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.
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.
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.
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.
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.
The dynamic behavior of metamaterials and metastructures is often modeled using finite elements; however, these models can become quite large and therefore computationally expensive to simulate. Traditionally, large models are made smaller using any of the array of model reduction methods, such as Guyan or Craig-Bampton reduction. The regularized nature of metamaterials makes them excellent candidates for reduced-order modeling because the system is essentially comprised of a repeating pattern of unit cell components. These unit cell components can be reduced and then assembled to form a reduced-order system-level model with equivalent dynamics. The process is demonstrated using a finite element model of a 1-D axially vibrating metamaterial bar using Guyan, SEREP, and Craig-Bampton reduction methods. The process is shown to provide substantial reduction in the time needed to simulate the dynamic response of a representative metamaterial while maintaining the dynamics of the system and resonators.
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.