Application of first-order sensitivity analysis to stabilisation of unstable continuous multi-degree-of-freedom parametric periodic systems - Wójcicki, Zbigniew (Oficyna Wydawnicza Politechniki Wrocławskiej, 2025)
From a practical point of view, the most important feature of a parametric periodic system is the instability phenomenon. Unlike in systems with constant coefficients, in which only points of instability exist, in parametric systems, whole areas of instability occur. This study presents a method of automatic stabilisation of unstable multi-degree-of-freedom linear continuous-in-time parametric systems. In this method, a parametric excitation can only be a continuous function of time. This paper concerns the sensitivity analysis of multipliers – complex eigenvalues of the monodromy matrix. The method is an alternative approach to that proposed in all other previous works on this subject. A procedure based on sensitivity analysis and directional derivative was used. The method's innovation is achieving the non-homogeneous parametric sensitivity equation by evaluating analytically the derivative of the homogeneous parametric equation of motion with respect to the design parameter. Then, by solving this sensitivity equation, evaluating the first derivative of the monodromy matrix, and finally, the first derivatives of multipliers. Ultimately, this method is based on a sensitivity analysis of the absolute values of multipliers. Furthermore, the sensitivity analysis method was improved and generalised to allow to correctly determine the eigenderivatives also with respect to those system parameters, on which the parametric excitation period depends. In particular, it becomes possible to use the parametric excitation period as a design parameter, which was not possible in the works of other authors. Examples of this method's implementation are also presented. This work continues the topics developed by the author in his earlier works.
Studia Geotechnica et Mechanica, Vol. 47, 2025, nr 3, s. 62-74
Poprzedni tytuł: Studia Geotechnica
Keywords: geotechnika - czasopisma; mechanika - czasopisma; first-order sensitivity; continuous; m.d.o.f; stabilisation; parametric