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

Abstract

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.The rapid development of analytical and semi-analytical methods being based at DSM idea is commented and justified in comparison with FEM analysis. Even if a certain overlap exists, the DSM approach represents an irreplaceable alternative of analysis especially in broadband dynamic problems stemming from area of the stochastic dynamics, like wind, earthquake or traffic engineering. Problems of dynamic stability are easier to be managed as well, when using DSM oriented approaches together with related algorithms of the analytical and semi-analytical method family. Moreover, many problems of physics at the nano-scale emerged recently where these procedures are preferable to be used.