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.