Static and dynamic snap-through behaviour of an elastic spherical shell

The deformation and snap-through behaviour of athin-walled elastic spherical shell statically compressed on aflat surface or impacted against a flat surface are studied theoretically and numerically in order to estimate the influenceof the dynamic effects on the response.A table tennis ballis considered as an example of a thin-walled elastic shell.Itis shown that the increase of the impact velocity leads to avariation of the deformed shape thus resulting in larger deformation energy.The increase of the contact force is causedby both the increased contribution of the inertia forces andcontribution of the increased deformation energy.The contact force resulted from deformation/inertia ofthe ball and the shape of the deformed region are calculated by the proposed theoretical models and compared withthe results from both the finite element analysis and somepreviously obtained experimental data.Good agreement isdemonstrated.     

具有弹性支座杆件的动力稳定性研究

研究了杆件具有弹性支座时直杆的动力稳定性问题，推导出了杆件的临界频率和动力不稳定边界，分析了弹簧刚度和阻尼对杆件动力稳定性的影响，研究表明杆件的侧向刚度愈大其结构的动力稳定性愈好，阻尼对杆的振动起有利作用，并提出了在实际工程中减小杆件振动的一些措施．     

On simplified solutions of contact problems for elastic foundations

In the paper, a method of averaging displacements in a circular area lying in a linearly elastic transversally isotropic foundation is developed for the four modes of motion: vertical, horizontal, rocking and torsional. The corresponding formulae are constructed in a general form which does not depend on the kind of Green functions. For vertical and horizontal modes, uniform load distribution are applied and the simple integral mean is considered, whereas for rotational modes the load proportional to the distance from an axis of rotation is used, and angles of rotation for individual points are averaged with weight of the distance squared. Along with the case of equal radii of circles of loading and averaging, the case of different radii is studied, which allows one to consider contact problems for embedded axisymmetric foundations having the radius varying with depth. As examples the following contact problems are studied: static stiffness for a cone embedded in a homogeneous isotropic half-space in vertical motion, and dynamic stiffness for a disk on a layer resting on a homogeneous half-space for four modes of motion. Comparisons with the corresponding exact solutions are carried out.     

Plastic dynamic stability of a column under nonconservative forces

I.Introduction'Whenadeformableobjectmovinginamedium,thedirectionsofinteractiveforcessuchasfrictionalforceshouldvarywiththedeformationoftheobject.Thisisakindofnonconservativeforces.Forexample,thedirectionofgasfrictionalforceactingontherocketsandthefossiles…     

Stability of elastic and viscoelastic systems under a random perturbations of their parameters

Summary The present paper is concerned with the investigation of the almost sure stability of elastic and viscoelastic systems, when their parameters assume a random wide-band stationary process. The parameters are parametric loads, characteristics of external damping and material viscosity. With the help of Liapunov's direct method, the sufficient condition of the almost sure asymptotic stability for distributed parameter systems with respect to perturbations of initial conditions of an arbitrary form is obtained. It is shown that, in some cases, this condition coincides with a similar condition derived from the assumption that the form of sure and required perturbations coincides with the first eigenfunction of system oscillations. However, an example is given for the stability of a viscoelastic rod, when the perturbations of initial conditions are more dangerous, if their form differs from the first eigenfunction.This research was sponsored by the Russian Foundation of Fundamental Research of the Russian Academy of Sciences under Grant 94-01-01522.     

Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates

The governing equation of motion of gradient elastic flexural Kirchhoff plates, including the effect of in-plane constant forces on bending, is explicitly derived. This is accomplished by appropriately combining the equations of flexural motion in terms of moments, shear and in-plane forces, the moment–stress relations and the stress–strain equations of a simple strain gradient elastic theory with just one constant (the internal length squared), in addition to the two classical elastic moduli. The resulting partial differential equation in terms of the lateral deflection of the plate is of the sixth order instead of the fourth, which is the case for the classical elastic case. Three boundary value problems dealing with static, stability and dynamic analysis of a rectangular simply supported all-around gradient elastic flexural plate are solved analytically. Non-classical boundary conditions, in additional to the classical ones, have to be utilized. An assessment of the effect of the gradient coefficient on the static or dynamic response of the plate, its buckling load and natural frequencies is also made by comparing the gradient type of solutions against the classical ones.     

旋转弹性体的有限元线法分析

本文将有限元线法应用于一般荷载作用下的旋转弹性体的分析。文中将任意荷载沿环向展开为Ｆｏｕｒｉｅｒ级数，利用正交性，将问题转变为一系列旋转子午面上二维问题的叠加。文中对任一环向谐波建立了旋转面上的曲线曲边单元，导出了相应的常微分方程体系，并对由柱坐标引起的ｒ＝０处的奇异性问题做了完备合理的处理。文中给出了具有代表性的数值算例，用以展示本法的出色的表现     

The regions of flutter and divergence instability of a column subjected to Beck's generalized load, taking into account the torsional flexibility of the loaded end of the column

Investigations into a slender system subjected to Beck's generalized load taking into account two rotational springs situated at each end of the system are presented in the paper. One spring models the finite rigidity of the mounting, while the second restricts the rotation of the loaded end of the system. The regions of divergence and flutter instability of the considered system were determined using the kinetic criterion of the stability. The boundary value of the rigidity of the spring situated at the loaded end of the column was also determined. The boundary value of the rigidity separates the regions of divergence and flutter instability. In respect of the problem of vibrations, the characteristic curves in the plane: load - natural frequency were presented. All computations were carried out using the parameters of the considered system, including the rigidity of the springs and the follower coefficient of the load.     

Stability of elastic bodies with nonmonotone multivalued boundary conditions of the quasidifferential type

Necessary conditions for the stability of elastic bodies subjected to nonmonotone multivalued boundary conditions are derived. These conditions are assumed to be derived from nonconvex and nonsmooth, quasidifferentiable energy functions. A difference convex approximation of the potential energy function is written based on an appropriate quasidifferential formulation. Under appropriate assumptions for the convex and the concave parts we prove the existence of at least one nontrivial solution to the nonlinear eigenvalue problem.     

Analysis of Stability and Instability Regions in Parametrically Excited Hamiltonian Systems

Qualitative analysis of parametrically excited linear Hamiltonian systems is carried out. It is proved that the stability and instability regions are convex in the excitation frequency. Lower bounds for the boundaries of some instability regions are obtained expressed in the natural frequencies of the system in the absence of a parametric excitation. It is shown that a dominant high-frequency excitation affects the stability regions similarly to an increase of the natural frequencies. Some of these findings extend known results, obtained by asymptotic methods under the assumption that the parametric excitation is small or its frequency is large, to finite values of the excitation and frequency.     

On stability boundary of linear multi-parameter Hamiltonian systems

In this paper an approximate equation is derived to describe smooth parts of the stability boundary for linear Hamiltonian systems, depending on arbitrary number of parameters. With this equation, we can obtain parameters corresponding to the stability boundary, as well as to the stability and instability domains, provided that one point on the stability boundary is known. Then differential equations describing the evolution of eigenvalues and eigenvectors along a curve on the stability boundary surface are derived. These equations also allow us to obtain curves belonging to the stability boundary. Applications to linear gyroscopic systems are considered and studied with examples. The project supported by the National Science Foundations of Russia and China (10072012)     

On the Proper Form of the Amplitude Modulation Equations for Resonant Systems

The complex amplitude modulation equations of a discrete dynamicalsystem are derived under general conditions of simultaneous internal andexternal resonances. Alternative forms of the real amplitude and phaseequations are critically discussed. First, the most popular polar formis considered. Its properties, known in literature for a multitude ofspecific problems, are here proven for the general case. Moreover, thedrawbacks encountered in the stability analysis of incomplete motions(i.e. motions containing some zero amplitudes) are discussed as aconsequence of the fact the equations are not in standard normal form.Second, a so-called Cartesian rotating form is introduced, which makesit possible to evaluate periodic solutions and analyze their stability,even if they are incomplete. Although the rotating form calls for theenlargement of the space and is not amenable to analysis of transientmotions, it systematically justifies the change of variables sometimesused in literature to avoid the problems of the polar form. Third, amixed polar-Cartesian form is presented. Starting from the hypothesisthat there exists a suitable number of amplitudes which do not vanish inany motion, it is proved that the mixed form leads to standard formequations with the same dimension as the polar form. However, if suchprincipal amplitudes do not exist, more than one standard form equationshould be sought. Finally, some illustrative examples of the theory arepresented.     

扰动法在结构分枝失稳分析中的应用

对结构进行平衡路径的跟踪分析，是全面了解该结构的受力性能所必须进行的一项工作。目前的结构非线性稳定分析技术一般仅对极值点失稳型问题较为有效，而对分枝点失稳型问题则困难较多。对于具有缺陷敏感性的结构，如拱结构、壳体等，在普通荷载作用下，其失稳路径常包含分枝点。文献[1]提出并认为位移扰动法和力扰动法在分析结构分枝失稳时具有很好的效果。本文采用多个不同类型的算例，对扰动法在结构分枝失稳问题中的应用进行了分析比较，表明该方法具有较强的跟踪能力。最后，就扰动法在结构分枝失稳问题中的应用提出几点建议。     

Stability of fluid-conveying periodic shells on an elastic foundation with external loads

This paper presents the beam-mode stability of a fluid-conveying periodic shell on an elastic foundation subjected to external loading. A transfer matrix (TM) method was developed to investigate the characteristics of steady-state waves in the system and the dynamic response of the periodic shell system. When subjected to external perturbations, including either a moving load or a stationary one, the shell may be subjected to instability for flow velocities exceeding a certain critical velocity. The system can also become unstable when a travelling load exceeds a certain critical value. The coupled effects of the speed of a moving load and the flow velocity of a fluid on the stability of the shell system were also investigated. A periodic structure was designed for such a shell system to enhance its dynamic stability. The periodic shell system produces innumerable velocity band gaps (VBGs), which could raise the critical velocity and extend the stable range for both the moving load and the flowing fluid. Finally, the formation mechanism of the VBGs was studied, as well as the effects of the thickness, length of the shell cells, Young׳s modulus and stiffness of the elastic foundation on modulating the VBGs.     

周边嵌入弹性圆薄板非线性自由振动直接摄动解法

采用弹性圆薄板中心无量纲振幅和板厚与半径的比值为参数。将挠度、应力函数对半径的导数以及自由振动频率展开为双参数的幂级数。用直接摄动法获得各级递推线性偏微分方程。应用变分法求得各级递推方程的近似解，从而给出弹性圆薄板非线性自由振动频率的基本公式。     

Instability of uniform flow

We consider uniform flow of a Newtonian fluid trasverse to a domain bounded by parallel planes. We investigate the possibility of introducing instabilities in this flow by the choice of inflow and outflow conditions. Some instabilities of this kind are found.     

On the Reconstitution Problem in the Multiple Time-Scale Method

Higher-order multiple-scale methods for general multiparameter mechanical systems are studied. The role played by the control and imperfection parameters in deriving the perturbative equations is highlighted. The definition of the codimension of the problem, borrowed from the bifurcation theory, is extended to general systems, excited either externally or parametrically. The concept of a reduced dynamical system is then invoked. Different approaches followed in the literature to deal with reconstituted amplitude equations are discussed, both in the search for steady-state solutions and in the analysis of stability. Four classes of methods are considered, based on the consistency or inconsistency of the approach, and on the completeness or incompleteness of the terms retained in the analysis. The four methods are critically compared and general conclusions drawn. Finally, three examples are illustrated to corroborate the findings and to show the quantitative differences between the various approaches.     

Instability of an elastic film on a viscous layer

A flat, compressed elastic film on a viscous layer is unstable. The film can form wrinkles to reduce the elastic energy. In this paper, we are interested in the two-dimensional models for thin films bonded to a viscous layer and in particular we focus on generic instabilities evidenced in this context by Suo and coworkers [Huang, Z., Hong, W., Suo, Z., 2005. Non linear analyses of wrinkles in a film bonded to a compliant substrate. J. Mech. Phys. Solids 53, 2101–2118; Lo, Y.H., 1991. New approach to grow pseudomorphic structures over the critical thickness. Appl. Phys. Lett. 59, 2311–2320]. We present a rigorous linear perturbation analysis for anisotropic materials, that allows the prediction of both the orientation of the corrugations of the thin film, and the wavelength that maximize the growth velocity. Finally, we compare our theoretical estimates to experimental results for a In_0.65Ga_0.35As alloy constraint to InP.     

Dynamic buckling of discrete structural systems under combined step and static loads

The dynamic instability of discrete, elastic, multiple degree of freedom (d.o.f.) systems under a combination of static and step loads is studied. Conservative, autonomous and holonomic systems are considered, in which the associated static response is a bifurcation under one load parameter, and a limit point under the second parameter. A review of different criteria and algorithms obtained from them for the computation of dynamic buckling loads is first presented, followed by a procedure derived from previous investigations on one d.o.f. systems. The different procedures are applied to a two d.o.f. problem under axial and lateral load, with quadratic and cubic non-linearities. The response in time shows that the system oscillates about the static equilibrium position before dynamic buckling is reached, with the kinetic energy tending to zero as assumed in the static (energy) procedures of stability.