Dynamic Substructures, Volume 4

Aerospace vehicles are complex systems comprised of interconnected components that must perform reliably together as an assembly. However, reliability tests are often performed at the component level, which can lead to poor results if the interconnections between components are not accurately represented during testing. In the ideal case, a system’s components would be tested in situ such that the boundary conditions of the components during testing would be the same as in operation. However, having access to the connecting components of a part can be difficult due to barriers such as inaccessible proprietary design specifications and different design timelines. Therefore, there is a need to simulate the boundary conditions provided by a connecting component without having direct access to the component. In this chapter, a process for designing a test fixture that accurately simulates the boundary conditions of a connecting component is proposed. It is assumed that the response of the connecting component is known, and the goal of the process is to design a fixture that replicates that response, without specific knowledge about the original component’s geometric dimensions or material properties. The design is determined using a genetic algorithm-based optimization process that chooses combinations of bounded design parameters until the response of the fixture matches the desired response. The optimization starts by subdividing a design volume, where each partition can be assigned a different density and Young’s modulus value. At each iteration, a different combination of parameter values, determined by a genetic algorithm, is chosen. The natural frequencies and mode shapes of the current design are determined and compared to the target values. Once a sufficient match is achieved, the design is complete.

To demonstrate the viability of the proposed approach, a simple component was designed and modeled. The component’s first five natural frequencies and mode shapes were determined and used as the basis for the objective function. The optimization algorithm was run and a component design was determined without assuming any a priori knowledge of the target component’s material properties or geometry, with the exception of its overall dimensions. The resulting component’s first five natural frequencies were within \(16\%\) of the target values and the Modal Assurance Criterion values for the first five mode shapes were all greater than \(0.79\).

The purpose of this work is to model the interface behavior of a bolted lap joint with dynamic substructuring techniques. Assembled structures are notable for their nonlinearities, energy dissipation, and uncertainties due to sliding and slapping of the substructures at the joint interface. The consequential linear and nonlinear effects make it imperative to accurately model the complex joint interface physics. A dual-assembly method is used to couple linear beam substructures and a nonlinear interface subdomain with Lagrange multipliers. Coupling methodologies that employ Lagrange multipliers allow the solvers to be tuned to the physics of each domain and subsequently increase the efficiency of the solution. Furthermore, this methodology enables the implementation of various joint models. The linear joint model employed in this work simulates the joint dynamics with springs and dampers, which are inserted between coincident nodes of the substructures. These elements inhibit the interface degrees of freedom from rigid coupling and define the contact stiffness and contact damping.

Nowadays, especially in the electric mobility field, noise and vibrations is becoming day by day a more important topic that requires accurate prediction. In the context of dynamic substructuring applied to the automotive field, a crucial aspect for determining the noise property of a vehicle is the evaluation of the transfer behavior of the rubber elements connecting the different components of a car. The goal of this research is to extract the frequency-dependent dynamic stiffness for different models of rubber bushings presenting a similar architecture and belonging to the rear drivetrain of a BMW i4 and iX. In particular, we will compare two different methods for characterizing the dynamic behavior of nonlinear elements. Firstly, for each type of bushings, a free-free measurement procedure will be considered. In comparison to literature, this approach will be adapted with a third metal bracket to account for the current bushing’s architecture. In this procedure, the dynamic properties are determined from triaxial accelerometers measurements with impact hammer excitation. The output results in terms of dynamic stiffness will be compared with dedicated measurements conducted on a Hydropulse machine inside a test bench of the BMW Group. Secondly, on the same bushings, an experiment will be performed by clamping the rubber mounts to the ground and creating fixed boundary conditions. This second approach should allow to directly extract dynamic stiffness properties, without first performing a system inversion. Since this approach is suitable both for hammer impacts or shaker excitations, both approaches will be explored and further analyzed. Both the free-free and the fixed boundary condition experiment implement the technique of virtual point transformation. Therefore, further attention will be given to error evaluations coming from such technique and how to improve the transformation in terms of accuracy and numerical stability. Future research will focus on how the extracted dynamic stiffness models can be integrated into a system-level simulation framework for electrical vehicle noise generation.

The phenomenon of multiple branches of nonlinear structures occurs due to complex nonlinear interactions. They are difficult to detect and mainly investigated by numerical simulation. Commonly, path-following strategies are applied to gradually lead the nonlinear system into the desired branch. Once steady-state conditions have been determined, phenomena like isolated branches can be reproduced in dynamic simulations. However, neither the path-following strategy nor the specification of arbitrary initial conditions is generally possible in experimental dynamic testing. Nevertheless, if the effect of a nonlinear absorber on a nonlinear host structure is studied, the concept of real-time hybrid testing allows separating absorber and host structure. In this work, the absorber is tested experimentally, whereas the host structure is simulated in real time. Assuming proper coupling, the absorber can be tested under very realistic conditions, and stable states on arbitrary branches can be obtained following a two-step approach. First, the experimental subsystem is driven very close to the expected steady-state oscillation, then the real-type hybrid testing loop is closed by coupling the absorber with the simulation model, again with proper initial conditions. If the configuration of the overall system is adjacent to a desired stable branch, the system will converge to the branch within several oscillations. Once a desired configuration is reached, neighboring points can be studied by adapting the excitation frequency or amplitude. 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.

Real-time hybrid substructuring (RTHS) is a promising approach to investigate the influence of foot prostheses on the gait pattern in versatile situations, while avoiding the necessity to model the prosthesis and its ground interaction. A numerical gait simulation is coupled in real-time to a prosthesis prototype on a test rig using sensors and actuators. However, synchronization errors make such experiments prone to instabilities. Thus, methods are required to ensure robust and stable RTHS experiments. In previous work, we developed promising methods using a one-dimensional contact RTHS experiment. In this contribution, we show our first experience with multidimensional contact RTHS in order to take the next step to enable foot prosthesis RTHS experiments. We present the design of a simplified planar foot prosthesis RTHS setup. Furthermore, we show the first results of a virtual RTHS test, where test stability is ensured through the application of normalized passivity control (NPC). From these results, we conclude that our approach is promising to extend our previously developed methods to the multidimensional case.

Real-time hybrid substructuring (RTHS) is proposed as a cyber-physical method, combining both experimental and numerical testing, to capture the system-level dynamic interaction between numerical and physical substructures. With RTHS, a structural dynamic system can be partitioned into separate experimental and numerical components or substructures and interfaced together as a cyber-physical system similar to hardware-in-the-loop (HWIL) testing. The substructures that are well understood are simulated in real time using analytical or numerical models, while the substructures of particular interest, overly complex, or nonlinear are tested experimentally using physical specimens. In an RTHS test, the experimental and numerical substructures communicate together in real time by transferring displacement and force signals through a feedback loop using controlled actuation and sensing.

In a typical RTHS configuration, a transfer system is used to impart numerically determined displacements onto the physical substructure, and force sensors are used to measure the resulting restoring forces. The measurements are feed back into the numerical model to determine the displacement response at the next integration time step. Depending on the structural system’s configuration, this traditional RTHS substructuring approach may not be desirable, and it may be required to apply forces or loads to the physical substructure through the transfer system and measure the specimen response (displacement or acceleration) for input to the numerical component. This might be the case for fluid–structure interaction problems or for physical testing of the bottom floor of a multistory building – as proposed here with use of an inertial shaker as the transfer system. In this study, an inertial shaker is utilized to transfer story shear force from the numerical substructure to the physical substructure.

Multi-axis testing has become a popular test method because it provides a more realistic simulation of a field environment when compared to traditional vibration testing. However, field data may not be available to derive the multi-axis environment. This means that methods are needed to generate “virtual field data” that can be used in place of measured field data. Transfer path analysis (TPA) has been suggested as a method to do this since it can be used to estimate the excitation forces on a legacy system and then apply these forces to a new system to generate virtual field data. This chapter will provide a review of using TPA methods to do this. It will include a brief background on TPA, discuss the benefits of using TPA to compute virtual field data, and delve into the areas for future work that could make TPA more useful in this application.

The high competition in the automotive industry has led to ever-shorter development cycles and the introduction of modular vehicle designs. To succeed in this environment, engineers need to be able to make quantitative design suggestions as early as possible. For NVH engineers, this means, for example, making an optimal choice for the suspension bushings to compromise between driving dynamics, ride comfort, noise, and durability. This is a task wherein decisions involve many stakeholders, design parameters, and targets, and these need to be made before the first physical prototype exists. Thus, early-phase insights are crucial.

In this chapter, we show how NVH engineers conquer these challenges by applying state-of-the-art methods from structural dynamics: first, the source excitation and noise propagation are separated using component TPA. Then, a model of the car suspension, including all the bushing degrees of freedom, is measured using the virtual points. The bushing stiffnesses are virtually modified using a frequency-based substructuring method called stiffness injection (SI). The bushing parameters are optimized using the genetic algorithm. Our results show how the combination of these technologies allows us to efficiently produce optimal design choices considering various driving conditions, target quantities, and design constraints. It furthermore shows how modern software design easily allows this to become an integral part of the standard vehicle development process.

In this chapter, a methodology to calculate equivalent forces by taking into account the possible time-varying dynamic behavior of the components under analysis is presented. This methodology is based on the use of the state-space realization of the in-situ component-based TPA method. To take into account possible time-varying dynamic behavior of the systems under study, a local linear parameter varying (LPV) model identification approach is used. This approach enables the computation of state-space models representative of the components at each time instant by interpolating a given set of linear time-invariant (LTI) state-space models representative of the dynamics of the components under study for fixed operating conditions. By exploiting a numerical example, it is found that when dealing with structures presenting time-varying behavior, accurate equivalent forces can be computed in time domain by using the approaches presented in this chapter. Furthermore, it is clearly demonstrated that ignoring the time dependency of the dynamic behavior of mechanical systems can lead to an important deterioration of the results.

Due to gyroscopic effects, the dynamics of rotating machinery depend on the rotational speed. In order to experimentally determine a machine’s dynamics, measurements for each rotational speed are required. A method using frequency-based substructuring to extrapolate the speed-dependent dynamics is presented for a system with gyroscopic forces. In an isolation step, the dynamics of the rotating system are decoupled from the dynamics of the nonrotating system, which yields the gyroscopic terms. In an expansion step, the isolated gyroscopic terms can be extrapolated to an arbitrary speed. This is coupled with the dynamics of the nonrotating system for the dynamics of the rotating system at an arbitrary speed. Here, the frequency response function must only be measured for the rotor at standstill and at one rotational speed. The FBS method gives good results as long as the interface is rigid enough. Also, another method that divides the dynamic stiffness into symmetric and skew-symmetric parts is presented, which only needs FRFs at one rotational speed. It gives slightly worse results than the FBS method in the current implementation.

The equations for a power flow sensitivity analysis are developed by means of describing the coupling interface between substructures with virtual points. This analysis describes how power flow between source and receiver structures changes with respect to modifications made to the real and imaginary components of the receiver impedance; termed the real and imaginary sensitivities, respectively. Previously developed power flow sensitivity equations have been applied to analytical and numerical models, but assume source and receiver interfaces have identical meshes, and both translational and rotational dynamics are accounted for. In an experimental setting, translational dynamics can easily be measured by use of accelerometers, but rotational measurements are more difficult to make as an array of accelerometers is often needed. However, the equations derived for numerical models are cast onto a domain in which physical and virtual coordinates are accounted for, where the virtual coordinates define the coupling interface with three translational and three rotational degrees of freedom (DOFs). It is shown that the properties of the real and imaginary sensitivity still hold after this coordinate transformation. Finally, retaining internal DOFs, and the need for defining a “synthetic impedance” in the derivation of the sensitivity equations, is discussed.

Some years ago, the Society for Experimental Mechanics’ (SEM) Dynamic Substructuring Technical Division (TD) recognized a need for a simpler, yet challenging benchmark structure for experimental–numerical substructuring exercises. That structure should replace the modified version of an Ampair 600 wind turbine as the common test object within the TD. 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 aluminum rectangular 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 a numerical representation of the thicker wing attached to the fuselage; a representation whose output is compared with test data stemming from the real structure counterpart.

Here, virtual points are used in the decoupling and coupling operations. Four attachment points, for example, four screws, are used. In addition, washers between the fuselage and the wings are used in the connections. The purpose is to avoid too challenging nonlinearities to start with. The component mode synthesis (CMS) technique is used.

This chapter will show the results of a study where component-based transfer path analysis was used to translate vibration environments between versions of the round-robin structure. This was done to evaluate a hybrid approach where the responses were measured experimentally, but the frequency response functions were derived analytically. This work will describe the test setup, force estimation process, response prediction (on the new system), and show comparisons between the predicted and measured responses. Observations will also be made on the applicability of this hybrid approach in more complex systems.

Resonant plate shock testing techniques have been used for mechanical shock testing at Sandia for several decades. A mechanical shock qualification test is often done by performing three separate uniaxial tests on a resonant plate to simulate one shock event. Multi-axis mechanical shock activities, in which shock specifications are simultaneously met in different directions during a single shock test event performed in the lab, are not always repeatable and greatly depend on the fixture used during testing. This chapter provides insights into various designs of a concept fixture that includes both resonant plate and angle bracket used for multi-axis shock testing from a modeling and simulation point of view based on the results of finite element modal analysis. Initial model validation and testing performed show substantial excitation of the system under test as the fundamental modes drive the response in all three directions. The response also shows that higher order modes are influencing the system, the axial and transverse response are highly coupled, and tunability is difficult to achieve. By varying the material properties, changing thicknesses, adding masses, and moving the location of the fixture on the resonant plate, the response can be changed significantly. The goal of this work is to identify the parameters that have the greatest influence on the response of the system when using the angle bracket fixture for a mechanical shock test for the intent of tunability of the system.