Nonlinear Conservative Systems Research Articles

Active particle motion in Poiseuille flow through rectangular channels.

We investigate the dynamics of a pointlike active particle suspended in fluid flow through a straight channel. For this particle-fluid system, we derive a constant of motion for a general unidirectional fluid flow and apply it to an approximation of Poiseuille flow through channels with rectangular cross- sections. We obtain a 4D nonlinear conservative dynamical system with one constant of motion and a dimensionless parameter describing the ratio of maximum flow speed to intrinsic active particle speed. Applied to square channels, we observe a diverse set of active particle trajectories with variations in system parameters and initial conditions which we classify into different types of swinging, trapping, tumbling, and wandering motion. Regular (periodic and quasiperiodic) motion as well as chaotic active particle motion are observed for these trajectories and quantified using largest Lyapunov exponents. We explore the transition to chaotic motion using Poincaré maps and show "sticky" chaotic tumbling trajectories that have long transients near a periodic state. We briefly illustrate how these results extend to rectangular cross-sectionswith a width-to-height ratio larger than one. Outcomes of this paper may have implications for dynamics of natural and artificial microswimmers in experimental microfluidic channels that typically have rectangular cross sections.

Mass-, Energy-, and Momentum-Preserving Spectral Scheme for Klein-Gordon-Schrödinger System on Infinite Domains

For the nonlinear conservative system, how to design an efficient scheme to preserve as manyinvariants as possible is a challenging task. The aim of this paper is to construct the finite difference/spectral method for the Klein–Gordon–Schrödinger (KGS) system on infinite domains ( , and 3) to conserve three kinds of the most important invariants, namely, the mass, the energy, and the momentum. Regarding the mass- and momentum-conservation laws as globally physical constraints, we elaborately combine the exponential scalar auxiliary variable (ESAV) approach and Lagrange multiplier approach to construct the ESAV/Lagrange multiplier reformulation of the KGS system to preserve its original energy-conservation law. When solving the ESAV/Lagrange multiplier reformulation, we employ the Hermite–Galerkin spectral method for the spatial approximation and apply the Crank–Nicolson scheme for the temporal discretization. At each time level, we only need to solve linearly algebraic systems with constant coefficients and a set of quadratic algebraic equations which can be efficiently solved by Newton iteration. We establish the conservation properties of the proposed method at the fullydiscrete level, which indicates that our scheme can preserve conserved quantities including mass, energy, and momentums of the KGS system. Numerical experiments are carried out to demonstrate the accuracy and conservation properties of the proposed scheme. As the applications of our scheme, we simulate the nonlinear interactions of vector solitons for KGS system in 2 dimensional/3 dimensional cases.

An improved analytical technique for a non-linear conservative system with inertia and static non-linearity

In this paper, an improved energy balance method has been presented to obtain an analytical approximate solution to investigate a conservative non-linear oscillator having inertia and static non-linearity. The method is valid for both small and large amplitudes of oscillation. Although the trial solution of the present paper is the same as Khan and Mirzabeigy (2014), the solution procedure is different. Moreover, the algebraic equations appeared in the present technique are solved analytically. The results obtained in this paper are compared with other existing results. The present method not only gives better result than other existing result but also gives perfect agreement with the numerical result (considered to be exact result).

Time-domain minimum residual method combined with energy balance for nonlinear conservative systems

This paper presents a new semi-analytical method that combines the time-domain minimum residual method with energy balance for nonlinear conservative systems. Due to work done only by the conservative force, the total energy of the nonlinear conservative system will remain constant. The response frequency is also the function of the initial conditions and it is difficult to achieve the semi-analytical solutions accurately. The proposed method aims to quickly obtain the higher-order approximate semi-analytical periodic or quasi-periodic solutions of both weakly and strongly nonlinear conservative systems. To this end, the approximate analytical solution is expressed as a group of trigonometric series with undetermined coefficients firstly. Next, by deriving the assumed trigonometric series and substituting the state variables back into the original nonlinear conservative equation, the goal of solving the semi-analytical solution is transformed into an optimization problem that minimizes the residual objective function in a period. Finally, the energy constraints are introduced into the optimization process, and the enhanced response sensitivity approach is called to resolve the nonlinear objective function iteratively. A conservative Duffing system and a nonlinear symmetrically conservative two-mass system are taken as two numerical examples to illustrate the performance of the proposed method.

Identifying phase-varying periodic behaviour in conservative nonlinear systems.

Nonlinear normal modes (NNMs) are a widely used tool for studying nonlinear mechanical systems. The most commonly observed NNMs are synchronous (i.e. single-mode, in-phase and anti-phase NNMs). Additionally, asynchronous NNMs in the form of out-of-unison motion, where the underlying linear modes have a phase difference of 90°, have also been observed. This paper extends these concepts to consider general asynchronous NNMs, where the modes exhibit a phase difference that is not necessarily equal to 90°. A single-mass, 2 d.f. model is firstly used to demonstrate that the out-of-unison NNMs evolve to general asynchronous NNMs with the breaking of the geometrically orthogonal structure of the system. Analytical analysis further reveals that, along with the breaking of the orthogonality, the out-of-unison NNM branches evolve into branches which exhibit amplitude-dependent phase relationships. These NNM branches are introduced here and termed phase-varying backbone curves. To explore this further, a model of a cable, with a support near one end, is used to demonstrate the existence of phase-varying backbone curves (and corresponding general asynchronous NNMs) in a common engineering structure.

Approximate periodic solution and qualitative analysis of nonnatural oscillators based on the restoring force

Non-natural oscillators are nonlinear conservative systems in which the kinetic energy is not a pure quadratic function of velocity and therefore, they do not possess a constant Hamiltonian function. They can be found in a lot of artificial systems and some predictive models for natural phenomena. The conventional form of their governing equation does not show the restoring force, which is important for analysis of the qualitative behavior of nonlinear oscillators and for obtaining their periodic solution. The purpose of this paper is to derive general solutions for the restoring force of non-natural systems and to demonstrate how the restoring force could be used to conduct qualitative analysis and derive periodic solutions. The general solutions for the restoring force were derived for two cases of non-natural oscillators namely: non-natural oscillators leading to (a) exact differential equation and (b) inexact differential equation. One example of each case of non-natural oscillator was solved and analysed using the general solutions for the restoring force and approximate analytical methods for periodic solution that rely on knowledge of the restoring force. The analysis showed that the general restoring force solutions and the continuous piecewise linearization method can be used to derive very accurate periodic solutions for non-natural oscillators.

Analytical computation method for steady-state stochastic response of a time-delay nonlinear automotive suspension system

Abstract A novel analytical computation method of steady-state probability density function (PDF) is presented to solve the stochastic response of a time-delay nonlinear automotive suspension system. In accordance with actual projects, a stochastic dynamics model of the automotive suspension system is established which includes cubic nonlinear restoring force, time-delay feedback control term and random road excitation simulated by white Gaussian noise. Based on the averaged angular frequency derived from Hamiltonian function of the nonlinear conservative system associated with the established stochastic dynamics model, the original system is transformed into a stochastic differential dynamics system without time-delay terms. By using generalized harmonic function-based stochastic averaging method and deterministic averaging method, the averaged Ito stochastic differential equation associated with the transformed system is obtained. Through solving the averaged Fokker-Planck-Kolmogorov equation established by drift and diffusion coefficients of the Ito equation, the steady-state PDFs of stochastic response amplitude and systematic energy as well as the steady-state joint PDF of stochastic response are obtained. All of the analytical results are verified by digital simulations.

Periodic orbits of a conservative 2-DOF vibro-impact system by piecewise continuation: bifurcations and fractals

The exact periodic orbits of a conservative 2-degree-of-freedom vibro-impact system with stereo-mechanical impact model is studied using a piecewise continuation method. Feasible initial guesses are extracted from grazing solution points so that each branch of solution can be initiated smoothly from previously solved ones. Frequency-energy plots (FEPs) are produced where an emphasis is placed on enumerating potential bifurcations. Extra critical points are discovered on multiple-period duplicates of existing stable solution branches and lead to cascades of period-multiplying bifurcations. The results indicate that the system’s complete FEP can be viewed through a process of infinite fractals toward zero frequency where pseudo-periodic or chaotic responses are approached. Finally, it is shown both mathematically and through the comparison of FEPs that the impacting system can be represented explicitly as the extreme case of nonlinear systems with an odd-order polynomial internal force. It is thus proposed that as the counterpart to the superposition of linear normal modes, the free responses of a general conservative nonlinear system can be tracked via bifurcations from its nonlinear normal modes.

Energy-preserving variational integrators for forced Lagrangian systems

The goal of this paper is to develop energy-preserving variational integrators for time-dependent mechanical systems with forcing. We first present the Lagrange-d’Alembert principle in the extended Lagrangian mechanics framework and derive the extended forced Euler-Lagrange equations in continuous-time. We then obtain the extended forced discrete Euler-Lagrange equations using the extended discrete mechanics framework and derive adaptive time step variational integrators for time-dependent Lagrangian systems with forcing. We consider three numerical examples to study the numerical performance of energy-preserving variational integrators. First, we consider the example of a nonlinear conservative system to illustrate the advantages of using adaptive time-stepping in variational integrators. In addition, we demonstrate how the implicit equations become more ill-conditioned as the adaptive time step decreases through a condition number analysis. As a second example, we numerically simulate the time-dependent example of a forced harmonic oscillator to demonstrate the superior energy performance of energy-preserving integrators for mechanical systems with explicit time-dependent forcing. Finally, we consider a damped harmonic oscillator using the adaptive time step variational integrator framework. The adaptive time step increases monotonically for the dissipative system leading to unexpected energy behavior.

A multi-harmonic generalized energy balance method for studying autonomous oscillations of nonlinear conservative systems

The Harmonic Balance Method (HBM) is a frequency-domain based approximation approach used for obtaining the steady state periodic behavior of forced dynamical systems. Intrinsically these systems are non-autonomous and the method offers many computational advantages over time-domain methods when the fundamental period of oscillation is known (generally fixed as the forcing period itself or a corresponding sub-harmonic if such behavior is expected). In the current study, a modified approach, based on He's Energy Balance Method (EBM), is applied to obtain the periodic solutions of conservative systems. It is shown that by this approach, periodic solutions of conservative systems on iso-energy manifolds in the phase space can be obtained very efficiently. The energy level provides the additional constraint on the HBM formulation, which enables the determination of the period of the solutions.The method is applied to the linear harmonic oscillator, a couple of nonlinear oscillators, the elastic pendulum and the Henon-Heiles system. The approach is used to trace the bifurcations of the periodic solutions of the last two, being 2 degree-of-freedom systems demonstrating very rich dynamical behavior. In the process, the advantages offered by the current formulation of the energy balance is brought out. A harmonic perturbation approach is used to evaluate the stability of the solutions for the bifurcation diagram.

Higher-order approximate analytical solutions to nonlinear oscillatory systems arising in engineering problems

In the current paper, a powerful approximate analytical approach namely the global residue harmonic balance method (GRHBM) is proposed for obtaining higher-order approximate frequency and periodic solution of nonlinear conservative oscillatory systems arising in engineering problems. The proposed method has a main difference with other traditional harmonic balance methods such that the residual errors obtained in pervious order approximation are used in the present one. Comparison of the obtained results with the exact and numerical solution as well as well-known analytical methods such as Hamiltonian approach, Max–Min approach, variational approach, and He’s amplitude–frequency formulation reveals the correctness and usefulness of the GRHBM. It is shown that the results are valid for different values of system parameters and both small and large amplitudes. Hence, the method can be easily applied to other strongly nonlinear conservative oscillatory systems. Furthermore, using the obtained analytical expressions, the effect of amplitude and system parameters on nonlinear frequency is studied.

Asynchronous modes of vibrations in linear conservative systems: an illustrative discussion of plane framed structures

The theme of asynchronous modes of vibration is recast in this paper to address the case of linear conservative discrete systems. In fact, it is known that linear non-conservative discrete systems such as those characterized by a non-symmetric stiffness matrix may undergo asynchronous free vibrations. Yet, non-conservativeness is not mandatory for the occurrence of asynchronous modes, as it is illustrated in this investigation of simple plane frames, provided the system parameters are properly chosen. Firstly, a one-storey-frame model, with consistent-mass and stiffness matrices, is studied and conditions for the occurrence of asynchronous modes are established. Next, a three-storey shear building is modelled, with lumped-mass matrix, and isolated asynchronous modes are not observed. However, non-isolated asynchronous modes are seen to exist, including a case in which the first floor is at rest. In this scenario, the upper floors play the role of vibration controllers, sparing the lower columns from oscillating. This might be welcome from the architectural viewpoint, so that pilotis would not be a priori ruled out in case of earthquake-borne vibration. Usually the participating masses of the asynchronous modes are low (and even null). To maximize the asynchronous-mode participating mass, a final model of the three-storey frame is discussed, this time taking into account the flexural stiffness of both the columns and the beams, although still using lumped-mass matrix. In this case, it is seen that much larger participating masses can be obtained for the asynchronous mode. Since asynchronous modes are associated with the vibration localization phenomenon, they can be potentially explored in the design of vibration controllers or, alternatively, energy harvesting systems.

A note on the complete instability of linear non-conservative undamped systems

The note is concerned with the problem of determining the completely unstable linear non-conservative undamped (circulatory) dynamical systems. Several conditions that provide the complete instability for such systems are derived using the direct method of Lyapunov and the concept of controllability. The conditions are expressed directly via the matrices describing the dynamical system.

Application of the Global Residue Harmonic Balance Method for Obtaining Higher-Order Approximate Solutions of a Conservative System

In the current paper, an iterative approximate analytical approach, namely global residue harmonic balance method (GRHBM), is introduced for investigation a nonlinear conservative oscillatory system. The equation of motion for the considered system has been derived as a nonlinear ordinary differential equation. The GRHBM is based on the ideas of homotopy perturbation and the residue harmonic balance method. The results of the presented method are compared with those of other analytical approaches as well as those predicted by the fourth-order Runge–Kutta numerical technique. The correctness of the obtained results reveals that the presented method is very effective, simple and exact and is valid for small and large amplitudes. Furthermore, the impact of system parameters on the ratio of nonlinear to linear frequency is investigated. Consequently, the presented method provides an effective tool to study the effects of material and geometrical parameters in designing of devices composed of nonlinear conservative oscillators.

Boundary control problems for quasilinear systems of hyperbolic equations

For quasilinear systems of hyperbolic equations, the nonclassical boundary value problem of controlling solutions with the help of boundary conditions is considered. Previously, this problem was extensively studied in the case of the simplest hyperbolic equations, namely, the scalar wave equation and certain linear systems. The corresponding problem formulations and numerical solution algorithms are extended to nonlinear (quasilinear and conservative) systems of hyperbolic equations. Some numerical (grid-characteristic) methods are considered that were previously used to solve the above problems. They include explicit and implicit conservative difference schemes on compact stencils that are linearizations of Godunov’s method. The numerical algorithms and methods are tested as applied to well-known linear examples.

Nonstationary energy localization vs conventional stationary localization in weakly coupled nonlinear oscillators

In this paper we study the effect of nonstationary energy localization in a nonlinear conservative resonant system of two weakly coupled oscillators. This effect is alternative to the well-known stationary energy localization associated with the existence of localized normal modes and resulting from a local topological transformation of the phase portraits of the system. In this work we show that nonstationary energy localization results from a global transformation of the phase portrait. A key to solving the problem is the introduction of the concept of limiting phase trajectories (LPTs) corresponding to maximum possible energy exchange between the oscillators. We present two scenarios of nonstationary energy localization under the condition of 1:1 resonance. It is demonstrated that the conditions of nonstationary localization determine the conditions of efficient targeted energy transfer in a generating dynamical system. A possible extension to multi-particle systems is briefly discussed.

On an efficient [formula omitted]-step iterative method for nonlinear equations

This paper is devoted to the construction and analysis of an efficient k-step iterative method for nonlinear equations. The main advantage of this method is that it does not need to evaluate any high order Fréchet derivative. Moreover, all the k-step have the same matrix, in particular only one LU decomposition is required in each iteration. We study the convergence order, the efficiency and the dynamics in order to motivate the proposed family. We prove, using some recurrence relations, a semilocal convergence result in Banach spaces. Finally, a numerical application related to nonlinear conservative systems is presented.

Asymmetric vibration characteristics of two-cylinder four-stroke single-piston hydraulic free piston engine

The structure and working principle of a two-cylinder four-stroke single-piston hydraulic free piston engine (HFPE) were introduced. The basic vibration equation of free piston assembly (FPA) was established based upon the energy conversion between the injected fuel and the friction together with the load. Both the theoretical and numerical results show that the vibration system of FPA is a nonlinear conservative autonomous system in one cycle. The FPA vibration is symmetric with constant amplitude when FPA is only driven by the compression pressure in the compression accumulator and that in the combustion chamber. When considering the friction and load, FPA could still achieve a stable vibration after a few cycles’ adjustment whether the input energy is equal to the consumed energy or not. The vibration characteristics are different when FPA vibrates in the compression stroke and the expansion stroke, which is the unique feature of the single-piston HFPE.

Power Balance Theorem of Frequency Domain and Its Application

This paper proves a power balance theorem of frequency domain. It becomes another circuit law concerning power conservation after Tellegen’s theorem. Moreover the universality and importance worth of application of the theorem are introduced in this paper. Various calculation of frequency domain in nonlinear circuit possess fixed intrinsic rule. There exists the mutual influence of nonlinear coupling among various harmonics. But every harmonic component must observe individually KCL, KVL and conservation of complex power in nonlinear circuit. It is a lossless network that the nonlinear conservative system with excited source has not dissipative element. The theorem proved by this paper can directly be used to find the main harmonic solutions of the lossless circuit. The results of solution are consistent with the balancing condition of reactive power, and accord with the traditional harmonic analysis method. This paper demonstrates that the lossless network can universally produce chaos. The phase portrait is related closely to the initial conditions, thus it is not an attractor. Furthermore it also reveals the difference between the attractiveness and boundedness for chaos.

A new type of bistable flow around circular cylinders with spanwise stiffening rings

The bistable flow condition around a single circular cylinders is a well-known fluid dynamic phenomenon in the critical range of Re. It is sensitive not only to small variations of the Reynolds number, but also to turbulence of the incoming flow and to surface roughness of the cylinder. Bistable flows are also common for side-by-side cylinders, depending on their centre-to-centre transverse pitch ratio.The paper reveals – through wind tunnel tests – the existence of a new type of bistable flow, induced around a single circular cylinder with a free-end by the presence of spanwise rings. There are some analogies with the aforementioned bistable phenomena, but the conditions of occurrence are profoundly different. The peculiarity of this phenomenon is that it does not disappear at moderately high Re. Its existence is confirmed by a cross-check of results in two different wind tunnels.In order to characterize such a bistable pressure field, the pressure is modeled in the paper as the output of a non-linear conservative system with asymmetric potential wells forced by a proper stochastic process. The identification of the system parameters is performed by the comparison between the theoretical distribution of the oscillator response and the histogram of the recorded pressure.