Nonlinear vibrations and dynamic instability analysis of a two-directional functionally graded conical shell under parametric and external excitations

The nonlinear vibration behavior and dynamic instability of Euler–Bernoulli nanobeams under thermo-magneto-mechanical loads is the main objective of the present paper. Firstly, a short Euler–Bernoulli nanobeam is modeled and exposed to an external parametric excitation. Based on the nonlocal continuum theory and nonlinear von Karman beam theory, the nonlinear governing differential equation of motion is derived. Secondly, to transport the partial differential equation to the ordinary differential equation, Galerkin method is applied. Then, multiple scales method, as an analytical approach, is used to solve the equation. At the end, modulation equation of Euler–Bernoulli nanobeams is obtained. Then, to evaluate the dynamic instability of the system, trivial and nontrivial steady-state solutions are discussed. Emphasizing the effect of parametric excitation, for considering the instability regions, bifurcation points are studied and investigated. As a result, it can be observed that the damping coefficient plays an effective role as well as parametric excitation in stability and frequency response of the system.

Nonlinear Vibrations and Dynamic Instability Analysis of Two-Directional Functionally Graded Conical Shell Under Parametric and External Excitations

Nonlinear vibration and dynamic instability analyses of laminated doubly curved panels in thermal environments

A novel technique for nonlinear dynamic instability analysis of FG-GRC laminated plates

Non-linear dynamic instability analysis of thin-walled stiffener beam subjected to uniform harmonic in-plane loading

Motivated by the lack of sufficient accuracy in investigation of nonlinear dynamics of graphene sheets (GS), nonlinear dynamic instability and frequency response of the pre-stressed single layered GS (SLGS) are investigated in the present paper. To achieve this aim, in the first step, SLGS embedded on a visco-Pasternak foundation is modeled while it is under an initial stress and subjected to a parametric axial force and magnetic field. Then, based on Eringen’s theory, nonlinear von Karman relations and Kelvin–Voigt model, the nonlinear governing equation of motion is derived. In the next step, Galerkin technique and multiple time scales method are employed to analyze and solve the equation of motion. Emphasizing the effect of parametric excitation, for considering the instability regions, bifurcation points are discussed. As a result, a parametric study is conducted to show the importance of damping coefficient and parametric excitation in dynamic instability of the system. Numerical examples are also treated which show various discontinuous bifurcations. Also, infinitely stable and unstable solutions are addressed.

Parametric excitation of Euler–Bernoulli nanobeams under thermo-magneto-mechanical loads: Nonlinear vibration and dynamic instability

Non-linear dynamic instability analysis of mono-symmetric thin walled columns with various boundary conditions

Nonlinear dynamic instability analysis of sandwich beams with integral viscoelastic core using different criteria

Size-dependent nonlinear vibration and instability of a damped microplate subjected to in-plane parametric excitation

Gear disengaging, back-side tooth contact or poor dynamic behavior during operating leads to dynamic instability in planetary gear trains (PGTs). A novel nonlinear dynamic model of PGTs with internal and external gear pairs considering multi-state engagement induced by backlash and contact ratio is established. An improved time-varying meshing stiffness model including temperature stiffness is analytically derived. The time-varying meshing stiffness with temperature effect, friction, backlash, time-varying pressure angle, and time-varying friction arm are integrated into the dynamic model of PGTs. Multi-state engaging behavior is efficiently identified by constructing different Poincaré mappings. A method to calculate dynamic instability is proposed in the time-domain trace. The intrinsic relationship between multi-state engaging and dynamic instability is investigated via multi-section bifurcation plots and phase trajectory topology. The global dynamic instability is revealed based on the bifurcation and evolution of coexistence behavior under the parameter-state synergy. The results show that the multistate engagement is heavily depending on bifurcation and phase trajectory topology, which whereby affects the dynamic instability. Two special phenomena, complete and incomplete bifurcations, are discovered under parameter-state synergy. Complete bifurcation causes global instability and incomplete bifurcation results in local instability and yields coexistence responses. Incomplete bifurcation brings about new bifurcation branches.

In this paper, the dynamic instability of thin laminated composite plates subjected to harmonic in-plane loading is studied based on nonlinear analysis. The equations of motion of the plate are developed using von Karman-type of plate equation including geometric nonlinearity. The nonlinear large deflection plate equations of motion are solved by using Galerkin’s technique that leads to a system of nonlinear Mathieu-Hill equations. Dynamically unstable regions, and both stable- and unstable-solution amplitudes of the steady-state vibrations are obtained by applying the Bolotin’s method. The nonlinear dynamic stability characteristics of both antisymmetric and symmetric cross-ply laminates with different lamination schemes are examined. A detailed parametric study is conducted to examine and compare the effects of the orthotropy, magnitude of both tensile and compressive longitudinal loads, aspect ratios of the plate including length-to-width and length-to-thickness ratios, and in-plane transverse wave number on the parametric resonance particularly the steady-state vibrations amplitude. The present results show good agreement with that available in the literature.

Non-linear dynamic instability analysis of laminated composite cylindrical shells subjected to periodic axial loads

Nonlinear vibration and instability of a randomly distributed CNT-reinforced composite plate subjected to localized in-plane parametric excitation

This paper presents a nonlinear dynamic analysis procedure used for the investigation of the response of a tensegrity bridge to a selected sudden cable rupture. In order to simulate a cable rupture, for the loaded or unloaded geometry of the tensegrity structure, a geometrical nonlinear analysis is performed, and the cable end tensions projected in the global coordinate system are determined. Next, these forces are applied as external nodal forces to the tensegrity structure, from which the selected cable has been omitted (damaged structure). Next, the nonlinear equation of motion of the tensegrity bridge subjected to dynamic loads is discretized and integrated in time using the unconditionally stable Newmark constant-average acceleration method combined with a Newton-Raphson iterative scheme. The dynamic simulation is initiated by cancelling the vector of external forces representing the damaged cable. For each case, the largest tension force in the cables, the largest compression force in the struts as well as the largest average midspan displacement are determined. The maximum tension obtained in all the bridge cables was way below their tension capacities for the unloaded bridge and exceeded them for only one case of the loaded one. However, the maximum compression forces obtained in the struts of the bridge were below their compression capacities. The limit deflection has been exceeded only for of the loaded bridge and for several cases of cable rupture. Nonlinear dynamic instabilities caused by cable slackening were observed in all simulations.