Singular value decomposition of dynamic response of structures - Yu. A. Kolotovichev (Vestnik MGSU)
Introduction. This paper presents the results of the decomposition of the synthesized free damped vibrations of a linear cantilever dynamic model into elementary components using singular value decomposition of the displacement matrix.Materials and methods. The well-known representation of a rectangular matrix in the form of a product of three matrix multipliers, called singular value decomposition (SVD), was used in the study.Results. Representing the dynamic response of a structure in a certain matrix form allows for the results of the singular value decomposition to have a clear physical interpretation: the left singular vectors approximate the eigenvectors, the singular values themselves determine the contribution of individual eigenmodes to the overall dynamic response, and the products of singular values and right singular vectors approximate the modal coordinates of the dynamic system at the time under consideration. Singular value decomposition allows to “automatically” obtain an a priori vector basis based on the external manifestations of the dynamic reaction (displacement, velocity or acceleration of the points of the structure), whereas the eigenvectors traditionally used for these purposes are a posteriori basis based on the study of the internal inertia, stiffness and damping properties.Conclusions. This study confirmed the possibility of using displacement matrix SVD to determine the major dynamic parameters of linear dynamic systems: eigenmodes, eigenfrequencies and quantitative damping parameters. All stages of obtaining and subsequent analysis of the dynamic response elementary components are easily automated, which allows to consider SVD as a basis for software development of automatic dynamic monitoring systems of structures under construction and in operation.