0.00 20.0 Linear Hz 0.00 87.0e-6 g 2 o o oo f o o o o f o f o o d o f s o d d f o o f d d d o o d d d d o v f f d d d o o v f f d d o d o s o f o fd d d f d o f s f f f df d d d f o o o s f d o ff d df o s o f v o f s o f f o d d f f d f o o d s f d f f d d f f f s o f o od s f d f o d o f f o f s f o o o fd d o f d f f dd d f f d f f s o dd d f f f f d f d v f s f f s f dd s o o f f d df f f f d f f s d dd s f f d f df s d f d s d d s o dd s f f d f f fd o f f fs d f o o dd o s f v d f d d f f f s d f o d dd f s f o o f f f d d f f f s f f oo d dd f s o f f f d d o d df f f d f s o d f fo d dd f s f f f s f dfd d f f s f d f v d ds f vo o f f d f o df f o f f f fs f f d os d dd f sv f f f f dff o f f d d f s f s d oo f dd v so o o f o o f d o d f f f d f d f f f f f d oo s dd f sv o f f o f f f f o f s ff d f f f f d f f d os d d f vv o f o f f d f f d sf f o d f f d f s f d d fo f d f sv o f d d f f f d f d f f f f s f o f f oo d fd d v o o f f f d d f d d f d f f f f f s d f d d fo d dd o d v f o v v d d d d f d f f d d d f ff d f f v d fv d ss f s s s f d f d d f d d d d d f s f f f s d f f s 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 100 (a) (b) Fig.6 An example of stabilization diagram (a), obtained by using the LMS operational PolyMAX algorithm, and of the Modal Assurance Criterion applied to the corresponding set of modes (b) The interpretation of the stabilization diagram yields a set of poles and corresponding operational reference factors. The mode shapes can then be found from ( ) { } { }* * * 1 LR j j UR j j j n i i i i yy i i i v g v g S ω ω ω ω λ ω λ + = 〈 〉 〈 〉 = + + + − − ∑ (6) in which all the present terms are now l m× matrices, with m the number of outputs selected as references and, in particular, LR, UR, respectively the lower and upper residuals, have been introduced to model the influence of the out-of-band modes in the considered frequency range [14]. A validation phase has then to follow, of course, to evaluate the quality of the estimated modal model, by using tools as the Modal Assurance Criterion (Fig.6b). 4 ANALYSIS OF THE RESULTS In the ride time frequency range from 0.2 to 20 Hz, recalled above, it is possible to identify a first sub-range, up to 3 Hz, where the heave, pitch and roll modes have to be found. It is well known that those modes and their eigenfrequencies are mainly related to the stiffness of suspension elastic elements and to the car body geometry and mass distribution. The damping ratios depend of course on damper coefficients. In the following sections, hence, the suspension operation will be analysed by OMA-estimating the modal parameters in the said 0 to 3 Hz frequency range. OMA leads, of course, also to the identification of higher-frequency deformation modes, that, for simplicity, are not considered and discussed in this paper. It is useful to stress from the beginning that during the random tests performed on the four poster test rig, not enough energy was delivered in the frequency range of interest (0 to 3 Hz) with the result that the said searched modes were not excited enough. In Fig. 7, for both the considered vehicles, a comparison is proposed of the auto-power spectrum of the vertical output signal at the driver seat rail location, during the different four poster tests. As one can clearly see the curves related to random excitation achieve lower values at the lower frequencies. In the case of road input, instead, the lower is the frequency, the higher is the energy delivered in the band of interest and, consequently, the better excited are the lower modes. More over, the higher is the car velocity, the higher is the excitation level, of course. 4.1 The semi-active suspension equipped car In this section the results obtained in the case of the vehicle equipped with the commercial semi-active suspension system are reported. In Tab.1 are, in particular, collected the modal parameters estimated by OMA-processing the data acquired and recorded during the four poster tests performed on this car. As it is possible to see, besides what anticipated regarding random excitation, the results got in the case of the two tracks, reproduced on the test rig, are in good agreement, as also confirmed by the modal validation reported in Fig.8. The modes identified in the case of the Florida track are represented in Fig. 9. In tab.2 are, then, collected the modal parameters estimated by OMA-processing the data coming from the proving ground tests, performed on the same car. 319
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