•A new filtering approach for FRF-based substructure decoupling is presented.•A modal model is used to reduce the impact of noise and mass loading.•The uncertainty observed in the original ...measurement is taken into account.•The spread on the results highlights the sensitivity or reliability of the results.•The new method allows to decrease the quality requirements for experimental data.
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.
•Vehicle subsystem dynamics can be characterized in the frequency domain by mechanical four-poles.•Four-pole coefficients are suitable to derive requirements to subsystem dynamics.•Subsystem ...identification is feasible in practice either experimentally or virtually.•The method is applicable to problems with limited knowledge about desirable subsystem and system dynamics.•The method is applicable to nonlinear systems whose dynamics depends on the input force level.
In a recent companion publication, we developed the basic theory for deriving requirements for the dynamic properties of vehicle components. These requirements correspond to targets at vehicle level, but they address stand-alone subsystems being developed simultaneously by different parties. For this purpose, the vehicle, i.e., the coupled system, is divided into subsystems such as the steering and the front axle. The substructuring method used for this was based on the four-pole theory, where frequency-dependent transfer matrices are needed. Thereby, the relevant transfer coefficients of each subsystem were assumed to be linear.
In this work, the method is extended to cope with nonlinear behavior of the subsystems. The basic idea is to characterize the nonlinear systems in the frequency domain by their mono-harmonic responses depending on the force level acting at their input. Both virtual and experimental methods can be used to identify the target-relevant four-pole model of the steering and the front axle. To consider nonlinearities in terms of amplitude-dependent behavior, each subsystem is investigated at multiple amplitude levels. As a main difference to the companion publication, the results are not limit curves, but rather limit surfaces in terms of the dynamics of each subsystem over frequency and force amplitude, which serve as envelope to subsystem design. Iterative algorithms are proposed to make the linear four-pole method still applicable to problems with this kind of nonlinearity where higher harmonics resulting from the nonlinearities can be neglected.
•Mechanical four-pole coefficients are useful to identify vehicle subsystem dynamics.•Four-pole coefficients are suitable to represent requirements to subsystem dynamics.•Subsystem identification is ...feasible in practice either experimentally or virtually.•Compared to other substructuring methods, fewer coefficients have to be identified.•A new virtual roller test rig (MBS model) for the front axle subsystem is presented.
Vehicle design with respect to steering feel and steering vibration is challenging for many reasons. One of them is that several subsystems need to be considered simultaneously, which are developed separately by different departments or external suppliers. Therefore, the requirements, which are usually imposed on the vehicle level, i.e. the coupled system, have to be reformulated on the level of subsystems. In this work, objective requirements on the steering subsystem are derived using mechanical four-poles. For this purpose, the vehicle system is divided into steering and front axle subsystems. Basic equations are derived in order to determine the relevant four-pole coefficients and to derive requirements to the subsystems by disassembling them from given vehicle system dynamics. Both virtual and experimental methods can be used to determine the relevant four-pole coefficients of the steering and the front axle during the design and verification stages. Vehicle targets are introduced, depending on vehicle speed or excitation frequency. Then, requirements in terms of necessary, sufficient and phase-exact limit values to selected subsystem dynamics are calculated. By assembling actual and permissible dynamics of the subsystems, the performance at vehicle level becomes predictable. It is shown that target mismatch can be detected already at subsystem level during the design phase, where corrective measures are still feasible. Reversely, vehicle targets are met if the subsystems fulfill their respective requirements.
The present paper is aimed at the application of a substructure methodology, based on the Frequency Response Function (FRF) simulation technique, to analyze the vibration of a stage reducer connected ...by a rigid coupling to a planetary gear system. The computation of the vibration response was achieved using the FRF-based substructuring method. First of all, the two subsystems were analyzed separately and their FRF were obtained. Then the coupled model was analyzed indirectly using the substructuring technique. A comparison between the full system response and the coupled model response using the FRF substructuring was investigated to validate the coupling method. Furthermore, a parametric study of the effect of the shaft coupling stiffness on the FRF was discussed and the effects of modal truncation and condensation methods on the FRF of subsystems were analyzed.
The frequency based substructuring method considering elastic joints (FBSM-CEJ) is reformulated according to Sherman-Morrison-Woodbury Formula (SMWF) in this paper, the derivation process of which is ...more concise and the order of the matrix that requires inversion in the corresponding derivation result is lower comparing to the existing FBSM-CEJ. Meanwhile, the reformulated FBSM-CEJ possesses more applicability and operability that can be used to directly and efficiently calculate the frequency response function (FRF) matrix of complex structure no matter the impedance matrix of the elastic joints is singular or not. Last but not least, via using none-mass spatial beam element to simulate the dynamic properties of elastic joints, the performance of the reformulated FBSM-CEJ is verified through numerical simulation. All the achievements obtained from this work will provide a theoretical basis for analysing the dynamic properties of a complex structure considering elastic joints.
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.
Dynamic substructuring (DS) is a research field that has gained a great deal of attention in both science and industry. The aim of DS techniques is to provide engineers in structural vibrations and ...sound practical solutions for analyzing the dynamic behavior of complex systems. This paper addresses the singularity problem that occurs when flexible joints are implemented as substructures into the Lagrange Multiplier Frequency-Based Substructuring (LM-FBS) coupling process. For illustration, we use rubber bushings from an automotive application. Considering the rubber isolators to exhibit hysteretic damping, we assume that only the property of the dynamic stiffness of material is given. To avoid singularity appearing in the admittance when inverting the impedance of a massless joint, we compare three different approaches to include rubber bushings in the framework of LM-FBS. One method consists in including the dynamic stiffness of material directly in the space of the interface constraints and add it to the assembled interface flexibility of the LM-FBS equation. This corresponds to a relaxation of the interface compatibility condition. In the second method, the rubber bushing is treated as a substructure by adding small masses to the equation of the joint. As a result, we obtain a nonsingular total dynamic stiffness matrix that can be included in the coupling process. The third method describes a novel extension of the LM-FBS approach, based on a solution for singular problems. If the applied forces are self-equilibrated with respect to the rigid body modes, a solution for the singular dynamic stiffness matrix exists. The methods are outlined, both mathematically and conceptually, based on a notation commonly used in LM-FBS. They facilitate the integration of connecting elements together with experimental or numerical determined system dynamics of substructures in order to predict the assembled system behavior.
The accuracy of the predicted dynamic behaviour of an assembled structure using the frequency based substructuring (FBS) method is often found to be diverged from the experimental counterparts. The ...divergence which has become the paramount concern and major issue for structural dynamicists is because of the unreliable experimental FRF data of the interfaces of substructures, arising from the limited resources of appropriate excitation points and accelerometer attachments in the vicinity of the interfaces. This paper presents an alternative scheme for FRF measurement of the experimental FRF data of substructures. In this study, an assembled structure consisting of two substructures were used, namely substructure A (Finite element model) and substructure B (Experimental model). The FE model of substructure A was constructed by using 3D elements and the FRFs were derived via the FRF synthesis method. Specially customised bolts were used to allow the attachment of accelerometers and excitation to be made at the interfaces of substructure B, and the FRFs were measured by using impact testing. Both substructures A and B were then coupled by using the FBS method and the coupled FRF was validated with the measured FRF counterparts. This work revealed that the proposed scheme with specially customized bolts has led to a significant enhancement and improvement in the FBS predicted results.
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