# Forward/Backward Decomposition for Dispersive Wave Propagation Measurements
## 4 Experimental Validation
### Experimental setup  
The test article, seen in Figure 2, is an aluminum 6061 rectangular beam with dimensions and material properties shown in Table 1. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Property** & **Symbol** & **Value** \\ \hline length & L & 1.75 m \\ width & \(W\) & 3.81 cm \\ thickness & \(T\) & 3.175 mm \\ cross-sectional area & \(A\) & \(W\cdot T\) \\ cross-sectional moment of & \(I\) & \(\frac{1}{12}W\cdot T^{3}\) \\ inertia & \(\rho\) & 2700 kg/m\({}^{3}\) \\ density & \(E\) & 69 GPa \\ Poisson ratio & \(\
u\) & 0.33 \\ flexural modulus & \(G\) & \(\frac{E}{2(1+\
u)}\) \\ Timoshenko correction & \(\kappa\) & \(\frac{\pi^{2}}{12}\) \\ factor & \(F_{0}\) & 0 N \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test article material and geometry. The experimental apparatus, shown in Figure 2, was developed by [PERSON] et al. [7]. It consists of a steel beam with two end brackets featuring a tensioning mechanism. In this work, no tension was applied, so the test article featured simple clamped-clamped boundary conditions. The actuator used to provide excitation pulses was a Macro Fiber Composite (MFC) 2814 adhered near the center of the beam. By virtue of being adhered to the side of the beam, displaced from the neutral axis, and expanding/contracting primarily in the longitudinal direction of the beam, the MFC provides a bending moment that has been found to primarily excite the flexural propagating modes of the beam. The actuator also may excite some longitudinal (bar) modes and torsional modes due to any asymmetries in the placement of the MFC or the alignment of the clamps at the boundary, but these modes are small in amplitude and can be ignored. A two-cycle sine wave with center frequency \(\Omega\) and amplitude modulated by a Hanning window is used as the excitation signal. The time domain representation, along with the magnitude of the amplitude spectrum in the frequency domain, can be seen in Figure 3.  
Center frequencies of 500 Hz and 1500 Hz were selected to demonstrate the algorithm's performance at two different frequency ranges with different amounts of dispersion. The excitation signal was generated in Matlab and output through an NI 9263 analog voltage output DAQ card. The NI DAQ used to send the excitation signal was housed in an NI cDAQ-9184 four-slot chassis, which was connected via Ethernet to a Windows 10 laptop with an AMD Ryzen 9 5900 HS CPU, 16 GB Ram, and running Matlab 2021a. A Trek 50/750 high-voltage amplifier with gain set to 15 V/V was used to amplify the signal to an amplitude of 150 Volts (zero to peak) for the actuator. This excitation level was found to provide a sufficiently strong excitation while avoiding pushing the actuator into its nonlinear regime. A PSV-400 laser Doppler vibrometer was used to obtain non-contact vibration measurements to avoid mass-loading the structure with accelerometers. Forty scan points with a spacing of approximately 2 cm were used, providing measurements on an 80 cm region of the beam between the actuator and one boundary. A PSV-A-420 geometry scan unit  
Figure 2: Experimental setup showing the view of the test article from the laser Doppler vibrometer. The scan region is labeled, along with stars highlighting the first and last scan points. The close-up pictures in call-outs depict, from left to right, the MFC actuator, two of the four strain gages, and the tensioning mechanism, which is present at both ends of the beam. was used to accurately measure the locations of the scan points on the beam, and the data acquisition was triggered by the excitation signal sent to the actuator. The PSV-400 sampling frequency for data acquisition was adjusted according to the center frequency of the excitation signal in order to obtain at least 20 samples per waveform, which is sufficient to capture the frequency content of interest without over- or under-sampling. Since the structure contains little damping, it continues ringing after the initial wave and subsequent reflections pass, so the best total sampling time is not obvious.