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.     

Stress and free vibration analysis of functionally graded beams using static Green's functions

In this paper static Green's functions for functionally graded Euler-Bernoulli and Timoshenko beams are presented. All material properties are arbitrary functions along the beam thickness direction. The closed-form solutions of static Green's functions are derived from a fourth-order partial differential equation presented in [2]. In combination with Betti's reciprocal theorem the Green's functions are applied to calculate internal forces and stress analysis of functionally graded beams (FGBs) under static loadings. For symmetrical material properties along the beam thickness direction and symmetric cross-sections, the resulting stress distributions are also symmetric. For unsymmetrical material properties the neutral axis and the center of gravity axis are located at different positions. Free vibrations of functionally graded Timoshenko beams are also analyzed [3]. Analytical solutions of eigenfunctions and eigenfrequencies in closed-forms are obtained based on reference [2]. Alternatively it is also possible to use static Green's functions and Fredholm's integral equations to obtain approximate eigenfunctions and eigenfrequencies by an iterative procedure as shown in [1]. Applying the Sensitivity Analysis with Green's Functions (SAGF) [1] to derive closed-form analytical solutions of functionally graded beams, it is possible to modify the derived static Green's functions and include terms taking cracks into account, which are modeled by translational or rotational springs. Furthermore the SAGF approach in combination with the superposition principle can be used to take stiffness jumps into account and to extend static Green's functions of simple beams to that of discontinuous beams by adding new supports. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Forced vibration analysis of a Timoshenko beam featuring bending-torsion on Pasternak foundation

Free and forced vibration analysis of a Timoshenko beam on viscoelastic Pasternak foundation featuring coupling between flapwise bending and torsional vibrations is studied in this article. The system motion is described through a coupled set of three partial differential equations. The differential transform method, DTM, as an efficient mathematical technique is adopted to obtain the natural frequencies and the mode shapes. The system force response is assessed for a moving concentrated load with a constant velocity. Two different methods are studied and applied in obtaining forced vibration response of the system: (1) the same time functions, STF, by setting out the orthogonality conditions derived in this article and (2) the different time functions, DTF. The difference between the responses of the system is assessed by applying STF and DTF for a constant moving load. The effects of some parameters on the system response are probed. A numerical example is solved to validate the results obtained here with the available ones and a close agreement is found. It is observed that the time functions in DTF and STF are almost identical for transverse displacement and bending angle and are significant for torsion angle, recommending the application of DTF when the bending-torsion coupling is of concern.     

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.     

Discrete kiguradze type inequalities

Solutions of nth order difference inequalities satisfying n boundary conditions of predetermined sign are compared to solutions of difference equalities satisfying n?1 homogenous boundary conditions. The inequalities obtained here apply to Green's functions of corresponding boundary value problems     

Bending solutions of FGM Timoshenko beams from those of the homogenous Euler–Bernoulli beams

By using mathematical similarity and load equivalence between the governing equations, bending solutions of FGM Timoshenko beams are derived analytically in terms of the homogenous Euler–Bernoulli beams. The deflection, rotational angle, bending moment and shear force of FGM Timoshenko beams are expressed in terms of the deflection of the corresponding homogenous Euler–Bernoulli beams with the same geometry, the same loadings and end constraints. Consequently, solutions of bending of the FGM Timoshenko beams are simplified as the calculation of the transition coefficients which can be easily determined by the variation law of the gradient of the material properties and the geometry of the beams if the solutions of corresponding Euler–Bernoulli beam are known. As examples, solutions are given for the FGM Timoshenko beams under S–S, C–C, C–F and C–S end constraints and arbitrary transverse loadings to illustrate the use of this approach. These analytical solutions can be as benchmarks in the further investigations of behaviors of FGM beams.     

Multi-scale modeling of beam-like structures: A new boundary condition concept for the RVE

In this work a coupled two-scale beam model using Timoshenko beam elements [1] with finite displacements on the macro scale and fully non-linear 3D brick elements on the micro scale is proposed. The calculation is carried out with the so-called FE² concept. To achieve the coupling between the beam and the brick elements, the algorithm from [2] is adapted. Within the degenerated concept of the Timoshenko beam, the introduction of a pure shear deformation leads to significant problems concerning the equilibrium condition on the micro scale. Applying this deformation mode on the RVE with periodic boundary conditions results in a rigid body rotation. Using linear displacement boundary conditions instead, the wrapping deformation is suppressed on the boundary, leading to a length dependency in the torsional deformation mode. In addition, the shear forces introduce a bending moment, which depends on the length of the RVE and adds spurious normal stresses and a length dependency of the shear stiffness. To overcome these problems, periodic boundary conditions are applied and the displacement assumptions are modified such that the shear deformation is achieved with force pairs on both ends of the RVE. The resulting model leads to length independent results in tension, bending and torsion and a domain which is able to produce a pure shear stress state. Consequently, only this domain of the model should be homogenized which can be accomplished by modifying the variations in the algorithm [2]. The concept is validated by simple linear and non-linear test problems. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lyapunov inequalities for nabla Caputo boundary value problems

ABSTRACT

We will establish uniqueness of solutions to boundary value problems involving the nabla Caputo fractional difference under two-point boundary conditions and give an explicit expression for the Green's functions for these problems. Using the Green's functions for specific cases of these boundary value problems, we will then develop Lyapunov inequalities for certain nabla Caputo BVPs.     

Comparison of Green's functions for a family of boundary value problems for fractional difference equations

ABSTRACT

In this paper, we obtain sign conditions and comparison theorems for Green's functions of a family of boundary value problems for a Riemann-Liouville type delta fractional difference equation. Moreover, we show that as the length of the domain diverges to infinity, each Green's function converges to a uniquely defined Green's function of a singular boundary value problem.     

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part I: Theoretical solutions

An extended displacement discontinuity (EDD) boundary integral equation method is proposed for analysis of arbitrarily shaped planar cracks in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack surface. Green's functions for unit point EDDs in an infinite three-dimensional medium of 2D hexagonal QC are derived using the Hankel transform method. Based on the Green's functions and the superposition theorem, the EDD boundary integral equations for an arbitrarily shaped planar crack in an infinite 2D hexagonal QC body are established. Using the EDD boundary integral equation method, the asymptotic behavior along the crack front is studied and the classical singular index of 1/2 is obtained at the crack edge. The extended stress intensity factors are expressed in terms of the EDDs across crack surfaces. Finally, the energy release rate is obtained using the definitions of the stress intensity factors.     

Green's functions for elliptic and parabolic systems with Robin-type boundary conditions

The aim of this paper is to investigate Green's function for parabolic and elliptic systems satisfying a possibly nonlocal Robin-type boundary condition. We construct Green's function for parabolic systems with time-dependent coefficients satisfying a possibly nonlocal Robin-type boundary condition assuming that weak solutions of the system are locally Hölder continuous in the interior of the domain, and as a corollary we construct Green's function for elliptic system with a Robin-type condition. Also, we obtain Gaussian bound for Robin Green's function under an additional assumption that weak solutions of Robin problem are locally bounded up to the boundary. We provide some examples satisfying such a local boundedness property, and thus have Gaussian bounds for their Green's functions.     

The diffraction in a class of unbounded domains connected through a hole

In this paper, the unique solvability, Fredholm property, and the principle of limiting absorption are proved for a boundary value problem for the system of Maxwell's equations in a semi‐infinite rectangular cylinder coupled with a layer by an aperture of arbitrary shape. Conditions at infinity are taken in the form of the Sveshnikov–Werner partial radiation conditions. The method of solution employs Green's functions of the partial domains and reduction to vector pseudodifferential equations considered in appropriate vectorial Sobolev spaces. Singularities of Green's functions are separated both in the domain and on its boundary. The smoothness of solutions is established. Copyright © 2003 John Wiley & Sons, Ltd.     

Application of the resonance method for determining the elastic and dissipative properties of an orthotropic polymer composite

Experimentally determined resonant frequencies and damping of flexural and torsional vibrations of rod-type rectangular test specimens made of an orthotropic GFRP fabric with different ratios of cross-sectional sizes are used for calculating six principal complex elastic and shear moduli. The application of the classical theories of flexural and torsional vibrations, the theory of flexural vibrations of a Timoshenko beam, and a refined theory of torsional vibrations of free-free orthotropic rods is analyzed.     

Multi-Point Boundary Value Problems for Nonlinear Fourth-Order Differential Equations with All Order Derivatives

By using fixed point theorem, multiple positive solutions for some fourth-order multi-point boundary value problems with nonlinearity depending on all order derivatives are obtained. The associated Green's functions are also given.     

Bending-torsion vibration of a partially submerged cylinder with an arbitrary cross-section

An analytical method is presented to investigate the bending-torsion vibration characteristics of a cylinder with an arbitrary cross-section and partially submerged in water. The compressibility and the free surface waves of the water are considered simultaneously in the analysis. The exact solution of structure–water interaction is obtained mathematically. Firstly, the analytical expression of the velocity potential of the water is derived by using the method of separation of variables. The unknown coefficients in the velocity potential are determined by the longitudinal and circumferential Fourier expansions along the outer surface of the cylinder and are expressed in the form of integral equations including the unknown dynamic bending deflection and torsional angle of the cylinder. Secondly, the force and torque acting on the cylinder per unit length, provided by the water, are obtained by integrating the water dynamic pressure along the circumference of the cylinder. The general solution of bending-torsion vibration of the cylinder under the water dynamic pressure is derived analytically. The integral equations included in the velocity potential of the water can be solved exactly. Finally, the eigenfrequency equation of cylinder–water interaction is obtained by means of the boundary conditions of the cylinder. Some numerical examples for elliptical columns partially submerged in water are provided to show the application of the present method.     

Unified theory of 2n+1 order size-dependent beams: Mathematical difficulty for functionally graded size-effect parameters solved

New insights on theoretical modeling of size-dependent functionally graded (FG) nanobeams are provided by establishing a unified theory of 2n+1 order shear deformable model with the aids of nonlocal strain gradient elasticity. The unified model covers Euler-type (n = 0), Reddy-type (n = 1), 5th (n = 2), 7th (n = 3) order beam and etc., and the limiting situation n → ∞ predicts nonlocal strain gradient Timoshenko model. The mathematical difficulty for FG nonlocal parameter is particularly emphasized, and an attempt is made for the first time to overcome the difficulty. Theoretically, the governing equations and boundary conditions of 2n+1 order nonlocal strain gradient beams, especially with FG nonlocal parameter and FG strain gradient parameter, are systematically formulated. The difficulty for FG nonlocal parameter is satisfactorily solved with by adopting the present 2n+1 order beam theory. Analytically, solutions to bending and buckling analyses within the unified model are obtained, from which the analytical solutions for Euler- and Timoshenko-type beam can be recovered. Numerically, bending deflection and buckling critical load for Euler beam, Reddy beam, 5th-11th order beam and Timoshenko beam are depicted, of which the benchmark solutions for the 5th to 11th order beam are given for the first time. Meanwhile, potential extensions of the unified model into fractional order is discussed, where benchmark solutions for n = 1.1, 0.88, 0.77, 0.4and0.2 are listed. The influences of FG nonlocal parameter, dimensionless height and Poisson's ratio (or the ratio E/G) on the bending deflection and buckling critical load are systematically studied. The present work mainly contributes to theoretical developments and greatly facilitates the mechanical analysis of beam-type structures.     

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part II: Numerical solutions

The extended displacement discontinuity (EDD) boundary element method is developed to analyze an arbitrarily shaped planar crack in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack face. Green's functions for uniformly distributed EDDs over triangular and rectangular elements for 2D hexagonal QCs are derived. Employing the proposed EDD boundary element method, a rectangular crack is analyzed to verify the Green's functions by discretizing the crack with rectangular and triangular elements. Furthermore, the elliptical crack problem for 2D hexagonal QCs is investigated. Normal, tangential, and thermal loads are applied on the crack face, and the numerical results are presented graphically.     

Aufspaltungen linearer gewöhnlicher Differentialoperatoren und der zugehörigen Greenschen Funktionen

Let M be a regular linear ordinary differential operator of the n-th order, associated with certain homogeneous boundary conditions. Suppose that M is invertable. We provide sufficient conditions to split M into a product of lower order operators M_k, which may be singular at the endpoints of the given interval. To these splittings-which depend on the given boundary conditions-there corresponds a splitting of the associated Green's function. The results are applied in the theory of inverse-positive operators and the theory of totally positive Green's functions. These applications, in general, require the operators M_k to be singular. Moreover, for special classes of operators the splittings can effectively be used for solving boundary value problems numerically.     

考虑非局部剪切效应的碳纳米管弯曲特性研究

基于Hamilton(哈密顿)变分原理和非局部连续介质弹性理论,建立了新型非局部Timoshenko(铁木辛柯)梁模型（ANT）,推导了碳纳米管（CNT）的ANT弯曲平衡方程以及两端简支梁、悬臂梁和简支 固定梁的边界条件表达式,分析了剪切变形效应和非局部微观尺度效应对碳纳米管弯曲特性的影响.数值计算结果显示,碳纳米管的弯曲刚度随着小尺度效应的增强而升高.其次,这种小尺度效应对自由端受集中力的悬臂梁碳纳米管有明显作用,其刚度变化规律和其它约束条件的碳纳米管一样,这一点是ANT模型区别于普通非局部纳米梁模型的主要特点.经分子动力学模拟验证,ANT模型是合理分析碳纳米管力学特性的有效方法.     

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.