Point attractors and dynamic buckling of autonomous systems under step loading

In this study the dynamic response of autonomous mainly dissipative multi D.O.F. systems under step loading is re-examined. Based on the geometrical point of view of the theory of non-linear dynamical systems and the rapidly developing theory of attractors, the investigation focuses on limit point like systems, with snapping as their salient feature. It is found that dynamic buckling (through a saddle or its neighborhood) , although leading to a large amplitude motion, may be associated with a point attractor response on the pre-buckling fixed point, depending on the amount of damping considered in close conjunction with the motion channel geometry and the total potential characteristics of all (stable and complementary) equilibria. For such systems, only a straightforward fully non-linear dynamic analysis can provide valid information on the global dynamic stability, since the shape of the total potential hypersurface may become very complicated, rendering energy aspects practically not applicable. A 2-D.O.F. model, simulating an asymmetric suspended roof is comprehensively analyzed to capture the above findings, and a parametric investigation is carried out, revealing a variety of new dynamic response types and leading to a more accurate insight of the stability of motion in the large.     

Nonlinear vibration of a two-mass system with nonlinear stiffnesses

An analytical approach is developed for nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of the proposed approach is twofold. First, it introduces the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment a nonlinear differential system by linearization coupled with Newton’s method and harmonic balance method. New and accurate higher-order analytical approximate solutions for the nonlinear system are established. After solving the nonlinear differential equation, the displacement of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, this higher-order Newton–harmonic balance (NHB) method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation unlike the classical harmonic balance method which results in complicated algebraic equations requiring further numerical analysis. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and the classical harmonic balance methods. Two examples of nonlinear two-degree-of-freedom mass–spring system are analyzed and verified with published result, exact solutions and numerical integration data.     

Stability of motion of a two-mass system with nonlinear friction

This paper concerns the motion of different elastically coupled masses. One of the masses is subjected to a motive force, while the other mass is acted upon by friction. The motive force decreases linearly, while friction changes nonlinearly. The differential equations of motion are derived and are reduced to the standard form (after Bogolyubov). The averaging method is used to find steady-state solutions, one of which agrees with the exact steady-state solution of the initial system of equations. It is found that the actual conditions of stability of the steady-state solution are differ greatly from the conditions calculated on the basis of avaraged equations. These differences are due to the difference in the degrees of the characteristic Rouse-Hurwitz polynomials calculated on the basis of the initial and averaged equations. The analytical results are illustrated by modeling on a microcomputer. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev; Regional Scientific Research and Experimental Design Institute, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 8, pp. 94–100, August, 1999.     

Chaotic phenomena in the dynamic buckling of an elastic-plastic column under an impact

The present study is concerned with the dynamic anomalous response of an elastic-plastic column struck axially by a massm with an initial velocityv ₀. This simple example is considered in order to clarify the influence of the impact characteristics and the material plastic properties on the dynamic buckling phenomenon and particularly on the final vibration amplitudes of the column when it shakes down to a wholly elastic behaviour. The material is assumed to have a linear strain hardening with a plastic with a plastic reloading allowed. These material properties are the reason a number of elastic-plastic cycles to be realized prior to any wholly elastic stable behaviour, which causes different amounts of energy to be absorbed due to the plastic deformations.The column exhibits two types of behaviour over the range of the impact masses — a quasi-periodic and a chaotic response. The chaotic behaviour is caused by the multiple equilibrium states of the column when any small changes in the loading parameters cause small changes in the plastic strains which result in large changes in the response behaviour. The two types of behaviour are represented by displacement-time and phase-plane diagrams. The sensitivity to the load parameters is illustrated by the calculation of a Lyapunov-like exponent. Poincaré maps are shown for three particular cases.Notation c elastic wave propagation speed - m impact mass - m _c column mass - s step of the spatial discretization - t time - u(x,t) axial displacement - v ₀ initial velocity - w ₀(x) initial imperfections - w(x,t)+w ₀(x) total lateral displacements - x axial axis - z axis along the column thickness - A cross-section areahb - E Young's modulus - E _t hardening modulus (Figure 2) - M(x,t) bending moment - N(x,t) axial force - impact mass ratiom/m _c - (x,z) strain - Lyapunov-like exponent - material density - (x,z) stress     

On dynamic buckling phenomena in axially loaded elastic-plastic cylindrical shells

Some characteristic features of the dynamic inelastic buckling behaviour of cylindrical shells subjected to axial impact loads are discussed. It is shown that the material properties and their approximations in the plastic range influence the initial instability pattern and the final buckling shape of a shell having a given geometry. The phenomena of dynamic plastic buckling (when the entire length of a cylindrical shell wrinkles before the development of large radial displacements) and dynamic progressive buckling (when the folds in a cylindrical shell form sequentially) are analysed from the viewpoint of stress wave propagation resulting from an axial impact. It is shown that a high velocity impact causes an instantaneously applied load, with a maximum value at t=0 and whether or not this load causes an inelastic collapse depends on the magnitude of the initial kinetic energy.     

A simple and efficient fourth-order approximation solution for nonlinear dynamic system

In the present paper, based on the classical Magnus expansion, a simple and efficient fourth-order integrator is given for an arbitrary nonlinear dynamic system, which can preserve the qualitative properties of the exact solution. The proposed method can be considered an averaging technique, and only requires evaluations of exponentials of simple unidimensional integrals. Finally, the numerical examples are given to demonstrate the validity and effectiveness of the method of this paper.     

Dynamic buckling of a simple elastic-plastic model under pulse loading

The dynamic buckling of an elastic-plastic imperfection-sensitive model subjected to rectangular- and triangular-shaped loading pulses is examined to provide some insight into the dynamic buckling behaviour of structures. The loading pulse is expressed as a function of the horizontal displacement, which allows an analytical method to be used for determining the stability domains for both pulses. The estimates obtained are compared with some previously published results on the dynamic elastic-plastic buckling of the same model under a step loading. It transpires that for the pulse loading of models with large imperfections dynamic instability occurs either elastically or plastically depending on the pulse duration, while for a step loading only an elastic instability is possible for the parameters examined.     

可变形电子元件的非线性动力屈曲行为分析

基于薄膜/基底模型,分析可变形电子元件结构的动力屈曲问题。用小变形平面应变理论描述基底,用Kirchhoff平板理论描述薄膜。定义Lagrange函数,包括薄膜应变能和动能,以及基底对薄膜所作的功。利用Euler-Lagrange方程导出薄膜的动力屈曲控制方程。计算线性荷载下薄膜的动力响应,并利用B-R准则确定临界屈曲荷载。动力屈曲的临界荷载较静力屈曲的大,波幅响应围绕静力屈曲的响应振荡。     

Nonlinear dynamic buckling of a viscoelastic model under impact loading

A simple nonlinear buckling analysis is applied to a one-degree-of-freedom arch under impact loading in which viscous damping may also be included. Such a loading consists of a falling body striking centrally the joint mass of the arch in such a way that a completely plastic impact can be postulated. When there is no damping the exact dynamic buckling load for such a kind of loading-associated with an unbounded motion can be established by using a static criterion (approach). More specifically, it was shown that the dynamic buckling load corresponds to that unstable equilibrium state where the total potential energy of the system is zero. Furthermore, it was proved that the second variation of the total potential energy at the foregoing unstable equilibrium state is negative definite. This implies that the curve loading versus displacement resulting by the vanishing of the total potential energy has always a maximum on the afore mentioned unstable state. It was also found that the system may become sensitive to initial conditions. If damping is included the foregoing static criterion yields lower bound buckling estimates. These findings were verified by employing a highly efficient approximate technique as well as the numerical scheme of Runge-Kutta for solving any nonlinear initial-value problem.     

Static and dynamic buckling modes of a cylindrical shell under external pressure

We consider the problem of static and dynamic buckling modes of thin shells under external hydrostatic pressure. If the statement of the problem uses the linearized equations of motion obtained in the moderately large bending theory of shells according to the classical or refined model, then part of terms related to the external load in these equations are assumed to be conservative, and the other terms are assumed to be nonconservative. In this connection, we study four statements of the elastic stability problem for a cylindrical shell with hinged faces. The first of them is the statement of the static boundary value problem in the sense of Euler, where the action of external pressure is assumed to be conservative. The second statement is used to study small vibrations near the static equilibrium by a dynamic method for the same conservative load. The third and fourth statements of the problem correspond to the action of a nonconservative load and are similar to the first and second statements, respectively. They use the linearized equations of equilibrium and motion constructed earlier in a consistent version on the basis of a Timoshenko type model and allowing one to reveal all classical and nonclassical shell buckling modes.     

Dynamic buckling of stiffened plates under fluid-solid impact load

A simple solution of the dynamic buckling of stiffened plates under fluid-solid impact loading is presented. Based on large deflection theory, a discretely stiffened plate model has been used. The tangential stresses of stiffeners and in-plane displacement are neglected. Applying the Hamilton‘ s principle, the motion equations of stiffened plates are obtained. The deflection of the plate is taken as Fourier series, and using Galerkin method, the discrete equations can be deduced, which can be solved easily by Runge-Kutta method. The dynamic buckling loads of the stiffened plates are obtained from Budiansky-Roth ( B-R ) curves.     

Response of a dynamic system to flow-induced load

An approximate analytical method is used to study in-line vibrations of a linear system induced by oscillatory flow. The hydrodynamic drag force is accounted for by an equivalent viscous dashpot. The obtained equivalent linear system is used to determine the amplitude and the phase of the oscillatory component, and the offset component of the steady-slate periodic response of the linear system. Several parametric studies are presented and discussed in detail. Particular attention is given to the magnitude of the effective viscous damping.     

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.     

Nonlinear vibration and buckling of circular sandwich plate under complex load

The nonlinear vibration fundamental equation of circular sandwich plate under uniformed load and circumjacent load and the loosely clamped boundary condi- tion were established by von Karman plate theory,and then accordingly exact solution of static load and its numerical results were given.Based on time mode hypothesis and the variational method,the control equation of the space mode was derived,and then the amplitude frequency-load character relation of circular sandwich plate was obtained by the modified iteration method.Consequently the rule of the effect of the two kinds of load on the vibration character of the circular sandwich plate was investigated.When circumjacent load makes the lowest natural frequency zero,critical load is obtained.     

Non-linear in-plane buckling of rotationally restrained shallow arches under a central concentrated load

This paper investigates the non-linear in-plane buckling of pin-ended shallow circular arches with elastic end rotational restraints under a central concentrated load. A virtual work method is used to establish both the non-linear equilibrium equations and the buckling equilibrium equations. Analytical solutions for the non-linear in-plane symmetric snap-through and antisymmetric bifurcation buckling loads are obtained. It is found that the effects of the stiffness of the end rotational restraints on the buckling loads, and on the buckling and postbuckling behaviour of arches, are significant. The buckling loads increase with an increase of the stiffness of the rotational restraints. The values of the arch slenderness that delineate its snap-through and bifurcation buckling modes, and that define the conditions of buckling and of no buckling for the arch, increase with an increase of the stiffness of the rotational end restraints.     

The torsional buckling of a cruciform column under compressive load with a vertex plasticity model

The torsional buckling of a plastically deforming cruciform column under compressive load is investigated. The problem is solved analytically based on the von Kármán shallow shell theory and the virtual work principle. Solutions found in the literature are extended for path-dependent incremental behaviour as typically found in the presence of the vertex effect that is present in metallic polycrystals.At the critical load for buckling the direction of straining changes by an additional shear component. It is shown that the incremental elastic–plastic moduli are spatially nonuniform for such situations, contrary to the classical J₂ flow and deformation theories. The critical shear modulus that governs the buckling equation is obtained as a weighted average of the incremental elastic–plastic moduli over the cross-section of the cruciform.Using a plasticity model proposed by the authors, that includes the vertex effect, the buckling-critical load is computed for a aluminium column both with the analytical model and a FEM-based eigenvalue buckling analysis. The stable post-buckling path is determined by the energy criterion of path-stability. A comparison with the experimentally obtained classical results by Gerard and Becker (1957) shows good agreement without relying on artificial imperfections as necessary in the classical J₂ flow theory.     

Contact interaction between crack edges under a dynamic load

Institute of Mechanics, Academy of Sciences of the Ukraine, Kiev. Kharlov Automobile and Highway Institute. Translated from Prikladnaya Mekhanika, Vol. 28, No. 7, pp. 3–11, July, 1992.