Dynamic Substructures, Volume 4

Experimental substructuring by means of virtual point transformation (VPT) can be utilized to implicitly account for rotational degrees of freedom (DOFs) at the coupling boundary of a given substructure. Measured frequency response functions (FRFs) are cast onto a “virtual” node containing three translational and three rotational DOFs via a projection matrix, which is determined by geometric relations between impact and response locations on the structure. In this study, the uncertainty of power flow due to measurement errors in the projection matrix and FRFs is quantified by means of Monte-Carlo simulation. This is performed on a numerical model of two beam structures and is then compared to experimentally obtained data using an impact modal test. Upper and lower bounds on the broadband power flow are created using these data. It is shown that closely space modes lead to high variability in the calculation of power flow near resonance even for small measurement errors. Metrics for analyzing the quality of the virtual point transformation are discussed, as well. This work is beneficial to understanding how experimental errors manifest in the calculation of power flow between coupled structures.

Measurements are often made during system field tests on a component of interest and the system. But the excitation forces are unknown. These tests are early in the development phase so the component may be of low fidelity. The environmental response of the high-fidelity component is what is desired to develop appropriate component qualification specifications. In this paper, the pseudo-force method from the field of transfer path analysis (TPA) is utilized in an analytical rocket and component example. A tractable number of frequency response function (FRF) measurements are made on the rocket and low-fidelity component. The low-fidelity component on a fixture is characterized by laboratory FRFs. Then the rocket and component are subjected to a random vibration loading in a flight test, with accelerations measured on the component and its base. Presumably at a later time, when the high-fidelity component is available, it is characterized with FRFs at the field measurement locations. The pseudo-force method is utilized to predict the flight response of the high-fidelity component without an additional flight test. A simple transmission simulator (fixture) is utilized in this work to minimize experimental errors at connection degrees of freedom (DOF).

In the previous paper (van der Seijs MV, Harvie JM, Song DP, Road noise NVH: embedding suspension test benches in NVH design using Virtual Points and the TPA framework. In: Proceedings of the thirty-ninth international modal analysis conference, 2021), we presented the feasibility of using a tire noise test bench in the engineering of full-vehicle NVH. Road noise at the driver’s ear was accurately simulated with blocked forces obtained using several different methods: with the suspension on two test benches (one rigid and one compliant) and in the full vehicle. All methods produced accurate results, using virtual points in DIRAC to ensure compatibility between the various test assemblies.

In this paper, we further investigate the capabilities of the framework using interface forces instead of blocked forces. Several methods are used to calculate interface forces: a matrix inverse method, a mount stiffness method, and with interface forces derived from blocked forces. Results are evaluated for the simulation of road noise at the driver’s ear in the vehicle.

Shock describes a rapid change in loading conditions and occurs in many mechanical, aerospace, and civil engineering systems. The shock response of these systems is of critical importance in their design and must therefore be studied. While experimental investigation of shock response offers accurate results, this approach is costly and requires highly specialized and unique facilities. In contrast, numerical investigation of shock events can be an effective alternative; however, modeling the systems accurately can be challenging. In this paper, the application of Real-Time Hybrid Substructuring (RTHS) to study the system response to a shock event is proposed. RTHS is a cyber-physical testing method, combining both experimental and numerical testing. The RTHS approach is intended to fully incorporate the dynamic interaction between the structure and the excitation source and realistically capture all dynamic phenomena. In this preliminary study of an RTHS shock test, the impact of a swinging pendulum on a mass–spring–damper system is investigated. This highly dynamic event requires precise actuator control and dynamics compensation. This work makes use of a model-based feedforward compensator, namely a minimum phase inverse compensator. To reduce any remaining frequency-dependent time delay or magnitude tracking errors, this compensator is combined with a P-type Iterative Learning Controller. The interaction force profile is studied for varying eigenfrequencies and mass ratios of the impacted mass–spring–damper system. The tests are able to replicate the free vibration response of the system accurately. Despite a good learning performance of the Iterative Learning Control, there are still tracking errors in the initial impact phase. Future work will look to improve actuator control and performance.

Periodic structures are found to exhibit band gaps which are frequency bandwidths where structural vibrations are absorbed. In this paper, meta-structures are built by dynamically linking oscillators in a periodic pattern, which are referred to as dynamically linked element grade oscillators or D-LEGOs. The location of the band gaps is numerically determined for a one-dimensional D-LEGO. The unit cell for the D-LEGO structure is considered to be made up of two longitudinal bar elements of different properties. For such a structure, the frequency response functions (FRFs) of a single unit cell are used to estimate the band gaps of a periodic-lattice structure by adapting the Bloch wave theory. Alternatively, the FRF of the multi-unit cell is determined using FRF-based substructuring (FBS) approach. The band gaps resulting from these two approaches are compared and verified.

Vibration analysis of shrouded bladed disk systems often becomes expensive due to friction nonlinearities and randomness stemming from mistuning phenomena. This implies a great demand for a highly efficient model order reduction technique to not only reduce the computational effort but, more importantly, provide reliable displacement predictions on certain degrees of freedom (shrouds). The latter becomes more critical in bladed disks with shroud contacts, since the promising results from contact models are limited by the accuracy of displacements predicted by reduced-order models for shroud degrees of freedom. In this study, some notable reduction of order methodologies based on substructuring, namely, fixed interface (Craig-Bampton), free interface (Rubin), and dual Craig-Bampton, and the mixed interface method, which is a combination of free and fixed interface methods, are investigated. The center of attention in this work is the modal contributions of components to the final result and the influence of modal characteristics of substructures on the efficiency of a particular reduction technique. To this end, the methods are examined by a different number of retained modes. The effect of adding up more vibration modes to the reduction basis on the accuracy and computational cost for each reduction technique is compared for predefined error tolerance. It is concluded that the physical characteristics of the blade and disk components significantly affect the forced response of the bladed disk system. Consequently, it can be capitalized on to find a more effective reduction technique for the specific geometry of shrouded blisks to address high computational cost and accurate forced response required in specific areas.

In this study, the dynamic substructuring of a benchtop three-storey building is realised where the two top storeys of the building are in real-time interaction with a simulated storey at the bottom of the structure. The challenging nature of this task is demonstrated by calculating the very limited stabilisable parameter domain if displacement-feedback control is used to connect the physical and virtual parts of the experiment. Instead of the direct feedback method, an iterative technique is applied to find the appropriate voltage input for the actuator that results in synchronous motion between the physical and virtual substructures and is free from the feedback-related instabilities.

The dynamic analysis of complex engineering systems can be performed by considering the assemblies as composed of subsystems and coupling them through substructuring techniques. A technique in the modal domain called Nonlinear Coupling Procedure (NLCP) has been recently defined to couple subsystems connected through nonlinear connections by using their Nonlinear Normal Modes (NNMs). It manages to capture the main dynamic features of the system, i.e., the backbone of each NNM, and it provides satisfactory results in terms of mode shape and resonance frequency as function of the excitation level of the assembled system, with a considerable reduction of the computational time. However the results may be inaccurate due to approximations either in the models of the subsystems or due to the considered coupling technique. Furthermore, nonlinear subsystems can be the cause of complex behaviors of the assembly and then their models need to be carefully characterized. Thus, it is necessary to evaluate the reliability of the method in terms of the accuracy of the solution. This is done by defining a reliability ratio based on energy concepts depending on the level of the excitation acting on the system. The effectiveness of the reliability ratio of nonlinear techniques is verified on the NLCP applied to a mechanical system with localized nonlinearities.

The methodology to divide complex systems into several subsystems is a common practice in the field of structural dynamics. Structural dynamic analyses can be carried out more efficiently if subsystems are analysed separately and later coupled using dynamic substructuring techniques. However a reliable experimental application of frequency based substructuring (FBS) remains a challenge as it requires highly accurate acquisition of subsystems’ frequency response functions (FRFs). Even a small error from the measurement campaign can yield erroneous coupling results. The measurement errors can be either random or systematic, with the latter often referred to as bias. Impact excitation is popular in dynamic substructuring due to the fast FRF calculation for each impact location. However, deviations in the location of excitation as a typical example of measurement bias affect the FRFs throughout the whole frequency range.

This paper proposes a novel methodology to characterize bias errors in FBS based on the small deviations in impact excitation from typical experimental measurements. The deviations are utilized to reconstruct a range of bias-affected FRFs. These are then used in the global sensitivity analysis in order to characterize how each impact location affects an arbitrary quality indicator, such as FRFs reciprocity or passivity. Therefore, the effect of bias can be evaluated directly from a single measurement campaign, without the need for a numerical model. The proposed approach is shown on a synthetic numerical example, where the advantages and limitations are outlined. Virtual point transformation is applied to obtain admittance matrix at the substructures’ interfaces. Hence, virtual point reciprocity is proposed as a criterion to quantify bias error influence. Finally, an application of frequency based substructuring on beam-like structure is depicted.

This tutorial offers an introductory overview of hybrid simulation (HS) and its basic variants, i.e., real-time (RT) and pseudo-dynamic (PsD) HS. Starting from a conceptual representation of the HS loop and the establishment of a toy example, a few application examples are provided and the distinct differences between RTHS and PsDHS are noted. Accordingly, the fundamental attributes of the numerical substructure and the control plant (i.e., the transfer system and the experimental substructure) are explained, and two fundamental challenges of the HS loop, namely the compensation of the transfer system’s dynamics and delay, are described. An adaptive, data-driven method for handling these challenges is then briefly presented.

Substructure decoupling techniques, defined in the frame of Frequency Based Substructuring, allow to identify the dynamic behaviour of a structural subsystem starting from the known dynamics of the coupled system and from information about the remaining components. The problem of joint identification can be approached in the substructuring framework by decoupling jointed substructures from the assembled system. In this case, information about the coupling DoFs of the assembled structure is necessary and this could be a problem if the interface is inaccessible for measurements. Expansion techniques can be used to obtain the dynamics on inaccessible (interface) DoFs starting from accessible (internal) DoFs. A promising technique is the System Equivalent Model Mixing (SEMM) that combines numerical and experimental models of the same component to obtain a hybrid model. This technique has been already used in an iterative coupling–decoupling procedure to identify the linear dynamic behaviour of a joint, with a Virtual Point description of the interface. In this work, a similar identification procedure is applied to the Brake Reus Beam benchmark to identify the linear dynamic behaviour of a three bolted connection at low levels of excitation. The joint is considered as a third independent substructure that accounts for the mass and stiffness properties of the three bolts, thus avoiding singularity in the decoupling process. Instead of using the Virtual Point Transformation, the interface is modelled by performing a modal condensation on remote points allowing deformation of the connecting surfaces between subcomponents. The purpose of the study is to highlight numerical and ill-conditioning problems that may arise in this kind of identification.

Accurate determination of the high-resolution dynamic response is a crucial step for the dynamic properties characterization in the development phase of a modern product. The conventional experimental procedure for the identification of the structure’s dynamic properties is an experimental modal analysis, where a high spatial resolution can be obtained by imposing a large number of response and excitation points at a structure. To avoid time-consuming experimental testing, measurements involving only a limited number of points at the structure can be expanded to the unmeasured points through model-based expansion techniques. They rely on the introduction of a numerical model, presenting the basics of the expansion. Recently, an expansion method called System Equivalent Model Mixing (SEMM) was proposed where a numerical DoF set is used to extend an experimental model consisting only of a limited number of measurement points. Using the dynamic substructuring approach, the equivalent experimental and numerical models are coupled so that the hybrid model inherits the dynamic properties of both. Although the method has been well adopted, there is still no comprehensive phenomenological analysis to determine the influence of the method’s parameters on the consistency of the hybrid model and thus on the accuracy of the expansion process.

This paper addresses the issue by evaluating the accuracy of the SEMM expansion process, focusing on the influence of the regularity of the so-called equivalent numerical model. The introduction of quasi-equivalent numerical models into SEMM is analysed here, which can differ not only with respect to the mass and stiffness properties but also in terms of the geometry and boundary conditions. The parametric study was carried out on a real component of a household appliance, and the most influential parameters in terms of accuracy of the SEMM expansion process were identified.

In structural dynamics, real-time hybrid simulation is used for physical testing of critical components. When compared to traditional testing methods, the approach is very efficient and reliable, if important surrounding effects can be included in a virtual model (simulation model) which is interacting with the physical component (physical model). The virtual model must be processed in real time, and proper coupling is essential for reliable and consistent test results. Consequently, the physical interface conditions must be satisfied, and therefore, a transfer system is generally required for component testing. In the work presented, a real-time hybrid simulation experiment is used to study the influence of a Duffing absorber on the dynamics of the fundamental mode of vibration of a nonlinear host structure. Only standard hardware components are used, e.g., electrodynamic shaker, force, displacement and acceleration sensors, as well as a real-time processing unit. The study is currently limited to periodic forcing, and consequently, only the steady-state response is studied. Since the physical absorber investigated behaves nonlinear elastic, all measurements can be collected sequentially to generate high-precision force response diagrams for different amplitudes and frequencies. So far, the experimental results agree well with theoretical predictions and confirm that nonlinear dynamic absorbers are particularly suited to mitigate vibrations of nonlinear host structures. As the properties of the nonlinear absorber are adjustable, existing design rules can be verified by the proposed experimental setup.

A substructuring-based method System Equivalent Model Mixing (SEMM) has recently been introduced as a novel expansion method. Originally it was implemented in the frequency domain and has been proven to have a great potential. The objective of this paper is to introduce M-SEMM, the modal domain formulation of system equivalent model mixing. Its basic formulation is presented, considering either physical or modal constraints between substructures. When modal constraints are applied, the derivation reveals that the resulting M-SEMM formulation is equivalent to System Equivalent Reduction/Expansion Process (SEREP), one of the most established reduction/expansion methods in the modal domain. Further, when considering physical constraints, the resulting formulation can be seen as a potentially useful novel modal expansion method.

Hybrid testing of a cantilever beam with two controlled degrees of freedom at the interface between virtual and physical structures is performed in this work. To circumvent instabilities caused by delays in the transfer system and inaccuracies caused by error in the characterization of the transfer system, an alternative open-loop method based on harmonic balance method is adopted that is suitable for steady-state motion. The periodic solution that ensures synchronous motion of the interface between the virtual and the physical structure is found via quasi-Newton iterations. Results show a strong agreement between the two sides of the interface, thus proving the capabilities of the method in dealing with continuous structures with two controlled degrees of freedom.

Experimental-analytical substructuring has been a popular field of research for several years and has seen many great advances for both frequency-based substructuring (FBS) and component mode synthesis (CMS) techniques. To examine these technical advances, a new benchmark structure has been designed through the SEM dynamic substructuring technical division to act as a benchmark study for anyone researching in the field. This work contains the first attempts at experimental dynamic substructuring using the new SEM testbed. Complete dynamic substructuring predictions will be presented along with an assessment of variability and nonlinear response in the testbed assembly. Systems will be available to check out through the authors beginning in December of 2021, and this paper intends to initiate in full the round-robin challenge.

Substructure decoupling allows identifying the unknown dynamic behavior of a subsystem starting from the dynamic behavior of the whole system and that of the residual part of the system. Recently, the coupling problem has been extended to deal with time or configuration-dependent coupling conditions. This approach is useful to numerically investigate the dynamics of a configuration-dependent system with a reduction of the computational burden. In this paper, we want to examine the feasibility of performing substructure decoupling when the coupling conditions among invariant mechanical subsystems are configuration-dependent. Typical examples of such systems could be a lifting crane or a Cartesian robot. Taking for granted that the dynamic behavior of the whole system must be known for each configuration, several further questions have to be addressed: if it is necessary to take FRF measurements on the connecting DoFs; if, for the residual system, it is necessary to consider a different set of FRF measurements for each configuration; if it is possible to take advantage of the redundancy of information provided by considering multiple configurations, i.e. multiple internal constraints. In the paper, we will try to answer these questions by starting from a well-known test bed, previously used for substructure decoupling.