Structural Dynamics with Coincident Eigenvalues: Modelling and Testing Elvio Bonisoli, Cristiana Delprete and Marco Esposito Politecnico di Torino, Corso Duca degli Abruzzi, 24 - 10129, Torino, Italy John E. Mottershead Department of Engineering, University of Liverpool, 1.19 Harrison-Hughes Building, The Quadrangle, Liverpool L69 3GH, United Kingdom ABSTRACT Theory of curve crossing and curve veering phenomena is well known in structural dynamics, but only few papers have used test bench to demonstrate and validate this eigenvalues behaviour. The aim of this paper is to present a theoretical and experimental analysis on a nonsymmetric experimental structure with eigenvalues curve veering and crossing phenomena. Starting from literature examples, detailed numerical models on lumped parameters systems and continuous systems with coincident and/or close eigenvalues are examined in order to developed a numerical FE model suitable to describe a tunable and simple test rig with coincident eigenvalues and curve veering phenomena without symmetric properties or completely uncoupled dynamic systems. The test bench is made of simple beams and masses properly linked together. The angle of an intermediate beam is used as tunable physical parameter to vary the eigenvalues of the system and to couple two bending modes or bending and torsional modes. Numerical and experimental results are compared, and sensitivity of mode shapes to variation of system parameters is discussed. INTRODUCTION There have been extensive research works on veering and crossing phenomena in dynamic systems. The behaviour is generally well understood. In the former, as the eigenvalues change under a parameter variation, converging roots loci get closer and then suddenly veer away. During this process all the properties of the involved modes are swapped, leading to a curious behaviour in the so called “transition zone”. In the latter eigenvalues loci do not veer away but intersect without any swap of modal properties. Therefore, when two eigenvalues loci approach each other, they can cross or abruptly diverge. Theoretical studies of this behaviour have been reported for half a century but despite this heritage, explicit references to experimental results are scarce. Moreover, literature presents mainly symmetric or uncoupled lumped systems with coincident eigenvalues properties and eigenvalues curve veering and crossing are not deeply analysed with experimental viewpoint. One of the earlier observer of these phenomena, in structural dynamics, was Leissa [1], that cited further examples to draw attention to the possibility of fallacious artefacts in numerical models, and demonstrated that veering could be artificially induced through inadequate approximations and discretizations. Furthermore, to explain that in the veering away region mode shapes and nodal patterns must undergo sudden changes, Leissa used this sentence: “figuratively speaking, a dragonfly one instant, a butterfly the next, and something indescribable in between”. The rapid change in the eigenfunctions during the veering has raised doubt on the validity of many approximate solutions. Later, Kuttler and Sigillito [2] used an example of fixed membrane problem on rectangle to confirm the existence of curve veering in accurate mathematical models. Perkins and Mote [3] presented an exact mathematical solution of elementary eigenvalue problem to confirm the existence of curve veering physical phenomena. Thus, the purpose of their study was to validate the existence of curve veering in continuous models by presenting an exact eigensolution, which veers, and to derive simple T. Proulx (ed.), Modal Analysis Topics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 6, 325 DOI 10.1007/978-1-4419-9299-4_29, © The Society for Experimental Mechanics, Inc. 2011
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