Stability analysis of nonlinear vibrations of a deploying flexible beam

Consider a rigid-flexible coupled system which consists of a central rigid body deploying a flexible appendage. The appendage is modeled as a finite deflection beam having linear constitutive equations. By taking the energy integral as Lyapunov function, it is proved that nonlinear transverse vibrations of the beam undergoing uniform extension or retrieval are stable when there are not controlling moment in the central rigid body and driving force on the beam, according to the partial stability theorem.     

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)     

Constitutive Modelling of Resins in the Compliance Domain

A rheological HWKK/H model for resins is developed taking into consideration the up-to-date analyses of experimental results. Constitutive compliance equations of linear are formulated for this model in the shear/bulk form, which describes, among other things, the first-rank reversible isothermal creep. The shear (distorsional) deformations are simulated with three independent stress history functions of fractional and normal exponential types. The volume deformations are simulated as perfectly elastic. The model is described by two elastic and six viscoelastic constants, namely three long-term creep coefficients and three retardation times.The constitutive compliance equations of viscoealsticity for resins are also formulated in the coupled form. Formulae for converting the constants of shear/bulk (uncoupled) viscoelasticity into the constants of coupled viscoelasticity are given too.An algorithm for identifying the material constants, based on the creep of uniaxially tensioned bar samples, is formulated in a way that gives unique results. The material constants are fiund for Epidian 53 epoxy and Polimal 109 polyester resins. The creep processes, simulated based on the experimental data, are presented graphically for both the resins examined.     

Constitutive Modelling of Resins in the Stiffness Domain

An analytic method for inverting the constitutive compliance equations of viscoelasticity for resins is developed. These equations describe the HWKK/H rheological model, which makes it possible to simulate, with a good accuracy, short-, medium- and long-term viscoelastic processes in epoxy and polyester resins. These processes are of first-rank reversible isothermal type. The time histories of deviatoric stresses are simulated with three independent strain history functions of fractional and normal exponential types. The stiffness equations are described by two elastic and six viscoelastic constants having a clear physic meaning (three long-term relaxation coefficients and three relaxation times). The time histories of axiatoric stresses are simulated as perfectly elastic.The inversion method utilizes approximate constitutive stiffness equations of viscoelasticity for the HWKK/H model. The constitutive compliance equations for the model are a basis for determining the exact complex shear stiffness, whereas the approximate constitutive stiffness equations are used for determining the approximate complex shear stiffness. The viscoelastic constants in the stiffness domain are derived by equating the exact and approximate complex shear stiffnesses. The viscoelastic constants are obtained for Epidian 53 epoxy and Polimal 109 polyester resins. The accuracy of the approximate constitutive stiffness equations are assessed by comparing the approximate and exact complex shear stiffnesses. The constitutive stiffness equations for the HWKK/H model are presented in uncoupled (shear/bulk) and coupled forms. Formulae for converting the constants of shear viscoelasticity into the constants of coupled viscoelasticity are given as well.     

Reduction of seismic vibration in multistorey structures retrofitted with nonlinear viscous dampers using mode summation method

Viscous dampers are the retrofitted devices used to reduce the vibration energy in mechanical systems. In the present paper an investigation is carried to see the effect of nonlinear viscous dampers (NVDs) in multistory buildings. A model of building consisting of NVDs in each floor is considered and is subjected to base shear excitation. The complete model is considered as an equivalent mechanical system and equation of motion is established for equilibrium condition. Using mode summation method analytical expressions are found for the displacement response of the building's floors due to shear waves. To see the role of NVDs different values of damping coefficients are considered and the result is compared with the response of building without NVDs. Analyzing the result it is found that NVDs present in the building's floors dissipate the vibration energy significantly.     

基于曲率插值的大变形梁单元

线性梁单元的形函数在单元大转动时会引起虚假应变,不适用于几何非线性分析．传统的几何非线性梁单元由于位移插值和转角插值的相干性,常常引起剪切闭锁等问题．该文 提出了一种平面大变形梁单元,通过单元域内的曲率插值以及曲率与节点位移之间的函数关系,将单元节点力和节点位移表示为节点曲率的函数．由于曲率插值本质上是对梁的应变进行插值,保证了单元任意刚体运动不会产生虚假的节点力;且将梁的截面形心位移表示为曲率的函数,避免了传统单元中的剪切闭锁问题．因而所提方法特别适用于梁的几何非线性分析．数值算例说明了所提方法的正确性和有效性.     

轴向运动导电导磁梁的磁弹性振动方程

针对磁场环境中轴向运动导电导磁梁磁弹性耦合振动的理论建模问题进行研究.基于Timoshenko(铁木辛柯)梁理论并考虑几何非线性因素,给出轴向运动弹性梁在横向双向振动下的形变势能、动能计算式以及电磁力和机械力的虚功表达式.应用Hamilton(哈密顿)变分原理,推得磁场中轴向运动Timoshenko梁的非线性磁弹性耦合振动方程,并给出了简化形式的Euler-Bernoulli(欧拉 伯努利)梁磁弹性振动方程.根据电磁理论和相应的电磁本构关系,得到载流导电弹性梁所受电磁力的表达式,基于磁偶极子-电流环路模型给出铁磁弹性梁所受磁体力和磁体力偶的表述形式.通过算例,分析了轴向运动导电弹性梁的奇点分布及其稳定性问题.     

Heat flux effect on the droplet entrainment and deposition in annular flow dryout

This paper discusses the internal mass transfer process in annular flow dryout. The emphasis is put on the order of magnitude estimation of respective hydrodynamic and thermal mechanisms and the analysis of the heat flux effect on droplet entrainment and deposition. A simple interfacial turbulence model is developed to characterize the turbulence intensity suppression due to interface tension. The heat flux effect on droplet entrainment depends on the competition between the shear force decrease due to vapor effusion and the bubble emission: in low flow condition, the bubble emission outweighs the shear force decrease, thus the net effect is to increase the droplet entrainment; in high flow condition, the situation may reverse. The heat flux impact on droplet deposition is significant only for very fine droplets (less than 1 μm) because of the coupled effect of interface turbulence damping and the radial vapor effusion due to evaporation, but for droplets of medium and large sizes the heat flux effect is negligible. The analysis is then used to develop constitutive equations for droplet entrainment and deposition rates to take into account the interaction between thermal and hydrodynamic mechanisms, which gives improved CHF prediction for limiting quality regime (LQR) CHF experimental data.     

Velocity and concentration profiles in uniform sediment-laden flow

This paper presents a theoretical model for computing the velocity and sediment concentration profiles in a uniform sediment-laden flow carrying all fine, medium and coarse sediments. The proposed model essentially includes the effect of sediment concentration in total turbulent shear stress and eddy diffusivity in addition to the modified mixing length derived by Umeyama and Gerritsen [J. Hydr. Engng., ASCE, 118 (2) (1992) 229–245] applied to Hunt’s diffusion equation. Numerical solution of coupled differential equations for velocity and sediment concentration is carried out. The theoretical results show quite good agreement with the experimental data.     

Static analysis of composite beams with weak shear connection

The objective of the present paper is to analyse the static behaviour of elastic two-layer beams with interlayer slip. The Euler–Bernoulli hypothesis is assumed to hold for each layer separately, and a linear constitutive equation between the horizontal slip and the interlaminar shear force is considered. The applied loads act in the plane of symmetry of the composite beam, and the material and geometrical properties do not depend on the axial coordinate. Closed-form solutions for displacements and interlayer slips are developed. A second order differential equation is derived for the interlayer slip whose solution is used to determine the deflections and slopes. Examples illustrate the application of the method presented.     

Boundary stabilization of a Bresse‐type system

In this paper, we are concerned with asymptotic stability of a class of Bresse‐type system with three boundary dissipations. The beam has a rigid body attached to its free end. We show that exponential stabilization can be achieved by applying force and moment feedback boundary controls on the shear, longitudinal, and transverse displacement velocities at the point of contact between the mass and the beam. Our method is based on the operator semigroup technique, the multiplier technique, and the contradiction argument of the frequency domain method. Copyright © 2015 John Wiley & Sons, Ltd.     

Formulation and evaluation of a finite element model for the linear analysis of stiffened composite cylindrical panels

A finite element model for linear static and free vibration analysis of composite cylindrical panels with composite stiffeners is presented. The proposed model is based on a cylindrical shell finite element, which uses a first-roder shear deformation theory. The stiffeners are curved beam elements based on Timoshenko and Saint-Venant assumptions for bending and torsion respectively. The two elements are developed in a cylindrical coordinate system and their stiffness matrices result from a hybrid-mixed formulation where the element assumed stress field is such that exact equilibrium equations are satisfied. The elements are free of membrane and shear locking with correct satisfaction of rigid body motions. Several examples dealing with stiffened isotropic and laminated plates and shells with eccentric as well as concentric stiffeners are analyzed showing the validity of the models.     

Extended multiscale FEM for design of beams and frames with complex topology

A numerical method for design of beams and frames with complex topology is proposed. The method is based on extended multi-scale finite element method where beam finite elements are used on coarse scale and continuum elements on fine scale. A procedure for calculation of multi-scale base functions, up-scaling and downscaling techniques is proposed by using a modified version of window method that is used in computational homogenization. Coarse scale finite element is embedded into a frame of a material that is representing surrounding structure in a sense of mechanical properties. Results show that this method can capture displacements, shear deformations and local stress-strain gradients with significantly reduced computational time and memory comparing to full scale continuum model. Moreover, this method includes a special hybrid finite elements for precise modelling of structural joints. Hence, the proposed method has a potential application in large scale 2D and 3D structural analysis of non-standard beams and frames where spatial interaction between structural elements is important.     

A nonlocal higher-order model including thickness stretching effect for bending and buckling of curved nanobeams

An analytical approach for static bending and buckling analyses of curved nanobeams using the differential constitutive law, consequent to Eringen’s strain-driven integral model coupled with a higher-order shear deformation accounting for through thickness stretching is presented. The formulation is general in the sense that it can be deduced to examine the influence of different structural theories, for static and dynamic analyses of curved nanobeams. The governing equations derived using Hamiltons principle are solved in conjunction with Naviers solutions. The formulation is validated considering problems for which solutions are available. A comparative study is made here by various theories obtained through the formulation. The effects various structural parameters such as thickness ratio, beam length, rise of the curved beam, and nonlocal scale parameter are brought out on bending and stability characteristics of curved nanobeams.     

Numerical analysis of two frictionless elastic-piezoelectric contact problems

In this work, we consider two frictionless contact problems between an elastic-piezoelectric body and an obstacle. The linear elastic-piezoelectric constitutive law is employed to model the piezoelectric material and either the Signorini condition (if the obstacle is rigid) or the normal compliance condition (if the obstacle is deformable) are used to model the contact. The variational formulations are derived in a form of a coupled system for the displacement and electric potential fields. An existence and uniqueness result is recalled. Then, a discrete scheme is introduced based on the finite element method to approximate the spatial variable. Error estimates are derived on the approximate solutions and, as a consequence, the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, some two-dimensional examples are presented to demonstrate the performance of the algorithm.     

The problem of the bending of a beam lying on an elastic base

The plane contact problem of the transmission of a normal force of specified strength onto an elastic anisotropic, wedge-shaped plate by an elastic beam of variable flexural stiffness is considered. The beam is coupled to one of the edges of the plate and its other edge is stress-free. The solution of the problem is obtained in closed form by reducing it to a Karleman boundary-value problem with shear for a strip. A conclusion is reached concerning the nature of the discontinuity of the normal contact stress at the vertex of the wedge.     

A homogeneous limit methodology and refinements of computationally efficient zigzag theory for homogeneous,laminated composite,and sandwich plates

The Refined Zigzag Theory (RZT) for homogeneous, laminated composite, and sandwich plates is revisited to offer a fresh insight into its fundamental assumptions and practical possibilities. The theory is introduced from a multiscale formalism starting with the inplane displacement field expressed as a superposition of coarse and fine contributions. The coarse displacement field is that of first‐order shear‐deformation theory, whereas the fine displacement field has a piecewise‐linear zigzag distribution through the thickness. The resulting kinematic field provides a more realistic representation of the deformation states of transverse‐shear‐flexible plates than other similar theories. The condition of limiting homogeneity of transverse‐shear properties is proposed and yields four distinct variants of zigzag functions. Analytic solutions for highly heterogeneous sandwich plates undergoing elastostatic deformations are used to identify the best‐performing zigzag functions. Unlike previously used methods, which often result in anomalous conditions and nonphysical solutions, the present theory does not rely on transverse‐shear‐stress equilibrium constraints. For all material systems, there are no requirements for use of transverse‐shear correction factors to yield accurate results. To model homogeneous plates with the full power of zigzag kinematics, infinitesimally small perturbations in the transverse shear properties are derived, thus enabling highly accurate predictions of homogeneous‐plate behavior without the use of shear correction factors. The RZT predictive capabilities to model highly heterogeneous sandwich plates are critically assessed, demonstrating its superior efficiency, accuracy, and a wide range of applicability. This theory, which is derived from the virtual work principle, is well‐suited for developing computationally efficient, C⁰ a continuous function of (x₁,x₂) coordinates whose first‐order derivatives are discontinuous along finite element interfaces and is thus appropriate for the analysis and design of high‐performance load‐bearing aerospace structures. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

An electromechanically coupled intrinsic,mixed variational formulation for geometrically nonlinear smart composite beam

The work presented in this article is the outcome of a combined strategy of a mathematical tool for 2D cross-sectional analysis, i.e., Variational Asymptotic Method (VAM) as well as the 1D exact beam analyzer, i.e., the intrinsic mixed variational formulation for modeling and analysis of Piezoelectric-laminated composite beams. This work talks about a novel approach of mixed variational formulation to analyze a two-way electromechanically coupled piezoelectric composite beam. In a classical intrinsic mixed variational approach for a passive structure, the 1D exact beam model deals only with mechanical degrees of freedom. In the present case, an extra 1D electrical degree of freedom has been incorporated. A computational code is developed based on the present theory to solve the two-way coupled electromechanical beam problem. In the present case, we have validated the static results for sensor application. Both linear and nonlinear results have been discussed. Results obtained are very promising and are helpful in building a platform where design, optimization and nonlinear analysis of composite ‘smart’ beams in a multibody framework can be done faster while maintaining acceptable accuracy.     

A two-grid method with backtracking for the mixed Navier–Stokes/Darcy model

In this paper, we consider the effect of adding a coarse mesh correction to the two-grid algorithm for the mixed Navier–Stokes/Darcy model. The method yields both L² and H¹ optimal velocity and piezometric head approximations and an L² optimal pressure approximation. The method involves solving one small, coupled, nonlinear coarse mesh problem, two independent subproblems (linear Navier–Stokes equation and Darcy equation) on the fine mesh, and a correction problem on the coarse mesh. Theoretical analysis and numerical tests are done to indicate the significance of this method.