Modified Parker-Sochacki method for solving nonlinear oscillators

A new approach is presented for solving nonlinear oscillatory systems. Parker-Sochacki method (PSM) is combined with Laplace-Padé resummation method to obtain approximate periodic solutions for three nonlinear oscillators. The first one is Duffing oscillator with quintic nonlinearity which has odd nonlinearity. The second one is Helmholtz oscillator which has even nonlinearity. The last one is a strongly nonlinear oscillator, namely; relativistic harmonic oscillator which has a fractional order nonlinearity. Solutions are also obtained using Runge-Kutta numerical method (RKM) and Lindstedt-Poincare method (LPM). However, the LPM could not be used to solve the relativistic harmonic oscillator since it is a strongly nonlinear oscillator. The comparison between these solutions shows that the convergence zone for the Parker-Sochacki with Laplace-Padé method (PSLPM) is remarkably increased compared to PSM method. It also shows that the PSLPM solutions are in excellent agreement with LPM solutions for Duffing oscillator and are superior to LPM solutions in case of Helmholtz oscillator. The PSLPM succeeded to give an accurate periodic solution for the relativistic harmonic oscillator. For a wide range of solution domain, comparing PSLPM with RKM prove the correctness of the PSLPM method. Hence, the PSLPM method can be used with satisfied confidence to solve a broad class of nonlinear oscillators.     

Analytical solutions for asymmetric periodic motions to chaos in a hardening Duffing oscillator

In this paper, the analytical dynamics of asymmetric periodic motions in the periodically forced, hardening Duffing oscillator is investigated via the generalized harmonic balance method. For the hardening Duffing oscillator, the symmetric periodic motions were extensively investigated with the aim of a good understanding of solutions with jumping phenomena. However, the asymmetric periodic motions for the hardening Duffing oscillators have not been obtained yet, and such asymmetric periodic motions are very important to find routes of periodic motions to chaos in the hardening Duffing oscillator analytically. Thus, the bifurcation trees from asymmetric period-1 motions to chaos are presented. The corresponding unstable periodic motions in the hardening Duffing oscillator are presented, and numerical illustrations of stable and unstable periodic motions are carried out as well. This investigation provides a comprehensive understanding of chaos mechanism in the hardening Duffing oscillator.     

Creation–annihilation process of limit cycles in the Rayleigh–Duffing oscillator

The present paper examines the creation?Cannihilation process of limit cycles in the Rayleigh?CDuffing oscillator with negative linear damping and negative linear stiffness. It is obtained by the perturbation method, in which the number of limit cycles in the Rayleigh?CDuffing oscillator varies with the linear damping and stiffness. Numerical simulations are performed in order to confirm the analytically obtained creation?Cannihilation process of limit cycles. Moreover, we compare the process of limit cycles in the Rayleigh?CDuffing oscillator to that of limit cycles in the van der Pol?CDuffing oscillator. The difference in these oscillator is only in nonlinear forces, which causes a qualitative difference in the creation?Cannihilation processes.     

Thermal stress measurement of quartz oscillator module packaging

The thermal stress of the quartz oscillator module packaging is investigated using digital-image correlation method (DICM), and the experimental results are given. Under the quartz oscillator module packaging, the quartz oscillator and the Fe−Sn−Cu alloy frame are joined together with the electroconductive adhesive (PI), and the electroconductive adhesive needs to be cured twice at 150°C and 275°C respectively. As the quartz oscillator and the Fe−Sn−Cu alloy frame have a distinct difference in both thermal expansion coefficients and mechanical properties and in process of packaging temperature rises or drops, the thermal stress is yielded easily. While temperature drops, the normal stress at the quartz oscillator edge is a tensile stress, which can make the quartz oscillator fracture. Project supported by the Natural Science Foundation of Tianjin (013605211).     

Marginal stability boundaries for some driven, damped, non-linear oscillators

A procedure is developed for averaging the differential equations for certain non-linear oscillators which are damped and externally driven. The procedure makes possible the obtaining of marginal stability boundaries for bifurcations in parameter space and is useful for systems with unperturbed solutions involving Jacobi elliptic functions. Specific cases of a driven, damped pendulum, an anharmonie oscillator, a Duffing oscillator, and a non-linear Helmholtz oscillator are examined.     

On the analysis of bipolar transistor based chaotic circuits: case of a two-stage colpitts oscillator

We perform a systematic analysis of a system consisting of a two-stage Colpitts oscillator. This well-known chaotic oscillator is a modification of the standard Colpitts oscillator obtained by adding an extra transistor and a capacitor to the basic circuit. The two-stage Colpitts oscillator exhibits better spectral characteristics compared to a classical single-stage Colpitts oscillator. This interesting feature is suitable for chaos-based secure communication applications. We derive a smooth mathematical model (i.e., sets of nonlinear ordinary differential equations) to describe the dynamics of the system. The stability of the equilibrium states is carried out and conditions for the occurrence of Hopf bifurcations are obtained. The numerical exploration reveals various bifurcation scenarios including period-doubling and interior crisis transitions to chaos. The connection between the system parameters and various dynamical regimes is established with particular emphasis on the role of both bias (i.e., power supply) and damping on the dynamics of the oscillator. Such an approach is particularly interesting as the results obtained are very useful for design engineers. The real physical implementation (i.e., use of electronic components) of the oscillator is considered to validate the theoretical analysis through several comparisons between experimental and numerical results.     

Bifurcation and path-following continuation analysis of periodic orbits of an extended Fermi oscillator model

In this research, we offer eigenvalue analysis and path following continuation to describe the impact, stick, and non-stick between the particle and boundaries to understand the nonlinear dynamics of an extended Fermi oscillator. The principles of discontinuous dynamical systems will be utilized to explain the moving process in such an extended Fermi oscillator. The motion complexity and stick mechanism of such an oscillator are demonstrated using periodic and chaotic motions. The major parameters are the frequency, amplitude in periodic excitation force, and the gap between the top and bottom boundary. We employ path-following analysis to illustrate the bifurcations that lead to solution destabilization. We present the evolution of the period solutions of the extended Fermi oscillator as the parameter varies. From the viewpoint of eigenvalue analysis, the essence of period-doubling, saddle-node, and Torus bifurcation is revealed. Numerical continuation methods are used to do a complete one- and two-parameter bifurcation analysis of the extended Fermi oscillator. The presence of codimension-one bifurcations of limit cycles, such as saddle-node, period-doubling, and Torus bifurcations, is shown in this work. Bifurcations cause all solutions to lose stability, according to our findings. The acquired results provide a better understanding of the extended Fermi oscillator mechanism and demonstrate that we may control the system dynamics by modifying the parameters.

     

A new model for the study of rain-wind-induced vibrations of a simple oscillator

In this paper, a model equation is presented for the study of rain-wind-induced vibrations of a simple oscillator. As will be shown the presence of raindrops in the wind-field may have an essential influence on the dynamic stability of the oscillator. In this model equation the influence of the variation of the mass of the oscillator due to an incoming flow of raindrops hitting the oscillator and a mass flow which is blown and shaken off is investigated. The time-varying mass is modeled by a time harmonic function whereas simultaneously also time-varying lift and drag forces are considered.     

Suppression of super-harmonic resonance response using a linear vibration absorber

Super-harmonic resonances may appear in the forced response of a weakly nonlinear oscillator having cubic nonlinearity, when the forcing frequency is approximately equal to one-third of the linearized natural frequency. Under super-harmonic resonance conditions, the frequency-response curve of the amplitude of the free-oscillation terms may exhibit saddle-node bifurcations, jump and hysteresis phenomena. A linear vibration absorber is used to suppress the super-harmonic resonance response of a cubically nonlinear oscillator with external excitation. The absorber can be considered as a small mass-spring-damper oscillator and thus does not adversely affect the dynamic performance of the nonlinear primary oscillator. It is shown that such a vibration absorber is effective in suppressing the super-harmonic resonance response and eliminating saddle-node bifurcations and jump phenomena of the nonlinear oscillator. Numerical examples are given to illustrate the effectiveness of the absorber in attenuating the super-harmonic resonance response.     

Synthesis of oscillators for exact desired periodic solutions having a finite number of harmonics

The synthesis of autonomous oscillators with exact desired periodic steady-state solution is described in this contribution. The vector field of the oscillator differential equation is built up with a conservative and a dissipative part. Both parts are synthesized using an algebraic function describing the desired limit cycle. The desired periodic motion is restricted by a finite numbers of harmonics, whereby the amplitude and the phase shift of every harmonic can be freely chosen, depending on the specific application. Afterwards the synthesis of a periodically driven oscillator with an exact desired periodic response is described. For this purpose, the differential equation of the autonomous oscillator is extended by a state-dependent compensation term that equals the excitation at the steady-state solution. Here the freely definable amplitudes and phase angles of the oscillator motion are restricted by the existence and stability conditions for synchronization.     

Periodic-impact motions and bifurcations in dynamics of a plastic impact oscillator with a frictional slider

A two-degree-of-freedom plastic impact oscillator with a frictional slider is considered. Dynamics of the plastic impact oscillator are analyzed by a three-dimensional map, which describes free flight and sticking solutions of two masses of the system, between impacts, supplemented by transition conditions at the instants of impacts. Piecewise property and singularity are found to exist in the impact Poincaré map. The piecewise property of the map is caused by the transitions of free flight and sticking motions of two masses immediately after the impact, and the singularity of the map is generated via the grazing contact of two masses immediately before the impact. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. The influence of piecewise property, grazing singularity and parameter variation on dynamics of the vibro-impact system is analyzed. The global bifurcation diagrams of before-impact velocity as a function of the excitation frequency are plotted to predict much of the qualitative behavior of the system. The global bifurcations of period-N single-impact motions of the plastic impact oscillator are found to exhibit extensive and systematic characteristics. Dynamics of the impact oscillator, in the elastic impact case, is also analyzed. This type of impact is modelled by using the conditions of conservation of momentum and an instantaneous coefficient of restitution rule. The differences in periodic-impact motions and bifurcations are found by making a comparison between dynamic behaviors of the plastic and elastic impact oscillators with a frictional slider. The best progression of the plastic impact oscillator is found to occur in period-1 single-impact sticking motion with large impact velocity. The largest progression of the elastic impact oscillator occurs in period-1 multi-impact motion. The simulative results show that the plastic impact feature for the impact-progressive oscillator is of a considerable importance in minimizing adverse effects such as high noise levels, wear and tear caused by impacts.     

A nonlinear oscillator concept for one-dimensional pulsating detonations

To interpret the results of direct numerical simulations for the one-dimensional pulsating detonation, a nonlinear oscillator model is proposed based on the integral conservation considerations. A gauging procedure is suggested in which the results from the direct numerical simulation are used to compute the coefficients of the oscillator equation. Various terms of the oscillator equation obtained are compared with those of mechanical nonlinear oscillators in order to illustrate their analogy. Among many important features captured by this nonlinear oscillator equation, the oscillatory behavior of the detonation wave front can be interpreted as resonant excitation of the chemical energy release. Received 24 March 1997 / Accepted 7 May 1998     

An analytical prediction of sliding motions along discontinuous boundary in non-smooth dynamical systems

This paper presents a method for the analytical prediction of sliding motions along discontinuous boundaries in non-smooth dynamical systems. The methodology is demonstrated through investigation of a periodically forced linear oscillator with dry friction. The switching conditions for sliding motions in non-smooth dynamical systems are given. The generic mappings for the friction-induced oscillator are introduced. From the generic mappings, the corresponding criteria for the sliding motions are presented through the force product conditions. The analytical prediction of the onset and vanishing of the sliding motions is illustrated. Finally, numerical simulations of sliding motions are carried out to verify the analytical prediction. This analytical prediction provides an accurate prediction of sliding motions in non-smooth dynamical systems. The switching conditions developed in this paper are expressed by the total force of the oscillator, and the nonlinearity and linearity of the spring and viscous damping forces in the oscillator cannot change such switching conditions. Therefore, the achieved force criteria can be applied to the other dynamical systems with nonlinear friction forces processing a C ⁰-discontinuity.     

Synchronization of a Pair of Opposed Facing Oscillators in a Side-by-Side Configuration

This study considers two opposed facing fluidic oscillators in a side-by-side layout. Each fluidic oscillator shares one of its feedback channels with the other oscillator thus this configuration provides a mean for communication between fluidic oscillators. As a result of this communication through the shared feedback channel, the exiting jets of the fluidic oscillator pair are synchronized. The details of this synchronization are revealed by means of hot-wire measurements, flow visualizations and validated numerical simulations. The oscillation frequencies for both jets were the same and varied from 123 Hz to 476 Hz for a flow rate range from 5 SLPM to 25 SLPM per inlet corresponding to Reynolds number range of 2,125 to 10,625. The calculated cross-correlation coefficients were over 0.88 for the considered flow rate range. The flow visualizations exposed that the exiting jet of each fluidic oscillator move away from the shared feedback channel and move toward the shared feedback channel simultaneously indicating motion in the same direction. Numerical results indicated that the shared feedback channel plays a pressure balancing role while allowing inter-oscillator flow through each side of the shared feedback channel thus controlling the flow direction of each exiting jet yielding a synchronized output from the fluidic oscillator nozzles.     

Analysis of Duffing oscillator with time-delayed fractional-order PID controller

The free vibration of Duffing oscillator with time-delayed fractional-order Proportional-Integral-Derivative (FOPID) controller based on displacement feedback is studied. The second-order approximate analytical solution is obtained by KBM asymptotic method. The effects of the parameters in FOPID controller on the dynamical properties are characterized by some equivalent parameters. The correctness of the approximate analytical results is verified by the numerical results. The effects of the time-delayed FOPID controller with displacement feedback on control performances of Duffing oscillator are analyzed in detail by time response, and the stability conditions of zero solution and periodic motions are also presented. Finally, the control performances on Duffing oscillator with large damping are further analyzed. And the results show that one could take the advantage of time delay, when the parameters of time-delayed FOPID controller are chosen reasonably.     

Transient response of a disk with discrete impedances from its decelerating boundary

Analyzed is transient response of an elastic disk from its decelerating boundary. Eigenfunctions of the bear disk are used as Galerkin trial functions to the disk with discrete masses and impedances in the form of a one-degree-of-freedom (1-dof) oscillator. The oscillator mass models a module connected to the disk and the spring models the connecting isolation stiffness. For a fixed oscillator mass, except for very weak springs, stiffness mostly raises transmissibility of deceleration from disk to mass.     

The codimension-two bifurcation for the recent proposed SD oscillator

In this paper, the complicated nonlinear dynamics at the equilibria of SD oscillator, which exhibits both smooth and discontinuous dynamics depending on the value of a parameter α, are investigated. It is found that SD oscillator admits codimension-two bifurcation at the trivial equilibrium when α=1. The universal unfolding for the codimension-two bifurcation is also found to be equivalent to the damped SD oscillator with nonlinear viscous damping. Based on this equivalence between the universal unfolding and the damped system, the bifurcation diagram and the corresponding codimension-two bifurcation structures near the trivial equilibrium are obtained and presented for the damped SD oscillator as the perturbation parameters vary.     

Non-Gaussianclosure techniques for stationary random vibration

The classical method of statistical linearization when applied to a non-linear oscillator excited by stationary wide-band random excitation, can be considered as a procedure in which the unknown parameters in a Gaussian distribution are evaluated by means of moment identities derived from the dynamic equation of the oscillator. A systematic extension of this procedure is the method of non-Gaussian closure in which an increasing number of moment identities are used to evaluate additional parameters in a family of non-Gaussian response distributions. The method is described and illustrated by means of examples. Attention is given to the choice of representations of non-Gaussian distributions and to techniques for generating independent moment identities directly from the differential equation of the non-linear oscillator. Some shortcomings of the method are pointed out.     

具有稳定数值解的三维谐振子

谐振子广泛应用于物理系统的描述和物理现象的数值模拟。由于二维或三维谐振子对于系统参数、初始条件和边界条件的高度敏感性,很多物理过程的动力学模拟都会出现数值解不稳定的现象。近年来发展的无网格法、物质点法和近场动力学法等数值模拟方法均绕开了对固体材料固有构形的量化描述。本文引入了定常耗散项和弹簧耗散项,考虑随机微扰效应,提出了一种三维耗散谐振子,构建了基于蛙跳法和边界松弛技术的数值积分算法。应用三维谐振子构建了耗散型弹簧摆、简化弦和简化梁三个模型,设定了13个定解问题进行动力学模拟。数值试验结果表明,三维谐振子是稳定的。基于简化弦模型,模拟了拉弦、放弦和重弦三个有界弦振动问题;其中,拉弦和放弦问题成功模拟了有界弦的三维振形;重弦问题模拟再现了悬链线在水平向的微幅振荡现象。基于简化梁模型,模拟了三维梁的拉伸、剪切和扭转行为,验证了三维谐振子对于非线性大变形问题动力学模拟的描述能力,及其对外部作用的高速响应能力。本文方法可以为弦振动问题和材料力学非线性大变形问题的动力学模拟提供一条可行的实现途径。     

Nonlinear dynamics and vibration reduction of a dry friction oscillator with SMA restraints

The dynamic behaviors of a dry friction oscillator with shape memory alloy (SMA) are investigated. Motion equations of the system are formulated by the restoring force of the oscillator in terms of a polynomial constitutive model dependent mainly on the temperature. The vibration response of the system and the influence of the temperature are investigated. It is shown that chaotic motions can be observed and dramatically changed by temperature characteristics. Moreover, some sliding bifurcations are also discovered and influenced by the temperature. Compared with conventional dry friction elastic oscillators, the dry friction SMA oscillator presents much richer dynamic behavior caused by pseudo-elasticity, and vibration reduction can be achieved through the shape memory property of SMA restraints.