This paper develops a novel method for reconstructing the full-field response of structural dynamic systems using sparse measurements. The singular value decomposition is applied to a frequency response matrix relating the structural response to physical loads, base motion, or modal loads. The left singular vectors form a non-physical reduced basis that can be used for response reconstruction with far fewer sensors than existing methods. The contributions of the singular vectors to measured response are termed singular-vector loads (SVLs) and are used in a regularized Bayesian framework to generate full-field response estimates and confidence intervals. The reconstruction framework is applicable to the estimation of single data records and power spectral densities from multiple records. Reconstruction is successfully performed in configurations where the number of SVLs to identify is less than, equal to, and greater than the number of sensors used for reconstruction. In a simulation featuring a seismically excited shear structure, SVL reconstruction significantly outperforms modal FRF-based reconstruction and successfully estimates full-field responses with as few as two uniaxial accelerometers. SVL reconstruction is further verified in a simulation featuring an acoustically excited cylinder. Finally, response reconstruction and uncertainty quantification are performed on an experimental structure with three shaker inputs and 27 triaxial accelerometer outputs.
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 24 shaker drives. This document describes the functionality, architecture, and usage of the Rattlesnake controller to perform combined environments testing.
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.
Proceedings of ISMA 2024 International Conference on Noise and Vibration Engineering and Usd 2024 International Conference on Uncertainty in Structural Dynamics
In general, multiple-input/multiple-output (MIMO) vibration testing utilizes a response-controlled test methodology where specifications are in the form of response quantities at various locations distributed on the device under test (DUT). There are some advantages to this approach, namely that DUT response could be measured in some field environment and directly used as MIMO specifications for subsequent MIMO vibration tests on similar DUTs. However, in some cases it may be advantageous to control the MIMO vibration test at the inputs rather than the responses. One such case is free-flight environments, where the DUT is unconstrained, and all loads come from aerodynamic pressures. In this case, the force-controlled test method is much more robust to system changes such as unit-to-unit variability as compared to a response-controlled test method. This could make force-controlled MIMO test specifications more generalizable and easier to derive. This is exactly akin to transfer path analysis, where pseudo-forces are applicable in special circumstances. This paper will explore the force-controlled test concept and demonstrate it with a numerical example, comparing performance under various conditions vs. the traditional response-controlled test method.
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 H1 and HSVD estimators.
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.
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.
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.
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.
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.
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.
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.
Summary: Sensor selection approach for MIMO testing was defined and demonstrated on a model of a practical system. Laboratory response obtained using the approach matched the field response well. Information about how many control sensors, and where to locate sensors can be gleaned from the approach.
Objectives: Introduce a sensor selection approach to assist test engineers with MIMO test design. Demonstrate the capability of the approach. Approach: Define a desired response from a field model. Supply the sensor selection technique to two lab models, with different boundary conditions than the “field” model. Compare the laboratory response to the field response using sensors selected from the approach.
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.
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.
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. Finally, 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.
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 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.