Free vibration analyses of axially loaded laminated composite beams based on higher-order shear deformation theory

The dynamic stiffness matrix method is introduced to solve exactly the free vibration and buckling problems of axially loaded laminated composite beams with arbitrary lay-ups. The Poisson effect, axial force, extensional deformation, shear deformation and rotary inertia are included in the mathematical formulation. The exact dynamic stiffness matrix is derived from the analytical solutions of the governing differential equations of the composite beams based on third-order shear deformation beam theory. The application of the present method is illustrated by two numerical examples, in which the effects of axial force and boundary condition on the natural frequencies, mode shapes and buckling loads are examined. Comparison of the current results to the existing solutions in the literature demonstrates the accuracy and effectiveness of the present method.     

Free vibration and dynamic stability of rotating thin-walled composite beams

The dynamic stability behavior of thin-walled rotating composite beams is studied by means of the finite element method. The analysis is based on Bolotin’s work on parametric instability for an axial periodic load. The influence of fiber orientation and rotating speeds on the natural frequencies and the unstable regions is studied for symmetrically balanced laminates. The regions of instability are obtained and expressed in non-dimensional terms. The “modal interchange” phenomenon arising in rotating beams is described. The dynamic stability problem is formulated by means of linearizing a geometrically nonlinear total Lagrangian finite element with seven degrees of freedom per node. This finite element formulation is based on a thin-walled beam theory that takes into account several non-classical effects such as anisotropy, shear flexibility and warping inhibition.     

Free vibration analysis of rotating composite Timoshenko beams with bending-torsion couplings

Meccanica - The free vibration of rotating bending-torsional composite Timoshenko beams (CTBs) with arbitrary boundary conditions is analyzed. The composite material coupled rigidity, Coriolis...     

Free vibration of axially loaded,rotating Timoshenko shaft systems by the wave-train closure principle

A systematic approach for the free vibration analysis of a rotating Timoshenko shaft system subjected to axial forces is presented in this paper. The system has multiple point discontinuities such as elastic supports, rotor masses, and cross-sectional changes. Wave reflection and transmission matrices are employed to characterize the wave motions between the sub-spans of the shaft system. These matrices are combined with the field transfer matrices expressed in wave forms to obtain the characteristic equation in a straightforward manner. The solutions are exact since effects of attenuating wave components are included in the formulation. The wave propagation-based matrix algebra leads to recursive algorithms which are suitable for computer coding. Three examples are presented to illustrate the numerical procedure.     

Coupled bending and torsional vibration of nonsymmetrical axially loaded thin-walled Bernoulli–Euler beams

The dynamic transfer matrix is formulated for a straight uniform and axially loaded thin-walled Bernoulli–Euler beam element whose elastic and inertia axes are not coincident by directly solving the governing differential equations of motion of the beam element. Bernoulli–Euler beam theory is used, and the cross section of the beam does not have any symmetrical axes. The bending vibrations in two perpendicular directions are coupled with torsional vibration and the effect of warping stiffness is included. The dynamic transfer matrix method is used for calculation of exact natural frequencies and mode shapes of the nonsymmetrical thin-walled beams. Numerical results are given for a specific example of thin-walled beam under a variety of end conditions, and exact numerical solutions are tabulated for natural frequencies and solutions calculated by the other method are also tabulated for comparison. The effects of axial force and warping stiffness are also discussed.     

Initial value method for free vibration of axially loaded functionally graded Timoshenko beams with nonuniform cross section

Free vibration of nonuniform axially functionally graded Timoshenko beams subjected to combined axially tensile or compressive loading is studied. An emphasis is placed on the effect of tip and distributed axial loads on the natural frequencies and mode shapes for an inhomogeneous cantilever beam including material inhomogeneity and geometric non-uniform cross section. The initial value method is developed to determine the natural frequencies. The method’s effectiveness is verified by comparing our results with previous ones for special cases. Natural frequencies of standing/hanging Timoshenko beams are calculated for four different cross sections. The influences of shear rigidity, taper ratio, gradient index, tip force, and axially distributed loading on the natural frequencies of clamped-free beams are discussed. Material inhomogeneity and geometric non-uniform cross-section strongly affect higher-order vibration frequencies and mode shapes.     

Free flapewise vibration analysis of rotating double-tapered Timoshenko beams

Free vibration analysis of a rotating double-tapered Timoshenko beam undergoing flapwise transverse vibration is presented. Using an assumed mode method, the governing equations of motion are derived from the kinetic and potential energy expressions which are derived from a set of hybrid deformation variables. These equations of motion are then transformed into dimensionless forms using a set of dimensionless parameters, such as the hub radius ratio, the dimensionless angular speed ratio, the slenderness ratio, and the height and width taper ratios, etc. The natural frequencies and mode shapes are then determined from these dimensionless equations of motion. The effects of the dimensionless parameters on the natural frequencies and modal characteristics of a rotating double-tapered Timoshenko beam are numerically studied through numerical examples. The tuned angular speed of the rotating double-tapered Timoshenko beam is then investigated.     

Non-uniform rational B-spline based free vibration analysis of axially functionally graded tapered Timoshenko curved beams

The free vibration of axially functionally graded (FG) tapered Timoshenko curved beams is studied with the numerical approach. By using the non-uniform rational B-spline (NURBS) basis functions, the exact geometry and the generalized displacement field are formulated. Variable geometric parameters and material properties, including the curvature, cross-sectional area, area moment of inertia, mass density, and Young’s modulus, are expanded as functions of the coordinate in a parametric domain. Based on Hamilton’s principle, the weak formulation is derived by applying a refined constitutive relation which considers the thickness effect. Natural frequencies and mode shapes are obtained from the eigenvalue equation. Circular, elliptic, and parabolic curved beams are considered in numerical examples. The obtained results are in good agreement with those in the existing studies and those calculated by the finite element software ANSYS. Moreover, the effects of the material gradient, taper ratio, slenderness ratio, and heightspan ratio on vibration behaviors are discussed.     

Free vibration analysis of axially functionally graded tapered Timoshenko beams using differential transformation element method and differential quadrature element method of lowest-order

The present paper investigates the free vibration characteristics of Timoshenko beams whose cross-sectional profile and material properties vary along the beam axis with any arbitrary functions. Free vibration analysis of these beams is carried out through solving the governing differential equations of motion. Since the application of differential transformation method (DTM) does not necessarily converge to satisfactory results, an element-based differential transformation method, namely differential transformation element method (DTEM), is introduced which significantly enhances the accuracy of the results. Furthermore, differential quadrature element of the lowest order (DQEL) is introduced which is based on differential quadrature element method (DQEM). DQEL formulates the problem on the basis of the interpolation of the first differential of the functions; therefore, in contrast with DQEM higher differentials of functions are not employed in DQEL. The competency of DQEL and DTEM in free vibration analysis is verified through several numerical examples. The effects of taper ratio and material non-homogeneity on natural frequencies are investigated.     

Free vibration of non-uniform axially functionally graded beams using the asymptotic development method

The asymptotic development method is applied to analyze the free vibration of non-uniform axially functionally graded(AFG) beams, of which the governing equations are differential equations with variable coefficients. By decomposing the variable flexural stiffness and mass per unit length into reference invariant and variant parts, the perturbation theory is introduced to obtain an approximate analytical formula of the natural frequencies of the non-uniform AFG beams with different boundary conditions.Furthermore, assuming polynomial distributions of Young's modulus and the mass density, the numerical results of the AFG beams with various taper ratios are obtained and compared with the published literature results. The discussion results illustrate that the proposed method yields an effective estimate of the first three order natural frequencies for the AFG tapered beams. However, the errors increase with the increase in the mode orders especially for the cases with variable heights. In brief, the asymptotic development method is verified to be simple and efficient to analytically study the free vibration of non-uniform AFG beams, and it could be used to analyze any tapered beams with an arbitrary varying cross width.     

Free vibration of nonhomogeneous Timoshenko nanobeams

Free vibration of nonhomogeneous nanobeams based on nonlocal Timoshenko beam theory has been studied using boundary characteristic orthogonal polynomial functions in the Rayleigh–Ritz method. Orthogonal polynomial functions satisfying essential boundary conditions have been generated with the help of Gram–Schmidt Process. Nonhomogeneity of nanobeams is assumed to arise due to linear and quadratic variations in Young’s modulus and density of the nanobeams with space coordinate. The lowest three frequency parameters of nanobeams subjected to different boundary conditions have been computed for various values of nonhomogeneous parameters to demonstrate the effect of each parameters on the frequency parameters. A detailed investigation has been reported for all the possible cases of variations in Young’s modulus and density to analyze the numerical results for different scaling effect parameters and four types of boundary conditions. Present results are compared with the results in special cases and are found to be in good agreement.     

Free vibration and spatial stability of non-symmetric thin-walled curved beams with variable curvatures

An improved formulation for free vibration and spatial stability of non-symmetric thin-walled curved beams is presented based on the displacement field considering variable curvature effects and the second-order terms of finite-semitangential rotations. By introducing Vlasov’s assumptions and integrating over the non-symmetric cross-section, the total potential energy is consistently derived from the principle of virtual work for a continuum. In this formulation, all displacement parameters and the warping function are defined at the centroid axis and also thickness-curvature effects and Wagner effect are accurately taken into account. For F.E. analysis, a thin-walled curved beam element is developed using the third-order Hermitian polynomials. In order to illustrate the accuracy and the practical usefulness of the present method, numerical solutions by this study are presented with the results analyzed by ABAQUS’ shell elements. Particularly, the effect of arch rise to span length ratio is investigated on vibrational and buckling behaviour of non-symmetric curved beams.     

Parametric resonance of axially moving Timoshenko beams with time-dependent speed

In this paper, parametric resonance of axially moving beams with time-dependent speed is analyzed, based on the Timoshenko model. The Hamilton principle is employed to obtain the governing equation, which is a nonlinear partial-differential equation due to the geometric nonlinearity caused by the finite stretch of the beam. The method of multiple scales is applied to predict the steady-state response. The expression of the amplitude of the steady-state response is derived from the solvability condition of eliminating secular terms. The stability of straight equilibrium and nontrivial steady-state response are analyzed by using the Lyapunov linearized stability theory. Some numerical examples are presented to demonstrate the effects of speed pulsation and the nonlinearity in the first two principal parametric resonances.     

Complex modes and traveling waves in axially moving Timoshenko beams

Complex modes and traveling waves in axially moving Timoshenko beams are studied. Due to the axially moving velocity, complex modes emerge instead of real value modes. Correspondingly, traveling waves are present for the axially moving material while standing waves dominate in the traditional static structures. The analytical results obtained in this study are verified with a numerical differential quadrature method.     

Dynamic response of axially moving Timoshenko beams: integral transform solution

The generalized integral transform technique (GITT) is used to find a semianalytical numerical solution for dynamic response of an axially moving Timoshenko beam with clamped-clamped and simply-supported boundary conditions, respectively. The implementation of GITT approach for analyzing the forced vibration equation eliminates the space variable and leads to systems of second-order ordinary differential equations (ODEs) in time. The MATHEMATICA built-in function, NDSolve, is used to numerically solve the resulting transformed ODE system. The good convergence behavior of the suggested eigenfunction expansions is demonstrated for calculating the transverse deflection and the angle of rotation of the beam cross-section. Moreover, parametric studies are performed to analyze the effects of the axially moving speed, the axial tension, and the amplitude of external distributed force on the vibration amplitude of axially moving Timoshenko beams.     

Free vibration of horizontally curved thin-walled beams with rectangular hollow sections considering two compatible displacement fields

A finite element is presented for vibration analyses of horizontally curved thin-walled rectangular hollow beams. Eight cross-section deformation modes are employed to describe the mid-surface contour displacement field with the modal superposition method. Focused on the in-plane moment equilibrium condition and the displacement continuity condition, two compatible displacement fields are constructed to calculate the strain energy and the kinetic energy of the beam, respectively. With the application of Hamilton’s principle the dynamic governing equations are formulated, and then approximated for the finite element implementation. Finally, numerical examples are illustrated to verify the validity of the present theory.     

Natural frequencies of nonlinear vibration of axially moving beams

Axially moving beam-typed structures are of technical importance and present in a wide class of engineering problem. In the present paper, natural frequencies of nonlinear planar vibration of axially moving beams are numerically investigated via the fast Fourier transform (FFT). The FFT is a computational tool for efficiently calculating the discrete Fourier transform of a series of data samples by means of digital computers. The governing equations of coupled planar of an axially moving beam are reduced to two nonlinear models of transverse vibration. Numerical schemes are respectively presented for the governing equations via the finite difference method under the simple support boundary condition. In this paper, time series of the discrete Fourier transform is defined as numerically solutions of three nonlinear governing equations, respectively. The standard FFT scheme is used to investigate the natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results are compared with the first two natural frequencies of linear free transverse vibration of an axially moving beam. And results indicate that the effect of the nonlinear coefficient on the first natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results also illustrate the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters.     

Non-linear stability analysis of imperfect thin-walled composite beams

The static non-linear behavior of thin-walled composite beams is analyzed considering the effect of initial imperfections. A simple approach is used for determining the influence of imperfection on the buckling, prebuckling and postbuckling behavior of thin-walled composite beams. The fundamental and secondary equilibrium paths of perfect and imperfect systems corresponding to a major imperfection are analyzed for the case where the perfect system has a stable symmetric bifurcation point. A geometrically non-linear theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field. An initial displacement, either in vertical or horizontal plane, is considered in presence of initial geometric imperfection. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton-Rapshon method. The numerical results are presented for a simply supported beam subjected to axial or lateral load. It is shown in the examples that a major imperfection reduces the load-carrying capacity of thin-walled beams. The influence of this effect is analyzed for different fiber orientation angle of a symmetric balanced lamination. In addition, the postbuckling response obtained with the present beam model is compared with the results obtained with a shell finite element model (Abaqus).