Randall L. Mayes

Epistemic Uncertainty in the Calculation of Margins

Epistemic uncertainty, characterizing lack-of-knowledge, is often prevalent in engineering applications. However, the methods we have for analyzing and propagating epistemic uncertainty are not as nearly widely used or well-understood as methods to propagate aleatory uncertainty (e.g. inherent variability characterized by probability distributions). In this paper, we examine three methods used in propagating epistemic uncertainties: interval analysis, Dempster-Shafer evidence theory, and second-order probability. We demonstrate examples of their use on a problem in structural dynamics, specifically in the assessment of margins. In terms of new approaches, we examine the use of surrogate methods in epistemic analysis, both surrogate-based optimization in interval analysis and use of polynomial chaos expansions to provide upper and lower bounding approximations. Although there are pitfalls associated with surrogates, they can be powerful and efficient in the quantification of epistemic uncertainty.

Fluid-structure interactions in compressible cavity flows

Experiments were performed to understand the complex fluid-structure interactions that occur during aircraft internal store carriage. A cylindrical store was installed in a rectangular cavity having a length-to-depth ratio of 3.33 and a length-to-width ratio of 1. The Mach number ranged from 0.6 to 2.5 and the incoming boundary layer was turbulent. Fast-response pressure measurements provided aeroacoustic loading in the cavity, while triaxial accelerometers provided simultaneous store response. Despite occupying only 6% of the cavity volume, the store significantly altered the cavity acoustics. The store responded to the cavity flow at its natural structural frequencies, and it exhibited a directionally dependent response to cavity resonance. Specifically, cavity tones excited the store in the streamwise and wall-normal directions consistently, whereas a spanwise response was observed only occasionally. The streamwise and wall-normal responses were attributed to the longitudinal pressure waves and shear layer vortices known to occur during cavity resonance. Although the spanwise response to cavity tones was limited, broadband pressure fluctuations resulted in significant spanwise accelerations at store natural frequencies. The largest vibrations occurred when a cavity tone matched a structural natural frequency, although energy was transferred more efficiently to natural frequencies having predominantly streamwise and wall-normal motions.

An Introduction to the SEM Substructures Focus Group Test Bed – The Ampair 600 Wind Turbine

Recent advances have provided renewed interest in the topic of experimental dynamic substructures. A focus group has been formed in the Society for Experimental Mechanics to advance the experimental dynamic substructures technology and theory. Sandia National Laboratories has developed two identical test beds to enable the focus group to advance the work. The system chosen was an Ampair 600 wind turbine with a fabricated tower and base. Some modifications were made to the system to make it more linear for initial studies. The test bed will be available for viewing in the technology booth of the IMAC exposition. A description of the turbine and modifications will be presented. Initial measurements on the full system will be described. Initial modal tests have been performed on six blades at the University of Massachusetts at Lowell [1]. Geometry and mass measurements for finite element modeling have been performed by the Atomic Weapons Establishment in the UK [2]. Initial efforts to quantify each blade as an experimental substructure are ongoing. One goal is to develop an experimental dynamic substructure of the blades and hub to couple with a finite element model of the nacelle and tower to predict parked system response.

Metrics for Diagnosing Negative Mass and Stiffness when Uncoupling Experimental and Analytical Substructures

Recently, a new substructure coupling/uncoupling approach has been introduced, called Modal Constraints for Fixture and Subsystem (MCFS) [Allen, Mayes, & Bergman, Journal of Sound and Vibration, vol. 329, 2010]. This method reduces ill-conditioning by imposing constraints on substructure modal coordinates instead of the physical interface coordinates. The experimental substructure is tested in a free-free configuration, and the interface is exercised by attaching a flexible fixture. An analytical representation of the fixture is then used to subtract its effects in order to create an experimental model for the subcomponent of interest. However, it has been observed that indefinite mass and stiffness matrices can be obtained for the experimental substructure in some situations. This paper presents two simple metrics that can be used by the analyst to determine the cause of indefinite mass or stiffness matrices after substructure uncoupling. The metrics rank the experimental and fixture modes based upon their contribution to offending negative eigenvalues. Once the troublesome modes have been identified, they can be inspected and often reveal why the mass has become negative. Two examples are presented to demonstrate the metrics and to illustrate the physical phenomena that they reveal.

Tutorial on Experimental Dynamic Substructuring Using the Transmission Simulator Method

Although analytical substructures have been used successfully for years, practical experimental substructures have been limited to special cases until recently. Many of the historical practical applications were based on a single point attachment. Since substructures have to be connected, theoretically, in both translation and rotation degrees of freedom, measurement translation responses and forces around the single point attachment could be used to estimate the rotational responses and moments. For multiple attachment points, often the rotations and moments have been neglected entirely. In addition, often the effect of the joint stiffness and damping is neglected. The translation simulator approach developed by Allen and Mayes captures the interface forces and motions through a fixture called the transmission simulator, overcoming the historical difficulties. The experimental free modes of the experimental substructure mounted to the transmission simulator and the finite element model of the transmission simulator are used to couple the experimental substructure to another substructure and subtract the transmission simulator. This captures the effects of the joint stiffness and damping. The experimental method and mathematics will be explained with examples. The tutorial assumes a basic understanding of the linear multi-degree of freedom equations of motion and the modal approximation.

Coupling of a Bladed Hub to the Tower of the Ampair 600 Wind Turbine Using the Transmission Simulator Method

This paper presents an example of the transmission simulator method of experimental dynamic substructuring combining two substructures of the Substructures Focus Group’s test bed, the Ampair 600 Wind Turbine. The two substructures of interest are the hub-and-blade assembly and the tower assembly that remains after the hub is removed. The hub-and-blade substructure was developed from elastic modes of a free-free test of the hub and blades, and rigid body modes were constructed from measured mass properties. Elastic and rigid body modes were extracted from experimental data for the tower substructure. A bladeless hub was attached to the tower to serve as the transmission simulator for this substructure. Modes up to the second bending mode of the blades and tower were extracted. Substructuring calculations were then performed using the transmission simulator method, and a model of the full test bed was derived. The combined model was compared to truth data from a test on the full turbine.

Coupling Experimental and Analytical Substructures with a Continuous Connection Using the Transmission Simulator Method

The transmission simulator method of experimental dynamic substructuring has the capability to couple substructures with continuous connections. A hardware example with continuous connections is presented in which the method is used to couple an experimental substructure with a finite element substructure to predict full system response. The predicted response is compared with frequency response functions measured on the full system hardware. The experimental substructure captures the motion of a component packed in foam. This is coupled to a finite element model of a cylindrical metal case which contains the foam and is attached through a flange to a plate and beam structure.

A Modal Model to Simulate Typical Structural Dynamic Nonlinearity

Some initial investigations have been published which simulate nonlinear response with almost traditional modal models: instead of connecting the modal mass to ground through the traditional spring and damper, a nonlinear Iwan element was added. This assumes that the mode shapes do not change with amplitude and there are no interactions between modal degrees of freedom. This work expands on these previous studies. An impact experiment is performed on a structure which exhibits typical structural dynamic nonlinear response, i.e. weak frequency dependence and strong damping dependence on the amplitude of vibration. Use of low level modal test results in combination with high level impacts are processed using various combinations of modal filtering, the Hilbert Transform and band-pass filtering to develop response data that are then fit with various nonlinear elements to create a nonlinear pseudo-modal model. Simulations of forced response are compared with high level experimental data for various nonlinear element assumptions.

Eliminating Indefinite Mass Matrices with the Transmission Simulator Method of Substructuring

The transmission simulator method of experimental dynamic substructuring captures the interface forces and motions through a fixture called a transmission simulator. The transmission simulator method avoids the need to measure connection point rotations and enriches the modal basis of the substructure model. The free modes of the experimental substructure mounted to the transmission simulator are measured. The finite element model of the transmission simulator is used to couple the experimental substructure to another substructure and to subtract the transmission simulator. However, in several cases the process of subtracting the transmission simulator has introduced an indefinite mass matrix for the experimental substructure. The authors previously developed metrics that could be used to identify which modes of the experimental model led to the indefinite mass matrix. A method is developed that utilizes those metrics with a sensitivity analysis to adjust the transmission simulator mass matrix so that the subtraction does not produce an indefinite mass matrix. A second method produces a positive definite mass matrix by adding a small amount of mass to the indefinite mass matrix. Both analytical and experimental examples are described.

Simultaneous Vibration and Acoustic Measurements of a Store in Compressible Open Cavity Flow.

To understand the complex fluid-structure interactions that occur during internal store carriage within a cavity, an experimental program has been developed to simultaneously measure the acoustic loading and store vibrations. A cylindrical store was installed in a cavity having a length-to-depth ratio of 3.33 and a length-to-width ratio of 1. Experiments were conducted at a freestream Mach number of 0.80 and the incoming boundary layer thickness was about 40% of the cavity depth. Fast-response pressures provided a measure of the aeroacoustic loading in the cavity, while triaxial accelerometers and laser Doppler vibrometry (LDV) were used for simultaneous store vibration measurements. Overall, the LDV and accelerometer data were in good agreement, but the LDV offered the advantage of increased spatial resolution, while the accelerometers were able to provide three-dimensional data. The simultaneous measurements demonstrated that cavity modes were able to excite the store, but with a directional dependence. Modal hammer tests were used to measure the natural frequencies of the store. The largest store accelerations were observed to occur along the spanwise direction at frequencies equal to the natural spanwise frequencies of the store.