Experimental Dynamic Substructuring Research Articles

On the experimental dynamic substructuring of rotating machine foundations

The dynamic behavior of a rotating machine consists not only of the response of the shaft, but it also contains information of all supporting structures and equipment interacting with the rotor. These support structures, also named foundations, can add substantial influence on the machine operational behavior, hence it becomes necessary to dynamically characterize its components. This study presents the application of substructuring techniques in assembling a supporting structure model for a rotating machine and presents a novel formulation of the mixed coordinates method, which is used to connect rotor and foundation. This foundation model is validated against the actual physical foundation and provides an efficient way to analyze and optimize the behavior of such systems. The robustness of the substructuring technique is demonstrated by comparing the responses of the global system solved with both the dynamically assembled and experimentally measured supporting structures in free-free and clamped conditions. The simulations reveal important phenomena of the complete system, including the presence of structural modes in the Campbell diagrams and in displacement responses. The technique allows flexibility in the design of numerical foundations without requiring physical mounting and measurements.

On the completeness of interface descriptions and the consistency of blocked forces obtained in situ

Blocked forces can be used to describe, independently, the operational characteristics of a vibratory source. Their use within a computational model avoids the need to represent explicitly the complex mechanisms that lead to vibratory excitation. To obtain and apply an experimental blocked force with confidence it is important that likely sources of error are known, and measures of their severity are available. In this paper we introduce the notions of completeness and consistency, and detail their role in the introduction of systematic errors in a blocked force characterisation. Their mathematical origins are described and criteria to quantify their severity are proposed; the Interface Completeness Criterion (ICC), and the Measurement Consistency Criterion (MCC). These are illustrated through numerical and experimental examples. Completeness is related to the interface description adopted in a source characterisation (i.e. the number of degrees of freedom used). The ICC represents the quality of an interface description and can be quantified from in situ measurements, i.e without having to remove the source from its assembly. Consistency is related to the underlying dynamics shared by active and passive quantities (whether measured or modelled). The issue of consistency is more general, completeness being a special case, and so a single criterion is hard to formulate. When an inconsistency arises between the blocked force of a vibration source and its corresponding free interface frequency response function matrix, the MCC provides a quantitative indication of its severity. Importantly, many of the concepts discussed apply equally in the context of experimental dynamic sub-structuring.

State-Space Dynamic Substructuring with the Transmission Simulator Method

Dynamic substructuring is a technique to simplify the analysis of complex structures. The vibrational problems of the constituent substructures are analysed and solved individually and their solutions are then assembled to form the global solution. In experimental dynamic substructuring, at least one of the constituent substructures is identified experimentally. The coupling interfaces are commonly simplified in such syntheses, which can result in poor prediction quality in many applications. The transmission simulator was introduced to address this problem. Transmission simulators are well-modelled parts attached to the interface of the substructures to be coupled. This allows for distributed interfaces and a relaxation of the coupling conditions by using the transmission simulator’s analytical modes as a basis for the coupling equations, at the cost of adding a decoupling step to the substructuring problem. In this paper, the transmission simulator method is translated to the state-space substructuring domain. The methodology is applied to the Society for Experimental Mechanics’ substructuring focus group’s Ampair A600 test bed in form of experimental-analytical substructuring. The Ampair wind turbine’s hub is used as the transmission simulator and is modelled with finite elements while the three blades, individually attached to the real hub, are experimentally identified. The three experimental blade hub systems are then coupled and two finite element hubs decoupled from the system, using the derived method. Finally, this system is compared to a directly measured hub with three blades by means of frequency response functions and modal properties.

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.

A composite experimental dynamic substructuring method based on partitioned algorithms and localized Lagrange multipliers

Abstract Successful online hybrid (numerical/physical) dynamic substructuring simulations have shown their potential in enabling realistic dynamic analysis of almost any type of non-linear structural system (e.g., an as-built/isolated viaduct, a petrochemical piping system subjected to non-stationary seismic loading, etc.). Moreover, owing to faster and more accurate testing equipment, a number of different offline experimental substructuring methods, operating both in time (e.g. the impulse-based substructuring) and frequency domains (i.e. the Lagrange multiplier frequency-based substructuring), have been employed in mechanical engineering to examine dynamic substructure coupling. Numerous studies have dealt with the above-mentioned methods and with consequent uncertainty propagation issues, either associated with experimental errors or modelling assumptions. Nonetheless, a limited number of publications have systematically cross-examined the performance of the various Experimental Dynamic Substructuring (EDS) methods and the possibility of their exploitation in a complementary way to expedite a hybrid experiment/numerical simulation. From this perspective, this paper performs a comparative uncertainty propagation analysis of three EDS algorithms for coupling physical and numerical subdomains with a dual assembly approach based on localized Lagrange multipliers. The main results and comparisons are based on a series of Monte Carlo simulations carried out on a five-DoF linear/non-linear chain-like systems that include typical aleatoric uncertainties emerging from measurement errors and excitation loads. In addition, we propose a new Composite-EDS (C-EDS) method to fuse both online and offline algorithms into a unique simulator. Capitalizing from the results of a more complex case study composed of a coupled isolated tank-piping system, we provide a feasible way to employ the C-EDS method when nonlinearities and multi-point constraints are present in the emulated system.

Predicting mobile machine tool dynamics by experimental dynamic substructuring

Predicting mobile machine tool dynamics prior to moving the machine to a new part and/or location is essential to guide first-time-right in situ machining solutions. This paper considers such a priori prediction of assembled dynamics under varying base/part/contact characteristics by applying dynamic substructuring procedures. Assembled dynamics are predicted by substructural coupling of the machine's known free-free response with the known response of any base/part measured at location. Since obtaining the machine's free-free response remains non-trivial, we instead extract the machine's dynamics using substructure decoupling procedures. Substructuring is carried out using measured frequency response functions. Methods are tested for robustness, and are experimentally validated.

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.

A novel hybrid testing approach for piping systems of industrial plants

The need for assessing dynamic response of typical industrial piping systems subjected to seismic loading motivated the authors to apply model reduction techniques to experimental dynamic substructuring. Initially, a better insight into the dynamic response of the emulated system was provided by means of the principal component analysis. The clear understanding of reduction basis requirements paved the way for the implementation of a number of model reduction techniques aimed at extending the applicability range of the hybrid testing technique beyond its traditional scope. Therefore, several hybrid simulations were performed on a typical full-scale industrial piping system endowed with a number of critical components, like elbows, Tee joints and bolted flange joints, ranging from operational to collapse limit states. Then, the favourable performance of the L-Stable Real-Time compatible time integrator and an effective delay compensation method were also checked throughout the testing campaign. Finally, several aspects of the piping performance were commented and conclusions drawn.

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.

Correcting indefinite mass matrices due to substructure uncoupling

The transmission simulator method of experimental dynamic substructuring captures the interface forces and motions through a fixture called a transmission simulator. The transmission simulator method avoids the need to measure connection point rotations, facilitates substructuring for systems with continuous connections and enriches the modal basis of the substructure model. To use this approach, one first attaches the transmission simulator to the experimental substructure and then measures the free modes of the assembly. A finite element model of the transmission simulator is then used to subtract the transmission simulator and to couple the experimental substructure to the assembly of interest. Unfortunately, in several cases the process of subtracting the transmission simulator has caused the mass matrix for the experimental substructure to become indefinite, making the substructure model difficult or impossible to use and potentially leading to erroneous results. The authors previously developed metrics that could be used to identify which modes of the experimental model led to the indefinite mass matrix. This work presents a method that utilizes those metrics with a sensitivity analysis to adjust the transmission simulator mass matrix so that the subtraction does not produce an indefinite mass matrix. A second method is presented in which the mass matrix is made positive definite by coupling additional mass to the substructure. The methods are evaluated using both numerical and experimental test cases, showing promising results.

Uncertainty quantification in experimental frequency based substructuring

Numerical conditioning of the subsystems’ interface flexibility matrix is an important issue in experimental frequency based substructuring (FBS) methods. As this matrix needs to be inverted, it is believed that ill numerical conditioning could severely magnify even small random errors in experimentally obtained subsystems, yielding erroneous FRFs of the coupled system. In this paper a method is introduced with which the uncertainty of the coupled system's FRFs can be quantified based on the uncertainties of the subsystem FRFs. This uncertainty propagation method is based on the statistical moment method and uses the measured (time) data of the subsystems as input. The method will be applied to a numerical example, where it is shown that the uncertainty on substructure FRFs can significantly influence the accuracy of the coupled system. Furthermore, the application shows that the method introduced is computationally efficient and highlights some interesting phenomena that can be encountered in experimental dynamic substructuring.

General Framework for Dynamic Substructuring: History, Review and Classification of Techniques

Four decades after the development of the first dynamic substructuring techniques, there is a necessity to classify the different methods in a general framework that outlines the relations between them. In this paper, a certain vision on substructuring methods is proposed, by recalling important historical milestones that allow us to understand substructuring as a domain decomposition concept. Thereafter, based on the dual and primal assembly of substructures, a general framework for the classification of the methods is presented. This framework allows us to indicate how the various classes of methods, proposed along the years, can be derived from a clear mathematical description of substructured problems. Current bottlenecks in experimental dynamic substructuring, as well as solutions found in literature, will also be briefly discussed.