Dynamics of Coupled Structures, Volume 4

When a vibration test is performed, the dynamic modes of a shaker or shaker table may be excited, changing the response of the parts depending on their location on the shaker and possibly invalidating the test. However, shaker systems are difficult to model because of joints with unknown properties, unknown stiffness and damping effects of the magnetic field and because drawings of the internal components are rarely available. This work explores the use of experimental/analytical substructuring to create a test based model of a shaker. A modal test was performed and a modal model of a shaker was obtained with an adapter plate attached. This was coupled to a finite element model of the adapter plate, and the Transmission Simulator Method was used to assemble the two and to remove one copy of the adapter plate. The model thus obtained can then be used to predict the variation in the environment across the top of the shaker as different components are connected.

Effective mass models are powerful tools that allow for a convenient means to calculate the energy associated with vibration response of a structure to a base input acceleration in a particular direction. This is useful for hardware qualification activities and margin assessment. Traditionally, these models are generated from purely analytical means such as a finite element model. However, experimental methods have recently been introduced as an intriguing alternative, particularly for applications where no finite element model is available. In this work, an effective mass modal model of a cable-connector assembly is desired, and neither component has a finite element model. Moreover, there can be multiple cable-connector combinations making analytical modeling as well as explicit testing of each combination impractical. This work develops the capability to combine an experimentally derived connector effective mass model with a simplified and easily extensible analytical cable model. The experimental connector effective mass model is generated through specialized modal testing. The simplified cable model is a Timoshenko beam finite element model whose properties are empirically derived from pinned-pinned cable modal data. The modeled length of the cable is appropriately adjusted for each configuration. Finally, the cable and connector component models can be combined to form the final assembly modal effective mass model for a given translational direction. This method lends itself to developing catalogues of connector and cable data, which can then be easily combined to form any number of assembly configurations without having to explicitly test/model them.

The classical Craig-Bampton method does not take any damping effects into account for the model order reduction of damped systems. There is generally no justification to neglect damping effects. If damping significantly influences the dynamic behavior of the system, the approximation accuracy can be very poor. One procedure to handle arbitrary damped systems is to transform the second-order differential equations into twice the number of first-order differential equations resulting in state-space representation of the system. Solving the corresponding eigenvalue problem allows the damped equations for the internal degrees of freedom of the substructures to be decoupled, but complex eigenmodes and eigenvalues occur.

Hasselman and Kaplan presented a coupling procedure for damped systems that employs complex component modes. Beliveau and Soucy proposed another version that modifies the classical Craig-Bampton method to include damping by replacing the real fixed interface normal modes of the second-order system with the corresponding complex modes of the first-order system. Additionally, they suggest an adaption of the method of Hasselman and Kaplan. A report of de Kraker gives another description of the Craig-Bampton method using complex normal modes and modified static modes. The derivation of all the different Craig-Bampton substructuring methods for viscously damped systems is presented in a comprehensible consistent manner. A comparison between the different formulations will be given. The presented theory and the comparison between the methods are illustrated by an example.

The FRF Decoupling Method for Nonlinear Systems (FDM-NS), recently proposed by the authors of this paper, is a technique based on predicting the dynamic behavior of a particular substructure of a coupled nonlinear structure from the knowledge of the measured FRFs of the coupled nonlinear structure and calculated or measured FRFs of the other substructure. The uncoupled substructure can be linear or nonlinear. The method is applicable to systems where the nonlinearity can be represented as a single nonlinear element. The method has been experimentally verified for a structure having a grounded nonlinear element. In this work, the applicability of the method to a structure having an internal nonlinearity is demonstrated. The test system used in this study is composed of two cantilever beams where their free ends are connected to each other with two identical thin beams which introduce an internal nonlinearity to the coupled structure. In this test, the FRFs of the coupled nonlinear assembly are measured in a frequency range for various different constant displacement levels of the nonlinear connection element. Tip point transverse FRFs of one of the cantilever beam, which is taken as the known subsystem, are also measured. By using the decoupling method proposed the modal parameters of the unknown nonlinear subsystem are calculated as a function of the relative displacement amplitude between ends of the nonlinear connection element, from which the dynamic response of the decoupled subsystem can be calculated for any harmonic excitation. In order to demonstrate the accuracy of the method, the decoupled system is connected to a cantilever beam with a different length, and firstly, the FRFs of the coupled new system are calculated for constant amplitude harmonic forcing. Then, the calculated FRF curves are compared with those which are directly measured.

Experimentally modeling the dynamics of substructures in a large assembly is of great benefit. This comes into play, if some components are too complex to model numerically, or certain effects, e.g. friction, within these parts cannot be modeled accurately. In engineering practice, this task also arises when building a product containing multiple supplier parts, with often no numerical models provided. When the dynamics of the substructures shall be assembled, it is well known that modeling the connection between them properly is essential for obtaining good results. For example, when coupling two pieces of a beam the interface between them must not be assumed as a point with only three translational degrees of freedom. Coupling only these, corresponds to a ball joint connection, which is clearly insufficient for coupling the pieces of a beam. For accurate results one needs to consider rotational degrees of freedom as well. One way of doing so is to apply a so called virtual point transformation. It is based on the assumption that a small region around a connection point is locally rigid. Measuring with sensors distributed around this connection point allows to calculate the rigid motions of the interface (including rotations), provided that their position and orientation is known. This paper focuses on the fact that in an actual experiment the sensor positions can only be known up to a certain measurement accuracy. We will show how a wrong estimate of sensor positions deteriorates the quality of the experimental model obtained from the virtual point transformation and thus its usefulness for dynamic substructuring. However, there are some measures for checking the quality of an experimental model. For instance if the assumption of rigidity on the interface is valid, then the sensors should move with negligible flexible motion between each other. The remaining residual in the transformation should thus be minimal. Another well known property from mechanics is the reciprocity principle. When exchanging the input and output on a linear structure one should get the same transfer function. In other words, the frequency response function (FRF) matrix should be symmetric. Those quality indicators can be used to formulate a cost function for an optimization, with the unknown positions as optimization variables. We will show how wrong position estimates of inputs and outputs can be removed to a large extend by the optimization. We will also show how removing these errors improves the quality of the experimental model and thus the results of dynamic substructuring.

This paper present a real-time hybrid substructuring (RTHS) shake table test to evaluate the seismic performance of a base isolated building. Significant experimental research has been conducted on base isolators and dampers toward developing high fidelity numerical models. Shake table testing where the building superstructure is tested while the isolation layer is numerically modeled can allow for a range of isolation strategies to be examined for a single shake table experiment. Further, recent concerns in base isolation for long period, long duration earthquakes necessitate adding damping at the isolation layer which can allow higher frequency energy to be transmitted into the superstructure and can result in damage to structural and nonstructural components that can be difficult to numerically model and accurately predict. As such, physical testing of the superstructure while numerically modeling the isolation layer may be desired. The RTHS approach has been previously proposed for base isolated buildings, however, to date it has not been conducted on base isolated structure isolated at the ground level and where the isolation layer itself is numerically simulated. This configuration provides multiple challenges associated with higher physical substructure frequencies and a low numerical to physical mass ratio. This paper demonstrates a base isolated RTHS test with a scaled idealized 4-story superstructure building model placed directly onto a shake table and the isolation layer simulated in MATLAB/Simulink using a dSpace real-time controller.

Two reduction methods for dynamic analysis of structures with local nonlinearity are compared. Dual and primal formulation have the same projection basis including flexibility residual attachment modes and free interface modes, but there are significant differences in their implementation. Both methods can be applied to nonlinear forced response analysis of turbine blades with contact interfaces in shroud. In this study, the shroud contact elements are employed using the adequate description of friction and 3D tangential coupled contact forces considering the effect of normal load variation. In order to examine and compare the accuracy of the two formulations, a rod and a simplified shrouded turbine blade was considered as case studies.

This work extends recent methods to calculate dynamic substructuring predictions of a weakly nonlinear structure using nonlinear pseudo-modal models. In previous works, constitutive joint models (such as the modal Iwan element) were used to capture the nonlinearity of each subcomponent on a mode-by-mode basis. This work uses simpler polynomial stiffness and damping elements to capture nonlinear dynamics from more diverse jointed connections including large continuous interfaces. The proposed method requires that the modes of the system remain distinct and uncoupled in the amplitude range of interest. A windowed sinusoidal loading is used to excite each experimental subcomponent mode in order to identify the nonlinear pseudo-modal models. This allows for a higher modal amplitude to be achieved when fitting these models and extends the applicable amplitude range of this method. Once subcomponent modal models have been experimentally extracted for each mode, the Transmission Simulator method is implemented to assemble the subcomponent models into a nonlinear assembled prediction. Numerical integration methods are used to evaluate this prediction compared to a truth test of the nonlinear assembly.

In this paper, the general framework for dynamic substructuring is extended to time-variant interfaces among connecting substructures. Specifically, a time-variant interface due to a sliding contact is considered. The contact can be without or with friction. The problem can be tackled in time domain using primal assembly and numerical time integration, and in time-frequency domain using dual assembly, thus obtaining a Time Dependent Frequency Response Function (TD-FRF) of the assembled structure. The method is applied to lumped parameter models of substructures, and, under some assumptions, it can be extended to simple finite element models.

A method named System Equivalent Model Mixing (SEMM) is presented. SEMM allows for a mixing of two equivalent frequency based models which can be created by either numerical simulation or direct measurements. The resulting hybrid FRF model is full-rank, consists of the DoF of both input models, and contains a physically relevant weighted mix of input dynamics. It is demonstrated that with SEMM a numerical DoF-set can be used to extend an experimental model with limited measurement points; specifically, it is shown how complete interface dynamics can be obtained with just a handful of sensors. The purpose of SEMM is similar to the well-known SEREP and VIKING concepts, yet instead applies Frequency Based Substructuring (FBS) techniques to form a hybrid dynamic model.

During the last years, a lot of research focusing on appropriate interfaces between substructures has been made; the transmission simulator method has become a tool in that strive. In this paper, the end effects on assembled structures consisting of finite element substructure models representing experimental setups with different levels of transmission simulator mass loadings at their interfaces are studied. Here, components of the Society of Experimental Mechanics, SEM, substructuring focus group’s benchmark; the Ampair A600 wind turbine, constitute the structure studied. Models of an A600 blade and bracket system attached to dummy masses of different sizes are coupled to models representing an A600 hub together with two blades attached to dummy masses of different sizes after numerical subtraction of the dummy masses on each of the substructures. The results are compared to data stemming from a model of the assembled system.

Orb weaving spiders and their webs have co-evolved into a highly efficient prey capture and retention system. Spiders also use their web as a sensory extension by “listening” to vibrations transmitted across the web such as struggling prey trapped in the web, courtship from potential mates, or the advance of a predator. What information is available to orb weaving spiders from web vibrations, and how might this information be used for localization and identification of cues? To better understand the information available to the spider in its web, we created both physical and computational models of spider webs. Enlarged, artificial webs suitable for physical measurements (1.2 m in diameter) were constructed from two types of parachute cord to mimic the different silks used by spiders in web construction. Accelerometers placed around the center of the web measured the vibration response of the artificial web. We formed a model for large networks of transversely vibrating strings, such as a web, using Frequency Based dynamic Substructuring (FBS). From the FBS model, we generated frequency response functions at locations corresponding to typical foot placements of orb weaving spiders for vibration sources at various points in the web. We explored the influence of web architecture on web frequency response by altering web composition, pre-tension, and geometry. Frequency response features sensitive to stimulus location were also identified from the FBS model. Our research indicates that localization of both bearing and range for a vibration stimulus is possible in the modeled webs.

In 2010, Allen & Mayes proposed to estimate the fixed-interface modes of a structure by measuring the modes of the structure bolted to a fixture and then applying constraints to the fixture using the transmission simulator method. While the method proved useful, and has indeed been used in studies since that point, a few peculiarities were noted. First, in some cases the estimated fixed-base natural frequencies were observed to converge very slowly to the true values (in simulated experiments) as the number of constraints was increased. To formulate these constraints, prior studies used only the free-interface modes of the fixture or the measured modes of the assembly. This work extends that to consider other sets of constraints, showing improved results. Furthermore, in some prior studies it has been observed that there were errors of more than 10% in the natural frequencies even when the fixture motion was hundreds of times smaller than the motion of the structure of interest (and so it had presumably been removed). This work explores this phenomenon, seeking to use the strain energy in the fixture, to the extent that it can be estimated using a test-analysis model for the fixture, as a metric to predict frequency error. The proposed methods are explored by applying them to simulated measurements from a beam and from the NASA Space Launch System coupled to the Mobile Launcher.

The implementation of both vibration source characterisation and sub-structure coupling/decoupling procedures rely on the complete description of a coupling interface, that is, the inclusion of coupling forces in all significant degrees of freedom (DoFs). However, it is not straight-forward to establish which DoFs are required in the description. E.g. is it necessary to include moments and/or in-plane forces? This is an important question as an incomplete description will lead to an erroneous representation of the dynamics. However, there are currently no methods of quantifying the completeness of an interface description. In this paper an experimental procedure is described for the assessment of interface completeness. Based on the theoretical blocking of DoF subsets, a relation is presented that allows for the contribution of an unknown DoF to be established. Further, a coherence style criterion is proposed to estimate the completeness of a given interface description. This criterion may be used to check whether sufficient coupling DoFs have been included in both source characterisation and sub-structure coupling/decoupling procedures. Numerical and experimental examples are provided to illustrate the concept.

Dynamic substructuring allows analysts to combine component structural dynamics models into system-level models. For purely structural systems, dynamic substructuring techniques are well established. Here, one such technique is adapted to solve the coupled response of elastic structures in contact with acoustic cavities. The inputs to the process are the component natural frequencies and the free-interface structural and acoustic modes. The approach is well suited for use with analytical, finite element and/or empirical component modes. Unlike alternative acoustic-structure interaction approaches, the present technique avoids the cumbersome calculation of coupling coefficients and can result in real-valued system natural frequencies and modes. The technique is applied to a simple acoustic-structure system and the results from the model are used to gain insight into acoustic-structure interaction phenomena that have been observed in test hardware.

Dynamic substructuring allows analysts to combine component structural dynamics models into a system-level model. An analogous process can be used to subtract off a component model from the system model. Here, this decoupling approach is adapted for use with acoustic-structure systems. This has applications to model validation where structural models are correlated to test data. When acoustic cavities are present, coupling with the acoustic subsystem can confound the test response and inhibit the model validation effort. Here, the Transmission Simulator method of Component Mode Synthesis is applied to extract the structure-only response from a coupled structural-acoustic system using the acoustic cavity modes as the subtracted component. This approach is demonstrated on a simple plate-box system using component modes from analytical and numerical models.

The component mode synthesis based on the Craig-Bampton method has two strong limitations that appear when the number of the interface degrees of freedom is large. First, the reduced-order model obtained is overweighed by many unnecessary degrees of freedom. Second, the reduction step may become extremely time consuming. Several interface reduction techniques addressed successfully the former problem, while the latter remains open. In this paper we tackle this latter problem through a simple interface-reduction technique based on an a-priory choice of the interface modes. An efficient representation of the interface displacement field is achieved adopting a set of orthogonal basis functions determined by the interface geometry.

In this contribution, Frequency Based Substructuring is applied to a well-known benchmark structure with special attention to correct modelling of the interface problem. Various approaches with the Virtual Point Transformation are used to incorporate the effect of interface flexibility that is present in the clamped connections. An extension of the rigid displacement basis with flexible interface displacement modes is proposed and evaluated. Furthermore, a practical approach using a Transmission Simulator is used in order to improve the quality of the interface description.