Multiple Input Multiple Output (MIMO) vibration testing provides the capability to expose a system to a field environment in a laboratory setting, saving both time and money by mitigating the need to perform multiple and costly large-scale field tests. However, MIMO vibration test design is not straightforward oftentimes relying on engineering judgment and multiple test iterations to determine the proper selection of response Degree of Freedom (DOF) and input locations that yield a successful test. This work investigates two DOF selection techniques for MIMO vibration testing to assist with test design, an iterative algorithm introduced in previous work and an Optimal Experiment Design (OED) approach. The iterative-based approach downselects the control set by removing DOF that have the smallest impact on overall error given a target Cross Power Spectral Density matrix and laboratory Frequency Response Function (FRF) matrix. The Optimal Experiment Design (OED) approach is formulated with the laboratory FRF matrix as a convex optimization problem and solved with a gradient-based optimization algorithm that seeks a set of weighted measurement DOF that minimize a measure of model prediction uncertainty. The DOF selection approaches are used to design MIMO vibration tests using candidate finite element models and simulated target environments. The results are generalized and compared to exemplify the quality of the MIMO test using the selected DOF.
Unlike traditional base excitation vibration qualification testing, multi-axis vibration testing methods can be significantly faster and more accurate. Here, a 12-shaker multiple-input/multiple-output (MIMO) test method called intrinsic connection excitation (ICE) is developed and assessed for use on an example aerospace component. In this study, the ICE technique utilizes 12 shakers, 1 for each boundary condition attachment degree of freedom to the component, specially designed fixtures, and MIMO control to provide an accurate set of loads and boundary conditions during the test. Acceleration, force, and voltage control provide insight into the viability of this testing method. System field test and ICE test results are compared to traditional single degree of freedom specification development and testing. Results indicate the multi-shaker ICE test provided a much more accurate replication of system field test response compared with single degree of freedom testing.
While research in multiple-input/multiple-output (MIMO) random vibration testing techniques, control methods, and test design has been increasing in recent years, research into specifications for these types of tests has not kept pace. This is perhaps due to the very particular requirement for most MIMO random vibration control specifications – they must be narrowband, fully populated cross-power spectral density matrices. This requirement puts constraints on the specification derivation process and restricts the application of many of the traditional techniques used to define single-axis random vibration specifications, such as averaging or straight-lining. This requirement also restricts the applicability of MIMO testing by requiring a very specific and rich field test data set to serve as the basis for the MIMO test specification. Here, frequency-warping and channel averaging techniques are proposed to soften the requirements for MIMO specifications with the goal of expanding the applicability of MIMO random vibration testing and enabling tests to be run in the absence of the necessary field test data.
Bayesian inference is a technique that researchers have recently employed to solve inverse problems in structural dynamics and acoustics. More specifically, this technique can identify the spatial correlation of a distributed set of pressure loads generated during vibroacoustic testing. In this context, Bayesian inference augments the experimenter’s prior knowledge of the acoustic field prior to testing with vibration measurements at several locations on the test article to update these pressure correlations. One method to incorporate prior knowledge is to use a theoretical form of the correlations; however, theoretical forms only exist for a few special cases, e.g., a diffuse field or uncorrelated pressures. For more complex loading scenarios, such as those arising in a direct-field acoustic test, utilizing one of these theoretical priors may not be able to accurately reproduce the acoustic loading generated during the experiment. As such, this work leverages the pressure correlations generated from an acoustic simulation as the Bayesian prior to increase the accuracy of the inference for complex loading scenarios.
Systems subjected to dynamic loads often require monitoring of their vibrational response, but limitations on the total number and placement of the measurement sensors can hinder the data-collection process. This paper presents an indirect approach to estimate a system's full-field dynamic response, including all uninstrumented locations, using response measurements from sensors sparsely located on the system. This approach relies on Bayesian inference that utilizes a system model to estimate the full-field response and quantify the uncertainty in these estimates. By casting the estimation problem in the frequency domain, this approach utilizes the modal frequency response functions as a natural, frequency-dependent weighting scheme for the system mode shapes to perform the expansion. This frequency-dependent weighting scheme enables an accurate expansion, even with highly correlated mode shapes that may arise from spatial aliasing due to the limited number of sensors, provided these correlated modes do not have natural frequencies that are closely spaced. Furthermore, the inherent regularization mechanism that arises in this Bayesian-based procedure enables the utilization of the full set of system mode shapes for the expansion, rather than any reduced subset. This approach can produce estimates when considering a single realization of the measured responses, and with some modification, it can also produce estimates for power spectral density matrices measured from many realizations of the responses from statistically stationary random processes. A simply supported beam provides an initial numerical validation, and a cylindrical test article excited by acoustic loads in a reverberation chamber provides experimental validation.
In recent years, the Boundary Condition Challenge has gained acceptance in the structural dynamics community. In this challenge problem, an example dynamic system known as the Box and Removable Component, or BARC, is subjected to a single point shock load. The BARC consists of a Removable Component mounted to a box-shaped fixture. The challenge problem specifies a shock load applied to the Box fixture. Here, an additional environment for the challenge problem is proposed. This new environment will be stationary random vibration due to multiple exciters on the Box fixture. In this work, the response of the BARC to this environment will be explored with mod/sim. The goal is to provide the structural dynamics community with all the pieces necessary to examine the various facets of the challenge problem in the context of random vibration and enable researchers to more easily explore problems in random vibration. A data set including input and output degrees of freedom, model modes, model frequency response functions, and input and output time histories and power spectral densities will be created and placed on the challenge problem shared site for others to download and use.
In general, existing methods to develop an effective input for multiple-input/multiple-output (MIMO) control do not offer flexibility to account for limitations in experimental test setups or tailor the control to specific test objectives. The work presented in this paper introduces a method to leverage global optimization approaches to define a MIMO control input to match a data set representing field data. This contrasts with traditional MIMO input estimation methods which rely on direct inverse methods. Efficacy of the iterative optimization method depends on the objective function and optimization method used as well as the definition of the format of the input cross-power spectral density (CPSD) matrix for the optimization routine. Various objective functions are explored in this work through sampling as well as implementation within the iterative optimization process and their impact on the resulting output CPSD. Performance of iterative optimization is assessed against the traditional, direct pseudoinverse method of obtaining the input CPSD as well as the buzz method and weighted least squares (LS). Constraints can be used within the optimization process to control the magnitude and other aspects of the input CPSD, which allows for shaker limitations to be accounted for, among other considerations. Iterative optimization can provide the best input CPSD possible for a test setup while accounting for any shortcomings the setup may have, including force and voltage constraints, which is not possible with traditional methods.
Rattlesnake is a combined-environments, multiple input/multiple output control system for dynamic excitation of structures under test. It provides capabilities to control multiple responses on the part using multiple exciters using various control strategies. Rattlesnake is written in the Python programming language to facilitate multiple input/multiple output vibration research by allowing users to prescribe custom control laws to the controller. Rattlesnake can target multiple hardware devices, or even perform synthetic control to simulate a test virtually. Rattlesnake has been used to execute control problems with up to 200 response channels and 12 drives. This document describes the functionality, architecture, and usage of the Rattlesnake controller to perform combined environments testing.
Expansion is useful for predicting response of un-instrumented locations and has traditionally been applied to structures alone. However, there are a range of hollow structures where the influence of the acoustic cavity affects the structural response, and the structural response affects the acoustic response. This structural-acoustic coupling results in a gyroscopically coupled system with complex modes. Though more complicated than modes of a structure alone, the modes of the coupled structural-acoustic system can be used as the basis vectors in an expansion process. In this work, complex modes of a model of a coupled structural-acoustic system are used to expand from a sparse set of structural and acoustic response degrees of freedom to a larger set of both structural and acoustic degrees of freedom. The expansion technique is demonstrated with a finite element model of a hollow cylinder with simulated displacement and pressure measurements, and expansion is studied for both modal and transient responses. Though more nuanced than traditional structure-only expansion problems, the displacement and pressure response of a coupled structural-acoustic system can be expanded using the coupled-system modes.
Multi-shaker testing is used to represent the response of a structure to a complex operational load in a laboratory setting. One promising method of multi-shaker testing is Impedance Matched Multi-Axis Testing (IMMAT). IMMAT targets responses at discrete measurement points to control the multiple shaker input excitations, resulting in a laboratory response representative of the expected operational response at the controlled measurement points. However, the relationship between full-field operational responses and the full-field IMMAT response has not been thoroughly explored. Poorly chosen excitation positions may match operational responses at the control points, but over or under excite uncontrolled regions of the structure. Additionally, the effectiveness of the IMMAT method on the whole test structure could depend on the type of operational excitation. Spatially distributed excitations, such as acoustic loading, may be difficult to reproduce over the whole test structure in a lab setting using the point force IMMAT excitations. This work will simulate operational and IMMAT responses of a lab-scale structure to analyze the accuracy of IMMAT at uncontrolled regions of the structure. Determination of the effect of control locations and operational locations on the IMMAT method will lead to better test design and improved predictive capabilities.
Traditional expansion techniques utilize a modal projection wherein modal response is estimated based on a generalized inverse of measurements at a sparse set of degrees of freedom. Those modal response estimates are then used to project out to a larger set of degrees of freedom, resulting in predicted responses at more points or even full- field. As with any generalized inverse problem, the results are sensitive to noise and conditioning of the inverted matrix. While much has been done to improve numerics of matrix inversion problems in the context of input estimation or source identification problems, little has been done to improve the numerics of inverse solutions in expansion problems. This work presents numerical correction or regularization techniques applied to expansion problems using both simple and complex example structures. The effects of degree of freedom selection and noise are explored. Improved expansion results are obtained using straightforward regularization techniques, meaning higher accuracy responses can be obtained at expansion degrees of freedom with no change in the sparse set of measurements.
Design of multi-shaker tests relies on locating shakers on the structure such that the desired vibration response is obtained within the shaker force, acceleration, voltage, and current requirements. While shaker electro-mechanical models can be used to relate the shaker force and acceleration to voltage and current requirements, they need to be integrated with a structural dynamics model of the device under test. This connection of a shaker to a structure is a substructuring problem, with the structure representing one component and the shaker representing a second component. Here, frequency based substructuring is used to connect a shaker electro-mechanical model to a model of device under test. This provides a straightforward methodology for predicting shaker requirements given a target vibration response in a multi-shaker test. Predictions of the coupled shaker-structure model yield the shaker force, acceleration, voltage and current requirements which can be compared with the shaker capabilities to choose optimal shaker locations.
Simple electro-mechanical models of electrodynamic shakers are useful for predicting shaker electrical requirements in vibration testing. A lumped parameter, multiple degree-of-freedom model can sufficiently capture most of the shaker electrical and mechanical features of interest. While several model parameters can be measured directly or obtained from a specifications sheet, others must be inferred from an electrical impedance measurement. Here, shaker model parameters are determined from electrical impedance measurements of a shaker driving a mass. Then, parameter sensitivity is explored to determine a model calibration procedure where model parameters are determined using manual and automated selection methods. The model predictions are then compared to test measurements. The model calibration procedure described in this work provides a simple, practical approach to developing predictive shaker electromechanical models which can then be used in test design and assessment simulations.
Expansion techniques have been used for many years to predict the response of un-instrumented locations on structures. These methods use a projection or transformation matrix to estimate the response at un-instrumented locations based on a sparse set of measurements. The transformation to un-instrumented locations can be done using modal projections or transmissibilities. Here, both expansion methods are implemented to demonstrate that expansion can be used for acoustic problems, where a sparse set of pressure measurements, say from a set of microphones in a cavity or room, are used to expand and predict the response at any location in the domain. The modal projection method is applied to a small acoustic cavity, where the number of active modes is small, and the transmissibility method is applied to a large acoustic domain, where the number of active modes is very large. In each case, expansion is shown to work well, though each case has its benefits and drawbacks. The numerical studies shown here indicate that expansion could be accurate and therefore useful for a wide range of interior acoustic problems where only sparse measurements are available, but full-field information is desired, such as field reconstruction problems, or model validation problems.
Multi-shaker vibration testing is gaining interest from structural dynamics test engineers as it can provide a much more accurate match to complicated field vibration responses than traditional single-axis shaker tests. However, the force capabilities of the small modal shakers typically used in multi-shaker vibration tests has limited the achievable response levels. To date, most multi-shaker vibration tests have been performed using a variety of standard, commercially-available control systems. While these control systems are adequate for a wide range of multiple-input/multiple-output tests, their control algorithms have not been tailored for the specific problem of multi-shaker vibration tests: efficiently coordinating the various shakers to work together to achieve a desired response. Here, a new input estimation algorithm is developed and demonstrated using simulations and actual test data. This algorithm, dubbed shape-constrained input estimation, is shown to effectively coordinate multiple shakers using a set of constraint vectors based on the deflection shapes of the test structure. This is accomplished by using the singular vector shapes of the system frequency response matrix, which allows the constraint vectors to automatically change as a function of frequency. Simulation and test results indicate a significant reduction in the input forces required to achieve a desired response. The results indicate that shape-constrained input estimation is an effective method to achieve higher response levels from limited shaker forces which will enable higher level multi-shaker vibration tests to be performed.
Design of multiple-input/multiple-output vibration experiments, such as impedance matched multi-axis testing and multi-shaker testing, rely on a force estimation calculation which is typically executed using a direct inverse approach. Force estimation can be performed multiple ways, each method providing some different tradeoff between response accuracy and input forces. Additionally, there are ways to improve the numerics of the problem with regularization techniques which can reduce errors incurred from poor conditioning of the system frequency response matrix. This paper explores several different force estimation methods and compares several regularization approaches using a simple multiple-input/multiple-output dynamic system, demonstrating the effects on the predicted inputs and responses.
Many in the structural dynamics community are currently researching a range of multiple-input/multiple-output problems and largely rely on commercially-available closed-loop controllers to execute their experiments. Generally, these commercially-available control systems are robust and prove adequate for a wide variety of testing. However, with the development of new techniques in this field, researchers will want to exercise these new techniques in laboratory tests. For example, modifying the control or input estimation method can have benefits to the accuracy of control, or provide higher response for a given input. Modification of the control methods is not typically possible in commercially-available control systems, therefore it is desirable to have some methodology available which allows researchers to synthesize input signals for multiple-input/multiple-output experiments. Here, methods for synthesizing multiply-correlated time histories based on desired cross spectral densities are demonstrated and then explored to understand effects of various parameters on the resulting signals, their statistics, and their relation to the specified cross spectral densities. This paper aims to provide researchers with a simple, step-by-step process which can be implemented to generate input signals for open-loop multiple-input/multiple-output experiments.
The ability to extrapolate response data to unmeasured locations has obvious benefits for a range of lab and field experiments. This is typically done using an expansion process utilizing some type of transformation matrix, which typically comes from mode shapes of a finite element model. While methods exist to perform expansion, it is still not commonplace, perhaps due to a lack of experience using expansion tools or a lack of understanding of the sensitivities of the problem setup on results. To assess the applicability of expansion in a variety of real-world test scenarios, it is necessary to determine the level of perturbation or error the finite element model can sustain while maintaining accuracy in the expanded results. To this end, the structure model’s boundary conditions, joint stiffness, and material properties were altered to determine the range of discrepancies allowable before the expanded results differed significantly from the measurements. The effect of improper implementations of the expansion procedure on accuracy is also explored. This study allows for better insights on prospective use cases and possible pitfalls when implementing the expansion procedure.
The Box Assembly with Removable Component (BARC) structure was developed as a challenge problem for those investigating boundary conditions and their effect on structural dynamic tests. To investigate the effects of boundary conditions on the dynamic response of the Removable Component, it was tested in three configurations, each with a different fixture and thus a different boundary condition. A “truth” configuration test with the component attached to its next-level assembly (the Box) was first performed to provide data that multi-axis tests of the component would aim to replicate. The following two tests aimed to reproduce the component responses of the first test through multi-axis testing. The first of these tests is a more “traditional” vibration test with the removable component attached to a “rigid” plate fixture. A second set of these tests replaces the fixture plate with flexible fixtures designed using topology optimization and created using additive manufacturing. These two test approaches are compared back to the truth test to determine how much improvement can be obtained in a laboratory test by using a fixture that is more representative of the compliance of the component’s assembly.
Multi-axis testing is growing in popularity in the testing community due to its ability to better match a complex three-dimensional excitation than a single-axis shaker test. However, with the ability to put a large number of shakers anywhere on the structure, the design space of such a test is enormous. This paper aims to investigate strategies for placement of shakers for a given test using a complex aerospace structure controlled to real environment data. Initially shakers are placed using engineering judgement, and this was found to perform reasonably well. To find shaker setups that improved upon engineering judgement, impact testing was performed at a large number of candidate excitation locations to generate frequency response functions that could be used to perform virtual control studies. In this way, a large number of shaker positions could be evaluated without needing to reposition the shakers each time. A brute force computation of all possible shaker setups was performed to find the set with the lowest error, but the computational cost of this approach is prohibitive for very large candidate shaker sets. Instead, an iterative approach was derived that found a suboptimal set that was nearly as good as the brute force calculation. Finally, an investigation into the number of shakers used for control was performed, which could help determine how many shakers might be necessary to perform a given test.
Acoustic-structure coupling can substantially alter the frequency response of air-filled structures. Coupling effects typically manifest as two resonance peaks at frequencies above and below the resonant frequency of the uncoupled structural system. Here, a dynamic substructuring approach is applied to a simple acoustic-structure system to expose how the system response depends on the damping in the acoustic subsystem. Parametric studies show that as acoustic damping is increased, the frequencies and amplitudes of the coupled resonances in the structural response undergo a sequence of changes. For low levels of acoustic damping, the two coupled resonances have amplitudes approximating the corresponding in vacuo resonance. As acoustic damping is increased, resonant amplitudes decrease dramatically while the frequency separation between the resonances tends to increase slightly. When acoustic damping is increased even further, the separation of the resonant frequencies decreases below their initial separation. Finally, at some critical value of acoustic damping, one of the resonances abruptly disappears, leaving just a single resonance. Counterintuitively, increasing acoustic damping beyond this point tends to increase the amplitude of the remaining resonance peak. These results have implications for analysts and experimentalists attempting to understand, mitigate, or otherwise compensate for the confounding effects of acoustic-structure coupling in fluid-filled test structures.
Finite element models are regularly used in many disciplines to predict dynamic behavior of a structure under certain loads and subject to various boundary conditions, in particular when analytical models cannot be used due to geometric complexity. One such example is a structure with an entrained fluid cavity. To assist an experimental study of the acoustoelastic effect, numerical studies of an enclosed cylinder were performed to design the test hardware. With a system that demonstrates acoustoelastic coupling, it was then desired to make changes to decouple the structure from the fluid by making changes to either the fluid or the structure. In this paper, simulation is used to apply various changes and observe the effects on the structural response to choose an effective decoupling approach for the experimental study.
Acoustoelastic coupling occurs when a hollow structure’s in-vacuo mode aligns with an acoustic mode of the internal cavity. The impact of this coupling on the total dynamic response of the structure can be quite severe depending on the similarity of the modal frequencies and shapes. Typically, acoustoelastic coupling is not a design feature, but rather an unfortunate result that must be remedied as modal tests are often used to correlate or validate finite element models of the uncoupled structure. Here, however, a test structure is intentionally designed such that multiple structural and acoustic modes are well-aligned, resulting in a coupled system that allows for an experimental investigation. Coupling in the system is first identified using a measure termed the magnification factor and the structural-acoustic interaction for a target mode is then measured. Modifications to the system demonstrate the dependency of the coupling with respect to changes in the mode shape and frequency proximity. This includes an investigation of several practical techniques used to decouple the system by altering the internal acoustic cavity, as well as the structure itself. Furthermore, acoustic absorption material effectively decoupled the structure while structural modifications, in their current form, proved unsuccessful. The most effective acoustic absorption method consisted of randomly distributing typical household paper towels in the acoustic cavity; a method that introduces negligible mass to the structural system with the additional advantages of being inexpensive and readily available.
Two novel and challenging applications of high-frequency pressure-sensitive paint were attempted for ground testing at Sandia National Labs. Blast tube testing, typically used to assess the response of a system to an incident blast wave, was the first application. The paint was tested to show feasibility for supplementing traditional pressure instrumentation in the harsh outdoor environment. The primary challenge was the background illumination from sunlight and time-varying light contamination from the associated explosion. Optimal results were obtained in pre-dawn hours when sunlight contamination was absent; additional corrections must be made for the intensity of the explosive illumination. A separate application of the paint for acoustic testing was also explored to provide the spatial distribution of loading on systems that do not contain pressure instrumentation. In that case, the challenge was the extremely low level of pressure variations that the paint must resolve (120 dB). Initial testing indicated the paint technique merits further development for a larger scale reverberant chamber test with higher loading levels near 140 dB.
A series of modal tests were performed on an acoustoelastic system to explore how changes to the air and structural components affect the acoustoelastic coupling. This work is a continuation of previous experimental and analytical efforts. Here, the test method and perturbations were much more controlled than in previous tests, resulting in more refined data. Outputs of interest here are the coupled system modes as well as the resulting frequency response for various perturbations of the coupled system. Perturbations explored in this work include mass loading the structure, changing the air damping, and changing the air boundary conditions. Results of these tests indicate that simply adding damping to the air component, using foam or other absorptive material, is not sufficient to fully decouple the system. Rather, it is preferred to employ a change to the air boundary conditions, in the form of volume inclusions or scatterers, to prevent formation of the acoustic coupled mode.
A phenomenon in which structural and internal acoustic modes couple is occasionally observed during modal testing. If the structural and acoustic modes are compatible (similar frequencies and shapes), the structural mode can split into two separate modes with the same shape but different frequencies; where one mode is expected, two are observed in the structural response. For a modal test that will inform updates to an analytical model (e.g. finite element), the test and model conditions should closely match. This implies that a system exhibiting strongly coupled structural-acoustic modes in test should have a corresponding analytical model that captures that coupling. However, developing and running a coupled structural-acoustic finite element model can be challenging and may not be necessary for the end use of the model. In this scenario, it may be advantageous to alter the test conditions to match the in-vacuo structural model by de-coupling the structural and acoustic modes. Here, acoustic absorption material was used to decouple the modes and attempt to measure the in-vacuo structural response. It was found that the split peak could be eliminated by applying sufficient acoustic absorbing material to the air cavity. However, it was also observed that the amount of acoustic absorbing material had an effect on the apparent structural damping of a second, separate mode. Analytical and numerical methods were used to demonstrate how coupled systems interact with changes to damping and mode frequency proximity while drawing parallels to the phenomena observed during modal tests.
Simulation of the response of a system to an acoustic environment is desirable in the assessment of aerospace structures in flight-like environments. In simulating a laboratory acoustic test a large challenge is modeling the as-tested acoustic field. Acoustic source inversion capabilities in Sandia’s Sierra/SD structural dynamics code have allowed for the determination of an acoustic field based on measured microphone responses—given measured pressures, source inversion optimization algorithms determine the input parameters of a set of acoustic sources defined in an acoustic finite element model. Inherently, the resulting acoustic field is dependent on the target microphone data. If there are insufficient target points, then the as-tested field may not be recreated properly. Here, the question of number of microphones is studied using synthetic data, that is, target data taken from a previous simulation which allows for comparison of the full pressure field—an important benefit not available with test data. By exploring a range of target points distributed throughout the domain, a rate of convergence to the true field can be observed. Results will be compared with the goal of developing guidelines for the number of sensors required to aid in the design of future laboratory acoustic tests to be used for model assessment.
Aero-acoustic loading has been established as the primary source of excitation for a Flight System at Sandia National Laboratories. However, flight data of this system does not exist, limiting estimations of system or component response in this environment. Therefore, an experimental acoustic simulation was performed on a heavily-instrumented Flight System, using the direct-field acoustic test (DFAT) method with a multi-input multi-output (MIMO) control system. The combination of DFAT and MIMO resulted in attaining uniform and gradient acoustic fields as high as 127 dB OASPL. This paper will discuss the design of the test, the speaker and controller configurations, and the test results of this unique test method. Additionally, an overview of the method used to apply the measured test data to the pressure-loading finite element simulations of the Flight System will be discussed as well.