## 3 Experimental and Numerical Setup  
The experimental equipment for penetration will be introduced in the following four modules: control, transmission, penetration, and data acquisition/output. As shown in Figure 1, the control module consists of a power supply, a programmable controller, a driver, an encoder, and several switches, and has stable low-speed torque characteristics, dynamic acceleration, and deceleration performance. The transmission module consists of a synchronous belt module, limit switches, and mounting feet. Compared to the ball screw drive and worm gear drive, the timing belt module has the advantages of a wider adjustable speed range, higher integration, safety, and less noise. Moreover, the forces along the axial direction and the two orthogonal lateral directions on the intruder are collected by the three-axis pressure sensor and transmitted to the computer via the multichannel digital transmitter to complete data storage, with the sensor's data precision of the three-axis direction able to reach a precision of 10\({}^{-6}\) N.  
The experimental setup is shown in Figure 2. A cylindrical container, with length 600 mm and inner diameter 230 mm, was filled to a depth of 535 mm with acrylic beads (diameter \(d_{\text{p}}=10\) mm and density \(\rho_{acr}=1190\) kg/cm\({}^{3}\)). The bed was prepared by pouring beads into a container, leveling it with a scraper. After each experimental run, the box was emptied and refilled. When the experiment started with an acrylic rod (diameter \(d_{\text{i}}=40\) mm; length \(L_{\text{i}}=450\) mm) penetrating the granular bed at a constant velocity of \(V\), the rod was attached to an L-shaped connector and driven by a servo motor. The maximum penetration depth was set to half the granular bed height to avoid the bottom boundary effect. The triaxial penetration resistances were measured by a force sensor as a function of time. Since the lateral forces caused by asymmetric particle distribution are much smaller than the axial forces, the lateral forces will not be discussed in this study. Figure 1: The four modules included in the experimental system. The DEM simulation setup is shown in Figure 3. Some key parameters were discussed in simulations to assess their impact. The simulations were performed as follows: particles were generated within a cylindrical domain and allowed to fall under the influence of gravity until the cylindrical container was filled to the same particle number and height as observed in the experiment. The initial solid volume fraction was maintained by using the same coefficient of friction as during compression. Once the particles had settled and the granular bed had reached a stable state, an intruder was inserted into the center of the particle bed at a constant vertical velocity. The simulation and experimental parameters are shown in Table 1.  
Figure 3: Schematic diagram of the simulation equipment. Figure 2: A constant speed penetration device and an experimental setup sketch. The drag force curves of five nose shapes exhibit similar trends at different constant speeds, as shown in Figure 5. The average drag force can be defined as  
\[F_{a}=\frac{\int_{0}^{Z_{0}}F(Z)dz}{Z_{0}} \tag{10}\]  
where \(Z_{0}\) represents the maximum distance of penetration, and we only research the drag force within 270 mm. Figure 6 illustrates a typical relationship between drag force and penetration velocity. It was observed that the drag force remains independent of velocity at lower speeds, exhibiting minor fluctuations across multiple datasets. However, as velocity increases, a distinct upward trend in the drag force is evident for the five different nose shapes examined. The experimental apparatus's constraints limit the penetration speed to a range of 10 to 1000 mm/s. Exploring beyond this velocity range to ascertain the presence of significant turning points in the relationship requires simulation. In this study, the DEM simulations and experiments of acrylic beads and glass beads penetrated by the same intruder were compared.