Keywords: mechanics, flexible structures, dynamic behaviour, simulation, linearisation, validity range, control.

Complex flexible mechanical systems often show nonlinear behaviour. Therefore a closed analytical solution of the system statics and dynamics is not possible for all system states. Only a numerical simulation provides the possibility of obtaining the requested information. Typically, for dynamic simulations a linearisation of the system model around a distinct working point has to be performed. This is true in particular, when any external interaction with the system is required as it occurs in control applications, for example. The possibilities of modelling dynamic external system interaction within finite element (FE) packages are limited. A different type of software, normally a multipurpose numerical package, has to be used for this task [1]. As the system model is linearised, obviously no hard limits exist for deflections during these simulation runs.

Therefore this paper addresses the question, in which deflection ranges a linearised system model is valid for further analyses. Following an explorative approach, it is possible to obtain an estimate for maximum tolerable deflections of a linearised system model. Hereby a combination of nonlinear static analyses and linear modal analyses of the system, in the working point and in its neighbourhood, is performed within the FE package. Based on these results, common criteria to compare the dynamic behaviour of general systems are defined. Extending its common use, the modal assurance criterion (MAC) is applied to system models, which differ geometrically. The method presented is demonstrated by its application to the dynamic mechanical analysis of a very large radio telescope [2].

The proposed method works for all types of mechanical nonlinearities in principle. Treatment of nonlinearities, which could take effect from small dynamic deflections, is not possible. Apart from that, the application of the approach is limited by the FE modelling and solving capabilities provided by the FE software only. One drawback is the limited level of accuracy of the validity range. This comes from several biases introduced by assumptions and the explorative nature of the approach. The latter can be partly compensated by increasing the number of configurations examined in the neighbourhood, to build a finer grid around the working point. Concerning the choice of a criterion to deduce a certain validity range, the results obtained show, that, in case of the MAC, only the main diagonal components of the corresponding matrix should be used. Mainly numerical issues lead to a bias for the other components of the MAC matrix. Nevertheless, applying the method presented enables a statement to be formulated concerning the limits of system linearisation under dynamic aspects.

1

U. Schönhoff, "Practical Robust Control of Mechatronic Systems with Structural Flexibilities", Shaker Verlag, Aachen et al., Germany, 2003.

2

B. Strah, S. Kern, F. Fomi Wamba, M. Lazanowski, H. Li, J. Sun, H. Kärcher, R. Nordmann, "Trajectory Control of Cable Suspended FAST Telescope Focus Cabin", in "Astronomical Instrumentation, SPIE 2008", M.C. Clampin, A.F. Moorwood, (Editors), Marseille, France, 2008. (To appear)

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