随机平均规范形方法

计算随机规范形系数是应用随机规范形方法的关键．提出一种应用随机平均计算随机规范形系数的方法．为了说明方法的有效性,对白噪声激励的Duffing系统,经过变换,对于相应的平均方程,比较了精确解、规范形方法解和能量包络方法解的稳态概率密度,结果表明,当非线性项系数较小时,三者完全一致．当非线性项系数较大时,规范形方法所得解与精确解相差不大．     

Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems

Approximations of the resonant non-linear normal modes of a general class of weakly non-linear one-dimensional continuous systems with quadratic and cubic geometric non-linearities are constructed for the cases of two-to-one, one-to-one, and three-to-one internal resonances. Two analytical approaches are employed: the full-basis Galerkin discretization approach and the direct treatment, both based on use of the method of multiple scales as reduction technique. The procedures yield the uniform expansions of the displacement field and the normal forms governing the slow modulations of the amplitudes and phases of the modes. The non-linear interaction coefficients appearing in the normal forms are obtained in the form of infinite series with the discretization approach or as modal projections of second-order spatial functions with the direct approach. A systematic discussion on the existence and stability of coupled/uncoupled non-linear normal modes is presented. Closed-form conditions for non-linear orthogonality of the modes, in a global and local sense, are discussed. A mechanical interpretation of these conditions in terms of virtual works is also provided.     

Data assimilation for non-linear tidal models

A data assimilation procedure to incorporate measurements into a non-linear tidal model by using Kalman-filtering techniques is developed. The Kalman filter is based on the two-dimensional shallow water equations. To account for the inaccuracies, these equations are embedded into a stochastic environment by introducing a coloured system noise process into the momentum equations. The continuity equation is assumed to be perfect. The deterministic part of the equations is discretized using an ADI method, the stochastic part using the Euler scheme. Assuming that the system noise is less spatially variable than the underlying water wave process, this stochastic part can be approximated on a coarser grid than the grid used to approximate the deterministic part. A Chandrasekhar-type filter algorithm is employed to obtain the constant-gain extended Kalman filter for weakly non-linear systems. The capabilities of the filter are illustrated by applying it to the assimilation of water level measurements into a tidal model of the North Sea.     

Non-linear normal modes,invariance, and modal dynamics approximations of non-linear systems

Non-linear systems are here tackled in a manner directly inherited from linear ones, that is, by using proper normal modes of motion. These are defined in terms of invariant manifolds in the system's phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach to which the theory developed for discrete systems can be applied-are simultaneously applied to the same study case-an Euler-Bernoulli beam constrained by a non-linear spring-and compared as regards accuracy and reliability. Numerical simulations of pure non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the non-linear normal modes are demonstrated, and it is also found that, for a pure non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differential equation.     

On the normal modes of non-linear dual-mass systems

The motions of a class of non-linear dual-mass systems are investigated. A new characterization of non-linear normal modes in terms of solutions of initial value problems is developed that leads to interesting properties both on the existence of such non-linear normal modes and their continuous dependence upon the energies.     

Approximate backbone curves for non-linear systems with several degrees of freedom

The concept or forced normal modes for linear damped systems is extended to non-linear systems. It is used together with the first harmonic method in order to establish an approximate but rapid method for the determination of backbone curves for non-linear systems having several degrees of freedom.     

Resonant non-linear normal modes. Part II: activation/orthogonality conditions for shallow structural systems

The general conditions, obtained in Lacarbonara and Rega (Int. J. Non-linear Mech. (2002)), for orthogonality of the non-linear normal modes in the cases of two-to-one, three-to-one, and one-to-one internal resonances in undamped unforced one-dimensional systems with arbitrary linear, quadratic and cubic non-linearities are here investigated for a class of shallow symmetric structural systems. Non-linear orthogonality of the modes and activation of the associated interactions are clearly dual problems. It is known that an appropriate integer ratio between the frequencies of the modes of a spatially continuous system is a necessary but not sufficient condition for these modes to be non-linearly coupled. Actual activation/orthogonality of the modes requires the additional condition that the governing effective non-linear interaction coefficients in the normal forms be different/equal to zero. Herein, a detailed picture of activation/orthogonality of bimodal interactions in buckled beams, shallow arches, and suspended cables is presented.     

Stochastic non-linear flutter of a panel subjected to random in-plane forces

—An analysis of non-linear flutter of a simply-supported panel exposed to supersonic gas flow and random in-plane forces is presented for two- and three-mode interactions. A first order quasi-steady state aerodynamic piston theory is used to model the aerodynamic loading. The Fokker-Planck equation is used to derive a general moment equation for two- and three-mode interactions. For stability analysis the moment equation is consistent and the mean square stability boundaries of the equilibrium are obtained in terms of the system parameters. The stability boundaries reveal common features to those predicted by the deterministic theory of panel nutter. For the non-linear response the moment equation is found inconsistent and a cumulant-neglect closure is used by setting cumulants of fifth and sixth orders to zero. This first order non-Gaussian closure is carried out to solve for the response statistics in terms of the air-to-plate mass ratio, aerodynamic pressure, modal damping, and in-plane random force spectral density. It is found that the non-Gaussian solution yields higher levels for the response statistics than those obtained by the Gaussian solution. The inclusion of more modes results in a reduction of the response levels and expands the stability region.     

Bifurcation and Chaos Analysis of Stochastic Duffing System Under Harmonic Excitations

The Chebyshev polynomial approximation is applied to the dynamic response problem of a stochastic Duffing system with bounded random parameters, subject to harmonic excitations. The stochastic Duffing system is first reduced into an equivalent deterministic non-linear one for substitution. Then basic non-linear phenomena, such as stochastic saddle-node bifurcation, stochastic symmetry-breaking bifurcation, stochastic period-doubling bifurcation, coexistence of different kinds of steady-state stochastic responses, and stochastic chaos, are studied by numerical simulations. The main feature of stochastic chaos is explored. The suggested method provides a new approach to stochastic dynamic response problems of some dissipative stochastic systems with polynomial non-linearity.     

拟可积哈密顿系统中噪声诱发的混沌运动

研究拟可积哈密顿系统在谐和与噪声激励联合作用下的混沌运动。通过对噪声性质的假定,并利用动力系统理论,给出了高维梅尔尼科夫方法应用于随机拟可积哈密顿系统的推广形式。根据这种推广的方法,研究了谐和与高斯白噪声激励联合使用下两自由度拟可积哈密顿系统 同宿分岔,得出了系统发生混沌运动的参数阈值,并由此讨论了噪声对系统的混沌运动的影响。蒙特-卡罗方法模拟、李雅普诺夫指数等数值结果表明,这种推广的方法是有效的。     

Static and Dynamic Analysis of Non-Linear Uncertain Structures

The procedures usually adopted in the evaluation of the stochastic response of structures with uncertain parameters, the so-called stochastic structures, are affected by some limits. Namely, the major drawbacks are: the conspicuous computational time required for the analysis of many degree of freedom systems and the loss of accuracy in the case of large uncertainty in the parameters. A method able to reduce the previous inconveniences was recently introduced in the study of statically loaded linear structures. The method, named improved perturbation method, was also extended to the field of linear dynamics. In both cases, the improved perturbation method provides a good approximation, even in the case of moderately large deviation of the uncertain parameters, and the computational time required is comparable to conventional first order perturbation. The present paper intends to apply the improved perturbation method in the second order analysis of geometrically non-linear uncertain systems subjected to static and dynamic deterministic forces.     

Long Asymptotic Correlation Time for Non-Linear Autonomous Itô's Stochastic Differential Equation

Itô's stochastic differential equations theory is a common approach to analysis of stochastic phenomena in various systems. In many applications, an important feature of the systems is the flicker effect. It is well known that it cannot be described with linear autonomous scalar equations of the above kind. The reason is that the flicker effect is usually associated with a correlation time which is much greater than the correlation time in the linear case. In the present work, we discuss modelling of the long correlation time with the help of non-linear autonomous scalar Itô's stochastic differential equation which includes non-linear drift. The expression for the asymptotic correlation time as time separation tends to zero is derived in terms of the equation. We formulate the condition for this time to be long in the above sense. It is pointed out that this condition can hold if the nonlinear damping is reduced compared to the linear case. These results are illustrated with an example of the equation with non-linear drift of a specific form.     

Examples of Analysis of Stochastic Processes Based on Time Series Data

Time series data from stochastic processes described by the Langevin equation are analyzed. Analysis is based on estimation of the deterministic and random terms of the Langevin equation from data. The terms are presented as fields and inspected visually. Forming a model of the process, the terms are also used to reconstruct the deterministic and stochastic process trajectories. The deterministic term is approximated by an analytical ansatz. The equations obtained by the approximation are used to generate deterministic process trajectories and to study their linear stability. Influence of measurement noise on the estimates is also discussed.     

On the stability of stochastic difference systems

In this paper the stability of linear stochastic difference equations and a class of weakly non-linear stochastic difference equations is considered. For the linear systems explicit criteria are derived for the stability of the moments of any order. We also show how the moments of a linear stochastic difference system can be computed when a certain Lie-algebraic condition is satisfied.     

On the coupling of non-linear normal modes

The phenomenon of internal resonance is known as the exchange of energy between the modes and the existence of coupled-mode response under a single-mode excitation. This phenomenon is observed whenever a non-linear normal mode loses its stability, called the modal coupling. The details of modal coupling are formulated in the free vibrations of two-degree-of-freedom systems, and compared with internal resonance. The theory is based on the structural change in Poincaré map due to the stability change of normal modes. It is shown that every change in stability of normal modes gives rise to a pitchfork or a period-doubling bifurcation. The functional form is derived to compute the coupled modes by the method of harmonic balance. Examples are given to describe the procedure of stability analysis of non-linear normal modes, to compute the coupled modes, and then to demonstrate that results of internal resonances can be derived by model coupling. Other examples are given to demonstrate that the results of some modal couplings cannot be obtained by internal resonances.     

Invariant linearization criteria for a three-dimensional dynamical system of second-order ordinary differential equations and applications

Second-order dynamical systems are of paramount importance as they arise in mechanics and many applications. It is essential to have workable explicit criteria in terms of the coefficients of the equations to effect reduction and solutions for such types of equations. One important aspect is linearization by invertible point transformations which enables one to reduce a non-linear system to a linear system. The solution of the linear system allows one to solve the non-linear system by use of the inverse of the point transformation. It was proved that the n-dimensional system of second-order ordinary differential equations obtained by projecting down the system of geodesics of a flat (n+1)-dimensional space can be converted to linear form by a point transformation. This is a generalization of the Lie linearization criteria for a scalar second-order equation. In this case it is of the maximally symmetric class for a system and the linearizing transformation as well as the solution can be directly written down. This was explicitly used for two-dimensional dynamical systems. The criteria were written down in terms of the coefficients and the linearizing transformation allowed for the general solution of the original system. Here the work is extended to a three-dimensional dynamical system and we find explicit criteria, including the linearization test given in terms of the coefficients of the cubic in the first derivatives of the system and the construction of the transformations, that result in linearization. Applications to equations of classical mechanics and relativity are given to illustrate our results.     

Excursion probabilities of non-linear systems

This paper suggests a procedure for estimating excursion probabilities for linear and non-linear systems subjected to Gaussian excitation processes. In this paper, the focus is on non-linear systems which might also have stochastic properties. The approach is based on the so-called “averaged excursion probability flow” which allows for a simple solution for the interaction in excursion problems. Simplifying, the dynamic reliability problem can be reduced to a simpler “static” problem by considering the probability flow at fixed time instances. The proposed approach is very general and can be applied to both linear and non-linear systems of which the response can be determined by deterministic methods. Hence, the procedure applies to arbitrary structures and any suitable mathematical model including large FE-models solved by deterministic FE-codes.     

Application of complex stochastic averaging to non-linear random vibration problems

The stochastic averaging procedure in a complex-variable setting, used previously by Ariaratnam and Tarn to analyze a linear system under random parametric excitation, is extended to non-linear systems under both parametric and external random excitations. It is shown that equations for the moments of the system response, while still constituting an infinite hierarchy, form a simpler pattern compared with the corresponding equations obtained from the usual amplitude and phase formulation. From this simpler pattern, it is possible to identify those moments which tend to zero at the stationary state. Furthermore, a much smaller number of equations needs to be solved when the infinite hierarchy is truncated to calculate approximately the non-zero moments.     

Nonlinear response of SDOF systems under combined deterministic and random excitations

Dynamical systems subjected to random excitations exhibit non-linear behavior for sufficiently large motion. The multiple time scale method has been extensively utilized in the framework of non-linear deterministic analysis to obtain two averaged first-order differential equations describing the slow time scale modulation of amplitude and phase response. In this paper the multiple time scale method, opportunely modified to take properly into account the correlation structure of the stochastic input process, is adopted to derive a stochastic frequency-response relationship involving the response amplitude statistics and the input power spectral density. A low-intensity noise is assumed to separate the strong mean motion from its weak fluctuations. The moment differential equations of phase and amplitude are derived and a linearization technique applied to evaluate the second order statistics. The theory is validated through digital simulations on a nonlinear single degree of freedom model for the transversal oscillation of a cantilever beam with tip force and to a Duffing-Rayleigh oscillator, to analyze non-linear damping effects.