Investigation of bar system modal characteristics using Dynamic Stiffness Matrix polynomial approximations

0465280 - ÚTAM 2017 RIV GB eng J - Journal Article
Náprstek, Jiří - Fischer, Cyril
Investigation of bar system modal characteristics using Dynamic Stiffness Matrix polynomial approximations.
Computers and Structures. Roč. 180, February (2017), s. 3-12. ISSN 0045-7949. E-ISSN 1879-2243
R&D Projects: GA ČR(CZ) GA15-01035S
Institutional support: RVO:68378297
Keywords : Dynamic Stiffness Matrix * lambda matrices * self-adjoint operators * approximation in frequency domain * Wittrick-Williams algorithm
OECD category: Construction engineering, Municipal and structural engineering
Impact factor: 2.887, year: 2017
http://www.sciencedirect.com/science/article/pii/S0045794916310495

The aim of this study is an alternative approach to structure response or modal analysis. The structure consists of one-dimensional bars with continuously distributed mass and stiffness. The analysis is considered on an abstract basis as a problem of a differential system on an oriented graph. This graph is a geometric representation of the investigated mechanical system, where elements of the graph are individual bars of the system, recti- or curvilinear. The system as a whole is fixed through boundary conditions or interconnected with other sub-systems. Hence the paper can be taken as a follow up to earlier works presented at the CC2013 and CST2014 Conferences, where full mathematical background dealing with a general problem has been discussed. This paper is focused on the problem of dynamics of a system with straight prismatic bars with uniformly distributed mass. Dissipation of energy is omitted in order to keep the formulation in the real domain. The detailed assembly procedure of the Dynamic Stiffness Matrix (DSM) and transformation from local to global coordinates is outlined and demonstrated. Conventional way of eigenvalue searching by means of discrete alternative of the Newton-Raphson method is sketched out and later two possibilities based on polynomial and hyperbolic approximations of the DSM elements are pointed out. Lambda matrices as a tool are introduced together with a couple of application possibilities. The Wittrick-Williams algorithm is discussed and applied to localize and facilitate the eigenvalues searching on the whole frequency interval investigated. Finally, an illustrative example of the eigenvalue analysis of a structure is included. Strengths and shortcomings of the approach are discussed. Some open problems and orientation of further investigation are briefly outlined.
Permanent Link: http://hdl.handle.net/11104/0263913