Application of second-order sensitivity analysis to stabilisation of unstable continuous multi-degree-of-freedom parametric periodic systems - Wójcicki, Zbigniew (Oficyna Wydawnicza Politechniki Wrocławskiej, 2025)
The work presents a method of automatic stabilisation of unstable multi-degree-of-freedom linear parametric systems. The publication is a continuation and extension of the subject matter described in Wójcicki’s earlier work ‘Application of first-order sensitivity analysis to stabilization of unstable continuous MDOF parametric periodic systems’ (Studia Geotechnica et Mechanica). While in that paper first-order sensitivity analysis was used, in this paper it was extended to the second-order sensitivity analysis. The algorithm of the presented method of stabilisation of an unstable continuous in time parametric system has become significantly more complicated, but the new formulas allow for better (nonlinear) prediction of extrapolated changes in the values of design parameters, which should accelerate the system stabilisation procedure. The obtained formulas were verified and validated using the same examples that were used in the study cited earlier. The method’s innovation is the idea to achieve the non-homogeneous parametric sensitivity equation by evaluating analytically the first and second derivatives of the parametric homogeneous equation of motion with respect to design parameter. Then, by solving the obtained sensitivity equation, the first and second derivatives of monodromy matrix and finally the first and second derivatives of multipliers are evaluated. Ultimately, this method is based on sensitivity analysis of 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.
Studia Geotechnica et Mechanica, Vol. 47, 2025, nr 3, s. 75-92
Poprzedni tytuł: Studia Geotechnica
Keywords: geotechnika - czasopisma; mechanika - czasopisma; second-order sensitivity; m.d.o.f.; continuous; stabilisation; parametric