Substructure Decoupling Research Articles

A method for substructure decoupling of mechanical systems by using frequency response functions

A method for substructure decoupling of mechanical systems by using frequency response functions

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.

Experimental decoupling of substructures by singular vector transformation

Substructure decoupling is the process of identifying the dynamic behavior of one component by removing the dynamic influence of the second component from the assembled system. In experimental practice, several techniques have been developed to address the decoupling problem. In this context, measurements errors of random and systematic nature remain a major hindrance to a successful implementation of the methodology. For this reason, approaches such as extended interface, Virtual Point Transformation and truncated Singular Value Decomposition are commonly adopted on top of a standard interface decoupling procedure. This paper introduces the Singular Vector Transformation. The idea is to weaken the interface problem by using the Singular Value Decomposition to extract reduction spaces directly from the measured dynamics. A least square smoothing minimizes random errors and outliers, thereby improving the conditioning of the interface matrix inversion. No geometrical or analytical model is required. The reduction basis are frequency-dependent and can include flexible interface behavior, if properly controlled and observed. Further understanding and interpretation of the interface problem in frequency-based decoupling is provided. Numerical and experimental examples show the potential of the proposed technique in comparison with state-of-the-art approaches.

A covariance based framework for the propagation of correlated uncertainty in frequency based dynamic sub-structuring

Abstract Dynamic sub-structuring (DS) is the procedure by which the passive properties (i.e. frequency response functions) of an assembled structure are predicted from those of its constituent sub-structures. In this paper we are concerned with the propagation of correlated uncertainty through such a prediction. In this work a first-order covariance based propagation framework is derived based on the primal and dual formulations of the sub-structuring problem and the complex bivariate description of FRF uncertainty. The proposed framework is valid also in the case of sub-structure decoupling, since the underlying equations are of an identical form. The present paper extends previous work into a more general framework by accounting for the presence of correlated uncertainty. This is important as recent work has demonstrated that the neglect inter-FRF correlation (i.e. the correlated uncertainty associated with impact-based FRF measurements) can lead to large errors in uncertainty estimates. Efficient algorithms are introduced for implementation of the proposed framework. Results are compared against Monte-Carlo simulations and shown to be in good agreement for both correlated, uncorrelated and mixed uncertainty. These results further illustrate that the neglect of inter-FRF correlation, when physically present, can lead to large over-estimations in the uncertainty of coupled structures. This result justifies use of the proposed framework.

Use of experimental dynamic substructuring to predict the low frequency structural dynamics under different boundary conditions

Flexible structural components can be attached to the rest of the structure using different types of joints. For instance, this is the case of solar panels or array antennas for space applications that are joined to the body of the satellite. To predict the dynamic behaviour of such structures under different boundary conditions, such as additional constraints or appended structures, it is possible to start from the frequency response functions in free-free conditions. In this situation, any structure exhibits rigid body modes at zero frequency. To experimentally simulate free-free boundary conditions, flexible supports such as soft springs are typically used: with such arrangement, rigid body modes occur at low non-zero frequencies. Since a flexible structure exhibits the first flexible modes at very low frequencies, rigid body modes and flexible modes become coupled: therefore, experimental frequency response function measurements provide incorrect information about the low frequency dynamics of the free-free structure. To overcome this problem, substructure decoupling can be used, that allows us to identify the dynamics of a substructure (i.e. the free-free structure) after measuring the frequency response functions on the complete structure (i.e. the structure plus the supports) and from a dynamic model of the residual substructure (i.e. the supporting structure). Subsequently, the effect of additional boundary conditions can be predicted using a frequency response function condensation technique. The procedure is tested on a reduced scale model of a space solar panel.

Reducing the impact of measurement errors in FRF-based substructure decoupling using a modal model

Abstract As the vibro-acoustic requirements of modern products become more stringent, the need for robust identification methods increases proportionally. Sometimes the identification of a component is greatly complicated by the presence of a supporting structure that cannot be removed during testing. This is where substructure decoupling finds its main applications. However, despite some recent advances in substructure decoupling, the number of successful applications has so far been limited. The main reason for this is the poor conditioning of the problem that tends to amplify noise and other measurement errors. This paper proposes a new approach that uses a modal model to filter the experimental frequency response functions (FRFs). This can reduce the impact of noise and mass loading considerably for decoupling applications and decrease the quality requirements for experimental data. Furthermore, based on the uncertainty of the observed eigenfrequencies, an arbitrary number of consistent (all FRFs exhibit exactly the same poles) FRF matrices can be generated that are all contained within the variation of the original measurement. This way, the variation that is observed within the measurement is taken into account. The result is a distribution of decoupled FRFs of which the average can be used as the decoupled FRF set while the spread on the results highlights the sensitivity or reliability of the obtained results. After briefly reintroducing the theory of FRF-based substructure decoupling, the main problems in decoupling are summarized. Afterwards, the new methodology is presented and tested on both numerical and experimental cases.

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.

Extraction of free body frequency response functions from flexibly constrained test

The traditional way of low stiffness suspension may no longer work for very large scale structures with extremely low first flexible natural frequency. In this paper, we replace the low stiffness suspension with a group of supports. After separate Frequency Response Function tests of the whole supported structure and the support structures, the influence of the supports could be removed from a later computation. To this end, the identification method and support design are critical. The existing substructure decoupling methods are briefly reviewed and some incomplete issues in the earlier works are discussed. Then the relationship among the methods is investigated. With these studies, some improvements on the pseudo-force method are made. The proper design of supports and measurement locations are also studied. Both numerical and experimental examples are described in the paper.

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.

A family of substructure decoupling techniques based on a dual assembly approach

In recent years, the structural dynamic community showed a renewed interest in dynamic substructuring (i.e. component coupling) techniques, especially in an experimental context. In this paper the reverse problem is addressed: the decoupling (or identification) of a substructure from an assembled system. This problem arises when substructures cannot be measured separately but only when coupled to neighboring substructures, a situation regularly encountered in practice. In this work we present a so-called dual approach to substructure (dis)assembly. In a transparent and straightforward manner, this dual framework allows imposing different equilibrium and compatibility conditions during decoupling in both the physical and modal space. Five substructure decoupling techniques will be derived and/or classified in a unified way by varying these conditions. Thereafter, the decoupling techniques will be applied to two case studies: the first is an academic example, the second a practical decoupling problem using measured data.