Control and exponential stabilization for the equation of an axially moving viscoelastic strip

The aim through this work is to suppress the transverse vibrations of an axially moving viscoelastic strip. A controller mechanism (dynamic actuator) is attached at the right boundary to control the undesirable vibrations. The moving strip is modeled as a moving beam pulled at a constant speed through 2 eyelets. The left eyelet is fixed in the sense that there is no transverse displacement (see Figure 1 ). The mathematical model of this system consists of an integro‐partial differential equation describing the dynamic of the strip and an integro‐differential equation describing the dynamic of the actuator. The multiplier method is used to design a boundary control law ensuring an exponential stabilization result.     

Numerical solution of an inverse problem for a two-dimensional mathematical model of sorption dynamics

We consider a two-dimensional mathematical model of sorption that allows for inner-diffusion kinetics as well as longitudinal and transverse diffusion. The inverse problem of determining the sorption isotherm from an experimental dynamic output curve is investigated for this model and stable solution methods are proposed for the inverse and the direct problem. The efficiency of the solution methods is explored in computer experiments.     

Dynamics of an Overhead Line and Pantograph System

The dynamic interaction between pantograph and catenary is studied. The overhead electrification system for high speed trains is modelled by visco‐elastically connected double homogeneous strings system, one finite suspended on rigid concentrated supports and the second of infinite length. The pantograph is represented by one degree of freedom oscillator moving along the string at a constant speed. The model is new in that it gives possibility to consider the influence of the locomotive transverse vibrations on railway overhead contact system. The dynamic state of the investigated system is described by a nonlinear set of coupled partial differential equations with complicated boundary conditions. Dynamic transverse displacements of the connected strings are determined. General results are illustrated by numerical examples.     

Enhancement of φ meson production in p-p and Pb-Pb collisions at SPS energy

The φ meson yield, rapidity and transverse mass distributions in p-p and Pb-Pb collisions at 158 A GeV were studied by the hadron-string cascade model, LUCIAE. By adjusting the width parameter in q-q Gaussian like transverse momentum distribution in string fragmentation to fit the NA49 data ofφ meson transverse mass distribution in p-p collisions, the obtained φ meson rapidity distribution in p-p collisions, the rapidity and transverse mass distributions in Pb-Pb collisions, and theφ meson enhancement factor in Pb-Pb relative to p-p collisions were compatible with corresponding NA49 data, which might be attributed to the collective effect in gluon emission of string and the reduction of strange quark suppression mechanisms involved in the LUCIAE model.     

On the determination of natural frequencies and mode shapes of cable-stayed bridges

An accurate analysis of the natural frequencies and mode shapes of a cable-stayed bridge is fundamental to the solution of its dynamic responses due to seismic, wind and traffic loads. In most previous studies, the stay cables have been modelled as single truss elements in conventional finite element analysis. This method is simple but it is inadequate for the accurate dynamic analysis of a cable-stayed bridge because it essentially precludes the transverse cable vibrations. This paper presents a comprehensive study of various modelling schemes for the dynamic analysis of cable-stayed bridges. The modelling schemes studied include the finite element method and the dynamic stiffness method. Both the mesh options of modelling each stay cable as a single truss element with an equivalent modulus and modelling each stay cable by a number of cable elements with the original modulus are studied. Their capability to account for transverse cable vibrations in the overall dynamic analysis as well as their accuracy and efficiency are investigated.     

Size-dependent dynamic analysis of rectangular nanoplates in the presence of electrostatic,Casimir and thermal forces

In the present study by considering the small-scale effects, the dynamic instability of fully clamped and simply supported nanoplates is studied in the attendance of electrostatic, Casimir as well as thermal forces. To this end, by applying the nonlocal elasticity theory of Eringen along with the classical plate theory, the dynamic equilibrium equation of nanoplates is obtained by incorporating the in-plane thermal and transverse intermolecular distributed loads. The solution of the obtained nonlinear governing equation is done using the Galerkin method and the dynamic pull-in instability voltage of the nanoplates is compared with the available experimental results. Finally, the simultaneous effects of thermal force as well as nonlocal parameter on the dynamic response of nanoplates are examined in the presence of Casimir force in detail.     

基于非约束模态的中心刚体-Timoshenko梁动力学建模与分析北大核心CSCD

梁的横向变形会导致梁纵向缩短,建模过程中考虑梁横纵变形二次耦合项则存在动力刚化现象,这说明梁的纵向变形会对模型的广义刚度造成影响.对于做旋转运动的梁结构,旋转运动时还会受到离心力的作用而产生轴向拉力,轴向拉力同样也会引起梁的轴向变形,这种影响对粗短梁更加明显.以大范围运动中心刚体-Timoshenko梁模型为研究对象:首先,运用Timoshenko梁理论以及Hamilton原理建立含离心力的动力学模型;其次,引入非约束模态概念,采用Frobenius方法求解非约束模态振型函数以及固有频率;最后,通过数值仿真探究不同恒定转速时非约束模态与约束模态广义刚度的差异和非约束模态条件下离心力对模型的影响.     

Non-conservative oscillations of a tool for deep hole honing

The dynamics of a rotating tool, commonly employed in deep hole honing, is considered. A mathematical model of the process including a dynamic representation of tool, workpiece surface and honing stones interaction is suggested and analyzed. It is shown that interaction forces are non-conservative. The honing tool is modeled as a rotating continuous slender beam with a mandrel attached at an intermediate cross-section. The transverse oscillations of tool shaft cause variation in expansion pressure and change the interaction forces on the surface of the workpiece. The interaction forces are nonlinear and non-conservative in the general case including delayed functions. The expansion pressure of stones turns out to be a critical parameter in a honing process. Since the removal of stock increases linearly with pressure, the productivity can be improved by increasing expansion pressure, but in certain conditions may cause dynamic instability of the tool shaft. On the other hand, the lower expansion pressure and feed rates increase accuracy. Another source of shaft instability is due to the asymmetry of the tool that leads to discrepancy of the system stiffness in the transverse directions. As a result, under certain conditions unstable parametric vibrations may occur. Needless to say, the dynamic behavior of the tool can considerably influence the machined surface formation. In this study a model of honing surface formation is introduced and integrated into the model of process simulation. The system response is studied for different parameters in a non-dimensional form. Thus it is possible to analyze a set of real processes by applying the similarity conditions. The variation in the machined surface having various initial discrepancies from ideal cylinder is also analyzed.     

Vibration behavior of simply supported inclined single-walled carbon nanotubes conveying viscous fluids flow using nonlocal Rayleigh beam model

A mathematical model is proposed to investigate the dynamic response of an inclined single-walled carbon nanotube (SWCNT) subjected to a viscous fluid flow. The tangential interaction of the inside fluid flow with the equivalent continuum structure (ECS) of the SWCNT is taken into account via a slip boundary condition. The dimensionless equations of motion describing longitudinal and lateral vibrations of the fluid-conveying ECS are obtained in the context of nonlocal elasticity theory of Eringen. The unknown displacement fields are expressed in terms of admissible mode shapes associated with the ECS under simply supported conditions with immovable ends. Using Galerkin method, the discrete form of the equations of motion is derived. The time history plots of the normalized longitudinal and transverse displacements as well as the nonlocal axial force and bending moment of the midspan point of the SWCNT are provided for different levels of the fluid flow speed, small-scale parameter, and inclination angle of the SWCNT. The effects of small-scale parameter, inclination angle, speed and density of the fluid flow on the maximum dynamic amplitude factors of longitudinal and transverse displacements as well as those of nonlocal axial force and bending moment of the SWCNT are then studied in some detail.     

Classical and Nonclassical Dynamic Problems of Sandwich Shells with a Transversely Soft Core

Based on the hypothesis of similarity of transverse displacements in thin-walled sandwich shells with a transversely soft core under dynamic and static loads, refined geometrically nonlinear dynamic equations of motion are constructed in the case of large variations in the parameters of the stress-strain state (SSS) in the tangential directions. For shells structurally symmetric across the thickness and loaded with initial static loads, linearized dynamic equations are derived, which, upon introducing the synphasic and antiphasic functions of displacements and forces, can be used to describe the synphasic and antiphasic buckling forms in the transverse and tangential directions. For nonshallow cylindrical and shallow spherical shells, the nonclassical problems on all possible vibration forms realized at zero indices of variability of the SSS parameters in the tangential directions are formulated and solved. For shallow shells of symmetric structure, the resolving equations are obtained by introducing, instead of tangential displacements and transverse tangential stresses in the core, the corresponding potential and vortex functions.     

Numerical analysis of 3D dynamic problems of the Cosserat elasticity theory subject to boundary symmetry conditions

In the framework of the Cosserat continuum model, one-dimensional solutions describing plane longitudinal waves, transverse (shear) waves with particle rotation, and torsional waves are analyzed. Boundary symmetry conditions for various types of loading are found. A parallel computational algorithm is worked out for solving 3D dynamic problems of the Cosserat elasticity theory on multiprocessor computer systems. Computations of the propagation of the stress and strain waves induced by a point impulse force in an elastic medium are performed.     

Analysis of the dynamic response of 3D-braided rectangular plates on an elastic foundation

A new micromacromechanical model of unit cells is suggested. In this model, a 3D-braided composite plate can be a cell system, and the geometry of each cell strongly depends on its position in the cross section of the plate. Based on the Reddy higher-order shear deformation theory of plates and the general von Karman-type equations, analytical solutions for the dynamic response of simply supported 3D-braided rectangular plates resting on a two-parameter (Pasternak-type) elastic foundation and subjected to transverse dynamic loads are found by using the perturbation method. The numerical illustrations concern the effects of geometric parameters, fiber volume fraction, braiding angle, and foundation stiffness. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 6, pp. 861–880, November–December, 2008.     

High-precision modeling of deformation of laminated structures

Precise three-dimensional solutions for homogeneous, two- and three-layer plates of symmetric and nonsymmetric structure over their thickness with orthotropic layers in transverse symmetric and antisymmetric loading normal to the surface of the plates are given. It is shown that the character of stressed states under flexural (antisymmetric) and nonflexural (symmetric transverse compression) loading differs greatly. It is noted that the known refined continual models, which take into account the transverse shear and compression, are all essentially flexural and therefore cannot describe the nonflexural deformations well. In particular, continual shear models in symmetric pressure loading lead to zero solutions. A refined nonflexural continual model of deformation of sandwich plates in bilateral symmetric compression is constructed. The general order of resolving differential equations for continual models does not depend on the number of layers. Approximation functions of the transverse coordinate are obtained with the help of well-founded hypotheses. A high-accuracy variant of the flexural continual model is proposed for antisymmetric loading with account of shear and transverse normal strains, as well as a version combining both models mentioned. A method of precise satisfation of all the constitutive relations for the layers, including the conditions of their contact, is proposed, whereas in the known continual models the dependence between the transverse normal stress and strain is satisfied only integrally, or else the Poisson effect is neglected.Ukrainian State Academy of Water Management, Rovno, the Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 1, pp. 13–28, January–February, 1999.     

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     

The effect of stiffness on the transverse and longitudinal motions of a musical string: Commemorating the 100th anniversary of the birth of Kh.A. Rakhmatulin

The effect of stiffness on the propagation of longitudinal and transverse waves and vibrations in prestretched strings is considered. The contribution of the longitudinal and transverse components to the dynamic load is of the same order. The longitudinal vibrations occur both at natural frequencies and at frequencies of the transverse vibrations. Resonance phenomena are possible. Low stiffness, which is characteristic for musical strings, leads to a small change in the frequencies of the whole spectrum of transverse and longitudinal vibrations, but to a considerable change in the shape of the string at strike and mounting points and on the transverse wave front.     

计及弯曲刚度的印刷运动薄膜横向振动控制研究

研究了不同边界条件下,计及弯曲刚度的轴向运动薄膜横向振动的主动控制问题.建立计及弯曲刚度的印刷运动薄膜的计算模型.利用有限差分法,对轴向运动薄膜的振动微分方程进行离散,推导出轴向运动矩形薄膜横向振动控制系统的状态方程.采用次最优控制法,对不同边界条件下轴向运动矩形薄膜横向振动进行主动控制研究.计算结果表明:采用次最优控制法能够在短时间内迅速、有效地降低运动薄膜的振动强度,并使之衰减趋近于0.作动器作用在固定位置点处时,对运动薄膜施加控制后,四边简支边界条件下的控制效果好.作动器作用在不同位置点处时,两种边界条件下中心点处的控制效果最好.计算证明次最优控制法能够有效地抑制印刷过程中计及弯曲刚度的轴向运动薄膜的横向振动,从而提高印刷套印精度,保证精密印刷质量.     

The modeling and dynamic analysis of two jointed beams with clearance

The dynamic behaviors of two beams connected by a joint with clearance are investigated. A new equivalent joint model using the transverse and torsion spring systems is developed to describe the dynamics of two jointed beams with clearance. Based on the finite element method (FEM), the equations of motion of the two jointed beams with clearance are established using Hamilton's principle. A modified numerical solution method is presented based on the Newmark integration method to solve the equations with the nonlinearity due to clearance. The contributions of the clearance size and stiffness on the amplitude-frequency response are discussed. The effects of the clearance on the vibration transfer between the two connected beams and the impact force of the joint are investigated.     

考虑横向剪切变形和旋转惯性的层合正交异性球壳的动态响应

本文建立了计及横向剪切变形和旋转惯性的复合材料轴对称层合圆柱正交异性球壳的运动方程．在此基础上，用有限差分法计算了球壳在轴对称动力载荷下的动态响应，并讨论了材料参数、结构参数和横向剪切变形的影响．     

The dynamic process of the inflation of thin elastomeric shells under the action of an excess pressure

A problem of the dynamic process of their deformation is formulated in the momentless approximation for thin shells made of rubber-like elastomers under the action of a time-varying excess hydrostatic pressure. A system of non-linear equations of motion is set up for the case of arbitrary displacements and deformations in which the true deformation of the transverse compression of the shell, corresponding to the use of the modified Kirchhoff–Love model proposed earlier, and the coordinates of the points of the middle surface with respect to a fixed Cartesian system of coordinates, are taken as the required unknown functions. Physical relations connecting the components of the true internal stresses with the elongation factors and the extent of the shear strain are constructed using relations proposed earlier by Chernykh. A finite-difference method is developed for solving the initial-boundary value problem and, on the basis of this, the dynamic process of the inflation of shells of revolution at different rates of pressure increase is investigated and the unstable stages of their deformation are established with a determination of the corresponding limiting (critical) pressure value. After this value has been reached, a further increase in the deformations occurs at decreasing values of the internal pressure.     

Local and extended models for high-speed,dynamic loading of metals

The purpose of this work is to compare and contrast two different modeling approaches for high-speed, dynamic loading of metals. A so-called “local” model including hardening, strain-rate hardening as well as temperature and ductile damage softening behaviour is extended by gradient plasticity. In particular, both models are compared in the context of their application to the modeling and simulation of dynamic shear-banding in the alloy Inconel 718. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)