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.     

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.     

Non-Gaussianclosure techniques for stationary random vibration

The classical method of statistical linearization when applied to a non-linear oscillator excited by stationary wide-band random excitation, can be considered as a procedure in which the unknown parameters in a Gaussian distribution are evaluated by means of moment identities derived from the dynamic equation of the oscillator. A systematic extension of this procedure is the method of non-Gaussian closure in which an increasing number of moment identities are used to evaluate additional parameters in a family of non-Gaussian response distributions. The method is described and illustrated by means of examples. Attention is given to the choice of representations of non-Gaussian distributions and to techniques for generating independent moment identities directly from the differential equation of the non-linear oscillator. Some shortcomings of the method are pointed out.     

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.     

Estimation of parameters of non-linear dynamical systems

A method is proposed to describe non-linear systems without memory, forced by random excitation. The concept of statistical linearization is extended to an approximation with a series of functions. The parameters of these functions describing the non-linearity are estimated in the presence of noise by using correlation techniques. The results apply also if the non-linearity cannot be separated from a linear dynamic model.     

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.     

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

The method of weighted residuals is applied to the reduced Fokker-Planck equation associated with a non-linear oscillator, which is subjected to both additive and multiplicative Gaussian white noise excitations. A set of constraints are deduced for obtaining an approximate stationary probability density for the system response. One of the constraints coincides with the previously proposed criterion of dissipation energy balancing, and the others are useful for calculating the equivalent conservative force. It is shown that these constraints imply certain relationships among certain statistical moments; their imposition guarantees that such moments computed from the approximate probability density satisfy the corresponding exact equations derived from the original equation of motion. Moreover, the well-known procedure of stochastic linearization and its improved version of partial linearization are shown to be special cases of this scheme, and they are less accurate since the approximations are not chosen from the entire set of the solution pool of generalized stationary potential. Applications of the scheme are illustrated by examples, and its accuracy is substantiated by Monte Carlo simulation results.     

Lyapunov exponents estimation for hysteretic systems

This article discusses the Lyapunov exponent estimation of non-linear hysteretic systems by adapting the classical algorithm by Wolf and co-workers [Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A., 1985. Determining Lyapunov exponents from a times series. Physica D 16, 285–317.]. This algorithm evaluates the divergence of nearby orbits by monitoring a reference trajectory, evaluated from the equations of motion of the original hysteretic system, and a perturbed trajectory resulting from the integration of the linearized equations of motion. The main issue of using this algorithm for non-linear, rate-independent, hysteretic systems is related to the procedure of linearization of the equations of motion. The present work establishes a procedure of linearization performing a state space split and assuming an equivalent viscous damping in order to represent hysteretic dissipation in the linearized system. The dynamical response of a single-degree of freedom pseudoelastic shape memory alloy (SMA) oscillator is discussed as an application of the proposed algorithm. The restitution force of the oscillator is provided by an SMA element described by a rate-independent, hysteretic, thermomechanical constitutive model. Two different modeling cases are considered for isothermal and non-isothermal heat transfer conditions, and numerical simulations are performed for both cases. The evaluation of the Lyapunov exponents shows that the proposed procedure is capable of quantifying chaos capturing the non-linear dissipation of hysteretic systems.     

Drifting response of hysteretic oscillators to stochastic excitation

Under a zero-mean, broad-band, stationary-random load, the symmetric elastic perfectly plastic oscillator and many similar hysteretic systems exhibit a Brownian-like displacement response: asymptotically, the displacement mean vanishes and the displacement variance linearly increases with time. This diverging behavior, often referred to as the drift, is observed even when the excitation power spectrum vanishes at zero frequency, an instance so far lacking a satisfactory modeling within the framework of statistical linearization. The paper presents a linearization-based method which captures the drift in such an instance without requiring any simulation-calibrated parameter. The method combines statistical linearization with stochastic averaging and a generalized van der Pol transformation comprising terms introduced to make allowance for the drift. Model predictions are compared with Monte Carlo estimates for an excitation whose power spectrum vanishes at zero frequency. Good agreement is found for a wide range of excitation levels despite the extremeness of the non-linearity.     

Response of Systems Under Non-Gaussian Random Excitations

The approach of nonlinear filter is applied to model non-Gaussian stochastic processes defined in an infinite space, a semi-infinite space or a bounded space with one-peak or multiple peaks in their spectral densities. Exact statistical moments of any order are obtained for responses of linear systems jected to such non-Gaussian excitations. For nonlinear systems, an improved linearization procedure is proposed by using the exact statistical moments obtained for the responses of the equivalent linear systems, thus, avoiding the Gaussian assumption used in the conventional linearization. Numerical examples show that the proposed procedure has much higher accuracy than the conventional linearization in cases of strong system nonlinearity and/or high excitation non-Gaussianity. An erratum to this article is available at .     

A novel statistical linearization solution for randomly excited coupled bending-torsional beams resting on non-linear supports

A novel statistical linearization technique is developed for computing stationary response statistics of randomly excited coupled bending-torsional beams resting on non-linear elastic supports. The key point of the proposed technique consists in representing the non-linear coupled response in terms of constrained linear modes. The resulting set of non-linear equations governing the modal amplitudes is then replaced by an equivalent linear one via a classical statistical error minimization procedure, which provides algebraic non-linear equations for the second-order statistics of the beam response, readily solved by a simple iterative scheme. Data from Monte Carlo simulations, generated by a pertinent boundary integral method in conjunction with a Newmark numerical integration scheme, are used as benchmark solutions to check accuracy and reliability of the proposed statistical linearization technique.

     

Generalization of the equivalent linearization method for non-linear random vibration problems

The method of equivalent linearization is generalized such that the response of a non-linear oscillator subject to both parametric and external random white-noise excitations can be determined approximately. The main objective of the new method is to obtain a closed system of differential equations for certain statistical moments of the response. This can be achieved by linearizing the drift term of the corresponding Itô-equations and by replacing the square of the diffusion term by a second degree polynomial. It is found that the accuracy of the mean square amplitudes is improved considerably compared with the original equivalent linearization.     

Stochastic linearization of dynamical systems under parametric Poisson white noise excitation

Several stochastic linearization techniques are derived for nonlinear systems under parametrical Poisson white noise excitation. The differential equations for the first and second order moments of the linearized systems are obtained and differences to the corresponding moment equations of the nonlinear system are discussed. It is shown that different linear models may lead to ‘true’ linearization coefficients in the sense of Kozin in: F. Ziegler, G.I. Schuëller (Eds.), Proceedings of the IUTAM Symposium on Nonlinear Stochastic Dynamic Engineering Systems, Springer, Berlin, Heidelberg, New York, 1988, pp. 45-56. For the Duffing oscillator and the van der Pol oscillator, the results are compared with Monte Carlo simulation.     

Elliptic averaging methods using the sum of Jacobian elliptic delta and zeta functions as the generating solution

The present paper describes an improved version of the elliptic averaging method that provides a highly accurate periodic solution of a non-linear system based on the single-degree-of-freedom Duffing oscillator with a snap-through spring. In the proposed method, the sum of the Jacobian elliptic delta and zeta functions is used as the generating solution of the averaging method. The proposed method can be used to obtain the non-odd-order solution, which includes both even- and odd-order harmonic components. The stability analysis for the approximate solution obtained by the present method is also discussed. The stability of the solution is determined from the characteristic multiplier based on Floquet’s theorem. The proposed method is applied to a fundamental oscillator in a non-linear system. The numerical results demonstrate that the proposed method is very effective for analyzing the periodic solution of half-swing mode for systems based on Duffing oscillators with a snap-through spring.     

Identification and parameter estimation of a bilinear oscillator using Volterra series with harmonic probing

Identification of non-linear systems is mainly limited to polynomial form non-linearities. Among the non-polynomial forms, bilinear oscillator constitutes an important class of non-linear systems and it has been used for modeling of various physical systems, particularly for structural elements with a breathing crack. An identification procedure is presented here for the class of bilinear oscillator, using higher order FRFs derived from Volterra series under harmonic excitation. The procedure addresses the problem of both; identification of the non-linearity structure as well as estimation of the bilinear parameter, which can be correlated to the crack severity and structural degradation. The procedure is illustrated with numerical simulation and the estimation results indicate that even a weakly bilinear state introduced by a small crack size can be accurately identified and measured.     

Control of the Duffing oscillator under non-Gaussian external excitation

The problem of suboptimal linear feedback control laws with mean-square criteria for the linear oscillator and the Duffing oscillator under external non-Gaussian excitations is considered. The input process is modeled as a polynomial of a Gaussian process or as a renewal driven impulse process. To determine the suboptimal control, a modified iterative procedure is proposed, where four criteria of statistical linearization are combined with an optimal control strategy. The results indicate that the obtained minima do not depend on the linearization criterion. The nonlinearity tends to reduce this minimum.     

EPC procedure for PDF solution of non-linear oscillators excited by Poisson white noise

The stationary probability density function (PDF) solution of the responses of non-linear stochastic oscillators subjected to Poisson pulses is analyzed. The PDF solutions are obtained by the exponential-polynomial closure (EPC) method. To assess the effectiveness of the solution procedure numerically, non-linear oscillators are analyzed with different impulse arrival rates, degree of oscillator non-linearity and excitation intensity. Numerical results show that the PDFs obtained with the EPC method yield good agreement with those obtained from Monte Carlo simulation when the polynomial order is 4 or 6. It is also observed that the EPC procedure is the same as the equivalent linearization procedure under Gaussian white noise in the case of the polynomial order being 2.     

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.     

The exact steady-state solution of a class of non-linear stochastic systems

In this paper exact steady state solutions are constructed for a class of non-linear systems subjected to stochastic excitation. The results are then applied to both classical and non-classical oscillator problems.     

Identification of the state equation in complex non-linear systems

Building on the basic idea behind the Restoring Force Method for the non-parametric identification of non-linear systems, a general procedure is presented for the direct identification of the state equation of complex non-linear systems. No information about the system mass is required, and only the applied excitation(s) and resulting acceleration are needed to implement the procedure. Arbitrary non-linear phenomena spanning the range from polynomial non-linearities to the noisy Duffing-van der Pol oscillator (involving product-type non-linearities and multiple excitations) or hysteretic behavior such as the Bouc-Wen model can be handled without difficulty. In the case of polynomial-type non-linearities, the approach yields virtually exact results for sufficiently rich excitations. For other types of non-linearities, the approach yields the optimum (in least-squares sense) representation in non-parametric form of the dominant interaction forces induced by the motion of the system. Several examples involving synthetic data corresponding to a variety of highly non-linear phenomena are presented to demonstrate the utility as well as the range of validity of the proposed approach.