Weak form quadrature elements for non-classical Kirchhoff plate theory

The problem of vibrations of double-beam systems is of technological interest and has been extensively studied in the literature. However, a general solution is yet to be developed for the general boundary conditions. In this paper, a general solution of the free and forced vibrations of double-beam systems under classical, nonclassical, and mixed boundary conditions is derived. The obtained mode-shape solutions are exact and explicit. The characteristic frequency equation of a double-beam system is of order-4 and order-8 for the classical boundary conditions and the nonclassical boundary conditions, respectively. The solution of forced vibrations is developed with the classical modal expansion technique. In addition to the theory, applications of the proposed method are illustrated by numerical examples. The obtained frequencies for special problems are compared with the literature, and the agreement between the proposed method and the literature is remarkably good. This shows that the proposed shape function method may be applied reliably to determine the transverse vibration mode shapes and natural frequencies of the double-beam system under the general boundary conditions. The solutions of free and forced vibrations of the double-beam systems presented in the paper could easily be programmed into a computer for vibration analyses.

Resonance behavior of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets

The classical continuum theory not only underestimates the stiffness of microscale structures such as microbeams but is also unable to capture the size dependency, a phenomenon observed in these structures. Hence, the non-classical continuum theories such as the strain gradient elasticity have been developed. In this paper, a Timoshenko beam finite element is developed based on the strain gradient theory and employed to evaluate the mechanical behavior of microbeams used in microelectromechanical systems. The new beam element is a comprehensive beam element that recovers the formulations of strain gradient Euler–Bernoulli beam element, modified couple stress (another non-classical theory) Timoshenko and Euler–Bernoulli beam elements, and also classical Timoshenko and Euler–Bernoulli beam elements; note that the shear-locking phenomenon will not happen for the new Timoshenko beam element. The stiffness and mass matrices of the new element are derived in closed forms by following an energy-based approach and using Hamilton’s principle. It is noted that unlike the classical beam elements, the stiffness matrix of the new element has a size-dependent nature that can capture the size-dependent behavior of microbeams. The shape functions of the newly developed beam element are determined by solving the equilibrium equations of strain gradient Timoshenko beams, which brings about a size-dependent characteristic for them. The new beam element is employed to evaluate the static deflection of a microcantilever, and the results are compared to the experimental data as well as the results obtained by using the classical beam element and the couple stress plane element. The new beam element is also implemented to calculate the static deflection, vibration frequency, and pull-in voltage of electrostatically actuated microbeams. The current results are compared to the experimental data as well as the classical FEM outcomes. It is observed that the results of the new element are in excellent agreement with the experimental data while the gap between the experimental and classical FEM results is significant.

Application and dynamical behavior of CNTs as fluidic nanosensors based on the nonlocal strain gradient theory

Modelling and evaluation of thermoelastic damping of FGM micro plates based on the Levinson plate theory

An accurate higher order plate theory for vibrations of cross-laminated timber panels

According to a generalized nonlocal strain gradient theory (NSGT), dynamic modeling and free vibrational analysis of nanoporous inhomogeneous nanoplates is presented. The present model incorporates two scale coefficients to examine vibration behavior of nanoplates much accurately. Porosity-dependent material properties of the nanoplate are defined via a modified power-law function. The nanoplate is resting on a viscoelastic substrate and is subjected to hygro-thermal environment and in-plane linearly varying mechanical loads. The governing equations and related classical and non-classical boundary conditions are derived based on Hamilton\'s principle. These equations are solved for hinged nanoplates via Galerkin\'s method. Obtained results show the importance of hygro-thermal loading, viscoelastic medium, in-plane bending load, gradient index, nonlocal parameter, strain gradient parameter and porosities on vibrational characteristics of size-dependent FG nanoplates.

In this study, the nonlinear free vibrations behaviour of nano-truncated conical shells was analysed, using the first-order shear deformable shell model. The analysis took into account the structure size through modified strain gradient theory, and differential quadrature and Fréchet derivative methods in von Kármán-Donnell-type approach to kinematic nonlinearity. The governing equations were obtained, utilizing Hamilton's principle. Partial differential equations plus the non-classical and classical boundary conditions were used to obtain the shells’ equations of motion. Discretizing the boundary conditions and equations of motion were performed based on a generalized differential quadrature analogy. The eigenvalue system was considered based on the harmonic balance technique. The Galerkin and Fréchet derivative approaches were used to determine the nonlinear free vibration behaviour of the carbon nano-cone, which was modelled in the simply- and clamped-supported boundary conditions. Comparisons were made between the findings from the new model versus the couple and classical stress theories, indicating that the classical and modified couple stress theories are distinct representations of modified strain gradient theory. The results also revealed that the degree of hardening of nano-truncated conical shells in the modified strain gradient theory is less than that of modified couple stress and classical theories. This led to a rise in the non-dimensional amplitude and frequency ratios. This study investigated the effect of size on free nonlinear vibrations of nano-truncated conical shells for various apex angles and lengths. Finally, we evaluated and compared our findings versus those reported by previous studies, which confirmed the precision and accuracy of our results.

A novel formulation of the weak form quadrature element method, referred to as the locally adaptive weak quadrature element method, is proposed to develop elements for nonlinear graded strain gradient Timoshenko and Euler–Bernoulli nanobeams. The equations of motion are obtained based on Hamilton principle while accounting for the position of the physical neutral axis. The proposed elements use Gauss quadrature points to ensure full integration of the variational statement. The proposed formulation develops matrices based on the differential quadrature method which employs Lagrange-based polynomials. These matrices can be modified to accommodate any number of extra derivative degrees of freedom including third-order beams and higher-order strain gradient beams without requiring an entirely new formulation. The performance of the proposed method is evaluated based on the free vibration response of the linear and nonlinear strain gradient Timoshenko and Euler–Bernoulli nanobeams. Both linear and nonlinear frequencies are evaluated for a large number of configurations and boundary conditions. It is shown that the proposed formulation results in good accuracy and an improved convergence speed as compared to the locally adaptive quadrature element method and other weak quadrature element methods available in the literature.

In this paper, differential transform method is applied for longitudinal vibration analysis of beams with non-uniform cross section. Natural frequencies and corresponding normalized mode shapes are calculated for different cases in cross section and boundary conditions. Comparison of results with the previous solutions proves the accuracy and versatility of the presented paper.     2 0 sin A xA a bx  . Li (4) combined the transfer matrix method and closed form solution of one step non-uniform rod to obtain a single frequency equation for a multi-step non-uniform rod. In another papers, he derived an exact solution for free longitudinal vibrations of one-step non-uniform rods with classical and non-classical boundary conditions (5) and an analytical solution for determine the longitudinal natural frequencies and mode shapes for a one step non-uniform bar with several boundary conditions (6). Li, et al. (7) found closed form solution for the longitudinal vibration of non-uniform structures with lumped masses and spring supports whose cross-section varies as   1 n

Consideration of spatial variation of the length scale parameter in static and dynamic analyses of functionally graded annular and circular micro-plates

Free vibration analysis of discrete-continuous functionally graded circular plate via the Neumann series method

The finite element method is widely used in strength calculations of machine-building structures and has a significant list of basic finite elements used to build the discrete finite element computable dynamic models of load-lifting cranes. The article describes static and dynamic characteristics of a computable dynamic model (CDM). CDM is designed to determine stiff features (developing stiffness matrix) and to define the deflected mode of the structures by using different means in structural mechanics, which have been chosen by a designer. Dynamic parameters are determined according to CDM to describe eigen and forced oscillations of crane structures under external action. The quality analysis of CDM of bearing metal structures of bridge cranes built on the basis of bar-shaped open and closed profiles and plate finite elements is based on comparison of the total flexural rigidity of the girders and eigen frequencies and eigenforms of oscillations of their CDM. The general methodology for building equations of motion for crane systems with many (n) degrees of freedom is based on their bar and plate finite elements, the latter are based on Kirchhoff theory of plates. Comparative analysis of the eigenforms of oscillations of the plate and bar CDM of crane with 130/32 t and 33.5m span has been given, the advantages of two types of CDM important for design analysis of strength and seismic resistance in designing bridge cranes are revealed. There has been substantiated the need to develop new methods of calculating finite element models taking into account the dimensional effect. The structures are recommended to regard as unified dimensional systems allowing for different types of non-linearity.

In the present paper, a new trigonometric two-variable shear deformation beam nonlocal strain gradient theory is developed and applied to investigate the combined effects of nonlocal stress and strain gradient on the bending, buckling and free vibration analysis of nanobeams. The model introduces a nonlocal stress field parameter and a length scale parameter to capture the size effect. The governing equations derived are solved employing finite element method using a 3-nodes beam element, developed for this purpose. The predictive capability of the proposed model is shown through illustrative examples for bending, buckling and free vibration of nanobeams. Comparisons with other higher-order shear deformation beam theory are also performed to validate its numerical implementation and assess its accuracy within the nonlocal context.

Free vibrations are analyzed for a flexible spinning disk / spindle system with axial translation and rigid-body tilting.Considering the effects of the rigid-body motions of the disk / spindle system, the coupled equations of motion are derived by using Hamilton's principle based upon the Kirchhoff plate theory and von Karman strain theory.The coupled equations are solved by the Galerkin approximation method in order to obtain natural frequencies of the disk / spindle system with axial translation or rigid-body tilting.This study puts a focus on the effects of the rigid-body motions such as axial translation and rigid-body tilting which are coupled with the flexible modes of the disk.