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

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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). 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.

Introduction

The paper considers the decoupling problem, i.e. the identification of the dynamic behaviour of a structural subsystem, starting from information about the remaining part of the structural system (residual subsystem) and from the known dynamic behaviour of the complete system. A trivial application of decoupling is mass cancellation, to get rid of the effect of the accelerometer mass on FRF measurements. Another application is joint identification, which is sometimes approached using specific techniques [1].

The decoupling problem can be seen as the reverse of the substructuring problem or as a structural modification problem with negative modification [2]. Due to modal truncation problems, in experimental dynamic substructuring, the use of FRFs (frequency based substructuring) is preferred with respect to the use of modal parameters. The main algorithm for frequency based substructuring is the improved impedance coupling [3] that involves just one matrix inversion with respect to the classical impedance coupling technique that requires three inversions. A general framework for dynamic substructuring is provided in [4], [5]: an interesting formulation is the so-called dual domain decomposition that allows to retain the full set of global DoFs by ensuring equilibrium at the interface between substructures.

Reliable solutions of the decoupling problem could lead to promising developments, both in the field of diagnostics (i.e. monitoring the dynamic behaviour of a critical subsystem which cannot be removed or accessed easily) and in the field of vibration control (i.e. identifying the dynamic behaviour of a coupled subsystem that can be changed to affect the dynamic behaviour of the complete system).

In view of such promising applications, several approaches have been proposed in the literature to tackle the decoupling problem: a state space approach [6] including a sensitivity analysis showing possible ill-conditioning due to inertia ratios at the interface; a modal based approach [7] exhibiting modal truncation problems; FRF based approaches [8], [9] showing ill-conditioning troubles. Whatever be the used approach, lack of information on rotational coupling DoFs is always a problem. There are several different reasons leading to ill-conditioning: inertia ratios at interface [6], different stiffnesses at interface [8], internal resonances of the residual subsystem with fixed interface [10], [11]. In [10], two FRF based approaches are considered: an impedance based approach and a mobility based approach. The latter is equivalent to the approach presented in [11].

In this paper, an approach derived through the dual formulation, within the general framework of frequency based substructuring, is developed and discussed. Typically, the FRF matrix of the coupled system is assumed to be known at the coupling DoFs (standard interface). To circumvent ill-conditioning due to internal resonances of the residual subsystems with fixed interface, the use of FRFs at some internal DoFs of the residual subsystem is suggested [10], [11]. By assuming that FRFs are curve-fitted from experimental tests, errors due to measurement noise and to identification can be expected. Information about the residual subsystem can consist either of measured FRFs or of a physical model. Here, the second assumption is considered because it seems unlikely to be able to perform experimental tests on the residual subsystem. 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 paper is organised as follows: after a short reminder on frequency based substructuring, the decoupling problem, based on dual domain decomposition, is presented and possible options for the choice of interface DoFs are analysed. 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.

Section snippets

Reminder on dynamic substructuring in the frequency domain

Let us consider a structural system consisting of n coupled subsystems. In the frequency domain, the equation of motion of a linear time-invariant subsystem r may be written as[Z(r)(ω)]{u(r)(ω)}={f(r)(ω)}+{g(r)(ω)}where:

  • [Z(r)] is the dynamic stiffness matrix of subsystem r;

  • {u(r)} is the vector of degrees of freedom of subsystem r;

  • {f(r)} is the external force vector;

  • {g(r)} is the vector of connecting forces with other subsystems (constraint forces associated with compatibility conditions).

For

The decoupling problem based on dual domain decomposition

The coupled structural system is assumed to be made by an unknown subsystem (A) and a residual subsystem (B) joined through a number of couplings (see Fig. 1). The residual subsystem (B) can be made by one or more substructures. The degrees of freedom (DoFs) of the coupled system can be partitioned into internal DoFs (not belonging to the couplings) of subsystem A (a), internal DoFs of subsystem B (b), and coupling DoFs (c).

It is required to find the FRF of the unknown substructure A starting

Application

A relatively simple application is considered on a torsional system that represents a model of a two-speed transmission. The complete system consists of three shafts: an input shaft, a layshaft and an output shaft (see Fig. 3). The layshaft is coupled both to the input shaft and to the output shaft by helical gears. Power flows through the gear that is locked to the output shaft by the shift collar (e.g. through gears 5 and 7 in Fig. 3). Input and output shafts are assumed to be fixed at the

Conclusion

In this paper, the role of interface DoFs in decoupling of substructures based on the dual domain decomposition is discussed.

Two options for interface DoFs are considered:

  • standard interface, including only the coupling DoFs between the unknown and the residual subsystems. The problem is ill-conditioned in the neighbourhood of the natural frequencies of the residual subsystem with coupling DoFs grounded;

  • extended interface, including also some internal DoFs of the residual substructure. In this

Acknowledgement

This research is supported by MIUR grants.

References (12)

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