Annalisa Fregolent

Direct decoupling of substructures using primal and dual formulation

Abstract 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). Substructure decoupling techniques can be classified as inverse coupling techniques or direct decoupling techniques. In inverse coupling, the equations written for the coupling problem are rearranged to isolate (as unknown) one of the substructures instead of the coupled structure. Examples of inverse coupling are impedance and mobility approaches. Direct decoupling consists in adding to the coupled system a fictitious subsystem which is the negative of the residual subsystem. Starting from the 3-field formulation (dynamic balance, compatibility and equilibrium at the interface), the problem can be solved in a primal or in a dual manner. Compatibility and equilibrium can be required either at coupling DoFs only, or at additional internal DoFs of the residual subsystem. Furthermore DoFs used to enforce equilibrium might be not the same as DoFs used for compatibility: this generates the so called non collocated approach. In this paper, direct decoupling techniques are considered: primal and dual formulation are compared in combination with collocated and non collocated interface.

Are Rotational DoFs Essential in Substructure Decoupling

Substructure decoupling consists in the identification of the dynamic behavior of a structural subsystem, starting from the known dynamic behavior of both the coupled system and the remaining part of the structural system (residual subsystem). The degrees of freedom (DoFs) of the coupled system can be partitioned into internal DoFs (not belonging to the couplings) and coupling DoFs. In direct decoupling, a fictitious subsystem that is the negative of the residual subsystem is added to the coupled system, and appropriate compatibility and equilibrium conditions are enforced at interface DoFs. Compatibility and equilibrium can be required either at coupling DoFs only (standard interface), or at additional internal DoFs of the residual subsystem (extended interface), or at some coupling DoFs and/or some internal DoFs of the residual subsystem (mixed interface). Using a mixed interface, rotational coupling DoFs could be eliminated and substituted by internal translational DoFs. This would avoid difficult measurements of rotational FRFs. This possibility is verified in this paper using simulated experimental data.

Regularisation Techniques for Dynamic Model Updating using Input Residual

The updating process to correct finite element models using experimental data is generally an ill conditioned problem. The method here presented is based on the use of the input residual, that presents some important advantages over other proposed techniques. The actions required to limit the solution instability due to ill conditioning are discussed. The importance of using regularisation techniques to minimise the influence of experimental errors is demonstrated. Several procedures are analysed through simulated and experimental tests, finally showing that the most reliable results are obtained using a priori information and the singular values truncation technique based on the minimisation of the output residual.

Dynamic model updating using virtual antiresonances

This paper considers an extension of the model updating method that minimizes the antiresonance error, besides the natural frequency error. By defining virtual antiresonances, this extension allows the use of previously identified modal data. Virtual antiresonances can be evaluated from a truncated modal expansion, and do not correspond to any physical system. The method is applied to the Finite Element model updating of the GARTEUR benchmark, used within an European project on updating. Results are compared with those previously obtained by estimating actual antiresonances after computing low and high frequency residuals, and with results obtained by using the correlation (MAC) between identified and analytical mode shapes.

Direct Hybrid Formulation for Substructure Decoupling

The paper considers the decoupling problem or subsystem subtraction, i.e. 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). Often it is necessary to combine numerical models (e.g. FEM) and test models (e.g. FRFs). In such cases, 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. Direct decoupling consists in adding to the coupled system a fictitious subsystem that is the negative of the residual subsystem. In this paper, starting from the 3-field formulation (dynamic balance, interface compatibility and equilibrium), a direct hybrid approach 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 simulated data from a discrete system.

Ignoring Rotational DoFs in Decoupling Structures Connected Through Flexotorsional Joints

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). The degrees of freedom (DoFs) of the coupled system can be partitioned into internal DoFs (not belonging to the couplings) and coupling DoFs. In direct decoupling, a fictitious subsystem that is the negative of the residual subsystem is added to the coupled system, and appropriate compatibility and equilibrium conditions are enforced at interface DoFs. Compatibility and equilibrium can be required either at coupling DoFs only (standard interface), or at additional internal DoFs of the residual subsystem (extended interface), or at some coupling DoFs and some internal DoFs of the residual subsystem (mixed interface). In this paper, a test bench is considered made by a cantilever column with two staggered short arms coupled to a horizontal beam. This involves both flexural and torsional DoFs, on which rotational FRFs are quite difficult to measure. Using a mixed interface, rotational DoFs are neglected and substituted by internal translational DoFs. Experimental results are presented and discussed.

Predicting the Dynamics of Flexible Space Payloads Under Different Boundary Conditions Through Substructure Decoupling

Flexible space payloads, such as solar panels or array antennas for space applications, can be attached to the body of the satellite using different types of joints. To predict the dynamic behaviour of such structures under different boundary conditions, it is convenient to start from their dynamic behaviour in free-free conditions. In fact, the effect of different boundary conditions, such as additional constraints or appended structures, can be taken into account starting from the frequency response functions in free-free conditions. In this situation, they would exhibit 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 flexible space payloads exhibit the first flexible modes at very low frequencies, the two sets of modes become coupled and the low frequency dynamics of the free-free structure cannot be estimated directly from measurements. To overcome this problem, substructure decoupling can be used, that allows to identify the dynamics of a substructure (i.e. the free-free panel) after measuring the FRFs on the complete structure (i.e. the panel with 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 an FRF condensation procedure. The procedure is tested on a reduced scale model of a space solar panel.

Optimal Replacement of Coupling DoFs in Substructure Decoupling

Substructure decoupling consists in the identification of a dynamic model of a structural subsystem, starting from an experimental dynamic model (e.g. FRFs) of the assembled system and from a dynamic model of a known portion of it (the so-called residual subsystem). The degrees of freedom (DoFs) of the assembled system are partitioned into internal DoFs (not belonging to the couplings) and coupling DoFs. To achieve decoupling, a negative structure opposite to the residual subsystem is added to the assembled system, and compatibility and equilibrium conditions are enforced at interface DoFs. Interface DoFs can include coupling DoFs only (standard interface), additional internal DoFs of the residual subsystem (extended interface), subsets of coupling DoFs and internal DoFs (mixed interface), or a subset of internal DoFs only (pseudo interface). As shown in previous papers, the use of a mixed interface allows to replace some coupling DoFs (e.g. rotational DoFs) with a subset of internal DoFs. Furthermore, qualitative criteria for an appropriate selection of the internal DoFs used to replace unwanted coupling DoFs are stated. In this paper, a procedure to optimally replace coupling DoFs with internal DoFs is developed, using either the Frequency Response Function (FRF) or the transmissibility between internal and coupling DoFs. The procedure is tested on an assembled structure made by a cantilever column with two staggered short arms (residual substructure) coupled to a horizontal beam (unknown substructure).

Selection of Interface DoFs in Hub-Blade(s) Coupling of Ampair Wind Turbine Test Bed

Substructure coupling is an important tool in several applications of modal analysis. It is particularly relevant in virtual prototyping of complex systems and responds to actual industrial needs, especially in an experimental context. Furthermore, the reverse problem, the decoupling of a substructure from an assembled system, arises when a substructure cannot be tested separately but only when coupled to neighboring substructures, a situation often encountered in practice. In this paper, the dynamic behavior of the Ampair test bed wind turbine rotor, made by three blades – each one bolted to the hub at three points – is analyzed. The aim is both to identify the dynamic behavior of the rotor starting from the frequency response functions (FRFs) of blades and hub, and to select a reduced set of relevant DoFs to represent the interface between blades and hub. FRFs to be used in the coupling procedure are obtained starting from FE model of each substructure, by using a super-element based computational approach. The decoupling problem, with the aim of identifying the dynamic behavior of each blade from the FRFs of the assembled rotor and of the hub, is also considered.

Experimental Dynamic Substructuring of the Ampair Wind Turbine Test Bed

In a recent paper, the authors discussed the selection of a reduced set of interface DoFs in order to describe the coupling between the blades and the hub of the Ampair test bed wind turbine rotor. The study was conducted using simulated FRFs obtained from Finite Element model of the blades and the hub, but in view of using experimental FRFs. In this paper, test data measured on the turbine by the UW-Madison participants in the IMAC Focus Group on Experimental Dynamic Substructuring, and posted on the Wiki page of the group, are used for dynamic substructuring of the wind turbine test bed.