Daniel C. Kammer

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.

Frequency Band Averaging of Spectral Densities for Updating Finite Element Models

The successful operation of proposed precision spacecraft will require finite element models that are accurate to much higher frequencies than the standard application. The hallmark of this mid-frequency range, between low-frequency modal analysis and high-frequency statistical energy analysis, is high modal density. The modal density is so high, and the sensitivity of the modes with respect to modeling errors and uncertainty is so great that test/analysis correlation and model updating based on traditional modal techniques no longer work. This paper presents an output error approach for finite element model updating that uses a new test/analysis correlation metric that maintains a direct connection to physical response. The optimization is gradient based. The metric is based on frequency band averaging of the output power spectral densities with the central frequency of the band running over the complete frequency range of interest. The results of this computation can be interpreted in several different ways, but the immediate physical connection is that it produces the mean-square response, or energy, of the system to random input limited to the averaging frequency band. The use of spectral densities has several advantages over using frequency response directly, such as the ability to easily include data from all inputs at once, and the fact that the metric is real. It is shown that the averaging process reduces the sensitivity of the optimization due to resonances that plague many output error model updating approaches.

Propagation of Uncertainty in Substructured Spacecraft Using Frequency Response

In many situations, it is either impossible to perform a spacecraft system level vibration test, or it is highly desirable to avoid one to conserve time and resources. If modeling and analysis are to replace system tests, it is imperative to have confidence in the results. Therefore, a probabilistic system correlation analysis must be performed by quantifying uncertainty in the substructures, and then propagating it into the system correlation metrics. A systematic procedure is presented for studying the effects of substructure uncertainty on system test-analysis correlation of spacecraft that are validated on a substructure-by-substructure basis. Covariance propagation is used to propagate uncertainty in a free-interface substructure frequency response into the expected frequency response uncertainty for the system using a component mode synthesis approach. The frequency response-based procedure is of special interest in systems with high-modal density, where modal correlation methods do not work. A simp...

Recyclable transmission line (RTL) and linear transformer driver (LTD) development for Z-pinch inertial fusion energy (Z-IFE) and high yield.

Z-Pinch Inertial Fusion Energy (Z-IFE) complements and extends the single-shot z-pinch fusion program on Z to a repetitive, high-yield, power plant scenario that can be used for the production of electricity, transmutation of nuclear waste, and hydrogen production, all with no CO{sub 2} production and no long-lived radioactive nuclear waste. The Z-IFE concept uses a Linear Transformer Driver (LTD) accelerator, and a Recyclable Transmission Line (RTL) to connect the LTD driver to a high-yield fusion target inside a thick-liquid-wall power plant chamber. Results of RTL and LTD research are reported here, that include: (1) The key physics issues for RTLs involve the power flow at the high linear current densities that occur near the target (up to 5 MA/cm). These issues include surface heating, melting, ablation, plasma formation, electron flow, magnetic insulation, conductivity changes, magnetic field diffusion changes, possible ion flow, and RTL mass motion. These issues are studied theoretically, computationally (with the ALEGRA and LSP codes), and will work at 5 MA/cm or higher, with anode-cathode gaps as small as 2 mm. (2) An RTL misalignment sensitivity study has been performed using a 3D circuit model. Results show very small load current variations for significant RTL misalignments. (3) The key structural issues for RTLs involve optimizing the RTL strength (varying shape, ribs, etc.) while minimizing the RTL mass. Optimization studies show RTL mass reductions by factors of three or more. (4) Fabrication and pressure testing of Z-PoP (Proof-of-Principle) size RTLs are successfully reported here. (5) Modeling of the effect of initial RTL imperfections on the buckling pressure has been performed. Results show that the curved RTL offers a much greater buckling pressure as well as less sensitivity to imperfections than three other RTL designs. (6) Repetitive operation of a 0.5 MA, 100 kV, 100 ns, LTD cavity with gas purging between shots and automated operation is demonstrated at the SNL Z-IFE LTD laboratory with rep-rates up to 10.3 seconds between shots (this is essentially at the goal of 10 seconds for Z-IFE). (7) A single LTD switch at Tomsk was fired repetitively every 12 seconds for 36,000 shots with no failures. (8) Five 1.0 MA, 100 kV, 100 ns, LTD cavities have been combined into a voltage adder configuration with a test load to successfully study the system operation. (9) The combination of multiple LTD coaxial lines into a tri-plate transmission line is examined. The 3D Quicksilver code is used to study the electron flow losses produced near the magnetic nulls that occur where coax LTD lines are added together. (10) Circuit model codes are used to model the complete power flow circuit with an inductive isolator cavity. (11) LTD architectures are presented for drivers for Z-IFE and high yield. A 60 MA LTD driver and a 90 MA LTD driver are proposed. Present results from all of these power flow studies validate the whole LTD/RTL concept for single-shot ICF high yield, and for repetitive-shot IFE.

Experimental Based Substructuring Using a Craig-Bampton Transmission Simulator Model

Recently, a new experimental based substructure formulation was introduced which reduces ill-conditioning due to experimental measurement noise by imposing the connection 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 transmission simulator. An analytical representation of the fixture is then used to subtract its effects from the experimental substructure. The resulting experimental component is entirely modal based, and can be attached in an indirect manner to other substructures by constraining the modal degrees of freedom of the transmission simulator to those substructures. This work explores a different alternative in which the transmission simulator is modeled with a Craig-Bampton model, a model that may be more appropriate when the interfaces are connected rigidly. The new method is compared to the authors’ previous approaches to evaluate the errors due to modal truncation using finite element models of several beam systems including one in which the transmission simulator is connected to the component of interest at two points, potentially producing an ill-conditioned inverse problem.

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.

Combining metamodels with rational function representations of discretization error for uncertainty quantification

Abstract Techniques for producing metamodels for the efficient Monte Carlo simulation of high consequence systems are presented. The bias of f.e.m mesh discretization errors is eliminated or minimized by extrapolation, using rational functions, rather than the power series representation of Richardson extrapolation. Examples, including estimation of the vibrational frequency of a one-dimensional bar, show that the rational function model gives more accurate estimates using fewer terms than Richardson extrapolation, an important consideration for computational reliability assessment of high-consequence systems, where small biases in solutions can significantly affect the accuracy of small-magnitude probability estimates. Rational function representation of discretization error enable the user to accurately extrapolate to the continuum from numerical experiments performed outside the asymptotic region of the usual power series, allowing use of coarser meshes in the numerical experiments, resulting in significant savings.

Formulation of a Craig-Bampton Experimental Substructure Using a Transmission Simulator

Recently, a new experimental based substructure formulation was introduced, called Modal Constraint for Fixture and Subsystem (MCFS). 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 transmission simulator. An analytical representation of the fixture is then used to subtract its effects from the experimental substructure. The resulting experimental component is entirely modal based, and can be attached in an indirect manner to other substructures using MCFS. In contrast, this work presents a formulation in which the analytical representation of the transmission simulator is in the form of a Craig-Bampton (CB) substructure including fixed-interface modal coordinates and physical interface coordinates. The negative of the analytical representation of the transmission simulator is constrained to the experimental modal model using MCFS by eliminating the fixed-interface modal coordinates. The resulting experimental substructure contains a hybrid set of coordinates, including modal coordinates and the physical interface degrees of freedom, analogous to a CB representation. This new formulation offers the improved conditioning of the MCFS approach, but can be directly connected through the physical interface coordinates to other finite element based substructures.

FEM Calibration with FRF Damping Equalization

A finite element model calibration procedure that uses frequency response function data and relies on damping equalization is presented. In this, the dampings of the finite element model and the corresponding experimental model are set equal before calibration. The damping equalization is made to avoid the mode pairing problem that normally needs to be solved in other model updating procedures. It is demonstrated that one particular use of frequency response data gives a calibration deviation metric that is smooth in the variation of model parameters and give a large radius of convergence to the calibration minimum. The method is combined with model reduction for increased speed and employs a minimizing procedure that employs randomized multiple starting points in the parameter space to get to the calibration solution. The performance of the calibration procedure is demonstrated by two numerical examples.

Energy Based Comparison of Test and Analysis Response in the Frequency Domain

Accepted modal based techniques for comparing finite element model and test data for test/analysis correlation and subsequent model updating are impossible to use in the high modal density midfrequency regime. A new approach is presented for comparing test and analysis representations using frequency-based response data instead of modal parameters. The new method uses frequency band averaging of the output power spectral densities with the central frequency of the band running over the complete frequency range of interest. The result of this computation can be interpreted in several different ways but the immediate physical connection is that it produces the mean-square response, or energy, of the system to random input limited to the averaging frequency band. The averaging process is consistent with the averaging done in statistical energy analysis for stochastic systems. The averaged response curves can be compared on a pointwise basis, or they can be compared within a running frequency band.