Nonlinear electromechanical dynamics of piezoelectric doubly-curved microsystem using modified strain gradient theory

Linear and nonlinear free vibration analysis of functionally graded porous nanobeam using stress-driven nonlocal integral model

Nonlinear free vibration of nanotube with small scale effects embedded in viscous matrix

Motivated by the evident lack of studies on the nonlinear thermal stability and vibration of functionally graded material (FGM) circular plates (CPs) including porosity and elastically restrained edge, this article investigates the combined influences of porosities, geometrical imperfection and tangential elastic constraint of boundary edge on the nonlinear axisymmetric free vibration and postbuckling behavior of porous FGM CPs subjected to uniform temperature rise. The pores are distributed into the FGM via even and uneven distribution types. Temperature dependence of material properties is included and effective properties of porous FGM are evaluated adopting a modified mixture rule. The CP is assumed to be under axisymmetric deformation and clamped at periphery. Motion and compatibility equations in terms of deflection and stress function are derived on the basis of the first-order shear deformation theory (FSDT) taking into account nonlinear strains in von Kármán sense and initial geometric imperfection. The derived equations are solved by using analytical solutions and Galerkin method. In the thermal postbuckling analysis, an iteration algorithm is employed to evaluate critical temperatures and postbuckling temperature–deflection curves. In the nonlinear free vibration analysis, fourth–order Runge–Kutta scheme is adopted to seek the frequencies of nonlinear free vibration. After verification, parametric studies are presented to assess numerous influences on the thermoelastic nonlinear vibration and postbuckling of porous FGM CPs. It is found that rigorous constraint of edge renders the thermal buckling resistance capacity weaker and the frequency nonlinearity stronger.

It has been proved that using pure nonlocal elasticity, especially in differential form, leads to inconsistent and unreliable results. Therefore, to obviate these weaknesses, Eringen’s two-phase local/nonlocal elasticity has been recently used by researchers to consider the nonlocal size dependency of nanostructures. Given this, for the first time, the size-dependent nonlinear free vibration of nanobeams is investigated in this article within the framework of two-phase elasticity by using the Galerkin method. Contrary to differential nonlocal elasticity, the size dependency of the axial tension force, due to von Karman nonlinearity, is considered, and its effect on the nonlinear vibration is examined. The correct procedure of using the Galerkin method for studying the nonlinear vibration of two-phase nanobeams is introduced. It is shown that, although it is possible to extract the linear mode shapes of two-phase nanobeams using its equal differential equation, integral form of two-phase elasticity should be considered in the Galerkin method. Furthermore, it is observed that in two-phase elasticity, applying classic mode shapes in the Galerkin method leads to significant errors, especially in higher nonlocal parameters and lower values of local phase fraction and amplitude ratios.

Size-dependent functionally graded beam model based on an improved third-order shear deformation theory

This paper is concerned with the analysis of the free and forced nonlinear vibrations of the suspended cable with thermal effects. By introducing the new thermal stressed configuration, the effect of temperature variation on the nonlinear vibration equations of motion is reflected by a non-dimensional cable tension variation factor. The partial differential equations of the planar motion are discretized to the ordinary equations via the Galerkin method, and the single-mode discretization is investigated. The Lindstedt–Poincare method and multiple scales method are applied to obtain the higher-order approximate solutions of the nonlinear free vibrations and primary resonances, respectively. Parametric investigations of temperature effects on the linear and nonlinear vibration characteristics of the suspended cable with different sag-to-span ratios and excitation amplitudes are performed. The results of the perturbation analysis show that the nonlinear free and forced vibration characteristics would be changed by the temperature variations qualitatively and quantitatively, depending on the sag-to-span ratio and the excitation amplitude. The asymmetric phenomena between the effects of warming and cooling conditions on the vibration characteristics can be observed. The crossover points between next two mode frequencies are shifted under the temperature variation, and these would significantly influence the internal resonances of the suspended cable. Finally, temperature effects on the time-history diagram of the cable axial total tension force are investigated.

Nonlinear forced oscillation of a rectangular orthotropic plate subjected to uniform harmonic excitation is solved using the method of multiple scales. The governing equations are based on the von Karman type geometrical nonlinearity, and the effect of damping is included. The general multimode solution is developed for simply supported boundary conditions, and the solution is specialized for two-symmetric modes analysis. The primary resonances and the subharmonic and superharmonic secondary resonances are studied in detail. HIN laminated composite panels subjected to transverse periodic loadings can encounter deflections of the order of panel thickness or even higher. The effect of these periodic excitations on the panel can be very severe. Responses of this kind cannot be predicted by linear theory. Consequently, the need to study large deflections using nonlinear methods of analysis is of paramount importance. The formulation of the equations governing the fundamen- tal kinematic behavior of the laminated composite plates in the presence of the von Karman geometrical nonlinearity is attributed to Whitney and Leissa.1 Based on these equations, various methods have been developed to solve nonlinear free and forced vibrations of composite panels. A good survey on mainly nonlinear free and forced vibrations of isotropic plates is given in a book by Nayfeh and Mook.2 The most compre- hensive work on geometrically nonlinear analysis of both static and dynamic behavior of the laminated panels through 1972 is collected in a book by Chia. 3 Bert4 has conducted a survey on the dynamics of composite panels for the period of 1979-81. A review of literature on linear vibrations of plates can be found in a review paper by Sathyamoorthy.5 Relatively few investigations have been reported on the nonlinear forced vibration of isotropic or composite panels under harmonic excitations. Yamaki6 presented a one-term solution for free and forced vibrations of the rectangular plates, using Galerkin's method. Lin7 studied the response of a nonlinear flat panel to periodic and randomly varying load- ings. Nonlinear forced vibrations of beams and rectangular plates were studied by Eisely8 using a single-mode Galerkin's method in conjunction with the Linstedt-Duffing perturbation technique. Free and forced response of beams and plates undergoing large-amplitude oscillations using the Ritz averag- ing method were studied by Srinivasan.9 Bennett10 studied the nonlinear vibration of simply supported angle-ply laminated plates by considering the instability regions of the response of such plates subjected to harmonic excitations. Nonlinear free and forced vibration of a circular plate with clamped bound-

Thickness-shear vibrations of a plate is one of the most widely used functioning modes of quartz crystal resonators. For an analysis of vibrations, the Mindlin and Lee plate theories based on the displacement expansion of the thickness coordinate have been used as the linear theories. However, due to lacking of available method and complexity of the problem, the nonlinear thickness-shear vibrations have been rarely studied with analytical methods. As a preliminary step for the research on nonlinear vibrations in a finite crystal plate, nonlinear thickness-shear vibrations of an infinite and isotropic elastic plate are studied. First, using the Galerkin approximation and forcing the weighted error to vanish, we have obtained a nonlinear ordinary differential equation depending on time. By assuming corresponding solution and neglecting the high-order nonlinear terms, the amplitude-frequency relation of the nonlinear vibrations is obtained. In order to verify the accuracy of our study, we have also employed the perturbation method to solve this ordinary differential equation and obtained the second-order amplitude-frequency relation. These equations and results are useful in verifying the available methods and improving our solution techniques for the coupled nonlinear vibrations of finite piezoelectric plates.

Closed-form solutions for nonlinear bending and free vibration of functionally graded microplates based on the modified couple stress theory

Nonlinear vibration and dynamic stability analysis of composite plates

This research concentrates on pull-in behavior of a micro-plate laminated with a novel material: light activated shape memory polymer (LASMP). Under light exposure, this material can realize dynamic Young’s modulus. With this feature, a LASMP laminated micro-plate model considering micro-scale effect and geometric nonlinearity is proposed. Dynamic equations of the micro-plate system are derived via Hamilton principle and solved by Galerkin method whose accuracy is validated by FEM. Effect of size dependency and dynamic Young’s modulus of LASMP on pull-in behavior of the micro-plate are studied. Present study provided a technique to manipulate pull-in phenomenon of electrically actuated micro-plate.

Non-linear vibration of initially stressed plates with initial imperfections

Two-degrees-of-freedom nonlinear free vibration analysis of magneto-electro-elastic plate based on high order shear deformation theory

Nonlinear free and forced vibrations of a fiber-reinforced dielectric elastomer-based microbeam

Imperfection sensitivity in the nonlinear vibration oscillations of initially stressed plates