Chaotic Vibrations Research Articles

Nonlinear and chaotic vibrations of rotating functionally graded GPL reinforced composite pre-twisted blade subjected to aerodynamic force

The nonlinear and chaotic vibrations of the rotating functionally graded graphene platelet reinforced composite (FG-GPLRC) pre-twisted blade subjected to the aerodynamic force are investigated by taking into account the cantilever boundary condition. The rotating FG-GPLRC pre-twisted blade is simplified to a rotating FG-GPLRC pre-twisted cylindrical panel. The graphene platelets (GPLs) are dispersed uniformly and oriented randomly in every layer while the GPL volume fractions vary from the layer to layer, which leads to four different GPL distribution patterns. Based on the strain–displacement relationship of the pre-twisted cylindrical panel derived by Green strain tensor, the natural frequencies of the rotating FG-GPLRC pre-twisted cylindrical panel are obtained by using the first-order shear deformation theory and Chebyshev–Ritz method. Lagrange’s formulation is used to derive the nonlinear ordinary differential equations of motion. The complex chaotic vibrations of the rotating FG-GPLRC pre-twisted cylindrical panel are investigated by performing numerical simulations on the basis of Runge–Kutta algorithm. The bifurcation diagrams, Lyapunov exponents, time histories, phase portraits and Poincare maps are depicted to detect the periodic and chaotic vibrations of the rotating FG-GPLRC pre-twisted cylindrical panel with different parameters, such as Coriolis force, GPL distribution pattern, steady-state rotating speed, perturbation rotating speed and aerodynamic force. The double-parameter maximum Lyapunov exponent is regarded as an effective method to detect the chaotic regions of the rotating FG-GPLRC pre-twisted cylindrical panel.

New Results of Some of the Conformable Models Arising in Dynamical Systems

This article investigates the new results of three nonlinear conformable models (NLCMs). To study such varieties of new soliton structures, we perform the generalized Kudryashov (GK) method. The obtained new results are defined in the styles of the exponential and rational functions. The derived new soliton structures are stable, serviceable, and fitting to embrace the conformable dynamics, chaotic vibrations, global bifurcations, optimal control problems, fluid mechanics, plasma physics, system identification, local bifurcations, control theory and resonances, and so many. The outcomes declare that the process is hugely valuable and accessible for investigating nonlinear conformable order models treated in theoretical physics.

Chaotic vibration of atomic force microscopes based on the modified couple stress theory

Chaotic vibration of atomic force microscopes based on the modified couple stress theory

Chaotic vibration analysis of auxetic nanocomposite structures

The present work deals with the study of nonlinear vibration behaviours of the carbon nanotube reinforced composite structures having negative Poisson’s ratio (CNTRC-NPR). Such types of flexible auxetic structures have many applications in various engineering fields. Based on classical laminated theory, auxetic structures can be designed by some special stacking sequence of layers in the laminated composites. A mathematical model is formulated for nonlinear vibration analysis of the clamped-clamped CNTRC-NPR beam under the time varying uniformly distributed transverse load. The present formulation is based on the Reddy’s third-order shear deformation theory and the governing equation of motion of such beam is derived by using the Hamilton’s principle. Based on the Galerkin’s method, the obtained governing partial differential equations are converted to an ordinary nonlinear second-order differential equation for the analysis of out plane vibration. The obtained vibrations of such auxetic structures are discussed using time series waveforms, phase portraits and Poincare maps, and the route of chaotic vibration of such structures is found and reported based on the bifurcation diagrams. The jump phenomena associated with the dynamics have also been characterized. Nonlinear vibrations of without and with auxetic nanocomposite structures have also been compared. It is found that the amplitude of transverse force of excitation is the controlling parameter of nonlinear dynamics of such CNTRC-NPR structures.

Piezoelectric energy harvester featuring a magnetic chaotic pendulum

Chaos is usually accompanied by broadband properties of nonlinear energy conversion. To investigate the effects and potential of chaotic vibration in enhancing energy harvesting technologies, this study presents a piezoelectric wind energy harvester with a magnetic chaotic pendulum. The proposed structure is mainly composed of a piezoelectric clamped-clamped beam and a magnetic chaotic pendulum attached to a radial wind turbine. A chaotic pendulum is a double pendulum consisting of two pendula that swing freely in the vertical plane and are connected by a pin joint. The magnet is fixed at the end of the second pendula. When the chaotic pendulum rotates with the wind turbine, the magnetic interaction between the magnet of the second pendula and the magnet in the middle of the clamped-clamped piezoelectric beam produces chaotic beam vibration for nonlinear energy harvesting. Since the energy harvesting system of the proposed structure is chaotic, particle swarm optimization combined with a neural network was adopted to optimize the output power of the proposed structure. The results showed that the proposed structure automatically adapts to the natural wind environment. The working flow velocity range can be increased compared to that of an energy harvester with a traditional single pendulum. An experimental maximum output power of 1.25 mW can be achieved at 557.31 rpm in the investigated flow velocity range.

Nonlocal chaotic analysis of double layered viscoelastic nanoplates embedded in Winkler–Pasternak elastic foundation and under in‐plane biaxial loads

AbstractIn this paper, based on the extended Melnikov method the homoclinic analysis for chaotic vibration of a double layered viscoelastic nanoplates (DLNPs) embedded in Winkler–Pasternak elastic foundation and under in‐plane excited biaxial loads using the Eringen's nonlocal elasticity theory has been investigated. The governing nonlinear coupled partial differential equations (PDEs) of motion for DLNP considering viscoelastic Kelvin–Voigt model in the Kirchhoff's thin plate theory in conjunction with the von‐Karman geometrical nonlinear strain–displacement relations have been derived. Then, applying the Galerkin method the governing nonlocal coupled PDEs of motion of DLNP have been converted into the nonlinear coupled ordinary differential equations (ODEs) using mode summation technique and discretized in terms of time. The criteria for the homoclinic transverse intersection for synchronous and asynchronous types of buckling are studied to predict the homoclinic orbits and chaotic motions applying the generalized Melnikov method. Besides, the critical buckling load has been determined for asynchronous and synchronous types of buckling. The obtained results are validated with comparing them with the results presented in the literature for a special case. Then, the effect of changes of different parameters including nonlocal small‐scale parameter, foundation coefficients, biaxial load ratio, geometric aspect ratio on the critical buckling load and ratio of nonlocal to local natural frequency have been studied. The critical threshold surface for the synchronous and asynchronous manifolds to predict transversely condition for the chaotic motion are discussed on the parametric space.

Cross-scale energy transfer of chaotic oscillator chain in stiffness-dominated range

The investigations on the dynamics of hierarchical oscillator chains provide basic theories for the fields of acoustic metamaterial and passive guidance of energy flow. In the present study, the characteristics of stiffness-dominated energy transfer of a reduced cross-scale oscillator chain that generates chaotic vibrations are studied. The semi-analytical solutions are obtained by the complexification-averaging method and the least-square method for thoroughly searching the branches of responses. Then, the energy transfer patterns of the oscillator chain are analyzed based on Runge–Kutta method. Moreover, chaotic responses are determined using the displacements and Lyapunov exponents of the system. The average energy and normalized energy flux are defined to evaluate the energy distribution in each oscillator and the energy transfer between oscillators, respectively. The semi-analytical results show that the oscillator chain produces complex unstable branches of response around the resonance frequency of the linear oscillator. It is found that the detached higher branches of response of the chain are similar to that of a system consisting of one linear oscillator and one cubic oscillator. When chaotic responses occur, the energy transfer patterns that reflected by average energy and normalized energy flux are similar in the same branch of response for different excitation frequencies. However, energy transfer patterns in different branches of response are quite different even at the same excitation frequency, and the higher branch generates the lower energy transfer. Furthermore, analyses of the input energy and the scale between cubic oscillators demonstrate that variations of the two parameters slightly affect the normalized energy flux.

Analysis of bifurcation and chaotic behavior for the flexspline of an electromagnetic harmonic drive system with movable teeth transmission

To develop the miniaturization of the electromagnetic harmonic movable tooth drive system, a novel configuration with an inner stator is proposed to create a great compressed structure. To avoid undesirable design parameters that lead to bifurcation and chaotic behavior of the flexspline, the nonlinear dynamic equations of the flexspline are established, and the Duffing equation and analysis results for the chaotic vibration of the flexspline are obtained by using the Donnell-Karman theory of thin-walled cylindrical shells with employing large deflection, the Bubnov-Galerkin principle, and the Melnikov function, separately. According to initial system parameters, based on the bifurcation diagrams, phase diagrams, displacement time course diagrams, and Poincare maps of the flexspline vibration, the dynamic behaviors are investigated, and the stability and chaos intervals are obtained. This study aims to reveal the influence of parameters on the chaotic phenomenon of the flexspline and to provide a theoretical reference for the design of the electromagnetic movable teeth drive.

Revisited chaotic vibrations in dielectric elastomer systems with stiffening

Revisited chaotic vibrations in dielectric elastomer systems with stiffening

Nonlinear dynamic analysis of constant-speed and variable-speed of autonomous vehicle passing uneven road

This paper focuses on the research about the comfort and safety of autonomous vehicles passing consecutive speed control humps (SCHs) at a highway, compares and analyzes the nonlinear dynamic characteristics of vehicles driving on uneven roads at a constant speed or with accelerations or decelerations. Firstly, a speed coupling excitation model of the consecutive SCHs and a four-degree-of-freedom (4-DOF) nonlinear vehicle suspension model are established, and then the numerical simulation is carried out by using the four-order fixed-step Runge-Kutta algorithm. The nonlinear dynamic characteristics are analyzed by means of a bifurcation diagram, phase portrait, Poincaré map, time history and power spectrum. Through the analysis, the relevant nonlinear motion characteristics and the speed critical conditions of chaotic vibration of autonomous vehicle suspension under these three conditions are obtained. The results show that chaotic motion may occur when the vehicle passes the uneven road in all three cases, and motion is not periodic during acceleration. Therefore, the vehicle should not accelerate passing continuous SCHs. The experimental results provide support for the follow-up researches of autonomous vehicle, because they give information on how to avoid chaotic motion, speed adaptive control, driving comfort and driving safety.

The effect of dynamic normal force on the stick–slip vibration characteristics

In the experiment, we observed such a phenomenon: the alternating normal force changes the vibration state of a friction system. A single-degree-of-freedom mathematical model was used in this paper to discuss the effects of a constant and alternating normal force on the stick–slip vibration characteristics for different dynamic and static friction coefficients. Under the condition that the applied constant normal force continues to increase, the vibration amplitude of the system, the amplitude of the limit cycle, and the adhesion time of the system increase. When the difference between the dynamic and static friction coefficients (DSFCs) is small, the system has a complete and clear limit cycle. When the dynamic friction coefficient is reduced, the difference between DSFCs increases, and the limit cycle of the system is deformed. The friction system has more abundant dynamic vibration characteristics under an alternating normal force than a constant normal force. The vibration state of the system presents a single-cycle stick–slip vibration when the alternating normal force excites the multi-order harmonic response of the friction system, and the excitation frequency of the alternating normal force is the same as the main response frequency of the system with the highest energy or the low-order even-order main frequency. In contrast, the system exhibits various vibration modes when the excitation frequency of the alternating normal force is dissimilar to the main frequency of the system's highest energy response or is consistent with the odd-order main frequency. In addition, increasing the difference between DSFCs or using very high excitation frequencies and excitation amplitudes increases the likelihood of the system entering a chaotic vibration state.

CALCULATION OF A TRAJECTORY FOR THE MOVEMENT OF A LOAD OF SWINGING SPRING

Pendulum oscillations in the vertical plane of a suspended weightless spring, while maintaining the straightness of its axis, are considered. In the literature, this type of pendulum is called a swinging spring. The required trajectory of the load of the swinging spring is modeled using a computer using the values ​​of the mass of the load, the stiffness of the spring and its length without load. In addition, the initial values ​​of the parameters for initiating oscillations of the oscillating spring are used: the initial angle of deviation of the spring axis from the vertical, the initial rate of change of this angle, as well as the initial parameter of the spring elongation and the initial rate of change of the elongation. The calculations were performed using the Lagrange equation of the second kind. Variants of finding conditionally periodic trajectories of movement of a point load of a swinging spring with a movable attachment point are considered.
 The relevance of the topic is determined by the need to research and improve new technological schemes of mechanical devices, which include springs. In particular, the study of the conditions for separating from chaotic vibrations of mechanical structure elements and determining the rational values ​​of parameters to ensure periodic trajectories of their vibrations.
 Calculation options are given to obtain periodic trajectories of cargo movement, when the given parameters are:
 - the length of the spring without load and its stiffness with an unknown value of the mass of the load;
 - the length of the spring without load and the value of the mass of the load with an unknown spring stiffness;
 - the value of the mass of the load and the stiffness of the spring with an unknown length of the spring without load.
 The results can be used as a paradigm for studying nonlinear coupled systems, as well as for calculating options for mechanical devices, when it is necessary to dissociate themselves from chaotic movements of loads, and provide periodic trajectories of their movement.

Nonlinear Vibration and Stability of a Dielectric Elastomer Balloon Based on a Strain-Stiffening Model

Limiting chain extensibility is a characteristic that plays a vital role in the stretching of highly elastic materials. The Gent model has been widely used to capture this behaviour, as it performs very well in fitting stress-stretch data in simple tension, and involves two material parameters only. Recently, Anssari-Benam and Bucchi (Int. J. Non. Linear. Mech. 128:103626, 2021) introduced a different form of generalised neo-Hookean model, focusing on the molecular structure of elastomers, and showed that their model encompasses all ranges of deformations, performing better than the Gent model in many respects, also with only two parameters. Here we investigate the nonlinear vibration and stability of a dielectric elastomer balloon modelled by that strain energy function. We derive the deformation field in spherical coordinates and the governing equations by the Euler-Lagrange method, assuming that the balloon retains its spherical symmetry as it inflates. We consider in turn that the balloon is under two types of voltages, a pure DC voltage and an AC voltage superimposed on a DC voltage. We analyse the dynamic response of the balloon and identify the influential parameters in the model. We find that the molecular structure of the material, as tracked by the number of segments in a single chain, can control the instability and the pull-in/snap-through critical voltage, as well as chaos and quasi-periodicity. The main result is that balloons made of materials exhibiting early strain-stiffening effects are more stable and less prone to generate chaotic nonlinear vibrations than when made of softer materials, such as those modelled by the neo-Hookean strain-energy density function.

Nonlinear and chaotic vibrations of FG double curved sandwich shallow shells resting on visco-elastic nonlinear Hetenyi foundation under combined resonances

In this study, the nonlinear and chaotic instability of functionally graded (FG) double curved shallow sandwich shells resting on a viscoelastic Hetenyi foundation, under simultaneous effect of in-plane and transverse excitations is studied. Employing the third-order Reddy theory and von-Karman relations and the Hamilton principle, the partial differential equations of motion under movable SS boundary conditions are derived. Introducing the trigonometric Airy stress function and applying the Galerkin’s method, the equations are reduced to a set of nonlinear ODEs with time. Then, under forcing resonance conditions and using the perturbation method, the modulation equations at the stationary conditions are derived and solved numerically. Then stability of non-trivial solutions for the resonance amplitude corresponding to the presence of limit cycle oscillation is investigated. Then by extracting the characteristic resonance amplitude curves, the effect of various parameters, including: frequency detuning parameter, in-plane excitation amplitude, linear, shear and nonlinear stiffness and damping parameters of the foundation on nonlinear response are analyzed. Finally, by extracting two degrees of freedom system time response, bifurcation and chaotic characteristic curves of the problem, the conditions for occurrence the periodic, double periodic, multi-periodic and chaotic behaviors under the simultaneous effect of parametric and internal resonances are studied comprehensively.

Relationship between chaotic vibrations and acoustic properties of percussion cymbals

Cymbals are a type of percussion instrument that have existed for a long time. However, the mechanism of sound generation has not yet been clarified, and there is no clear standard for the production process. Therefore, cymbals have been manufactured and processed using the experience of craftsmen to the present day. In this study, we evaluated and compared the vibration and acoustic characteristics of cymbals manufactured by different manufacturers and processing methods to clarify the sound generation mechanism of percussion cymbals. The cymbals were vibrated in the horizontal direction using a shaker, and various vibration evaluations using the Poincaré section and correlation dimensions, along with various psychoacoustic evaluations using loudness curves and sharpness, were used to evaluate and compare the cymbals. Consequently, it was clarified that the route to the chaotic vibration of cymbals was determined and that there was a correlation between the correlation dimension and sharpness.

Effects of upstream wake on vortex-induced vibrations and wake patterns of side-by-side circular cylinders

In this paper, vortex-induced vibrations and wake patterns of two side-by-side circular cylinders in the wake of a stationary upstream cylinder are numerically investigated for a uniform free stream flow at the Reynolds number of 100. The three cylinders are in equilateral triangle arrangements with a center-to-center spacing ratio of 2.0–5.0. The considered mass ratio is 2.0 and the reduced velocity range is 0.5–15.0. The side-by-side cylinders are allowed to vibrate only in the transverse direction. Variations of response amplitudes, frequencies, mean position shifts and wake patterns with the reduced velocity and spacing ratio are investigated using direct numerical simulations. In comparison with the case without an upstream cylinder, amplitudes, frequencies, mean position shifts and hydrodynamic forces of two side-by-side cylinders are significantly impacted by the upstream cylinder wake and the associated wake-vortex-structure interactions. Key observations include (a) vibrations initiated at higher reduced velocities for a small to medium spacing ratio, (b) a wider range of large-amplitude vibrations, (c) a lower-frequency component, (d) appearances of periodic, quasi-periodic, chaotic and asymmetric vibrations, and (e) a hysteresis phenomenon. The asymmetric vibration is characterized by dissimilar vibration amplitudes of the two cylinders whereas the hysteresis is related to a transition of wake patterns when increasing versus decreasing the reduced velocities. Behind two side-by-side circular cylinders, the flip-flopping, anti-phase synchronized, in-phase synchronized, and chaotic patterns are identified.

Nonlinear supersonic flutter of sandwich truncated conical shell with flexible honeycomb core manufactured by fused deposition modeling

The self-sustained vibrations of sandwich conical shell with a flexible honeycomb core manufactured by fused deposition modeling are studied. The interaction of the sandwich thin-walled structure with supersonic gas flow is analyzed. The geometrical nonlinearity of the shell structure is taken into account to predict the self-sustained vibrations. Vibrations of the structure layer are described by three displacements of the middle surface of each layer and two rotations of the normal to the middle surface. The higher-order shear deformation theory is used to describe the strain–displacement relationships. The self-sustained vibrations of the sandwich conical shell are described by a system of nonlinear ordinary differential equations with respect to the generalized coordinates. The assumed mode method is used to derive the equations.The bifurcations of the nonlinear self-sustained vibrations are analyzed numerically using a continuation technique. Quasi-periodic and chaotic self-sustained vibrations are numerically studied for the shell subjected to different boundary conditions. Numerical results show that the amplitudes of quasi-periodic and chaotic vibrations are significantly larger than the amplitudes of periodic vibrations for this shell.

Nonlinear vibration analysis of CNT-reinforced functionally graded composite cylindrical shells resting on elastic foundations

Nonlinear vibration analysis of carbon-nanotube reinforced functionally graded composite (FG-CNTRC) cylindrical shells resting on elastic foundations is carried out. Four carbon nanotube distributions, namely UD, FGV, FGΛ and FGX, are considered and the elastic foundation is described using the Pasternak model. According to the first-order shear deformation shell theory, the nonlinear governing equations of FG-CNTRC cylindrical shells are derived from Lagrange equations. The forced nonlinear vibrations of FG-CNTRC cylindrical shells with/without surrounding elastic media are then investigated using the harmonic balance method combined with the arc length continuation technique, and chaotic vibrations due to the 1:1 internal resonance are further studied through direct integration. Finally, the effects of the distribution patterns and the stiffness of elastic foundations on the nonlinear behaviors of FG-CNTRC cylindrical shells are presented in detail.

Mathematical modeling of nonlinear thermodynamics of nanoplates

The theory of nonlinear statics and dynamics of flexible plates, taking into account the modified couple stress theory and temperature field, is developed. The theory is based on the classical Kirchhoff model for an isotropic elastic body. Geometric nonlinearity is taken into account according to the von Kármán model. Variational differential equations are yielded by the Hamilton principle, and partial differential equation (PDE's) related to displacements of the middle surface and deflection are derived. The hypotheses of the modified couple stress theory implied an increase of the order of the system of partial differential equations due to the fact that moments of higher order appeared. The resolved PDE's are reduced to the Cauchy problem by the method of finite differences of the second order accuracy, which is solved by methods of the Runge-Kutta type (from the fourth to the eighth order of accuracy) and the Newmark method. No restrictions are imposed on the temperature field, and it determined from the solution of the three-dimensional heat equation using the finite element method (FEM). The convergence of the proposed algorithms is investigated depending on the number of finite elements and the time step. Static problems are solved by the dynamic approach. Nonlinear dynamics is analyzed based on (Fourier spectrum, wavelets of different types and phase portraits). The analysis of Lyapunov exponents obtained by different methods (Kantz, Wolf, Rosenstein, Sano-Sawada and the authors method) is carried out, which validated occurrence of chaotic vibrations. Hyperchaotic vibrations have been detected and studied. It is also illustrated how increase of the temperature field influences localization of the chaotic zones for size-dependent plates.

Nonlinear vibrations and dynamic snap-through behaviors of four-corner simply supported bistable asymmetric laminated composite square shell

The nonlinear vibrations and dynamic snap-through behaviors are studied for a four-corner simply supported bistable asymmetric laminated composite square shell under the foundation excitation by using the theoretical, Abaqus finite element (FE) and experiment approaches. The first-order shear deformation shell theory and Hamilton principle are adopted to derive the partial differential governing equations of motion. A novel type of nonlinear strain-displacement relations is given by Sander’s strain. Galerkin approach is utilized to discretize the partial differential equations of motion into a three-degree-of-freedom system. The natural frequencies and vibration modes of the bistable shell are calculated by using Chebyshev polynomial method. The nonlinear vibrations and dynamic snap-through are investigated by using the maximum Lyapunov exponent, bifurcation diagrams, time histories, phase portraits, 3D phase portraits and Poincare maps. The theoretical results of the vibrations are well compared with those of the FE and experiment. The effects of the excitation and structural parameters on the dynamic snap-through and nonlinear vibrations are fully discussed. The influences of the initial radius of the curvature on the critical load of the dynamic snap-through behaviors and rule of the chaotic vibrations are analyzed for the bistable asymmetric laminated composite shell.