# Application of Tooth Gear Impact-Echo System for Repeated and Rapid Data Acquisition
## 2 Theoretical Background  
### Impact-Echo Vibration  
The IE method provides the thickness information of a concrete slab or its delamination by measuring the dominant frequency of vibration. After applying an impact source onto the surface of the concrete, the measured vibration is analyzed in the frequency domain where the frequency at the highest amplitude corresponds to the thickness mode. The relationship between the measured frequency (\(f_{thickness}\)) and the thickness (\(h\)) is  
\[f_{thickness}=\frac{\beta V_{L}}{2h} \tag{1}\]  
where \(\beta\) is the correction factor (usually taken to be 0.96) and \(V_{L}\) is the longitudinal wave velocity in the concrete. [PERSON] and [PERSON] demonstrated that the thickness mode frequency was at a zero-group velocity motion of the first symmetric vibration (S1-ZGV) governed by plate guided waves [4]. Moreover, a correction factor was required to compensate for the shear motion of the plate, depending on a ratio between the shear and longitudinal waves or Poisson's ratio. Many studies have utilized the concept of the S1-ZGV to detect internal defects using depth information [13; 14; 15; 16]. Using Equation (1), the thickness of internal delamination-like defects can be estimated using information of the apparent velocity \((\beta V_{L})\).  
However, in the case of shallow delamination, the thickness mode frequency is difficult to measure. This is because the impact source excitation frequencies have limited range up to the thickness mode frequency of shallow delamination. Instead, low frequencies from the flexural vibration of the plate are much more dominant compared to the thickness mode frequency. [PERSON] investigated the excitability of thickness mode frequencies based on a ratio between the diameter (\(a\)) and thickness (\(h\)) of the plate under certain boundary conditions, demonstrating that the thickness mode frequency becomes dominant when the ratio (\(a/h\)) is lower than 2. Therefore, the IE testing results from shallow delamination mostly exhibit low frequencies [17]. The fundamental flexural vibration mode frequency (\(f_{flexural}\)) of rectangular delamination is presented as  
\[f_{thickness}=\frac{k_{DL}^{2}\pi}{2h^{2}}\sqrt{\frac{Eh^{2}}{12\rho(1-v^{2})}} \tag{2}\]where \(E\) is the elastic modulus, \(v\) is the Poisson's ratio, \(\rho\) is the density, and \(k_{DL}^{2}\) is the dimensionless frequency based on width to depth ratio of the delamination [18].  
The excitation frequencies are directly related to the contact duration \((t_{c})\) of impactors onto the concrete surface. The shorter contact duration generates a higher frequency range of excitation. Based on contact mechanics [19], the contact duration is calculated by the sphere (impactor) and half-space (concrete) interaction, as follows:  
\[t_{c}=2.536D\rho_{i}^{2/5}V_{i}^{-1/5}\Bigg{[}\frac{\left(1-v_{i}^{2}\right)}{ E_{i}}+\frac{\left(1-v_{c}^{2}\right)}{E_{c}}\Bigg{]}^{2/5} \tag{3}\]  
where \(D\) is the diameter of the impactor, \(\rho_{i}\) is the density of the impactor, \(V_{i}\) is the impact velocity, \(v_{i}\) and \(E_{i}\) are the Poisson's ratio and elastic modulus of the impactor, respectively, and \(v_{c}\) and \(E_{c}\) are the Poisson's ratio and elastic modulus of concrete, respectively. Equation (3) expresses that the contact duration is directly related to the size of the impactor, beyond its material properties. The smaller size of the impactor reduces the contact duration, corresponding to a higher frequency range of excitation.