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Identification of mass, damping and stiffness matrices of multi degree of freedom system subjected to kinematic excitations

Majcher Krzysztof

Studia Geotechnica et Mechanica

2025

Vol. 47, nr 1

89--102

This paper presents theoretical considerations relating to the possibility of fully identifying the parameters of a numerical model describing a building structure. An input–output method with system momentum change is proposed for this purpose, thanks to which the basic matrices describing the system were identified, that is, the mass matrix M, the damping matrix C and the stiffness matrix K. The proposed way of system identification is based on the knowledge of the vibration excitation (the input signal) and the structure’s dynamic response (the output signal) to the applied excitation, and the analyses are performed in the time domain. The reverse problem defined in this way consists of determining the coefficients of matrices M, C and K at any discrete point of time. In the case when the vibrations of the system are excited by kinematic excitation (ground motion), in order for the inverse problem to be solvable, either knowledge of the mass matrix or a known modification of the system masses is required. This is due to the representation of excitation forces, which in the case of kinematic excitation contains a mass matrix in their full description. This paper presents a method based on an inertial modification, that is, adding known masses to the analysed system, which entails a change in system momentum. The addition of known masses to the system being identified results in the introduction of additional known forces into the system. In this way, a heterogenous linear algebraic system of equations is obtained in the reverse problem and the coefficients of the particular matrices M, C and K are calculated from this system of equations. Moreover, considering the fact that the input signal and the output signal are known in many time points, the proposed procedure leads to a set of systems of equations. In order to verify the correctness and effectiveness of the proposed system identification method, numerical analyses for a shear building model were carried out as part of this study. Model vibrations were induced kinematically, and the functions describing the displacements of the subgrade were assumed in the form of a harmonic, the sum of three asynchronous harmonic functions, and a real earthquake. The results of numerical analyses confirmed the effectiveness of the proposed method in each of the three excitations.

Wykorzystanie sprężystego podłoża Winklera do analizy dynamicznej mostu pontonowego

Czajkowski Jakub, Majcher Krzysztof

Przegląd Budowlany

2025

R. 96, nr 5

29--32

W artykule przedstawiono uproszczony model analizy mostu pontonowego, oparty na sprężystym jednoparametrowym podłożu Winklera. Na podstawie prawa Archimedesa wyznaczono parametr zastępczej więzi sprężystej podłoża, co przy jednoczesnym wykorzystaniu programu SOFiSTiK (model numeryczny MES) umożliwiło szybkie i efektywne określenie reakcji konstrukcji pod wpływem obciążeń statycznych i dynamicznych. Metoda została zweryfikowana na przykładzie realizacji rzeczywistej przeprawy przez rzekę Odrę.

This paper presents a simplified model for the analysis of a floating bridge, based on an elastic one-parameter Winkler substrate. Based on Archimedes’ law, the substitute elastic link parameter of the substrate was determined, which, while using the SOFiSTiK program (FEM numerical model), allowed to quickly and efficiently determine the response of the structure under static and dynamic load. The method was verified on the example of the implementation of an actual crossing over the Oder River.