Behavior Of Nonlinear Oscillators Research Articles

Advanced Numerical Simulation of Pendulum Dynamics: A Comprehensive Analysis of Environmental Influences and Non-Linear Behavior

This study explores the intricate dynamics of pendulum motion by numerically simulating the influence of environmental factors, including temperature, pressure, and humidity. Employing a high-precision Euler method with a time step of 0.0001 seconds, a 1-meter-long pendulum was modeled under varying conditions: temperatures ranging from 0°C to 40°C, pressures between 950 hPa and 1050 hPa, and humidity levels from 20% to 80%. The simulation incorporated key factors such as thermal expansion, air resistance, and non-linear oscillatory behavior for initial displacements up to 30°. Results reveal that a 40°C rise in temperature induces a 0.0007-second change in period and a 0.001 m/s² variation in calculated gravitational acceleration, predominantly due to rod expansion. Conversely, the effects of pressure and humidity were found to be negligible. Non-linear analysis at a 30° initial displacement indicated a 0.5% increase in period compared to 5°, underscoring the impact of the initial angle on pendulum dynamics. The model demonstrated remarkable accuracy for small-angle oscillations, aligning within 0.01% of theoretical predictions and 0.1% of experimental data. These findings offer valuable insights into pendulum behavior under diverse environmental conditions, providing a robust foundation for advancing the design and calibration of precision instruments and timekeeping mechanisms.

EXPERIMENTAL STUDY ON TWO-PHASE NONLINEAR OSCILLATION BEHAVIOR OF MINIATURIZED GRAVITATIONAL HEAT PIPE

An experimental study on oscillation behavior of vapor-liquid two-phase flow in an ethanol-filled aluminum miniaturized gravitational heat pipe with parallel rectangular channels is carried out. The nonlinear oscillation behavior and oscillation characteristics of two-phase flow in the oscillation state are qualitatively analyzed. In addition, the nonlinear oscillation characteristic of the wall temperature is revealed. Furthermore, the influence of heat load on temperature oscillation characteristics is further analyzed. The research shows that the independent oscillation in a single channel characterized by the random formation and oscillation of liquid plug in a single channel is the typical oscillation behavior of two-phase flow under medium heat load (<i>Q</i> = 28 W). In addition, the interaction on oscillation of liquid plugs between adjacent channels is small. For high heat load (<i>Q</i> = 45 W), the vapor pressure difference at both ends of the liquid plug has a more important role in promoting the liquid plug, the superposition of the liquid plug oscillation in single channel and the oscillation between adjacent channels forms the interference oscillation in adjacent channels. For interference oscillation in adjacent channels, under coupling erosion of the liquid plug in a single channel and the liquid plugs in adjacent channels, with increase of heat load (<i>Q</i> increases from 38 W to 45 W), the dominant frequency of oscillation for independent oscillation in a single channel increases (from 2.5 Hz to 4.2 Hz), and the larger driving force caused by heat load increase enhances the oscillation of liquid plug in a single channel and weakens the oscillation of liquid plug in different channels for interference oscillation in adjacent channels.

Optimization-Oriented EPC Approach for Analyzing the Stochastic Nonlinear Oscillators with Displacement-Multiplicative and Additive Excitations

This paper presents a new approach named optimization-oriented exponential–polynomial-closure (OEPC) to study the behavior of stochastic nonlinear oscillators under both displacement-multiplicative and additive excitations that are Gaussian white noise. In addition to the original projection exponential–polynomial-closure (PEPC) solution procedure, the OEPC method provides an alternative procedure for solving the Fokker–Planck–Kolmogorov (FPK) equation. The OEPC method is formulated by constructing an objective function with the residue of FPK equation. By minimizing the objective function with a gradient-based method, the parameters in the exponential polynomial can be determined and the approximate PDF solution of the nonlinear random oscillator can be obtained. The solutions obtained through the OEPC approach are highly consistent with the available true solutions in special case or the Monte Carlo simulations (MCS). They are much more accurate than those obtained using the Gaussian closure method when the nonlinearity is strong. In addition, the OEPC method is computationally more efficient than MCS. The difference of the results from the OEPC and PEPC is compared and the advantage of the OEPC over PEPC is also shown when the system nonlinearity is strong and complicated.

Effects of medium viscoelasticity on bubble collapse strength of interacting polydisperse bubbles

Due to its physical and/or chemical effects, acoustic cavitation plays a crucial role in various emerging applications ranging from advanced materials to biomedicine. The cavitation bubbles usually undergo oscillatory dynamics and violent collapse within a viscoelastic medium, which are closely related to the cavitation-associated effects. However, the role of medium viscoelasticity on the cavitation dynamics has received little attention, especially for the bubble collapse strength during multi-bubble cavitation with the complex interactions between size polydisperse bubbles. In this study, modified Gilmore equations accounting for inter-bubble interactions were coupled with the Zener viscoelastic model to simulate the dynamics of multi-bubble cavitation in viscoelastic media. Results showed that the cavitation dynamics (e.g., acoustic resonant response, nonlinear oscillation behavior and bubble collapse strength) of differently-sized bubbles depend differently on the medium viscoelasticity and each bubble is affected by its neighboring bubbles to a different degree. More specifically, increasing medium viscosity drastically dampens the bubble dynamics and weakens the bubble collapse strength, while medium elasticity mainly affects the bubble resonance at which the bubble collapse strength is maximum. Differently-sized bubbles can achieve resonances and even subharmonic resonances at high driving acoustic pressures as the elasticity changes to certain values, and the resonance frequency of each bubble increases with the elasticity increasing. For the interactions between the size polydisperse bubbles, it indicated that the largest bubble generally has a dominant effect on the dynamics of smaller ones while in turn it is almost unaffected, exhibiting a pattern of destructive and constructive interactions. This study provides a valuable insight into the acoustic cavitation dynamics of multiple interacting polydisperse bubbles in viscoelastic media, which may offer a potential of controlling the medium viscoelasticity to appropriately manipulate the dynamics of multi-bubble cavitation for achieving proper cavitation effects according to the desired application.

Collective behavior of nonlinear dynamical oscillators

This topical issue collects contributions of recent achievements and scientific progress related to the collective behavior of nonlinear dynamical oscillators. The individual papers focus on different questions of present-day interest in this topic.

Lie point symmetry infinitesimals, optimal system, power series solution, and modulational gain spectrum to the mathematical Noyes–Field model of nonlinear homogeneous oscillatory Belousov–Zhabotinsky reaction

Introduction:The chemical oscillators are identified as open system that demonstrate periodic changes in the concentration of some reaction species as a result of intricate physico-chemical mechanisms which can lead to bi-stability, the occurrence of limit cycle attractors, the emergence of spiral waves and turing patterns, and finally, deterministic chaos. Objectives:The main objective of this paper is to analyze the simple Noyes–Field governing system of differential equations for the nonlinear Belousov–Zhabotinsky reaction which delineates the non-linear oscillatory behavior of chemical systems that occurs in the homogeneous media. Methodology:The Lie symmetry invariance analysis performed to extract the symmetries infinitesimal generators and the adjoint representation carried out to develop optimal system for the obtained Lie vectors. The significant power series approach applied to obtain the analytical solution. The modulation instability criteria ensured the stability of nonlinear oscillatory Belousov–Zhabotinsky reaction process. Results:The one-dimensional Lie symmetry generators algebra of the mathematical Noyes–Field governing system for oscillatory reaction is established. Furthermore, similarity reductions are carried out as well as the development of an optimal system of the sub-algebras. The similarity transformation technique converted the controlling system to ordinary differential equations and generates the large quantity of analytical traveling wave solutions. Moreover, the closed-form analytical solution for the proposed homogeneous nonlinear oscillatory chemical process is secured. The (MI) gain spectrum graphically visualized with the suitable choice of arbitrary parameters. Conclusion:The graphical performance of the Noyes–Field model solution at various settings reveals new perspectives and fascinating model phenomena. The attained outcomes have significant applications and have opened up innovative development areas for research across numerous scientific fields.

Study of Gravitational Force Effects, Magnetic Restoring Forces, and Coefficients of the Magnetic Spring-Based Nonlinear Oscillator System

The aim of this article is to analyze the design of a magnetic spring-based nonlinear oscillator system and investigate its magnetic properties, the effect of gravitational force, the accuracy of the modeled magnetic restoring force, the coefficient of the magnetic restoring force, and the nonlinear response of the proposed system. The proposed magnetic spring-based nonlinear oscillator system consists of a nonmagnetic shaft, floating magnet, and two fixed magnets where all magnets are placed in such a way that each magnet can repel the other. The magnetic properties of the proposed system are studied numerically and analytically to provide insight into its role in the nonlinear oscillatory behavior of the system. The effect of the gravitational force, which changes the equilibrium position and the nonlinear behavior of the magnetic spring, is studied. The numerical analysis of the magnetic flux density and magnetic restoring force is validated using an analytical and experimental analysis. Different orders of polynomial curve fittings, such as cubic and quintic, are used to model the magnetic restoring force between the moving magnet and the two fixed end magnets. Finally, the linear and nonlinear coefficients of the magnetic spring-based system for different positions of the floating magnet are investigated.

Nonlinear oscillations of a one-dimensional dielectric elastomer generator system

The nonlinear oscillations of a dielectric elastomer generator (DEG) consisting of an oscillator and two hyperelastic membranes are investigated in this paper. Due to the finite deformations of the membranes and the Maxwell stress induced by external electric fields, the DEG can be considered as a damped nonlinear oscillator. We then develop a theoretical model for the DEG system to describe the nonlinear oscillatory behavior, which can be used to elucidate the effects of the pre-stretch, membrane size, electric charge, and forcing amplitude on the DEG. From our theoretical analysis, it is found that the Maxwell stress plays a significant role in stabilizing the oscillations of the system. In practical applications, nonlinear elastic DE films exhibit strong oscillations possibly associated with a jumping phenomenon which may be harmful to the system. To describe the jumping phenomenon, an asymptotic solution on the primary resonance is derived based on the averaging method, and the obtained asymptotic solutions are further validated by the corresponding numerical results. The evolutions of the jumping phenomena with respect to the relevant factors are further analyzed. It is found that by tuning the membrane size, electric charge amount, or forcing amplitude, the jumping phenomenon can be avoided. Finally, the onsets of the jumping phenomenon are acquired from the graphic representation of the evolutions of the system resonance characteristics. It is anticipated that the current analysis could provide useful insights into designs of DE generators.

Study on nonlinear dynamic behavior and stability of aviation pressure servo valve-controlled cylinder system

In this paper, aiming at the phenomenon of self-excited oscillation caused by nonlinear factors in a large aircraft wheel brake control system, the nonlinear dynamic behavior of the pressure servo valve-controlled cylinder system (PSVCS) is studied, and the influence law of the key parameters on the nonlinear self-excitation behavior is obtained. On this basis, the stability of the PSVCS is analyzed both in time domain and frequency domain, and it is proved in principle that the PSVCS is a stable self-closed-loop control system. Firstly, the nonlinear dynamics model of the PSVCS is established in this paper. Secondly, using the method of phase plane analysis, the nonlinear dynamic behavior of the PSVCS and the influence law of key parameters on the system are studied. Thirdly, the nonlinear system of the PSVCS is transformed into a segmented local linear system, and the stability of the prestage and the power stage is analyzed, respectively. Finally, through a performance test platform, which is used to simulate the load of PSVCS, the theoretical analysis results of this paper are verified experimentally under different working conditions. The final experimental results show that both the nonlinear dynamic model established in this paper and the influence law of the key parameters obtained by the phase plane analysis on the nonlinear self-excited oscillation behavior are correct, and the relevant conclusions can provide a reference for the design of the braking system control system.

Surface stress size dependency in nonlinear free oscillations of FGM quasi-3D nanoplates having arbitrary shapes with variable thickness using IGA

The present study aims to propose a computational package for analyzing the geometrically nonlinear large-amplitude vibrations of nanoplates having arbitrary shapes with variable thickness in the presence of surface stress type of size effect. Accordingly, isogeometric analysis (IGA) is employed to achieve exact geometrical description as well as higher-order efficient smoothness with no meshing difficulty. The associated size-dependent plate model is constructed for the first time via implementation of the Gurtin–Murdoch surface elasticity to a quasi-3D plate model having the capability to take the thickness stretching into consideration with only four variables. The nanoplates are assumed made of functionally graded materials (FGMs) through the variable thickness, the displacement along which is calculated independently via a trigonometric normal shape function. The variation of plate thickness obeys three different schemes including linear, concave, and convex ones. It is highlighted that by changing the pattern of the thickness variation from convex type to linear one, and then from linear type to concave one, in spite of the higher classical flexural stiffness, the surface elastic-based flexural stiffness of FGM composite nanoplates gets lower, which results in a lower nonlinear frequency corresponding to a very thin nanoplates. Moreover, it is portrayed that by increasing the value of material gradient index, the role of stiffer character of the surface stress in the nonlinear free oscillation behavior of FGM composite nanoplates is pronounced. This anticipation is the same for the both initial and deeper parts of the nonlinear frequency-deflection response.

Nonlinear Vibration of Functionally Graded Graphene Nanoplatelets Polymer Nanocomposite Sandwich Beams

We provide an analytical investigation of the nonlinear vibration behavior of thick sandwich nanocomposite beams reinforced by functionally graded (FG) graphene nanoplatelet (GPL) sheets, with a power-law-based distribution throughout the thickness. We assume the total amount of the reinforcement phase to remain constant in the beam, while defining a relationship between the GPL maximum weight fraction, the power-law parameter, and the thickness of the face sheets. The shear and rotation effects are here considered using a higher-order laminated beam model. The nonlinear partial differential equations (PDEs) of motion are derived from the Von Kármán strain-displacement relationships, here solved by applying an expansion of free vibration modes. The numerical results demonstrate the key role of the amplitudes on the vibration response of GPL-reinforced sandwich beams, whose nonlinear oscillation behavior is very important in the physical science, mechanical structures and other mathematical analyses. The sensitivity of the response to the total amount of GPLs is explored herein, along with the possible effects related to the power-law parameter, the structural geometry, and the environmental conditions. The results indicate that changing the nanofiller distribution patterns with the proposed model can remarkably increase or decrease the effective stiffness of laminated composite beams.

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.

Non periodic oscillations, bistability, coexistence of chaos and hyperchaos in the simplest resistorless Op-Amp based Colpitts oscillator

In the framework of a project on simple circuits with unexpected high degrees of freedom, we report an autonomous microwave oscillator made of a CLC linear resonator of Colpitts type and a single general purpose operational amplifier (Op-Amp). The resonator is in a parallel coupling with the Op-Amp to build the necessary feedback loop of the oscillator. Unlike the general topology of Op-Amp-based oscillators found in the literature including almost always the presence of a negative resistance to justify the nonlinear oscillatory behavior of such circuits, our zero resistor circuit exhibits chaotic and hyperchaotic signals in GHz frequency domain, as well as many other features of complex dynamic systems, including bistability. This simplest form of Colpitts oscillator is adequate to be used as didactic model for the study of complex systems at undergraduate level. Analog and experimental results are proposed.

Uncovering Droop Control Laws Embedded Within the Nonlinear Dynamics of Van der Pol Oscillators

This paper examines the dynamics of power-electronic inverters in islanded microgrids that are controlled to emulate the dynamics of Van der Pol oscillators. The general strategy of controlling inverters to emulate the behavior of nonlinear oscillators presents a compelling time-domain alternative to ubiquitous droop control methods which presume the existence of a quasistationary sinusoidal steady state and operate on phasor quantities. We present two main results in this paper. First, by leveraging the method of periodic averaging, we demonstrate that droop laws are intrinsically embedded within a slower time scale in the nonlinear dynamics of Van der Pol oscillators. Second, we establish the global convergence of amplitude and phase dynamics in a resistive network interconnecting inverters controlled as Van der Pol oscillators. Furthermore, under a set of nonrestrictive decoupling approximations, we derive sufficient conditions for local exponential stability of desirable equilibria of the linearized amplitude and phase dynamics.

Behavior of Nonlinear Flutter-Type Oscillations of Flexible Plates in Supersonic Gas Flow

AbstractThe behavior of nonlinear oscillations of isotropic rectangular plate fluttering in a supersonic gas flow is examined. The study was conducted taking into account both types of nonlinearity...

Nonlinear behavior of different sized granular bead media on a vibrating clamped circular elastic plate

This experiment uses a soil plate oscillator (SPO), or a circular acrylic (elastic) plate clamped into place using flanges, to study the nonlinear oscillatory behavior of a column of granular media (with selected diameters) that is supported by the plate. Secured on the underside of the plate's center is a rare earth magnet that is driven by an AC coil which is controlled by an amplified swept sinusoidal chirp. An accelerometer is fastened to the magnet and a spectrum analyzer measures the plate's acceleration versus frequency. In separate measurements of nonlinear tuning curve response, the plate is loaded with 350 grams of spherical soda lime beads of diameters 2, 3, 4, 5, 6, 7, and 10 mm. The driver current is changed in 40 evenly spaced increments, holding other parameters fixed, during each sweep. A study of the backbone curve (rms acceleration a(f) peak versus frequency f) for each bead diameter exhibits a frequency “softening” vs. amplitude—similar to the nonlinear mesoscopic elasticity observed in resonance experiments in geo-materials. After some initial curvature, the plot has a negative linear slope with increasing amplitude. Smaller beads show significantly more acceleration response than larger beads while the slopes have somewhat comparable values.

Geometrically nonlinear dynamic behavior on detection sensitivity of carbon nanotube-based mass sensor using finite element method

The principle of mass detection using carbon nanotube (CNT) resonators is based on the detection of the resonant frequency shift due to an attached mass. Although CNT resonators can easily show a nonlinear oscillation behavior, there is a lack of studies on the influence of the design parameters and the nonlinear dynamic behavior on the detection sensitivity of CNT-based mass sensors. In addition, most of the finite element method (FEM) analysis models that are used to predict the resonant frequency shift due to attached masses have been implemented in the linear oscillation regime. In order to enhance the sensing performance of the CNT-based mass sensor, a parametric study of the resonant frequency shift is conducted herein with respect to the attached mass, the electrostatic force, the initial tension and the CNT length, using an FEM-based nonlinear analysis model. The FE model is applied to solve the nonlinear dynamic behavior of CNT resonators using direct time integration and the solution is verified by its comparison to the corresponding analytical solution that had been validated in previous studies. The analysis results of the nonlinear dynamic behavior of the CNT resonator indicate that the CNT length plays a key role in the detection sensitivity, and the amount of electrostatic force determines the linear or nonlinear oscillation behavior of the resonator. It is shown that the detection sensitivity can be improved using the nonlinear oscillation behavior and this improvement is more effective with longer CNTs. This study's results justify the possibility and validity of FEM use for the analysis of the nonlinear behavior of CNT resonators and elucidate the relationship between the design parameters and the nonlinear behavior of the CNT-based mass sensor in enhancing the sensing performance.

Thermal effects on nonlinear vibration of a carbon nanotube-based mass sensor using finite element analysis

In the present study we carried out a dynamic analysis of a CNT-based mass sensor by using a finite element method (FEM)-based nonlinear analysis model of the CNT resonator to elucidate the combined effects of thermal effects and nonlinear oscillation behavior upon the overall mass detection sensitivity. Mass sensors using carbon nanotube (CNT) resonators provide very high sensing performance. Because CNT-based resonators can have high aspect ratios, they can easily exhibit nonlinear oscillation behavior due to large displacements. Also, CNT-based devices may experience high temperatures during their manufacture and operation. These geometrical nonlinearities and temperature changes affect the sensing performance of CNT-based mass sensors. However, it is very hard to find previous literature addressing the detection sensitivity of CNT-based mass sensors including considerations of both these nonlinear behaviors and thermal effects. We modeled the nonlinear equation of motion by using the von Karman nonlinear strain–displacement relation, taking into account the additional axial force associated with the thermal effect. The FEM was employed to solve the nonlinear equation of motion because it can effortlessly handle the more complex geometries and boundary conditions. A doubly clamped CNT resonator actuated by distributed electrostatic force was the configuration subjected to the numerical experiments. Thermal effects upon the fundamental resonance behavior and the shift of resonance frequency due to attached mass, i.e., the mass detection sensitivity, were examined in environments of both high and low (or room) temperature. The fundamental resonance frequency increased with decreasing temperature in the high temperature environment, and increased with increasing temperature in the low temperature environment. The magnitude of the shift in resonance frequency caused by an attached mass represents the sensing performance of a mass sensor, i.e., its mass detection sensitivity, and it can be seen that this shift is affected by the temperature change and the amount of electrostatic force. The thermal effects on the mass detection sensitivity are intensified in the linear oscillation regime and increase with increasing CNT length; this intensification can either improve or worsen the detection sensitivity.

A Circuit for Automatic Measurement of Bifurcation Diagram in Nonlinear Electronic Oscillators

Experimental bifurcation diagram is very helpful to describe the global behaviour of nonlinear electronic oscillators. The diagram allows insight gain of the oscillators behaviour for a parameter variation. However, when the bifurcation parameter is a resistance, experimental bifurcation diagrams are hard to achieve. In this case, if the potentiometer resistance has been used to change the gain, it is possible to replace the potentiometer by an alternative circuit using operational transconductance amplifier. This kind of circuit is presented in this work and experimental results are provided to piecewise affine Rossler electronic oscillator. The results obtained show a very good matching with the theoretical diagram. Then, if the bifurcation parameter is a resistance, voltage or current, we have implemented a circuit to extract the experimental bifurcation diagram of all nonlinear electronic oscillators.

Semi-analytic solutions to oscillatory behavior of initially curved micro/nano systems

In this study, Homotopy analysis method (HAM) is used to investigate the nonlinear oscillatory behavior of clamped-clamped initially curved micro/nano beam, under electrostatic force actuation. The electrostatic actuation, midplane stretching, and geometric nonlinearity effects are considered. Initially, through Galerkin’s decomposition method, the partial differential governing equation is converted to an ordinary differential equation. Thereafter, HAM is used to find a semi-analytic solution for the nonlinear oscillatory behavior of an initially curved microbeam. To ensure the convergence of the problem, the convergence region of solutions is investigated. Furthermore, the influence of initial elevation, input voltage, midplane stretching, and order of the HAM approximation are examined. Analytic results achieved by HAM are in good agreement with numerical solutions and published literature.