Many engineering signals can be modeled as stationary random processes. The distributions of these signals are defined by covariance functions in the time domain and power spectral densities (PSDs) in the frequency domain. A common problem is to synthesize time series from a random model with desired autospectra, coherence, and phase across multiple signals. For a finite number of randomly sampled signal partitions, the calculated sample PSD does not generally match the distribution PSD. We describe a method for synthesizing waveforms with exactly prescribed auto-and cross-spectral densities. Furthermore, we prove that the random variable induced by the waveform generator is the closest in distribution to the original distribution among all random variables satisfying the PSD constraint. Numerical studies using a Gaussian model demonstrate convergence in practice, and the measured kurtosis and skewness of synthesized waveforms are similar to those obtained by Gaussian sampling. The synthesis method also improves control accuracy in a random vibration control simulation.

In random vibration testing with multiple control channels, existing control laws require specification of a complete spectral density matrix at each control frequency. Spectral density matrices include autospectral densities on the diagonal and cross-spectral densities on the off-diagonal. In practice, the off-diagonal terms are often unknown, and recent vibration testing research has focused on fixing the diagonal and specifying the off-diagonal to minimize the required control energy, subject to a constraint that the target matrix is positive semidefinite. This paper shows that, even with a fixed diagonal, off-diagonal terms strongly affect control residuals. This overlooked effect occurs in both square and rectangular systems. By jointly considering input energy and control residuals, open-loop inputs are derived directly from the diagonal without specifying the off-diagonal terms. Vibration targets that can be used in closed-loop control are then derived using the optimal inputs, with positive semidefinite constraints applied during the derivation. The result is a set of Pareto-optimal control solutions. For each solution in the set, any other possible solution produces greater control error, greater input energy, or both. A balanced solution is selected automatically, though others can be chosen based on test needs. Simulations and experiments show that the proposed method outperforms state-of-the-art energy-minimizing approaches, achieving significant reductions in both control error and input energy.

This paper studies Tikhonov regularization (ridge regression) parameter selection for problems in vibrations and acoustics. The selection method is based on a popular Bayesian method, but it incorporates measurements of sensor noise. The regularization parameter is closely related to the ratio of system input energy to noise energy, so noise measurements inform the inference procedure and improve parameter identification. In cases where standard Bayesian regularization identifies zero as the optimal regularization parameter, noise measurements guarantee a unique nonzero optimum. Sufficient theoretical criteria are developed for this guarantee. The method is verified in even-determined and under-determined configurations in an acoustic source localization simulation and a vibration load identification experiment. It is shown to yield significant improvements over existing empirical Bayesian regularization. Improvements are larger in the even-determined case and smaller in the under-determined case, wherein the inverse solution is less sensitive to the regularization parameter.

Multi-input, multi-output (MIMO) testing is used in component qualification to reproduce operational responses in the laboratory. It is often preferred to single-input and base-shake testing because of the potential for equivalent or better tests using smaller actuators and shorter test suites. Given a target response, two key steps in MIMO test design are selecting actuator locations and solving for input loads. Actuator locations are often manually selected using expert judgment. If an automatic method is used, locations are usually determined by simulating the vibration control problem and minimizing a combination of the input energy and control residuals. To select a configuration, the relative importance of input energy and residuals must be specified. Specifying relative weights is, in general, a manual and subjective process. This paper develops an objective function that compares actuator configurations based on control accuracy and required input energy without any manual parameter tuning. The objective function uses an optimally selected tradeoff parameter for each candidate configuration. To choose actuator locations using the new objective function, a pivoting algorithm for integer programming problems is developed. Starting with an initial configuration (such as the one generated by a greedy algorithm), the pivoting algorithm guarantees an objective function decrease in each iteration until convergence is reached. In a simulation featuring a structure excited by a diffuse acoustic field, electrodynamic shaker locations and regularized inputs are solved for without any analyst-specified parameters. Simulations are performed in MIMO configurations where the number of target responses is less than, equal to, and greater than the number of actuators.

The frequency response function (FRF) is an essential means by which dynamic systems are qualified. In recent years, local modeling approaches have been extensively researched and shown to significantly outperform traditional FRF estimators. However, the standard local modeling approach assumes a perfectly-known system input, which results in biased FRF estimates in the presence of input noise. This paper derives a simple adjustment that can be used to improve FRF estimation for systems subjected to random excitation with noisy input data. This improvement can be implemented with little modification to standard local modeling algorithms and with little additional computational burden. The adjustment is coupled with a model selection procedure to avoid underfitting and overfitting. The methods presented in this paper are validated on a simulation, and they are shown to reduce bias due to input noise.

Accurate measurement of frequency response functions is essential for system identification, model updating, and structural health monitoring. However, sensor noise and leakage cause variance and systematic errors in estimated FRFs. Low-noise sensors, windowing techniques, and intelligent experiment design can mitigate these effects but are often limited by practical considerations. This chapter is a guide to implementation of local modeling methods for FRF estimation, which have been extensively researched but are seldom used in practice. Theoretical background is presented, and a procedure for automatically selecting a parameterization and model order is proposed. Computational improvements are discussed that make local modeling feasible for systems with many input and output channels. The methods discussed herein are validated on a simulation example and two experimental examples: a multi-input, multi-output system with three inputs and 84 outputs and a nonlinear beam assembly. They are shown to significantly outperform the traditional H-1 and H-SVD estimators.

A typical constraint in multiple-input/multiple-output (MIMO) vibration testing is the number of available control gauges and associated MIMO-compatible specifications. This can arise due to the difficulty in obtaining field response data with high channel counts which would be used to create MIMO-compatible specifications at multiple control gauge locations. As MIMO vibration control is typically posed as an over-determined (i.e., more outputs than inputs) regression problem, having limited controls (outputs) limits the number of inputs that can be used. To allow a MIMO vibration test to be run with a sufficient number of inputs, it may be necessary to utilize fewer outputs than inputs, which switches the control to an under-determined regression problem. Under-determined regression problems have different behavior than over-determined problems, and it is not immediately clear how those differences translate to quantities of interest in MIMO vibration control. This paper aims to explore under-determined problems in the context of MIMO vibration control using examples to provide guidance to the MIMO vibration practitioner.

The importance of user-accessible multiple-input/multiple-output (MIMO) control methods has been highlighted in recent years. Several user-created control laws have been integrated into Rattlesnake, an open-source MIMO vibration controller developed at Sandia National Laboratories. Much of the effort to date has focused on stationary random vibration control. However, there are many field environments which are not well captured by stationary random vibration testing, for example shock, sine, or arbitrary waveform environments. This work details a time waveform replication technique that uses frequency domain deconvolution, including a theoretical overview and implementation details. Example usage is demonstrated using a simple structural dynamics system and complicated control waveforms at multiple degrees of freedom.

Multiple-input/multiple-output (MIMO) vibration control often relies on a least-squares solution utilizing a matrix pseudo-inverse. While this is simple and effective for many cases, it lacks flexibility in assigning preference to specific control channels or degrees of freedom (DOFs). For example, the user may have some DOFs where accuracy is very important and other DOFs where accuracy is less important. This chapter shows a method for assigning weighting to control channels in the MIMO vibration control process. These weights can be constant or frequency-dependent functions depending on the application. An algorithm is presented for automatically selecting DOF weights based on a frequency-dependent data quality metric to ensure the control solution is only using the best, linear data. An example problem is presented to demonstrate the effectiveness of the weighted solution.

Flexible structures adjacent to acoustic cavities are commonly encountered. In some instances, coupling between the structure and cavity causes non-negligible changes to the system frequency response. Here, we describe the use of modern dynamic substructuring techniques in coupling and decoupling structure-cavity systems. As input, free-interface in vacuo natural frequencies and modes of the structure are used along with an uncoupled modal representation of the cavity. In the acoustic model, the fluid-structure interface is typically modeled as pressure release, though faster frequency convergence is possible by enriching the representation with rigid-wall modes. The subsystems are assembled in modal coordinates with continuity at the interface enforced via Lagrange multipliers. This approach circumvents the calculation of coupling coefficients, and results in real-valued system natural frequencies and modes; however, the approach presents challenges when large numbers of interface degrees of freedom are present. Attempts to overcome these challenges with interface reduction techniques are described. Finally, this research led to the derivation of a formula approximating the in vacuo natural frequencies of test structures that are well coupled to adjacent acoustic cavities. The formula was successfully applied to an air-filled cylindrical test structure for which the frequency response was confounded by structure-cavity coupling.

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.

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.

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.(c) 2021 Elsevier Ltd. All rights reserved.

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.

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.