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)     

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.     

A revisit of spinning disk models. Part I: derivation of equations of motion

Previous models of spinning disks have focused on modelling the disk as a spinning membrane. The effect of bending stiffness was then incorporated by adding the appropriate term to the previously derived spinning membrane equation. A pure spinning plate model does not exist in the literature. Furthermore, in both existing linear and nonlinear models of spinning disks, the in-plane inertia and rotary inertia of the disk have been ignored. This paper revisits the derivation of the equations of motion of a spinning plate. The derivation focuses on the use of Hamilton's principle with linear Kirchhoff and nonlinear von Karman strain expressions. In-plane and rotary inertias of the plate are automatically taken into account. The use of Hamilton's principle guarantees the correct derivation of the corresponding boundary conditions. The resulting equations and boundary conditions are discussed.     

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.     

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.     

LQR Control for Vibration Suppression of Piezoelectric Integrated Smart Structures

In the present paper, the equations of motion for piezoelectric integrated smart beams are derived by applying Hamilton's principle and the Finite Element (FE) method, based on the First-Order Shear Deformation (FOSD) theory. Then, a state space model is constructed from the equations of motion for control design. A Linear Quadratic Regulator (LQR) control for vibration suppression is implemented on the mathematical model of the piezoelectric bonded beam. Finally, a numerical application is performed to testify the applicability and effectiveness of the present method. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Integrators for Thermo-Viscoelastic Discrete Systems

Variational integrators are modern time-integration schemes based on a discretization of the underlying variational principle. In this paper, Hamilton's principle is approximated by an action sum, whose vanishing variation results in discrete Euler-Lagrange equations or, equivalently, in discrete evolution equations for the position and momentum. In order to include the viscous and thermal virtual work (mechanical and thermal virtual dissipation), Hamilton's principle is extended by D'Alembert terms, which account for the time evolution equation of the internal variable and Fourier's law. From this variational formulation, variational integrators using different orders of approximation of the state variables as well as of the quadrature of the action integral are constructed and compared. A thermo-viscoelastic double pendulum comprised of two discrete masses connected by generalized Maxwell elements, and subject to heat conduction between them serves as a discrete model problem. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic stability of a porous cylindrical shell

This paper is devoted to a closed cylindrical shell made of a porous-cellular material. The mechanical properties vary continuously on the thickness of a shell. The mechanical model of porosity is as described as presented by Magnucki, Stasiewicz. A shell is simply supported on edges. On the ground of assumed displacement functions the deformation of shell is defined. The displacement field of any cross section and linear geometrical and physical relationships are assumed in cylindrical coordinate system. The components of deformation and stress state were found. Using the Hamilton's principle the system of differential equations of dynamic stability is obtained. The forms of unknown functions are assumed and the system of a differential equations is reduced to a simple ordinary equation of dynamic stability of shell (Mathieu's equation). The derived equation are used for solving a problem of dynamic stability of porous-cellular shell with intensity of load directed in generators of shell. The critical loads are derived for a family of porous shells. The unstable space of family porous shells is found. The influence a coefficient of porosity on the stability regions in Figures is presented. The results obtained for porous shell are compared to a homogeneous isotropic cylindrical shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Piezoelectric energy harvesting from vibrations of a beam subjected to multi-moving loads

This study is intended to investigate piezoelectric energy harvesting from vibrations of a beam induced by multi-moving loads. Various multi-moving loads are analyzed by considering various parameters. The system of equations for electro-mechanical materials is derived by using the generalized Hamilton's principle under the assumptions of the Euler–Bernoulli beam theory. The electromechanical behavior of piezoelectric harvesters in a unimorph configuration is analyzed using finite element method. The Newmark's explicit integration technique is adopted for the transient analysis. The predictions of the results of the finite element models are verified by that of the available solutions. The effects of piezoelectric bonding location, velocity and number of moving loads as well as time lags between moving loads on the produced power are investigated. The numerical results show that the investigated parameters have significant effects on the energy harvesting from a vibration of beams under the action of multi-moving loads.     

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.     

Newton's second law in field theory

In this article we present a natural generalization of Newton's Second Law valid in field theory, i.e., when the parameterized curves are replaced by parameterized submanifolds of higher dimension. For it we introduce what we have called the geodesic k-vector field, analogous to the ordinary geodesic field and which describes the inertial motions (i.e., evolution in the absence of forces). From this generalized Newton's law, the corresponding Hamilton's canonical equations of field theory (Hamilton-De Donder-Weyl equations) are obtained by a simple procedure. It is shown that solutions of generalized Newton's equation also hold the canonical equations. However, unlike the ordinary case, Newton equations determined by different forces can define equal Hamilton's equations.     

A transformation of the equations of mechanics

The structure of the equations obtained from Hamilton's equations by a coordinate Legendre transformation of the Hamilton function is discussed. Examples are considered.     

Forms of Hamilton's principle in quasi-coordinates

In a previous paper [1], certain conditions, due to Hp̈lder, Voronets and Suslov in the case of linear constraints, for deriving three forms of Hamilton's principle in generalized coordinates and velocities for the general case of non-linear non-holonomic constraints were analysed. It was shown that these three forms are equivalent and transform into one another. As a sequel to that analysis, similar issues are investigated for the case of non-linear quasi-coordinates and quasi-velocities and, in addition, the three forms of Hamilton's principle are exhibited in the case of a Legendre transformation, which transforms the equation of motion to canonical form in quasi-coordinates.     

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.     

A Note on a Nonlinear Model of a Piezoelectric Rod

If piezoceramics are excited by weak electric fields a nonlinear behavior can be observed, if the excitation frequency is close to a resonance frequency of the system. To derive a theoretical model nonlinear constitutive equations are used, to describe the longitudinal oscillations of a slender piezoceramic rod near the first resonance frequency. Hamilton's principle is used to receive a variational principle for the piezoelectric rod. Introducing a Rayleigh Ritz ansatz with the eigenfunctions of the linearized system to approximate the exact solution leads to nonlinear ordinary differential equations. These equations are approximated with the method of harmonic balance. Finally it is possible to calculate the amplitudes of the displacements numerically. As a result it is shown, that the Duffing type nonlinearities found in measurements can be described with this model.     

The modeling and dynamic analysis of two jointed beams with clearance

The dynamic behaviors of two beams connected by a joint with clearance are investigated. A new equivalent joint model using the transverse and torsion spring systems is developed to describe the dynamics of two jointed beams with clearance. Based on the finite element method (FEM), the equations of motion of the two jointed beams with clearance are established using Hamilton's principle. A modified numerical solution method is presented based on the Newmark integration method to solve the equations with the nonlinearity due to clearance. The contributions of the clearance size and stiffness on the amplitude-frequency response are discussed. The effects of the clearance on the vibration transfer between the two connected beams and the impact force of the joint are investigated.     

Dynamic modeling for rotating composite Timoshenko beam and analysis on its bending-torsion coupled vibration

A formulation is presented for steady-state dynamic responses of rotating bending-torsion coupled composite Timoshenko beams (CTBs) subjected to distributed and/or concentrated harmonic loadings. The separation of cross section's mass center from its shear center and the introduced coupled rigidity of composite material lead to the bending-torsion coupled vibration of the beams. Considering those two coupling factors and based on Hamilton's principle, three partial differential non-homogeneous governing equations of vibration with arbitrary boundary conditions are formulated in terms of the flexural translation, torsional rotation and angle rotation of cross section of the beams. The parameters for the damping, axial load, shear deformation, rotation speed, hub radius and so forth are incorporated into those equations of motion. Subsequently, the Green's function element method (GFEM) is developed to solve these equations in matrix form, and the analytical Green's functions of the beams are given in terms of piecewise functions. Using the superposition principle, the explicit expressions of dynamic responses of the beams under various harmonic loadings are obtained. The present solving procedure for Timoshenko beams can be degenerated to deal with for Rayleigh and Euler beams by specifying the values of shear rigidity and rotational inertia. Cantilevers with bending-torsion coupled vibration are given as examples to verify the present theory and to illustrate the use of the present formulation. The influences of rotation speed, bending-torsion couplings and damping on the natural frequencies and/or shape functions of the beams are performed. The steady-state responses of the beam subjected to external harmonic excitation are given through numerical simulations. Remarkably, the symmetric property of the Green's functions is maintained for rotating bending-torsion coupled CTBs, but there will be a slight deviation in the numerical calculations.     

On wave propagation in two-dimensional functionally graded porous rotating nano-beams using a general nonlocal higher-order beam model

This paper studies the wave propagation of two-dimensional functionally graded (2D-FG) porous rotating nano-beams for the first time. The rotating nano-beams are made of two different materials, and the material properties of the nano-beams alter both in the thickness and length directions. The general nonlocal theory (GNT) in conjunction with Reddy's beam model are employed to formulate the size-dependent model. The GNT efficiently models the dispersions of acoustic waves when two independent nonlocal fields are modelled for the longitudinal and transverse acoustic waves. The governing equations of motion for the 2D-FG porous rotating nano-beams are established using Hamilton's principle as a function of the axial force due to centrifugal stiffening and displacement. The analytic solution is applied to obtain the results and solve the governing equations. The effect of the features of different parameters such as functionally graded power indexes, porosity, angular velocity, and material variation on the wave propagation characteristics of the rotating nano-beams are discussed in detail.     

Dynamic behaviors of single- and multi-span functionally graded porous beams with flexible boundary constraints

This paper investigates the dynamic behaviors of single- and multi-span functionally graded porous (FGP) beams with flexible boundary constraints modelled by a combination of two-dimensional translational springs and a rotational spring. It is assumed that the pores are distributed either non-uniformly or uniformly according to four porosity distributions and that the material properties vary smoothly along the thickness direction of the beam. The dynamic governing equations are derived from Hamilton's principle within the framework of Timoshenko beam theory and solved by using discrete singular convolution element method (DSCEM) in conjunction with Taylor series expansion (TSE) method. To validate the accuracy of the proposed method, the present results are compared with those in open literature and obtained by finite element method (FEM). A comprehensive parametric study is conducted to investigate the effects of spring constants, boundary condition, porosity distribution, porosity coefficient and beam span ratio on the dynamic behaviors.