On using non-Gaussian distributions to perform statistical linearization

In statistical linearization non-linear elements are approximated by equivalent linear elements according to recipes proposed by the pioneers of the procedure. The recipes require the evaluation of certain statistics which, ideally, should be evaluated using the exact probability distribution of the non-linear response. Because the exact non-linear response distribution is unknown it has become traditional to use a Gaussian distribution as an approximation to the exact distribution. With the modern computing tools now available it is easy to use non-Gaussian distributions which can provide better approximations in cases where Gaussian distributions are not appropriate. Examples are displayed for power-law oscillators with stiffening and softening springs, and for the Duffing oscillator, and for a double-well oscillator. Two families of probability distributions with varying shape are studied.     

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.     

Trispectrum for the response of a non-linear oscillator

Simulation is used to obtain information about non-Gaussian aspects of the absolute response acceleration of a bi-linear hysteretic oscillator with an excitation which is Gaussian white noise. Attention is focused on the frequency content of the fourth cumulant of the response. This frequency content is studied by consideration of the trispectrum and also by the simplified technique of looking at the coefficient of excess for the response of a narrowband linear system mounted on the non-linear oscillator. Attempts are also made to model the non-Gaussian response of the non-linear oscillator by a filtered delta correlated (FDC) process, but it is shown that no process of this type can exhibit some of the significant characteristics of the non-linear response. In particular, the trispectrum of the non-linear response appears to be more narrowband than the power spectral density, and also it sometimes does not have the same sign at every point in the three-dimensional frequency space, and this behavior is distinctly different from that of any FDC process. Modifications of the FDC model are suggested in order to obtain improved approximations of the non-linear response.     

On statistical quasi-linearization

In carrying out the statistical linearization procedure to a non-linear system subjected to an external random excitation, a Gaussian probability distribution is assumed for the system response. If the random excitation is non-Gaussian, however, the procedure may lead to a large error since the response of bother the original non-linear system and the replacement linear system are not Gaussian distributed. It is found that in some cases such a system can be transformed to one under parametric excitations of Gaussian white noises. Then the quasi-linearization procedure, proposed originally for non-linear systems under both external and parametric excitations of Gaussian white noises, can be applied to these cases. In the procedure, exact statistical moments of the replacing quasi-linear system are used to calculate the linearization parameters. Since the assumption of a Gaussian probability distribution is avoided, the accuracy of the approximation method is improved. The approach is applied to non-linear systems under two types of non-Gaussian excitations: randomized sinusoidal process and polynomials of a filtered process. Numerical examples are investigated, and the calculated results show that the proposed method has higher accuracy than the conventional linearization, as compared with the results obtained from Monte Carlo simulations.     

An approach to the problem of closure in non-linear stochastic mechanics

An approach combining the method of moment equations and the statistical linearization technique is proposed for analysis of the response of non-linear mechanical systems to random excitation. The adaptive statistical linearization procedure is developed for obtaining a more accurate mean square of responses. For these, a Duffing oscillator and an oscillator with cubic non-linear damping subject to white noise excitation are considered. It is shown that the adaptive statistical linearization proposed yields good accurate results for both weak and strong non-linear stochastic systems.Presented at the First European Solid Mechanics Conference, September 9–13, 1991. Munich, Germany     

Multiple solutions generated by statistical linearization and their physical significance

Three cases are examined where the statistical linearization (SL) procedure can yield multiple solutions for the first and second moments of the response. The first is an oscillator with a hardening non-linear stiffness excited by a narrow-band random excitation, the second is an oscillator with two potential wells excited by wide-band random excitation, and the third is an oscillator where the non-linear features present in the first two problems are combined. The results of an SL analysis are quantitatively compared with the behaviour of digitally simulated sample functions of the displacement response. In all cases a definite correspondence is found between the occurrence of multiple solutions generated by the SL method and the appearance of noticeable jumps in sample functions of the response. In some cases a quantitative agreement exists between the first and second moment values of the multiple solutions and the magnitude of “local” moments of the response.     

Equivalent linearization for systems subjected to non-stationary random excitation

The method of eauivalent linearization is applied to the general problem of the response of non-linear discrete systems to non-stationary random excitation. Conditions for minimum equation difference are determined which do not depend explicitly on lime but only on the instantaneous statistics of the response process. Using the equivalent linear parameters, a deterministic non-linear ordinary differential equation for the covariance matrix is derived. An example is given of a damped Duffing oscillator subjected to modulated white noise.     

Planar and non-planar non-linear dynamics of suspended cables under random in-plane loading-I. Single internal resonance

The non-linear interaction of the in-plane and out-of-plane motions of a suspended cable in the neighbourhood of 2:1 internal resonance under random loading is studied. The random loading acts externally on the in-plane mode, while the out-of-plane mode is non-linearly coupled with the in-plane mode. Any non-trivial motion of the out-of-plane mode is mainly due to this non-linear coupling, which becomes significant in the neighbourhood of internal resonance. The response statistics are estimated by employing the Fokker-Planck equation together with Gaussian and non-Gaussian closures. Monte-Carlo simulation is also used for numerical verification. Away from the internal resonance condition, the response is governed by the inplane motion, and the non-Gaussian closure solution is found to be in good agreement with numerical simulation results. The stochastic bifurcation of the out-of-plane mode is predicted by Gaussian and non-Gaussian closures, and by Monte-Carlo simulation. The non-Gaussian closure can only predict bounded solutions within a limited region. The on-off intermittency of the second mode is observed in the Monte-Carlo simulation over a small range of excitation level. The influence on response statistics of excitation level and cable parameters, such as internal detuning, damping ratios, and sag-to-span ratio, is reported.     

Effects of excitation probability distribution on system responses

For a system subjected to a random excitation, the probability distribution of the excitation may affect behaviors of the system responses. Such effects are investigated for a variety of dynamical systems, including a linear oscillator, an oscillator of cubic non-linearity in both damping and stiffness, and a non-linear oscillator of the van der Pol type. The random excitations are assumed to be stationary stochastic processes, sharing the same spectral density, but with different probability distributions. Each excitation process is generated by passing a Brownian motion process through a non-linear filter, which is governed by an Ito stochastic differential equation. Monte Carlo simulations are carried out to obtain the transient and stationary properties of the system response in each case. It is shown that, under different excitations, the transient behaviors of the system response can be markedly different. The differences tend to reduce, however, as time of exposure to the excitations increases and the system reaches the stationary state.     

Probabilistic solution of non-linear random ship roll motion by path integration

This paper investigates the probability density function (PDF) of non-linear random ship roll motion using a previously developed path integration method. The mathematical model of ship rolling motion consists of a linear-plus-cubic damping and a non-linear restoring moment in the form of odd-order polynomials up to fifth-order terms. In the path integration method, the interpolation scheme is based on the Gauss–Legendre quadrature integration rule and the short-time transition probability density function is formulated by short-time Gaussian approximation. The present work extends the path integration method to the case of non-linear random ship roll motion. Different values of non-linearity coefficient and excitation intensity are used to examine the effectiveness of the path integration method. Numerical analysis shows that the results of the path integration method agree well with the simulation results, even in the tail region. The path integration method is effective and it is simply implemented in the examined cases. Due to the presence of non-linear damping terms and non-linear restoring moment terms, the PDFs of roll angle and angular velocity exhibit highly non-Gaussian behaviors.     

Investigation of a non-linear system under partially prescribed random excitation

The problem of non-linear systems excited by random forces with known power spectral density functions and unspecified probability structure is considered. Sufficient, but not necessary, conditions on the input under which the response can be a Gaussian process are investigated. The approach is illustrated by investigating the hardening spring cubic oscillator under wide and narrow band excitations. The non-Gaussian probability density of the input that leads to Gaussian response is determined.     

Marginal stability boundaries for some driven, damped, non-linear oscillators

A procedure is developed for averaging the differential equations for certain non-linear oscillators which are damped and externally driven. The procedure makes possible the obtaining of marginal stability boundaries for bifurcations in parameter space and is useful for systems with unperturbed solutions involving Jacobi elliptic functions. Specific cases of a driven, damped pendulum, an anharmonie oscillator, a Duffing oscillator, and a non-linear Helmholtz oscillator are examined.     

Higher order linearization in non-linear random vibration

In this paper a higher order linearization method for analyzing non-linear random vibration problems is presented. The non-linear terms of the given equation are replaced by unknown linear terms. These are in turn described by extra non-linear differential equations. The combined system of equations is then linearized to arrive at a higher degree-of-freedom equation for the original system. The method is illustrated by considering the Duffing oscillator under white noise input. The equivalent two d.o.f linear system is derived by the present method. Numerical results on steady state variance and PSD functions are obtained. These are found to be better than the simple linearization results.     

Survival probability of non-linear oscillators subjected to broad-band random disturbances

An analytical method is developed for examining the first-passage problem formulated in context with the response of a class of lightly damped non-linear oscillators to broad-band random excitations. A circular (E-type) barrier is considered. The amplitude of the oscillator response is modeled as a Markovian process. This modeling leads to a backward Kolmogorov equation which governs the evolution of the survival probability of the oscillator. The Kolmogorov equation is solved approximately by using the Galerkin technique and a perturbation technique. A set of confluent hypergeometric functions are used as an orthogonal basis for the expansions which are involved in the application of the Galerkin technique and the perturbation technique. The proposed method is exemplified by considering the response of the classical Van der Pol oscillator to white noise excitation. The reliability of the derived analytical solution is assessed by comparison with digital data obtained by a Monte Carlo simulation.     

Magnetic torque attitude control of a satellite using the state-dependent Riccati equation technique

A non-linear attitude control method for a satellite with magnetic torque rods using the state-dependent Riccati equation (SDRE) technique has been developed. The magnetic torque caused by the interaction with the Earth's magnetic field and the magnetic moment of torque rods plays a role of the control torque. The detailed equations of motion for this system are presented using angular velocity and quaternions. The SDRE controller is developed for the non-linear systems which can be formed in pseudo-linear representations using the state-dependent coefficient (SDC) method without linearization procedure. The aim of this control system is to achieve a stable attitude within 5°, and minimize the control effort. The stability regions for the SDRE controlled satellite system are estimated through the investigation of the stability conditions developed for pseudo-linear systems and the application of Lyapunov's theorem. For comparisons, the Linear Quadratic Regulator (LQR) method using the solution of the algebraic Riccati equation (ARE) is also applied to this non-linear system. The performance of the SDRE non-linear control system demonstrates more robustness and stability than the LQR control system when subjected to a wide range of initial conditions.     

考虑非高斯特性的结构首次失效时间计算

本文发展了一个计算具有非高斯特性的结构首次失效时间的解析方法.该方法利用Hemite矩模型将非高斯结构反应映射为标准高斯过程,由此计算反应的平均超越率、成群超越以及初始状态的影响,并最终给出结构的首次失效时问概率.二次力函数激励下线性单自由度系统的首次失效时间分析说明了该方法的使用过程,同时对该方法的计算结果与Monte Carlo模拟结果进行了对比.     

Response and stability of strongly non-linear oscillators under wide-band random excitation

A new stochastic averaging procedure for single-degree-of-freedom strongly non-linear oscillators with lightly linear and (or) non-linear dampings subject to weakly external and (or) parametric excitations of wide-band random processes is developed by using the so-called generalized harmonic functions. The procedure is applied to predict the response of Duffing–van der Pol oscillator under both external and parametric excitations of wide-band stationary random processes. The analytical stationary probability density is verified by digital simulation and the factors affecting the accuracy of the procedure are analyzed. The proposed procedure is also applied to study the asymptotic stability in probability and stochastic Hopf bifurcation of Duffing–van der Pol oscillator under parametric excitations of wide-band stationary random processes in both stiffness and damping terms. The stability conditions and bifurcation parameter are simply determined by examining the asymptotic behaviors of averaged square-root of total energy and averaged total energy, respectively, at its boundaries. It is shown that the stability analysis using linearized equation is correct only if the linear stiffness term does not vanish.     

Multi-physics dynamics of a mechanical oscillator coupled to an electro-magnetic circuit

The dynamics of a non-linear electro-magneto-mechanical coupled system is addressed. The non-linear behavior arises from the involved coupling quadratic non-linearities and it is explored by relying on both analytical and numerical tools. When the linear frequency of the circuit is larger than that of the mechanical oscillator, the dynamics exhibits slow and fast time scales. Therefore the mechanical oscillator forced (actuated) via harmonic voltage excitation of the electric circuit is analyzed; when the forcing frequency is close to that of the mechanical oscillator, the long term damped dynamics evolves in a purely slow timescale with no interaction with the fast time scale. We show this by assuming the existence of a slow invariant manifold (SIM), computing it analytically, and verifying its existence via numerical experiments on both full- and reduced-order systems. In specific regions of the space of forcing parameters, the SIM is a complicated geometric object as it undergoes folding giving rise to hysteresis mechanisms which create a pronounced non-linear resonance phenomenon. Eventually, the roles played by the electro-magnetic and mechanical components in the resulting complex response, encompassing bifurcations as well as possible transitions from regular to chaotic motion, are highlighted by means of Poincaré sections.     

Dynamics of a linear oscillator connected to a small strongly non-linear hysteretic absorber

The present investigation deals with the dynamics of a two-degrees-of-freedom system which consists of a main linear oscillator and a strongly non-linear absorber with small mass. The non-linear oscillator has a softening hysteretic characteristic represented by a Bouc-Wen model. The periodic solutions of this system are studied and their calculation is performed through an averaging procedure. The study of non-linear modes and their stability shows, under specific conditions, the existence of localization which is responsible for a passive irreversible energy transfer from the linear oscillator to the non-linear one. The dissipative effect of the non-linearity appears to play an important role in the energy transfer phenomenon and some design criteria can be drawn regarding this parameter among others to optimize this energy transfer. The free transient response is investigated and it is shown that the energy transfer appears when the energy input is sufficient in accordance with the predictions from the non-linear modes. Finally, the steady-state forced response of the system is investigated. When the input of energy is sufficient, the resonant response (close to non-linear modes) experiences localization of the vibrations in the non-linear absorber and jump phenomena.     

A new approximate solution technique for randomly excited non-linear oscillators

A new technique is proposed to obtain an approximate probability density for the response of a non-linear oscillator under Gaussian white noise excitations. The random excitations may be either multiplicative (also known as parametric) or additive (also known as external), or both. In this new technique, the original non-linear oscillator is replaced by another oscillator belonging to the class of generalized stationary potential for which the exact solution is obtainable. The replacement oscillator is selected on the basis that the average energy dissipation remains unchanged. Examples are given to illustrate the application of the new procedure. In one of the examples, the new procedure leads to a better approximation than that obtained by stochastic averaging.