Nonlinear free vibration of piezoelectric semiconductor doubly-curved shells based on nonlinear drift-diffusion model

In this paper, the nonlinear free vibration behaviors of the piezoelectric semiconductor(PS) doubly-curved shell resting on the Pasternak foundation are studied within the framework of the nonlinear drift-diffusion(NLDD) model and the first-order shear deformation theory. The nonlinear constitutive relations are presented, and the strain energy, kinetic energy, and virtual work of the PS doubly-curved shell are derived.Based on Hamilton's principle as well as the condition of charge continuity, the nonlinear governing equations are achieved, and then these equations are solved by means of an efficient iteration method. Several numerical examples are given to show the effect of the nonlinear drift current, elastic foundation parameters as well as geometric parameters on the nonlinear vibration frequency, and the damping characteristic of the PS doublycurved shell. The main innovations of the manuscript are that the difference between the linearized drift-diffusion(LDD) model and the NLDD model is revealed, and an effective method is proposed to select a proper initial electron concentration for the LDD model.

Nonlinear free vibration of spinning cylindrical shells with arbitrary boundary conditions

The aim of the present study is to investigate the nonlinear free vibration of spinning cylindrical shells under spinning and arbitrary boundary conditions.Artificial springs are used to simulate arbitrary boundary conditions.Sanders’shell theory is employed,and von Kármán nonlinear terms are considered in the theoretical modeling.By using Chebyshev polynomials as admissible functions,motion equations are derived with the Ritz method.Then,a direct iteration method is used to obtain the nonlinear vibration frequencies.The effects of the circumferential wave number,the boundary spring stiffness,and the spinning speed on the nonlinear vibration characteristics of the shells are highlighted.It is found that there exist sensitive intervals for the boundary spring stiffness,which makes the variation of the nonlinear frequency ratio more evident.The decline of the frequency ratio caused by the spinning speed is more significant for the higher vibration amplitude and the smaller boundary spring stiffness.

Nonlinear free vibration analysis of piezoelastic laminated plates with interface damage

This paper presents a nonlinear model for piezoelastic laminated plates with damage effect of the intra-layers and inter-laminar interfaces. Discontinuity of displacement and electric potential on the interfaces are depicted by three shape functions. By using the Hamilton variation principle, the three-dimensional nonlinear dynamic equations of piezoelastic laminated plates with damage effect are derived. Then, by using the Galerkin method, a mathematical solution is presented. In the numerical studies, effects of various factors on the natural frequencies and nonlinear amplitude-frequency response of the simply-supported peizoelastic laminated plates with interfacial imperfections are discussed. These factors include different damage models, thickness of the piezoelectric layer, side-to-thickness ratio, and length-to-width ratio.

Nonlinear free vibration of reticulated shallow spherical shells taking into account transverse shear deformation

This paper deals with nonlinear free vibration of reticulated shallow spherical shells taking into account the effect of transverse shear deformation. The shell is formed by beam members placed in two orthogonal directions. The nondimensional fundamental governing equations in terms of the deflection, rotational angle, and force function are presented, and the solution for the nonlinear free frequency is derived by using the asymptotic iteration method. The asymptotic solution can be used readily to perform the parameter analysis of such space structures with numerous geometrical and material parameters. Numerical examples are given to illustrate the characteristic amplitudefrequency relation and softening and hardening nonlinear behaviors as well as the effect of transverse shear on the linear and nonlinear frequencies of reticulated shells and plates.

Size-Dependent Geometrically Nonlinear Free Vibration of First-Order Shear Deformable Piezoelectric-Piezomagnetic Nanobeams Using the Nonlocal Theory

This article investigates the geometrically nonlinear free vibration of piezoelectric-piezomagnetic nanobeams subjected to magneto-electro-thermal loading taking into account size effect using the nonlocal elasticity theory.To this end,the sizedependent nonlinear governing equations of motion and corresponding boundary conditions are derived according to the nonlocal elasticity theory and the first-order shear deformation theory with von K´arm´an-type of kinematic nonlinearity.The effects of size-dependence,shear deformations,rotary inertia,piezoelectric-piezomagnetic coupling,thermal environment and geometrical nonlinearity are taken into account.The generalized differential quadrature(GDQ)method in conjunction with the numerical Galerkin method,periodic time differential operators and pseudo arclength continuation method is utilized to compute the nonlinear frequency response of piezoelectric-piezomagnetic nanobeams.The influences of various parameters such as non-dimensional nonlocal parameter,temperature change,initial applied electric voltage,initial applied magnetic potential,length-to-thickness ratio and different boundary conditions on the geometrically nonlinear free vibration characteristics of piezoelectric-piezomagnetic nanobeams are demonstrated by numerical examples.It is illustrated that the hardening spring effect increases with increasing the non-dimensional nonlocal parameter,positive initial applied voltage,negative initial applied magnetic potential,temperature rise and decreases with increasing the negative initial applied voltage,positive initial applied magnetic potential and length-tothickness ratio.

Nonlinear vibration of functionally graded graphene platelet-reinforced composite truncated conical shell using first-order shear deformation theory

In this study,the first-order shear deformation theory(FSDT)is used to establish a nonlinear dynamic model for a conical shell truncated by a functionally graded graphene platelet-reinforced composite(FG-GPLRC).The vibration analyses of the FG-GPLRC truncated conical shell are presented.Considering the graphene platelets(GPLs)of the FG-GPLRC truncated conical shell with three different distribution patterns,the modified Halpin-Tsai model is used to calculate the effective Young’s modulus.Hamilton’s principle,the FSDT,and the von-Karman type nonlinear geometric relationships are used to derive a system of partial differential governing equations of the FG-GPLRC truncated conical shell.The Galerkin method is used to obtain the ordinary differential equations of the truncated conical shell.Then,the analytical nonlinear frequencies of the FG-GPLRC truncated conical shell are solved by the harmonic balance method.The effects of the weight fraction and distribution pattern of the GPLs,the ratio of the length to the radius as well as the ratio of the radius to the thickness of the FG-GPLRC truncated conical shell on the nonlinear natural frequency characteristics are discussed.This study culminates in the discovery of the periodic motion and chaotic motion of the FG-GPLRC truncated conical shell.

Energetics and conserved quantity of an axially moving string undergoing three-dimensional nonlinear vibration

Nonlinear three-dimensional vibration of axially moving strings is investigated in the view of energetics. The governing equation is derived from the Eulerian equation of motion of a continuum for axially accelerating strings. The time-rate of the total mechanical energy associated with the vibration is calculated for the string with its ends moving in a prescribed way. For a string moving in a constant axial speed and constrained by two fixed ends, a conserved quantity is proved to remain unchanged during three-dimensional vibration, while the string energy is not conserved. An approximate conserved quantity is derived from the conserved quantity in the neighborhood of the straight equilibrium configuration. The approximate conserved quantity is applied to verify the Lyapunov stability of the straight equilibrium configuration. Numerical simulations are performed for a rubber string and a steel string. The results demonstrate the variation of the total mechanical energy and the invariance of the conserved quantity.

Effect of different geometrical non-uniformities on nonlinear vibration of porous functionally graded skew plates: A finite element study

This article presents the investigation of nonlinear vibration analysis of tapered porous functionally graded skew(TPFGS)plate considering the effects of geometrical non-uniformities to optimize the thickness in the structural design.The TPFGS plate is analyzed considering linearly,bi-linearly,and exponentially varying thicknesses.The plate’s effective material properties are tailor-made using a modified power-law distribution in which gradation varies along the thickness direction of the TPFGS plate.Incorporating the non-linear finite element formulation to develop the kinematic equation’s displacement model for the TPFGS plate is based on the first-order shear deformation theory(FSDT)in conjunction with von Karman’s nonlinearity.The nonlinear governing equations are established by Hamilton’s principle.The direct iterative method is adopted to solve the nonlinear mathematical relations to obtain the nonlinear frequencies.The influence of the porosity distributions and porosity parameter indices on the nonlinear frequency responses of the TPFGS plate for different skew angles and variable thicknesses are studied for various geometrical parameters.The influence of taper ratio,variable thickness,skewness,porosity distributions,gradation,and boundary conditions on the plate’s nonlinear vibration is demonstrated.The nonlinear frequency analysis reveals that the geometrical nonuniformities and porosities significantly influence the porous functionally graded plates with varying thickness than the uniform thickness.Besides,exponentially and linearly variable thicknesses can be considered for the thickness optimizations of TPFGS plates in the structural design.

Nonlinear Vibration Analysis of Functionally GradedNanobeam Using Homotopy Perturbation Method

In this paper, He’s homotopy perturbation method is utilized to obtainthe analytical solution for the nonlinear natural frequency of functionally gradednanobeam. The functionally graded nanobeam is modeled using the Eringen’s nonlocalelasticity theory based on Euler-Bernoulli beam theory with von Karman nonlinearityrelation. The boundary conditions of problem are considered with both sidessimply supported and simply supported-clamped. The Galerkin’s method is utilizedto decrease the nonlinear partial differential equation to a nonlinear second-order ordinarydifferential equation. Based on numerical results, homotopy perturbationmethodconvergence is illustrated. According to obtained results, it is seen that the second termof the homotopy perturbation method gives extremely precise solution.