Publications

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Normal tolerance limits are frequently used in dynamic environments specifications of aerospace systems as a method to account for aleatory variability in the environments. Upper tolerance limits, when used in this way, are computed from records of the environment and used to enforce conservatism in the specification by describing upper extreme values the environment may take in the future. Components and systems are designed to withstand these extreme loads to ensure they do not fail under normal use conditions. The degree of conservatism in the upper tolerance limits is controlled by specifying the coverage and confidence level (usually written in “coverage/confidence” form). Moreover, in high-consequence systems it is common to specify tolerance limits at 95% or 99% coverage and confidence at the 50% or 90% level. Despite the ubiquity of upper tolerance limits in the aerospace community, analysts and decision-makers frequently misinterpret their meaning. The misinterpretation extends into the standards that govern much of the acceptance and qualification of commercial and government aerospace systems. As a result, the risk of a future observation of the environment exceeding the upper tolerance limit is sometimes significantly underestimated by decision makers. This note explains the meaning of upper tolerance limits and a related measure, the upper prediction limit. So, the objective of this work is to clarify the probability of exceeding these limits in flight so that decision-makers can better understand the risk associated with exceeding design and test levels during flight and balance the cost of design and development with that of mission failure.

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Sandia National Laboratories has developed an experimental procedure for characterizing the input and transfer impedance functions of flight test units. Admittance relations that describe the force-acceleration relations can be derived from this procedure and used for a variety of purposes ranging from environmental response prediction to substructure coupling and numerical model validation. The theoretical developments described herein have been developed using substructure coupling methods. The method allows characterization in six degrees of freedom as well as the removal of fixture effects from the measured frequency response functions. This paper describes the basic theory, presents a numerical validation of the procedure, and illustrates a complete experimental example. © The Society for Experimental Mechanics, Inc. 2014.

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This paper develops deformational response power descriptions of multiple degree-of-freedom systems due to stationary random vibration excitation. Two new concepts are developed. The deformational response power density (DRPD) can be computed when a structure's natural frequencies and modal masses are available. The DRPD shows the spectral content of the deformational power delivered to a specific structure by the stationary, random excitation. This function can be found through a weighted windowing of the power spectrum of the input acceleration excitation. Deformational response input power spectra (DRIPS), similar to the input energy spectrum and shock response spectrum, give the power delivered to single-degree-of-freedom systems as a function of natural frequency. It is shown that the DRIPS is simply a smoothed version of the power spectrum of the input acceleration excitation. The DRIPS gives rise to a useful power-based data smoothing operation. © 2009 - IOS Press.

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A relatively new concept in the field of mechanical shock analysis has been introduced whereby an analysis is made on the work done on structures by the excitation force. The energy imparted to a structure by the excitation can then be divided into various storage and loss mechanisms within the structure. These energies can be used to both evaluate shock response severity and characterize the underlying excitation. Previous work has illustrated the many advantages of the energy methods over traditional shock response spectrum techniques. This work will show that the energy delivered to a MDOF system is uncoupled between modes. Therefore, the total deformational energy delivered to a MDOF system is a weighted sum of the uncoupled modal contributions. This leads to the ability to compute input energy on a modal basis using uncoupled, SDOF calculations. Further, the internal storage and loss energies are also uncoupled. When the input excitation is broadband, the energy input into a MDOF structure by ground motion is dominated by that mode with the largest fraction of participating mass, often the fundamental mode of the system. This leads to the justification for treating complex structures as SDOF oscillators when using energy methods to evaluate both the underlying excitation and the structural response.

By far the most widely used tool in shock data analysis is the shock response spectrum (SRS). The SRS has gained popularity because of several primary considerations. It has physical significance, it is simple to understand and it is believed to indicate shock severity. Despite its popularity, the SRS has limitations. Foremost among them is the underlying assumption that shock severity is proportional to a time derivative of position, which does not agree with accepted material failure models. Also, the SRS cannot distinguish between naturally occurring, complex shocks and the chirps sometimes used to achieve a desired SRS using electrodynamic shakers with inadequate force capabilities. Thirdly, SODF models used in the computation of the SRS do not accurately predict accelerations in MDOF structures. A relatively new concept has been introduced whereby an analysis is made on the work done on structures by the excitation force. Since work is equal to the change in the energy of a system, this quantity is closely related to failure models based on strain energy such as the Von Mesis criterion. This paper is the first in a series exploring the use of energy-based description of shock motion and structural response. The input energy spectrum has attractive properties which include intuitive physical significance, insensitivity to system parameters such as viscous damping or hysteretic loss, the ability to distinguish between realistic shocks and chirps, and a close relation to accepted material failure models. Input energy spectra can be calculated using SDOF models and, in many cases, accurately predict the energy input to MDOF structures. Finally, this paper gives an introduction to these methods, derives the equations for relevant energy measures and presents relationships to several other shock analysis tools.

During the course of processing acceleration data from mechanical systems it is often desirable to integrate the data to obtain velocity or displacement waveforms. However, those who have attempted these operations may be painfully aware that the integrated records often yield unrealistic residual values. This is true whether the data has been obtained experimentally or through numerical simulation such as Runge-Kutta integration or the explicit finite element method. In the case of experimentally obtained data, the integration errors are usually blamed on accelerometer zero shift or amplifier saturation. In the case of simulation data, incorrect integrations are often incorrectly blamed on the integration algorithm itself. This work demonstrates that seemingly small aliased content can cause appreciable errors in the integrated waveforms and explores the unavoidable source of aliasing in both experiment and simulation-the sampling operation. Numerical analysts are often puzzled as to why the integrated acceleration from their simulation does not match the displacement output from the same simulation. This work shows that these strange results can be caused by aliasing induced by interpolation of the model output during sampling regularisation. © 2005 Elsevier Ltd. All rights reserved.

The summary of this report is: (1) The Kernal Density Estimator (KDE) model using log data provides the most conservative estimates; (2) The Empirical Tolerance Limit (ETL) model provides the least conservative estimates; (3) The results for the Karhunen-Loeve (K-L) and Normal Tolerance Limit (NTL) models lie in between the extremes; (4) The NTL results ended up being as credible as any of the other methods. This may be related to the fact that the data appeared to fit a lognormal distribution for higher values of {beta}; (5) The discrepancy between these methods appears to widen for higher values of {beta} and {gamma}; (6) The reasons for the extreme difference in the KDE results depending on whether one uses the raw data or the log of the data is not clear at this time; and (7) Which model will best suit our needs is not clear at this time.