Non-linear theories of beams and plates accounting for moderate rotations and material length scales

The primary objective of this paper is to formulate the governing equations of shear deformable beams and plates that account for moderate rotations and microstructural material length scales. This is done using two different approaches: (1) a modified von Kármán non-linear theory with modified couple stress model and (2) a gradient elasticity theory of fully constrained finitely deforming hyperelastic cosserat continuum where the directors are constrained to rotate with the body rotation. Such theories would be useful in determining the response of elastic continua, for example, consisting of embedded stiff short fibers or inclusions and that accounts for certain longer range interactions. Unlike a conventional approach based on postulating additional balance laws or ad hoc addition of terms to the strain energy functional, the approaches presented here extend existing ideas to thermodynamically consistent models. Two major ideas introduced are: (1) inclusion of the same order terms in the strain–displacement relations as those in the conventional von Kármán non-linear strains and (2) the use of the polar decomposition theorem as a constraint and a representation for finite rotations in terms of displacement gradients for large deformation beam and plate theories. Classical couple stress theory is recovered for small strains from the ideas expressed in (1) and (2). As a part of this development, an overview of Eringen׳s non-local, Mindlin׳s modified couple stress theory, and the gradient elasticity theory of Srinivasa–Reddy is presented.     

几何非线性摩擦阻尼隔振系统动力学行为研究

非线性隔振系统由于具有较线性系统更优的隔振性能,因此在工程中应用广泛.本文通过配置与被隔振对象的运动方向相垂直的库伦摩擦阻尼器,构建了几何非线性摩擦阻尼模型.由于引入了几何非线性,因此其摩擦力与位移正相关,这与传统与位移无关摩擦力模型有显著不同.首先,建立了具有几何非线性摩擦阻尼的数学模型以及隔振系统的受迫振动方程;然后,使用谐波平衡法求解了动力学方程,并使用数值仿真方法验证了谐波平衡法求解的准确性;最后,研究了几何非线性摩擦阻尼隔振器的绝对位移传递率和相对位移传递率.研究结果表明,在库伦摩擦阻尼选择适当,非线性摩擦阻尼系统可以在保持高频振动衰减效果的前提下,显著降低系统共振峰,其性能优于传统的恒定摩擦阻尼隔振模型.同时,几何非线性摩擦阻尼系统能够避免传统摩擦阻尼系统中的“锁定”现象,从传递率角度来说,不利于共振峰控制;但从激励环境改变引发隔振系统失效的角度来看,几何非线性摩擦阻尼系统可以拓宽系统对激励幅值的适应范围,避免隔振系统失效.本文的研究结果对此类隔振系统的设计和摩擦阻尼参数的选择具有通用的指导意义.      

On the use of generalized Hamilton׳s principle for the derivation of the equation of motion of a pipe conveying fluid

In this paper, first, some pertinent studies are reviewed, undertaken to extend Hamilton׳s principle or Lagrange׳s equations to open systems, and continua including so-called non-material volumes; then, the corresponding mathematical formulations are reproduced and discussed using a uniform nomenclature. A typical flexible pipe conveying fluid system is considered next, and it is shown that the expression given by McIver for Hamilton׳s principle remains valid even in the presence of non-material volumes; the extra term due to the flux of kinetic energy through the control surface in the expression obtained by Casetta and Pesce vanishes.     

A unifying model for fluid flow and elastic solid deformation: A novel approach for fluid–structure interaction

Fluid–structure coupling is addressed through a unified equation for compressible Newtonian fluid flow and elastic solid deformation. This is done by introducing thermodynamics within Cauchy׳s equation through the isothermal compressibility coefficient that is experimentally measurable for both fluids and solids. The vectorial resolution of the governing equation, where every component of velocity vectors and displacement variation vectors is calculated simultaneously in the overall multi-phase system, is characteristic of a monolithic resolution involving no iterative coupling. For system equation closure, mass density and pressure are both re-actualized from velocity vector divergence, when the shear stress tensor within the solid phase is re-actualized from the displacement variation vectors. This novel approach is first validated on a two-phase system, involving a plane fluid–solid interface, through the two following test cases: (i) steady-state compression and (ii) longitudinal and transverse elastic wave propagations. Then the 3D study of compressive fluid injection towards an elastic solid is analyzed from initial time to steady-state evolution.     

Structural analogies on systems of deformable bodies coupled with non-linear layers

The paper is addressed at phenomenological mapping and mathematical analogies of oscillatory regimes in systems of coupled deformable bodies. Systems consist of coupled deformable bodies like plates, beams, belts or membranes that are connected through visco-elastic non-linear layer, modeled by continuously distributed elements of Kelvin–Voigt type with nonlinearity of third order. Using the mathematical analogies the similarities of structural models in systems of plates, beams, belts or membranes are obvious. The structural models consist by a set of two coupled non-homogenous partial non-linear differential equations. The problems to solve are divided into space and time domains by the classical Bernoulli–Fourier method. In the time domains the systems of coupled ordinary non-linear differential equations are completely analog for different systems of deformable bodies and are solved by using the Krilov–Bogolyubov–Mitropolskiy asymptotic method. This paper presents the beauty of mathematical analytical calculus which could be the same even for physically different systems.The mathematical numerical calculus is a powerful and useful tool for making the final conclusions between many input and output values. The conclusions about nonlinear phenomena in multi-body systems dynamics have been revealed from the particular example of double plate׳s system stationary and non-stationary oscillatory regimes.     

A comparison of limited-stretch models of rubber elasticity

In this paper we describe various limited-stretch models of non-linear rubber elasticity, each dependent on only the first invariant of the left Cauchy–Green strain tensor and having only two independent material constants. The models are described as limited-stretch, or restricted elastic, because the strain energy and stress response become infinite at a finite value of the first invariant. These models describe well the limited stretch of the polymer chains of which rubber is composed. We discuss Gent׳s model which is the simplest limited-stretch model and agrees well with experiment. Various statistical models are then described: the one-chain, three-chain, four-chain and Arruda–Boyce eight-chain models, all of which involve the inverse Langevin function. A numerical comparison between the three-chain and eight-chain models is provided. Next, we compare various models which involve approximations to the inverse Langevin function with the exact inverse Langevin function of the eight-chain model. A new approximate model is proposed that is as simple as Cohen׳s original model but significantly more accurate. We show that effectively the eight-chain model may be regarded as a linear combination of the neo-Hookean and Gent models. Treloar׳s model is shown to have about half the percentage error of our new model but it is much more complicated. For completeness a modified Treloar model is introduced but this is only slightly more accurate than Treloar׳s original model. For the deformations of uniaxial tension, biaxial tension, pure shear and simple shear we compare the accuracy of these models, and that of Puso, with the eight-chain model by means of graphs and a table. Our approximations compare extremely well with models frequently used and described in the literature, having the smallest mean percentage error over most of the range of the argument.     

Numerical simulation of the fluid–structure interaction in a tube array under cross flow at moderate and high Reynolds number

The unsteady loads in a tube bundle are studied at moderate and high Reynolds number by means of URANS and hybrid (DDES) modelling. The onset of fluid-elastic instability is analysed for different structural parameters, Scruton number and reduced velocity. The simulations have been carried out with the code NSMB (Navier–Stokes Multi Block) by using turbulence modelling methods URANS and DDES (Delayed Detached Eddy Simulation). The CEA-DIVA configuration is considered for the cylinders array for an inter-tube Reynolds number 60 000. The study is carried out for a configuration of (4×5) cylinders in static conditions as well as for the vertical free motion of one of the central cylinders in one DOF (Degree Of Freedom).The inter-tube Reynolds number is 60 000. It is found that this cylinder spontaneously displays an oscillatory motion which first corresponds to Vortex Induced Vibration (VIV), associated to a lock-in mechanism for low values of the reduced velocity and secondly develops Movement Induced Vibration, MIV, for higher values of the reduced velocity. The variation of the cylinder׳s oscillations frequency, of the unsteady loads and the structure׳s displacement are studied as a function of the reduced velocity for low and high values of the Scruton number. The increase of the phase-lag between the forces and the displacement is predicted and discussed for different Scruton number values and reduced velocities.     

Diffusion Mass Transfer in Miscible Oil Recovery: Visual Experiments and Simulation

Co-and counter-current type transfers due to diffusion and -free- convection caused by the buoyant forces between fracture and matrix were studied experimentally using 2-D glass-bead models. Mineral oil and kerosene were used as the displaced phase. The model saturated with oil was exposed to solvent phase (pentane) under static conditions (no flow in fracture) to mimic matrix-fracture interaction during gas or liquid solvent injection in naturally fractured reservoirs. Displacement fronts and patterns were analyzed and quantified using fractal techniques to obtain correlations between the fractal properties and displacement type. Displacements resulted in a mixture of bulk diffusion and -free- convection mainly depending on the interaction type (co- or counter-current), oil type, and displacement direction (horizontal and vertical). Conditions yielding different types of displacement patterns were identified. Finally, a stochastic model that was inspired from invasion percolation and diffusion limited aggregation algorithms was developed for the horizontal displacement cases. The experimental observations were matched to the displacement patterns obtained through the stochastic modeling.     

Tilted guides with friction in web conveyance systems

One challenge in designing web conveyance systems is controlling the displacement and vibration of the webs by guides without introducing instabilities or higher frequency disturbances from flange impacts. A solution to this problem is to use an actively or passively tilted guide or roller to steer the web. In this paper, a model of tilted guides with friction is developed, and it is shown that tilted guides produce a change in the web’s displacement, slope, bending moment, and shear force. When the web is conceptually unwrapped from its path, the normal force between the web and a tilted guide has a component that acts in the direction of the web’s lateral displacement, resulting in an equivalent force and bending moment acting on the web. The model is validated by measurements, and is compared to a previously existing model of guide tilt. In the configurations studied, the displacement of the web near the guide is linearly dependent on the tilt angle and tension and it increases exponentially with the web’s span length. When the guide’s tilt is oriented towards the center of the web’s wrap around the guide, the equivalent bending moment is zero in the absence of friction, and there is good agreement between the model developed in this paper and the previously existing model. However, when the center of the web’s wrap is oriented 90° away from the guide’s tilt orientation, the equivalent force is zero in the absence of friction, and measurements demonstrate the necessity of the equivalent bending moment.     

Local and nonlocal conserved vectors of the system of two-dimensional generalized inviscid Burgers equations

The concept of non-linear self-adjointness for the construction of conservation laws has attracted a lot of interest in recent years. The most noteworthy aspect of it is the likelihood of explicitly constructing the conservation laws for any arbitrary systems of differential equations, in particular for those for which Noether׳s theorem is not applicable. In this study, we shall use both Noether׳s theorem and the non-linear self-adjoint method to construct local and nonlocal conserved vectors of the system of two-dimensional Burgers equations under consideration. The first integrals obtained not only give more credence to obtained results due to their generality with respect to any arbitrary functions of the velocity components but are also independent, nontrivial and infinitely many.     

Influence of incident vortex street on separated flow around a finite blunt plate: PIV measurement and POD analysis

This study investigated the influence of incident vortex street on the spatial characteristics of separated shear layers around finite blunt plates through use of planar particle image velocimetry. Three systems with different chord-to-thickness ratios (c/t) were chosen for the comparative study (i.e., c/t=3.0, 6.0 and 9.0). The Reynolds number, based on the plates׳ thickness (t), was Re_t=1000. The incident vortex street was generated by placing a circular cylinder (D=t) far upstream of the plate. For the systems without incident vortex street, the separated shear layers around the shortest (c/t=3.0), the median-length plate (c/t=6.0) and the longest (c/t=9.0) plates had no re-attachment, periodic re-attachment and faithful re-attachment on the plate׳s surface, respectively. However, the separated shear layers subjected to incident vortex street were restrained at the leading edges by the upstream vortical structures, which were less influenced by increases in chord-to-thickness ratio. Contour plots of the spatial v–v correlation coefficient revealed that the wakes behind plates longer than c/t=6.0 were not severely influenced by incident vortex street. Distributions of the spatial v–v correlations and the POD eigenmodes revealed that the incident vortical structures were split by the leading edges of the plates in all systems, resulting in two vortices with the same rotating direction. Subsequently, a further phase-averaged analysis convincingly demonstrated the splitting process of the incident vortical structures by the leading edges.     

A fringe-compensation technique for stress analysis by reflection holographic interferometry

The use of holographic interferometry for stress analysis of nontransparent objects is limited by rigid-body displacements of the object. These displacements can alter the fringe patterns and often cause the fringes to disappear completely. A technique of compensation for this deterioration of the fringe pattern forreal-time holographic interferometry is described in this paper. It is especially designed to permit the accurate measurement of the out-of-plane component of strain near regions of stress concentration in plates that are subjected to in-plane loading. It is first shown that the fringes caused by a pure rigid-body displacement can be eliminated almost completely by translations of the hologram and rotation of the illumination wave. This procedure is first described when the displacement is known; then when it is unknown. A method to estimate the error made in the correction is presented. In actual stress-analysis problems, the object is both rigidly displaced and strained. Assuming the rigid displacement is known and corrected as previously, the analysis is developed to relate the fringe pattern to the strain-related displacement. This analysis takes into account the optical modifications of the system that are necessary to achieve the rigid-body-displacement correction. When the rigid-body displacement is unknown, the method is shown still to be workable through the use of various symmetries and boundary conditions. Two sample interferograms are presented as illustrations. Quantitative treatment of data from one of these are presented in a companion paper.     

SH波与界面多圆孔的散射及动应力集中

研究了平面SH波对相邻多个界面圆孔的散射及其动应力集中,为了求解,首先利用复变函数和多极坐标方法构造了在含有多个半圆形缺口的弹性半空间,水平面上任一点随时间谐和出平面线源载荷作用时失主移,邓Green函数,且采用“契合”模型,推导了SH波对相邻多个界面圆孔散射的定解积分方程组,进而求得圆孔附近的动应力系数,作为算例,讨论了具有两个界面圆孔对SH波的散射及其相互影响,给出了孔附近的动应力分布曲线。     

Non-local free and forced vibrations of graded nanobeams resting on a non-linear elastic foundation

We consider in this paper the free and forced vibration response of simply-supported functionally graded (FG) nanobeams resting on a non-linear elastic foundation. The two-constituent Functionally Graded Material (FGM) is assumed to follow a power-law distribution through the beam thickness. Eringen׳s non-local elasticity model with material length scales is used in conjunction with the Euler–Bernoulli beam theory with von Kármán geometric non-linearity that accounts for moderate rotations. Non-linear natural frequencies of non-local FG nanobeams are obtained using He׳s Variational Iteration Method (VIM) and the direct and discretized Method of Multiple Scales (MMS), while the primary resonance analysis of an externally forced non-local FG nanobeam is performed only using the MMS. The effects of the non-local parameter, power-law index, and the parameters of the non-linear elastic foundation on the non-linear frequency-response are investigated.     

Flexure analysis of circular elastic layers bonded between rigid plates

An elastic layer of circular cross-section which is bonded between rigid plates and subjected to pure bending moment is analyzed through a theoretical approach. Based on two kinematic assumptions, the governing equations for the two horizontal displacement functions are established from the equilibrium equations. The horizontal displacements are then solved by satisfying the stress boundary conditions in the elastic layer. Through these solved displacements, the vertical stress in the elastic layer, the shear stress on the bonding surfaces, and the tilting stiffness of the bonded layer are derived in closed-forms and are also compared with the results of finite element analysis.     

On the reduction of Blackstock׳s model of thermoviscous compressible flow via Becker׳s assumption

Issues regarding the derivation and applicability of the weakly non-linear model presented by Brunnhuber and Kaltenbacher in their article [Well-posedness and asymptotic behavior of solutions for the Blackstock–Crighton–Westervelt equation, Discret. Contin. Dyn. Syst. Ser. A 34 (2014) 4515–4535] are revisited and clarified; in particular, its links to both Blackstock׳s thermoviscous compressible flow model and Becker׳s assumption are established.     

Numerical and experimental investigation of flow-acoustic resonance of side-by-side cylinders in a duct

The phenomenon of flow-excited acoustic resonance is a design concern in many engineering applications, especially when wakes of bluff bodies are encountered in ducts, piping systems, heat exchangers, and other confined systems. In this paper, the case of self-excited acoustic resonance of two side-by-side cylinders in a duct with cross-flow is investigated both numerically and experimentally for a single spacing ratio of T/D=2.5, where D is the diameter of the cylinders and T is the centre-to-centre distance between them. The numerical investigation is performed using a finite-volume method at a Reynolds number of 3.0×10⁴ to simulate the unsteady flow field, which is then coupled with an imposed resonant sound field of the first acoustic cross-mode of the duct calculated through the use of Finite Element Analysis (FEA). The experimental investigation has been performed using phase-locked Particle Image Velocimetry (PIV) of the flow field during the occurrence of a self-excited acoustic resonance condition in the duct. The results of both methods reveal that the flow-excited acoustic resonance produces a strong oscillatory flow pattern in the cylinder wakes, with strong in-phase vortex shedding being synchronized by the acoustic resonance. The distribution and strength of the aeroacoustic sources and sinks within the flow field have been computed by means of Howe׳s theory of aerodynamic sound for both the experimental and numerical cases, with the results of the two methods comparing favourably, showing comparable trends in the oscillating flow fields, and very similar trends in the distribution of net acoustic power.     

A Note on a Model Hopping Machine

A model of a statically unstable (one-legged) hopping machine is discussed. It is used to analyze whether and how well the control algorithms for systems with variable constraints can be employed in the engineering implementation of this machine. The results obtained indicate that the control algorithms make it possible to create a functioning machine with relatively simple components. The dynamical characteristics of this hopping machine are such that it can, for example, double the horizontal speed in two to three hops. This fact is interesting because a linear controller controls a nonlinear object.     

安装方向对角接触球轴承保持架动态特性的影响

利用二次开发后的有限元软件MSC Patran/Ls-dyna实现角接触球轴承的参数化建模与动态性能分析. 通过与试验以及文献结果进行对比,验证了建立的有限元模型的正确性及仿真结果的可靠性. 在此基础上研究了角接触球轴承不同安装方向对保持架动态特性的影响. 结果表明:由于重力作用,保持架水平安装较竖直安装时更易形成圆形的质心轨迹. 低速工况下,保持架竖直安装时最大应力发生在过梁位置,水平安装时最大应力发生在侧梁位置. 高速工况下,保持架在两种安装方向的最大应力均发生在过梁位置. 不同转速下保持架在轴承竖直安装时最大应力值均更大.      

Theoretical and Experimental Investigations of Miscible Displacement in Fractured Porous Media

In this paper a mathematical model for miscible displacement in fractured porous media is developed. The model takes into account mechanisms of mass transfer between fracture and matrix. The model is normalized by using the dimensionless parameters, which characterize the process, and the analytical solutions of the resulting system of equations are provided by utilizing the method of characteristics. For comparison the results of model with experimental results, laboratory displacement tests have been performed in fractured systems under miscible displacement. The porous media used were cylindrical Asmari cores from Iranian reservoirs containing an artificially vertical fracture. Normal heptane and kerosene were two miscible fluid used. There is very good agreement between experiments and model prediction.