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.     

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.

     

An analytical approximate technique for a class of strongly non-linear oscillators

An analytical approximate technique for large amplitude oscillations of a class of conservative single degree-of-freedom systems with odd non-linearity is proposed. The method incorporates salient features of both Newton's method and the harmonic balance method. Unlike the classical harmonic balance method, accurate analytical approximate solutions are possible because linearization of the governing differential equation by Newton's method is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of non-linear algebraic equations without analytical solution. With carefully constructed iterations, only a few iterations can provide very accurate analytical approximate solutions for the whole range of oscillation amplitude beyond the domain of possible solution by the conventional perturbation methods or harmonic balance method. Three examples including cubic-quintic Duffing oscillators are presented to illustrate the usefulness and effectiveness of the proposed technique.     

A new approximate solution of nonlinear diffusion equation

In this paper,the same problem in ref.[1] is studied.The author’s solutionapproximately satisfies the whole fundamental equations (1.1)and(1.2)and the wholebound values conditions (1.3-1.5).But the Liu’s solution does not satisfy theequation of continuity(1.2).     

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.     

A new numerical technique for the analysis of parametrically excited nonlinear systems

A new computational scheme using Chebyshev polynomials is proposed for the numerical solution of parametrically excited nonlinear systems. The state vector and the periodic coefficients are expanded in Chebyshev polynomials and an integral equation suitable for a Picard-type iteration is formulated. A Chebyshev collocation is applied to the integral with the nonlinearities reducing the problem to the solution of a set of linear algebraic equations in each iteration. The method is equally applicable for nonlinear systems which are represented in state-space form or by a set of second-order differential equations. The proposed technique is found to duplicate the periodic, multi-periodic and chaotic solutions of a parametrically excited system obtained previously using the conventional numerical integration schemes with comparable CPU times. The technique does not require the inversion of the mass matrix in the case of multi degree-of-freedom systems. The present method is also shown to offer significant computational conveniences over the conventional numerical integration routines when used in a scheme for the direct determination of periodic solutions. Of course, the technique is also applicable to non-parametrically excited nonlinear systems as well.     

A new analytical technique to find periodic solutions of non-linear systems

Based on the classical harmonic balance method a new technique is presented to determine higher approximate periodic solutions of the non-linear differential equations. The new method is systematic and simple. The solution covers the general initial value problem (i.e., for while the existing solution is determined for a particular case, especially for . The solution is easily transformed to perturbation solution. The method is used in various non-linear problems possessing second and more than second derivatives.     

A new method for predicting the maximum vibration amplitude of periodic solution of non-linear system

An original method based on the proposed framework for calculating the maximum vibration amplitude of periodic solution of non-linear system is presented. The problem of determining the worst maximum vibration is transformed into a non-linear optimization problem. The harmonic balance method and the Hill method are selected to construct the general non-linear equality and inequality constraints. The resulting constrained maximization problem is then solved by using the MultiStart algorithm. Finally, the effectiveness of the proposed approach is illustrated through two numerical examples. Numerical examples show that the proposed method can, at much lower cost, give results with higher accuracy as compared with numerical results obtained by a parameter continuation method.     

On an approximate analytical solution to a nonlinear vibrating problem,excited by a nonideal motor

This paper analyzes through Multiple Scales Method a response of a simplified nonideal and nonlinear vibrating system. Here, one verifies the interactions between the dynamics of the DC motor (excitation) and the dynamics of the foundation (spring, damper, and mass). We remarked that we consider cubic nonlinearity (spring) and quadratic nonlinearity (DC motor) of the same order of magnitude according to experimental results. Both analytical and numerical results that we have obtained had good agreement.     

A simple approximate solution for horizontal infiltration in a Brooks-Corey medium

A simple approximate solution is derived for the problem of one-dimensional absorption in a porous medium characterized by the Brooks-Corey equations. A piecewise-linear saturation profile, which satisfies flux continuity at the edge of the tension-saturated zone as well as an integrated form of the Richards' equation, is assumed. The predicted sorptivity agrees very well with the results of numerical simulations.     

Identification of stiffness, dissipation and input parameters of randomly excited non-linear systems: Capability of restricted potential models (RPM)

A dynamic identification technique in the time domain for time invariant systems under random external forces is presented. This technique is based on the use of the class of restricted potential models (RPM), which are characterized by a non-linear stiffness and a special form of damping, that is a product of the input power spectral density (PSD) matrix and the velocity gradient of a non-linear function of the total mechanical energy. By applying stochastic differential calculus and by specific analytical manipulations, some algebraic equations, depending on the response statistics and on the mechanic parameters that characterize RPM, are obtained. These equations can be used for the dynamic identification of the above mechanic parameters once the response statistics of the system to be identified are evaluated. The proposed technique allows one to identify single-degree-of-freedom or multi-degrees-of-freedom systems in the case of unmeasurable input. Further, the probabilistic characteristics of the external forces can be completely estimated in terms of PSD matrix.     

Projectile motion in a resistant medium Part II: approximate solution and estimates

The motion of a particle which is projected into a resistant medium and subjected to a uniform gravitational field is considered. The drag force that acts upon the particle within the medium is proportional to the particle speed squared. The problem is formulated in terms of particle-speed and local-path-angle variables, and the equations of motion that result are non-linear and coupled. An exact solution to these equations can be obtained but involves quadratures which cannot be analytically evaluated in terms of standard functions. An approximate solution that is remarkably accurate is presented. This solution is based upon the so-called cubic law, which is motivated by certain properties of the exact solution. This solution is also utilized to obtain estimates for the maximum projectile range, optimal projection angle, and other quantities of interest related to the particle motion.     

A perturbation solution for non-linear vibration of uniformly curved pipes conveying fluid

The first-order non-linear interactions between the pipe structure and the flowing fluid are considered to formulate the governing equations of motion for the in-plane vibration of a circular-arc pipe containing flowing fluid. The forces and moments induced in a pipe element by the flowing fluid are analyzed as functions of the instantaneous local curvature of the pipe. The flow field is assumed to be one-dimensional, incompressible and of uniform flow, and to remain independent of pipe motion. For a fixed-end circular-arc pipe with arbitrary arc angle, the non-linear governing equations are solved by the method of multiple scales in conjunction with the Bubnov-Galerkin method. The non-linear solutions indicate that the vibrational behavior of the system can differ substantially from that predicted by a linear analysis.     

New approximate solution for N-point correlation functions for heterogeneous materials

Statistical N-point correlation functions are used for calculating properties of heterogeneous systems. The strength and the main advantage of the statistical continuum approach is the direct link to statistical information of microstructure. Two-point correlation functions are the lowest order of correlation functions that can describe the morphology and the microstructure-properties relationship. Experimentally, statistical pair correlation functions are obtained using SEM or small x-ray scattering techniques. Higher order correlation functions must be calculated or measured to increase the precision of the statistical continuum approach. To achieve this aim a new approximation methodology is utilized to obtain N-point correlation functions for non-FGM (functional graded materials) heterogeneous microstructures. Conditional probability functions are used to formulate the proposed theoretical approximation. In this approximation, weight functions are used to connect subsets of (N?1)-point correlation functions to estimate the full set of N-point correlation function. For the approximation of three and four point correlation functions, simple weight functions have been introduced. The results from this new approximation, for three-point probability functions, are compared to the real probability functions calculated from a computer generated three-phase reconstructed microstructure in three-dimensional space. This three-dimensional reconstruction was based on an experimental two-dimensional microstructure (SEM image) of a three-phase material. This comparison proves that our new comprehensive approximation is capable of describing higher order statistical correlation functions with the needed accuracy.     

A new method for detecting non-linear damping and restoring forces in non-linear oscillation systems from transient data

The purpose of this study is to recover the functional form of both non-linear damping and non-linear restoring forces in the non-linear oscillatory motions of an autonomous system. Using two sets of measured motion response data of the system, an inverse problem is formulated for recovering (or identification): the differential equation of motion is transformed into an equivalent integral equation of motion. The identification, which is non-linear, is shown to be one-to-one. However, the inverse problem formulated herein is concerned with the Volterra-type of non-linear integral equation of the first kind. This leads to numerical instability: solutions of the inverse problem lack stability properties. In order to overcome the difficulty, a regularization method is applied to the identification process. In addition, an L-curve criterion, combined with regularization, is introduced to find an optimal choice for the regularization parameter (i.e., the number of iterations), in the presence of noisy data. The workability of the identification is investigated for simultaneously recovering the functional form of the non-linear damping and the non-linear restoring forces through a numerical experiment.     

A perturbation solution of non-linear vibration of rectangular orthotropic plates

Classical perturbation theory is applied to the non-linear dynamic response of orthotropic plates. Expressions are derived for the ratio of non-linear to linear frequency, membrane stress, and the ratio of the maximum total stress to the maximum bending stress. Where possible the analysis is compared to other available numerical solutions, and excellent agreement is shown.     

A non-linear model for the dynamics of open cross-section thin-walled beams—Part II: forced motion

The discrete equations developed in Part I are here used to analyze the non-linear dynamics of an inextensional shear indeformable beam with given end constraints. The model takes into account the non-linear effects of warping and of torsional elongation. Non-linear 3D oscillations of a beam with a cross-section having one symmetry axis is examined. Only terms of higher magnitude are retained in the equations, which exhibit quadratic, cubic and combination resonances. A harmonic load acting in the direction of the symmetry axis and in resonance with the corresponding natural frequency, is considered. Steady-state solutions and their stability are studied; in particular the effects of non-linear warping and of torsional elongation on the response are highlighted.     

A constitutive equation for a dilute solution of Hookean dumbbells with approximate hydrodynamic interaction

In a recent series of papers, Öttinger's consistently averaged hydrodynamic interaction has been shown to yield shear-rate dependent viscosity and normal stress coefficients in steady shear flow for dilute solutions of elastic dumbbells and chains. Even more recently, Fan has numerically solved the diffusion equation for the Hookean dumbbell with complete hydrodynamic interaction and he has compared his results with those of the Öttinger model.In this paper, a new approximation¹ for the Oseen—Burgers tensor is proposed where the configuration-dependent terms are replaced by appropriate averages rather than averaging the Oseen—Burgers tensor as a whole as in the Öttinger model. The proposed model leads to a differential constitutive equation which at low shear rates is similar to the Giesekus constitutive equation for a Hookean dumbbell with anistropic drag and anisotropic Brownian motion. The steady shear viscosity and normal stress coefficients for the proposed model are shear-rate dependent and are in close agreement with Fan's numerical calculations. Elongational viscosity for both positive and negative elongation rates are calculated.