A revisit of spinning disk models. Part II: linear transverse vibrations

In this paper, two recently derived linear models for the transverse vibrations of a spinning plate are considered. The disk is modelled as a pure plate with no membrane effects. Furthermore, the effect of the rotary inertia of the plate is taken into account. The first model is based on the assumption of linear (Kirchhoff) strains. The second model is based on the assumption of non-linear (von Karman) strains. The merits of both models are considered and their predictions are compared with those of the traditional linear model.     

Analysis and modeling of a flexible rectangular cantilever plate

Flexible plate structures with large deflection and rotation are commonly used structures in engineering. How to analyze and solve the cantilever plate with large deflection and rotation is still an unsolved problem. In this paper, a general nonlinear flexible rectangular cantilever plate considering large deflection and rotation angle is modeled, solved and analyzed. Hamilton's principle is applied to obtain the nonlinear differential dynamic equations and boundary conditions by introducing a coordinate transformation between the Cartesian coordinate system and the deformed local coordinate system. Stress function relating to in-plane force resultants and shear forces is given for the first time for complex coupling equations caused by coordinate transformation. The nonlinear equations and the solving method are validated by experiments. Then, harmonic balance method is adopted to get the nonlinear frequency-response curves, which shows strong hardening spring characteristic of this system. Runge–Kutta methods are used to reveal complex nonlinear behaviors such as 5 super-harmonic resonance, bifurcations and chaos for general nonlinear flexible rectangular cantilever plate.     

Nonlinearly coupled in-plane and transverse vibrations of a spinning disk

Previous nonlinear spinning disk models neglected the in-plane inertia of the disk since this permits the use of a stress function. This paper aims to consider the effect of including the in-plane inertia of the disk on the resulting nonlinear dynamics and to construct approximate solutions that capture the new dynamics. The inclusion of the in-plane inertia results in a nonlinear coupling between the in-plane and transverse vibrations of the spinning disk. The full nonlinear partial differential equations are simplified to a simpler nonlinear two degrees of freedom model via the method of Galerkin. A canonical perturbation approach is used to derive an approximate solution to this simpler nonlinear problem. Numerical simulations are used to evaluate the effectiveness of the approximate solution. Through the use of these analytical and numerical tools, it becomes apparent that the inclusion of in-plane inertia gives rise to new phenomena such as internal resonance and the possibility of instability in the system that are not predicted if the in-plane inertia is ignored. It is also demonstrated that the canonical perturbation approach can be used to produce an effective approximate solution.     

Dynamic stability of a porous rectangular plate

The study is devoted to a axial compressed porous-cellular rectangular plate. Mechanical properties of the plate vary across is its thickness which is defined by the non-linear function with dimensionless variable and coefficient of porosity. The material model used in the current paper is as described by Magnucki, Stasiewicz papers. The middle plane of the plate is the symmetry plane. First of all, a displacement field of any cross section of the plane was defined. The geometric and physical (according to Hook's law) relationships are linear. Afterwards, the components of strain and stress states in the plate were found. The Hamilton's principle to the problem of dynamic stability is used. This principle was allowed to formulate a system of five differential equations of dynamic stability of the plate satisfying boundary conditions. This basic system of differential equations was approximately solved with the use of Galerkin's method. The forms of unknown functions were assumed and the system of equations was reduced to a single ordinary differential equation of motion. The critical load determined used numerically processed was solved. Results of solution shown in the Figures for a family of isotropic porous-cellular plates. The effect of porosity on the critical loads is presented. In the particular case of a rectangular plate made of an isotropic homogeneous material, the elasticity coefficients do not depend on the coordinate (thickness direction), giving a classical plate. The results obtained for porous plates are compared to a homogeneous isotropic rectangular plate. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the linearization of the equations of motion of a rotating disk

In this paper, we present a consistent approach to reduce the fully nonlinear equations of a rotating disk to the classical linear equation derived by Lamb and Southwell and the nonlinear equations derived by Nowinski. The approach recognizes the fact that the out-of-plane deflection and the in-plane deflections are of different orders of magnitude. By using the ratio between the plate thickness and the outer radius as a measurement and carefully examining the reasonable magnitudes of all the variables involved, the fully nonlinear equations can be non-dimensionalized with all the terms being sorted according to their orders of magnitude. It is found that the classical linear equation derived by Lamb and Southwell can be recovered if all the terms of the lowest order of magnitude in the fully nonlinear equations are retained. If all the terms of the lowest two orders of magnitude are retained, Nowinski’s equations can then be recovered. Furthermore, the terms arising from in-plane deformation and rotary inertia are of the highest order and can be ignored in most of the applications.     

Vibration suppression of self-excited oscillations by parametric inertia excitation

The main objective of this contribution is to show the phenomenon of full vibration suppression of a simple two degrees of freedom rotary oscillator by interaction between self-excitation and parametric excitation. One disk is under the influence of self-excitation, modelled by a negative damping coefficient, while the moment of inertia of the second disk is periodically varied in time within an open-loop control with a fixed frequency. Both disks are coupled by a linear spring-element. Parametric excitation develops equations of motion with time-periodic coefficients. Using the averaging method for a firstorder approximation general conditions for full vibration suppression are analytically derived for the two degrees of freedom system with harmonic inertia variation. The approximated analytical stability predictions are verified and compared to results obtained from numerical time integration of the original equations of motion. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vibration and stability analysis of a simply-supported Rayleigh beam with spinning and axial motions

The vibration and stability of a simply supported beam are analyzed when the beam has an axially moving motion as well as a spinning motion. When a beam has spinning and axial motions, rotary inertia plays an important role on the lateral vibration. Compared to previous studies, the present study adopts the Rayleigh beam theory and derives more exact kinetic energy and equations of motion. The rotary inertia terms derived by the present study are compared to those of the previous studies. We investigate the natural frequencies between the present and previous studies. In addition, the critical speed and stability boundary for the spinning and moving speeds are also analyzed. It can be observed from the computed natural frequencies and dynamic responses that the present equations of motion are more reliable than the previous equations because the present equations fully consider the rotary inertia terms.     

Electro-mechanical shear buckling of piezoelectric nanoplate using modified couple stress theory based on simplified first order shear deformation theory

This paper studies the electro-mechanical shear buckling analysis of piezoelectric nanoplate using modified couple stress theory with various boundary conditions.In order to be taken electric effects into account, an external electric voltage is applied on the piezoelectric nanoplate. The simplified first order shear deformation theory (S-FSDT) has been employed and the governing differential equations have been obtained using Hamilton's principle and nonlinear strains of Von-Karman. The modified couple stress theory has been applied to considering small scale effects. An analytical approach was developing to obtain exact results with various boundary conditions. After all, results have been presented by change in some parameters, such as; aspect ratio, effect of various boundary conditions, electric voltage and length scale parameter influences. At the end, results showed that the effect of external electric voltage on the critical shear load occurring on the piezoelectric nanoplate is insignificant.     

Mathematical Modeling and Simulation of a Flexible Shaft-Flexible Link System With End Mass

In this study, the equation of motion of a single link flexible robotic arm with end mass, which is driven by a flexible shaft, is obtained by using Hamilton's principle. The physical system is considered as a continuous system. As a first step, the kinetic energy and the potential energy terms and the term for work done by the nonconservative forces are established. Applying Hamilton's principle the variations are calculated and the time integral is constructed. After a series of mathematical manipulations the coupled equations of motion of the physical system and the related boundary conditions are obtained. Numerical solutions of equations of motion are obtained and discussed for verification of the model used.     

Application of hetero junction CNTs as mass nanosensor using nonlocal strain gradient theory: An analytical solution

This paper aims to propose an analytical solution for dynamic analysis of the hetero junction carbon nanotubes (HJCNTs)-based mass nanosensors using a nonlocal strain gradient Timoshenko beam model. To have a more precise nanosensor, it is necessary to have deep information about the vibration characteristics of the nanostructure. So, two main goals are followed in this paper. At first, the vibration of HJCNTs with general (elastic) boundary conditions and without attached mass are studied using a proposed analytical solution. Afterward, the HJCNT is applied as a cantilever mass sensor for sensing light as well as heavy masses attached to its tip. For the large and heavy masses, the rotary inertia of the attached mass is also considered in the analysis. The governing differential equations are derived based on the Hamilton's principle and solved by an analytical method, which is based on the modified Fourier series. The weighted residual method is employed for obtaining the variationally consistent boundary conditions using the known equations of motion of the structure. The field quantities are obtained in the closed forms. The convergence and accuracy of the proposed solution are validated through some special cases available in the literature. The effects of small scale parameters and the elastic boundaries on the frequency and mode shapes of HJCNTs are studied. Moreover, the factors that affect the frequency shift of HJCNT-mass sensor are discussed. The obtained results introduce HJCNTs as new mass nanosensors that can operate more efficiently than uniform CNTs. This paper can be greatly useful in designing HJCNT-mass sensors and may serve as a benchmark for the future research in this field.     

Vibration analysis of Nano-Rotor's Blade applying Eringen nonlocal elasticity and generalized differential quadrature method

This study investigates the small scale effect on the flapwise bending vibrations of a rotating nanoplate. The nanoplate is modeled with a classical plate theory and considering cantilever and propped cantilever boundary conditions. Due to the rotation, the axial forces are included in the model as true spatial variation. Hamilton's principle is used to derive the governing equation and boundary conditions of the classical plate theory based on Eringen's nonlocal elasticity theory. The generalized differential quadrature method is employed to solve the governing equation. The effect of small-scale parameter, non-dimensional angular velocity, non-dimensional hub radius, aspect ratio, and different boundary conditions in the first four non-dimensional frequencies is discussed. Due to considering rotating effects, results of this study are applicable in nano-machines such as nano-motors and nano-turbines and other nanostructures.     

Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects

In this paper, the linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams are studied based on the Gurtin–Murdoch surface stress theory. Firstly, the constitutive equations of fractional viscoelasticity theory are considered, and based on the Gurtin–Murdoch model, stress components on the surface of the nanobeam are incorporated into the axial stress tensor. Afterward, using Hamilton's principle, equations governing the two-dimensional vibrations of fractional viscoelastic nanobeams are derived. Finally, two solution procedures are utilized to describe the time responses of nanobeams. In the first method, which is fully numerical, the generalized differential quadrature and finite difference methods are used to discretize the linear part of the governing equations in spatial and time domains. In the second method, which is semi-analytical, the Galerkin approach is first used to discretize nonlinear partial differential governing equations in the spatial domain, and the obtained set of fractional-order ordinary differential equations are then solved by the predictor–corrector method. The accuracy of the results for the linear and nonlinear vibrations of fractional viscoelastic nanobeams with different boundary conditions is shown. Also, by comparing obtained results for different values of some parameters such as viscoelasticity coefficient, order of fractional derivative and parameters of surface stress model, their effects on the frequency and damping of vibrations of the fractional viscoelastic nanobeams are investigated.     

Geometrically nonlinear analysis of Timoshenko piezoelectric nanobeams with flexoelectricity effect based on Eringen's differential model

This study analyzes the nonlinear free vibration and post-buckling of nanobeams with flexoelectric effect based on Eringen's differential model. The nanobeam is modeled based on Timoshenko beam's theory. The von-Kármán strain–displacement relation together with the electrical Gibbs free energy and Hamilton's principle are employed to derive equations of motion. The nonlinear free vibration frequencies are obtained for pinned–pinned (P–P) and clamped–clamped (C–C) boundary conditions. Multiple scales method is employed to obtain the closed-form solution for the nonlinear governing equations. By employing this methodology, the natural frequencies of nanobeams are obtained and their post-buckling behavior is examined. The influence of nonlocal parameter, amplitude ratio, and input voltage on the top surface and flexoelectricity constant on nonlinear free vibration and post-buckling characteristics of nanobeam is investigated. In this paper, it is concluded that the flexoelectricity has a significant effect on free vibration of the beams in nano-scale and its effect has to be considered in designing nano-electro-mechanical systems (NEMS) such as nano- generators and nano-sensors.     

Influence of nonlinear terms on dynamical behavior of graphene reinforced laminated composite plates

This research is focused on the effects of nonlinear terms on the dynamical behavior of graphene reinforced laminated composite plates. Firstly, the governing equations of the graphene reinforced composite thin plate subjected to transverse excitations are derived by using the Hamilton's principle and the von Karman deformation theory. Then numerical method is applied to investigate the nonlinear behaviors of graphene reinforced composite plates. Bifurcation diagram, waveform and phase portrait are demonstrated to analyze the nonlinear dynamics of the graphene reinforced laminated composite plates. Furthermore, the effects of nonlinear terms on the dynamical behavior are discussed in detail, where both the stronger and weaker nonlinear characteristics of lower modes of the plate are presented. Moreover, some interesting phenomena are obtained in numerical simulation.     

Characterization of a nonlinear MEMS-based piezoelectric resonator for wideband micro power generation

Micro-scale piezoelectric unimorph beams with attached proof masses are the most prevalent structures in MEMS-based energy harvesters considering micro fabrication and natural frequency limitations. In doubly clamped beams a nonlinear stiffness is observed as a result of midplane stretching effect which leads to amplitude-stiffened Duffing resonance. In this study, a nonlinear model of a doubly clamped piezoelectric micro power generator, taking into account geometric nonlinearities including stretching and large curvatures, is investigated. The governing nonlinear coupled electromechanical partial differential equations of motion are determined by exploiting Hamilton's principle. A semi-analytical approach implementing the perturbation method of multiple scales is used to solve the nonlinear coupled differential equations and analyze the primary and superharmonic resonances. Results indicate that operational bandwidth of the nonlinear harvester is enhanced considerably with respect to linear models. Moreover considerable amount of power is generated due to occurrence of superharmonic resonances. This yields to extraction of energy at subharmonics of the natural frequency which is crucially important in MEMS-based harvesters.     

Nonlinear modeling and analysis of electrically actuated viscoelastic microbeams based on the modified couple stress theory

A nonclassical nonlinear continuum model of electrically actuated viscoelastic microbeams is presented based on the modified couple stress theory to consider the microstructure effect in the framework of viscoelasticity. The nonlinear integral-differential governing equation and related boundary conditions of are derived based on the extended Hamilton's principle and Euler–Bernoulli hypothesis for viscoelastic microbeams with clamped-free, clamped-clamped, simply-supported boundary conditions. The proposed model accounts for system nonlinearities including the axial residual stress, geometric nonlinearity due to midplane stretching, electrical forcing with fringing effect. The behavior of the microbeam is simulated using generalized Maxwell viscoelastic model. A new generalized differential/integral quadrature method is developed to solve the resulting governing equation. The developed model is verified against elastic behavior and a favorable agreement is obtained. Efficiency of the developed model is demonstrated by analyzing the quasistatic pull-in phenomena of electrically actuated viscoelastic microbeams with different boundaries at various material length scale parameters and axial residual stresses in the framework of linear viscoelasticity.     

A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells

Nonlinear vibration analysis of circular cylindrical shells has received considerable attention from researchers for many decades. Analytical approaches developed to solve such problem, even not involved simplifying assumptions, are still far from sufficiency, and an efficient numerical scheme capable of solving the problem is worthy of development. The present article aims at devising a novel numerical solution strategy to describe the nonlinear free and forced vibrations of cylindrical shells. For this purpose, the energy functional of the structure is derived based on the first-order shear deformation theory and the von–Kármán geometric nonlinearity. The governing equations are discretized employing the generalized differential quadrature (GDQ) method and periodic differential operators along axial and circumferential directions, respectively. Then, based on Hamilton's principle and by the use of variational differential quadrature (VDQ) method, the discretized nonlinear governing equations are obtained. Finally, a time periodic discretization is performed and the frequency response of the cylindrical shell with different boundary conditions is determined by applying the pseudo-arc length continuation method. After revealing the efficiency and accuracy of the proposed numerical approach, comprehensive results are presented to study the influences of the model parameters such as thickness-to-radius, length-to-radius ratios and boundary conditions on the nonlinear vibration behavior of the cylindrical shells. The results indicate that variation of fundamental vibrational mode shape significantly affects frequency response curves of cylindrical shells.     

A Nonlinear Sandwich Shell Theory Accounting for Transverse Core Compressibility

The present study is concerned with an advanced theory of sandwich shells with transversely compressible core. The model is based on the standard Kirchhoff‐Love hypothesis for the face sheets and a third‐order displacement expansion for the core. Consistent equations of motion and boundary conditions are derived by means of Hamilton's variational principle. The model is applied to a postbuckling analysis of cylindrical shells under axial compression.     

Vibration of fluid-conveying nanotubes subjected to magnetic field based on the thin-walled Timoshenko beam theory

In this article, the flutter vibrations of fluid-conveying thin-walled nanotubes subjected to magnetic field is investigated. For modeling fluid structure interaction, the nonlocal strain gradient thin-walled Timoshenko beam model, Knudsen number and magnetic nanoflow are assumed. The Knudsen number is considered to analyze the slip boundary conditions between the fluid-flow and the nanotube's wall, and the average velocity correction parameter is utilized to earn the modified flow velocity of nano-flow. Based on the extended Hamilton's principle, the size-dependent governing equations and associated boundary conditions are derived. The coupled equations of motion are transformed to a general eigenvalue problem by applying extended Galerkin technique under the cantilever end conditions. The influences of nonlocal parameter, strain gradient length scale, magnetic nanoflow, longitudinal magnetic field, Knudsen number on the eigenvalues and critical flutter velocity of the nanotubes are studied.     

Development of size-dependent consistent couple stress theory of Timoshenko beams

In this paper, a linear size-dependent Timoshenko beam model based on the consistent couple stress theory is developed to capture the size effects. The extended Hamilton's principle is utilized to obtain the governing differential equations and boundary conditions. The general form of boundary conditions and the concentrated loading are employed to determine the exact static/dynamic solution of the beam. Utilizing this solution for the beam's deformation and rotation, the exact shape functions of the consistent couple stress theory (C-CST) is extracted, which leads to the stiffness and mass matrices of a two-node C-CST finite element beam. Due to the complexity and high computational cost of using the exact solution's shape functions, in addition to the Ritz approximate solution, a two primary variable finite element model of C-CST is proposed, and the corresponding general deformation and rotation fields, shape functions, mass and stiffness matrices are calculated. The C-CST is validated by comparing the prediction of different beam models for a benchmark problem. For the fully and partially clamped cantilever, and free-free beams, the size dependency of the formulations is investigated. The static solutions of the classical and consistent couple stress Timoshenko beam models are compared, and a criterion for selecting the proper model is proposed. For a wide range of material properties, the relation between the beam length and length scale parameter is derived. It is shown that the validity domain of the consistent couple stress Timoshenko model barely depends on the beam's constituent material.