The nonlinear system’s output signal generally does not preserve input frequencies. Their sum and difference produce the new spectral components in the output signal that are not presented in the input signal. These wave interactions have to be considered when we carry out the nonlinear system modeling. Higher order spectra, which are defined as the multiple Fourier transform of the higher order statistics such as the higher order moments or the cumulant of the stochastic signals, provide a means of detecting and quantifying the nonlinear wave interactions. The bispectra and the trispectra are typical higher order spectra, which can be obtained by the Fourier transform of the third order statistics and the fourth order statistics of the stationary signal. For the non-stationary nonlinear system output signal, the Wigner higher order spectrum can be used to detect the system nonlinear characteristics. To show the nonlinear frequency interaction, Vortex induced vibration experiment is carried out. The strains of the structure are measured to visualize the nonlinear phenomena of the nonlinear coupling vibration systems such as fluid and structure oscillation systems. The higher order spectrum such as bispectrum and trispectrum are calculated to show the quadratic and the cubic frequency interactions with the stationary data. The Wigner bispectrum and the Wigner trispectrum are also obtained to verify the time varying nonlinear interaction effects with the non-stationary data.
The Volterra series, which is the functional generalization of the Taylor series, is a useful tool to understand the nonlinear system because it expresses the system by the sum of linear and higher order nonlinear terms of input. Mathematically, every term of Volterra functional responses can be calculated by the multiple convolution-integrals in the time domain. The measurements of the Volterra kernels, however, have been the main means to identify the nonlinear systems. The assumption of input signal, which possesses Gaussian statistics, is used to have the substantial simplification of mathematical calculations. This assumption is possible when the input can be controlled in the laboratory. However, in many practical cases, input excitation is not under the direct control of the experimentalist. Thus one must use the measured input data as they exist. The general input data such as non-Gaussian have to be used to have the realistic model, because the modeling of the nonlinear system is varied according to the input data statistics.
The novelty of this paper is to provide a convenient means to analyze the variation of the physical characteristics of nonlinear models under the influence of the input signal statistics using signal vectors, which were introduced in the linear multiple-input problem. The output signal vectors of the quadratic Volterra model provide the favorable visualizations and make it easy to understand the variation of the nonlinear system responses according to the input data statistics such as Gaussian and non-Gaussian. Using the signal vector notation, the wave interactions are explained in term of the higher order frequency functions. The vector expressions certainly have advantages to express the nonlinear frequency models, which exhibit the nonlinear interactions. One of the advantages is that it geometrically explains the nonlinear characteristics, and therefore provides a better way to understand. The frequency response functions of a linear and a nonlinear system in terms of spectral vectors are obtained. These vector notations convey the system characteristics into physically realizable measures. One of these is the expression of the coherence function using the vector notation. The visualization of the coherence function demonstrates the degree of dependence for the linear and nonlinear output spectral vectors of the Volterra model according to the input signal statistics.
The ER(electro-rheological) fluid damper, which has the distinctive nonlinear characteristics, is used to verify the Volterra model in frequency domain. The bicoherence is used to detect the quadratic nonlinearities of the ER damper. The Volterra linear and quadratic frequency response functions are obtained according to the Gaussian and Non-Gaussian input signals. The orthogonal relationships are investigated between the linear and quadratic component output signals of the Volterra models with the Gaussian and Non-Gaussian input signals. With the non-Gaussian input signals, the linear and quadratic transfer functions can not be calculated independently because the linear and quadratic component output signals of the Volterra models do not satisfy the orthogonal relationship. The multiple coherence functions, which show the contributions between the linear and quadratic output signals of the Volterra model, are investigated according to the Gaussian and non-Gaussina input signals. The orthogonal relationships are also analyzed with the coherence between the linear and quadratic output signals of the Volterra models. The Gaussian input signal showed the orthogonal relationships between the linear and quadratic output signals of the Volterra models.
