Nonlinear Oscillatory Systems Research Articles

EXCITATION OF PARAMETRIC OSCILLATIONS WITH RANDOM EXTERNAL SIGNALS

Currently, the main technical means of communication systems are receiving and transmitting radio devices, which in their structure contain various converters and amplifiers operating in nonlinear modes. In the transmission paths themselves, interference, delays, mismatches and parametric effects can occur, which leads to the appearance of small amplitude signals in the information channels. These factors have a significant impact on the characteristics of the telecommunications system.Theoretical investigations of the influence of random signals (oscillations) on the operating modes of a nonlinear oscillatory system is an urgent task and aimed at expanding the possibilities of its use. In general, the analysis of random excitation of oscillations in nonlinear parametric systems is of interest for many branches of science and technology.The goal of the work is to research the factors that influence the possibility of excitation of oscillations in nonlinear systems and possible modes of their operation in the presence of small in amplitude random signals.The analysis of parametric systems with significant nonlinearity in the presence of random external signals. The most important cases of excitation of the inductive nonlinear system are considered. The relations allowing to control the parameters of the regenerate signal depending on the mode of excitation are got.It is shown that in the case of a quasiharmonic input signal, the noise reduces the amplitude of the useful output signal. In the case of a broadband input signal (modeled by white noise), a limited response value is possible only for nonzero losses (the nonlinearity of the system is not enough for this).

Semi-Analytical Solution for Nonlinear Maglev Oscillation using the Lindstedt's Perturbation Method

This paper focuses on the development of an approach for solving nonlinear mechanical maglev oscillation systems. The original physical information included in the governing equations is mostly transferred into analytical and numerical solutions. The analytical method based on the Lindstedt’s perturbation method is used and compared to general numerical solution using the direct Runge-Kutta integration. General procedures for nonlinear oscillation problems are formulated in detail for allocation in the dynamic analysis. Nonlinear oscillation systems with cubic nonlinearities are solved to demonstrate the applications of the present approach. The works reveal that the employed analytical solution can be used for the nonlinear analysis as good as the numerical Runge-Kutta method. The studies also provide an explanation on the behavior of the oscillation system with high nonlinearity. The secondary amplitudes of the higher nonlinearities only occur at the minimum/maximum amplitude of the systems with small or no nonlinearities.

An efficient reliability analysis method combining adaptive Kriging and modified importance sampling for small failure probability

A vital challenge problem of structural reliability analysis is how to estimate the small failure probability with a minimum number of model evaluations. The Adaptive Kriging combined with Importance Sampling method (AK-IS) which is developed from the adaptive Kriging combined with Monte Carlo simulation (AK-MCS) is a viable method to deal with this challenge. The aim of this paper is to reduce the number of model evaluations of the existing AK-IS algorithm. Firstly, we use a contributive weight function to divide the candidate samples of model input variables generated in AK-IS. The candidate samples are used to select the best next sample to update the Kriging model in AK-IS. Secondly, select the best next sample only in the important area obtained according to the contributive weight value to failure probability to update the Kriging model until the stopping condition is satisfied. Thirdly, use the Kriging model constructed in the important area to predict the other area and update the important area by adding the point with the maximum contributive weight value in the area except the important area ceaselessly until the probability of the accurate identification on the limit state function’s signs (positive limit state value or negative limit state value) of all the importance sampling points satisfies a criterion. Finally, the updated Kriging model is used to estimate the failure probability especially for the small failure probability. The proposed method uses a thought from local to global in order to reduce the computational cost of AK-IS and simultaneously guarantees the accuracy of estimation. A non-linear oscillator system, a roof truss structure and a planar ten-bar structure are analyzed by the proposed method. The results demonstrate the efficiency and accuracy of the proposed method in structural reliability analysis especially for small failure probability.

Analytic methods for solving the cylindrical nonlinear Maxwell equations

Cylindrical electromagnetic waves propagation in inhomogeneous and nonlinear media is of great interest for theory and application, and is an extremely complicated problem which can be described by the cylindrical nonlinear Maxwell equations. In this paper, we propose a method to exactly solve the cylindrical Maxwell equations in various types of inhomogeneous nonlinear media without dispersion. The obtained solutions can be used to describe the evolution of nonlinear oscillatory system and the interaction between two radiation waves in nonlinear media. We also show some nonlinear effects, such as second harmonic generation, and sum- and difference-frequency generation, result quite naturally from the solutions.

Forecasting critical points and post-critical limit cycles in nonlinear oscillatory systems using pre-critical transient responses

A unique method for forecasting Hopf bifurcations and the complete post-bifurcation dynamics of nonlinear oscillatory systems is proposed. The method is model-less and uses measurements of the system only in the pre-bifurcation regime. The forecasts are three-dimensional bifurcation diagrams using a few measurements of the system response to perturbations in the pre-bifurcation regime. To demonstrate the forecasting method and its advantages, it is employed to predict bifurcation diagrams of a 3-DOF fluid-structural system. Surrogate measurement data is obtained from simulations and used as input for the forecasting algorithm. Results show that the bifurcation diagrams are forecasted accurately in both supercritical and subcritical bifurcations. The method accurately forecasts the bifurcation point, the bifurcation type and the post-bifurcation limit cycle amplitudes despite the fact that only pre-bifurcation regime data are used for forecasting.

Approximate Prediction-Based Control Method for Nonlinear Oscillatory Systems with Applications to Chaotic Systems

The approximate Prediction-Based Control method (aPBC) is the continuous-time version of the well-known Prediction-Based Chaos Control method applied to stabilize periodic orbits of nonlinear dynamical systems. The method is based on estimating future states of the free system response of continuous-time systems using the solution from the Runge-Kutta implicit method in real time. Some aspects of aPBC are evaluated in the present work, particularly its robustness to low future states estimation precision is exemplified.

Active vibration suppression of a nonlinear electromechanical oscillator system with simultaneous resonance

In this paper, we explored the nonlinear dynamics behavior and vibration suppression of a nonlinear electromechanical oscillator system under harmonic and parametric excitation. The model comprises of an electrical part coupled to mechanical part and displayed by a coupled nonlinear ordinary differential equations. The analytical up to second order approximate solutions are sought applying the method of multiple scales method. We utilized the time-series and method of averaging to analyze the response and stability of the solutions at the worst resonance cases. We checked the results of perturbation solution through numerical simulations and the effects of different system parameters have been reported. Comparison between analytical and numerical solutions is obtained. Also, the numerical results are obtained using MAPLE and MATLAB algorisms.

A parallel time integrator for noisy nonlinear oscillatory systems

In this paper, we adapt a parallel time integration scheme to track the trajectories of noisy non-linear dynamical systems. Specifically, we formulate a parallel algorithm to generate the sample path of nonlinear oscillator defined by stochastic differential equations (SDEs) using the so-called parareal method for ordinary differential equations (ODEs). The presence of Wiener process in SDEs causes difficulties in the direct application of any numerical integration techniques of ODEs including the parareal algorithm. The parallel implementation of the algorithm involves two SDEs solvers, namely a fine-level scheme to integrate the system in parallel and a coarse-level scheme to generate and correct the required initial conditions to start the fine-level integrators. For the numerical illustration, a randomly excited Duffing oscillator is investigated in order to study the performance of the stochastic parallel algorithm with respect to a range of system parameters. The distributed implementation of the algorithm exploits Massage Passing Interface (MPI).

Emergence of Soft-bodied Robot Motion by Pattern Formation of Nonlinear Coupled Oscillator System

Emergence of Soft-bodied Robot Motion by Pattern Formation of Nonlinear Coupled Oscillator System

Power Series -Aftertreatment Technique for Nonlinear Cubic Duffing and Double-Well Duffing Oscillators

Modeling of large amplitude of structures such as slender, flexible cantilever beam and fluid-structure resting on nonlinear elastic foundations or subjected to stretching effects often lead to strongly nonlinear models of Duffing equations which are not amendable to exact analytical methods. In this work, explicit analytical solutions to the large amplitude nonlinear oscillation systems of cubic Duffing and double-well Duffing oscillators are provided using power series-aftertreatment technique. The developed analytical solutions are valid for both small and large amplitudes of oscillation. The accuracy and explicitness of the analytical solutions are carried out to establish the validity of the method. Good agreements are established between the solution of the new method and established exact analytical solution. The developed analytical solutions in this work can serve as a starting point for a better understanding of the relationship between the physical quantities of the problems as it provides continuous physical insights into the problems than pure numerical or computation methods.

An equivalent linearization method for nonlinear Van der Pol oscillator subjected to random vibration using orthogonal functions

This paper deals with the idea of the orthogonal functions in the equivalent linearization of the nonlinear systems. Block Pulse (BP) function gives effective tools to approximate complex problems. The aim of this work is on using properties of the BP function as an orthogonal function in process of linearization. The BP functions have been used to propose an equivalent linearization method in the time domain to determine the unknown linearization coefficients. The accuracy of the proposed method compared with the other equivalent linearization approaches, including the regulation linearization and the dual criterion linearization methods. This study exploited the nonlinear Van der Pol oscillator system under stationary random excitation to demonstrate the feasibility of the proposed method. The validity of the analytical method is verified by applying different values of nonlinearity and intensity of excitation. Besides, by comparing the mean-square responses and frequency response functions of the linearized systems for a wide range of nonlinearity depicted the present method is in agreement with other methods.

On the Inverse Problem of Competitive Modes and the Search for Chaotic Dynamics

Generalized competitive modes (GCM) have been used as a diagnostic tool in order to analytically identify parameter regimes which may lead to chaotic trajectories in a given first order nonlinear dynamical system. The approach involves recasting the first order system as a second order nonlinear oscillator system, and then checking to see if certain conditions on the modes of these oscillators are satisfied. In the present paper, we will consider the inverse problem of GCM: If a system of second order oscillator equations satisfy the GCM conditions, can we then construct a first order dynamical system from it which admits chaotic trajectories? Solving the direct inverse problem is equivalent to finding solutions to an inhomogeneous form of the Euler equations. As there are no general solutions to this PDE system, we instead consider the problem for restricted classes of functions for autonomous systems which, upon obtaining the nonlinear oscillatory representation, we are able to extract at least two of the modes explicitly. We find that these methods often make finding chaotic regimes a much simpler task; many classes of parameter-function regimes that lead to nonchaos are excluded by the competitive mode conditions, and classical knowledge of dynamical systems then allows us to tune the free parameters or functions appropriately in order to obtain chaos. To find new hyperchaotic systems, a similar approach is used, but more effort and additional considerations are needed. These results demonstrate one method for constructing new chaotic or hyperchaotic systems.

Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations.

In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the smoothest invariant manifolds tangent to linear modal subspaces of an equilibrium. Amplitude–frequency plots of the dynamics on SSMs provide the classic backbone curves sought in experimental nonlinear model identification. We develop here, a methodology to compute analytically both the shape of SSMs and their corresponding backbone curves from a data-assimilating model fitted to experimental vibration signals. This model identification utilizes Taken’s delay-embedding theorem, as well as a least square fit to the Taylor expansion of the sampling map associated with that embedding. The SSMs are then constructed for the sampling map using the parametrization method for invariant manifolds, which assumes that the manifold is an embedding of, rather than a graph over, a spectral subspace. Using examples of both synthetic and real experimental data, we demonstrate that this approach reproduces backbone curves with high accuracy.

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.

A simple new iterative method for solving strongly nonlinear oscillator systems having a rational and an irrational force

In this paper, a new noble modified iterative method is proposed to obtain the approximate solution of strongly nonlinear oscillator systems having a rational and an irrational forces. The approximate frequency and the corresponding periodic solution can easily be determined by the present proposed method. Two examples are considered to illustrate the effectiveness and convenience of this procedure. The results obtained in this paper show good agreement with the corresponding numerical solution (considered to be exact) for both small and large amplitudes of oscillation. Furthermore, the method provides better result than other existing results (those results obtained by several authors). The main advantage of the present method is its simplicity which contains up to third harmonic terms. These harmonic terms make the solution rapidly converge. The method is important and powerful for solving nonlinear oscillator systems arising in nonlinear science and engineering.

Method of the direct linearization of mixed nonlinearities

This paper describes the method of the direct linearization of mixed nonlinear forces, which is used to analyze nonlinear oscillatory systems. This method has been compared with other methods in order to demonstrate their resemblance in terms of the practical results. In contrast to well-known methods of nonlinear system analysis, direct linearization methods allow computational cost (time and effort) to be reduced by several orders, which considerably increases their effectiveness.

A physically meaningful new approach to parametric excitation and attenuation of oscillations in nonlinear systems

Parametric excitation of a nonlinear physical pendulum by modulation of its moment of inertia is analyzed in terms of physics as an example of the suggested approach. The modulation is provided by a redistribution of auxiliary masses. The system is investigated both analytically and with the help of computer simulations. The threshold and other characteristics of parametric resonance are found and discussed in detail. The role of nonlinear properties of the physical system in restricting the resonant swinging is emphasized. Phase locking between the drive and oscillations of the pendulum and the phenomenon of parametric autoresonance are investigated. The boundaries of parametric instability are determined as functions of the modulation depth and the quality factor. The feedback providing active optimal control of amplification and attenuation of oscillations is analyzed. An effective method of suppressing undesirable rotary oscillations of suspended constructions is suggested.

Nonlinear Dynamic Behavior of a Heavy Articulated Vehicle With Magnetorheological Dampers

The chaotic vibration analysis of a heavy articulated vehicle (HAV) under consecutive speed control humps (SCHs) excitation is studied. The vehicle is modeled as a nonlinear half-truck oscillatory system with three axles. The suspension system between the truck bodies and axles is equipped with passive viscous damper and magnetorheological (MR) damper. The consecutive SCHs-speed coupling excitation function is presented by a half-sine wave with constant amplitude and variable frequency. The nonlinear dynamic behavior of the system is investigated by special respective techniques. Also, the ride comfort is assessed by the RMS value of truck bodies' accelerations. The results reveal that the quasi-periodic motion is observed at lower speeds when the truck moves on SCHs without load; while in the presence of the load, the dynamic characteristics of the system confirm the chaotic vibration possibility in a widespread range at higher speeds. Further studies indicate that the chaotic behaviors can directly affect on driving comfort and lead to the ride comfort becoming lower. The obtained results can be helpful in designing the oscillatory system for the heavy vehicles to preserve the comfort of drivers and the protection of load.

Robust Generation of Limit Cycles in Nonlinear Systems: Application on Two Mechanical Systems

This paper studies inducing robust stable oscillations in nonlinear systems of any order. This goal is achieved through creating stable limit cycles in the closed-loop system. For this purpose, the Lyapunov stability theorem which is suitable for stability analysis of the limit cycles is used. In this approach, the Lyapunov function candidate should have zero value for all the points of the limit cycle and be positive in the other points in the vicinity of it. The proposed robust controller consists of a nominal control law with an additional term that guarantees the robust performance. It is proved that the designed controller results in creating the desirable stable limit cycle in the phase trajectories of the uncertain closed-loop system and leads to induce stable oscillations in the system's output. Additionally, in order to show the applicability of the proposed method, it is applied on two practical systems: a time-periodic microelectromechanical system (MEMS) with parametric errors and a single-link flexible joint robot in the presence of external disturbances. Computer simulations show the effective robust performance of the proposed controllers in generating the robust output oscillations.

Spring analogy of non-linear oscillations of a bubble in a liquid at resonance

Abstract Two non-linear oscillatory systems are considered. The first is a point mass on a spring with vertical vibration of the suspension point with a frequency that coincides with the frequency of free vertical oscillations and is two times greater than the frequency of free horizontal oscillations. The friction force in the spring is taken into account. For an initial deviation of the point mass from the vertical, after a long enough time the energy of the vertical oscillations is almost completely transferred into the energy of horizontal oscillations. Using an averaging method, an asymptotic solution is constructed, describing the transient process setting up a periodic solution. Comparison of the analytical solution with the numerical one demonstrates its high accuracy. The second system is an axisymmetrical bubble in a liquid under the variable pressure. An analogy between this system and the previous one is established. Vibration of the suspension point of a spring pendulum corresponds to variable liquid pressure, and the vertical and horizontal oscillation modes of the swinging spring correspond to the radial and deformational oscillation modes of the bubble, and the ratio of the frequencies of these modes is also taken to be equal to two. The friction force in the spring corresponds to energy dissipation under radial oscillations of the bubble. In our calculations of energy dissipation, we take into account the liquid viscosity, thermal dissipation, and acoustic radiation due to liquid compressibility. During transfer of the energy of the radial oscillations, the amplitude of the resonant deformational mode of the bubble oscillations grows anomalously, which makes it possible for the bubble to break up with small energy dissipation under the action of a time-varying external pressure field.