A tutorial on a multi-mode identification procedure based on the complex-curve fitting method

Modal identification is a key step in modal analysis. It enables the researcher to extract modal parameters, such as natural frequency, amplitude, and damping from a given structure. There are a considerable number of techniques in the state of the art aiming to address this problem, where multi-mode approaches arise as an appealing choice due to their ability to deal with mode coupling. This tutorial paper focuses on the complex-curve fitting technique, originally conceived for an application distinct from modal analysis. It aims at guiding other researchers by providing a tutorial-like and in-depth analysis of this important method, associated with a nonlinear weighting procedure for improved precision. Additionally, this paper fills a gap on the original technique, which is limited to the ratio of two polynomials, by proposing an automatic parameter extraction technique. The original and improved methods are applied on both simulated and experimental data, highlighting the effectiveness of the proposed changes. The proposed procedure is also compared with the rational fraction polynomial method.

On a half-discrete Hilbert-type inequality related to hyperbolic functions

By the introduction of a new half-discrete kernel which is composed of several exponent functions, and using the method of weight coefficient, a Hilbert-type inequality and its equivalent forms involving multiple parameters are established. In addition, it is proved that the constant factors of the newly obtained inequalities are the best possible. Furthermore, by the use of the rational fraction expansion of the tangent function and introducing the Bernoulli numbers, some interesting and special half-discrete Hilbert-type inequalities are presented at the end of the paper.

Audio cross-fade implementation via rational functions

Implementation of the cross-fade audio effect requires shaping the fade profile for a certain audio content that is to be faded out, as well as customizing the audio fade for an additional audio content, which is to be faded in, with the purpose of achieving a smooth transition between the two different audio contents. Similar to the case of applying adjustable fades, the audio cross-fades are usually carried out in the off-line mode, by employing various transcendental functions to shape the audio fades (both fade-out and fade-in effects). To improve the computational capabilities by minimizing the delay between receiving the position within the cross-fade effect and returning both the volume of the faded out section and the volume of the faded in one, we consider that, during the cross-fade effect, the audio volume of each of the two overlapping sections is the output of a rational function i.e. a mapping defined by a rational fraction. A plain HTML5/JavaScript implementation, prepared to be tested in any major browser, is advanced in the paper in order to highlight the suitability of the suggested approach to audio cross-fade customization with real-time computing.

Modal control based on direct modal parameters estimation

Modal active control is based on a state model that requires the identification of modal parameters. This identification can typically be done through a rational fraction polynomial algorithm applied in the frequency domain. This method generates numerical problems when estimating high-order models, particularly when moving from the basis of orthogonal polynomials for the modal basis. This algorithm must therefore be applied independently on multiple frequency ranges with a low order for each range. In this case, the controller design cannot be automated and requires a lot of human intervention, especially to build the state space model. To address this issue, this paper presents the application of the direct modal parameters estimation (DMPE) algorithm for active modal control design. The identification algorithm is presented in a simplified version with only positive frequencies. Unlike other classical identification methods in the frequency domain, the DMPE algorithm provides a solution with a great numerical stability and allows estimating models with a higher order. Using this method, the design of the controller can be largely automated and requires a minimum of human intervention. After a theoretical presentation, the proposed method is experimentally validated by controlling the vibration modes of a suspended plate.