# Experimental and Numerical Study on the Low Velocity Impact Behavior of ABR Pipe
## 3 Experimental Results and Discussion
### Impact Force  
In the whole impact process, various parameters will affect the impact force. In this study, for all specimens, the hammer head shape of the drop hammer, the boundary conditions of the specimens and the materials of the specimens are the same. Therefore, only the geometric parameters of the specimen, the weight of the drop weight and the height of the drop weight can significantly affect the peak impact force. The peak impact force of each specimen was extracted according to the impact time curve, as shown in Table 3. This paper will use a dimensional analysis method to analyze the peak impact force obtained from the test. The dimensional harmony principle as the basic principle of dimensional analysis, also called dimensional consistency principle or the principle of dimensional homogeneity, describes the relationship between complete physical equations in both their differential and integral forms. Its various dimensions must be consistent, that is only two types of the same physical quantity for addition and subtraction. In the analysis process of this paper, we choose the falling weight (m, dimension \([M]\)), the initial impact velocity (\(v_{0}\), dimension \([L][T]^{-1}\)), the ring stiffness (\(S_{p}\), \([M][T]^{-2}\)), peak impact force (\(F_{\max}\), dimension \([M][L][T]^{-2}\)), pipe quality (\(m_{2}\), The dimension \([M]\)) as \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Specimen** & \(W\mathbf{\delta_{a}}\) & \(\mathbf{E_{a}}\) & \(\mathbf{E_{sa}}\) & **Specimen** & \(W\mathbf{\delta_{a}}\) & \(\mathbf{E_{a}}\) & \(\mathbf{E_{sa}}\) \\ \hline M1-H1 & 12.876 & 0.136 & 93.749 & M3-H1 & 0.448 & 24.578 & 54.844 \\ M1-H2 & 25.670 & 0.190 & 134.284 & M3-H2 & 0.623 & 48.784 & 78.310 \\ M1-H3 & 51.241 & 0.281 & 181.429 & M3-H3 & 0.924 & 97.152 & 105.171 \\ M1-H4 & 76.769 & 0.329 & 231.982 & M3-H4 & 1.204 & 145.464 & 120.832 \\ M1-H5 & 102.303 & 0.383 & 265.241 & M3-H5 & 1.355 & 193.481 & 142.792 \\ M2-H1 & 19.409 & 0.299 & 64.109 & M4-H1 & 0.689 & 29.941 & 43.466 \\ M2-H2 & 38.640 & 0.420 & 91.273 & M4-H2 & 0.954 & 59.398 & 62.282 \\ M2-H3 & 77.044 & 0.604 & 126.241 & M4-H3 & 1.337 & 117.688 & 88.013 \\ M2-H4 & 115.423 & 0.763 & 149.109 & M4-H4 & 1.949 & 177.120 & 90.858 \\ M2-H5 & 153.733 & 0.853 & 177.975 & M4-H5 & 2.244 & 234.843 & 104.673 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The energy absorption capacity of specimens. the basic physical quantities. It is assumed that basic physical quantities meet physical equations, as shown in Formula (10), and their unit relations are shown in Formula (12). \[f\big{(}m,v_{0},S_{p},F_{\max},m_{2}\big{)}=0 \tag{10}\]  
\[m^{y_{1}}.v_{0}^{y_{2}}.S_{p}^{y_{3}}.F_{\max}^{y_{4}}.m_{2}^{y_{5}}=0 \tag{11}\]  
\[[M]^{y_{1}+y_{4}+y_{5}}.[L]^{y_{2}+y_{4}}.[T]^{-y_{2}-2y_{3}-2y_{4}}=0 \tag{12}\]  
The dimensional matrix of the equation is:  
\[ \begin{bmatrix}1&0&1&1&1\\ 0&1&0&1&0\\ 0&-1&-2&-2&0\end{bmatrix} \]  
The basic solution of the linear homogeneous system is:  
\[ \begin{bmatrix}-\frac{1}{2}&-1&-\frac{1}{2}&1&0\\ -1&0&0&0&1\end{bmatrix} ^{T}\]  
Five physical quantities, involving three basic dimensions, are selected for the process of low velocity impact. This physical and mechanical process can be described by two dimensionless relations. \[F(\pi)=0 \tag{13}\]  
\[\pi_{1}=m^{-\frac{1}{2}}.v_{0}^{-1}.S_{p}^{-\frac{1}{2}}.F_{\max}^{1} \tag{14}\]  
\[\pi_{2}=m^{-1}.m_{2}^{1} \tag{15}\]  
According to Formula (14), the function relation Formula (16) of peak load can be rewritten:  
\[F_{\max}=\frac{1}{\pi_{1}}.\sqrt{m}.v_{0}.\sqrt{S_{p}} \tag{16}\]  
The constant \(\pi_{1}\) has some corresponding relation with \(\pi_{2}\) and is related to \(m_{1}\) and \(m_{2}\). By introducing the relation \(\varphi\), Formulas (15) and (16) are transformed into Formula (17). \[F_{\max}=\varphi\Big{(}\frac{m_{2}}{m}\Big{)}.\sqrt{m}.v_{0}.\sqrt{S_{p}} \tag{17}\]  
The expression of impact coefficient \(\varphi\) is supposed as the Formula (18), where a and b are undetermined constants.