Narrow Results

The chaos and bifurcations in transverse motion of an axially moving thin plate under external and parametric excitations are studied herein. The geometric nonlinearity is introduced by using the von Karman large deflection theory. The coupled partial differential equations of transverse deflection and stress are truncated into a set of ordinary differential equations. By using the Poincaré map and the largest Lyapunov exponent, the dynamical behaviors including chaos are identified based on numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented for different parameters, such as axially moving velocity, damping, external and parametric excitation amplitudes. The chaos is detected in both cases of external and parametric excitations. The interesting relevance between onset of chaos with the corresponding linear instability range are indicated in the external and parametric responses.

The nonlinear inherent vibration of an axially moving ferromagnetic thin plate under the action of the armature air-gap magnetic field is investigated. Based on the nonlinear elasticity theory, the energy relationship equations of the thin plate are given. Based on the electromagnetic theory and the solution of Laplace’s magnetic potential equation, magnetic force on the ferromagnetic rectangular plate under air-gap magnetic field environment is deduced. The Hamilton variational principle is applied to derive the magneto-solid coupled nonlinear vibration equation of the axially moving ferromagnetic thin plate. The two-degree-of-freedom nonlinear vibration differential equations containing static load terms with boundary conditions of SSSS and SSCC are obtained by discretizing through the Galerkin method. The multi-scale method is applied to solve the second-order approximation for deriving the first two orders’ intrinsic frequency of the nonlinear system. The variation laws of first two orders’ nonlinear inherent vibration with axial velocity, magnetic potential, initial air-gap thickness, and initial value are given through numerical examples, and the comparative analysis is performed. The results show that the first-order and second-order inherent vibration frequencies decrease with the increase of axial velocity and magnetic potential and increase with the increase of initial air-gap magnetic field thickness. Different materials and different boundary conditions have greater influence on the first-order and second-order inherent frequencies, and show obvious nonlinear characteristics. The results can provide references for analyzing and controlling the vibration behavior of moving structures in electromagnetic environment.

The subharmonic resonances of an axially moving graphene-reinforced laminated composite plate are studied based on the Galerkin and multiscale methods. Graphene nanoplatelets (GPLs) are added into matrix material which acts as the basic layer of the plate, and a graphene-reinforced nanocomposite plate is thus obtained. Different GPL distribution patterns in the laminated plate are considered. The Halpin–Tsai model is selected to predict the physical properties of the nanocomposite. Hamilton’s principle is utilized to conduct the dynamic modeling of the plate and the von Kármán deformation theory is used. The velocity is assumed to be a combination of constant and harmonically varied velocities. The natural frequencies of the linear system with constant velocity can be obtained using the eigenvalues of the coefficient matrix of the ordinary differential equations after the governing partial differential equations of motion are discretized through the Galerkin method. The instability regions of the linear system and the amplitude–frequency relations of the nonlinear system considering the harmonically varied velocity are obtained based on the multiscale analysis. The effect of GPL reinforcement on the subharmonic resonances of the linear and nonlinear systems is analyzed in detail.

This research presents theoretical investigation to analyze vibration of axially moving sandwich plate floating on fluid. This plate is composed of balsa wood core and two nanocomposite face sheets where the three layers vibrated as an integrated sandwich. The fluid–structure interaction (FSI) effects on the stability of moving plate are considered for both ideal and viscous fluid. Halpin–Tsai model is utilized to determine the material properties of two-phase composite consist of uniformly distributed and randomly oriented carbon nanotubes (CNTs) through the PmPV (poly{(m-phenylenevinylene)-co-[(2,5-dioctoxy-p-phenylene)vinylene]}) matrix. The governing equations are derived based on sinusoidal shear deformation plate theory (SSDT) which is more accurate than the conventional theories, and significantly, it does not require a shear correction factor. Employing Hamilton’s principle, the equations of motion are obtained and solved by the semi-analytical method. Results indicated that the dimensionless frequencies of moving sandwich plate decrease rapidly with increasing the water level and they are almost independent of fluid level when it is higher than 50% of the plate length. The results of this investigation can be used in design and manufacturing of marine vessels and aircrafts.