Publications

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A previous study in the UK demonstrated that vibration response on a scaled-down model of a missile structure in a wind tunnel could be replicated in a laboratory setting with multiple shakers using an approach dubbed as impedance matching. Here we demonstrate on a full scale industrial structure that the random vibration induced from a laboratory acoustic environment can be nearly replicated at 37 internal accelerometers using six shakers. The voltage input to the shaker amplifiers is calculated using a regularized inverse of the square of the amplitude of the frequency response function matrix and the power spectral density responses of the 37 internal accelerometers. No cross power spectral density responses are utilized. The structure has hundreds of modes and the simulation is performed out to 4000 Hz.

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.

Experimental dynamic substructuring is a means whereby a mathematical model for a substructure can be obtained experimentally and then coupled to a model for the rest of the assembly to predict the response. Recently, various methods have been proposed that use a transmission simulator to overcome sensitivity to measurement errors and to exercise the interface between the substructures; including the Craig-Bampton, Dual Craig-Bampton, and Craig-Mayes methods. This work compares the advantages and disadvantages of these reduced order modeling strategies for two dynamic substructuring problems. The methods are first used on an analytical beam model to validate the methodologies. Then they are used to obtain an experimental model for structure consisting of a cylinder with several components inside connected to the outside case by foam with uncertain properties. This represents an exceedingly difficult structure to model and so experimental substructuring could be an attractive way to obtain a model of the system.

Experimental dynamic substructuring is a means whereby a mathematical model for a substructure can be obtained experimentally and then coupled to a model for the rest of the assembly to predict the response. Recently, various methods have been proposed that use a transmission simulator to overcome sensitivity to measurement errors and to exercise the interface between the substructures; including the Craig-Bampton, Dual Craig-Bampton, and Craig-Mayes methods. This work compares the advantages and disadvantages of these reduced order modeling strategies for two dynamic substructuring problems. The methods are first used on an analytical beam model to validate the methodologies. Then they are used to obtain an experimental model for structure consisting of a cylinder with several components inside connected to the outside case by foam with uncertain properties. This represents an exceedingly difficult structure to model and so experimental substructuring could be an attractive way to obtain a model of the system.

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Qualification of complex systems typically involves testing the components individually in shock and vibration environments before assembling them into the system. When the components are secured to a fixture on the shaker table, the mechanical impedance of the boundary condition is quite different from that of the next level of assembly. Thus the modes of the component under test are not excited in the same way that they are excited in the system using the typical methods for defining input specifications. Here, the boundary condition impedance is investigated and quantified using substructuring techniques. Also, fixture inputs are derived to overcome the impedance differences and excite a component in the same way it is excited in the next level of assembly.

This work proposes a means whereby weak nonlinearity in a substructure, as typically arises due to friction in bolted interfaces, can be captured experimentally on a mode-by-mode basis and then used to predict the nonlinear response of an assembly. The method relies on the fact that the modes of a weakly nonlinear structure tend to remain uncoupled so long as their natural frequencies are distinct and higher harmonics generated by the nonlinearity do not produce significant response in other modes. Recent experiments on industrial hardware with bolted joints has shown that this type of model can be quite effective, and that a single degree-of-freedom (DOF) system with an Iwan joint, which is known as a modal Iwan model, effectively captures the way in which the stiffness and damping depend on amplitude. Once the modal Iwan models have been identified for each mode of the subcomponent(s) of interest, they can be assembled using standard techniques and used with a numerical integration routine to compute the nonlinear transient response of the assembled structure. The proposed methods are demonstrated by coupling a modal model of a 3DOF system with three discrete Iwan joints to a linear model for a 2DOF system.

Experimental-analytical substructuring is attractive when there is motivation to replace one or more system subcomponents with an experimental model. This experimentally derived substructure can then be coupled to finite element models of the rest of the structure to predict the system response. The transmission simulator method couples a fixture to the component of interest during a vibration test in order to improve the experimental model for the component. The transmission simulator is then subtracted from the tested system to produce the experimental component. The method reduces ill-conditioning by imposing a least squares fit of constraints between substructure modal coordinates to connect substructures, instead of directly connecting physical interface degrees of freedom. This paper presents an alternative means of deriving the experimental substructure model, in which a Craig-Bampton representation of the transmission simulator is created and subtracted from the experimental measurements. The corresponding modal basis of the transmission simulator is described by the fixed-interface modes, rather than free modes that were used in the original approach. These modes do a better job of representing the shape of the transmission simulator as it responds within the experimental system, leading to more accurate results using fewer modes. The new approach is demonstrated using a simple finite element model based example with a redundant interface.

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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.

A collaborative research institute was organized and held at Sandia Albuquerque for a period of six weeks. This research institute brought together researchers from around the world to work collaboratively on a set of research projects. These research projects included: developing experimental guidelines for studying variability and repeatability of nonlinear structures; decoupling aleatoric and epistemic uncertainty in measurements to improve dynamic predictions; a numerical round robin to assess the performance of five different numerical codes for modeling systems with strong nonlinearities; and an assessment of experimentally derived and numerically derived reduced order models. In addition to the technical collaborations, the institute also included a series of seminars given by both Sandians and external experts, as well as a series of tours and field trips to local places of scientific and engineering importance. This report details both the technical research and the programmatic organization of the 2014 Sandia Nonlinear Mechanics and Dynamics Summer Research Institute.

This work was motivated by a desire to transform an experimental dynamic substructure derived using the transmission simulator method into the Craig-Bampton substructure form which could easily be coupled with a finite element code with the Craig-Bampton option. Near the middle of that derivation, a modal Craig-Bampton form emerges. The modal Craig-Bampton (MCB) form was found to have several useful properties. The MCB matrices separate the response into convenient partitions related to (1) the fixed boundary modes of the substructure (a diagonal partition), (2) the modes of the fixture it is mounted upon, (3) the coupling terms between the two sets of modes. Advantages of the MCB are addressed. (1) The impedance of the boundary condition for component testing, which is usually unknown, is quantified with simple terms. (2) The model is useful for shaker control in both single degree of freedom and multiple degree of freedom shaker control systems. (3) MCB provides an energy based framework for component specifications to reduce over-testing but still guarantee conservatism.

Experimental dynamic substructures in both modal and frequency response domains using the transmission simulator method have been developed for several systems since 2007. The standard methodology couples the stiffness, mass and damping matrices of the experimental substructure to a finite element (FE) model of the remainder of the system through multi-point constraints. This can be somewhat awkward in the FE code. It is desirable to have an experimental substructure in the Craig-Bampton (CB) form to ease the implementation process, since many codes such as Nastran, ABAQUS, ANSYS and Sierra Structural Dynamics have CB as a substructure option. Many analysts are familiar with the CB form. A square transformation matrix is derived that produces a modified CB form that still requires multi-point constraints to couple to the rest of the FE model. Finally the multi-point constraints are imported to the modified CB matrices to produce substructure matrices that fit in the standard CB form. The physical boundary degrees-of-freedom (dof) of the experimental substructure matrices can be directly attached to physical dof in the remainder of the FE model. This paper derives the new experimental substructure that fits in the CB form, and presents results from an analytical and an industrial example utilizing the new CB form.