Extended Melnikov Method Research Articles

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.

Resonant responses and double-parameter multi-pulse chaotic vibrations of graphene platelets reinforced functionally graded rotating composite blade

The applications of the new graphene platelet reinforced functionally graded (GPR-FG) composite materials are helpful to improve the overall performance of the rotating blade. However, the new GPR-FG rotating composite blade may produce the strong nonlinear vibrations under the coupling effects of the aerodynamic force and disturbed speed, which are one of the main reasons leading to the overall failure of the rotating blade. Therefore, it is of great value to analyze the nonlinear dynamics of the GPR-FG rotating composite blade under the aerodynamic force and varying speed. Based on the nonlinear dynamic model of the GPR-FG rotating composite blade, the resonant responses, global bifurcations and double-parameter multi-pulse chaotic vibrations of the rotating composite blade are studied for the first time. The prediction-correction continuation algorithm is used to obtain the amplitude-frequency response curves and force-amplitude curves of the GPR-FG rotating composite blade. It is found that there are the sudden changes in the resonant responses and complex nonlinear vibrations in the GPR-FG rotating composite blade. The extended Melnikov method is applied to study the global bifurcations and double-parameter multi-pulse chaotic vibrations of the GPR-FG rotating composite blade under the coupling excitations of aerodynamic force and disturbed speed. Through theoretical analysis and numerical simulation, we find that there are the complex double-parameter multi-pulse chaotic vibrations of the GPR-FG rotating composite blade. These results have the theoretical significance for the safe operation of the new GPR-FG rotating composite blade.

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.

Theoretical and experimental studies of global dynamics for a class of bistable nonlinear impact oscillators with bilateral rigid constraints

A new bistable impact oscillator with bilateral rigid constraints under periodic excitations is established and the global dynamics are studied in detail respectively by the extended analytical Melnikov method for non-smooth systems and dynamical experiments. Firstly, the Melnikov method is extended from a new viewpoint of geometry for an abstract non-smooth dynamical system denoting a class of bistable impact oscillators with bilateral rigid constraints. Secondly, the analytical expression of homoclinic orbits is difficult to obtain, a semi-analytical and semi-numerical method for calculating the Melnikov function is applied to obtain the threshold of parameters for the global bifurcations and chaotic oscillations of the established impact oscillator. Then, numerical simulations and dynamical experiments are carried out together to show intra-well periodic oscillations, inter-well periodic or chaotic oscillations under different amplitudes of periodic excitations, which furthermore validates the reliability of the extended Melnikov method for this class of non-smooth systems.

Multi-pulse jumping double-parameter chaotic dynamics of eccentric rotating ring truss antenna under combined parametric and external excitations

We study the multi-pulse jumping double-parameter homoclinic orbits and chaotic dynamics of the eccentric rotating ring truss antenna under combined the parametric and external excitations for the first time. Considering combined the parametric and external excitations, the averaged equations of the eccentric rotating circular cylindrical shell are obtained in the case of the primary parametric resonance-1/2 sub-harmonic resonance and 1:2 internal resonance by using the perturbation analysis. From the averaged equation, the theory of normal form is applied to find the explicit expression of the eccentric rotating circular cylindrical shell under combined the parametric and external excitations. Based on the extended Melnikov method, we study the multi-pulse double-parameter homoclinic bifurcation and chaos of the eccentric rotating circular cylindrical shell under combined the parametric and external excitations and also estimate the double-parameter chaotic threshold. Numerical simulations are utilized to study double-parameter chaotic dynamic behaviors of the eccentric rotating circular cylindrical shell under combined the parametric and external excitations based on the double-parameter Lyapunov exponents. When one of the excitations is constant, it is found that the temperature parametric excitation directly decides the occurrence of the chaotic motion and the external excitation determines the paths to chaos. It is also known that the topology shapes of the chaotic motions are similar when the excitations corresponding to the chaotic motions change in a small range. Otherwise, there exists the obvious difference of the shapes for the topological structure of the double-parameter chaotic motions in the eccentric rotating circular cylindrical shell.

Bifurcation and number of subharmonic solutions of a 2n-dimensional non-autonomous system and its application

This paper is devoted to consider the existence and bifurcation of subharmonic solutions of two types of 2n-dimensional nonlinear systems with time-dependent perturbations. When the unperturbed system is a Hamiltonian system, we obtain the extended Melnikov function by means of performing the curvilinear coordinate frame and constructing a Poincare map. Then some conditions of the bifurcation of subharmonic solutions are obtained. The results obtained in this paper contain and improve the existing results for $$n=2,3$$ . When the unperturbed system contains an isolated invariant torus, we investigate the bifurcation of subharmonic solutions by analyzing the Poincare map. We apply the extended Melnikov method to study the bifurcation and number of subharmonic solutions of the ice-covered suspension system. The maximum number of subharmonic solutions of this system is 2, and the relative parameter control condition is obtained.

Nonlocal nonlinear chaotic and homoclinic analysis of double layered forced viscoelastic nanoplates

The homoclinic phenomena and chaotic motions of the forced double layered nanoplates (DLNP) are investigated. By the nonlocal theory, the nonlinear equations of motion of DLNP subjected to transverse harmonic excitation are established. The buckled DLNP considered herein means that the parameter regime of the first mode are always unstable. It should be emphasized that the homoclinic Melnikov function is related to the nonlinear term which is affected by the boundary conditions. Hence two different boundary conditions, i.e. simply supported with movable and immovable edges are compared herein. The extended Melnikov method is employed to discuss the homoclinic phenomena and chaotic motions of the DLNP system. The criteria for the homoclinic motions of the four buckling cases are established. Then the results by the above global perturbation analyses are verified by the numerical integration evidences including Lyapunov exponential spectrums, Fourier spectrums and Poincaré sections. The influences of the structural parameters such as small scale effect and boundary conditions on the homoclinic behaviors are mainly discussed. From the results, the most remarkable fact can be seen that the rotary inertia term could break the symplectic symmetry of the unperturbed vibration system. The parametric regime where the chaotic motion may appear shrink with the increase of the nonlinear term r1, which means chaotic motion more likely appear for immovable edges than those with movable edges. Parametric regime where the transverse homoclinic phenomenon appears will decrease with the increase of the small scale parameter. And the homoclinic phenomena more likely appear in higher mode vibration.

Multipulse Orbits and Chaotic Dynamics of an Aero-Elastic FGP Plate Under Parametric and Primary Excitations

The multipulse global bifurcations and chaotic dynamics of a simply supported Functionally Graded Piezoelectric (FGP) rectangular plate with bonded piezoelectric layer are investigated with the case of 1:2 internal resonance and primary parametric resonance. Based on the averaged equations obtained, the theory of normal form is utilized to obtain the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. According to the explicit expressions of normal form, the extended Melnikov method developed by Camassa et al. is employed to study the Shilnikov-type multipulse homoclinic bifurcations and chaotic dynamics of the aero-elastic FGP plate. The analytical results indicate that there exists the Shilnikov-type multipulse chaotic dynamics for the FGP plate. Numerical simulations are presented to show that for the FGP plate, the Shilnikov-type multipulse chaotic motions can occur. The influence of the in-plane excitation and the piezoelectric voltage excitation to the system dynamic behaviors is also discussed by numerical simulations. The results obtained here imply the existence of chaos in the sense of the Smale horseshoes for the FGP plate.

Nonlocal effect on the nonlinear dynamic characteristics of buckled parametric double-layered nanoplates

The homoclinic phenomena in double-layered nanoplates (DLNP) are investigated. Based on the nonlocal continuum theory, the nonlinear dynamical equations for DLNP subjected to in-plane excitation are derived by double-mode Galerkin truncation. The extended Melnikov method is utilized to discuss the homoclinic phenomena and chaotic motion for the buckled DLNP system. The criterions for the existence of transverse homoclinic orbits are established under different four buckling cases (i.e., the first- and second-type synchronous buckling as well as asynchronous buckling). And the results derived by the above-mentioned analysis are verified by molecular dynamics and Lyapunov exponent spectrum. Small-scale effect on the homoclinic motion is mainly inspected. From the result, it is rather novel that the transversality condition is independent of the nonlinear terms in the equations. This fact means that it is not necessary to distinguish whether the boundaries are movable or immovable for the in-plane parametric excitation DLNP. The parametric regime where homoclinic phenomena appear shrinks with the augment of nonlocal parameter for the first-type synchronous buckling case. However, this trend is just opposite for the other three buckling cases. Finally, it can be seen that homoclinic phenomena more likely occur on Mode I (i.e., lower mode) for the second-type synchronous and asynchronous buckling cases. However, the homoclinic motion for the first-type asynchronous buckling most likely takes place on Mode III.

Homoclinic behaviors and chaotic motions of double layered viscoelastic nanoplates based on nonlocal theory and extended Melnikov method

The nonlinear dynamical equations are established for the double layered viscoelastic nanoplates (DLNP) subjected to in-plane excitation based on the nonlocal theory and von Kármán large deformation theory. The extended high dimensional homoclinic Melnikov method is employed to study the homoclinic phenomena and chaotic motions for the parametrically excited DLNP system. The criteria for the homoclinic transverse intersection for both the asynchronous and synchronous buckling cases are proposed. Lyapunov exponents and phase portraits are obtained to verify the Melnikov-type analysis. The influences of structural parameters on the transverse homoclinic orbits and homoclinic bifurcation sets are discussed for the two buckling cases. Some novel phenomena are observed in the investigation. It should be noticed that the nonlocal effect on the homoclinic behaviors and chaotic motions is quite remarkable. Hence, the small scale effect should be taken into account for homoclinic and chaotic analysis for nanostructures. It is significant that the nonlocal effect on the homoclinic phenomena for the asynchronous buckling case is quite different from that for the synchronous buckling case. Moreover, due to the van der Walls interaction between the layers, the nonlocal effect on the homoclinic behaviors and chaotic motions for high order mode is rather tiny under the asynchronous buckling condition.

Chaotic dynamics of the vibro-impact system under bounded noise perturbation

In this paper, chaotic dynamics of the vibro-impact system under bounded noise excitation is investigated by an extended Melnikov method. Firstly, the Melnikov method in the deterministic vibro-impact system is extended to the stochastic case. Then, a typical stochastic Duffing vibro-impact system is given to application. The analytic conditions for occurrence of chaos are derived by using the random Melnikov process in the mean-square-value sense. In addition, the numerical simulations confirm the validity of analytic results. Also, the influences of interesting system parameters on the chaotic dynamics are discussed.

Multipulse Chaotic Dynamics of Six-Dimensional Nonautonomous Nonlinear System for a Honeycomb Sandwich Plate

This paper studies the global bifurcations and multipulse chaotic dynamics of a four-edge simply supported honeycomb sandwich rectangular plate under combined in-plane and transverse excitations. Based on the von Karman type equation for the geometric nonlinearity and Reddy's third-order shear deformation theory, the governing equations of motion are derived for the four-edge simply supported honeycomb sandwich rectangular plate. The Galerkin method is employed to discretize the partial differential equations of motion to a three-degree-of-freedom nonlinear system. The six-dimensional nonautonomous nonlinear system is simplified to a three-order standard form by using the normal form method. The extended Melnikov method is improved to investigate the six-dimensional nonautonomous nonlinear dynamical system in a mixed coordinate. The global bifurcations and multipulse chaotic dynamics of the four-edge simply supported honeycomb sandwich rectangular plate are studied by using the improved extended Melnikov method. The multipulse chaotic motions of the system are found by using numerical simulation, which further verifies the result of theoretical analysis.

Multipulse Chaotic Dynamics for Nonautonomous Nonlinear Systems and Application to a FGM Plate

The extended Melnikov method is improved to investigate the nonautonomous nonlinear dynamical system in Cartesian coordinate. The multipulse chaotic dynamics of a simply supported functionally graded materials (FGM) rectangular plate subjected to transversal and in-plane excitations is investigated in this paper for the first time. The formulas of the FGM rectangular plate are two-degree-of-freedom nonautonomous nonlinear system with coupling of nonlinear terms including several square and cubic terms. The extended Melnikov method is improved to enable us to analyze directly the nonautonomous nonlinear dynamical system of the simply-supported FGM rectangular plate. The results obtained here indicate that multipulse chaotic motions can occur in the simply-supported FGM rectangular plate. Numerical simulation is also employed to find the multipulse chaotic motions of the simply-supported FGM rectangular plate.

Bifurcations and chaotic thresholds for the spring-pendulum oscillator with irrational and fractional nonlinear restoring forces

Nonlinear dynamical systems with irrational and fractional nonlinear restoring forces often occur in both science and engineering, and always lead to a barrier for conventional nonlinear techniques. In this paper, we have investigated the global bifurcations and the chaos directly for a nonlinear system with irrational and fractional nonlinear restoring forces avoiding the conventional Taylor's expansion to retain the natural characteristics of the system. By introducing a particular dimensionless representation and a series of transformations, the two-degree-of-freedom system can be transformed into a perturbed Hamiltonian system. The extended Melnikov method is directly used to detect the chaotic threshold of the perturbed system theoretically, which overcomes the barrier caused by solving theoretical solution for the homoclinic orbit of the unperturbed system. The numerical simulations are carried out to demonstrate the complicated dynamics of the nonlinear spring-pendulum system, which show the efficiency of the criteria for chaotic motion in the system.

Multi-Pulse Chaotic Dynamics of a Composite Laminated Plate with Strong Coupling

In this paper, the multi-pulse chaotic dynamics of a simply-supported symmetric cross-ply composite laminated rectangular plate with the parametric and forcing excitations is investigated by using the extended Melnikov method. The two-degree-of-freedom non-autonomous nonlinear dynamical system of the plate with strong coupling is considered. The results obtained here indicate that multi-pulse chaotic motions can occur in the plate. Numerical simulation is also employed to find the multi-pulse chaotic motions of the plate based on the theoretical analysis.

Multi-pulse homoclinic orbits and chaotic dynamics of a parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities

In this paper, the complicated dynamics and multi-pulse homoclinic orbits of a two-degree-of-freedom parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities are studied. The damping, parametrical excitation and the nonlinearities are regarded as weak. The averaged equation depicting the fast and slow dynamics is derived through the method of multiple scales. The dynamics near the resonance band is revealed by doing a singular perturbation analysis and combining the extended Melnikov method. We are able to determine the criterion for the existence of the multi-pulse homoclinic orbits which can form the Shilnikov orbits and give rise to chaos. At last, numerical results are also given to illustrate the nonlinear behaviors and chaotic motions in the nonlinear nano-oscillator.

Modeling and Chaotic Dynamics of the Laminated Composite Piezoelectric Rectangular Plate

This paper investigates the multipulse heteroclinic bifurcations and chaotic dynamics of a laminated composite piezoelectric rectangular plate by using an extended Melnikov method in the resonant case. According to the von Karman type equations, Reddy’s third-order shear deformation plate theory, and Hamilton’s principle, the equations of motion are derived for the laminated composite piezoelectric rectangular plate with combined parametric excitations and transverse excitation. The method of multiple scales and Galerkin’s approach are applied to the partial differential governing equation. Then, the four-dimensional averaged equation is obtained for the case of 1 : 3 internal resonance and primary parametric resonance. The extended Melnikov method is used to study the Shilnikov type multipulse heteroclinic bifurcations and chaotic dynamics of the laminated composite piezoelectric rectangular plate. The necessary conditions of the existence for the Shilnikov type multipulse chaotic dynamics are analytically obtained. From the investigation, the geometric structure of the multipulse orbits is described in the four-dimensional phase space. Numerical simulations show that the Shilnikov type multipulse chaotic motions can occur. To sum up, both theoretical and numerical studies suggest that chaos for the Smale horseshoe sense in motion exists for the laminated composite piezoelectric rectangular plate.

Using the extended Melnikov method to study multi-pulse chaotic motions of a rectangular thin plate

This paper investigates the multi-pulse global heteroclinic bifurcations and chaotic dynamics for the nonlinear vibrations of a simply supported rectangular thin plate by using an extended Melnikov method in the resonant case. The rectangular thin plate is subjected to spatially and temporally varying transversal and in-plane excitations, simultaneously. The equations of motion for the rectangular thin plate are derived from the von Kármán equation. Applying the method of multiple scales and Galerkin’s approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1:2 internal resonance and primary parametric resonance. From the averaged equations obtained, the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary Eigenvalues. Based on the explicit expressions of normal form, the extended Melnikov method is used to analyze the multi-pulse heteroclinic bifurcations and chaotic dynamics of the rectangular thin plate. The contribution of the paper is the simplification of the extended Melnikov method. The extended Melnikov function can be simplified in the resonant case and does not depend on the perturbation parameter. The necessary conditions of the existence for the Shilnikov type multi-pulse chaotic dynamics of the rectangular thin plate are analytically obtained. Numerical simulations also display that the Shilnikov type multi-pulse chaotic motions can occur in the rectangular thin plate. Overall, both theoretical and numerical studies demonstrate that the chaos for the Smale horseshoe sense exists in the rectangular thin plate.

Multi-pulse chaotic motions of high-dimension nonlinear system for a laminated composite piezoelectric rectangular plate

This paper investigates the multi-pulse global bifurcations and chaotic dynamics of the high-dimension nonlinear system for a laminated composite piezoelectric rectangular plate by using an extended Melnikov method in the resonant case. Using the von Karman type equations, Reddy’s third-order shear deformation plate theory and Hamilton’s principle, the equations of motion are derived for the laminated composite piezoelectric rectangular plate with combined parametric excitations and transverse excitation. Applying the method of multiple scales and Galerkin’s approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1:2 internal resonance and primary parametric resonance. From the averaged equations obtained, the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on the explicit expressions of normal form, the extended Melnikov method is used for the first time to investigate the Shilnikov type multi-pulse homoclinic bifurcations and chaotic dynamics of the laminated composite piezoelectric rectangular plate. The necessary conditions of the existence for the Shilnikov type multi-pulse chaotic dynamics of the laminated composite piezoelectric rectangular plate are analytically obtained. Numerical simulations also illustrate that the Shilnikov type multi-pulse chaotic motions can also occur in the laminated composite piezoelectric rectangular plate. Overall, both theoretical and numerical studies demonstrate that the chaos in the Smale horseshoe sense exists for the laminated composite piezoelectric rectangular plate.

Multi-pulse chaotic dynamics of six-dimensional non-autonomous nonlinear system for a composite laminated piezoelectric rectangular plate

Global bifurcations and multi-pulse chaotic dynamics are studied for a four-edge simply supported composite laminated piezoelectric rectangular plate under combined in-plane, transverse, and dynamic electrical excitations. Based on the von Karman type equations for the geometric nonlinearity and Reddy’s third-order shear deformation theory, the governing equations of motion for a composite laminated piezoelectric rectangular plate are derived. The Galerkin method is employed to discretize the partial differential equations of motion to a three-degree-of-freedom nonlinear system. The six-dimensional non-autonomous nonlinear system is simplified to a three-order standard form by using the method of normal form. The extended Melnikov method is improved to investigate the six-dimensional non-autonomous nonlinear dynamical system in mixed coordinate. The global bifurcations and multi-pulse chaotic dynamics of the composite laminated piezoelectric rectangular plate are studied by using the improved extended Melnikov method. The multi-pulse chaotic motions of the system are found by using numerical simulation, which further verifies the result of theoretical analysis.