Chaotic Vibrations Research Articles

Controlling Chaotic Vibrations of a Quarter-Car Model Excited by Road Surface Profile using Parametric Excitation.

Chaotic Vibrations are considered for a quarter-car model excited by the road surface profile. The equation of motion is obtained in the form of a classical Duffing equation and it is modeled with deliberate introduction of parametric excitation force term to enable us manipulate the behavior of the system. The equation of motion is solved using the Method of Multiple Scales. The steady-state solutions with and without the parametric excitation force term is investigated using NDSolve MathematicaTM Code and the nonlinear dynamical system’s analysis is by a study of the Bifurcations that are observed from the analysis of the trajectories, and the calculation of the Lyapunov. In making the system more strongly nonlinear the excitation amplitude value is artificially increased to various multiples of the actual value. Results show that the system’s response can be extremely sensitive to changes in the amplitude and the that chaos is evident as the system is made more nonlinear and that with the introduction of parametric excitation force term the system’s motion becomes periodic resulting in the elimination of chaos and the reduction in amplitude of vibration.

New multipoint algorithm for eliminating chaotic vibrations in complex non-linear systems

Wydawnictwo SIGMA-NOT wydaje czasopisma fachowe informujące swoich czytelników o najnowszych osiągnięciach naukowych i nowoczesnych rozwiązaniach technicznych w Polsce i na świecie, popularyzuje problemy techniczne oraz poszerza wiedzę i kulturę techniczną.

Experiment and analysis on chaotic characteristics vibration mill with variable rotational speed

In an attempt to improve the current low efficiency and high consumption situation of vibration mills, this paper analyses the chaotic motion characteristics of the system and the movement of vibration mill. The complex stiffness-dispersion coupling of the system is also studied, so as to investigate the effect of the system’s chaotic motion characteristics on the efficiency improvement and energy consumption reduction. Based on the ADAMS software, this paper establishes a simplified vibration mill mechanical model, analyzes the singularity and stability of the system, and determines the critical speed at which the vibration motor becomes chaotic according to the bifurcation diagram. Then the chaotic state of the grinding machine with sinusoidal variation in its motor speed is studied based on the Poincaré principle, singular attractor and maximum Lyapunov exponent. Lastly, a 200[Formula: see text]h vibration test on diamond powder with an average particle size of 10 [Formula: see text]m was carried out. Test results under the two operating conditions of variable and constant speeds are compared and analyzed. Our results show that with variable speed the vibration mill achieved higher grinding efficiency but smaller particle grain size. The research elaborated in this paper provides a valuable reference for the engineering application of the chaotic characteristics of vibration mill.

Double mode model of size-dependent chaotic vibrations of nanoplates based on the nonlocal elasticity theory

In this paper vibrations of the isotropic micro/nanoplates subjected to transverse and in-plane excitation are investigated. The governing equations of the problem are based on the von Kármán plate theory and Kirchhoff–Love hypothesis. The small-size effect is taken into account due to the nonlocal elasticity theory. The formulation of the problem is mixed and employs the Airy stress function. The two-mode approximation of the deflection and application of the Bubnov–Galerkin method reduces the governing system of equations to the system of ordinary differential equations. Varying the load parameters and the nonlocal parameter, the bifurcation analysis is performed. The bifurcations diagrams, the maximum Lyapunov exponents, phase portraits as well as Poincare maps are constructed based on the numerical simulations. It is shown that for some excitation conditions the chaotic motion may occur in the system. Also, the small-scale effects on the character of vibrating regimes are illustrated and discussed.

Chaos and subharmonic bifurcation of a composite laminated buckled beam with a lumped mass

With analytical and numerical methods, dynamic response of a composite laminated buckled beam with a lumped mass and an axial excitation is investigated in this paper. The dynamic equation of the first mode for this model, which is a perturbation of a Hamilton system with both quadratic and cubic nonlinearities as well as external and parametric excitations, is obtained with the Galerkin method. The distribution of the equilibriums, homoclinic orbits and periodic orbits for the unperturbed system under different system parameters are obtained analytically. Chaotic vibrations arising from homoclinic intersections are studied with Melnikov method. Chaotic feature on the location of the lumped mass is investigated in detail. It is presented that there exist controllable frequencies decreasing monotonously with the location of the lumped mass for this system. Subharmonic bifurcation is also studied with the subharmonic Melnikov method. It is numerically proved that the system may undergo chaotic motions through finite subharmonic bifurcations. Numerical simulations including the phase portraits, time histories, Poincare´ sections, Lyapunov spectrums and bifurcation diagram are given, which confirm the analytical results.

Experimental and numerical studies on nonlinear vibrations and dynamic snap-through phenomena of bistable asymmetric composite laminated shallow shell under center foundation excitation

This paper experimentally and numerically investigates the nonlinear vibrations and dynamic snap-through behaviors of the bistable asymmetric carbon fiber-reinforced [90n/0n] composite laminated shallow shell. In the experimental research, in order to produce the dynamic snap-through phenomena, the center of a bistable asymmetric composite laminated shallow shell is clamped on an electromechanical shaker. The shaker provides the controlled frequency and amplitude of the foundation excitation. A laser displacement testing system collects the data of the vibration signals for the bistable asymmetric composite laminated shallow shell. A high-speed camera captures the steady-state vibrations of the bistable asymmetric composite laminated shallow shell. The experimental results demonstrate the influence of the structural parameters on the dynamic snap-through phenomena and nonlinear vibrations of the bistable asymmetric composite laminated shallow shell. The amplitude-frequency response curves are obtained by the experimental results. The bifurcation diagrams, phase portraits, time histories, power spectrums, and Poincaré maps are obtained to experimentally and numerically illustrate the single-well periodic, chaotic and double-well chaotic vibrations of the bistable asymmetric composite laminated shallow shell under the center foundation excitation. In numerical research, ABAQUS/CAE is used to simulate the dynamic responses of the bistable asymmetric composite laminated shallow shell. The subspace iteration method is utilized to give the vibration modal analysis. The vibration modes in the experimental research are qualitatively consistent with the results of the ABAQUS numerical simulation. The phenomenon of the energy transfer from the high-order to low-order primary resonance is discovered. The results of the experiment and numerical simulation are well verified with each other.

Onset of chaos in nano-resonators based on strain gradient theory: Numerical analysis

In recent years, experimental and theoretical studies have shown that size effect is non-negligible in mechanical micro- and nano-structures. As a result, classical continuum theory cannot model these structures. To accurately predict the behavior of micro- and nano-structures, non-classical continuum theories should be used. In this paper, the effect of size on the chaotic region of resonators is studied by comparing the results of classical and strain gradient theories. A bifurcation diagram and Lyapunov exponent are used for detecting chaos in nano-resonators.It is shown that the effect of size on the chaotic region in bifurcation diagram varies significantly depending on the bias voltage. Lyapunov exponent is utilized to validate the chaotic region and it is in best agreement with bifurcation diagram. For some DC voltage values, size effect can eliminate chaotic vibrations while in higher DC voltages, it can result in chaos at much lower amplitudes than those predicted by classical theories. Dynamic behaviors of nano-resonators are categorized based on their bias voltages and their chaotic behaviors are analyzed based on classical and non-classical theories in detail. The obtained numerical results are compared with Melnikov and Maximum Velocity methods and it is shown that all methods predict the lower critical actuation amplitude for strain gradient theory.

Coexistence of double-parameter nonlinear dynamics and metastable chaos for bistable asymmetric composite laminated square panel under combined external and parametric excitations

This paper studies the double-parameter multi-pulse jumping chaotic vibrations and metastable chaos of the bistable asymmetric composite laminated square panel under combined external and parametric excitations for the first time. The double-parameter multi-pulse jumping chaotic motions are studied by using the extended Melnikov method for the bistable asymmetric composite laminated square panel. It is indicated that there exist the Shilnikov type multi-pulse jumping orbits. The occurred mechanism of the metastable chaos is first analyzed. A theoretical explanation is given for coexisting in the double-parameter multi-pulse jumping chaotic vibrations and metastable chaotic behaviors. Using the double-parameter Lyapunov exponents, the double-parameter nonlinear dynamics of the bistable asymmetric composite laminated square panel are analyzed. According to the numerical simulation results, the in-depth investigation of the topological changes is obtained for the double-parameter multi-pulse jumping chaotic vibrations. It is found that the dynamic snap-through phenomena and the coupled effects of the external and parametric excitations on the double-parameter nonlinear dynamic behaviors are obtained for the bistable asymmetric composite laminated square panel. It is also observed that the change of the external excitation can only affect the complexity of the chaotic vibrations. However, the parametric excitations determine the types of the chaotic vibrations. There exists a direct connection between the occurrence of the metastable state chaos and the increase of parametric excitations for the bistable asymmetric composite laminated square panel.

Нестационарное запаздывание установления нелинейных колебаний в системе двух связанных осцилляторов. Часть 1. Общие положения. Формирование упрощенной системы.

Нестационарное запаздывание установления нелинейных колебаний в системе двух связанных осцилляторов. Часть 1. Общие положения. Формирование упрощенной системы.

Нестационарное запаздывание установления нелинейных колебаний в системе двух связанных осцилляторов. Часть 4. Кольцевое взаимодействие.

Нестационарное запаздывание установления нелинейных колебаний в системе двух связанных осцилляторов. Часть 4. Кольцевое взаимодействие.

Нестационарное запаздывание установления нелинейных колебаний в системе двух связанных осцилляторов. Часть 2. Воздействие осцилляторов друг на друга.

Нестационарное запаздывание установления нелинейных колебаний в системе двух связанных осцилляторов. Часть 2. Воздействие осцилляторов друг на друга.

Нестационарное запаздывание установления нелинейных колебаний в системе двух связанных осцилляторов. Часть 3. Определяющее уравнение.

Нестационарное запаздывание установления нелинейных колебаний в системе двух связанных осцилляторов. Часть 3. Определяющее уравнение.

Chaotic Dynamics of a 2D Hyperbolic PDE with the Boundary Conditions of Superlinear Type

The study of the chaotic dynamics of partial differential equation (PDE) system has become the focus of dynamical system. While important progress has been made for the research of chaos theory of one-dimensional wave PDE, the understanding of chaotic vibrations for two-dimensional (2D) hyperbolic PDE is still incomplete. This paper is concerned with a 2D hyperbolic PDE system governed by a linear equation with two suplinear boundary conditions. Based on the chaotic mapping theory and method of characteristics, sufficient conditions for the existence of nonisotropic chaotic vibrations are obtained for such system under three different system parameters. In addition, three numerical examples are provided to illustrate the effectiveness of our theoretical results.

Simulation Analysis of Emulsion Droplet Excited by Chaotic Frequency Pulse Electric Field

The frequency of the chaotic pulse electric field is random in a limited range, which can be adjusted and controlled, thus can effectively cover the harmonic resonance frequency of the droplets in the oil, and realize the efficient coalescence and demulsification of the droplets. In this study, the chaotic frequency pulse electric field is constructed by embedding the chaotic sequence into the pulse interval with the Chaotic-Pulse-Position Modulation method, and the vibration dynamics simulation model of the emulsion droplet in the chaotic frequency electric field is established. The vibration dynamics response of the emulsion droplet under the action of the chaotic frequency pulse electric field is obtained by numerical calculation, and the dynamic analysis, phase diagram analysis and maximum Lyapunov exponent calculation of the response output are carried out. It is found that under the action of chaotic frequency pulse electric field, the emulsion droplet can obtain large tensile deformation at high frequency, but relatively small at low frequency, and the fluctuation is stable, which can qualitatively and quantitatively identify the chaotic vibration characteristics of the droplet.

Dynamic analysis and chaos control of spur gear transmission system with idler

This study aims to analyze the chaotic dynamics and present a chaos controller for spur gear transmission systems with idler. The chaotic dynamics of the spur gear mechanism has already been investigated. However, the presence of idler gears affects the chaotic behavior and the route to chaos for the nonlinear model of spur gears. This is investigated through the derivation of dimensionless dynamics, defining a Poincare’ section, and extracting the bifurcation diagrams of the system for variations of several parameter models. A nonlinear time-varying dynamic model of a spur gear transmission system with idler is established where backlash, time-varying stiffness, static transmission error, and external excitation are included and a region for the occurrence of chaos is found. The chaotic vibration suppression of the system is done by detecting the unstable periodic orbits embedded in the strange attractors and developing control law by employing sliding mode and adaptive sliding mode control strategy. The controller transmits a chaotic trajectory into the detected unstable periodic orbits. Numerical simulations including phase plane portrait, time histories diagrams, Poincare’ sections, and bifurcation diagrams demonstrate the behavior of the system and confirm the performance of the proposed controller.

Nonlinear dynamics of rectangular nano-shells

Mathematical model of non-linear vibrations of shallow, elastic, isotropic nano-shells with rectangular base subjected to transverse sign-variable load are constructed. Based on Kirchhoff-Love thin shell theory with von Kármán nonlinear strains and the modified couple stress theory (MCST), size-dependent governing equations and corresponding boundary conditions are established through Hamilton’s principle. The governing PDEs are reduced to ODEs by the second-order Finite Difference Method (FDM). The obtained system of equations is solved by Runge–Kutta methods of second order accuracy. The Cauchy problem is solved by the Runge–Kutta fourth-order method. We analyzed the convergence of these solutions depending on the step of integration over time and spatial coordinate. It was revealed, that taking into account nano-effects increases area of harmonic vibrations and leads to the appearance of as chaotic and hyperchaotic vibrations. The carried out numerical experiment shows, that the transition of vibrations from harmonic to chaotic follows to Feigenbaum’s scenario. In particular, to analyze the character type of vibration computation of largest Lyapunov exponents are employed. We found that hyperchaotic vibrations are characterized by two positive Lyapunov exponents and chaotic vibration by one positive Lyapunov exponent. For nano-shells, this phenomenon was discovered for the first time. Lyapunov exponents spectra estimated by different algorithms, including Wolf’s, Rosenstein’s, Kantz’s, and Sawada. Numerical examples of the theoretical investigations are given.

On the chaotic and hyper-chaotic dynamics of nanobeams with low shear stiffness.

We construct a mathematical model of non-linear vibration of a beam nanostructure with low shear stiffness subjected to uniformly distributed harmonic transversal load. The following hypotheses are employed: the nanobeams made from transversal isotropic and elastic material obey the Hooke law and are governed by the kinematic third-order approximation (Sheremetev-Pelekh-Reddy model). The von Kármán geometric non-linear relation between deformations and displacements is taken into account. In order to describe the size-dependent coefficients, the modified couple stress theory is employed. The Hamilton functional yields the governing partial differential equations, as well as the initial and boundary conditions. A solution to the dynamical problem is found via the finite difference method of the second order of accuracy, and next via the Runge-Kutta method of orders from two to eight, as well as the Newmark method. Investigations of the non-linear nanobeam vibrations are carried out with a help of signals (time histories), phase portraits, as well as through the Fourier and wavelet-based analyses. The strength of the nanobeam chaotic vibrations is quantified through the Lyapunov exponents computed based on the Sano-Sawada, Kantz, Wolf, and Rosenstein methods. The application of a few numerical methods on each stage of the modeling procedure allowed us to achieve reliable results. In particular, we have detected chaotic and hyper-chaotic vibrations of the studied nanobeam, and our results are authentic, reliable, and accurate.

Design and Dynamic Analysis of a Four‐Degree‐of‐Freedom Chaotic Vibrating Screen

In order to address the problems of low screening efficiency, easy blocking of screen holes, and short service life of key parts commonly used in vibrating screen equipment, the TRIZ (Theory of the Solution of Inventive Problems) was applied in the present work to design a four‐degree‐of‐freedom (4‐DOF; three translational and one rotational movements) chaotic vibrating screen with a chaotic vibration exciter as the main power source and a 3‐DOF (three translational movements) parallel kinetic chain as the kinematic constraint mechanism of the outer screen frame. Based on the topological structure theory, a hybrid mechanism with structure [4SOC{−Ci1∥Ri2∥Ri3−} + R, (i = 1,2,3,4)] was constructed as the kinematic constraint mechanism of the inner screen box of the chaotic vibrating screen to solve the freedom of motion and POC (position and orientation characteristic) equations of parallel kinematic constraint mechanism of the outer screen frame and hybrid constraint mechanism of the inner screen box. The dynamic simulation of a virtual prototype of the chaotic vibrating screen was carried out in ADAMS software, and MATLAB was used to chaos recognition of the simulation results. It was found that the chaotic exciter moved aperiodically in X‐, Y‐, and Z-directions when the chaotic exciter motor rotated at uniform speed, and the amplitude, velocity, and acceleration of the outer screen frame of the vibrating screen had characteristics of reciprocating aperiodic and irregular changes. Through the phase trajectories of the eccentric block and inner screen box of the exciter in all directions, it was observed that the motion output of the vibrating screen was a chaotic vibration. Therefore, the present paper can provide an important reference for the design and application of chaotic vibrating screens.

Нелинейное возбуждение гиперзвуковых колебаний в ферритовой пластине в условиях комбинированного воздействия на двух частотах. Часть 2. Вариация постоянного поля

Нелинейное возбуждение гиперзвуковых колебаний в ферритовой пластине в условиях комбинированного воздействия на двух частотах. Часть 2. Вариация постоянного поля

Chaotic dynamics of the vertical model in vehicles and chaos control of active suspension system via the fuzzy fast terminal sliding mode control

In this paper, the control of chaotic vibrations in vehicle suspension system is studied via a fuzzy fast terminal sliding mode method. Therefore, the nonlinear equations of motion in the vehicle half model are derived initially through the Newton-Euler laws and then these equations are solved by the fourth-order Runge-Kutta method. For analysis of chaos, some techniques including bifurcation diagrams, frequency response, power spectrum, phase plane trajectories, and Poincare section are used to identify the chaotic behaviors. The chaotic zones are depicted with critical values on the uncontrolled model under the road surface force. In order to eliminate chaotic behavior, the control signals in the active suspension system are generated using the novel fuzzy fast terminal sliding mode control algorithm. The simulation results of the feedback system show that the suspension system can be stabilized the vibrations by efficient fuzzy fast terminal SMC while eliminating irregular chaotic behaviors.