A (3,2) reduced degree-of-freedom unified zigzag laminated beam theory

A (3,2) unified zigzag beam theory is developed with a reduced number of degree-of-freedom. Comparing to previous methods in the field of zigzag beam theory, the main novelty in this paper's method is that a more general non-vanishing top/bottom surface's shear stress boundary conditions are satisfied automatically in strong form. The bottom surface shear stress condition and the interface shear stress continuity conditions are used to uniquely determine the coefficients of zigzag functions. For the top surface shear stress condition, it is used to eliminate one degree-of-freedom, changing the 7°-of-freedom (3,2) zigzag beam to a 6°-of-freedom (3,2) zigzag beam. The zigzag coefficients are derived with an explicit formulation. Since the proposed method's formula is based on the unified beam theory, the formulation can be applied to any specific beam theory. The corresponding zigzag coefficients are also dependent on the specific beam theory's thickness basis function.In the numerical test section, several benchmark problems are solved to verify the accuracy. It is observed that the proposed beam has accurate solution for both thick and thin beams. The shear stress accuracy is also good for both vanishing and non-vanishing shear stress boundary conditions on top/bottom surfaces.     

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.     

Green's functions for forced vibration analysis of bending-torsion coupled Timoshenko beam

A method based on Green's functions is proposed for the analysis of the steady-state dynamic response of bending-torsion coupled Timoshenko beam subjected to distributed and/or concentrated loadings. Damping effects on the bending and torsional directions are taken into account in the vibration equations. The elastic boundary conditions with bending-torsion coupling and damping effects are derived and the classical boundary conditions can be obtained by setting the values of specific stiffness parameters of the artificial springs. The Laplace transform technology is employed to work out the Green's functions for the beam with arbitrary boundary conditions. The Green's functions are obtained for the beam subject to external lateral force and external torque, respectively. Coupling effects between bending and torsional vibrations of the beam can be studied conveniently through these analytical Green's functions. The direct expressions of the steady-state responses with various loadings are obtained by using the superposition principle. The present Green's functions for the Timoshenko beam can be reduced to those for Euler–Bernoulli beam by setting the values of shear rigidity and rotational inertia. In order to demonstrate the validity of the Green's functions proposed, results obtained for special cases are given for a comparison with those given in the literature and they agree with each other exactly. The influences of external loading frequency and eccentricity on Green's functions of bending-torsion coupled Timoshenko beam are investigated in terms of the numerical results for both simply supported and cantilever beams. Moreover, the symmetric property of the Green's functions and the damping effects on the amplitude of Green's functions of the beam are discussed particularly.     

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.     

Bending of a two-layer beam with non-rigid contact between the layers

The bending, under plane stress state conditions, of a two-layer beam-strip with identical isotropic linearly elastic layers with non-rigid contact between them is considered. The effect of the contact interaction between the layers, simulated by an elastic or elastoplastic gasket of negligibly small thickness with a finite shear stiffness, on the deflection of the beam is studied. Absolute slippage and rigid contact between the layers are the two limiting values of the shear stiffness. The values of the flexural stiffness of the beam differ by a factor of four in these limiting situations. The problem is reduced to a one- dimensional problem in the case of harmonic external load and an asymptotic solution is constructed for it. In the case of a load of general form, the Kirchhoff - Love hypotheses are used to construct an approximate solution and the problem is reduced to a one-dimensional problem. The difficulties which arise in simulating of the interaction forces between the layers using Coulombic dry friction forces are discussed.     

A mathematical model for beams on geosynthetic reinforced earth beds under strip loading

In this study, an attempt has been made to analyze a beam on geosynthetic reinforced earth beds subjected to strip loading. Geosynthetic layer has been assumed to have finite bending stiffness and therefore idealized as a beam. The foundation beam has been placed on compacted granular soil layer overlying the geosynthetic layer below which lies on the original weak/loose soil deposit. The upper dense and lower loose soil layers have been idealized as Winkler springs of different stiffnesses. Governing differential equations for the flexural response of the system have been derived and presented in non-dimensional form. These equations have been solved using appropriate boundary and continuity conditions. It was possible to obtain a closed form analytical solution for such a foundation system.     

Wave propagation in,layered, transversally isotropic,elastic media

Wave propagation in a transversally isotropic, elastic medium consisting of plane-parallel layers and half spaces is considered. A generalized matrix method is used to derive the dispersion equation of this medium and to find the coefficients of reflection and refraction. This method makes it possible to consider dispersion curves and the coeffients of reflection and refraction in a broader domain than with Haskell's method. The results obtained generalize to layers in which the elastic characteristics vary with depth according to an arbitrary law. For such layers it is possible to find matrices in the form of series which converge rapidly for low and high frequencies. Moreover, a rule is formulated which makes it possible on the basis of a known field in an isotropic medium to find the field in the corresponding transversally isotropic medium.     

Study on vibration and stability of an axially translating viscoelastic Timoshenko beam: Non-transforming spectral element analysis

Engineering systems, such as rolled steel beams, chain and belt drives and high-speed paper, can be modeled as axially translating beams. This article scrutinizes vibration and stability of an axially translating viscoelastic Timoshenko beam constrained by simple supports and subjected to axial pretension. The viscoelastic form of general rheological model is adopted to constitute the material of the beam. The partial differential equations governing transverse motion of the beam are derived from the extended form of Hamilton's principle. The non-transforming spectral element method (NTSEM) is applied to transform the governing equations into a set of ordinary differential equations. The formulation is similar to conventional FFT-based spectral element model except that Daubechies wavelet basis functions are used for temporal discretization. Influences of translating velocities, axial tensile force, viscoelastic parameter, shear deformation, beam model and boundary condition types are investigated on the underlying dynamic response and stability via the NTSEM and demonstrated via numerical simulations.     

Asymmetric Green's functions for a functionally graded transversely isotropic tri-material

This paper considers the elastic analysis of a functionally graded transversely isotropic tri-material solid under the arbitrary distribution of applied static loads. Using two displacement potential functions, for three-dimensional point-load and patch-load configurations, Green's functions for displacement and stress components are generated in the form of infinite line-integrals. These solutions are shown to be analytically reducible to the special cases of exponentially graded bi-material, exponentially graded half-space and homogeneous tri-material Green's functions. It also encompasses a functionally graded finite layer on a rigid base with surface loading with two cases of interfacial conditions, rigid-bonded and rigid-frictionless. Finally, for the special case of a functionally graded layer sandwiched between two homogeneous layers, using several numerical displays, the effect of material inhomogeneity on the responses is studied and the accuracy of numerical scheme is verified.     

Mechatronic system with shunted piezoelectric damper modelled with structural damping

Paper presents analysis of an one-dimension flexural vibrating mechatronic system. The considered system is a cantilever beam with a piezoelectric transducer bonded to the beam's surface. An external electric circuit is adjoined to the transducer's clamps in order to damp vibrations. System was analyzed on the basis of an approximate Galerkin method. Verification and assumptions of the approximate method were described in the previous papers where analysis of the mechatronic system with piezoelectric shunt damper was presented. Structural damping of all system's components was being taken into consideration. Rheological properties were introduced using Kelvin-Voigt model of materials. Influences of component's structural damping coefficients values on the system's dynamic flexibility were defined. Obtained results were presented on 3D graphs as dynamic flexibility dependence on the structural damping coefficient and frequency of an external force that was applied to the system. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite element based nonlinear dynamic response of elastically supported piezoelectric functionally graded beam subjected to moving load in thermal environment with random system properties

The second order statistics in terms of mean and standard deviation (SD) of normalized nonlinear transverse dynamic central deflection (NTDCD) response of un-damped elastically supported functionally graded materials (FGMs) beam with surface-bonded piezoelectric layers under the action of moving load are investigated in this paper. The random system properties such as Young's modulus, Poisson's ratio, density, thermal expansion coefficients, piezoelectric materials, volume fraction exponent and external loading are modeled as uncorrelated random variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics combined with Newton–Raphson technique through Newmark's time integrating scheme using finite element method (FEM). The non-uniform temperature distribution with temperature dependent material properties is taken into consideration for consideration of thermal loading. The one parameter Pasternak elastic foundation with Winkler cubic nonlinearity is considered as an elastic foundation. The stochastic based second order perturbation technique (SOPT) and direct Monte Carlo simulation (MCS) are adopted for the solution of nonlinear dynamic governing equation. The influences of volume fraction exponents, temperature increments, moving loads and velocity, nonlinearity, slenderness ratios, foundation parameters and external loadings with random system properties on the NTDCD are examined. The capability of present stochastic model in predicting the NTDCD statistics are compared by studying their convergence with the existing results those available in the literature.     

Vector field reconstruction via quaternionic setting

We propose the reconstruction of the solenoidal part of a vector field supported in the unit ball in 3 dimensions by using cone beam data from a curve surrounding it, and this curve satisfies the Tuy's condition of order 3. We use the quaternionic inversion formula to decompose the solenoidal part of a vector field into 2 parts. To recover the first one, which is the main part of the solenoidal component, another definition of a cone beam transform containing both Doppler and transverse data will be introduced. The second part will be reconstructed by using information from the first part as in Katsevich and Schuster's work with less data.     

Asymptotic analysis of the nonsteady micropolar fluid flow through a curved pipe

ABSTRACT

We consider the nonsteady flow of a micropolar fluid in a thin (or long) curved pipe via rigorous asymptotic analysis. Germano's reference system is employed to describe the pipe's geometry. After writing the governing equations in curvilinear coordinates, we construct the asymptotic expansion up to a second order. Obtained in the explicit form, the asymptotic approximation clearly demonstrates the effects of pipe's distortion, micropolarity and the time derivative. A detailed study of the boundary layers in space is provided as well as the construction of the divergence correction. Finally, a rigorous justification of the proposed effective model is given by proving the error estimates.     

Swelling and tilting in smectic layers

Smectic liquid crystals are quasi-solid materials, in that they possess microstructure both of the material and the local type (the nematic microstructure and the lamellae). In the standard approach, the rod-like molecules are supposed to be strictly normal to the layers; here we abandon this constraint and admit the possibility for the molecules to change orientation with respect to the layers, paying in energy. As a consequence, we obtain two equations: one expresses the balance of micromomentum, the other is Cauchy's equation. We derive also their form in the linear approximation and solve two typical problems.     

Active control of dynamic behaviors of graded graphene reinforced cylindrical shells with piezoelectric actuator/sensor layers

This work investigates the active vibration control and vibration characteristics of a sandwich thin cylindrical shell whose intermediate layer is made of the graphene reinforced composite that is bonded with integrated piezoelectric actuator and sensor layers at its outer and inner surfaces. The volume fraction of graphene platelets in the intermediate layer varies continuously in the shell's thickness direction, which generates position-dependent effective material properties. The constitutive relations of the graphene reinforced composite and piezoelectric materials are given by taking one-dimensional steady thermal field into account. Considering Donnell's shell theory, a final equation of motion in terms of the generalized radial displacement is derived by using Hamilton's principle and Galerkin method. Shell's natural frequencies are derived considering influences of the thermo-electro-elastic field. Introducing a constant velocity feedback control algorithm, active vibration control of the sandwich cylindrical shell is presented by employing the Runge-Kutta method. The feedback control gain has a pronounced effect on the damping, as well as the inertia of the system. Comparisons between the present results and those in other papers are done to validate the present solutions. Influences of weight fractions, distribution patterns and geometrical sizes of graphene platelets, temperature variations, thicknesses of layers and the feedback control gain on the vibration characteristics and active vibration control behaviors of the novel sandwich cylindrical shell are discussed.     

The geometrically nonlinear dynamic responses of simply supported beams under moving loads

This paper presents a method for determining the nonlinear dynamic responses of structures under moving loads. The load is considered as a four degrees-of-freedom system with linear suspensions and tires flexibility, and the structure is modeled as an Euler–Bernoulli beam with simply supported at both ends. The nonlinear dynamic interaction of the load–structure system is discussed, and Kelvin−Voigt material model is employed for the beam. The nonlinear partial differential equations of the dynamic interaction are derived by using the von Kármán nonlinear theory and D'Alembert's principle. Based on the Galerkin method, the partial differential equations of the system are transformed into nonlinear ordinary equations, which can be solved by using the Newmark method and Newton−Raphson iteration method. To validate the approach proposed in this paper, the comparison are performed using a moving mass and a moving oscillator as the excitation sources, and the investigations demonstrate good reliability.     

The discrete-continuous model of the one-dimension vibrating mechatronic system

This work presents the analysis of the flexural vibrating one-dimension mechatronic system – the cantilever beam and the piezoelectric transducer bonded with the beam's surface by means of a connection layer. The external RC circuit is adjoined to the transducer's clamps. Dynamic equations of motion of the considered mechatronic system were written down using discrete – continuous mathematical model, taking into consideration the influence of the connection layer and the external electric circuit. The dynamic flexibility of the mechatronic system was assigned on the basis of the approximate Galerkin's method. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)