**5. Experimental Validation**

In this section, the theoretical results are compared with experimental data. The experimental setup is illustrated in Figure 11. A cart with rollers is used as a moving pin along the beam. Accelerometers are attached at locations 300 mm and 400 mm. The mass of the accelerometers is assumed to have a negligible effect. Each accelerometer weighs 1.7 gm, not including cables, which is 0.2% of the weight of the beam. At the free end, an electromagnet is used to simulate the point mass. The specifications of the experiment are listed in Table 4.

**Figure 11.** Illustration of the experimental setup.


**Table 4.** Specifications of the experimental setup.

The accelerometers are single axis PCB 353B17. They are connected to a NI-9234 IEPE analog input module seated in an NI cDAQ-9172 chassis. A PCB 086C01 modal hammer with a white ABS plastic tip is used to excite the beam at 300 mm. Five tests under each condition are conducted and averaged in the frequency domain to generate frequency response functions (FRFs) using the *Hv* algorithm [15]. The FRFs for the different tests are plotted in Figures 12–15. The vertical red dashed lines represent the theoretical modes computed from the proposed model. To better understand the differences, the modes are extracted and tabulated in Tables 5–8.

For the four test conditions evaluated, the difference between the theoretical calculations and experimental results for modes 4 and 5 are non-trivial. Three possible explanations for these differences are (1) the electromagnet vibrates separate from the beam, (2) the beam vibrates within the rollers, and (3) the rotational inertia from a large mass at the end of a long beam impacts the higher frequencies. In Figures 13 and 15, the coherence for mode 5 drops significantly such that it cannot be said with certainty that the frequencies are correct. Percent difference is used to quantify how different the theoretical frequencies are from the experimental. All frequencies fall below 20% difference with the exception of mode 4 in Table 5.

**Figure 13.** FRF for pinned at *a* = 100 mm and *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.2.

**Figure 14.** FRF for pinned at *a* = 150 mm and *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.2.

**Figure 15.** FRF for pinned at *a* = 200 mm and *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.2.


**Table 5.** Pinned at *a* = 50 mm and *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.2.

**Table 6.** Pinned at *a* = 100 mm and *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.2.


**Table 7.** Pinned at *a* = 150 mm and *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.2.


**Table 8.** Pinned at *a* = 200 mm and *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.2.


#### **6. Conclusions**

A high-rate experimental testbed is studied. The testbed is characterized as being a clamped-pinned-free beam with a mass at the free end. Euler-Bernoulli beam theory is applied to derive the transcendental equation for a general case applicable to the system pinned at an arbitrary location and with an arbitrary mass. The eigenvalues and mode shapes were presented under various test conditions. Experimental tests were conducted and results compared with the theoretical calculations of the first five natural frequencies. The comparison of results exhibited a good match in frequency values for the first three modes. The errors increase with the higher modes. The difference in higher modes can be attributed to the electromagnet vibrating separate to the beam, the beam vibrating within the rollers, and the rotational inertia of the mass not taken into consideration. Nevertheless, the percent difference of all modes between the theoretical and experimental values fell below 20% except for one case. These results confirm that within reason, the theory matches the experimental results.

The analytical model developed here can be useful in the design and numerical assessment of structural health monitoring solutions designed for systems operating in high-rate dynamic environments. Furthermore, the experimental quantification of the error in the higher modes validates and defines the bounds for which the high-rate experimental testbed is best utilized. Future work will include the evaluation of algorithms and methodologies for the SHM of structures experiences high-rate dynamic events.

**Author Contributions:** J.H. performed the analytical work, numerical simulations, and experimental activities, and led the write-up; J.D. and S.L. co-led the investigation; A.D. assisted in the collection of experimental data and validation of the analytical work. All authors contributed to preparing and proofreading the manuscript.

**Funding:** The material in this paper is based upon work supported by the Air Force Office of Scientific Research (AFOSR) award numbers FA9550-17-1-0131 and FA9550-17RWCOR503, and AFRL/RWK contract number FA8651-17-D-0002. Opinions, interpretations, conclusions and recommendations are those of the authors and are not necessarily endorsed by the United States Air Force. Additionally, this work was partially supported by the National Science Foundation Grant No. 1850012. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

**Conflicts of Interest:** The authors declares no conflict of interest.
