On the nonlinear dynamic buckling mechanism of autonomous dissipative/nondissipative discrete structural systems

Summary Nonlinear dynamic buckling of nonlinearly elastic dissipative/nondissipative multi-mass systems, mainly under step load of infinite duration, is studied in detail. These systems, under the same loading applied statically, experience a limit point instability. The analysis can be readily extended to the case of dynamic buckling under impact loading. Energy, topological and geometrical aspects for the total potential energyV, which is constrained to lie in a region of phase-space whereV0, allow conclusions to be drawn directly regarding dynamic buckling. Criteria leading to very good, approximate and lower/upper bound dynamic buckling estimates are readily established without solving the highly nonlinear set of equations of motion. The theory is illustrated with several analyses of a two-degree-of-freedom model.     

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.     

On a stationarity principle for discrete non-linear dissipative dynamic systems

A new function depending upon the Lagrangian, and upon the Rayleigh dissipation function is introduced. It is shown that for a certain class of discrete systems the requirement of stationarity of the new function with respect to generalized velocities is tantamount to the setting up of differential equations of motion for the system.     

Geometric stability criteria for certain non-linear time varying systems

Explicit geometric criteria for the exponential stability of a non-linear feedback system (consisting of a time-invariant block G in the forward path and a non-linear time varying gain φ.k(t) in the feedback path) are presented when φ(.) belongs to certain classes of non-linear functions. The resulting bound on [$\text{(}\text{dk}\text{dt}\text{)}\text{k}$] is less stringent than those found in the literature.     

Conservation laws in non-linear dissipative systems

In Hamiltonian theory, Noether's theorem commonly is used to show the conservation of linear momentum and energy as a consequence of symmetry properties. The possibility of enclosing Hamiltonian theory in a wider context by use of Gibbs-Falkian thermodynamical methods, offering the opportunity to cover mechanical and thermodynamical systems with the same mathematical tools, is considered. Consequently it is shown how Noether's identity can be extended for dissipative systems which are appropriate to describe real life phenomena. By use of the principle of least action an extended version of Noether's theorem is calculated, from which the conservation of linear momentum and total energy can be derived. Additionally, the condition of absolute invariance is shown to be too restrictive for physical applications.     

A geometric approach for establishing dynamic buckling loads of autonomous potential N-degree-of-freedom systems

Non-linear dynamic buckling of autonomous non-dissipative N-degree-of-freedom systems whose static instability is governed either by a limit point or by an unstable symmetric bifurcation is thoroughly discussed using energy and geometric considerations. Characteristic distances associated with the geometry of the zero level total potential energy “hypersurface” in connection with total energy-balance equation lead to dynamic (global) instability criteria. These criteria allow the determination of “exact” dynamic buckling loads without solving the non-linear initial-value problem. The reliability and efficiency of the proposed geometric approach is demonstrated via several dynamic buckling analyses of 3-degree-of-freedom systems which subsequently are compared with corresponding numerical analyses based on the Verner–Runge–Kutta scheme.     

Stochastic stability of non-linear SDOF systems

A semi-analytical procedure for obtaining stability conditions for strongly non-linear single degree of freedom system (SDOF) subjected to random excitations is presented using stochastic averaging technique. The method is useful for finding stability conditions for systems having highly irregular non-linear functions which cannot be integrated in closed form to yield analytical expressions for averaged drift and diffusion coefficients. In spite of numerical methods available for finding stability of SDOF system by determining Lyapunov exponent, the proposed technique may have to be adopted (i) when the excitation is non-white; and (ii) when numerical integration fails due to convergence problem. The method is developed in such a way that it lends itself to a numerical computational scheme using FFT for obtaining numerical values of drift and diffusion coefficients of Its differential equation and the corresponding FPK equation for the system. These values of averaged drift and diffusion coefficients are then fit into polynomial form using curve fitting technique so that polynomials can be used for stability analysis. Two example problems are solved as illustrations. The first one is the Van der Pol oscillator having non-linearities which can be treated purely analytically. The example is considered for the validation of the proposed method. The second one involves non-linearities in the form of signum function for which purely analytical solution is not possible. The results of the study show that the proposed method is useful and efficient for performing stability analysis of dynamic systems having any type of non-linearities.     

On stability of non-linear differential systems

The object of this paper is to study the stability and asymptotic stability of solutions of the non-linear differential equation $\text{dx}\text{dt}\text{= A(t)x + f(t,x)}$ by using the method of equivalent inner products. This method enables one to determine a stability region without the ingenuity in constructing a Lyapunov function. It shows also that for an unstable linear system it is possible to choose a non-linear function so that the non-linear system is stable or asymptotically stable. Both global and regional stability are discussed.     

含高阶余项的非线性动力系统数值计算法

建立了一种求解非线性动力系统高精度数值计算的新方法,重构了等价的非线性动力系统方程,该方程考虑了非线性函数的任意高阶项,并给出了该方程的Duhamel积分表达式,在时间步长内用Newton-Raphson法进行数值迭代求解,该方法能连续满足微分方程而不只是在离散的步长端点满足方程,从而打破了传统的Euler型有限差分法。计算实例表明,该方法计算精度高于传统的Runge-Kutta,Newmark-β和Wilson-θ等方法。     

A quasi-static stability analysis for Biot's equation and standard dissipative systems

In this paper, an extended version of Biot's differential equation is considered in order to discuss the quasi-static stability of a response for a solid in the framework of generalized standard materials. The same equation also holds for gradient theories since the gradients of arbitrary order of the state variables and of their rates can be introduced in the expression of the energy and of the dissipation potentials. The stability of a quasi-static response of a system governed by Biot's equations is discussed. Two approaches are considered, by direct estimates and by linearizations. The approach by direct estimates can be applied in visco-plasticity as well as in plasticity. A sufficient condition of stability is proposed and based upon the positivity of the second variation of energy along the considered response. This is an extension of the criterion of second variation, well known in elastic buckling, into the study of the stability of a response. The linearization approach is available only for smooth dissipation potentials, i.e. for the study of visco-elastic solids and leads to a result on asymptotic stability. The paper is illustrated by a simple example.     

On stability of moving autonomous mechanical systems under uncertainty

Conditions of stability with respect to two measures of autonomous mechanical systems under uncertainty are established by decomposing the phase space of the system, a space of fuzzy sets, and using the natural monotonicity of fuzzy differential equations __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 119–131, February 2008.     

Global invariant manifolds delineating transition and escape dynamics in dissipative systems: an application to snap-through buckling

Nonlinear Dynamics - Invariant manifolds play an important role in organizing global dynamical behaviors. For example, it is found that in multi-well conservative systems where the potential energy...     

On the non-linear dynamic stability problem for thin elastic plates

In this paper the non-linear response of thin elastic plates under parametric excitation is investigated. A new analytical method is proposed. It gives the possibility to obtain all the characteristic features of the phenomenon considered, which are known from experiments—the existing of beats, their dependence on the excitation parameter, the influence of the initial conditions, the typical character of the vibrations in the different regions. Analog computer studies are carried out, and they show clearly the influence of different parameters on the output of the problem considered.     

On damped non-linear dynamic systems with many degrees of freedom

Non-linear mass-spring-damper systems with many degrees of freedom are studied; all springs and/or all dampers may be strongly non-linear. It is shown that the ultimate state of completely damped systems is always rest, that of incompletely damped systems may be either rest or a periodic normal mode motion. Necessary and sufficient conditions are given for the existence of classical normal mode motion in completely or incompletely damped systems. When the system is linear, these reduce to known results found by Rayleigh and generalized by Caughey and O'Kelly.     

On non-linear stability in unconstrained non-linear elasticity

A method of obtaining a full three-dimensional non-linear Hadamard stability analysis of inhomogeneous deformations of arbitrary, unconstrained, hyperelastic materials is presented. The analysis is an extension of that given by Chen and Haughton (Proc. Roy. Soc. London A 459 (2003) 137) for two-dimensional incompressible problems. The process that we present replaces the second variation condition expressed as an integral involving a quadratic in three arbitrary perturbations, with an equivalent sixth-order system of ordinary differential equations. The positive definiteness condition is thereby reduced to the simple numerical evaluation of zeros of a well-behaved function. The general theory is illustrated by applying it to the problem of the inflation of a thick-walled spherical shell. The present analysis provides a simpler alternative approach to bifurcation problems approached by using the incremental equations of non-linear elasticity.