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A Method to Expand Sparse Set Acceleration Data to Full Set Strain Data

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Dynamic Environments Testing, Volume 7 (SEM 2023)

Abstract

Expansion methods are commonly used to compute the response at locations not measured during physical testing. The System Equivalent Reduction Expansion Process (SEREP) produces responses at finite-element degrees of freedom through the mode shapes of that model. Measurements used for expansion are often acceleration, strain, and displacement and operate only on like sets of data. For example, expanding acceleration data produces only additional acceleration data and does not provide insight into the test article’s stress or strain state. Stress and strain are often desired to evaluate yield limits and create fatigue models. The engineer may have acceleration measurements available but desire a component’s stress and strain state. This chapter evaluates a physical experiment from which acceleration is measured and a full set of strain, stress, and displacement data is obtained through SEREP and integration. Honeywell Federal Manufacturing & Technologies, LLC, operates the Kansas City National Security Campus for the United States Department of Energy/National Nuclear Security Administration under Contract Number DE-NA0002839.

Notice: This manuscript has been authored by Honeywell Federal Manufacturing & Technologies, LLC, under Contract No. DE-NA-0002839 with the U.S. Department of Energy/National Nuclear Security Administration. The United States Government retains, and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a nonexclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.

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References

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Correspondence to Jonathan Hower .

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Appendices

Appendix 1: Displacement Mode Shapes from Finite-Element Model

Displacement mode shapes for the first six flexible modes of the structure are shown in Figs. 3.203.25.

Fig. 3.20
A 3 Dimensional illustration of a bent color-coded bobble head structure inclined towards the Y axis.

First flexible mode. First-order bending in Y direction at 91 Hz

Fig. 3.21
A 3 Dimensional color-coded illustration of a bent bobble head inclined towards the X axis.

Second flexible mode. First-order bending in X direction at 120 Hz

Fig. 3.22
A 3 Dimensional color-coded illustration of a bobble head inclined towards the Z axis.

Third flexible mode. First-order torsion in Z at 317 Hz

Fig. 3.23
A 3 Dimensional color-coded illustration of a bobble head structure. The column is inclined towards the Y axis and the head inclination is towards the X axis..

Fourth flexible mode. Second-order bending in Y direction at 976 Hz

Fig. 3.24
A 3 Dimensional color-coded illustration of a bobble head structure. The column is inclined towards the Y axis and the head inclination is towards the X axis..

Fifth flexible mode. Second-order bending in X direction at 1265 Hz

Fig. 3.25
A 3 Dimensional color-coded illustration of a bobble head structure without any inclination.

Sixth flexible mode. First-order axial mode at 2462 Hz

Fig. 3.26
A 3 Dimensional strain mode shape of the bobble head inclined towards the Y axis.

First flexible mode. First-order bending in Y direction at 91 Hz

Fig. 3.27
A 3 Dimensional strain mode shape of the bobble head inclined towards the X axis.

Second flexible mode. First-order bending in X direction at 120 Hz

Appendix 2: Strain Mode Shapes from Finite-Element Model

Strain mode shapes for the first six flexible modes of the structure are shown in Figs. 3.263.31.

Fig. 3.28
A 3 Dimensional strain mode illustration of the bobble head inclined towards the Z axis.

Third flexible mode. First-order torsion in Z at 317 Hz

Fig. 3.29
A 3 Dimensional strain mode illustration of the bobble head. The column is inclined towards the Y axis and the head inclination is towards the X axis.

Fourth flexible mode. Second-order bending in Y direction at 976 Hz

Fig. 3.30
A 3 D strain mode illustration of the bobble head. The column is inclined towards the Y axis and the head inclination is towards the X axis.

Fifth flexible mode. Second-order bending in X direction at 1265 Hz

Fig. 3.31
A 3 D strain mode illustration of the bobble head. The surface and head are towards the Z axis and the column is along the Z axis.

Sixth flexible mode. First-order axial mode at 2462 Hz

Appendix 3: Supplementary Error Metrics

3.1.1 Time and Frequency Response Assurance Criterion

The time response assurance criterion (TRAC) and frequency response assurance criterion (FRAC) are both metrics that qualitatively compare two signals in the time and frequency domains, respectively. The equation for TRAC is shown as Eq. 3.14, and the equation FRAC is shown as Eq. 3.15 per dynamic design solutions [10]. A TRAC or FRAC of 1 indicates perfect consistency and 0 indicates inconsistency or orthogonal signals.

$$\displaystyle \begin{aligned} {} TRAC=\frac{(|{\overline{X}_{t,meas}}^\top||\overline{X}_{t,pred}|)^2}{(|{\overline{X}_{t,pred}}^\top||\overline{X}_{t,meas}|)(|{\overline{X}_{t,meas}}^\top||\overline{X}_{t,pred}|)} \end{aligned} $$
(3.14)
$$\displaystyle \begin{aligned} {} FRAC=\frac{(|{\overline{X}_{f,meas}}^\top||\overline{X}_{f,pred}|)^2}{(|{\overline{X}_{f,pred}}^\top||\overline{X}_{f,meas}|)(|{\overline{X}_{f,meas}}^\top||\overline{X}_{f,pred}|)}. \end{aligned} $$
(3.15)

3.1.2 Root Mean-Squared Error

The root mean-squared error (RMSE) compares two signals by computing the deviation between them. It may be computed in the time or frequency domain, as shown in Eq. 3.16 and 3.17, respectively. A low RMSE indicates good agreement between measured and predicted values.

$$\displaystyle \begin{aligned} {} RMSE_{time} = \sqrt{\frac{\sum_{t}(\overline{X}_{t,meas}-\overline{X}_{t,pred})^2}{n_t}} \end{aligned} $$
(3.16)
$$\displaystyle \begin{aligned} {} RMSE_{freq} = \sqrt{\frac{\sum_{f}(\overline{X}_{f,meas}-\overline{X}_{f,pred})^2}{n_f}}. \end{aligned} $$
(3.17)

Appendix 4: Supplementary Acceleration Expansion Results

The time- and frequency-domain responses of a few of the signals are studied in greater detail in Figs. 3.32 and 3.33, respectively. For succinctness, only three signals are shown. These signals were selected to represent a worst, average, and best expansion of the total 24 signals based on the various error metrics discussed above. The TRAC and FRAC values are listed generally as SAC, which stands for Signature Assurance Criterion.

Fig. 3.32
3 sets of 2 line graphs plot the acceleration versus time in seconds. The graphs illustrate a normal and zoomed graph. The lines for measured and expanded R M S and for S A C, M A E, and R M S E, for channels 31, 27, and 39 densely fluctuate.

Acceleration time history compared between measured and expanded data of worst (top), average (middle), and best (bottom) channels

Fig. 3.33
3 sets of 2 line graphs plot P S D versus frequency with decreasing trends. The lines for measured and expanded R M S, and the lines for S A C, M A E, and R M S E for channels 31, 27, and 39 move with fluctuations.

Acceleration power spectral density compared between measured and expanded data of worst (top), average (middle), and best (bottom) channels

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Hower, J., Joshua, R., Schoenherr, T. (2024). A Method to Expand Sparse Set Acceleration Data to Full Set Strain Data. In: Harvie, J. (eds) Dynamic Environments Testing, Volume 7. SEM 2023. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-031-34930-0_3

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  • DOI: https://doi.org/10.1007/978-3-031-34930-0_3

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