Free vibration analysis of functionally graded beams with non-uniform cross-section using the differential transform method

Free vibrations of non-uniform cross-section and axially functionally graded Euler–Bernoulli beams with various boundary conditions were studied using the differential transform method. The method was applied to a variety of beam configurations that are either axially non-homogeneous or geometrically non-uniform along the beam length or both. The governing equation of an Euler–Bernoulli beam with variable coefficients was reduced to a set of simpler algebraic recurrent equations by means of the differential transformations. Then, transverse natural frequencies were determined by requiring the non-trivial solution of the eigenvalue problem stated for a transformed function of the transverse displacement with appropriately transformed its high derivatives and boundary conditions. To show the generality and effectiveness of this approach, natural frequencies of various beams with variable profiles of cross-section and functionally graded non-homogeneity were calculated and compared with analytical and numerical results available in the literature. The benefit of the differential transform method to solve eigenvalue problems for beams with arbitrary axial geometrical non-uniformities and axial material gradient profiles is clearly demonstrated.     

A higher-order micro-beam model with application to free vibration

The free vibration of micro-beams is analyzed employing three different beam models. The previously obtained equations of motion for Bernoulli–Euler and Timoshenko models are solved analytically. A higher-order model is devised, which satisfies the lateral boundary conditions of micro-beams. The equations of motion with associated boundary conditions are derived by means of Hamilton's principle. The generalized differential quadrature method is used for the solution of the equations. The first five natural frequencies are obtained for micro-beams with three length-scale parameter/height ratios and five different boundary conditions.     

Free Vibration of the Axially Loaded System of Two Beams Connected by a Two-Directional Viscoelastic Interlayer

In this paper, the analytical method has been used for solving a problem of free vibration with clamping of an axially loaded sandwich beam. The sandwich beam consists of two external layers connected by a viscoelastic two-directional Winkler interlayer. The upper external layer is described by the Bernoulli–Euler model, which is loaded by a constant axial force. The lower external layer is modeled as the Timoshenko model. The phenomenon of free vibration has been described using a homogenous system of conjugate partial differential equations. After separation of variables in the system of differential equations, the boundary problem has been solved and three complex equations for the definition of frequency and modes of free vibration have been obtained. The free-vibration problem for arbitrarily assumed initial conditions and various axial forces has been considered.     

Shear deformation effect in flexural–torsional vibrations of beams by BEM

In this paper, a boundary element method is developed for the general flexural–torsional vibration problem of Timoshenko beams of arbitrarily shaped cross section taking into account the effects of warping stiffness, warping and rotary inertia and shear deformation. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Both free and forced vibrations are examined. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy.     

Optimal design of dynamically loaded continuous structures

A computational algorithm is developed and applied for optimization of beam and plate structures, subject to constraints on transient dynamic response. A continuous design formulation is retained, with dynamic response governed by partial differential operator equations. Adjoint equations are employed for sensitivity analysis and a function space gradient projection optimization approach is presented. Finite element analysis methods are applied for solution of the system dynamic and adjoint differential equations. Displacement constrained beam and plate minimum weight examples are solved, with a variety of boundary conditions.     

Similarity solutions of unsteady boundary layer equations of a special third grade fluid

Two-dimensional, unsteady, laminar boundary layer equations of a special model of non-Newtonian fluids are considered. The fluid can be considered as a special type of power-law fluid. The problem investigated is the flow over a moving surface, with suction of injection. Two different type of ordinary differential equations system are found using the transformations. Using scaling and translation transformations, equations and boundary conditions are transformed into a partial differential system with two variables. Using translation and a more general transformation, the boundary value problem is transformed into an ordinary differential equations system. Finally, we numerically solve two different ordinary differential equations, separately.     

High-order free vibration analysis of sandwich beams with a flexible core using dynamic stiffness method

In this paper, high-order free vibration of three-layered symmetric sandwich beam is investigated using dynamic stiffness method. The governing partial differential equations of motion for one element are derived using Hamilton’s principle. This formulation leads to seven partial differential equations which are coupled in axial and bending deformations. For the harmonic motion, these equations are divided into two ordinary differential equations by considering the symmetrical sandwich beam. Closed form analytical solutions of these equations are determined. By applying the boundary conditions, the element dynamic stiffness matrix is developed. The element dynamic stiffness matrices are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by use of numerical techniques and the known Wittrick–Williams algorithm. Finally, some numerical examples are discussed using dynamic stiffness method.     

Heat transfer during fluidized-bed coating

A fully implicit numerical method for linear parabolic free boundary problems with coupled and integral boundary conditions is described. The partial differential equation and the boundary conditions are time discretized with the method of lines. An auxiliary function is introduced to remove the coupled and integral boundary conditions from the resulting free boundary problem for ordinary differential equations. Once separated boundary conditions are obtained, invariant imbedding is used to solve the free boundary problem numerically. The method is illustrated by solving the heat transfer equations for the fluidized-bed coating of a thin-walled cylinder.     

Analyses of Circular Magnetoelectroelastic Plates with Functionally Graded Material Properties

A meshless method based on the local Petrov–Galerkin approach is proposed for plate bending analysis with material containing functionally graded magnetoelectroelastic properties. Material properties are considered to be continuously varying along the plate thickness. Axial symmetry of geometry and boundary conditions for a circular plate reduces the original 3D boundary value problem into a 2D problem in axial cross section. Both stationary and transient dynamic conditions for a pure mechanical load are considered in this article. The local weak formulation is employed on circular subdomains in the axial cross section. Subdomains surrounding nodes are randomly spread over the analyzed domain. The test functions are taken as unit step functions in derivation of the local integral equations (LIEs). The moving least-squares (MLS) method is adopted for the approximation of the physical quantities in the LIEs. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. That system is solved numerically by the Houbolt finite-difference scheme as a time-stepping method.     

Free vibration analysis of beams by using a third-order shear deformation theory

In this study, free vibration of beams with different boundary conditions is analysed within the framework of the third-order shear deformation theory. The boundary conditions of beams are satisfied using Lagrange multipliers. To apply the Lagrange’s equations, trial functions denoting the deflections and the rotations of the cross-section of the beam are expressed in polynomial form. Using Lagrange’s equations, the problem is reduced to the solution of a system of algebraic equations. The first six eigenvalues of the considered beams are calculated for different thickness-to-length ratios. The results are compared with the previous results based on Timoshenko and Euler-Bernoulli beam theories.     

Free vibration analysis of sandwich beams using improved dynamic stiffness method

In this paper, free vibration of three-layered symmetric sandwich beam is investigated using dynamic stiffness and finite element methods. To determine the governing equations of motion by the present theory, the core density has been taken into consideration. The governing partial differential equations of motion for one element contained three layers are derived using Hamilton’s principle. This formulation leads to two partial differential equations which are coupled in axial and bending deformations. For the harmonic motion, these equations are combined to form one ordinary differential equation. Closed form analytical solution for this equation is determined. By applying the boundary conditions, the element dynamic stiffness matrix is developed. They are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by the use of numerical techniques and the known Wittrick–Williams algorithm. After validation of the present model, the effect of various parameters such as density, thickness and shear modulus of the core for various boundary conditions on the first natural frequency is studied.     

Logarithmic stress singularities at clamped-free corners of a cantilever orthotropic beam under flexure

The rectangular orthotropic beam under flexure is studied by decomposing the problem into an interior problem and a boundary problem. The interior problem is required to satisfy field equations and boundary conditions at the lateral faces, whereas the boundary conditions at the beam ends are imposed in the average sense. The boundary problem, which reestablishes the pointwise boundary conditions at the beam ends, is solved via eigenfunction expansion under the assumption of transverse inextensibility. It is shown that logarithmic stress singularities are present at the corners of the clamped end section and the corresponding stress-intensity factor is computed.     

Postbuckling characteristics of nanobeams based on the surface elasticity theory

In the present paper, an attempt is made to numerically investigate the postbuckling response of nanobeams with the consideration of the surface stress effect. To accomplish this, the Gurtin–Murdoch elasticity theory is exploited to incorporate surface stress effect into the classical Euler–Bernoulli beam theory. The size-dependent governing differential equations are derived and discretized along with various end supports by employing the principle of virtual work and the generalized differential quadrature (GDQ) method. Newton’s method is applied to solve the discretized nonlinear equations with the aid of an auxiliary normalizing equation. After solving the governing equations linearly, to obtain each eigenpair in the nonlinear model, the linear response is used as the initial value in Newton’s method. Selected numerical results are given to show the surface stress effect on the postbuckling characteristics of nanobeams. It is found that by increasing the thickness of nanobeams, the postbuckling equilibrium path obtained by the developed non-classical beam model tends to the one predicted by the classical beam theory and this anticipation is the same for all selected boundary conditions.     

Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory

Dynamic stability of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and Timoshenko beam theory. This non-classical Timoshenko beam model contains a material length scale parameter and can interpret the size effect. The material properties of FGM microbeams are assumed to vary in the thickness direction and are estimated though Mori–Tanaka homogenization technique. The higher-order governing equations and boundary conditions are derived by using the Hamilton’s principle. The differential quadrature (DQ) method is employed to convert the governing differential equations into a linear system of Mathieu–Hill equations from which the boundary points on the unstable regions are determined by Bolotin’s method. Free vibration and static buckling are also discussed as subset problems. A parametric study is conducted to investigate the influences of the length scale parameter, gradient index and length-to-thickness ratio on the dynamic stability characteristics of FGM microbeams with hinged–hinged and clamped–clamped end supports. Results show that the size effect on the dynamic stability characteristics is significant only when the thickness of beam has a similar value to the material length scale parameter.     

The Timoshenko beam model of the differential quadrature element method

A new numerical approach for solving Timoshenko beam problems is proposed. The approach uses the differential quadrature method (DQM) to discretize the Timoshenko beam equations defined on all elements, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of Timoshenko beam structures. The resulting overall discrete equation can be solved by using a solver of the linear algebra. Numerical results of the DQEM Timoshenko beam model are presented. They demonstrate the DQEM numerical method.     

Flexural-torsional Vibrations of Beams by BEM

In this paper a boundary element method is developed for the general flexural-torsional vibrations of Euler–Bernoulli beams of arbitrarily shaped constant cross-section. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved using the analog equation method, a BEM based method. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. The general character of the proposed method is verified from the fact that all basic equations are formulated with respect to an arbitrary coordinate system, which is not restricted to the principal one. Both free and forced vibrations are examined. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The range of applicability of the thin-tube theory is also investigated through examples with great practical interest.     

Group method analysis of unsteady free-convective laminar boundary-layer flow on a nonisothermal vertical flat plate

The transformation group theoretic approach is applied to present an analysis of the problem of unsteady laminar free convection from a non-isothermal vertical flat plate. The application of two-parameter groups reduces the number of independent variables by two, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The possible forms of surface-temperature variations with position and time are derived. The ordinary differential equations are solved numerically using a fourth-order Runge-Kutta scheme and the gradient method. The heat-transfer characteristics for finite values of the Prandtl number Pr are presented, as temperature and velocity distributions.     

Nonlinear free vibration analysis of generic coupled induced strain actuated piezo-laminated beams

Piezo-laminated thin beams have been analyzed with induced strain actuation using Kirchhoff’s hypothesis and von Kármán strain displacement relations. Extremizing the Lagrangian of the system derives the governing nonlinear partial differential equations for the beam. Eliminating the in-plane displacement, an integro-partial differential equation of motion is obtained in terms of the transverse displacement. A deflection function that satisfies the simply supported boundary conditions is assumed to get the system equation as a nonlinear second order ordinary differential equation in time, which is of Duffing’s type. The solution of the problem is obtained through exact integration. Results are presented for frequency and amplitude for surface bonded PZT-5A layer in composite beams with various stacking sequences.     

Nonlinear analysis of non-uniform beams on nonlinear elastic foundation

In this paper a boundary integral equation solution to the nonlinear problem of non-uniform beams resting on a nonlinear triparametric elastic foundation is presented, which permits also the treatment of nonlinear boundary conditions. The nonlinear subgrade model which describes the foundation includes the linear and nonlinear Winkler (normal) parameters and the linear Pasternak (shear) foundation parameter. The governing equations are derived in terms of the displacements for nonlinear analysis in the deformed configuration and for linear analysis in the undeformed one. Moreover, as the cross-sectional properties of the beam vary along its axis, the resulting coupled nonlinear differential equations have variable coefficients which complicate the mathematical problem even more. Their solution is achieved using the analog equation method of Katsikadelis. Several beams are analyzed under various boundary conditions and load distributions, which illustrate the method and demonstrate its efficiency and accuracy. Finally, useful conclusions are drawn from the investigation of the nonlinear response of non-uniform beams resting on nonlinear elastic foundation.