Chapter 2 Prediction of the Coupled Impedance from Frequency Response Data Ramona Fagiani, Elisabetta Manconi, and Marcello Vanali Abstract This work deals with the common case of large mechanical systems which can be modelled considering two main sub-structures: a source structure, corresponding for example to the engine, and a receiving structure, as the accommodation area/cabin in these machines. Numerical models are highly desired for predicting the vibroacoustic behaviour of the receiver at a design stage and for design optimisation. In many cases the mechanical systems must be studied using sub-structuring techniques Here a standard frequency response coupling technique (or impedance coupling technique) is applied for the prediction of the complex power transmitted between a source structure and a receiving structure in a multi-point-connection case. The data required in this case are the frequency response functions (FRFs) of both the source and the receiver and the velocity at the connecting points under operating conditions. Experimental aspects and issues associated with this technique are investigated considering a simplified mechanical model. This consists of a driven beam connected in two or more points to a source. Results are discussed and compared with those obtained from a numerical model. Keywords Coupled impedance • Coupled structures • Modal analysis 2.1 Introduction Determining the vibro-acoustic behaviour of complex mechanical structures can be very challenging, since several components be simultaneously involved [1, 2]. In order to quantify the force transmitted through substructures for structureborn noise estimation, the knowledge of mechanical mobility and impedance at the connecting points is necessary [3]. In this regard, Dynamic Substructuring (DS) [4] plays a significant role for investigating a structural system by the analysis of the single sub-structure dynamic features and the way in which they combine. The substructuring approach has some significant advantages over the consideration of the full structure, in particular it allows evaluation of the dynamic behaviour of large and complex structures when the analysis of the all structure is unrealistic. Moreover, combining analytically, numerically or experimentally single modelled parts of the whole structure, local dynamic behaviour can be recognized more easily and neglected when it has no significant impact on the assembled system. In many cases this results in a simplified and clearer representation of the structure overall dynamics and in a significant reduction of the analysis time. Despite all the attention received in the last years, DS still requires additional research, especially concerning a number of challenges associated with the coupling of experimentally obtained substructures. Just to cite a few works, a review of the difficulties encountered in experimental DS are summarised in [4], errors related to truncation of modal degrees of freedom in [5] and the estimation of rotational degrees of freedom in [6], inclusion of rigid body modes in [7], non-linear mechanical coupling in [8], random measurement noise which may result in low signal-to-noise ratio, especially in the case of lightly damped structures in [9], inexact determination of the antiresonance modes in [10]. In this paper a frequency response coupling technique (or impedance coupling technique) is revised for the prediction of the complex impedance. Data required are the frequency response functions (FRFs) of the source and the receiver and the velocity/accelerations at the connecting points under operating conditions. These can be either experimental data or numerical data obtained from finite element analysis or modal analysis. The FRF coupling technique is first applied to a simple case study concerning two beams for validating the method numerically and experimentally and for investigating its accuracy according to the frequency range of interest. R. Fagiani • E. Manconi • M. Vanali ( ) Industrial Engineering Department, Università degli Studi di Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy e-mail: marcello.vanali@unipr.it © The Society for Experimental Mechanics, Inc. 2016 J. De Clerck, D.S. Epp (eds.), Rotating Machinery, Hybrid Test Methods, Vibro-Acoustics & Laser Vibrometry, Volume 8, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-30084-9_2 19
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