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.     

Controlling lü-system using partial linearization

Partial linearization method is proposed for controlling Lü-system. Through partially cancelling the nonlinear cross-coupling terms the stabilization of the resulting system was realized. This method can be easily realized. The robust behavior was proved with respect to an uncertain system. Numerical simulation are provided to show the effectiveness and feasibility of the method. Foundation item: the National Natural Science Foundation of China (10171099) Biography: Yu Yong-guang (1976≈)     

Simulation of non-Gaussian multivariate stationary processes

Mathematical justifications are given for a simulation technique of multivariate non-Gaussian random processes and fields based on Rosenblatt's transformation of Gaussian processes. Different types of convergences are given for the approaching sequence. Moreover an original numerical method is proposed in order to solve the functional equation yielding the underlying Gaussian process autocorrelation function.     

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.     

A probabilistic linearization method for non-linear systems subjected to additive and multiplicative excitations

A general method to obtain approximate solutions for the random response of non-linear systems subjected to both additive and multiplicative Gaussian white noises is presented. Starting from the concept of linearization, the proposed method of “Probabilistic Linearization” (PL) is based on the replacement of the Fokker–Planck equation of the original non-linear system with an equivalent one relative to a linear system subjected to additive excitation only. By means of the general scheme of the weighted residuals, the unknown coefficients of the equivalent system are determined. Assuming a Gaussian probability density function of the response process and by choosing the weighting functions in a suitable way, the equivalence of the proposed method, called “Gaussian Probabilistic Linearization” (GPL), with the “Gaussian Stochastic Linearization” (GSL) applied to the coefficients of the Itô differential rule is evidenced. In addition, the generalization of the proposed method, called “Generalized Gaussian Probabilistic Linearization” (GGPL), is presented. Numerical applications show as, varying the choice of the weighting functions, it is possible to obtain different linearizations, with a variable degree of accuracy. For the two examples considered, different suitable combinations of the weighting functions lead to different equivalent linear systems, all characterized by the exact solution in terms of variance.     

Direct evaluation of the equivalent linear damping for TLCD systems in random vibration for pre-design purposes

Passive control of structural vibrations has received in recent years a great attention from researchers concerned with vibration control. Several types of devices have been proposed in order to reduce the dynamic responses of different kinds of structural systems. Among them, the Tuned Liquid Column Damper (TLCD) proved to be very effective in reducing vibration of structures. Since the increasing use of TLCDs in practical realizations, this paper aims at developing an approximate formulation, by means of a statistical linearization technique, able to estimate the parameters of a structure equipped with a TLCD subjected to random loads for pre-design purposes. Moreover, it is shown that results obtained via Monte Carlo approach on the non-linear system are in good agreement with those obtained by means of the proposed formulation.     

EXACT LINEARIZATION BASED MULTIPLE-SUBSPACE ITERATIVE RESOLUTION TO AFFINE NONLINEAR CONTROL SYSTEM

To the optimal control problem of affine nonlinear system, based on differential geometry theory, feedback precise linearization was used. Then starting from the simulative relationship between computational structural mechanics and optimal control, multiple-substructure method was inducted to solve the optimal control problem which was linearized. And finally the solution to the original nonlinear system was found. Compared with the classical linearizational method of Taylor expansion, this one diminishes the abuse of error expansion with the enlargement of used region.     

MHD flow of upper-convected Maxwell fluid over porous stretching sheet using successive Taylor series linearization method

This paper investigates the magnetohydrodynamic(MHD) boundary layer flow of an incompressible upper-convected Maxwell(UCM) fluid over a porous stretching surface.Similarity transformations are used to reduce the governing partial differential equations into a kind of nonlinear ordinary differential equations.The nonlinear problem is solved by using the successive Taylor series linearization method(STSLM).The computations for velocity components are carried out for the emerging parameters.The numerical values of the skin friction coefficient are presented and analyzed for various parameters of interest in the problem.     

Analysis of nonlinear sliding structures by modified stochastic linearization methods

Probabilistic characteristics of a sliding structure is investigated by using new versions of stochastic linearization technique. The structure is composed of base part and upper part, which are connected to each other in a spring-damping system. Coulomb friction between the base structure and earth ground is considered. Two alternative versions of stochastic linearization approach, suggested by X. Zhang and I. Elishakoff, respectively, are applied to such a sliding structure to evaluate its statistical properties. Compared with the results of Monte Carlo simulation, the two new approaches are performing much better than the conventional one in their applications to the sliding structure. Moreover, numerical results indicate that the criterion proposed by Elishakoff turns out to be superior to all other versions in the problem under study. Numerical results also suggest that the entire structure may be replaced by the rigid body in the sliding problem as long as the difference of velocity responses are considered less important than those of displacement responses.     

Exact linearization of one-dimensional jump-diffusion stochastic differential equations

Necessary and sufficient conditions for the linearization of the one-dimensional Itô jump-diffusion stochastic differential equations (JDSDE) are given. Stochastic integrating factor has been introduced to solve the linear JDSDEs. Exact solutions to some linearizable JDSDEs have been provided.     

Stochastic averaging of n-dimensional non-linear dynamical systems subject to non-Gaussian wide-band random excitations

The averaged generalized Fokker-Planck-Kolmogorov (GFPK) equation for response of n-dimensional (n-d) non-linear dynamical systems to non-Gaussian wide-band stationary random excitation is derived from the standard form of equation of motion. The explicit expressions for coefficients of the fourth-order approximation of the averaged GFPK equation are given in series form. Conditions for convergences of these series are pointed out. The averaged GFPK equation is then reduced to that for 1-d dynamical systems derived by Stratonovich and compared with the closed form of GFPK equation for n-d dynamical systems subject to Poisson white noise derived by Di Paola and Falsone. Finally, this averaged GFPK equation is further reduced to that for quasi linear system subject to non-Gaussian wide-band stationary random excitation. Stationary probability density for quasi linear system subject to filtered Poisson white noise is obtained. Theoretical results for an example are confirmed by using Monte-Carlo simulation for different parameter values.     

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.     

Random dynamics of structures with gaps: Simulation and spectral linearization

This paper deals with the prediction of the response of strutures with gaps excited by a white noise. This model is frequently introduced for practical analysis of dynamic systems such as pipings having a clearance into the supporting device, mechanical equipments subjected to impact disturbance with the basements, and so on. It is impliccitly assumed throughout that the systems have been discretized, so that its motion can be represented in terms of a finlte number of coordinates. Some DOF will be subjected to limitations represented by a hard spring without damping.The first section deals with the numerical aspects of the problem. Two classical algorithms for MDOF systems are considered: the pseudo-static formulation, and the modal decomposition, Comparison of the results leads to the selection of the more effective method.In the second section some limitations of the statistical linearization method are established and a new linearization method based upon an ARMA representation of the response is proposed. The main point is in the relation between the coefficients of the ARMA model and the parameters of the system which is studied in this paper.The third section is the core of the paper. It presents a method for identifying a linear structure whose displacement response is close to the response of the lumped structure with gaps in terms of power spectra. Results are illustrated by some examples on MDOF oscillators. Only cases with a single excitation (corresponding to scismic analysis) are considered.     

Modelling heavy tails and asymmetry using ARCH-type models with stable Paretian distributions

Several approaches have been considered to model the heavy tails and asymmetric effect on stocks returns volatility. The most commonly used models are the Exponential Generalized AutoRegressive Conditional Heteroskedasticity (EGARCH), the Threshold GARCH (TGARCH), and the Asymmetric Power ARCH (APARCH) which, in their original form, assume a Gaussian distribution for the innovations. In this paper we propose the estimation of all these asymmetric models on empirical distributions of the Standard & Poor’s (S&P) 500 and the Financial Times Stock Exchange (FTSE) 100 daily returns, assuming the Student’s t and the stable Paretian with (α < 2) distributions for innovations. To the authors’ best knowledge, analysis of the EGARCH and TGARCH assuming innovations with α-stable distribution have not yet been reported in the literature. The results suggest that this kind of distributions clearly outperforms the Gaussian case. However, when α-stable and Student’s t distributions are compared, a general conclusion should be avoided as the goodness-of-fit measures favor the α-stable distribution in the case of S&P 500 returns and the Student’s t distribution in the case of FTSE 100.     

Some observations on two piecewise-linear dynamic systems with induced hysteretic damping

Significant hysteretic damping can be introduced into dynamical systems for vibration control purpose by means of semi-active control implementation. The dynamic behavior of two such models with a single-degree-of-freedom are examined in this paper. Control performance is found to be not only dependent on the stiffness ratio between the primary spring and auxiliary spring, but also on the forcing frequency of the harmonic excitation. Some problems associated with the equivalent linearization of these types of dynamical systems are also discussed.     

The robustness of an efficient probabilistic data-based tool for simulating the non-stationary response of non-linear systems

The stochastic response of dynamic systems has been an area of considerable interest for some time in the analysis of risk and structural reliability. The authors, in previous work, have developed a method which can analyze the response of linear multi-degree-of-freedom systems to completely general data-based non-stationary excitations in a highly efficient and analytical form. The authors extended this work to non-linear system response by using equivalent linearization techniques. This paper explores the range of application of the extension to the analysis of non-linear systems through the use of real and simulated data-sets. In particular, sensitivity issues of non-Gaussianity of the excitation data, and the degree of response non-linearity of the dynamic system, are investigated for their effect on the estimated response using this equivalent linearization-based approach.     

On the statistical analysis of local fracture stresses in notched bars

Test results for critical local fracture stresses are analysed statistically for both “as-received” and “degraded” pressure-vessel weld metal. The values were determined from the fracture loads of blunt-notch four-point-bend specimens fractured over a range of low test temperatures, making use of results from a finite-element stress analysis of the stress-strain distributions ahead of the notch root. The “degraded” material tested in this work has been austenitized at a high temperature, followed by both prestraining and temper embrittlement. This has led to a situation in which the fracture stress for the “degraded” material is reduced significantly below that for the “as-received” material. The fracture mechanisms are different in that the “degraded” material shows evidence of intergranular fracture as well as cleavage fracture (in coarse grain size) whereas the “as-received” material shows only cleavage fracture (in fine grain size). The critical stress (σ_F) distributions plotted on normal probability paper show that the experimental cumulative distribution function (CDF) is linear for each condition with different mean values: for “as-received” material and for “degraded” material. The values of standard deviation are small and almost identical (33-). The decrease of the local fracture stress after degradation is related to the local fracture micro-mechanisms. Statistical analysis of the results for the two conditions supports the hypothesis that the values of σ_F are essentially single valued, within random experimental errors. A similar analysis of the data treating both conditions as a single population reveals some interesting points relating to statistical modelling and lower-bound estimation for mechanical properties. Comparisons are made with Weibull analysis of the data. A further conclusion is that it is extremely important to base any statistical model on inferences drawn from micro-mechanical modelling of processes, and that examination of “normal” CDFs can often provide good indications of when it is necessary to subject data to further statistical and physical analysis.     

Controlling chaotic oscillations of visco-elastic plates by the linearization via output feedback

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Effective conductivity in isotropic heterogeneous media using a strong-contrast statistical continuum theory

The strong-contrast formulation is used to predict the effective conductivity of a porous material. The distribution, shape and orientation of the two phases are taken into account using two- and three-point probability distribution functions. A new approximation for the three-point probability function appropriate for two-phase media is proposed and discussed. Computed results for the effective conductivity using the strong-contrast formulation are compared to the Voigt and the Hashin-Shtrikman upper-bound estimates. These results show that the predicted effective conductivity is lower than both Voigt and Hashin-Shtrikman bounds. Compared to previous results using the weak-contrast formulation, the strong-contrast formulation seems to provide a better estimate for the effect of the microstructure on the conductivity.     

Sensitivity of probabilistic fracture mechanics analysis to variations in statistical parameters

Probability of failure (p_f) of a structure is usually calculated for a specified set of statistical parameters (mean, standard deviation, and probability distribution) that characterize random variables. This approach may not be efficient in cases where one would like to know the effect of variations in statistical parameters on the probability of failure. A method based on generating and analyzing randomly selected statistical parameters is proposed. The method consists of generating databases of mean and coefficient of variation (COV = mean/standard deviation) values of relevant fracture mechanics variables through a random process. The method was applied to surface cracks in a flat wide plate loaded under elastic conditions. Probability of failure was calculated for each database record using the first-order reliability method (FORM). Multiple linear regression analyses of the database records were performed with p_f as dependent variable and statistical parameters as independent variables. The predicted p_f values were in very good agreement with the directly calculated p_f values for the specified variations of statistical parameters (±10%, ±15%, and ±20%), except those for fracture toughness and tensile stress, where variations should be limited to ±10% and ±15% ranges.