## 2 Modeling of the Proder-Object System and the Numerical Analysis Method  
As shown in Figure 1a, the prodder that is described in Ref. [26] is composed of seven components, namely, (1) electromechanical actuator; (2) adjusting load spring; (3) aluminum cylinder for mechanical connection between springs; (4) aluminum cylinder for mechanical connection between spring and steel rod; (5) steel rod; (6) triaxial accelerometer with the y-axis aligned with the steel rod; and, (7) shell of the prodder. The air cavity between a thin upper casing and the explosive makes the landmine structure more compliant than other buried clutters, such as rocks, tree roots, metal shreds, etc. The prodder is designed to distinguish landmines from those clutters by sensing buried objects' resonance behavior, including vibration intensity and frequency response. In the detection procedure, the rigid steel rod is inserted into soil to get direct contact with the buried object to form a prodder-object coupling system. Subsequently, a programmable data acquisition board generates a chirp signal to control a pulsewidth-modulated (PWM) power driver that excites the electromechanical actuator of the prodder to vibrate. The steel rod stimulates the buried object with the mechanical force output from the actuator resultantly. For safety consideration, the output force of the prodder should be controlled below 10 N by adjusting the PWM power driver current. Meanwhile, the acceleration of the contact point is measured by a light silicon MEMS accelerometer that is mounted on the steel rod and then transmitted to the acquisition board for data analysis. After Analogue-to-Digital (A/D) conversion, filtering and Fast Fourier Transformation (FFT) processing of the measured data, the frequency response of the buried object can be acquired. Landmine structure is typically much more compliant than those rigid clutters, thus the vibration intensity of landmine would be much larger and its natural frequency would focus on the low frequency band when compared to other rigid clutters. Since these two resonance behaviors are the key features in distinguishing landmines from other rigid clutters, the resonance mechanism of the prodder-object coupling system should be further studied to make the influencing factors and their influencing mechanism clear. The kinetic model of this prodder-object system can be obtained by lumped parameter method; this method has been used to investigate acoustic landmine detection [27; 28]. As indicated in Figure 1b, \(k_{1}\) and \(c_{1}\) represent the stiffness and damping constants of adjusting spring respectively; and, \(k_{2}\) and \(c_{2}\) represent the viscoelastic parameters of the moving part of buried object's upper casing; Stiffness constant \(k_{3}\) and damping constant \(c_{3}\) of the prodder-object coupling part are equivalent to the viscoelastic constants of steel rod. While \(m_{1}\) and \(m_{2}\) represent the mass of the moving parts of prodder and buried object respectively. Figure 1: **(a)** Mechanical sketches of the sensorized prodder; **(b)** Kinetic model of “prodder-object” system. According to the lumped model that is shown in Figure 1b and linear vibration theory, we can obtain the following kinetic equations:  
\[ \begin{array}{l}m_{1}\ddot{x}_{1}\!=F-k_{1}x_{1}\!-\!c_{1}\dot{x}_{1}\!-\!k_{3 }(x_{1}\!-\!x_{2})\!-\!c_{3}\big{(}\dot{x}_{1}\!-\!\dot{x}_{2}\big{)}\\ m_{2}\ddot{x}_{2}\!=k_{3}(x_{1}\!-\!x_{2})\!+\!c_{3}\big{(}\dot{x}_{1}\!-\!\dot {x}_{2}\big{)}-\big{(}k_{2}x_{2}\!+\!c_{2}\dot{x}_{2}\big{)}\end{array}  \tag{1}\]  
The above equations in matrix form can be written, as follows:  
\[\textbf{K}\textbf{X}+\textbf{C}\dot{\textbf{X}}+\textbf{M}\ddot{\textbf{X}}= \textbf{F} \tag{2}\]  
where \(\textbf{X}=\text{diag}(x_{1},x_{2})\) is the displacement matrix dependent on the moving part of prodder and buried object, thus \(\dot{\textbf{X}}\) and \(\ddot{\textbf{X}}\) are the corresponding velocity and acceleration matrixes, respectively; while \(\textbf{K}=\left[