MS1.2 - Advances in Computational Structural Dynamics

The standard Newton–Raphson scheme suffers from divergence problems due to softening, negative tangent stiffness, bifurcations or snap-back when modelling fracture of quasi-brittle materials, such as masonry and concrete. Convergence can be achieved by applying sequentially linear analysis (SLA) in some cases. However, it has difficulties in dealing with cases in which the displacement history matters such as dynamic damage modelling due to its total approach. Recently, incremental sequentially linear analysis (ISLA) which combines the merits of the N–R method and SLA, has been proposed. The solution search path follows damage cycles sequentially with secant stiffness corresponding to local damage increments, which traces both damage history (explicit) and displacement history (implicit). This work focuses on dynamic damage modelling of quasi-brittle materials accounting for physical non-linearity as well to address the transient effects in the fracture process. The standard ISLA, similar to the large time increment (LATIN) method, the extended finite element method (X-FEM) and strong discontinuity approaches (SDA) which are proposed to capture the crack propagations for quasi-brittle materials, is not suitable for dynamic damage modelling. The objective of this work is therefore to propose a damage integration scheme (damage-driven instead of time integration scheme) based on ISLA for non-linear transient analysis of structures under dynamic loading while retaining the merits of the previous ISLA. The proposed damage integration scheme has been compared with the standard time integration scheme in a dynamically loaded bar example, showing good agreements and sufficient accuracy. The proposed method has been validated further for a notched beam under tension and four-point bending. Since all physical non-linearity is linearized in damage cycles with the explicit secant stiffness of the reduced elastic material model, the algorithm possesses high robustness.

When considering industrial models for aircraft braking systems, the number of degrees of freedom (DoF) limits the type of analysis that can be performed. It is therefore sometimes necessary to simplify the model by introducing some hypotheses ; this allows to have satisfactory simulation times. However, these operations can affect the accuracy of the model, in particular its capacity to appropriately represent instabilities due to friction-induced vibrations. This paper investigates and discusses the hypothesis of non-deformation of the disks (i.e. rotors and stators in frictional contact) on the stability of an aircraft braking system at low frequencies. In order to conduct such a study, both finite element models (by considering rigid rotor and stator disks and non-rigid disks respectively) are developed. The non-rigid disk model is introduced and its performance is analysed in order to assess the improvement of the instability prediction. Besides, the convergence with regards to the number of points per surface contact of the non-rigid disks model will be verified. Regarding the non-rigid disk model, the study also presents an efficient strategy to reduce the number of DoF through a model based on a Double Modal Synthesis. This numerical strategy is developed to perform relevance prediction of unstable vibration modes for an aircraft braking system subjected to friction- induced vibration. Particular attention is brought to validating the convergence of the reduced model, specifically on the prediction of the appearance of instabilities as well as its characteristics (i.e. the prediction of both the real and imaginary parts, as well as eigenvectors of the unstable modes generated). It is finally verified that the numerical results via the Double Modal Synthesis are in good agreement with the experiments.

Machines are typically an interconnection of components, i.e., they are assemblies. To analyze the dynamics of these assemblies, their components are generally modeled by large structural, finite element models. The large order of these component models, and of their interconnection, necessitates the use of model reduction techniques to allow further analysis. Model reduction of the assembly is often performed by reduction of the individual components, because direct reduction of the assembly is not computationally tractable. Component mode synthesis, for example, is one such popular method for component-level reduction. However, by reducing the individual components, the dynamics that is retained in the reduced component model might not be relevant to the assembled model. Stated differently, if the reduction of a component model does not take the dynamics of the component’s environment into account, the accuracy of the interconnected, reduced-order model, cannot be guaranteed. In this work, we propose a new model reduction method called Passive Interconnected Balanced Truncation (PIBT). PIBT can improve the accuracy of the assembly model with respect to component-level reduction, while guaranteeing passivity and stability of both the components and the assembly. PIBT is based on the reduction method of Balanced Truncation (BT), which focuses on the accurate description of (selected) input-output behavior. BT consists of two steps: first the coordinates are transformed to a balanced realization, where states are sorted by their ‘importance', followed by a truncation of the least important states. This `importance' of the balanced states is determined based on two so-called Gramians, which depend on the realization of the model. To aim for higher accuracy, while guaranteeing passivity, PIBT is based on a combination of one local (component-level) and one global (assembly-level) Gramian. The local Gramian is the minimal solution to the Positive Real Lemma and relates to passivity. The global Gramian is the controllability Gramian of the assembly and relates to its input-output behavior. By using this combination of Gramians, both the reduced component and reduced assembly models are guaranteed to be passive, while the assembly’s input-output behavior tends to be approximated accurately. To validate PIBT, it is tested on a numerical example of two interconnected Euler beam models. In a comparison to two existing methods from literature, reduction with PIBT results in a superior reduced order model, showing both passivity preservation and an accurate approximation of the assembly.