Frequency–energy plot and targeted energy transfer analysis of coupled bistable nonlinear energy sink with linear oscillator

The nonlinear energy sink (NES), which is proven to perform rapid and passive targeted energy transfer (TET), has been employed for vibration mitigation in many primary small- and large-scale structures. Recently, the feature of bistability, in which two nontrivial stable equilibria and one trivial unstable equilibrium exist, is utilized for passive TET in what is known as bistable NES (BNES). The BNES generates a nonlinear force that incorporates negative linear and multiple positive or negative nonlinear stiffness components. In this paper, the BNES is coupled to a linear oscillator (LO) where the dynamic behavior of the resulting LO-BNES system is studied through frequency–energy plots (FEPs), which are generated by analytical approximation using the complexification-averaging method and by numerical continuation techniques. The effect of the length and stiffness of the transverse coupling springs is found to affect the stability and topology of the branches and indicates the importance of the exact physical realization of the system. The rich nonlinear dynamical behavior of the LO-BNES system is also highlighted through the appearance of multiple symmetrical and unsymmetrical in- and out-of-phase backbone branches, especially at low energy levels. The superimposed wavelet frequency spectrums of the LO-BNES response on the FEP have verified the robustness of the TET mechanism where the role of the unsymmetrical NNM backbones in TET is clearly observed.

     

Response analysis of a pitch–plunge airfoil with structural and aerodynamic nonlinearities subjected to randomly fluctuating flows

This study focuses on numerically investigating the response dynamics of a pitch–plunge airfoil with structural nonlinearity under dynamic stall conditions. The aeroelastic responses are investigated for both deterministic and randomly time varying flow conditions. To that end, a pitch–plunge airfoil under dynamic stall condition is considered and the nonlinear aerodynamic loads are computed using a Leishman–Beddoes formulation. It is shown that the presence of structural nonlinearities can give rise to a variety of dynamical responses in the pre-flutter regime. Next, a response analysis under the presence of a randomly fluctuating wind is carried out. It is demonstrated that the route to flutter occurs via a regime of pre-flutter oscillations called intermittency. Finally, the manifestation of these stochastic responses is characterized by invoking stochastic bifurcation concepts. The route to flutter via intermittency is presented in terms of topological changes occurring in the joint-probability density function of the state variables.     

Dynamical behavior of a Bouc–Wen type oscillator coupled to a nonlinear energy sink

We try to enlighten the time multi-scale energy exchange between a main oscillator of Bouc–Wen type and a nonlinear energy sink. A general methodology is presented to detect the invariant manifold of the system at fast time scale and then to search equilibrium points and fold singularities at the slow time scale. A numerical example is enclosed to show the application of the coupled nonlinear energy sink in passive control of Bouc–Wen type main structures.     

Higher-order rogue wave solutions of a general coupled nonlinear Fokas–Lenells system

Nonlinear Dynamics - In this paper, we construct a generalized Darboux transformation (DT) of the general coupled nonlinear Fokas–Lenells system. And using this generalized DT, new exact...     

Experimental and numerical analysis of nonlinear dynamics of rotor–bearing–seal system

Nonlinear dynamics and stability of the rotor–bearing–seal system are investigated both theoretically and experimentally. An experimental rotor–bearing–seal device is designed and corresponding tests are carried out. The experimental rotor system is simplified as the Jeffcott rotor. The nonlinear oil–film forces are obtained under the short bearing theory and Muszynska nonlinear seal force model is used. Numerical method is utilized to solve the nonlinear governing equations. Bifurcation diagrams, waterfall plots, Poincaré maps, spectrum plots and rotor orbits are drawn to analyze various nonlinear phenomena and system unstable processes. Theoretical results from numerical analysis are in good agreement with results from experiments. Conclusions are drawn and prove that this study will contribute to the further understanding of nonlinear dynamics and stability of the rotor system with the fluid-induced forces from oil–film bearings and the seals.     

Singularity analysis of response bifurcation for a coupled pitch–roll ship model with quadratic and cubic nonlinearity

Nonlinear Dynamics - Based on the coupling of roll and pitch motion of ships, a mathematical model with quadratic and cubic nonlinear terms is presented. Primary resonance is discussed by the...     

Modulation instability analysis and soliton solutions of an integrable coupled nonlinear Schrödinger system

Nonlinear Dynamics - In this paper, an integrable coupled nonlinear Schrödinger system is investigated, which is derived from the integrable Kadomtsev–Petviashvili system, and can be...     

Elastic cables–rigid body coupled dynamics: asymptotic modeling and analysis

An elastic cables–rigid body coupled model is proposed for investigating dynamic interactions between cables’ nonlinear transversal vibrations and boundary tower’s torsional dynamics, arising in large transmission line–tower systems and suspended cable–bridge tower systems. By introducing a weak torsion assumption and a large moment of inertia for the tower, an asymptotic expansion of cables–tower coupled dynamics is conducted in a weakly nonlinear framework, and a cables–tower reduced coupled model is eventually established. After model’s validations using direct numerical simulations, two distinct kinds of coupled dynamics are fully investigated. The first is that an external torque is applied to the tower and the two cables would both be indirectly excited, asymmetrically, by the torsional/oscillating tower. The two cables’ responses are the same in this case. The second is that only one of the two cables, i.e., the leader cable, is directly excited, and the other cable, i.e., the follower one, is only indirectly excited through cables–tower dynamic interactions. In such kind of leader–follower dynamics, the leader cable is quite different from the follower one. Nonlinear coupled frequency response diagrams for both systems are constructed using numerical continuation algorithms, mainly focused on the coupled steady solutions’ stabilities and bifurcations. Furthermore, the dynamic effects of tower’s moment of inertia, wing span and damping are thoroughly investigated.

     

Hopf bifurcation of a predator–prey system with predator harvesting and two delays

In this paper, we consider the differential-algebraic predator–prey model with predator harvesting and two delays. By using the new normal form of differential-algebraic systems, center manifold theorem and bifurcation theory, we analyze the stability and the Hopf bifurcation of the proposed system. In addition, the new effective analytical method enriches the toolbox for the qualitative analysis of the delayed differential-algebraic systems. Finally, numerical simulations are given to show the consistency with theoretical analysis obtained here.     

Periodic solutions and homoclinic bifurcation of a predator–prey system with two types of harvesting

In this paper, a predator–prey model with both constant rate harvesting and state dependent impulsive harvesting is analyzed. By using differential equation geometry theory and the method of successor functions, the existence, uniqueness and stability of the order one periodic solution have been studied. Sufficient conditions which guarantee the nonexistence of order k (k≥2) periodic solution are given. We also present that the system exhibits the phenomenon of homoclinic bifurcation under some parametric conditions. Finally, some numerical simulations and biological explanations are given.     

Atomistic–continuum coupled model for nonlinear analysis of single layer graphene sheets

In this paper, atomistic–continuum coupled model for nonlinear flexural response of single layer graphene sheet is presented considering von-Karman geometric nonlinearity and material nonlinearity due to atomic interactions. The strain energy density function at continuum level is established by coupling the deformation at continuum level to that of at atomic level through Cauchy–Born rule. Strain and curvature dependent tangent in-plane extensional, bending–extension coupling, bending stiffness matrices are derived from strain energy density function constructed through Tersoff–Brenner potential. The finite element method is used to discretize the graphene sheet at continuum level and nonlinear bending response with and without material nonlinearity is studied. The present results are also compared with Kirchhoff plate model and significant differences at higher load are observed. The effects of other parameters like number of atoms in the graphene sheet, boundary conditions on the central/maximum deflection of graphene sheet are investigated. It is also brought out that the occurrence of bond length exceeding cutoff distance initiates at corners for CFCC, CFCF, SFSS, SFSF graphene sheets and near center for SSSS and CCCC graphene sheets.     

Thermal–mechanical coupled analysis of a brake disk rotor

The main purpose of this study is to analyze the thermomechanical behavior of the dry contact between the brake disk and pads during the braking phase. The simulation strategy is based on computer code ANSYS11. The modeling of transient temperature in the disk is actually used to identify the factor of geometric design of the disk to install the ventilation system in vehicles The thermal–structural analysis is then used with coupling to determine the deformation and the Von Mises stress established in the disk, the contact pressure distribution in pads. The results are satisfactory when compared to those of the specialized literature.     

Nonlinear modeling and dynamic analysis of the rotor-bearing system

To overcome the shortcomings of extreme time-consuming in solving the Reynolds equation, two efficient calculation methods, based on the free boundary theory and variational principles for the unsteady nonlinear Reynolds equation in the condition of Reynolds boundary, are presented in the paper. By employing the two mentioned methods, the nonlinear dynamic forces as well as their Jacobians of the journal bearing can be calculated saving time but with the same accuracy. Of these two methods, the one is called a Ritz model which manipulates the cavitation region by simply introducing a parameter to match the free boundary condition and, as a result, a very simple approximate formulae of oil-film pressure is being obtained. The other one is a one-dimensional FEM method which reduces the two-dimensional variational inequality to the one-dimensional algebraic complementary equations, and then a direct method is being used to solve these complementary equations, without the need of iterations, and the free boundary condition can be automatically satisfied. Meanwhile, a new order reduction method is contributed to reduce the degrees of freedom of a complex rotor-bearing system. Thus the nonlinear behavior analysis of the rotor-bearing system can be studied time-sparingly. The results in the paper show the high efficiency of the two methods as well as the abundant nonlinear phenomenon of the system, compared with the results obtained by the usual numerical solution of the Reynolds equation.     

Piezoresistive fiber-reinforced composites: A coupled nonlinear micromechanical–microelectrical modeling approach

Piezoresistive composites are materials that exhibit spatial and effective electrical resistivity changes as a result of local mechanical deformations in their constituents. These materials have a wide array of applications from non-destructive evaluation to sensor technology. We propose a new coupled nonlinear micromechanical-microelectrical modeling framework for periodic heterogeneous media. These proposed micro-models enable the prediction of the effective piezoresistive properties along with the corresponding spatial distributions of local mechanical–electrical fields, such as stress, strain, current densities, and electrical potentials. To this end, the high fidelity generalized method of cells (HFGMC), originally developed for micromechanical analysis of composites, is extended for the micro-electrical modeling in order to predict their spatial field distributions and effective electrical properties. In both cases, the local displacement vector and electrical potential are expanded using quadratic polynomials in each subvolume (subcell). The equilibrium and charge conservations are satisfied in an average volumetric fashion. In addition, the continuity and periodicity of the displacements, tractions, electrical potential, and current are satisfied at the subcell interfaces on an average basis. Next, a one way coupling is established between the nonlinear mechanical and electrical effects, whereby the mechanical deformations affect the electrical conductivity in the fiber and/or matrix constituents. Incremental and total formulations are used to arrive at the proper nonlinear solution of the governing equations. The micro-electrical HFGMC is first verified by comparing the stand-alone electrical solution predictions with the finite element method for different doubly periodic composites. Next, the coupled HFGMC is calibrated and experimentally verified in order to examine the effective piezoresistivity of different composites. These include conductive polymeric matrices doped with carbon nano-tubes or particles. One advantage of the proposed nonlinear coupled micro-models is its ability to predict the local and effective electro-mechanical behaviors of multi-phase periodic composites with different conductive phases.     

Bifurcation and chaos of a cable–beam coupled system under simultaneous internal and external resonances

The bifurcation and chaos of a cable–beam coupled system under simultaneous internal and external resonances are investigated. The combined effects of the nonlinear term due to the cable’s geometric and coupled behavior between the modes of the beam and the cable are considered. The nonlinear partial-differential equations are derived by the Hamiltonian principle. The Galerkin method is applied to truncate the governing equation into a set of ordinary differential equations. The bifurcation diagrams in three separate loading cases, namely, excitation acting on the cable, on the beam and simultaneously on the beam and cable, are analyzed with changing forcing amplitude. Based on careful numerical simulations, bifurcations and possible chaotic motions are represented to reveal the combined effects of nonlinearities on the dynamics of the beam and the cable when they act as an overall structure.     

Head-on collision of solitary waves in coupled Korteweg–de Vries systems modeling nonlinear transmission lines

The extended Poincaré–Lighthill–Kuo (PLK) method is applied to characterize head-on collisions of solitary waves in a coupled Korteweg–de Vries (KdV) system that has multiple modes supporting solitons. As a simple physically realizable system, we investigate two coupled electrical nonlinear transmission lines (NLTLs), and the proposed method successfully leads to the collision-induced phase shifts and the wave equation that governs the dynamics of the pulses generated by colliding solitary waves.     

Microstructural modeling of asphalt concrete using a coupled moisture–mechanical constitutive relationship

The coupled effect of moisture diffusion and mechanical loading on the microstructure of asphalt concrete is studied. The traditional Continuum Damage Mechanics (CDM) framework is modified to model detrimental effects of moisture and mechanical loading. Adhesive/cohesive moisture-induced damage constitutive relationships are proposed to describe the time-dependent degradation of material properties due to moisture. X-ray two-dimensional (2D) computed tomography-imaging technique is used to construct finite element (FE) microstructural representation of a typical dense-graded asphalt concrete. After being calibrated against pull-off experiments, the proposed moisture-induced damage constitutive relationship, which is coupled to thermo-viscoelastic–viscoplastic–viscodamage mechanisms, is used to simulate the microstructure of asphalt concrete. Simulation results demonstrate that the generated 2D FE microstructural representation along with the coupled moisture–mechanical constitutive relationship can be effectively used to model the overall thermo-hygro-mechanical response of asphalt concrete.     

Order reduction and nonlinear behaviors of a continuous rotor system

An isotropic flexible shaft, acted by nonlinear fluid-induced forces generated from oil-lubricated journal bearings and hydrodynamic seal, is considered in this paper. Dimension reductions of the rotor system were carried out by both the standard Galerkin method and the nonlinear Galerkin method. Numerical simulations provide bifurcation diagrams, spectrum cascade, orbits of the disk center and Poincaré maps, to demonstrate the dynamical behaviors of the system. The results reveal transitions, or bifurcations, of the rotor whirl from being synchronous to non-synchronous as the unstable speed is exceeded. The non-synchronous oil/seal whirl is a quasi-periodic motion. In the regime of quasi-periodic motion, the “windows” of multi-periodic motion were found. The investigation shows that the nonlinear Galerkin method has an advantage over the standard one with the same order of truncations, because the influences of higher modes are considered by the former.