## Abstract

Vibration testing of complex aerospace structures requires substantial pretest planning. Ground and flight testing of structures can be costly to execute in terms of time and money, so it is pertinent that tests are properly set up to capture mode shapes or dynamics of interest. One of the most important planning tasks is the placement of sensors to acquire measurements for control and characterization of the results. In this paper, we will examine two techniques that can leverage available output from finite element modeling to intelligently place accelerometers for a vibration test to capture the structural dynamics throughout a specified frequency range with a data acquisition channel budget. These two techniques are effective independence (EI) and optimal experimental design (OED). Both methods will be applied to an aerospace structure. Effects of the chosen sets on system equivalent reduction and expansion process (SEREP) are detailed alongside simpler comparison metrics, like the auto-modal assurance criterion (auto-MAC). In addition to comparing the resulting instrumentation sets, the application of the two approaches will be compared in terms of the inputs required, the information obtained from their application, and the computation time requirements. Both OED and EI offer an effective method for selecting an instrumentation set for a given vibration test. EI is a straightforward, computationally inexpensive approach that provides effective instrumentation sets. OED provides an effective alternative that is less sensitive to the impact of local modes and leads to a natural ranking of importance for each chosen degree-of-freedom (DOF).

## 1 Introduction

Properly instrumenting an assembly for vibration testing is paramount to the resulting data quality and post-test analyses. Various methods exist to choose a possible instrumentation set for a vibration test. A popular approach introduced by Kammer is effective independence (EI) [1]. This method can be computationally inexpensive and aims to minimize the condition number of the Fisher information matrix for the modes of interest. This method has been studied by several groups in various applications. Meo and Zumpano leveraged EI for health monitoring in civil engineering structures and compared it to other approaches [2]. Papadopoulos and Garcia compared EI to other placement strategies for dynamic testing of a Euler–Bernoulli beam and structural frame [3]. Penny et al. compared EI to a method based on Guyan reduction [4]. Friswell and Castro-Triguero demonstrated the clustering issue that can occur with EI [5]. Castro-Triguero et al. also examined the performance of EI and other selection methods under parametric uncertainty [6]. Finally, Kim et al. extended the method by developing a stochastic version of the approach [7]. The EI method is straightforward to implement, and a workflow can be developed that leverages the tools available for an analyst or test engineer.

Various other methods have been proposed and demonstrated throughout the aerospace and civil engineering communities. Suryanarayana et al. demonstrated a data-based method leveraging the Hilbert–Schmidt independence criterion and a grey-box building model to develop the approach [8]. Clark et al. demonstrated the use of a method based on a QR decomposition method to minimize the reconstruction error in problems using a linear map for reconstruction with the measured values at the sensors for various problem types [9]. Finally, Beale et al. demonstrated the use of optimal experimental design (OED) to place sensors for a multiple-input/multiple-output vibration test [10]. This method was selected for designing an instrumentation set in this paper due to its compatibility with existing workflows as an alternative approach, using Sierra structural dynamics (Sierra/SD) to generate the required input data and matlab to complete further analyses [11,12].

Section 2 defines the structural dynamic model of the aerospace structure and the analyses completed to support the application of the EI and OED instrumentation selection approaches. Section 3 introduces the two approaches and explores their application to the chosen structure. Section 4 provides various comparisons between the sets chosen with EI and OED. Finally, Sec. 5 provides conclusions and possible future work.

## 2 Definition of Structure for Analysis

The structure of interest for this work is an assembled configuration of previous research articles studied at Sandia National Laboratories. The assembled structure is termed the wedding cake and removable component assembly (WRCA) and consists of the wedding cake (WC) structure from Ref. [13] and the removable component (RC) subassembly from Ref. [14]. The RC was designed to investigate approaches to design impendence matched test fixtures. The two subassemblies are attached to a shell structure, which contains a base support for the WC and a plate for the RC. An image of the assembly is shown in Fig. 1. The WRCA was developed as an exemplar for investigating instrumentation design approaches for assembled aerospace structures with dynamics of interest in the bandwidth of 20–2000 Hz.

### 2.1 Finite Element Model of the Dynamic System.

The structure was modeled using in-house software packages at Sandia National Laboratories, including CUBIT for the meshing [15] and the Sierra/SD linear finite element code for the dynamic analysis [12]. Modal and random vibration analyses were performed, and the results were used for subsequent instrumentation selection studies.

### 2.2 Modal Solution.

Sierra/SD was used to perform modal analyses of the system and its subassemblies. Sixty modes were computed between 0 and 2000 Hz, with a free-free boundary condition for the instrumentation design with EI. A fixed-base version of the structure was created for random vibration loading and definition of the frequency response function (FRF) arrays for the OED implementation. A free-free boundary condition was used for EI since it is a typical starting point for analysis and model calibration, and it allows for direct comparison to experimental data that is available during test design. A selection of elastic modes is shown in Fig. 2 and the natural frequencies associated with all computed modes are provided in Table 8 in the Appendix. The contours in Fig. 2 correspond to the magnitude of the modal displacement vector for each mode.

In addition, modes of each subassembly (WC, RC, and shell) were computed up to 2 kHz. Figure 3 shows a selection of elastic modes for each subassembly as well as the modal displacement magnitude contours. Modes from both the assembly and subassembly configurations were utilized to compare the different instrumentation selection strategies. Considering the subassemblies separately can provide more flexibility in a test plan as well as provide individual instrumentation sets that can be used for testing articles separately. Section 3 explains how these analyses as well as the random vibration solution detailed next were used in the instrumentation selection.

### 2.3 Random Vibration Solution.

Random vibration was selected as the loading condition of interest for this work since it is a common environment for aerospace structures. A 0.1 g^{2}/Hz flat random, Gaussian white noise from 20 to 1000 Hz was used as the input, shown in Fig. 4. This bandwidth was selected to limit the effects of modal truncation using the structure modes out to 2 kHz.

The dynamic analysis was conducted with Sierra/SD, using a seismic mass and rigid elements for base excitation. A transient simulation was performed with a force applied to the seismic mass via rigid elements. The resulting acceleration time history of the random vibration was applied to all points on the base of the structure. Loading was applied in the transverse *x*-axis direction, illustrated in Fig. 5. This modeling approach simulates single axis fixed-base vibration testing typically conducted on a shaker table.

The results from the random vibration analysis were subsequently used for expansion studies with system equivalent reduction and expansion process (SEREP) and to evaluate the effectiveness of the various instrumentation sets. Once an instrumentation set is defined and test data collected, SEREP allows engineers to expand from the instrumented locations to other points within a test article leveraging analysis data. A good instrumentation set allows for an effective expansion that can provide a complete picture of the response of a test article with limited measurement points.

## 3 Instrumentation Selection Strategies

Two major techniques were used for the instrumentation selection: EI and OED. The instrumentation selection workflow for each approach is outlined in Fig. 6.

First, a finite element mesh of the structure of interest is generated using CUBIT. Candidate surfaces are selected based on available locations for accelerometers. Next, candidate accelerometer locations on each surface are generated using a subset of the nodes. These are typically determined based on an evenly spaced Cartesian or cylindrical grid dependent on the surface and part geometry. Local coordinate systems are then created for each candidate point. An example of the resulting outputs is shown in Fig. 7. The outputs include a mesh file that contains the coordinate locations chosen as well as a channel table that can be read into either matlab or Microsoft Excel.

After the candidate set has been determined, a modal solution is performed with Sierra/SD, with mode shapes output at each candidate node location. The subsequent steps vary based on the desired instrumentation selection technique. As stated before, Fig. 6 provides an overview of the workflow leveraged for both the EI and OED approaches. Compared to EI, OED requires an additional analysis with Sierra/SD that outputs the transfer functions from the input location of the test to the candidate gauge locations. Sections 3.1 and 3.2 detail the two selection strategies used within matlab with the intermediary outputs from Sierra/SD. The optimization problem for OED is also completed within the Sierra environment.

The computation time associated with EI corresponds to a modal analysis of the test article of interest in Sierra/SD as well as the EI implementation in matlab with output data of Sierra/SD. In comparison, OED requires the modal analysis in addition to a modal FRF solution to obtain the required transfer functions. Finally, the OED approach is implemented in the Sierra environment. Sierra simulations can be run in parallel to help improve speed.

For the analyses performed in this paper, the EI implementation in matlab was fast enough to not require parallelization, taking under 3 min for the assembly level set. Table 1 provides time requirements for the computationally intensive steps for EI and OED instrumentation selection normalized to the time required for the modal analysis. The computation times were dependent on how many processors were used. The modal run required approximately 1/3 the computation time of the six degrees-of-freedom (DOF) modal FRF runs, where the modal run required 5 min. In addition, OED can be anywhere from 5 to more than 100 times the computation time required for the modal run. These run times are associated with different machines with multiple processors, so direct time comparisons are difficult. OED does not require significantly more setup time than EI; however, it does require additional computational resources.

### 3.1 Effective Independence Introduction.

EI is a commonly used method to determine placement of accelerometers for vibration testing. Kammer introduced EI for identifying instrumentation locations for testing of aerospace structures [1]. EI is a sub-optimal approach since it is implemented iteratively to determine the least impactful sensor location during each iteration [1,4]. EI is built upon the foundation of the Fisher information matrix (*Q*):

^{T}the transpose of the mode shape matrix.

*Q*

^{−1}denotes the inverse of the Fisher information matrix. In this process, the minimum diagonal term of

*E*is identified and the associated instrumentation location and degree-of-freedom is removed from the candidate set. This process can be defined such that uniaxial accelerometers, triaxial accelerometers, or a combination of both (hybrid) can be used. This is implemented by either grouping degrees-of-freedom or considering them individually. The iterations continue until the required channel budget is met.

*i*and

*j*is given as

_{i}is the modal vector for mode

*i*and Φ

_{j}is the modal vector for mode

*j*. The auto-MAC measures the independence of the modes of the system of interest and has a value from 0 to 1, where 1 denotes a significant similarity between mode shapes [16]. Off-diagonal MAC values are used to measure the quality of a chosen set since a low value signifies independence of the modes measured with the instrumentation set allowing them to be captured in testing. This implies that the instrumentation set will perform well for modal analysis as well as capture the overall dynamic response of the structure. Section 3.3 details the instrumentation set chosen with EI.

### 3.2 Optimal Experimental Design Introduction.

*T*(

*θ*) =

*Q*

_{O}(

*θ*)

*H*and

*y*

_{M}is a set of measurements from experiment or finite element model at

*n*points in either the complex or real domain.

*Q*

_{O}(

*θ*) is an observation operating with a parameter set

*θ*signifying the possible degrees-of-freedom at the various instrumentation locations.

*H*denotes the transformation from the unknown parameters,

*s*, to the measurement locations,

*y*

_{M}. The dimension of

*H*is

*n*×

*p*in the complex or real domain.

^{*}denotes the complex transpose. The minimization process seeks to minimize an uncertainty-based metric, which can also be viewed as maximizing the acquired information [10].

*C*(

*θ*) denotes the covariance of the estimated parameters and is defined by

*σ*is the variance [10].

*H*

_{i}is the

*i*th column of the previously discussed

*H*, which is the FRF array for the WRCA assembly with a fixed base. This array is acquired, as outlined in Fig. 6, using Sierra/SD. The optimal instrumentation, or parameter set, implies that the model’s average prediction variance is minimized, denoted as $var(yi^)$ [10]. Since OED is convex, the optimal solution is the global minimizer.

The output of the optimization problem is an instrumentation set alongside a probability measure of the chosen sensor location. Each probability is between 0 and 1. When the optimization constraints are met and the solution converges, the sum of all probabilities will be equal to 1. These probabilities allow one to see the importance of each chosen instrumentation location, which provides additional utility beyond EI [10]. However, OED provides slightly less control over the number of locations chosen as well as how many locations are chosen for each subassembly. This is an important consideration since it is often desirable to be able to perform subassembly level design while considering overall channel budget constraints. It is difficult to enforce total sensor budget constraints for OED since these constraints are inherently non-convex. Current research is focusing on how to implement these types of constraints. Section 3.4 provides a brief overview of the instrumentation set chosen through OED. Section 4 provides comparisons between the OED set and the sets chosen with EI.

### 3.3 Application of Effective Independence.

The EI method was used to generate instrumentation sets for both the assembly and subassembly configurations. A predefined instrumentation budget of 80 channels or DOF was established for the WRCA to adequately cover the 60 modes up to 2 kHz. For the subassemblies, 8, 40, and 32DOF were split between the RC, WC, and shell, respectively. The budget used is higher than would be considered for a physical test. Often, budget constraints are limited due to available space or data acquisition (DAQ) channels. Ongoing work focuses on methods to reduce the channel budget while maintaining the performance of the chosen set. Instrumentation studies with EI were performed for cases with triaxial, uniaxial, and a hybrid of triaxial and uniaxial sensors. Figure 8 shows the resulting down-selected instrumentation sets for the hybrid approach and Figs. 9–11 display the individual subassembly sets.

The instrumentation sets for each EI case appeared visually similar and both assemblies had the same number of DOF, although not equally split between subassemblies. As mentioned previously, one popular method used to evaluate the quality of an instrumentation set is by computing an auto-MAC, given in (3), which computes the similarity of modes. The auto-MAC for the WRCA assembly designed and subassembly designed sets is provided in Figs. 12(a) and 12(b), respectively.

The vast majority of the 60 WRCA modes were independent and well-captured by each set, indicated by the low off-diagonals in the auto-MAC. There were a few modes with a high off-diagonal MAC, even more so for the subassembly designed set. Overall, using an assembly or subassembly based approach appeared to provide similar instrumentation sets using EI. Subsequent sections will use expansion and other metrics to further compare the sets in detail.

### 3.4 Application of Optimal Experimental Design.

Figure 13 shows the instrumentation set chosen with OED. The same candidate set used with EI was leveraged in the OED process. The FRF array was formed by taking the input location as the seismic mass shown in Fig. 5. All FRFs were generated from 100 Hz to 2000 Hz at a frequency step of 1 Hz in Sierra/SD. The OED process can be expensive computationally; therefore, a coarser set of frequency lines can be used for the FRF array, such as 3 or 6 points per octave. The coarser frequency line definition can result in some modes that are dominated by local motion being overlooked, especially if the mode does not significantly alter the FRFs. Additionally, the subassemblies could be considered separately by dividing up the FRF array and candidate sets for each subassembly and then conducting the optimization for the subassemblies individually. A complete instrumentation set could then be generated by combining the final solutions. This can also provide additional flexibility in the final chosen set.

Optimal experimental design selected 86DOF for the total assembly. The set was divided into 38DOF for the WC, 29 for the shell, and 19 for the RC. The auto-MAC for the first 22 modes of the WRCA is given in Fig. 14 for the OED instrumentation set. The set shown in Fig. 13 was generated with a relatively coarse frequency line step with three points per octave from 100 Hz to 2000 Hz. The set was generated by considering each subassembly separately with the FRF array defined using the complete WRCA assembly. Observing Fig. 14, the auto-MAC was comparable to EI; however, the auto-MAC for the OED set deteriorated for modes 23 through 29 as shown in Fig. 15. Even though the auto-MAC measure was slightly worse for the OED set, the expansion metrics shown in Sec. 4 were better for the OED set compared to the EI sets. Modes 23 through 29 may not contribute significantly in aggregate to the system response and FRFs used in the OED process, so placements that can capture these modes may be overlooked in lieu of areas that contribute more to the overall system response.

Optimal experimental design was also applied to a fixed-based version of the RC mesh to generate a FRF array for just the RC component. Figure 16 compares the OED set generated for the RC with the assembly and subassembly FRF arrays. The two RC sets were comparable, with the subassembly set requiring three fewer DOF for convergence of the optimization problem.

## 4 Comparison of Instrumentation Sets and Associated Analyses

To quantitatively compare the resulting instrumentation sets from the EI and OED methods, the following metrics were utilized:

Auto-MAC

Condition number

Mean absolute peak error

Root-mean-square (RMS) error

As stated before, the auto-MAC provides both a visual and numeric measure of the independence of each mode in the set relative to the others. A lower off-diagonal auto-MAC means that the modes can be distinguished from each other, for example while extracting modal parameters with experimental data. Mathematically, the condition number evaluates the sensitivity of an output to changes in the input. In our case, it also gives a numerical measure of mode independence, where a low value (often below 100) is desired for processing using expansion or other techniques such as sub-structuring [1].

The error metrics were selected to give a quantitative measure of the expansion results. The mean absolute peak error measures the accuracy of the peak response, and the RMS error identifies low or high overall bias in the expanded vibration response.

### 4.1 Auto-Modal Assurance Criterion and Other Direct Metrics.

First, the quality of the various EI instrumentation sets will be compared using the auto-MAC and condition number. Table 2 lists the max off-diagonal auto-MAC and condition numbers of the full candidate set and subassembly sets. Considering the subassemblies individually increased the condition number over an order of magnitude, from 53.4 to 829.1. One possible explanation is the high displacement gradients in the local RC modes affecting the condition number. As a result, choosing slightly different DOF in the computation could end up having a large effect on the condition number. Another factor could also be due to the fixed-base approximation of the RC used in the process. In the full assembly, there are higher-order modes near 2 kHz which contain significant motion of the RC mounting plate. One example is mode 59, shown in Fig. 18. This relative motion would not be adequately captured by fixing the bottom of the RC legs, as was done in the subassembly instrumentation design. This could negatively impact the response recorded for the RC and is one example of a possible detriment that could arise from considering subassemblies separately. This impact will depend on how much boundary conditions of the assembly do not align with the simple subassembly boundary conditions that could be imposed. Additionally, the overall impact of such a mode will also depend on the excitations of interest to analyze during testing.

Case | Max off-diagonal auto-MAC | Condition number |
---|---|---|

Full candidate set | 0.81 | 9.18 |

Assembly set | 0.97 | 53.4 |

Subassembly set | 0.98 | 829.1 |

Subassembly set without mode 59 | 0.93 | 587.7 |

Subassembly set without modes 53, 54, 56, 59 | 0.88 | 76.7 |

Case | Max off-diagonal auto-MAC | Condition number |
---|---|---|

Full candidate set | 0.81 | 9.18 |

Assembly set | 0.97 | 53.4 |

Subassembly set | 0.98 | 829.1 |

Subassembly set without mode 59 | 0.93 | 587.7 |

Subassembly set without modes 53, 54, 56, 59 | 0.88 | 76.7 |

The effect of these local RC leg modes can be further demonstrated by removing them from the set and re-evaluating the condition number. Table 2 shows that removing these modes from the subassembly set can result in auto-MAC and condition numbers close to the assembly set. The relationship between condition number and number of modes kept was studied for the EI sets and the resulting plot is provided in Fig. 19. As more modes are added, the condition number increased, as expected. However, the increase for the subassembly set was drastic after mode 51. Again, this is likely due to the poor approximation of RC legs being fixed since modes 50–60 have significant motion of the plate the RC is mounted on.

The sensitivity of the condition number on each DOF in the set was also evaluated as shown for each case in Fig. 20. Evaluating the removal of any individual DOF emphasizes the relative stability and importance of the DOF in the instrumentation set. Sets with uniaxial accelerometers appeared to be more sensitive to a given DOF while the assembly case with only triaxial accelerometers resulted in locations with high insensitivity to loss. This sensitivity is important to consider during experiments and would inform possible backups if a sensor was lost during testing, for example. The triaxial sensor case also provided the lowest condition number as shown in Table 3, although results were comparable to the other methods.

Case | Max off-diagonal auto-MAC | Condition number |
---|---|---|

Hybrid method | 0.97 | 53.4 |

Uniaxial only | 0.97 | 53.2 |

Triaxial only | 0.96 | 46.0 |

Case | Max off-diagonal auto-MAC | Condition number |
---|---|---|

Hybrid method | 0.97 | 53.4 |

Uniaxial only | 0.97 | 53.2 |

Triaxial only | 0.96 | 46.0 |

Additionally, EI results can be used to warm start the OED process or to provide additional information about the importance of each chosen sensor. OED was able to be implemented without warm starting and it provided a better final set using the full candidate set. Using the EI subset limits the possible selection of gauges for OED. This can create an issue with the overall instrumentation choice since channel budget constraints cannot be enforced directly. Therefore, it is possible for OED to converge with a relatively small set of gauges that can impact overall performance in terms of the metrics considered. A possible alternative to make warm starting more effective is to use EI to down-select to a relatively large possible instrumentation set and use OED to further reduce the set. Adding in an effective channel budget constraint to OED, either minimum or maximum channel definition, would further increase the applicability of warm starting the OED process with EI or other methods.

Using the EI output to define a smaller candidate set for OED was done to determine the importance of the chosen candidate sets in the uniaxial EI case. The top five gauges based on normalized DOF probability are contained in Table 4. The probability values are normalized to the maximum value observed in the OED instrumentation set. DOF indices 14, 62, and 71 were all pertinent for the uniaxial EI case and 2 and 30 are elevated in the EI assembly case. Therefore, OED also can be used to help determine the importance of gauges once selected and could supplement information from EI.

DOF index | Normalized probability values |
---|---|

2 | 1 |

14 | 0.86 |

62 | 0.80 |

71 | 0.67 |

30 | 0.65 |

DOF index | Normalized probability values |
---|---|

2 | 1 |

14 | 0.86 |

62 | 0.80 |

71 | 0.67 |

30 | 0.65 |

Table 5 provides the max off-diagonal auto-MAC and condition number for the instrumentation set chosen with OED. The condition numbers for the OED sets, with the RC locations defined with assembly and subassembly data, were higher than both EI sets for 22 and 60 modes. Off-diagonal auto-MAC values were comparable between the EI and OED sets for 60 modes.

Case | Max off-diagonal auto-MAC | Condition number |
---|---|---|

OED assembly (22 modes) | 0.61 | 153.24 |

OED assembly with RC DOF (22 modes) | 0.36 | 140.78 |

OED assembly (60 modes) | 0.98 | 1931.70 |

OED assembly with RC DOF (60 modes) | 0.99 | 1680.30 |

Case | Max off-diagonal auto-MAC | Condition number |
---|---|---|

OED assembly (22 modes) | 0.61 | 153.24 |

OED assembly with RC DOF (22 modes) | 0.36 | 140.78 |

OED assembly (60 modes) | 0.98 | 1931.70 |

OED assembly with RC DOF (60 modes) | 0.99 | 1680.30 |

### 4.2 Impact of Selection Techniques Applied on Modal Expansions.

SEREP was used to expand the random vibration response from a point in each instrumentation set to a new point on the RC not chosen with either instrumentation selection strategy. A comparison of the expanded response was performed with the different instrumentation selection strategies for both EI and OED. The point, highlighted in Fig. 21, was part of the original candidate set and was chosen as an example where response is desired on a component or subassembly, in a location that was not instrumented.

Simulated test data were generated by adding Gaussian white noise to the responses obtained from the Sierra/SD simulations. A signal to noise ratio of 20 dB was chosen, which is a minimum value recommended by Csurcsia et al. [17] for dynamic measurements. In addition, all data were filtered using a band-pass filter from 20 to 1000 Hz.

The expansion results for the EI instrumentation sets over a 50-millisecond portion of the random vibration loading are plotted in Fig. 22 and the error metrics are summarized in Table 6. The plot and metrics are for the response along the *x*-axis. In addition, the acceleration spectral densities (ASDs) of the response are provided in Fig. 23. Immediately, both the assembly and subassembly responses at the RC point appeared very close to the simulation-generated test response, with some noticeable differences in the spectrum above 800 Hz. For the time history, the error magnitude was low, about 1 g, and most of the discrepancies were at the peaks. Still, the mean absolute peak error was 30% lower for the assembly set relative to the instrumentation set designed with subassembly data. This result was consistent with the higher quality observed earlier in the condition number and auto-MAC, but the changes were not as drastic for the expanded response.

Metric | Assembly set | Subassembly set |
---|---|---|

Mean absolute peak error (g) | 0.788 | 1.130 |

RMS error (g) | −0.962 | −0.963 |

Metric | Assembly set | Subassembly set |
---|---|---|

Mean absolute peak error (g) | 0.788 | 1.130 |

RMS error (g) | −0.962 | −0.963 |

The RC expansion for the OED sets with the RC instrumentation locations defined with assembly and subassembly data is provided in Fig. 24. The corresponding error metrics are provided in Table 7. Both the mean absolute peak and RMS errors were lower for the OED sets, using the assembly and subassembly RC locations, compared to the EI instrumentation choices. The subassembly OED case is denoted with [RC] in Table 7. Figure 25 provides the ASDs of the response for the OED instrumentation sets compared to the EI assembly set. The OED also performed well for frequencies up to 800 Hz for the simulation-generated data representing a set of test data. At 800 Hz, the ASD deviated slightly from what the simulation-test data were, which could indicate some deficiencies in the instrumentation set. This difference was consistent with the deviations observed in the auto-MAC for both the OED and EI sets for modes in the bandwidth of 800–1000 Hz.

Metric | OED | OED [RC] |
---|---|---|

Mean absolute peak error (g) | 0.003 | 0.354 |

RMS error (g) | 0.005 | −0.426 |

Metric | OED | OED [RC] |
---|---|---|

Mean absolute peak error (g) | 0.003 | 0.354 |

RMS error (g) | 0.005 | −0.426 |

Compared to the auto-MAC and condition number, the expansion results were much more consistent for the instrumentation sets. Both quantitatively and qualitatively, each method provided an expanded response that was close to the simulated test response. This is a stark contrast to the condition number, which varied nearly two orders of magnitude depending on the set. One major factor is that the responses used in expansion were filtered to 1 kHz and the instrumentation sets perform much better in this bandwidth, as mentioned in Sec. 3.4. A different expansion point may not perform as well, but these results indicate that all the instrumentation sets adequately capture the dynamics of the RC.

## 5 Conclusion

Both OED and EI offer an effective method for selecting an instrumentation set for a given vibration test. EI is a straightforward, computationally inexpensive approach that provides effective instrumentation sets for vibration testing in terms of modal analysis and SEREP expansion of the results. It also offers the flexibility to use uniaxial gauges, triaxial gauges, or a combination of both. OED is also an effective approach that is more computationally expensive. EI only requires a modal analysis whereas OED requires a FRF array to be developed from the input location for a test to the prospective measurement points. Additionally, the OED optimization implementation can be more expensive than the iterative optimization process used for EI. OED is less sensitive to the impact of local modes in the definition of the instrumentation set. OED provides a natural ranking of importance of each chosen DOF, which can be extremely helpful. EI does not automatically provide this ranking but can be leveraged in such a way to provide arbitrary numbers of redundant, or additional, gauges to fit a specific channel budget. OED provides an effective alternative that can be improved with some small changes. Future work could consider implementing channel budget constraints that would provide needed flexibility to reduce required OED runs. Additionally, using EI to warm start the OED selection approach could be further examined to improve benefits. Also, improving the efficiency of OED relative to EI would help increase its applicability.

## Footnote

## Acknowledgment

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & 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-NA0003525.

This article has been authored by an employee of National Technology & Engineering Solutions of Sandia, LLC under Contract No. DE-NA0003525 with the U.S. Department of Energy (DOE). The employee owns all right, title, and interest in and to the article and is solely responsible for its contents. The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan.^{3}

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

### Appendix: WRCA Vibration Mode Frequencies

Mode number | Frequency (Hz) | Mode number | Frequency (Hz) |
---|---|---|---|

7 | 221.404 | 35 | 1625.64 |

8 | 221.96 | 36 | 1626.09 |

9 | 255.244 | 37 | 1640.83 |

10 | 280.597 | 38 | 1655 |

11 | 282.241 | 39 | 1764.49 |

12 | 477.357 | 40 | 1767.47 |

13 | 596.614 | 41 | 1775.52 |

14 | 674.978 | 42 | 1781.09 |

15 | 809.248 | 43 | 1781.38 |

16 | 824.767 | 44 | 1783.63 |

17 | 833.891 | 45 | 1783.69 |

18 | 874.089 | 46 | 1791.35 |

19 | 885.444 | 47 | 1792.03 |

20 | 942.987 | 48 | 1801.55 |

21 | 982.563 | 49 | 1827.79 |

22 | 1011.34 | 50 | 1837.16 |

23 | 1011.8 | 51 | 1852.93 |

24 | 1022.15 | 52 | 1857.9 |

25 | 1022.4 | 53 | 1874.15 |

26 | 1112.02 | 54 | 1876.09 |

27 | 1206.06 | 55 | 1894.03 |

28 | 1328.69 | 56 | 1958.37 |

29 | 1344.64 | 57 | 1963.51 |

30 | 1439.56 | 58 | 1963.85 |

31 | 1490.66 | 59 | 1976.84 |

32 | 1508.59 | 60 | 1982.53 |

33 | 1509.49 | 61 | 2143.12 |

34 | 1549.44 |

Mode number | Frequency (Hz) | Mode number | Frequency (Hz) |
---|---|---|---|

7 | 221.404 | 35 | 1625.64 |

8 | 221.96 | 36 | 1626.09 |

9 | 255.244 | 37 | 1640.83 |

10 | 280.597 | 38 | 1655 |

11 | 282.241 | 39 | 1764.49 |

12 | 477.357 | 40 | 1767.47 |

13 | 596.614 | 41 | 1775.52 |

14 | 674.978 | 42 | 1781.09 |

15 | 809.248 | 43 | 1781.38 |

16 | 824.767 | 44 | 1783.63 |

17 | 833.891 | 45 | 1783.69 |

18 | 874.089 | 46 | 1791.35 |

19 | 885.444 | 47 | 1792.03 |

20 | 942.987 | 48 | 1801.55 |

21 | 982.563 | 49 | 1827.79 |

22 | 1011.34 | 50 | 1837.16 |

23 | 1011.8 | 51 | 1852.93 |

24 | 1022.15 | 52 | 1857.9 |

25 | 1022.4 | 53 | 1874.15 |

26 | 1112.02 | 54 | 1876.09 |

27 | 1206.06 | 55 | 1894.03 |

28 | 1328.69 | 56 | 1958.37 |

29 | 1344.64 | 57 | 1963.51 |

30 | 1439.56 | 58 | 1963.85 |

31 | 1490.66 | 59 | 1976.84 |

32 | 1508.59 | 60 | 1982.53 |

33 | 1509.49 | 61 | 2143.12 |

34 | 1549.44 |