## 2 Principle of Variable-Frequency Hammering  
Acoustic investigations of hammering tests have often concentrated on the analysis of single-frequency hammering's acoustic features [14], determining damage on the basis of variations in the excitation effect obtained by hammering the detection area. The excitation effect obtained by hammering the detection area is reflected acoustically using the power spectral density (PSD) [15], as depicted in Figure 1, which is a PSD schematic diagram of single-frequency and variable-frequency hammering to determine the excitation effect of the detection area.  
During the study of hammering detection, we discovered that the excitation effect of the damage area differs for different hammering frequencies when the detection area contains just one type of damage, as observed on the left side of Figure 1. Furthermore, the highest power, which corresponds to the hammering frequency at which the shaded portion is the largest, is also the optimal excitation frequency. The effective information gleaned from a single-frequency hammering; however, is insufficient to discern between the various types of damage when more than one type of damage is present in a hammering area.  
Unlike with the traditional hammering method, the proposed variable-frequency hammering method does not start with acoustic processing, which improves the useful information that can be used to identify the damage by activating the detection area with various hammering frequencies. On the right-hand side of Figure 1, a schematic representation of the acoustic data obtained using the variable frequency hammering method is illustrated. To identify more intricate damages, the excitation effect produced by altering the hammering frequency allows the highest number of shaded sections to be obtained. This provides the biggest informational contrast with other detection areas.  
Therefore, an identification method based on variable-frequency hammering is proposed in this study. As shown in Figure 2, for the existing single-frequency hammering method, the acoustic data obtained from the single-frequency hammering and extracted acoustic features were used for machine learning. In this study, the acoustic data of multiple hammer frequencies obtained with variable-frequency hammering were used to increase the effective information in the vertical direction, as shown in Figure 2. The acoustic features were obtained by acoustic feature extraction, and the dimension was then reduced to the same dimension as that of the single-frequency hammering through feature selection; finally, the selected features were used for machine learning. This study aims to increase the valid acoustic information for concrete damage classification without increasing the acoustic feature dimensionality or causing a dimensional disaster.  
Figure 1: Schematic diagram of the excitation effect under different hammering methods.  
Figure 2: Diagram of the difference between variable-frequency hammering and single-frequency hammering.  
The flowchart of the proposed method is shown in Figure 3. After recording the acoustic signal using a microphone, it was divided into two processes: training and testing. The first process is the training process, including the use of Mel-frequency cepstrum coefficients (MFCCs) were used for feature extraction of acoustic signals obtained from single-frequency and variable-frequency hammering and feature selection of MFCCs for variable-frequency hammering. Subsequently, SVMs were used to model the MFCCs for single-frequency and variable-frequency hammering. During the testing process, the MFCCs of single-frequency and variable-frequency hammering after feature selection were matched with those of the established model. In this study, two different models were established and compared based on single-frequency and variable-frequency hammering.