*3.2. Mechanical Model*

The reluctance coil gun mechanical system shown in Figure 15 has been described in a former work [12]. This model takes into account the transmission of the movement from the plunger to the ball through an aluminium lever. Movement can be split into three phases as shown in Figure 21:

**Figure 21.** Plunger speed in m·s−<sup>1</sup> over the time in *<sup>s</sup>*.

• Phase 1: Acceleration of the plunger without contact on the lever, due to the magnetic force as shown in Equation (2) without contact on the lever. Force *Fmagneto* depends non-linearly on plunger position and current *I*. A look-up table (LUT) interpolates linearly the value of *F* using the simulations performed with *FEMM 4.2*.

$$m\_P \ddot{\mathbf{x}} = F\_{\text{Magneto}}(\mathbf{x}, I) \tag{2}$$

• Phase 2: Impact on the lever corresponding to an elastic shock when plunger hits it at a distance *R*<sup>1</sup> from its rotation centre. Kinetic energy is conserved as described in Equation (3).

$$\frac{1}{2}m\_p \mathbf{x}\_{\text{Initial}}^{\cdot \cdot \cdot} = \frac{1}{2}m\_p \mathbf{x}\_{\text{Final}}^{\cdot \cdot \cdot} + \frac{1}{2}m\_B \frac{R\_2^2}{R\_1^2} \mathbf{x}\_{\text{Final}}^{\cdot \cdot \cdot} + \frac{1}{2}I\_{Lverv} \frac{\mathbf{x}\_{\text{Final}}^{\cdot \cdot}}{R\_1^2} \tag{3}$$

where *JLever* is the inertial moment of the lever, *R*<sup>1</sup> and *R*<sup>2</sup> the distances between the lever axis and respectively the plunger impact point and the ball impact point as shown in Figure 15. This leads to a plunger speed just after the shock equal to the *xfinal* given in Equation (4).

$$\mathbf{x}\_{Final} = \sqrt{\frac{m\_p}{m\_p + m\_B \frac{R\_2^2}{R\_1^2} + \frac{J\_{Lver}}{R\_1^2}}} \mathbf{x}\_{limit} \tag{4}$$

• Phase 3: the plunger is accelerated in contact with the lever, which is also in contact with the ball. This means that the lever applies a force on the plunger in subtraction of the magnetic force as shown in Equation (5). This force is an inertial one due to the acceleration of the ball and the lever as shown in Equation (6). It is important to note that theoretically, the speeds of the ball, lever and plunger are not equal after the shock, but in reality they are due to the elastic deformation of the ball as shown in the slow motion picture in Figure 22.

$$m\_p \pounds = F\_{Magnetic}(\mathbf{x}\_\prime \, I) - F\_{Lver} \tag{5}$$

where

$$F\_{Lerv} = \frac{f\_{Lver} + m\_B R\_2^2}{R\_1} \theta \tag{6}$$

with:

$$J\_{Lerror} = \frac{m\_{Lerror}R\_2^2}{3}$$

For small *θ* angles, ¨ *θ x*¨ *R*1 , this leads to :

$$\frac{m\_p R\_1^2 + f\_{L \text{rev}} + m\_B R\_2^2}{R\_1^2} \ddot{\mathbf{x}} = F\_{\text{Magnetic}}(\mathbf{x}, I) \tag{7}$$

**Figure 22.** Ball deformation after phase 2.

Implementation of this three-phase mechanical model has been done using Matlab Simulink. Figure 23 shows the mechanical part model of a 1 coil electromagnetic launcher, whereas Figure 24 shows the mechanical part model of a 4 coils electromagnetic launcher. It is important to note that, as shown in Figure 22, plunger, lever and ball are in contact after the shock. This is due to the softness of the ball, and because the ball is close to be in contact with the lever before the impact. Thus, the hypothesis of a perfect elastic shock is almost verified except for a transitional short period of less than one millisecond after the shock of the rod on the lever.

**Figure 23.** Mechanical part model of a 1 coil electromagnetic launcher.

**Figure 24.** Mechanical part model of a 4 coils electromagnetic launcher.

However, in order to understand more accurately what is going on during this transition, impact of the plunger on the lever and impact of the lever on the soft ball will be modelled in another work (for example using MSC ADAMS software), but this is out of the scope of this paper.

As shown in Figures 23 and 24, the only difference between both models is the number of inputs of the force look-up table (LUT) block interpolating linearly the value of *F* using the simulations performed with *FEMM 4.2*. In the case of a four-coil EML, there are five inputs: plunger position, and the currents on each of the four coils.

## **4. Reluctance Coil Gun Simulations**
