This review article starts by addressing the mathematical principles of the perturbation method of multiple scales in the context of mechanical systems which are defined by weakly nonlinear ordinary differential equations. At this stage the paper investigates some different forms of typical nonlinearities which are frequently encountered in machine and structural dynamics. This leads to conclusions relating to the relevance and scope of this popular and versatile method, its strengths, its adaptability and potential for different variant forms, and also its weaknesses. Key examples from the literature are used to develop and consolidate these themes. In addition to this the paper examines the role of term-ordering, the integration of the so-called small (ie, perturbation) parameter within system constants, nondimensionalization and time-scaling, series truncation, inclusion and exclusion of higher order nonlinearities, and typical problems in the handling of secular terms. This general discussion is then applied to models of the dynamics of space tethers given that these systems are nonlinear and necessarily highly susceptible to modelling accuracy, thus offering a rigorous and testing applications case-study area for the multiple scales method. The paper concludes with comments on the use of variants of the multiple scales method, and also on the constraints that the method can bring to expectations of modelling accuracy.

Multiple scales analyses of the dynamics of weakly nonlinear mechanical systems

Chatter is considered as one of the most important causes of instability in the precision grinding process. Suffering from the influence of nonlinearity and changed machining condition, regenerative chatter signal is nonlinear and non-stationary in nature, which makes the linear representations and stationary assumed methods not appropriate. Wavelet bicoherence is considered as an effective method for these nonlinear signals analysis. However, current wavelet bicoherence is often estimated by integrating over the finite time interval. This procedure may result to the loss of the time information; hence, it is not appropriate for chatter signal that cannot be assumed to be stationary. In this paper, an instantaneous nonlinearity indicator-based method is proposed for regenerative chatter identification in grinding by using the servomotor current signal. Firstly, the nonlinearity of servomotor current affected by regenerative chatter is discussed. After that, a non-stationary wavelet bicoherence is proposed for the quadratic nonlinear coupling detection of these nonlinear and non-stationary signals. On this basis, an instantaneous nonlinearity indicator is further established for regenerative chatter identification. The effectiveness of the proposed method is verified by simulations and experiments, and the results show that, compared with the commonly used wavelet bicoherence and continuous wavelet transform, this method can not only describe the non-stationary chatter signal in time-frequency domain but also extract the true nonlinear coupling component resulted from regenerative chatter.

Regenerative chatter identification in grinding using instantaneous nonlinearity indicator of servomotor current signal

A nonlinear system identification technique exploiting the dynamic response features of fully nonlinear physics-based plate models extracted by Higher-Order Spectral (HOS) analysis tools is developed. The changes induced by an imperfection in the dynamics through the structural nonlinearities are used as key detection mechanism. The differences in dynamic response of a baseline and a modified/imperfect structure are enhanced by the local nonlinearities induced by the structural modification which thus represent the specific objective of identification. The validation of the procedure and the developed algorithms is carried out through extensive experimental testing employing various plates, including isotropic and composite lay-ups, and excitation sources, including White Gaussian Noise and a train of impulses.Copyright © 2011 by ASME

Advanced system identification of plates using a Higher-Order-Spectral approach: Theory and experiment

Higher-order spectral (HOS) analysis tools are employed to extract the nonlinear dynamic response features of elastic and laminated plates by using both physics-based mechanical plate models and experimental data. Bispectral and trispectral densities are computed to highlight the presence and relative importance of quadratic and cubic nonlinearities. The former are associated with the presence of asymmetry either in the excitation or in the mechanical response of predeflected plates while the latter are due to midplane stretching effects. Besides the detection of these structural nonlinearities in perfect (baseline) fully clamped plates, the changes of such nonlinearities induced by the presence of small inertial imperfections (i.e., lumped masses) are identified and exploited to localize the imperfections. The numerical and experimental investigations are carried out both on isotropic and laminated composite plates subject to Gaussian white noise excitation. The effectiveness of the HOS-based procedure for detection of the nonlinearities is fully demonstrated for both types of plates.

Detection of Nonlinearities in Plates Via Higher-Order-Spectra: Numerical and Experimental Studies

A geometrically non-linear Spectral Finite Flement Model (SFEM) including hysteresis, internal friction and viscous dissipation in the material is developed and is used to study non-linear dissipative wave propagation in elementary rod under high amplitude pulse loading. The solution to non-linear dispersive dissipative equation constitutes one of the most difficult problems in contemporary mathematical physics. Although intensive research towards analytical developments are on, a general purpose cumputational discretization technique for complex applications, such as finite element, but with all the features of travelling wave (TW) solutions is not available. The present effort is aimed towards development of such computational framework. Fast Fourier Transform (FFT) is used for transformation between temporal and frequency domain. SFEM for the associated linear system is used as initial state for vector iteration. General purpose procedure involving matrix computation and frequency domain convolution operators are used and implemented in a finite element code. Convergnence of the spectral residual force vector ensures the solution accuracy. Important conclusions are drawn from the numerical simulations. Future course of developments are highlighted.

K.A. Khan

Papers

Spectral Analyses of Non-Linear Interactions