Algebraic Parameter Identification of Nonlinear Vibrating Systems and Non Linearity Quantification Using the Hilbert Transformation

A novel algebraic scheme for parameters’ identification of a class of nonlinear vibrating mechanical systems is introduced. A nonlinearity index based on the Hilbert transformation is applied as an effective criterion to determine whether the system is dominantly linear or nonlinear for a specific operating condition. The online algebraic identification is then performed to compute parameters of mass and damping, as well as linear and nonlinear stiffness. The proposed algebraic parametric identification techniques are based on operational calculus of Mikusiński and differential algebra. In addition, we propose the combination of the introduced algebraic approach with signals approximation via orthogonal functions to get a suitable technique to be applied in embedded systems, as a digital signals’ processing routine based on matrix operations. A satisfactory dynamic performance of the proposed approach is proved and validated by experimental case studies to estimate significant parameters on the mechanical systems. The presented online identification approach can be extended to estimate parameters for a wide class of nonlinear oscillating electric systems that can be mathematically modelled by the Duffing equation.

Acoustics Source Identification of Diesel Engines Based on Variational Mode Decomposition, Fast Independent Component Analysis, and Hilbert Transformation

Diesel engines are widely used in railway systems, particularly in freight trains. Despite their high efficiency in energy conversion, they usually generate high levels of acoustics pollution during operation. In order to mitigate this problem, a series of active/passive acoustics control methods are used to reduce noise. Most of these methods are only effective if the prior knowledge of sources is given. In other words, it is essential to recognize the acoustics source. Variational mode decomposition (VMD) is a signal processing method that enhances the signal corrupted by background noise. However, the decomposed results of VMD depend on their mode parameter and penalty parameter. Therefore, an evaluation method based on system modal parameters (natural frequency and damping ratio) is proposed to select the mode parameter, and the penalty parameter can be selected from the power spectra of signals. In order to increase the accuracy of decomposition for diesel engines and find out the sources of acoustics, a method combining VMD, fast independent component analysis, and Hilbert transformation (VMD-FastICA-HT) is proposed for the separation and identification of different sources for diesel engines. The optimization results indicate that when the penalty parameter value is 1.5 to 16 times the maximum signal amplitude, better decomposition results can be achieved. Therefore, the separated independent acoustics are more accurate in source identification. Furthermore, both simulation data and in situ operational data of diesel engines for vehicles are used to validate the effectiveness of the proposed method.

Model of multicomponent narrow-band periodical non-stationary random signal

The correlation and spectral properties of a multicomponent narrowband periodical non-stationary random signal (PNRS) and its Hilbert transformation are considered. It is shown that multicomponent narrowband PNRS differ from the monocomponent signal. This difference is caused by correlation of the quadratures for the different carrier harmonics. Such features of the analytic signal must be taken into account when we use the Hilbert transform for the analysis of real time series.

An Exact Realization of a Modified Hilbert Transformation for Space-Time Methods for Parabolic Evolution Equations

We present different possibilities of realizing a modified Hilbert type transformation as it is used for Galerkin–Bubnov discretizations of space-time variational formulations for parabolic evolution equations in anisotropic Sobolev spaces of spatial order 1 and temporal order \frac{1}{2}. First, we investigate the series expansion of the definition of the modified Hilbert transformation, where the truncation parameter has to be adapted to the mesh size. Second, we introduce a new series expansion based on the Legendre chi function to calculate the corresponding matrices for piecewise polynomial functions. With this new procedure, the matrix entries for a space-time finite element method for parabolic evolution equations are computable to machine precision independently of the mesh size. Numerical results conclude this work.

A note on the efficient evaluation of a modified Hilbert transformation

In this note we consider an efficient data–sparse approximation of a modified Hilbert type transformation as it is used for the space–time finite element discretization of parabolic evolution equations in the anisotropic Sobolev space H1,1/2(Q). The resulting bilinear form of the first order time derivative is symmetric and positive definite, and similar as the integration by parts formula for the Laplace hypersingular boundary integral operator in 2D. Hence we can apply hierarchical matrices for data–sparse representations and for acceleration of the computations. Numerical results show the efficiency in the approximation of the first order time derivative. An efficient realisation of the modified Hilbert transformation is a basic ingredient when considering general space–time finite element methods for parabolic evolution equations, and for the stable coupling of finite and boundary element methods in anisotropic Sobolev trace spaces.