Linearization generally represents a valid approximation in the modelling process of complex dynamic systems, but in several cases it might affect the reliability of the results. Although all systems in nature are nonlinear, accounting for all the existing nonlinearities is practically impossible since the computational time would increase indefinitely. Therefore, in some cases the most relevant nonlinear phenomena should be included. In many engineering systems, significant nonlinear effects are due to the connections between coupled subsystems, such as bolted joints and wire rope isolators. The present research aims at predicting the nonlinear behaviour of coupled systems using a modal substructuring technique. This project focuses on the effects of nonlinear connections on the dynamics of an assembly in which the coupled subsystems can be considered as linear. A first experimental setup has been built at the University of Madison-Wisconsin to measure the dynamic response of the assembly and compare it to the numerical results. A particular focus is given to the realization of the connecting elements between the linear substructures, specifically designed to have a cubic nonlinear behaviour. First results show an interesting correlation between the mode shapes and a good accuracy in the resonance frequency at several excitation levels. However, since some problems arose during the measurement campaign, a new experimental setup needs to be designed. The interfaces between the substructures and the nonlinear elements should be carefully adjusted to avoid loosening of the bolts. Also, the design of the connecting element needs to be updated, for example by employing compliant mechanism, to consider frictionless connections and have a better control on the dynamics of the assembly. Eventually, it could be possible to use real connections and, by measuring their dynamic response in terms of NNMs, perform an experimental substructuring based only on experimental data.
The present research is meant to extend the current substructuring methods to the analysis of systems jointed through nonlinear connections, whose effects cannot be neglected when the whole system is considered. Currently, substructuring techniques are very spread for analysing linear systems considering reduced order models to speed up the analysis and obtaining reliable results. However, these methods have been adapted in order to take into consideration the intrinsic nonlinearities of the substructures that have significant effects on the dynamic of the complete assembly, as it is for the wing-fuselage system.
The purpose of this project is to use the same technique to tackle those cases in which the effects due to nonlinearities localised at the common interface between two substructures are stronger than those of the substructures to be coupled, such that they can be considered as linear. The nonlinearity at the connection can be considered as a substructure itself and included in the substructuring process, following the same modal approach defined by Allen and Kuether [1], evaluating the nonlinear effects. This method was already used to couple lumped parameter models connected by single or multiple nonlinear springs [2].
The method has been significantly enhanced by enforcing a better control of the total energy at each step of procedure through an adjustment of the energy distribution among substructures. Also, the initial guess and the convergence criterion have been modified in order to speed up the procedure.
The method has been applied to a system made of two continuous linear subsystems coupled by nonlinear connections. The numerical results of the coupling are compared to the ones obtained by using the HB on the model of the complete assembly to evaluate its effectiveness and understand the effects of modal truncation.
The results are very encouraging, since they allow to relate the error in frequency with the number of modes retained in the modal basis, and also to understand that there is a correlation between the error for a mode and the energy level at which it is considered. The evaluation of Modal Assurance Criterion (MAC) alone has highlighted that it does not suffice in providing an understanding of how the mode shapes are affected by the truncation, but it has to be integrated with the information regarding how the energy is distributed among substructures.
Preliminary experimental tests have been performed on a real system. A nonlinear connecting element has been designed and manufactured in order to have a nearly cubic hardening behaviour. An experimental setup, composed by two beams joined by two of the aforementioned connecting elements, was built.
Tests have been performed on both the beams and the nonlinear connecting elements to identify the parameters of the corresponding numerical models, to be used in the coupling procedure.
Substructuring results are compared to experimental results measured on the assembled system, in order to evaluate the correlation between mode shapes and the accuracy in the resonance frequency at several excitation levels.
Further studies are meant to solve some issues related to the experimental setup in order to avoid some undesirable effects at the interface between the beams and the connecting elements for example by using compliant mechanisms.
Also, since with this setup only one NNM has been measured, the objective is to experimentally identify at least two NNMs in order to have more data to use as a comparison with the numerical results of the substructuring procedure. On the other hand, non-conservative systems are meant to be analysed by introducing damping effects either on the linear substructures or on the nonlinear ones, evaluating the possible outcomes. This step is necessary in order to allow this method to be used for the analysis of real complex systems in which it is not possible to neglect the presence of damping effects.
[1] R. J. Kuether, M. S. Allen, Nonlinear modal substructuring of systems with geometric nonlinearities, in: 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2013, p. 1521.
[2] F. Latini, J. Brunetti, W. D'Ambrogio, A. Fregolent, Substructures' coupling with nonlinear connecting elements, Nonlinear Dynamics (2019) 1--16.