# Modeling and Optimization of an Indirect Coil Gun for Launching Non-Magnetic Projectiles

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## Abstract

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## 1. Introduction

- In Section 2, the principles of coil guns are presented with limitations concerning their modeling;
- In Section 3, a mechanic and electromagnetic model is proposed and implemented using finite elements simulation tools for the electromagnetic part of the coil gun, and using Matlab Simulink for the mechanical and electrical aspects;
- In Section 4, the simulation results are discussed in order to optimize the parameters of an existing indirect coil gun.

## 2. Principles of Coil Guns

#### 2.1. Physical Concept

#### 2.2. Electromagnetic Theory and Simulation Software

- N: number of turns of the coil;
- I: current in the coil (A);
- R: reluctance (${H}^{-1}$);
- $\Phi $: flux (Wb).

#### 2.3. Electrical Model

#### 2.4. Mechanical Model

## 3. Mixed Electrical and Mechanical Model of the Indirect Coil Gun

#### 3.1. Electrical Model

#### 3.2. Mechanical Model

- Phase 1: the plunger is accelerated without contact on the lever. Acceleration is caused by the magnetic force only, as shown in Equation (8). Force ${F}_{magneto}$ is a function of the plunger position and current I. Again, this dependence is modeled using an LUT interpolating linearly the value of F using the simulations performed with FEMM 4.2.$${m}_{p}\ddot{x}={F}_{Magneto}(x,I).$$
- Phase 2: an elastic shock occurs when plunger hits the lever. Kinetic energy is conserved as described in Equation (9). In theory, the speeds of the ball, lever, and plunger are not equal after the shock, but in reality they are all moving together thanks to the deformation of the ball which ensures that the contact is permanent after the shock, as shown in the slow motion picture in Figure 12.$$\frac{1}{2}{m}_{p}{\dot{{x}_{Init}}}^{2}=\frac{1}{2}{m}_{p}{\dot{{x}_{Final}}}^{2}+\frac{1}{2}{m}_{B}\frac{{R}_{2}^{2}}{{R}_{1}^{2}}{\dot{{x}_{Final}}}^{2}+\frac{1}{2}{J}_{Lever}\frac{{\dot{{x}_{Final}}}^{2}}{{R}_{1}^{2}}$$This leads to a plunger speed just after the shock equal to the ${x}_{final}$, given in Equation (10).$$\dot{{x}_{Final}}=\sqrt{{\displaystyle \frac{{m}_{p}}{{\displaystyle {m}_{p}+{m}_{B}\frac{{R}_{2}^{2}}{{R}_{1}^{2}}+\frac{{J}_{Lever}}{{R}_{1}^{2}}}}}}\dot{{x}_{Init}}$$
- Phase 3: plunger is accelerated in contact with the lever, which one 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 (11). This force is inertial due to the acceleration of the ball and the lever, as shown in Equation (12).$${m}_{p}\ddot{x}={F}_{Magneto}(x,I)-{F}_{Lever}$$$${F}_{Lever}=\frac{{J}_{Lever}+{m}_{B}{R}_{2}^{2}}{{R}_{1}}\ddot{\theta}$$$${J}_{Lever}={\displaystyle \frac{{m}_{Lever}{R}_{2}^{2}}{3}}.$$For small $\theta $ angles, $\ddot{\theta}\simeq \frac{\ddot{x}}{{R}_{1}}$, this leads to$$\frac{{m}_{p}{R}_{1}^{2}+{J}_{Lever}+{m}_{B}{R}_{2}^{2}}{{R}_{1}^{2}}\ddot{x}={F}_{Magneto}(x,I).$$

## 4. Indirect Coil Gun Simulation Results

- Distance from lever axis to plunger touch point: ${R}_{1}$ = 13 cm;
- Distance from lever axis to ball touch point: ${R}_{2}$ = 24 cm;
- Coil length: ${L}_{Coil}=11.5$ cm;
- Coil number of turns: ${N}_{Coil}=1050$ turns;
- Plunger iron rod diameter: ${D}_{Plunger}=25$ mm;
- Plunger iron rod length: ${L}_{Plunger}=11.5$ cm;
- Plunger iron rod mass: ${m}_{Plunger}=690$ g;
- Plunger extension diameter: ${D}_{Ext}=18$ mm;
- Plunger extension length: ${L}_{Ext}$ cm;
- Plunger extension mass: ${m}_{Ext}=0.68\ast {L}_{Ext}$ (in m);
- Distance from coil to lever: ${D}_{Lever}=4$ cm;
- Vertical lever mass: ${m}_{Lever}=80$ g;
- Ball mass: ${m}_{Ball}=450$ g;
- Capacitor value: 4700 $\mathsf{\mu}$F;
- Capacitor charge voltage: 450 V;
- Coil resistance: 2.5 $\Omega $.

#### Comparison with Previous Experimental Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Example of magnetic field saturation [3].

**Figure 14.**Impact of initial plunger position and plunger extension length on the ball kicking speed.

**Figure 16.**Simulation using the Tech United [5] one phase configuration.

**Figure 17.**3D diagram of a system with multiple coils [8].

**Table 1.**Inductance of the coil depending on the plunger position with a coil current equal to $10A$.

Position (mm) | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 | 130 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Inductance (mH) | 96 | 92 | 86 | 80 | 73 | 66 | 58 | 51 | 43 | 34 | 26 | 22 | 20.3 | 19.8 |

Position (mm) | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 | 130 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Force (N) | 0.25 | 421 | 565 | 646 | 682 | 700 | 712 | 709 | 703 | 686 | 644 | 558 | 378 | 76 |

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**MDPI and ACS Style**

Gies, V.; Soriano, T.
Modeling and Optimization of an Indirect Coil Gun for Launching Non-Magnetic Projectiles. *Actuators* **2019**, *8*, 39.
https://doi.org/10.3390/act8020039

**AMA Style**

Gies V, Soriano T.
Modeling and Optimization of an Indirect Coil Gun for Launching Non-Magnetic Projectiles. *Actuators*. 2019; 8(2):39.
https://doi.org/10.3390/act8020039

**Chicago/Turabian Style**

Gies, Valentin, and Thierry Soriano.
2019. "Modeling and Optimization of an Indirect Coil Gun for Launching Non-Magnetic Projectiles" *Actuators* 8, no. 2: 39.
https://doi.org/10.3390/act8020039