Dynamic Substructures, Vol. 4

The Society of Experimental Mechanics’ (SEM’s) Technical Division (TD) on Dynamic Substructuring recognized a need for a simpler yet challenging benchmark structure for experimental-numerical substructuring exercises. Some years ago, representatives from several research institutes formed a group that defined several desirable properties for the new benchmark structure. The outcome is a frame structure together with different plates. Together, they can represent various structures such as automotive frames, wing-fuselage structures, and building floors. The frame is made as a one-piece structure with many 10/32 tapped holes that can be used to attach other components, sensors, or excitation devices.

Sandia National Labs has manufactured the benchmark structure’s components, an aluminum frame together with two rectangular aluminum wings. An exercise/challenge has been formulated. The components have been shipped to the ones that have shown interest in participating in the exercise. The idea of the exercise is to compare different strategies to tackle an experimental substructuring task, containing both decoupling and coupling, thereby learning from each other.

In the exercise, the participants start with an assembly built up by the frame and the thinner of the rectangular wings. That wing should then be numerically decoupled from the fuselage. To that numerical representation of the fuselage, the thicker wing should be coupled numerically. These decoupling and coupling operations render in a numerical representation of the thicker wing attached to the fuselage—a representation which output should be compared with test data stemming from the real structure counterpart.

The success of the decoupling and coupling exercises is dependent on the quality of the models of the components that are subtracted and added. Here, models of finite element representing components building up the Technical Division’s Substructuring Benchmark structure are developed. Their convergences are studied, and they are validated by test data stemming from vibration tests of the models’ hardware counterparts.

In many aerospace applications, it is impossible to measure the forces, both aerodynamic and structure-borne, applied to a structure in its service environment. Furthermore, collecting service environment data requires a modified flight structure which accommodates telemetry equipment. Due to the different mechanical impedances of the shaker table and payload, the test structure experiences different loads in the test environment which can lead to over- and undertesting of the structure. Consequently, force-controlled testing may be a more desirable testing procedure as it more accurately reflects the service environment and eliminates the adverse effects caused by the differences in mechanical impedance. As part of this procedure, a numerical model is required to derive the test forces to replace the acceleration control. Determining these test forces from the model will introduce uncertainty into the flight structure’s response profile. This paper aims to quantify the uncertainty in the determination of the test forces used in the force-controlled test and the uncertainty in the test structure’s response.

Overfitting appears to be a common problem in inverse force estimation, which leads to forces that have erroneously high amplitudes. As such, overfitting negatively impacts the predictive capability of the transfer path analysis (TPA) model, especially in component-based TPA. Sparse inverse solutions have recently been shown as a potential method for mitigating these overfitting errors when traditional methods (i.e., Tikhonov regularization and singular value truncation) fail. Overfitting is reduced via the variable selection feature of the sparse solution, where “inappropriate” inputs (which contribute to overfitting) are identified and eliminated from the inverse problem. However, sparse solutions come at the cost of apparent nonphysicality in the estimated forces (where forces are activated/deactivated on a frequency-by-frequency basis), leading to suspicions about the applicability of the methods. This paper will discuss the apparent nonphysicality inherent to sparse inverse solutions while making comparisons to how the traditional methods also modify the inverse source-estimation problem in potentially nonphysical ways. Further, the advantages and disadvantages of the different inverse methods will be discussed to help understand the applicability of sparse inverse solutions in TPA.

Accurate estimation of stiffness and damping of linear bolted joints is a challenge. Commonly employed approaches using frequency-based substructuring techniques can fail if the measurement quality is not good enough. Even a minimal noise level can significantly impact the accuracy of estimations due to matrix inversions. This study proposes a novel approach that combines an optimization framework utilizing frequency-based substructuring with the emerging concept of Dynamic Disturbance (DyDis) to enhance the robustness of linear bolted joint parameter estimation.

Hubs, bearings, and other rotor components can be connected to rotor shafts with connections that are mechanically locked, friction-based, bonded, or a combination of these. In order to create accurate, predictive models of rotor systems, the stiffness and damping or the dynamics of these connections must be known in advance.

Substructuring techniques provide methods for the identification of linear joints. Linear joint identification techniques have been presented for some engineering connections like bolted joints or rubbers.

This contribution presents a workflow to identify shaft-to-hub connection dynamics on the example of a friction-based connection via cone clamping elements. A system with two parts connected by the clamping element is designed, and frequency response functions (FRFs) are measured on the assembly and on the individual parts. Using a virtual point transformation, the dynamics are projected in a collocated connection point in 6 degrees of freedom, and quasistatic and dynamic substructuring is used to isolate the connection element. Stiffness is identified from the isolated joint. The methodology is validated by comparing resynthesized FRFs on the test structure, giving good agreement for some directions.

Complex interactions of nonlinear structures often lead to diverse and unexpected effects. Phenomena like isolated branches typically occur at moderate to high vibration amplitudes. Even in numerical simulations, they are difficult to detect, and commonly path following strategies are used to determine a bifurcation point on the isolated branch. The majority of numerical investigations assume steady state conditions, and modifications in the system or excitation parameters are carried out carefully. However, many common concepts for numerical simulations are not available in experimental dynamic testing. Even for single degree of freedom systems, laboratory experiments of nonlinear structures are rather demanding because clearly defined nonlinearities are difficult to provide and identify. In this work, a Duffing type absorber is constructed and the required nonlinear forces are provided by permanent magnets. It is coupled to a host structure, which is also modeled as single degree of freedom Duffing oscillator. However, the model of the host structure is purely virtual, and realistic testing of the absorbing effects is obtained by a real-time hybrid testing approach. Assuming proper coupling, the absorber can be tested under very realistic conditions, and the concepts known from numerical investigations can be adapted to the proposed experimental setup. It is important that the coupled systems remain in steady state and therefore adjustments in the excitation are made slowly. Once a desired configuration is reached, neighboring points can be studied by adapting either the excitation amplitude or the excitation frequency. However, the physical system must remain on stable branches; thus, only the stable part of an isolated region can be reached. The experimental results indicate a very high sensitivity with respect to changes in amplitude, forcing, and nonlinear parameters, and therefore clearly defined laboratory conditions are essential. So far, the real-time hybrid testing results agree well with theoretical predictions and confirm that stable branches of nonlinear dynamic systems can be investigated using the proposed method.

Vehicles in the upper atmosphere travel through increasingly rarefied media. As acoustic radiation is dependent upon its media, acoustic radiation losses of vibrating structures in lower density air are investigated. An aluminum Euler beam is placed in a vacuum chamber, supported by thin nylon wires at known nodal positions, according to the excited mode. These strings are laced across a large cavity in a small steel table, effectively minimizing losses due to boundary supports. An impulse hammer excites the beam into its flexural state, while a laser vibrometer measures the velocity response. Extraction of mechanical loss factors occurs as the vacuum pressure increases (atmospheric pressure decreases) in experimental increments, thus determining the dependence of acoustic radiation losses on the rarefied media. Analysis of experimental results is presented as a topic of discussion for vibrations of launch vehicles and satellites. Other inferences and inductions are also considered.

When determining critical paths for transmission of sound and vibration in assembled products, transfer path analysis (TPA) is a reliable and effective tool. TPA represents a source with a set of forces that replicate the operational responses. The indirect determination of the forces at the interface is commonly performed using an inverse procedure; however, admittance-based TPA methods are often strongly influenced by imperfect measurements. Given that the condition number of the transfer path admittance is high, this can lead to severe error amplification in the equivalent (also known as blocked) forces. In order to overcome this problem, regularization techniques such as singular value truncation or Tikhonov regularization are usually suggested. These techniques generally improve the accuracy of the determined interface forces but provide little insight into what is the actual source of errors.

In this chapter, we investigate the benefits of projecting measured displacements into various representative subspaces in the scope of admittance-based TPA methods. In particular, a comparative investigation of three established reduction bases using singular, physical, and interface deflection modes is conducted. The definition of the reduced subspace using different sets of modes assures only dynamic information, which is relevant and dominant for the measured configuration, is retained after the reduction. Hence, badly observed dynamic, commonly dominated by measurement errors, is effectively filtered out. The main point of interest of this study is the effect of projecting measured displacements to the reduced domain on the transferability of the equivalent forces. The feasibility of all approaches is supported by an experimental case study, which can guide the reader in selecting a suitable approach for their specific needs.

System equivalent model mixing (SEMM) is a substructuring-based dynamic expansion method that has recently undergone reformulation for its application in the modal domain, resulting in M-SEMM. This chapter aims to present the practical aspect of the method, beginning with a concise overview of the implementation-relevant equations. Subsequently, we introduce a substructuring laboratory testbench, for which a full-field response is obtained by laser scanning vibrometer. Different expansion case studies are examined, showcasing the capabilities of M-SEMM. Additionally, the performance of the method is compared to one of the most established expansion methods in the modal domain, system equivalent reduction/expansion process (SEREP). The results highlight the practicality and potential of M-SEMM as a novel modal expansion method.

Dynamic substructuring enables to analyze the dynamics of complex systems on a substructure level. In experimental context, a successful substructuring prediction relies on reliable and accurate measurement acquisitions, as well as proper design of experiments and description of the interface dynamics. Depending on the quality and quantity of information stored, system dynamics, frequency range of interest, and interface connection, several strategies based on directly estimated transfer functions or identified modal properties may be adopted. In this chapter, a frequency-based substructuring and a modal-based substructuring approach are compared on a coupling prediction of a continuous/flexible-like connection on the SEM dynamic substructuring round robin testbed. The focus is on measurement setup (e.g., design of experiments, etc.), experimental modeling of the interface (e.g., discretization, reduction basis, additional fixtures, etc.), and data processing (e.g., filtering, selection, reconstruction\(\ldots \)) to ensure a robust substructuring prediction given the assembly configuration and dynamics. Benefits and drawbacks of the applied strategies, as well as key assumptions and methodological differences, are highlighted. Both a numerical and an experimental application are presented.

Research focus on experimental dynamic substructuring has grown in recent years in both academic and industrial interests. Progress continues for both component mode synthesis and frequency based substructuring methods with a heightened interest around the nonlinearities that come about in interfaces often found in dynamic substructuring examples. Many examples of substructuring decouple structures at the interface where sources of nonlinear damping and stiffness may occur, and some, like the transmission simulator method, instead mass-load the interface. A clear path to incorporate these interface nonlinearities is a true challenge for the substructuring community.

In recent years, the Society of Experimental Mechanics’ (SEM’s) Technical Division on Dynamic Substructuring recognized a need for a simple yet challenging benchmark structure for experimental-analytical substructuring collaborations as compared to previous benchmark structures (Mayes, An introduction to the SEM substructures focus group test bed – The Ampair 600 Wind Turbine. In topics in experimental dynamics substructuring & wind turbine dynamics, Jacksonville, FL, 2012; Harvie and Avitable, Comparison of some wind turbine blade tests in various configurations. In Proceedings from the 30th International Modal Analysis Conference, Orlando, FL, 2012; Roettgen and Mayes, Ampair 600 wind turbine 3-bladed assembly substructuring using the transmission simulator method. In Proceedings from the XXXIII IMAC Conference, Orlando, FL, 2015). A team with members from many research institutes set out from several desirable properties and a unit-frame structure was designed as a benchmark for current collaborative efforts detailed in (Roettgen and Linderholt, Planning of a black-box benchmark structure for dynamic substructuring. In International Modal Analysis Conference 37, 2019). The benchmark structure is built up by a frame with threaded inserts that is bolted to plates of varying thickness and materials. When assembled, this structure can span a diverse application space of substructuring techniques. Previous endeavors with this benchmark have been focused on testing different methods of linear substructuring as well as collaborating in the development of different methods. The next steps in this challenge aim to direct the community at different nonlinear substructuring challenges that can be studied using the four-unit frame from the benchmark structure. This abstract highlights three potential nonlinear adaptations of the benchmark structure, where drawings and data for these options will be made available on the dynamic substructuring Wiki prior to the IMAC conference.

pyFBS is a Python package for frequency-based substructuring, transfer path analysis, and also, as new additions, multi-reference modal identification and a variety of different expansion techniques. It enables the user to use state-of-the-art dynamic substructuring methodologies in an intuitive manner. With the package also basic and application examples are provided, together with real datasets, so you can directly try out the capabilities of the pyFBS. It is not only currently being used by several undergraduate students and postgraduate researchers but also offers great potential in various industrial projects. Within this chapter, a short recap of pyFBS capabilities is presented, and plans for the future goals are set. The pyFBS team plans to share the most advanced tools and methods in the field of dynamic substructuring and transfer path analysis. The goal is to create a common working base for scientists interested in this topic.

SDynPy is an open-source Python package that was introduced to the community at IMAC XLI. To help mature the package for transfer path analysis and MIMO test simulation, several additions have been made to the inverse source estimation tools. Another goal was to make inverse methods more approachable for less experienced engineers. The main additions to the package include a more complete suite of inverse methods, time domain methods, and visualization tools to understand the effects of matrix conditioning schemes on the frequency response functions. This paper will go over the available methods and provide a brief tutorial for using the tools.

Complex aerospace systems are often more than just the sum of their parts. When tested individually, each component of the system has a certain dynamic response characterized by natural frequencies, mode shapes, and, for durability testing, failure modes. However, when assembled, this response often changes due to the added boundary conditions and interfaces. This change makes durability testing at the component level difficult because the failure modes observed may not be representative of those that will occur at the system level. Performing durability testing at the system level can be an even greater challenge due to difficulties accessing the connecting components, inaccessible proprietary design specifications, and different design timelines for the various components of the system. To overcome these challenges and perform meaningful, accurate durability testing, there is a need to simulate system-level boundary conditions during component-level testing. In this chapter, a genetic algorithm-based optimization approach is proposed to design a fixture for a component-level test that preserves the dynamics of the component determined while it is in the full assembly. It is assumed that the system-level dynamics of the component are known, but the boundary conditions and dynamics of the connecting parts are unknown.

To demonstrate the potential of this approach, the Box Assembly with Removable Component (BARC) is used to represent a system, where the top component is the component of interest. The dynamics of the top component were determined while it was installed in the full BARC using modal analysis. These natural frequencies and mode shapes were the target values used to evaluate an optimized fixture design. A finite element model of the BARC was developed, where the large box component was replaced with a “fixture,” represented by a generic design volume. The properties (density and modulus values) throughout the design volume were the parameters of the optimization problem, where the goal was to match the dynamics of the top component when attached to the optimized fixture to the target values, determined while the top component was installed in the BARC. The result of this approach produced a fixture design in which the first five natural frequencies of the top component were all within \(11\%\) of the target values, and three of the first five modal assurance criterion (MAC) values were greater than 0.8.

Prior works [i.e., Lacayo and Allen, MSSP, 2019] have used discrete four-parameter Iwan elements to capture the localized energy dissipation and loss of stiffness that is experienced near bolted interfaces. Those works have validated that approach by inserting these joint models into a structural model and then verifying that it accurately captures how the natural frequency and damping of one mode of the structure vary with vibration amplitude. Those studies were limited to only a single level of preload in the joint. If this were to be used as an empirical means of modeling structures, one would need to know how the Iwan parameters of the joints vary with preload and to verify that the method is effective in a wider range of scenarios. This work explores this issue in more detail, seeking to ascertain whether a library of Iwan elements can be identified that capture a structure with two bolts when those bolts have a range of different preloads. Measurements were acquired from the S4 beam [Singh et al., IMAC, 2018] at various preloads and with a few permutations of the preloads. A finite-element model was then created with rigid-bar spiders that reduce the joint to a single pair of nodes, and Iwan elements were inserted between those to model the slip in the joint. The model was tuned to capture each set of measurements, to understand how the Iwan elements vary with bolt preload. The results presented show how the Iwan parameters evolve as preload is increased and also how the frequency and damping versus amplitude evolve over a wide range of preload and response amplitude.