Coupling DoFs Research Articles

Efficient compression of reduced impedance matrices for trimmed structural models: a novel methodology and industrial validation

In response to increasingly strict vibration and noise regulations, software frameworks such as Nastran and Actran have stepped up to the challenge, providing sophisticated solutions to efficiently model damping treatments within numerical methods. One of the available techniques is the representation of trim components as frequency-dependent reduced impedance matrices (RIM) in a frequency response analysis of fully trimmed models. While RIM enables coupling between physical or modal-based components, challenges arise due to large dense matrices. Each matrix is reduced either on physical coupling DOFs or modes of vibration of the component and stored on disk. Since each matrix is dense and large in industrial models, it leads to storage and efficiency issues related to computational and input/output (I/O) operations. This paper introduces a methodology to compress RIMs, particularly for modal components, by selecting the most contributing modes to represent the coupling relationship of each surface. Numerical results on an industrial car body model demonstrate that this approach effectively eliminates low-effect contributions, minimizes storage requirements, and reduces computational time. General Motors' involvement in validating the method adds real-world applicability and reliability to the findings, enhancing its practical engineering value.

A critical evaluation of SEMM-based joint identification procedure to reduce the error propagation effects

Mechanical joints play a critical role in many engineering structures, but identifying their characteristics can be challenging, particularly when the joint interface (coupling DoFs) cannot be directly measured. In these cases, an existing iterative procedure based on the use of the System Equivalent Model Mixing (SEMM) expansion technique and substructure decoupling can be used to identify the joint properties by taking measurement on other accessible DoFs (internal DoFs). Despite the potential of this procedure, it is prone to large error propagation. In addition, in the SEMM expansion a weighted pseudo-inverse operation is needed to ensure the convergence of the iterative procedure when both coupling and internal DoFs (extended interface) are involved in the decoupling.This paper focuses on the detection of the error sources in the process and on the definition of some strategies to limit error propagation. The use of internal DoFs only (pseudo-interface) in the decoupling is proposed. This avoids the use of coupling DoFs affected by the expansion error. Furthermore, two strategies are proposed to improve the conditioning of the procedure when using the extended interface. Both strategies are based on the Truncated Singular Values Decomposition (TSVD). It is shown that the weights clearly indicate the number of singular values to be retained in the matrices to be inverted.The proposed improvements are validated on a laboratory benchmark. Measurements on the benchmark are performed to validate the strategies with experimental data. In addition, the Monte Carlo method is applied using noise-polluted numerical data to evaluate the potential of the proposed strategies to mitigate the error propagation in the SEMM-based joint identification procedure.

Coupling of Component Models with Mismatching Interfaces for an Efficient NVH Vehicle Design

<div class="section abstract"><div class="htmlview paragraph">The NVH optimization of new vehicle models can in principle only be carried out in a relatively late stage of the development process, when the geometrical data (CAD) are available and can be used to generate detailed Finite Element (FE) models of the car body. Unfortunately, in this stage of the development process most of the geometrical data are already fixed and countermeasures are limited and expensive. In order to be able to evaluate design concepts in an earlier conceptual stage of the development process existing models of similar predecessor vehicles must be used leading to techniques such as “mesh-morphing” or “concept modelling” (see for instance [<span class="xref">1</span>, <span class="xref">2</span>]). Here, a different approach is investigated based on a substructuring technique. In principle the coupling of the component-structures coming from different models would require post-processing in order to obtain compatible interface degrees-of-freedom (DOFs), an operation which in most cases must be carried out manually and is very time consuming. This paper presents a novel substructuring approach that tackles a continuous interface by only considering a small set of coupling DoFs. This approach employs the discretisation of interfaces between structural parts or structural-acoustic domains in terms of pivotal points and patches. In this manner, the requirement of spatial continuity between the models of the separate substructures is no longer needed and subsystems with incompatible interfaces can be coupled. Thus, an efficient vibro-acoustic design optimisation procedure for components shared by different vehicle models, such as the car floor, is enabled. It will be shown that a single car floor model can be successfully coupled with different upper-body structures, even when they have a different location of coupling DoFs.</div></div>

Replacement of unobservable coupling DoFs in substructure decoupling

In this paper, suitable criteria are sought for an optimal replacement of unobservable coupling DoFs when performing substructure decoupling, that is the identification of a dynamic model of a substructure embedded in a known structure. The need arises since coupling DoFs are often difficult to observe, either because they cannot be easily accessed or because they include rotational DoFs. The substitution must be carried out both to preserve the information that would be lost when some coupling DoFs are not taken into account, and to avoid or minimize ill-conditioning problems.As shown in previous papers, coupling DoFs can be effectively replaced by internal DoFs for the sake of substructure decoupling. Furthermore, criteria for an appropriate selection of the internal DoFs used to replace unobservable coupling DoFs are sketched, which involve either the Frequency Response Function (FRF) or the transmissibility between internal and coupling DoFs.Here, previously introduced FRF and transmissibility criteria are combined with the condition number of the interface flexibility matrix to develop a procedure to optimally replace some coupling DoFs with a subset of internal DoFs. The procedure is tested using both simulated and experimental data of a tree structure (known structure), made by a cantilever beam with two offset short arms, coupled to another beam (structure to be identified).

Substructure decoupling as the identification of a set of disconnection forces

The objective of this paper is to achieve a better understanding of the substructure decoupling problem by analyzing different sets of disconnection forces. Substructure decoupling consists in the identification of a dynamic model of an unknown substructure embedded in a known structure (coupled structure). Disconnection forces can be imagined to be applied to the coupled structure in order to reproduce the dynamic behavior of the substructure to be identified. The disconnection forces are such as to cancel the effect of constraint forces exerted at the coupling DoFs of the unknown substructure by the remaining part of the structural system. Several sets of disconnection forces can be devised: the trivial set, consisting of disconnection forces acting at the coupling DoFs and opposite to the constraint forces; non trivial sets of disconnection forces acting at different DoFs but such as to cancel the effect of the constraint forces. Here, a procedure to determine disconnection forces is recalled in the framework of substructure decoupling. Different sets of disconnection forces are compared using both simulated and experimental data of a tree structure (known structure). Indications provided by disconnection forces are in agreement with the results of the decoupling procedure.

Substructure decoupling without using rotational DoFs: Fact or fiction?

In the framework of experimental dynamic substructuring, substructure decoupling consists in the identification of the dynamic behaviour of a structural subsystem, starting from the dynamic behaviour of both the assembled system and the residual subsystem (the known portion of the assembled system). On the contrary, substructure coupling identifies an assembled system starting from the component subsystems. The degrees of freedom (DoFs) of the assembled system can be partitioned into internal DoFs (not belonging to the couplings) and coupling DoFs.In substructure coupling, whenever coupling DoFs include rotational DoFs, the related rotational FRFs must be obtained experimentally. Does this requirement holds for substructure decoupling too, as it is commonly believed?Decoupling can be ideally accomplished by adding the negative of the residual subsystem to the assembled system (direct decoupling) and by enforcing compatibility and equilibrium at enough interface DoFs. Ideally, every DoF of the residual subsystem belongs to the interface between the assembled system and the residual subsystem. Hopefully, not all the coupling DoFs are necessary to enforce compatibility and equilibrium. This may allow us to skip coupling DoFs and specifically rotational DoFs.The goal of the paper is indeed to establish if rotational FRFs at coupling DoFs can be neglected in substructure decoupling. To this aim, after highlighting the possibility of avoiding the use of coupling DoFs from a theoretical standpoint, a test bed coupled through flexural and torsional DoFs is considered. Experimental results are presented and discussed.

Indirect Inverse Substructuring Method for Multibody Product Transport System with Rigid and Flexible Coupling

The aim of this paper is to develop a new frequency response function- (FRF-) based indirect inverse substructuring method without measuring system-level FRFs in the coupling DOFs for the analysis of the dynamic characteristics of a three-substructure coupled product transport system with rigid and flexible coupling. By enforcing the dynamic equilibrium conditions at the coupling coordinates and the displacement compatibility conditions, a closed-form analytical solution to inverse substructuring analysis of multisubstructure coupled product transport system is derived based on the relationship of easy-to-monitor component-level FRFs and the system-level FRFs at the coupling coordinates. The proposed method is validated by a lumped mass-spring-damper model, and the predicted coupling dynamic stiffness is compared with the direct computation, showing exact agreement. The method developed offers an approach to predict the unknown coupling dynamic stiffness from measured FRFs purely. The suggested method may help to obtain the main controlling factors and contributions from the various structure-borne paths for product transport system.

Inverse dynamic substructuring using the direct hybrid assembly in the frequency domain

The paper deals with the identification of the dynamic behaviour of a structural subsystem, starting from the known dynamic behaviour of both the coupled system and the remaining part of the structural system (residual subsystem). This topic is also known as decoupling problem, subsystem subtraction or inverse dynamic substructuring. Whenever it is necessary to combine numerical models (e.g. FEM) and test models (e.g. FRFs), one speaks of experimental dynamic substructuring. Substructure decoupling techniques can be classified as inverse coupling or direct decoupling techniques. In inverse coupling, the equations describing the coupling problem are rearranged to isolate the unknown substructure instead of the coupled structure. On the contrary, direct decoupling consists in adding to the coupled system a fictitious subsystem that is the negative of the residual subsystem. Starting from a reduced version of the 3-field formulation (dynamic equilibrium using FRFs, compatibility and equilibrium of interface forces), a direct hybrid assembly is developed by requiring that both compatibility and equilibrium conditions are satisfied exactly, either at coupling DoFs only, or at additional internal DoFs of the residual subsystem. Equilibrium and compatibility DoFs might not be the same: this generates the so-called non-collocated approach. The technique is applied using experimental data from an assembled system made by a plate and a rigid mass.

The role of interface DoFs in decoupling of substructures based on the dual domain decomposition

The paper considers the decoupling problem, i.e. the identification of the dynamic behaviour of a structural subsystem, starting from the known dynamic behaviour of the coupled system, and from information about the remaining part of the structural system (residual subsystem). Typically, the FRF matrix of the coupled system is assumed to be known at the coupling DoFs (standard interface). To circumvent ill-conditioning around particular frequencies, some authors suggest the use of FRFs at some internal DoFs of the residual subsystem. In this paper, the decoupling problem is revisited in the general framework of frequency based substructuring. Specifically, the dual domain decomposition is used by adding a fictitious subsystem, which is the negative of the residual subsystem, to the coupled system. In this framework, the use of internal DoFs of the residual subsystem, in addition to coupling DoFs, appears quite natural (extended interface). The effects of using an extended interface are widely discussed: the main drawback is that the problem becomes singular at any frequency. However, this singularity is easily removed by using standard smart inversion techniques. The approach is tested on a discrete system describing a two-speed transmission, using simulated data polluted by noise. Results are compared with those obtained from existing approaches.