On the computation of stable coupled state-space models for dynamic substructuring applications

TL;DR

The methodology to compute stable coupled state-space models for dynamic substructuring applications is introduced and it is shown that the coupled state-space models obtained using this methodology are suitable to be exploited in time-domain analyses and simulations.

Abstract

This paper aims at introducing a methodology to compute stable coupled state-space models for dynamic substructuring applications by introducing two novel approaches targeted to accomplish this task: a) a procedure to impose Newtons's second law without relying on the use of undamped RCMs (residual compensation modes) and b) a novel approach to impose stability on unstable coupled state-space models. The enforcement of stability is performed by dividing the unstable model into two different models, one composed by the stable poles (stable model) and the other composed by the unstable ones (unstable model). Then, the poles of the unstable state-space model are forced to be stable, leading to the computation of a stabilized state-space model. Afterwards, to make sure that the Frequency Response Functions (FRFs) of the stabilized model well match the FRFs of the unstable model, the Least-Squares Frequency Domain (LSFD) method is exploited to update the modal parameters of the stabilized model composed by the pairs of complex conjugate poles. The validity of the proposed methodologies is presented and discussed by exploiting experimental data. Indeed, by exploiting the FRFs of a real system, accurate state-space models respecting Newton's second law are computed. Then, decoupling and coupling operations are performed with the identified state-space models, no matter the models resultant from the decoupling/coupling operations are unstable. Stability is then imposed on the computed unstable coupled model by following the approach proposed in this paper. The methodology proved to work well on these data. Moreover, the paper also shows that the coupled state-space models obtained using this methodology are suitable to be exploited in time-domain analyses and simulations.

Figures (14)

• Figure 1: Test set up used to experimentally characterize the isolated crosses and assemblies RD_MSSP_2022.
• Figure 2: Locations of measurement accelerometers (red), hammer impact directions (black arrows) and virtual point (yellow) RD_MSSP_2022.
• Figure 3: Comparison of interface FRFs: i) obtained by applying VPT on the measured FRFs (black solid line - colour version only) (the amplitudes of these FRFs must be evaluated by using the left y axis); ii) from the identified SSM (red dashed line - colour version only) (the amplitudes of these FRFs must be evaluated by using the left y axis); iii) from the identified SSM transformed into UCF (green dotted line - colour version only) (the amplitudes of these FRFs must be evaluated by using the left y axis); iv) from the passive model computed according to the approach presented in AL_14 (blue dotted line - colour version only) (the amplitudes of these FRFs must be evaluated by using the right y axis): a) FRF of the aluminum cross A, whose output is $v_{1}^{y}$ and the input is $m_{1}^{y}$; b) FRF of the aluminum cross B, whose output is $v_{2}^{R_{x}}$ and the input is $m_{2}^{R_{x}}$.
• Figure 4: Comparison of interface FRFs: i) obtained by applying VPT on the measured FRFs (black solid line - colour version only) (the amplitudes of these FRFs must be evaluated by using the left y axis); ii) from the identified SSM (red dashed line - colour version only) (the amplitudes of these FRFs must be evaluated by using the left y axis); iii) from the identified SSM transformed into UCF (green dotted line - colour version only) (the amplitudes of these FRFs must be evaluated by using the left y axis); iv) from the passive model computed according to the approach presented in AL_14 (blue dotted line - colour version only) (the amplitudes of these FRFs must be evaluated by using the right y axis): a) FRF of the steel cross A, whose output is $v_{1}^{R_{x}}$ and the input is $m_{1}^{R_{x}}$; b) FRF of the steel cross B, whose output is $v_{1}^{z}$ and the input is $m_{1}^{z}$.
• Figure 5: Comparison of interface FRFs: i) obtained by applying VPT on the measured FRFs (black solid line - colour version only) (the amplitudes of these FRFs must be evaluated by using the left y axis); ii) from the identified SSM (red dashed line - colour version only) (the amplitudes of these FRFs must be evaluated by using the left y axis); iii) from the identified SSM transformed into UCF (green dotted line - colour version only) (the amplitudes of these FRFs must be evaluated by using the left y axis); iv) from the passive model computed according to the approach presented in AL_14 (blue dotted line - colour version only) (the amplitudes of these FRFs must be evaluated by using the right y axis): a) FRF of the assembly A, whose output is $v_{2}^{z}$ and the input is $m_{2}^{z}$; b) FRF of the assembly A, whose output is $v_{1}^{R_{y}}$ and the input is $m_{2}^{R_{y}}$.