Periodic Steady-state Solution Research Articles

Multimodal Resonance Response of Incompressible Hyperelastic Moderately Thick Cylindrical Shells

Based on the modified third-order shear deformation theory, the harmonic balance method, and the pseudo-arclength continuation method with two-point prediction, the nonlinear forced vibration response of incompressible hyperelastic moderately thick cylindrical shells subjected to a concentrated harmonic load at mid-span and simply supported boundary conditions at both ends is investigated. The algorithmic procedure for solving steady-state periodic solutions of strongly nonlinear systems of differential equations is presented. The structural response characteristics of shells under different excitation amplitudes and structural parameters are analyzed. The numerical results indicate that the aspect ratio of moderately thick hyperelastic cylindrical shells has a significant effect on the natural frequency ratio. Different frequency ratios lead to varying nonlinear mode coupling effects. The coupling effects among modes result in complex nonlinear behavior in the vibration response of each mode, leading to abundant multi-valued phenomena in the response curve.

Dynamic perturbation analysis of fractional order differential quasiperiodic Mathieu equation

The paper investigates the influence of parameters on the stability of fractional order differential quasiperiodic Mathieu equations. First, we use the perturbation method to obtain approximate expressions (i.e., transition curves) for the stability and unstable region boundaries of the equation. After obtaining the approximate expression of the transition curve, we use Lyapunov's first method to analyze the stability of the fractional order differential quasiperiodic Mathieu system, thereby obtaining the conditions for the stability of the fractional order differential quasiperiodic Mathieu equation system. Second, by comparing the approximate expressions of the transition curve of the steady-state periodic solution of the quasiperiodic Mathieu oscillator under different parameter conditions, we obtained the conclusion that the fractional order differential term exists in the form of equivalent stiffness and equivalent damping in the fractional order differential quasiperiodic Mathieu system. By comparison, we have summarized the general forms of equivalent linear damping and equivalent stiffness of the system. Through this general form, we can define an approximate expression for the thickness of unstable regions to better study the characteristics of fractional order differential quasiperiodic Mathieu systems. Finally, the influence of the parameters of the fractional order differential quasiperiodic Mathieu equation on the transition curve of the equation was intuitively analyzed through numerical simulation, to analyze the stability changes in the equation.

A migration-selection model in genetic engineering

We investigate a migration-selection system arising from CRISPR-Cas9 genetic engineering, which describes the evolution of the frequencies of a wild allele O, a drive allele D, and a brake allele B. The purpose is to see whether the drive allele D can persist in the population and whether its spread can be limited or stopped by the brake allele B when necessary. We give a complete classification of the dynamics of this system when there is no migration. We further show that migration may cause complex spatiotemporal patterns by demonstrating the existence of spatially inhomogeneous periodic solutions and steady state solutions.

Stability and Bifurcation of a Delayed Reaction–Diffusion Model with Robin Boundary Condition in Heterogeneous Environment

In this paper, we investigate a reaction–diffusion model with delay and Robin boundary condition in heterogeneous environment. The existence, multiplicity and stability of spatially nonhomogeneous steady-state solutions and periodic solutions are studied by employing the Lyapunov–Schmidt reduction method. Moreover, the Hopf bifurcation direction is derived. It is observed that Robin boundary condition plays a crucial role in the Hopf bifurcation. More precisely, when the boundary effect is stronger than the interaction of the species within the region, there is no Hopf bifurcation no matter how the time delay [Formula: see text] changes. Finally, we illustrate our general theoretical results by an application to the Nicholson’s blowflies model.

The periodic steady-state solution for queues with Erlang arrivals and service and time-varying periodic transition rates

We study a queueing system with Erlang arrivals with k phases and Erlang service with m phases. Transition rates among phases vary periodically with time. For these systems, we derive an analytic solution for the asymptotic periodic distribution of the level and phase as a function of time within the period. The asymptotic periodic distribution is analogous to a steady-state distribution for a system with constant rates. If the time within the period is considered part of the state, then it is a steady-state distribution. We also obtain waiting time and busy period distributions. These solutions are expressed as infinite series. We provide bounds for the error of the estimate obtained by truncating the series. Examples are provided comparing the solution of the system of ordinary differential equation with a truncated state space to these asymptotic solutions involving remarkably few terms of the infinite series. The method can be generalized to other level independent quasi-birth-death processes if the singularities of the generating function are known.

Directed transport of a particle on a horizontal surface under asymmetric vibrations

Asymmetric unbiased fluctuations coupled with the effect of nonlinear dry (Coulombic) friction are observed to generate a directed transport even in the absence of any net external force or gradient. This phenomenon, called the ratchet effect, occurs when the driving force does not overcome the static friction threshold symmetrically. An elementary model is used to generate directed and unidirectional motion of a particle placed on top of a horizontal rigid surface which is given a zero mean spatially asymmetric motion through a multi-frequency in-plane excitation. We obtain steady-state periodic solutions of the nondimensional transport equations analytically to assess the distinct single/multiple sticking and sliding phases of particle motion and various parametric conditions for phase transitions. These analytical results are then validated by solutions obtained semi-analytically using Galerkin projection and numerically using the fourth-order Runge–Kutta method (RK4) and Newton–Raphson method with a fixed arc-length based continuation scheme. Also, a multibody dynamics model of this particle-on-plate system is built in MSC Adams to help us visualize the actual particle behavior under real dimensional system parameters and compare it with the foregoing analyses. A highly counter-intuitive phenomenon is encountered as the particle drift velocity increases non-monotonically with the increase in friction whereas a very high or a very low amount of friction impedes motion. Maximum transport velocity is found to be independent of the amplitude ratio of the vibratory input for low values of kinetic friction and is observed to manifest at a critical static friction threshold beyond which the particle motion becomes unidirectional. The effect of viscous drag of the ambient fluid on transport is assessed to be quite insignificant. Optimal vibratory input parameters to achieve the best transport current under varying friction conditions are determined. At low kinetic friction, a fixed set of system parameters is observed to generate distinct solution branches that coexisted just beyond the static friction threshold corresponding to a motion transition from “slip” to “stick–slip”.

FINITE ELEMENT METHOD FOR NONLINEAR EDDY CURRENT PROBLEMS IN POWER TRANSFORMERS

An efficient finite element method to take account of the nonlinearity of the magnetic materials when analyzing three dimensional eddy current problems is presented in this paper. The problem is formulated in terms of vector and scalar potentials approximated by edge and node based finite element basis functions. The application of Galerkin techniques leads to a large, nonlinear system of ordinary differential equations in the time domain. The excitations are assumed to be time-periodic and the steady state periodic solution is of interest only. This is represented in the frequency domain as a Fourier series for each finite element degree of freedom and a finite number of harmonics is to be determined, i.e. a harmonic balance method is applied. Due to the nonlinearity, all harmonics are coupled to each other, so the size of the equation system is the number of harmonics times the number of degrees of freedom. The harmonics would be decoupled if the problem were linear, therefore, a special nonlinear iteration technique, the fixed-point method is used to linearize the equations by selecting a timeindependent permeability distribution, the so called fixed-point permeability in each nonlinear iteration step. This leads to uncoupled harmonics within these steps resulting in two advantages. One is that each harmonic is obtained by solving a system of algebraic equations with only as many unknowns as there are finite element degrees of freedom. A second benefit is that these systems are independent of each other and can be solved in parallel. The appropriate selection of the fixed point permeability accelerates the convergence of the nonlinear iteration. The method is applied to the analysis of a large power transformer. The solution of the electromagnetic field allows the computation of various losses like eddy current losses in the massive conducting parts (tank, clamping plates, tie bars, etc.) as well as the specific losses in the laminated parts (core, tank shielding, etc.). The effect of the presence of higher harmonics on these losses is investigated.

Semi-analytical solution to the steady-state periodic dynamic response of an infinite beam carrying a moving vehicle

Vehicle-induced vibration is among the most important aspects for riding comfort and structural durability. It is well known, however, that the span of structure is much larger than the size of vehicle, resulting in a huge computational cost for dynamic analysis, and meanwhile, the multi-degree-of-freedom (MDOF) vehicle and the nonlinear interaction between beam and vehicle are difficult to be involved in an explicit solution. In the present work, a joint harmonic balance and Fourier transform (HBFT) method is proposed for the steady-state periodic dynamic response of an infinite beam carrying a vehicle taking into account the surface roughness and the nonlinear interaction between beam and vehicle, and the semi-analytical steady-state periodic (SASP) solution is then obtained. First, the vehicle and beam are separated as two subsystems, where the infinite beam is supported by a viscoelastic foundation and the vehicle is described by a MDOF discrete model. The response of the beam is analytically solved from the Fourier transform approach and the response of the vehicle is explicitly formulated considering the Fourier expansion of the nonlinear interaction forces. Next, the Fourier coefficients for such interaction forces are numerically solved from the harmonic balance method based on the compatibility between the subsystems. By substituting the solved interaction forces back into the closed-form analytical solutions, the responses of both the subsystems are evaluated explicitly. This allows for accurate and efficient analysis for the steady-state periodic response of vehicle–beam system considering the infinite beam, the discrete vehicle and the nonlinear interaction between them. After validating the SASP solution by comparing with the assumed modal method, the effect from the vehicle speed on the response of the vehicle–beam system is discussed, indicating the existence of a few resonant speeds, having potential value for the optimal design of vehicles and road pavements. Apart from the present example, the SASP solution can be applied in broader cases and the proposed HBFT method is generally feasible to a variety of dynamic coupling problems consisting of a continuum and a discrete model.

On the stability of thin-walled circular cylindrical shells under static and periodic radial loading

In this paper, we study the dynamic behavior of the cylindrical shells subjected to static and time-varying (periodic) radial loading. Flügge-Luré-Byrne shell theory has been used to derive dynamic equations in the form of nonlinear partial differential equations (PDEs). Linearization of PDEs about the steady-state periodic solutions, followed by Galerkin projection, yields a system of ordinary differential equations (ODEs) with time-varying periodic coefficients. Those are similar to the higher-order Mathieu-Hill equations and their stability analyses have been ascertained using Floquet theory. Dynamic instabilities of the cylindrical shells have been presented through stability charts in the plane of forcing parameters, namely, the static component of the pressure, the amplitude, and frequency of the fluctuating component of the pressure. Also, possible sub and super harmonic resonances have been explored through the stability charts. Four different boundary conditions, viz., simply supported, clamped–free, pinned–free (axial constraint at the pinned end) and pinned–free (axial constraint at the free end) have been considered in the present analysis. Presented charts can be very useful during the design of the shell structures subjected to static/dynamic loading.

Steady state solution of NFC model with nonlinear load using PEEC

PurposeThe purpose of this paper is to show that the computation of time-periodic signals for coupled antenna-circuit problems can be substantially accelerated by means of the single shooting method. This allows an efficient analysis of nonlinearly loaded coupled loop antennas for near field communication (NFC) applications.Design/methodology/approachFor the modelling of electrically small coupled field-circuit problems, the partial element equivalent circuit (PEEC) method shows to be very efficient. For analysing the circuit-like description of the coupled problem, this paper developed a generalised modified nodal analysis (MNA) and applied it to specific NFC problems.FindingsIt is shown that the periodic steady state (PSS) solution of the resulting differential-algebraic system can be computed very time efficiently by the single shooting method. A speedup of roughly 114 to conventional transient approaches can be achieved.Practical implicationsThe proposed approach appears to be an efficient alternative for the computation of time PSS solutions for nonlinear circuit problems coupled with discretised conductive structures, where the homogeneous solution is not of interest.Originality/valueThe present paper explores the implementation and application of the shooting method for nonlinearly loaded coupled antenna-circuit problems based on the PEEC method and shows the efficiency of this approach.

Direct steady-state solutions for circuit models of nonlinear electromagnetic devices

PurposeThis paper aims to omit the difficulties of directly finding the periodic steady-state solutions for electromagnetic devices described by circuit models.Design/methodology/approachDetermine the discrete integral operator of periodic functions and develop an iterative algorithm determining steady-state solutions by a multiplication of matrices only.FindingsAn alternative method to creating finite-difference relations directly determining steady-state solutions in the time domain.Research limitations/implicationsReduction of software and hardware requirements for determining steady-states of electromagnetic.Practical implicationsA unified approach for directly finding steady-state solutions for ordinary nonlinear differential equations presented in the normal form.Originality/valueEliminate the necessity of solving high-order finite-difference equations for steady-state analysis of electromagnetic devices described by circuit models.

Bifurcation in a reaction-diffusion model with nonlocal delay effect and nonlinear boundary condition

In this paper, the existence, stability, and multiplicity of steady-state solutions and periodic solutions for a reaction-diffusion model with nonlocal delay effect and nonlinear boundary condition are investigated by using Lyapunov-Schmidt reduction. When the interior reaction term is weaker than the boundary reaction term, it is found that there is no Hopf bifurcation no matter how either of the interior reaction delay and the boundary reaction delay changes. When the interior reaction term is stronger than the boundary reaction term, it is the interior reaction delay instead of the boundary reaction delay that determines the existence of Hopf bifurcation. Moreover, the general results are illustrated by applications to models with either a single delay or bistable boundary condition.

Periodic traveling waves and asymptotic spreading of a monostable reaction-diffusion equations with nonlocal effects

This article concerns the dynamical behavior for a reaction-diffusion equation with integral term. First, by using bifurcation analysis and center manifold theorem, the existence of periodic steady-state solution are established for a special kernel function and a general kernel function respectively. Then, we prove the model admits periodic traveling wave solutions connecting this periodic steady state to the uniform steady state u=1 by applying center manifold reduction and the analysis to phase diagram. By numerical simulations, we also show the change of the wave profile as the coefficient of aggregate term increases. Also, by introducing a truncation function, a shift function and some auxiliary functions, the asymptotic behavior for the Cauchy problem with initial function having compact support is investigated.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/22/abstr.html

Nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading in the subcritical buckling regime

Equations of motion of a hyperelastic beam under time-varying axial loading are derived via the extended Hamilton’s principle in this work, where the transverse vibration is coupled with the longitudinal vibration, and nonlinear vibrations of the beam in the subcritical buckling regime are investigated. Complex nonlinear boundary conditions of the beam are determined under some geometric constraints. The critical buckling load is first determined through linear bifurcation analysis. Effects of material and geometric parameters on the forced longitudinal vibration of the beam are numerically investigated. Steady harmonic shapes of the beam at different times under harmonic axial loading are determined. The beam is in the barreling deformation state even when the axial load is not in excess of the critical buckling load. The governing equation for the nonlinear transverse vibration of the beam is obtained by decoupling its equations of motion. Natural frequencies of the free linearized transverse vibration of the beam are studied. By applying the eigenfunction expansion method, the governing equation for the nonlinear transverse vibration of the beam transforms to a series of strongly nonlinear ordinary differential equations (ODEs). Two-to-one internal resonance of the beam is studied by the numerical integration method and its phase-plane portraits are obtained. The harmonic balance method and pseudo arc-length method are used to determine steady-state periodic solutions of the beam from the strongly nonlinear ODEs, and amplitude-frequency responses of the beam are determined. Effects of the external mean axial load, excitation amplitude, and damping coefficient on the amplitude-frequency response of the beam are numerically investigated. Combined effects of the external excitation amplitude and frequency on response amplitudes are also investigated.

Comparative Analysis of State-Space and Companion-Circuit Methodologies for the Periodic Steady-State Solution in Time-Domain of Nonlinear Electric Networks

This contribution reports a comparative analysis of two methodologies for the periodic steady-state solution of linear and nonlinear electric networks in time-domain (TD), based on the state-space model (SSM) and companion-circuit analysis (CCA), respectively. Newton methods based on a conventional Numerical Differentiation (ND) and Enhanced Numerical Differentiation (END) process, respectively, are used. The application of the SSM and CCA methods is done using a brute force (BF) procedure and the application of advanced numerical techniques (ANT) through the forward Euler (FE) and Trapezoidal Rule (TR) numerical integration, respectively. Related analysis show comparisons of each case study in terms of efficiency, speed, computer effort and accuracy between the combination of SSM, CCA and Newton methods, e.g. SSM-ND, SSM-END, CCA-ND and CCA-END, respectively. In particular, the analysis is shown in terms of time-steps per period, CPU processing time and a computer platform used are shown. The results obtained from the reported case studies are validated against the Power Systems Computer Aided Design/Electromagnetic Transients Program including Direct Current (PSCAD/EMTDC®) response, widely accepted by the power industry.

Steady-State Process Analysis of DC Converter Based on Equations Expansion

The paper deals with processes analysis in circuits of converter working on a time-varying load. A control of inverter and load switches are realised by signals with incommensurable frequencies. Processes in such a system are described by differential equations with periodical coefficients. Steady-state periodic solutions can be obtained by the extension of ordinary differential equations with one independent time variable into partial differential equations with two independent variables of time. These equations are solved by use of the Galerkin method with trigonometric basis and weight functions. The results of calculations of the steady-state process for a buck-boost converter are presented in form of the double Fourier series. They are compared with results obtained in the way of numerical calculation of differential equations for a transient process. Extended equations are also solved by a generalized state-space averaging method. A balance of active power in circuits of converter with the time-varying load is shown.

Dynamics of an SIS network model with a periodic infection rate

Seasonal forcing and contact patterns are two key features of many disease dynamics that generate periodic patterns. Both features have not been ascertained deeply in the previous works. In this work, we develop and analyze a non-autonomous degree-based mean field network model within a Susceptible-Infected-Susceptible (SIS) framework. We assume that the disease transmission rate being periodic to study synergistic impacts of the periodic transmission and the heterogeneity of the contact network on the infection threshold and dynamics for seasonal diseases. We demonstrate both analytically and numerically that (1) the disease free equilibrium point is globally asymptotically stable if the basic reproduction number is less than one; and (2) there exists a unique global periodic solution that both susceptible and infected individuals coexist if the basic reproduction number is larger than one. We apply our framework to Scale-free contact networks for the simulation. Our results show that heterogeneity in the contact networks plays an important role in accelerating disease spreading and increasing the amplitude of the periodic steady state solution. These results confirm the need to address factors that create periodic patterns and contact patterns in seasonal disease when making policies to control an outbreak.

Enumeration and Analysis of DC–DC Converter Implementations Based on Piezoelectric Resonators

Demand for power electronics with smaller volume, lighter weight, and lower cost will eventually require new converter energy storage technologies with fundamentally higher power density and efficiency limits. This motivates investigation into piezoelectric resonators (PRs), which offer very high power density and efficiency capabilities and significantly improved scaling properties compared to magnetics. PRs have been used in power conversion previously, but the realm of possible converter implementations using only PRs for energy storage has seen little exploration. In this work, we enumerate and evaluate dc–dc converter topologies and switching sequences that best utilize a PR as the only energy storage component, and that enable low-loss resonant “soft charging” of the PR's input capacitance with voltage regulation capability. To compare these implementations, we present analysis techniques for their operation and periodic steady-state solutions considering practical constraints. In addition, we provide useful techniques for estimating PR utilization and efficiency, which we validate experimentally in a 200–100-V, 25-W rated prototype. This prototype exhibits peak efficiency $>$ 99% as well as high efficiency ( $\geq$ 96%) across a wide range of operating conditions, illustrating the promise of PR-based converters for high voltage, low power applications. This article is accompanied by an instructional video for selecting and analyzing PR-based dc–dc converter implementations.

Enhanced harmonic state estimation in unbalanced three-phase electrical grids based on the Kalman filter and physical scale-down implementation

A time-domain methodology is proposed based on Kalman filter, to determine the harmonic state estimation in three-phase electrical grids. Time-domain methodologies based on the Kalman filter need an initial solution to begin estimating the harmonic state in electrical grids. Previous methods have found this solution using simulated electrical grids; thus, the measurements taken from the simulations are synchronised only with the simulated grid model. In practical power networks, measurements taken from the grid need to be synchronised with the theoretical grid model. Synchronization between the model and measurements using the Kalman filter is proposed. This paper also proposes an alternative method to obtain the periodic steady-state solution in power grids. The method adequately processes the information given by a limited number of measurements to find the required solution. An experimental case study for a three-phase unbalanced condition with a nonlinear load and a limited number of measurements is used to validate the proposed methodology, i.e. practical data has been used instead of synthetic data. Hence, the results are validated by direct comparison of the state estimation response against the actual data recorded from the experimental laboratory implementation. The results show that using the proposed methodology, the time-domain harmonic state estimation can be efficiently applied to the analysis of practical power grids.

An improved quasi-zero stiffness vibration isolation system utilizing dry friction damping

Quasi-zero stiffness (QZS) isolators, a nonlinear vibration isolation technique, enhance the isolation performance, by lowering the natural frequencies of an isolation system while providing higher static load bearing capacity compared to similarly performing linear isolators. Despite its performance improvement, the challenge of implementing QZS isolation systems is due to their highly nonlinear stiffness characteristics as a result of cubic like behavior of stiffness elements used. Although increasing linear damping in the isolation alleviates the input dependency of the QZS isolation systems by reducing the resonance amplitudes, it results in increased transmissibility in the isolation region, which is an adverse effect. Therefore, in this study, in order to overcome this, dry friction damping is implemented on the QZS isolation system. Hysteresis loop for the new QZS dry friction element is obtained and a mathematical model is introduced. For the nonlinear isolation system, harmonic balance method is used to transform the nonlinear differential equations into a set of nonlinear algebraic equations. For single harmonic motion, analytical expressions of Fourier coefficients are obtained in terms of elliptic integrals. Numerical solution of the resulting set of nonlinear algebraic equations is obtained via Newton’s method with arc-length continuation. Performance of the isolation system under periodic base excitation is studied for different base excitation levels and the stability of the periodic steady-state solutions is investigated.