Composite Timoshenko Beam Research Articles

Equations of motion of rotating composite beam with a nonconstant rotation speed and an arbitrary preset angle

In the presented paper the equations of motion of a rotating composite Timoshenko beam are derived by utilising the Hamilton principle. The non-classical effects like material anisotropy, transverse shear and both primary and secondary cross-section warpings are taken into account in the analysis. As an extension of the other papers known to the authors a nonconstant rotating speed and an arbitrary beam’s preset (pitch) angle are considered. It is shown that the resulting general equations of motion are coupled together and form a nonlinear system of PDEs. Two cases of an open and closed box-beam cross-section made of symmetric laminate are analysed in details. It is shown that considering different pitch angles there is a strong effect in coupling of flapwise bending with chordwise bending motions due to a centrifugal force. Moreover, a consequence of terms related to nonconstant rotating speed is presented. Therefore it is shown that both the variable rotating speed and nonzero pitch angle have significant impact on systems dynamics and need to be considered in modelling of rotating beams.

Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams

This research deals with the forced vibration behavior of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on the Timoshenko beam theory along with von Kármán geometric nonlinearity. For the carbon-nanotube reinforced composite (CNTRC) beams, uniform distribution (UD) and three types of functionally graded (FG) distribution patterns of SWCNT reinforcements are considered. It is assumed that the material properties of FG-CNTRC beams are graded in the thickness direction and estimated through the rule of mixture. The nonlinear governing equations and corresponding boundary conditions are derived based on the Hamilton principle and discretized by means of the generalized differential quadrature (GDQ) method. After that, a Galerkin-based numerical technique is employed to reduce the set of nonlinear governing equations into a time-varying set of ordinary differential equations of Duffing type. Since the nanobeam responds periodically to harmonic excitations, a set of periodic differential matrix operators is introduced to discretize the Duffing equations on the time domain using the derivatives of a periodic base function. The vectorized form of final nonlinear parameterized equations is then solved through the use of pseudo-arc length continuum technique. Numerical results are presented to examine the effects of different parameters such as nanotube volume fraction, slenderness ratio, dimensionless damping parameter, dimensionless transverse force, CNT distributions and boundary conditions on the natural frequencies and frequency responses of FG-CNTRC beams.

Energy Derivation and Extension-Flapwise Bending Vibration Analysis of a Rotating Piezolaminated Composite Timoshenko Beam

In this study, extension and flapwise bending vibrations of a rotating piezolaminated composite Timoshenko beam are analyzed. Energy expressions are derived using several explanatory tables and figures. Parameters for the hub radius, rotational speed, rotary inertia, shear deformation, slenderness ratio, and ply orientation are incorporated into the equations of motion. The differential transform method is used to transform the differential equations of motion and the boundary conditions into simple analytical expressions. Numerical results are obtained to investigate the effects of the applied voltage, ply orientation, rotational speed, and hub radius on the natural frequencies and tip deflection.

Variational Principles for Bending and Vibration of Partially Composite Timoshenko Beams

Variational principles are established for the partially composite Timoshenko beam using the semi-inverse method. The principles are derived directly from governing differential equations for bending and vibration of the beam considered. It is concluded that the semi-inverse method is a powerful tool for searching for variational principles directly from the governing equations. Comparison between our results and the results reported in literature is given.

Stochastic response of an axially loaded composite Timoshenko beam exhibiting bending–torsion coupling

A spectral finite element method is proposed to investigate the stochastic response of an axially loaded composite Timoshenko beam with solid or thin-walled closed section exhibiting bending–torsion materially coupling under the stochastic excitations with stationary and ergodic properties. The effects of axial force, shear deformation (SD) and rotary inertia (RI) as well as bending–torsion coupling are considered in the present study. First, the damped general governing differential equations of motion of an axially loaded composite Timoshenko beam are derived. Then, the spectral finite element formulation is developed in the frequency domain using the dynamic shape functions based on the exact solutions of the governing equations in undamped free vibration, which is used to compute the mean square displacement response of axially loaded composite Timoshenko beams. Finally, the proposed method is illustrated by its application to a specific example to investigate the effects of bending–torsion coupling, axial force, SD and RI on the stochastic response of the composite beam.

Exact solution for nonlinear thermal stability of hybrid laminated composite Timoshenko beams reinforced with SMA fibers

Post-buckling behavior of Shape Memory Alloy (SMA) hybrid composite laminated beams under uniform heating is investigated in this paper. Properties of the constituents are temperature-dependent. The von-Karman type of geometrical non-linearity suitable for small strains and moderate rotations is taken into account. Displacement field through the beam obeys the kinematics of first order shear deformation theory of Timoshenko. To model the SMA fibers, the one-dimensional thermomechanical constitutive law proposed earlier by Brinson (1993) [1], is implemented. These basic assumptions are incorporated with the static version of virtual displacements to extract the nonlinear governing equations of the beam. The resulting system of nonlinear equations are uncoupled and solved analytically. Closed-form expressions are presented to trace the deflection-temperature as well as bending moment-temperature and end-shortening force-temperature in heating process. After validating the proposed approach, various parametric studies are performed to study the influence of lay-up, SMA volume fraction, SMA prestrain, boundary condition, and thickness of the layer in which SMA fibers are embedded. Fascinating results are extracted due to the recovery stress of SMA fibers.

Dynamic modeling and analysis of the PZT-bonded composite Timoshenko beams: Spectral element method

The health of thin laminated composite beams is often monitored using the ultrasonic guided waves excited by wafer-type piezoelectric transducers (PZTs). Thus, for the smart composite beams which consist of a laminated composite base beam and PZT layers, it is very important to develop a very reliable mathematical model and to use a very accurate computational method to predict accurate dynamic characteristics at very high ultrasonic frequency. In this paper, the axial–bending–shear–lateral contraction coupled differential equations of motion are derived first by the Hamilton’s principle with Lagrange multipliers. The smart composite beam is represented by a Timoshenko beam model by adopting the first-order shear deformation theory (FSDT) for the laminated composite base beam. The axial deformation of smart composite beam is improved by taking into account the effects of lateral contraction by adopting the concept of Mindlin–Herrmann rod theory. The spectral element model is then formulated by the variation approach from coupled differential equations of motion transformed into the frequency domain via the discrete Fourier transform. The high accuracy of the present spectral element model is verified by comparing with other solution methods: the finite element model developed in this paper and the commercial FEA package ANSYS. Finally the dynamics and wave characteristics of some example smart composite beams are investigated through the numerical studies.

Nonlinear modal interaction in rotating composite Timoshenko beams

Nonlinear free vibration analysis of rotating composite Timoshenko beams featuring internal resonance is studied in this paper. Three nonlinear coupled equations of motion for flapping, shear and axial motions, are based on the assumptions of Timoshenko theory and the nonlinear von Karman strain–displacement relationships. Due to the small magnitude of low-order flapping/shear frequencies ratio, usually a big gap exists between the aforementioned frequencies especially in isotropic beams, while this gap reduces for composite beams which exhibit more characteristics of Timoshenko beams. Different material, geometrical and operational parameters effects on the frequencies of rotating beams are investigated. For the first time the possibility of internal resonance occurrence between low-order flapping and shear modes is proved. The direct multiple scales method is implemented for construction of the flapping nonlinear normal modes. Results for the flapping nonlinear normal modes are validated via comparison with the results of the fourth-order Runge–Kutta method. Depend on the amplitude ratios besides the nearness of the frequencies of the interacting modes, four bifurcation regions are defined through the nonlinear normal mode stability analysis; (a) one stable coupled mode, (b) two stable and one unstable coupled modes, (c) three stable coupled modes and (d) one stable coupled mode.

Nonlinear free vibration analysis of rotating composite Timoshenko beams

Linear and nonlinear free vibrations of rotating composite Timoshenko beams are investigated in this paper. The formulation is based on the assumptions of Timoshenko beam theory and the nonlinear von-Karman strain–displacement relationships. The updated equations of motion about the prestressed configuration are obtained from the three-coupled partial differential equations of motion. The differential transform method is implemented to find the prestressed configuration due to centrifugal forces. The Galerkin discretization approach is applied to the linearized updated equations of motion to determine the linear normal modes and the associated natural frequencies of the rotating composite Timoshenko beam. The rotation speed and number of layers variation effects on the flapping natural frequencies are studied and the necessity of employing the Timoshenko beam model for the composite beams is illustrated. The direct multiple scales method is applied to the three-coupled nonlinear updated partial differential equations for the nonlinear vibration study especially on the flapping backbone curves. The number of layers variation effects is investigated on the three lowest flapping backbone curves. The present results are validated with the previous literature results.

Dynamics of a composite Timoshenko beam with delamination

In this work dynamic behaviour of a composite beam having delamination is presented. The model of delamination takes into account a contact interaction between sublaminates including normal forces, shear forces and additional damping. Numerical calculations are performed in order to estimate the influence of the new terms included into the model.

Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation

This study deals with free vibrations and buckling analysis of nanocomposite Timoshenko beams reinforced by single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation. The SWCNTs are assumed to be aligned and straight with a uniform layout. Four different carbon nanotubes (CNTs) distributions including uniform and three types of functionally graded distributions of CNTs through the thickness are considered. The rule of mixture is used to describe the effective material properties of the nanocomposite beams. The governing equations are derived through using Hamilton's principle and then solved by using the generalized differential quadrature method (GDQM). Natural frequencies and critical buckling load are obtained for nanocomposite beams with different boundary conditions. Effects of several parameters, such as nanotube volume fraction, foundation stiffness parameters, slenderness ratios, CNTs distribution and boundary conditions on both natural frequency and critical buckling load are investigated. The results indicate that the above-mentioned parameters play a very important role on the free vibrations and buckling characteristics of the beam.

Semi-analytical solution for the free vibration analysis of generally laminated composite Timoshenko beams with single delamination

A rather new semi-analytical method towards investigating the free vibration analysis of generally laminated composite beam (LCB) with a delamination is presented. For the first time the combined effects of material couplings (bending–tension, bending–twist, and tension–twist couplings) with the effects of shear deformation, rotary inertia and Poisson’s effect are taken into account. The semi-analytical solution for the natural frequencies and mode shapes are presented by incorporating the constraint conditions using the method of Lagrange multipliers. To verify the validity and the accuracy of the obtained results, they were compared with the results from other available references. Very good agreements were observed. Furthermore, the effects of some parameters such as slenderness ratio, the rotary inertia, the shear deformation, material anisotropy, ply configuration, and delamination parameters on the dynamic response of the delaminated beam are examined.

Spectral element analysis of the axial-bending-shear coupled vibrations of composite Timoshenko beams

This article presents a spectral element model for the axially loaded axial-bending-shear coupled vibrations of composite laminated beams, which are represented by the Timoshenko beam models based on the first-order shear deformation theory. The variation approach is used to formulate the frequency-dependent spectral element matrix (often called exact dynamic stiffness matrix) for the present spectral element model. As the spectral element matrix is formulated from exact wave solutions satisfying the frequency domain governing equations of motion transformed by the use of discrete Fourier transform theory, the present spectral element model cannot only provide extremely accurate solutions with using only a minimum number of degrees of freedom but also contribute to improving the computation efficiency. The high accuracy of the present spectral element model is numerically verified by comparing its solutions with exact analytical solutions available from references as well as with the solutions obtained by conventional finite element method. The effects of the axial loading and damping are also numerically investigated. For the numerical verification, the finite element model is also provided for the axially loaded axial-bending-shear coupled composite laminated Timoshenko beams.

An evolutionary search technique to determine natural frequencies and mode shapes of composite Timoshenko beams

Natural frequencies and mode shapes of composite Timoshenko beams are determined by a diversity guided evolutionary algorithm (DGEA) with different boundary conditions. After applying boundary conditions, frequency equation is obtained in determinant form. Then, natural frequencies and consequently mode shapes are obtained using DGEA where the absolute value of determinant is the subject of optimization. Advantages of employing DGEA are: first, all natural frequencies are produced in a simple run, second, its simplicity for implementation and third, the procedure is not computationally prohibitive. Results clearly show the applicability of the proposed method for obtaining natural frequencies and mode shapes.

Free vibration of axially loaded thin-walled composite Timoshenko beams

Based on shear-deformable beam theory, free vibration of thin-walled composite Timoshenko beams with arbitrary layups under a constant axial force is presented. This model accounts for all the structural coupling coming from material anisotropy. Governing equations for flexural-torsional-shearing coupled vibrations are derived from Hamilton’s principle. The resulting coupling is referred to as sixfold coupled vibrations. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results are obtained for thin-walled composite beams to investigate the effects of shear deformation, axial force, fiber angle, modulus ratio on the natural frequencies, corresponding vibration mode shapes and load–frequency interaction curves.

Spectral element model for axially loaded bending–shear–torsion coupled composite Timoshenko beams

Abstract The use of frequency-dependent spectral element matrix (or exact dynamic stiffness matrix) in structural dynamics is known to provide extremely accurate solutions, while reducing the total number of degrees-of-freedom to resolve the computational and cost problems. Thus, in this paper, the spectral element model is developed for an axially loaded bending–shear–torsion coupled composite laminated beam which is represented by the Timoshenko beam model based on the first-order shear deformation theory. The high accuracy of the spectral element model is then numerically verified by comparing with exact theoretical solutions or the solutions obtained by conventional finite element method. For the numerical verification, the finite element model is also provided for the composite laminated beam.

Imposing boundary conditions in Sinc method using highest derivative approximation

Sinc approximate methods are often used to solve complex boundary value problems such as problems on unbounded domains or problems with endpoint singularities. A recent implementation of the Sinc method [Li, C. and Wu, X., Numerical solution of differential equations using Sinc method based on the interpolation of the highest derivatives, Applied Mathematical Modeling 31 (1) 2007 1–9] in which Sinc basis functions are used to approximate the highest derivative in the governing equation of the boundary value problem is evaluated for structural mechanics applications in which interlaminar stresses are desired. We suggest an alternative approach for specifying the boundary conditions, and we compare the numerical results for analysis of a laminated composite Timoshenko beam, implementing both Li and Wu’s approach and our alternative approach for applying the boundary conditions. For the Timoshenko beam problem, we obtain accurate results using both approaches, including transverse shear stress by integration of the 3D equilibrium equations of elasticity. The beam results indicate our approach is less dependent on the selection of the Sinc mesh size than Li and Wu’s SIHD. We also apply SIHD to analyze a classical laminated composite plate. For the plate example, we experience difficulty in obtaining a complete system of equations using Li and Wu’s approach. For our approach, we suggest that additional necessary information may be obtained by applying the derivatives of the boundary conditions on each edge. Using this technique, we obtain accurate results for deflection and stresses, including interlaminar stresses by integration of the 3D equilibrium equations of elasticity. Our results for both the beam and the plate problems indicate that this approach is easily implemented, has a high level of accuracy, and good convergence properties.

Shear deformation effect in non-linear analysis of composite beams of variable cross section

In this paper the non-linear analysis of a composite Timoshenko beam with arbitrary variable cross section undergoing moderate large deflections under general boundary conditions is presented employing the analog equation method (AEM), a BEM-based method. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson's ratio and are firmly bonded together. The beam is subjected in an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, the axial displacement and to two stress functions and solved using the AEM. Application of the boundary element technique yields a system of non-linear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. Numerical examples are worked out to illustrate the efficiency, the accuracy, the range of applications of the developed method and the influence of the shear deformation effect.

Flexural–torsional-coupled vibration analysis of axially loaded closed-section composite Timoshenko beam by using DTM

This study introduces the differential transform method (DTM) to analyse the free vibration response of an axially loaded, closed-section composite Timoshenko beam which features material coupling between flapwise bending and torsional vibrations due to ply orientation. The governing differential equations of motion are derived using Hamilton's principle and solved by applying DTM. The mode shapes are plotted, the natural frequencies are calculated and the effects of the bending–torsion coupling, the axial force and the slenderness ratio on the natural frequencies are investigated using the computer package, Mathematica. Wherever possible, comparisons are made with the studies in open literature.

Free and Forced Wave Vibration Analysis of Axially Loaded Materially Coupled Composite Timoshenko Beam Structures

In this paper, wave vibration analysis of axially loaded bending-torsion coupled composite beam structures is presented. It includes the effects of axial force, shear deformation, and rotary inertia; namely, it is for an axially loaded composite Timoshenko beam. The study also includes the material coupling between the bending and torsional modes of deformations that is usually present in laminated composite beam due to ply orientation. From a wave standpoint, vibrations propagate, reflect, and transmit in a structure. The transmission and reflection matrices for various discontinuities on an axially loaded materially coupled composite Timoshenko beam are derived. Such discontinuities include general point supports, boundaries, and changes in section. The matrix relations between the injected waves and externally applied forces and moments are also derived. These matrices can be combined to provide a concise and systematic approach to vibration analysis of axially loaded materially coupled composite Timoshenko beams or complex structures consisting of such beam components. The systematic approach is illustrated through numerical examples for which comparative results are available in the literature.