analysis of periodic Sturm–Liouville problem with transmission condition. Since we are dealing with a singular problem, the characteristic function we obtained is a piecewise function. At the end of the study, the existence of eigenvalues of investigated problem by using Prüfer transformation is given.

The nonlinear traveling-wave vibration of a ring-stringer stiffened cylindrical shell is analyzed. Using Donnell’s nonlinear shell theory and Lagrange equations, the nonlinear dynamic model of the ring-stringer stiffened cylindrical shell is derived. Galerkin’s method based on multi-mode instead of single-mode approximation is used to discretize the shell’s displacements. Two types of orthogonal circumferential modes with same frequency are used and the interaction between them is considered in the analysis of the shell’s nonlinear traveling-wave vibration. The harmonic balance (HB) method, along with the pseudo-arc length continuation algorithm, is adopted to solve the forced vibration responses of the shell. The stability of the solution is determined by the Floquet theory. Through comparison with the results available in the literature, the correctness of the present nonlinear dynamic model and its solution process are validated first. Next, the mode selection rules are determined through a convergence study. Finally, the nonlinear traveling-wave vibration of the ring-stringer stiffened cylindrical shell is studied. Also, the paper investigates, in detail, the effects of stiffener parameters on the nonlinear dynamic characteristics of the stiffened shell.

A hybrid method is presented to obtain the analytical approximate solution to the primary resonance of harmonically forced strongly nonlinear oscillators. This hybrid method combines the classical perturbation method and the classical harmonic balance method. With the proposed splitting procedure some free parameters are introduced, more accurate and reliable analytical approximation compared to the results obtained by the classical harmonic balance method are presented. The proposed method is not based on the small parameter assumption when perturbation method is applied. It is found that the corrections to erroneous solution when harmonic balance method and Floquet theory are adopted in stability analysis is necessary. The proposed method gives excellent stability results compared to those obtained by using harmonic balance method and Floquet theory. Two examples are presented to illustrate the applicability, validity and convergence of the proposed method. The convergence of the solution in stability analysis by the proposed hybrid method are compared with that obtained by using the Floquet theory and the harmonic balance method. The results obtained by the proposed method are verified by the numerical simulations.

In this paper, the approximate analytical solutions obtained by using the constrained parameter-splitting-multiple-scales (C-PSMS) method to the primary and $1/3$ sub-harmonic resonances responses of a cantilever-type energy harvester are presented. The C-PSMS method combines the multiple-scales (MS) method with the harmonic balance (HB) method. Different from the erroneous stability results obtained by using the Floquet theory and the classical HB method, accurate stability results are obtained by using the C-PSMS method. It is found that the correction to the erroneous solution when the HB method and Floquet theory are adopted in the stability analysis of the primary and $1/3$ sub-harmonic resonances of a largely deflected cantilever-type energy harvester is necessary. On the contrary, the C-PSMS method gives much improved results compared to those obtained by using Floquet theory and HB method when the numbers of terms in each response expression are the same. The frequency response curves of the primary resonance and the $1/3$ sub-harmonic resonance of the harvester obtained by the C-PSMS method are compared to those obtained by the HB method and verified by those obtained by the fourth-order Runge–Kutta method. Moreover, the basin of attraction based on the fourth-order Runge–Kutta method is presented to confirm the inaccurate stability results obtained by using the HB method and Floquet theory. The convergence examinations on the stability analysis carried out by the HB method and Floquet theory show that enough terms in the response assumption are needed to achieve relatively accurate stability results when studying the stability of the primary and sub-harmonic resonances of a cantilever by using the HB method and the Floquet theory. However, the low-order C-PSMS method is able to give an accurate frequency-amplitude response and accurate stability results of the primary and sub-harmonic resonances of a largely deflected cantilever-type energy harvester.

We consider the electrons bound in the conduction band of a double quantum well interacting with a strong monochromatic electromagnetic field in the terahertz (THz) region. Using a variant of the stationary perturbation theory, we have obtained the quasienergies and Floquet eigenvectors. The main features of power broadening and the dynamic Stark shift are retained. The sensitivity to the field parameters, such as intensity and frequency of the electric field on the state populations, can be used in various optical semiconductor device applications.

In this paper, we study the existence problem of periodic first integrals for periodic Hamiltonian systems of Lie type. From a natural ansatz for time-dependent first integrals, we refer their existence to the existence of periodic solutions for a periodic Euler equation on the Lie algebra associated to the original system. Under different criteria based on properties for the Killing form or on exponential properties for the adjoint group, we prove the existence of Poisson algebras of periodic first integrals for the class of Hamiltonian systems considered. We include an application for a nonlinear oscillator having relevance in some modern physics applications.

The rotors working in real technological devices are always slightly imbalanced. This excites their lateral vibrations and generates forces that are transmitted to the rotor casing. These effects can be significantly reduced if damping devices are added to the support elements. The possibility of controlling the damping, in order to achieve their optimum performance, is offered by magnetorheological squeeze film dampers. In this paper, a computational modeling method is used to investigate the dynamical behavior of a rigid flexibly supported rotor loaded by its unbalance and equipped with two short magnetorheological dampers. The equations of motion of the rotor are nonlinear due to the damping forces. Computational procedures were developed to verify the applicability of such dampers by simulating their behavior and analyzing their effect on the amplitude of the rotor vibration, on the magnitude of the forces transmitted to the rotor casing and on the amount of the power dissipated in the magnetorheological films. The proposed approach to study the optimum performance of semiactive magnetorheological dampers applied in rotor systems, in terms of vibration amplitudes and transmitted forces, together with the developed efficient computational methods to calculate the system steady state response and to evaluate its stability represent the new contributions of this paper.

We investigate the response of electrons confined in a quantum wire in the presence of intense terahertz (THz) electric field. An exact and powerful nonperturbative fundamental approach of Floquet theory is employed to solve the equation of motion for resonantly driven intersubband transitions. Several interesting features namely dynamic Stark shift, power broadening and hole-burning are observed with the variation in electric field strength. In addition, the degeneracy between several excited states is found to be removed in the presence of high electric field.