HHT-based nonlinear signal processing method for parametric and non-parametric identification of dynamical systems

This paper presents a time–frequency signal processing method based on Hilbert–Huang transform (HHT) and a sliding-window fitting (SWF) technique for parametric and non-parametric identification of nonlinear dynamical systems. The SWF method is developed to reveal the limitations of conventional signal processing methods and to perform further decomposition of signals. Similar to the short-time Fourier transform and wavelet transform, the SWF uses windowed regular harmonics and function orthogonality to extract time-localized regular and/or distorted harmonics. On the other hand, HHT uses the apparent time scales revealed by the signal's local maxima and minima to sequentially sift components of different time scales, starting from high- to low-frequency ones. Because HHT does not use pre-determined basis functions and function orthogonality for component extraction, it provides more accurate time-varying amplitudes and frequencies of extracted components for accurate estimation of system characteristics and nonlinearities. Methods are developed to reduce the end effect caused by Gibbs’ phenomenon and other mathematical and numerical problems of HHT analysis. For parametric identification of a nonlinear one-degree-of-freedom system, the method processes one free damped transient response and one steady-state response and uses amplitude-dependent dynamic characteristics derived from perturbation analysis to determine the type and order of nonlinearity and system parameters. For non-parametric identification, the method uses the maximum displacement states to determine the displacement–stiffness curve and the maximum velocity states to determine the velocity-damping curve. Moreover, the SWF method and a synchronous detection method are used for further decomposition of components extracted by HHT to improve the accuracy of parametric and non-parametric estimations. Numerical simulations of several nonlinear systems show that the proposed method can provide accurate parametric and non-parametric identifications of different nonlinear dynamical systems.     

Nonlinear transient response analysis for double walls with a porous material supported by nonlinear springs using FEM and MSKE method

In this paper, we newly propose a fast computation method for the nonlinear transient responses including coupling between nonlinear springs and sound proof structures having porous materials using FEM. In this method, we extend our numerical method named as Modal Strain and Kinetic Method (i.e. MSKE method proposed previously by Yamaguchi who is one of the authors) from linear damping analysis to nonlinear dynamic analysis. We assume that the restoring force of the spring has cubic nonlinearity and linear hysteresis damping. To calculate damping properties for soundproof structures including elastic body, viscoelastic body and porous body, displacement vectors as common unknown variable are solved under coupled condition. The damped sound fields in the porous materials are defined by complex effective density and complex bulk modulus. The discrete equations in physical coordinate for this system are transformed into nonlinear ordinary coupled differential equations using normal coordinates corresponding to linear natural modes. Further, using MSKE method, modal damping can be derived approximately under coupled conditions between hysteresis damping of viscoelastic materials, damping of the springs and damping due to flow resistance in porous materials. The modal damping is used for the nonlinear differential equation to compute nonlinear transient responses.Moreover, using the proposed method, we demonstrate new vibration phenomena including nonlinear coupling between nonlinear springs and soundproof structures by use of a simplified model. As a typical numerical example of the soundproof structure, we adopt double walls with a porous material. The double walls are supported by nonlinear concentrated springs. We clarify influences of amplitude of the impact force on nonlinear transient responses. We focused on the vibration modes, which magnify the amplitudes of the double walls. In these modes, the internal air of the porous material played a role of a pneumatic spring. Under a very large impact force as a severe condition, there exist the complicated nonlinear couplings between these modes and the super harmonic components of the rigid modes of the whole structure with large deformations in the nonlinear springs.     

Evaluation of transmissibility for a class of nonlinear passive vibration isolators

In this study, the concept of Output Frequency Response Functions (OFRFs) is applied to represent the transmissibility of nonlinear isolators in frequency domain. With the OFRFs estimated from numerical simulation responses, an explicit analytical relationship between the transmissibility and the nonlinear characteristic parameters is derived for a wide class of nonlinear isolators that have nonlinear anti-symmetric damping characteristics and a comprehensive pattern about how the nonlinear damping characteristic parameters might affect the force and displacement transmissibility is built for the vibration isolators. The results reveal that it is reasonable to analyze the force and displacement transmissibility of the nonlinear isolators by simply investigating the fundamental harmonic components of the force and displacement outputs of the nonlinear isolators, and the introduction of a nonlinear anti-symmetric damping into vibration isolators can significantly suppress both the force and displacement transmissibility over the resonant frequency region, but has almost no effect on the transmissibility at non-resonant regions. These conclusions are of significant importance in the analysis and design of the nonlinear vibration isolators with nonlinear anti-symmetric damping.     

Nonlinear parameter estimation for multi-degree-of-freedom nonlinear systems using nonlinear output frequency-response functions

Based on the concept of nonlinear output frequency-response functions (NOFRFs), a novel algorithm is developed in this paper to estimate the nonlinear stiffness and damping parameters for multi-degree-of-freedom (MDOF) nonlinear systems. The validity of this algorithm is demonstrated by numerical studies. This work is an extension and complement of the authors’ previous studies, which involve the study of the properties of the NOFRFs of MDOF nonlinear systems, and the estimation of the system linear stiffness and damping parameters. These results build up a NOFRF-based systematic approach for the analysis of MDOF nonlinear systems.     

A new method for measuring nonharmonic periodic excitation forces in nonlinear damped systems

The purpose of this study is to identify an external loading of long time duration, which is nonharmonic but periodic, acting on a nonlinear dynamic system with nonlinear restoring as well as nonlinear damping. A new procedure is proposed for the force identification through an inverse formalism. However, this involves a Volterra-type nonlinear integral equation of the first kind, which lacks solution stability. Therefore, the nonlinear dynamic system under investigation is transformed into a linear relation between forces and pseudo-displacements. The lack of solution stability is resolved by applying available regularization methods. The feasibility of the force identification is demonstrated through a numerical example.     

Identification of uncertain nonlinear systems for robust fuzzy control

In this paper, we consider fuzzy identification of uncertain nonlinear systems in Takagi-Sugeno (T-S) form for the purpose of robust fuzzy control design. The uncertain nonlinear system is represented using a fuzzy function having constant matrices and time varying uncertain matrices that describe the nominal model and the uncertainty in the nonlinear system respectively. The suggested method is based on linear programming approach and it comprises the identification of the nominal model and the bounds of the uncertain matrices and then expressing the uncertain matrices into uncertain norm bounded matrices accompanied by constant matrices. It has been observed that our method yields less conservative results than the other existing method proposed by S?krjanc et al. (2005) [11] and [12]. With the obtained fuzzy model, we showed the robust stability condition which provides a basis for different robust fuzzy control design. Finally, different simulation examples are presented for identification and control of uncertain nonlinear systems to illustrate the utility of our proposed identification method for robust fuzzy control.     

Observer-based robust control of one-sided Lipschitz nonlinear systems

This paper presents an observer-based controller design for the class of nonlinear systems with time-varying parametric uncertainties and norm-bounded disturbances. The design methodology, for the less conservative one-sided Lipschitz nonlinear systems, involves astute utilization of Young’s inequality and several matrix decompositions. A sufficient condition for simultaneous extraction of observer and controller gains is stipulated by a numerically tractable set of convex optimization conditions. The constraints are handled by a nonlinear iterative cone-complementary linearization method in obtaining gain matrices. Further, an observer-based control technique for one-sided Lipschitz nonlinear systems, robust against L₂-norm-bounded perturbations, is contrived. The proposed methodology ensures robustness against parametric uncertainties and external perturbations. Simulation examples demonstrating the effectiveness of the proposed methodologies are presented.     

Identification and quantification of nonlinear stiffness and nonlinear damping in resonant circuits

This paper presents a study of nonlinear vibrating mechanical structures whose resonances have nonlinearities due to stiffness and/or damping. A method is presented to detect the presence of nonlinear distortions qualitatively and quantitatively. Basically, a device under test is configured as a second-order, single-input–single-output closed-loop feedback system where static nonlinearities are confined to the feedback path, and the dynamic linear part is modeled as the forward gain. The system is excited by a random phase multisine, which allows for the modeling of the linear part by its frequency response function, thus facilitating the characterization of the nonlinear part. The identification is formulated in a linear-in-parameters framework, using input and output regressors. A good agreement between the estimated data and the measurements indicates the presence of nonlinearities due to stiffness and damping with distinguishable levels.     

IDENTIFICATION OF TIME-VARYING MODAL PARAMETERS USING LINEAR TIME-FREQUENCY REPRESENTATION

A new method of parameter identification based on linear time-frequency representation andHilbert transform is proposed to identify modal parameters of linear time-varying systems frommeasured vibration responses. Using Gabor expansion and synthesis theory measured responses arerepresented in the time-frequency domain and modal components are reconstructed by time-frequencyfiltering. The Hilbert transform is applied to obtain time histories of the amplitude and phase angle ofeach modal component, from which time-varying frequencies and damping ratios are identified. The     

Coupled bending and torsional vibration of a rotor system with nonlinear friction

Unacceptable vibrations induced by the nonlinear friction in a rotor system seriously affect the health and reliability of the rotating machinery. To find out the basic excitation mechanism and characteristics of the vibrations, a coupled bending and torsional nonlinear dynamic model of rotor system with nonlinear friction is presented. The dynamic friction characteristic is described with a Stribeck curve, which generates nonlinear friction related to relative velocity. The motion equations of unbalance rotor system are established by the Lagrangian approach. Through numerical calculation, the coupled vibration characteristics of a rotor system under nonlinear friction are well investigated. The influence of main system parameters on the behaviors of the system is discussed. The bifurcation diagrams, waterfall plots, the times series, orbit trails, phase plane portraits and Poincaré maps are obtained to analyze dynamic characteristics of the rotor system and the results reveal multiform complex nonlinear dynamic responses of rotor system under rubbing. These analysis results of the present paper can effectively provide a theoretical reference for structural design of rotor systems and be used to diagnose selfexcited vibration faults in this kind of rotor systems. The present research could contribute to further understanding on the self-excited vibration and the bending and torsional coupling vibration of the rotor systems with Stribeck friction model.     

Robust fast controller design via nonlinear fractional differential equations

A new method for linear system controller design is proposed whereby the closed-loop system achieves both robustness and fast response. The robustness performance considered here means the damping ratio of closed-loop system can keep its desired value under system parameter perturbation, while the fast response, represented by rise time of system output, can be improved by tuning the controller parameter. We exploit techniques from both the nonlinear systems control and the fractional order systems control to derive a novel nonlinear fractional order controller. For theoretical analysis of the closed-loop system performance, two comparison theorems are developed for a class of fractional differential equations. Moreover, the rise time of the closed-loop system can be estimated, which facilitates our controller design to satisfy the fast response performance and maintain the robustness. Finally, numerical examples are given to illustrate the effectiveness of our methods.     

Identification of nonlinear hysteretic systems by artificial neural network

An identification method is developed for nonlinear hysteretic systems by use of artificial neural network in the paper. Employing the Bouc–Wen differential model widely used for memory-type nonlinear hysteretic systems, the approach sets up a Bouc–Wen model-based neural network. The weights of the designed specifically network correspond to the Bouc–Wen model parameters and are thus physical ones. Taking advantage of powerful function approximation capability of neural network, the nonlinear hysteretic systems can be identified with the proposed approach by network training. The identification scheme is validated by a simulated case and thereafter applied to modeling of a wire cable vibration isolation experimental system. The results show that the presented identification method can identify the nonlinear hysteretic systems with high accuracy.     

多自由度系统非线性参数识别的灵敏度方法

为了识别多自由度系统的非线性参数（3次阻尼系数、3次刚度系数和干摩擦力），提出一种基于灵敏度分析用测量响应（位移、速度或加速度）识别非线性参数的方法。将结构抽象为由线性和非线性阻尼、刚度连接的集中质量系统，用四阶Runge—Kutta法求解多自由度非线性系统的振动响应和振动响应对非线性参数的灵敏度。假定非线性参数为零初值，根据计算响应与测试响应的差值，由基于正则化的阻尼最小二乘法迭代识别系统的非线性参数值。算例表明，该方法可以有效地识别多自由度系统的非线性参数。     

A refined nonlinear vibration absorber

Presented here is a theoretical study on how to use saturation phenomena to design nonlinear vibration absorbers and how to improve their stability and effective frequency bandwidth. The so-called original saturation control method uses 2 : 1 internal resonances and saturation phenomena to suppress steady-state vibrations of a dynamical system by connecting it to a second-order controller using quadratic position coupling terms, which do not really suppress vibration to zero as a linear vibration absorber does. However, a linear vibration absorber uses direct position feedbacks and splits one natural frequency of the original system into two and hence spill-over effects exist when the system is subjected to broad-band and/or transient excitations. Although a saturation controller does not split one natural frequency into two, one large-amplitude nonlinear solution coexists with a small-amplitude linear solution outside of the resonance area. Hence, the existence of spillover effects depends on initial conditions. A refined nonlinear vibration absorber is designed by using a quadratic velocity coupling term in the controller and adding a negative velocity feedback to the system. It is shown that the quadratic velocity coupling term enables a saturation controller to suppress system vibrations to zero. Moreover, the linear velocity feedback enhances the capability of suppressing transient vibrations and prevents the system from having the large-amplitude nonlinear response. Two equations describing the first-mode vibration of a stainless-steel beam and a saturation controller from the authors’ previous experimental work are used in this theoretical study. Both perturbation and direct numerical integration solutions are presented. Guidelines for designing nonlinear vibration absorbers are derived.     

Analysis of the non-resonance of nonlinear vibration isolation system with dry friction

We studied the properties of cubic nonlinear systems with dry friction damping on the prospect of ‘non-resonance’. An approximate method was used to get the frequency-response function. And the critical dry friction which ensures no displacement transmissibility amplified in whole frequency range was determined through the frequency-response function. We also researched those characters which may influence the effect of the non-resonance in this system, and proved those results by empirical experiments. The above conclusions have a theoretical significance for the design of nonlinear vibration isolators.     

An investigation on effects of traveling mass with variable velocity on nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions

In this paper, the nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions subjected to a traveling mass with variable velocity is investigated. The nonlinear coupled partial differential equations of motion for the bending rotation of cross-section, longitudinal and transverse displacements are derived using Hamilton’s principle. These nonlinear coupled PDEs are solved by applying Galerkin’s method to obtain dynamic response of the beam under the act of a moving mass. The appropriate parametric studies by taking into account the effects of the magnitude of the traveling mass, the velocity of the traveling mass with a constant acceleration/deceleration and effect of different beam’s boundary conditions are carried out. The beams’ large deflection has been captured by including the stretching effect of its mid-surface. It was seen that the existence of quadratic-cubic nonlinear terms in the governing coupled PDEs of motion renders hardening/stiffening behavior on the dynamic responses of the beam when traversed by a moving mass. In addition, the obtained nonlinear results are compared with those from the linear analysis.     

Linear parameter estimation for multi-degree-of-freedom nonlinear systems using nonlinear output frequency-response functions

The Volterra series approach has been widely used for the analysis of nonlinear systems. Based on the Volterra series, a novel concept named nonlinear output frequency-response functions (NOFRFs) was proposed by the authors. This concept can be considered as an alternative extension of the classical frequency-response function for linear systems to the nonlinear case. In this study, based on the NOFRFs, a novel algorithm is developed to estimate the linear stiffness and damping parameters of multi-degree-of-freedom (MDOF) nonlinear systems. The validity of this NOFRF-based parameter estimation algorithm is demonstrated by numerical studies.     

A wavelet-based approach for the identification of damping in non-linear oscillators

A special class of non-linear damping models is studied in which the damping force is proportional to the product of positive powers of the absolute values of displacement and velocity. For a single degree of freedom system, the Krylov–Bogoliubov averaging method is used to determine the approximate free response. The wavelet transform of this response is used as a time-scale representation for parameter identification: two methods based on this wavelet transform are presented to estimate instantaneous frequency, damping and envelope of the system. The first method uses cross-sections of the wavelet transform. The second method uses ridges and skeletons of the wavelet transform. This second method is general and gives accurate results in the case of noisy non-linear oscillators. These methods are illustrated using a simulated example.     

On the computation of the slow dynamics of nonlinear modes of mechanical systems

A novel method for the numerical prediction of the slowly varying dynamics of nonlinear mechanical systems has been developed. The method is restricted to the regime of an isolated nonlinear mode and consists of a two-step procedure: In the first step, a multiharmonic analysis of the autonomous system is performed to directly compute the amplitude-dependent characteristics of the considered nonlinear mode. In the second step, these modal properties are used to construct a two-dimensional reduced order model (ROM) that facilitates the efficient computation of steady-state and unsteady dynamics provided that nonlinear modal interactions are absent.The proposed methodology is applied to several nonlinear mechanical systems ranging form single degree-of-freedom to Finite Element models. Unsteady vibration phenomena such as approaching behavior towards an equilibrium point or limit cycles, and resonance passages are studied regarding the effect of various nonlinearities such as cubic springs, unilateral contact and friction. It is found that the proposed ROM facilitates very fast and accurate analysis of the slow dynamics of nonlinear systems. Moreover, the ROM concept offers a huge parameter space including additional linear damping, stiffness and near-resonant forcing.     

Load-carrying capacity of a journal bearing under dynamic loading

A method of obtaining the simultaneous solution of the Reynolds equation for dynamically loaded bearings by using a suitable numerical method is briefly outlined. It is assumed that the relation between the relative radial velocity and the hydrodynamically effective angular velocity of the journal is known. The solution is illustrated by graphs and a table. Examination of the application of separate solutions of the Reynolds equation in Holland's method for solving the motion of the journal centre suggests that an error exists which is not always negligible. Therefore a modification of Holland's method based on the simultaneous solution derived in this paper is proposed.