Vibration analysis of harmonically excited non-linear system using the method of multiple scales

An analytical method is presented for evaluation of the steady state periodic behavior of non-linear systems. This method is based on the substructure synthesis formulation and a multiple scales procedure, which is applied to the analysis of non-linear responses. A complex non-linear system is divided into substructures, of which equations are approximately transformed to modal co-ordinates including non-linear term under the reasonable procedure. Then, the equations are synthesized into the overall system and the solution of the non-linear system can be obtained. Based on the method of multiple scales, the proposed procedure reduces the size of large-degree-of-freedom problem in solving the non-linear equations. Feasibility and advantages of the proposed method are illustrated by the application of the analytic procedure to the non-linear rotating machine system as a large mechanical structure system. Results obtained are reported to be an efficient approach with respect to non-linear response prediction when compared with other conventional methods.     

Equivalence conditions for classes of linear and non-linear distributed parameter systems

Equivalence of certain classes of second-order non-linear distributed parameter systems and corresponding linear third-order systems is established through a differential transformation technique. As linear systems are amenable to analysis through existing techniques, this study is expected to offer a method of tackling certain classes of non-linear problems which may otherwise prove to be formidable in nature.     

Time-parallel multiscale/multiphysics framework

We introduce the time-parallel compound wavelet matrix method (tpCWM) for modeling the temporal evolution of multiscale and multiphysics systems. The method couples time parallel (TP) and CWM methods operating at different spatial and temporal scales. We demonstrate the efficiency of our approach on two examples: a chemical reaction kinetic system and a non-linear predator–prey system. Our results indicate that the tpCWM technique is capable of accelerating time-to-solution by 2–3-orders of magnitude and is amenable to efficient parallel implementation.     

A new approach with piecewise-constant arguments to approximate and numerical solutions of oscillatory problems

This paper is devoted to the development of a novel approximate and numerical method for the solutions of linear and non-linear oscillatory systems, which are common in engineering dynamics. The original physical information included in the governing equations of motion is mostly transferred into the approximate and numerical solutions. Therefore, the approximate and numerical solutions generated by the present method reflect more accurately the characteristics of the motion of the systems. Furthermore, the solutions derived are continuous everywhere with good accuracy and convergence in comparing with Runge-Kutta method. An approximate solution is developed for a linear oscillatory problem and compared with its corresponding exact solution. A non-linear oscillatory problem is also solved numerically and compared with the solutions of Runge-Kutta method. Both the graphical and numerical comparisons are provided in the paper. The accuracy of the approximate and numerical solutions can be controlled as desired by the number of terms in the Taylor series and the value of a single parameter used in the present work. Formulae for numerical computation in solving various linear and non-linear oscillatory problems by the new approach are provided in the paper.     

Application of Laplace transform technique to the solution of certain third-order non-linear systems

A number of papers have appeared on the application of operational methods and in particular the Laplace transform to problems concerning non-linear systems of one kind or other. This, however, has met with only partial success in solving a class of non-linear problems as each approach has some limitations and drawbacks. In this study the approach of Baycura has been extended to certain third-order non-linear systems subjected to non-periodic excitations, as this approximate method combines the advantages of engineering accuracy with ease of application to such problems. Under non-periodic excitations the method provides a procedure for estimating quickly the maximum response amplitude, which is important from the point of view of a designer. Limitations of such a procedure are brought out and the method is illustrated by an example taken from a physical situation.     

Experimental and Numerical Study of Structural Identification Using Non-Linear Resonant Decay Method

One of the practical approaches in identifying structures is the non-linear resonant decay method which identifies a non-linear dynamic system utilizing a model based on linear modal space containing the underlying linear system and a small number of extra terms that exhibit the non-linear effects. In this paper, the method is illustrated in a simulated system and an experimental structure. The main objective of the non-linear resonant decay method is to identify the non-linear dynamic systems based on the use of a multi-shaker excitation using appropriated excitation which is obtained from the force appropriation approach. The experimental application of the method is indicated to provide suitable estimates of modal parameters for the identification of non-linear models of structures.     

Use of the Hilbert transform in modal analysis of linear and non-linear structures

An initial study into the application of the Hilbert transform in modal analysis procedures is presented. It is shown that typical structural non-linearities such as non-linear damping and stiffness can be detected and identified directly without the need to generate explicit models. No assumptions regarding the degree of non-linearity are made, which is a restriction in the classical methods for dealing with non-linearities. The properties of the Hilbert transform are discussed with respect to linear and non-linear dynamical systems, and a discrete transform, developed from the continuous functions, is derived in the frequency domain and adapted to modal analysis data in the form of mobility transfer functions. Truncation effects arising from limited frequency ranges of the mobility transfer functions are accounted for by employing correction terms in the frequency domain. Several examples are studied of single and multi-mode systems with non-linearities such as friction, clearance and non-linear stiffness. These examples indicate that the Hilbert transform offers a new method for extending modal analysis to the domains of non-linear systems.     

STATISTICAL SEISMIC RESPONSE ANALYSIS AND RELIABILITY DESIGN OF NONLINEAR STRUCTURE SYSTEM

Industrial structure systems may have non-linearity, and are also sometimes exposed to the danger of earthquake. In the design of such system, these factors should be accounted for from the viewpoint of reliability. This paper proposes a method to analyze seismic response and reliability design of a complex non-linear structure system under random excitation. The actual random excitation is represented to the corresponding Gaussian process for the statistical analysis. Then, the non-linear system is subjected to this random process. The non-linear structure system is modelled by substructure synthesis method (SSM) procedure. The non-linear equations are expanded sequentially. Then, the perturbed equations are solved in probabilistic method. Several statistical properties of a random process that are of interest in random vibration applications are reviewed in accordance with the non-linear stochastic problem. The system performance condition in the design of system is that responses caused by random excitation be limited within safe bounds. Thus, the reliability of the system is considered according to the crossing theory. Comparing with the results of the numerical simulation proved the efficiency of the proposed method.     

A reliable treatment of the homotopy analysis method for viscous flow over a non-linearly stretching sheet in presence of a chemical reaction and under influence of a magnetic field

The similarity solution for the steady two-dimensional flow of an incompressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretching sheet in the presence of a chemical reaction and under the influence of a magnetic field gives a system of non-linear ordinary differential equations. These non-linear differential equations are analytically solved by applying a newly developed method, namely the Homotopy Analysis Method (HAM). The analytic solutions of the system of non-linear differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. Graphical results are presented to investigate the influence of the Schmidt number, magnetic parameter and chemical reaction parameter on the velocity and concentration fields. It is noted that the behavior of the HAM solution for concentration profiles is in good agreement with the numerical solution given in reference [A. Raptis, C. Perdikis, Int. J. Nonlinear Mech. 41, 527 (2006)].      

EXPERIMENTAL VALIDATION OF THE CONSTANT LEVEL METHOD FOR IDENTIFICATION OF NON-LINEAR MULTI-DEGREE-OF-FREEDOM SYSTEMS

System identification for non-linear dynamical systems could find use in many applications such as condition monitoring, finite element model validation and determination of stability. The effectiveness of existing non-linear system identification techniques is limited by various factors such as the complexity of the system under investigation and the type of non-linearities present. In this work, the constant level identification approach, which can identify multi-degree-of-freedom systems featuring any type of non-linear function, including discontinuous functions, is validated experimentally. The method is shown to identify accurately an experimental dynamical system featuring two types of stiffness non-linearity. The full equations of motion are also extracted accurately, even in the presence of a discontinuous non-linearity.     

Differential transform method for solving linear and non-linear systems of partial differential equations

In this Letter, we propose a reliable algorithm to develop exact and approximate solutions for the linear and non-linear systems of partial differential equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and non-linear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.     

Applications of a generalized Galerkin's method to non-linear oscillations of two-degree-of-freedom systems

A generalized Galerkin's method is formulated for multi-degree-of-freedom holonomic systems. Two autonomous non-linear two-d.o.f. oscillators are used as the examples for the applications of the method: (i) free oscillations of a non-linear mass-spring system with two d.o.f., (ii) two weakly non-linearly coupled identical van der Pol oscillators. The accuracy of the approximate solutions is discussed. The effect of different time intervals of integration on the results is also investigated.     

Statistical errors for non-linear system measurements involving square-law operations

This paper develops normalized random error formulas for special bispectra estimates and associated frequency response function estimates in finite memory square-law systems. Error formulas are also derived for output spectrum estimates from these non-linear systems and for associated non-linear coherence functions. These formulas are useful to evaluate such measured non-linear results as well as to design experimental programs.     

A dynamic Lagrangian frequency-time method for the vibration of dry-friction-damped systems

A new frequency-time domain procedure, the dynamic Lagrangian mixed frequency-time method (DLFT), is proposed to calculate the non-linear steady state response to periodic excitation of structural systems subject to dry friction damping. In this formulation, the dynamic Lagrangians are defined as the non-linear contact forces obtained from the equations of motion in the frequency domain, with the adjunction of a penalization on the difference between the interface displacements calculate by the non-linear solver in the frequency domain and those calculated in the time domain from the non-linear contact forces, thus accounting for Coulomb friction and non-penetration conditions. The dynamic Lagrangians allow one to solve for the non-linear forces between two points in contact without using artifacts such as springs. The new DLFT method is thus particularly well suited to handling finite element models of structures in frictional contact, as it does not require a special model for the contact interface. Dynamic Lagrangians are also better suited to frequency-domain friction problems than the traditional time-domain method of augmented Lagrangians. Furthermore, a reduction of the non-linear system to relative interface displacements is introduced to decrease the computation time. The DLFT method is validated for a beam in contact with a flexible dry friction element connected to ground, for frictional constraints that feature two-dimensional relative motion. Results are also obtained for a large-scale structural system with a large number of one-dimensional dry-friction dampers. The DLFT method is shown to be accurate and fast, and it does not suffer from convergence problems, at least in the examples studied.     

Spectral analysis of non-linear systems involving square-law operations

This paper contains a number of useful theoretical formulas to analyze the frequency domain properties of Gaussian input data passing through non-linear square-law systems. Special bispectral density functions are defined and applied that are functions of a single variable. From measurements of input data and output data only, results are obtained to identify the separate frequency response functions for two models of linear systems in parallel with non-linear square-law systems. Non-linear coherence functions are defined from these models which determine the proportion of the output spectrum due to the non-linear operations. Together with ordinary coherence functions, a measured output spectrum for these models can be decomposed into three components representing the linear operations, the non-linear operations, and the remaining uncorrelated noise effects. This material indicates also how to analyze other types of non-linear models by employing similar techniques.     

Non-linear dynamics and geometrical properties of localized wave structures

The present state of investigations of geometric properties of spatially localized configurations of electromagnetic wave fields with TE and TM polarizations is reviewed. The spatial structures of these field configurations are determined by their discrete amplitude spectra. The possibilities of controlled formation of localized field structure with different symmetries are analysed. The non-linear dynamics of such distributions, depending upon their initial geometric properties, are considered for series of realistic models of non-linear media. The utilization of adequate analytical methods, including a geometric analysis in special space, a generalized variational approach, an inverse scattering method for non-stationary processes, is illustrated. The new tendencies in non-linear dynamics of short polarized signals in directional systems are outlined.     

Analytical method for steady state vibration of system with localized non-linearities using convolution integral and Galerkin method

The analytical method using transfer function or impulse response is very effective for analyzing non-linear systems with localized non-linearities. This is because the number of non-linear equations can be reduced to that of the equations with respect to points connected with the non-linear element. In the present paper, analytical method for the steady state vibration of non-linear system including subharmonic vibration is proposed by utilizing convolution integral and the impulse response. The Galerkin method is introduced to solve the non-linear equations formulated by the convolution integral, and then the steady state vibration is obtained. An advantage of the present method is that stability or instability of the steady state vibration can be discriminated by the transient analysis from convolution integral. The three-degree-of-freedom mass-spring system is shown as a numerical example and the proposed method is verified by comparing with the result by Runge-Kutta-Gill method.     

A Field Method for Integrating Equations of Motion of Nonlinear Mechanico-Electrical Coupling Dynamical Systems

We deal with the generalization of the field method to weakly non-linear mechanico-electrical coupling systems. The field co-ordinates and field momenta approaches are combined with the method of multiple time scales in order to obtain the amplitudes and phase of oscillations in the first approximation. An example in mechanico-electrical coupling systems is given to illustrate this method.     

PERIODIC SOLUTIONS OF STRONGLY NON-LINEAR OSCILLATORS BY THE MULTIPLE SCALES METHOD

The multiple scales method, developed for the systems with small non-linearities, is extended to the case of strongly non-linear self-excited systems. Two types of non-linearities are considered: quadratic and cubic. The solutions are expressed in terms of Jacobian elliptic functions. Higher order approximations, of solution as well as modulations of amplitude and phase, are derived. Comparisons to numerical simulations are provided and discussed.