## 2 A Helicopter with Multiple Landing Gears  
The helicopter referenced in this research is a light military model called the Eurocopter Fennec (see Figure 1). This helicopter has a maximum takeoff weight of 2250 kg [13]. It is equipped with skids that were designed and tested with an initial sink-speed of 2.54 m/s based on the standards outlined in FAA Part 27 [14]. Landing gears with wheels are better than those with skids during contact with the ground in differing landing scenarios [2]. Moreover, light helicopters with three landing gears are more popular than those with four. Hence, we modeled a helicopter following Fennec but with three landing gears instead of the original skids in this study. In particular, we assumed that the landing gears were equipped with MR dampers, which have been regarded as a promising candidate to improve landing gear performance by controlling the damping force [9; 15; 16; 17]. In our previous research, a single landing gear equipped with an MR damper for a small aircraft was designed [16; 17], as can be seen in Figure 2. This landing gear was designed for a maximum sprung weight of 680 kg and an initial sink-speed of 3 m/s. Accordingly, the total maximum allowable weight of the aircraft with three landing gears of this type is 2040 kg. This maximum weight was slightly less than our helicopter weight (2040 kg \(<\) 2250 kg). However, the designed initial sink speed of this aircraft was larger than that of the helicopter (3 m/s \(>\) 2.54 m/s). Thus, the landing gear equipped with the MR damper in our previous research was able to be used in our helicopter model with similar parameters that can be referred to in [16; 17]. The 3D model of our helicopter equipped with three landing gears was drawn using RECURDYN (Figure 3). In this model, the 6-DOF helicopter body was assumed to be rigid and was set up for the initial parameters--i.e., the aircraft mass, moment of inertia, initial sink speed, initial pitch, and roll angle. It was assumed that there was no input power into the rotary wings, and there was no lift force on the helicopter. Hence, the power-off landing,  
Figure 1: Eurocopter Fennec. Figure 2: Structure of a landing gear equipped with an MR damper. which, as mentioned earlier, is the most dangerous case, was simulated in this research. The landing gear has mainly two parts: the upper part consists of the cylinder of the MR damper and a structure that connects to the bottom of the helicopter with fixed joints; and the lower part connects the damper piston and wheel assembly, which has 1-DOF movement along the cylinder. Thus, the helicopter with the three landing gears has 9-DOF in total. In this research, the damping force (\(F^{d}\)) of the \(i\)th landing gear (\(i\) = {1,2,3}: nose landing gear, left main landing gear, and right main landing gear, respectively) is the combination of hydraulic force (\(F^{v}\)), air force (\(F^{u}\)), and MR force (\(F^{MR}\)) which is described below:  
\[F^{d}_{i}=F^{p}_{i}+F^{d}_{i}+F^{MR}_{i} \tag{1}\]  
\[F^{p}_{i}=C\dot{s}_{i} \tag{2}\]  
\[F^{a}_{i}=A_{P}\bigg{(}p_{0}\bigg{(}\frac{V_{0}}{V_{0}-A_{PSi}}\bigg{)}^{n}-p_ {ATM}\bigg{)} \tag{3}\]  
\[F^{MR}_{i}=f(u_{i},\dot{s}_{i}) \tag{4}\]  
where \(s,\dot{s}\), are the stroke and stroke velocity, respectively. Without any the magnetic field applied, the MR fluid can be modeled as a Newtonian fluid. However, it behaves like a Bingham fluid if magnetic field is present and is able to sustain some shear stress in the direction normal to the magnetic flux. In the Bingham plastic model for MR fluid in [18], the yield shear stress of MR fluid, which is the origin of the MR force (\(F^{MR}\)), is related directly to the magnetic field intensity. The MR damper model in this paper generates the magnetic field by the electrical current \(u\) through the coil of the damper core at the top of the piston. The detailed model of the MR core is given in [15]. Moreover, the relationship between the tire and ground is set up by using \"Tire Tool-Kit\" inside RECURDYN.