Electromechanical coupled chaotic vibration for electromagnetic railgun

Abstract

The chaotic vibration has more important effect on the dynamic performance of the railgun system. It will cause violent vibrations and noise of the mechanical system and also cause large current oscillation in the electric system for the railgun. In this paper, the nonlinear electromechanical coupled dynamics equations for the railgun system are proposed. Using the equations, the nonlinear dynamic performance of the railgun system is investigated. Results show that under some conditions, quasi-periodic vibration and chaotic vibration could occur in the railgun system. The rail current, the distance between two rails, the rail thickness and rail width have effects on the nonlinear vibration of the railgun system. To obtain good dynamic performance of the railgun system, these parameters should be chosen properly. These results are useful in maximizing the power density of the railgun and ensuring launch accuracy.

Keywords

Railgun chaotic vibration bifurcation electromechanical coupled

1. Introduction

The railgun is an attractive electromagnetic gun due to its apparent simple design. The muzzle velocities up to 2.5 km/s for masses of several hundred grams have been demonstrated experimentally [1, 2, 3] The railgun includes two parallel copper rails across which an armature makes electrical contact (see Fig. 1). The rails are copper strips $h\times b$ and length $L$ . The distance between the two rails is $d$ . A large current $I$ pass through the rails and the armature (projectile), and the projectile current interacts with the strong magnetic fields generated by the rails. It produces a strong force to accelerate the armature together with the projectile along the rails. Meanwhile, a mutually repulsive force occurs between the two rails.

The dynamics behavior of the railgun is of great importance for the system’s performance. In the railgun, the mechanical response of the rail to the transient magnetic load may lead to disturbances of the projectile trajectory because the armature performance is very sensitive to variations of normal forces at the contact interface. Therefore, the evaluation of the dynamics performance of railgun is a mandatory task for the railgun system. For the dynamic behavior of the electromagnetic railguns, the first model is to take the rail as one-dimensional beam on an elastic foundation [4]. The dynamic response of the railgun to the moving magnetic excitation was investigated [5, 6]. The axis-symmetric shell and two-dimensional solid models were used to simulate the railgun and numerically study the transient resonance at critical velocities of the projectile [7]. The transient elastic waves in electromagnetic launchers and their influence on armature contact pressure were studied [8, 9]. A 2D plane stress finite element model resting upon discrete elastic supports was developed and the transient analysis for a set of constant loading velocities was performed [10]. The equations for the forced responses of the rail to constant velocity load and the acceleration load were developed and the dynamic displacements of the rail under the running electromagnetic force for various accelerations was studied [11]. The vibration experiment of a railgun with discrete supports was done [12]. The rails of rectangular electromagnetic rail gun were simplified as a double layer elastic foundation beam, and the dynamic response of the rails was investigated [13].

Figure 1.

Schematic of railgun.

In railgun, the electromagnetic field can cause an essential change of the dynamics behaviors of the railgun. The linear electromechanical coupled effects were considered, and the natural frequencies of the railgun were investigated [14]. In operation, the rail current is quite large which causes the strong electromagnetic nonlinearity in the railgun system. So, the electromechanical coupled nonlinear free vibration of the railgun system was studied [15]. Using Euler beam model of rails for electromagnetic railgun, the nonlinear free vibration frequency of the railgun and the nonlinear forced responses of the rail to the electromagnetic excitation were investigated [16]. However, the bifurcation and chaotic vibration will occur under some conditions. The chaotic vibration has more important effect on the dynamic performance of the railgun system. It will cause violent vibrations and noise of the mechanical system for the railgun, and also cause large current oscillation in the electric system for the railgun. This can result in loss of the load-carrying ability and launch accuracy of the drive system. The studies on the chaotic vibration of the railgun system have not been developed yet. To design, evaluate and control dynamics behavior of the railgun system better, the chaotic vibration of the railgun system should be investigated.

In this paper, the nonlinear electromagnetic force equation on the rail for the railgun is given. Based on the electromagnetic force equation, the nonlinear electromechanical coupled dynamics equations for the railgun are proposed. Using the equations, the nonlinear dynamic performance of the railgun system is investigated. Results show that under some conditions, quasi-periodic vibration and chaotic vibration occur in the railgun system. The rail current, the distance between two rails, and the rail thickness have important effects on the nonlinear vibration of the railgun system. To obtain good dynamic performance of the railgun system, these parameters should be chosen properly. These results can be used to predict the noise and dynamic load of the railgun system and are useful in maximizing the power density of the railgun and ensuring launch accuracy.

2. Electromechanical coupled nonlinear dynamics equations of the rail

When the current flows in the rails, a magnetic flux is produced between the rails. It interacts with the current flowing in the armature to cause Lorentz force which accelerates the armature and the projectile. Meanwhile, a nonlinear magnetic force occurs between the two rails. The nonlinear magnetic force per unit length between the two rails is [14]

$\displaystyle q_{r0}=K_{r}I^{2}$ (1)

$\displaystyle K_{r}=\frac{\mu_{0}}{4\pi h^{2}b^{2}}\int_{d}^{d+b}\int_{-h/2}^{% h/2}\int_{-b}^{0}\int^{h/2}_{-h/2}\frac{(z-z^{\prime})}{[(y-y^{\prime})^{2}+(z% -z^{\prime})^{2}]}\left[\frac{x}{\sqrt{\left({y-y^{\prime}}\right)^{2}+\left({% z-z^{\prime}}\right)^{2}+x^{2}}}-\frac{x-l}{\sqrt{\left({y-y^{\prime}}\right)^% {2}+\left({z-z^{\prime}}\right)^{2}+\left({x-l}\right)^{2}}}\right]dy^{\prime}% dz^{\prime}dydz$ (2)

where $\mu_{0}$ is the permeability, $I$ is the current intensity in the rail, $l$ is the running position of the armature, $x$ is the position coordinate in the direction of the rail length, $b$ is the thickness of the rail, $h$ is the width of the rail, $d$ is the distance between the two rails, $P^{\prime}(z^{\prime},y^{\prime})$ is one point on the left rail, $P(z,y)$ is one point on the right rail.

By means of the regressive interpolation, the nonlinear electromagnetic force as a function of the distance $d$ , the width $h$ , and the thickness $b$ can be given as below

$\displaystyle K_{r}=\frac{A}{\left({d+2w+0.01343}\right)}$ (3)

where

$A=\frac{1.2101\times 10^{-3}\cdot\left({0.01244h^{3}-0.001221h^{2}-4.296\times 1% 0^{-6}h+6.809\times 10^{-6}}\right)}{\left({b+0.02622}\right)}.$

The dynamic equation of the rail can be given as

$\displaystyle\rho_{l}\frac{\partial^{2}\Delta w}{\partial t^{2}}+EI\frac{% \partial^{4}\Delta w}{\partial x^{4}}+k\Delta w=\Delta q_{r}$ (4)

where $\Delta w$ is the dynamic displacement of the rail, $\rho_{l}$ is the mass coefficient per unit length of the rail, $E$ is the modulus of elasticity of the rail material, $I_{z}$ is the sectional modular of the rail, $k$ is the stiffness of the elastic foundation, $\Delta q_{r}$ is the dynamic electromagnetic force on the rail.

The dynamic electromagnetic force can be expressed in series form as

$\displaystyle\Delta q_{r}=\left.\frac{dq_{r}}{dw}\right|_{w_{0}}\Delta w+\frac% {1}{2!}\left.\frac{d^{2}q_{r}}{dw^{2}}\right|_{w_{0}}\left({\Delta w}\right)^{% 2}+\left.\frac{dq_{r}}{dI}\right|_{I_{0}}\Delta I+\ldots$ (5)

where $\Delta I$ is the dynamic current in the rail, $I_{0}$ is the static current in the rail.

Substituting Eqs (1) and (3) into Eq. (5), the dynamic electromagnetic force $\Delta q_{r}$ can be expressed as

$\displaystyle\Delta q_{r}=-\frac{AI_{0}^{2}}{\left({d+2w_{0}+0.01343}\right)^{% 2}}\Delta w+\frac{2AI_{0}^{2}}{\left({d+2w_{0}+0.01343}\right)^{3}}\Delta w^{2% }+\frac{4AI_{0}^{2}}{\left({d+2w_{0}+0.01343}\right)^{3}}\Delta w^{3}+\frac{2% AI_{0}}{\left({d+2w_{0}+0.01343}\right)}\Delta I\ldots$ (6)

Neglecting higher order terms and substituting Eq. (6) into Eq. (4), yields

$\displaystyle\rho_{l}\frac{\partial^{2}\Delta w}{\partial t^{2}}+2\xi\omega% \frac{\partial\Delta w}{\partial t}+EI\frac{\partial^{4}\Delta w}{\partial x^{% 4}}+k\Delta w=-\frac{AI^{2}}{\left({d+2w_{0}+0.01343}\right)^{2}}\Delta w$ $\displaystyle\quad+\frac{2AI^{2}}{\left({d+2w_{0}+0.01343}\right)^{3}}\left({% \Delta w}\right)^{2}-\frac{4AI^{2}}{\left({d+2w_{0}+0.01343}\right)^{4}}\left(% {\Delta w}\right)^{3}$ (7) $\displaystyle\quad+\frac{2AI}{\left({d+2w_{0}+0.01343}\right)}\Delta I$

Let $\Delta w=\phi(x)q(t)$ , substituting it into Eq. (2), yields

$\displaystyle\frac{\ddot{q}(t)+Q_{0}\dot{q}(t)}{q(t)}-Q_{1}q(t)+Q_{2}q^{2}(t)-% \frac{Q_{3}\Delta I}{q(t)}=-\frac{EI\phi^{(4)}(x)+P\phi(x)}{\rho_{l}\phi(x)}$ (8)

where

$\displaystyle P=k+\frac{AI^{2}}{(d+2w_{0}+0.01343)^{2}},Q_{0}=\frac{2\xi\omega% }{\rho_{l}},Q_{1}=\frac{2AI^{2}}{\rho_{l}(d+2w_{0}+0.01343)^{3}}\bar{\phi},$ $\displaystyle Q_{2}=\frac{4AI^{2}}{\rho_{l}(d+2w_{0}+0.01343)^{4}}\bar{\phi}^{% 2},Q_{3}=\frac{2AI}{\rho_{l}\bar{\phi}(d+2w_{0}+0.01343)},\bar{\phi}=\frac{1}{% L}\int_{0}^{L}{\phi(x)dx}.$

Let Eq. (8) equal constant $-\omega^{2}$ , thus

$\displaystyle\ddot{q}(t)+Q_{0}\dot{q}(t)+\omega^{2}q(t)-Q_{1}q^{2}(t)+Q_{2}q^{% 3}(t)=Q_{3}\Delta I$ (9)

$\displaystyle EI\phi^{(4)}(x)+P\phi(x)-\omega^{2}\rho_{l}\phi(x)=0$ (10)

From Eq. (10) plus the boundary conditions and the continuity conditions of the rail, the natural frequencies and the mode functions of the rail can be obtained.

Equation (10) can be resolved with numerical method and can be changed into following form

$\displaystyle\left\{{\begin{array}[]{l}\dot{q}(1)=q(2)\\ \dot{q}(2)=-Q_{0}q(2)-\omega^{2}q(1)+Q_{1}q^{2}(1)-Q_{2}q^{3}(1)+Q_{3}\Delta I% \\ \end{array}}\right.$ (11)

Using ode45 function in matlab, Eq. (11) can be resolved and the function $q(t)$ can be obtained. Substituting $q(t)$ and mode function $\phi(x)$ into $\Delta w=\phi(x)q(t)$ , the dynamic displacements of the rail can be given.

3. Results and discussions

Using above method, the dynamic displacements of the rail for the railgun system and their changes along with the system parameters are investigated. The time step is $T$ /100, the total time is 400 $T$ . At the beginning running time, the unstable initial response of 300 periods is eliminated, and the response of the final 100 $T$ is selected, here $T$ is the time period of the rail vibration. The parameters of the numerical example are shown in Table 1.

Table 1
Parameters of the example system

$h$ (mm)	$w$ (mm)	$b$ (mm)	$I$ (KA)	$I_{0}$ (KA)	$\omega$ (HZ)	$\xi$
20	5	10	125	25	31000	0.07

Figure 2 shows the bifurcation diagram of the displacement $\Delta w$ with rail width $w$ varying from 3 mm to 23 mm (here, $h=$ 20 mm, $b=$ 10 mm, $I_{0}=$ 25 kA and $I=$ 125 kA). Figure 4 shows that the displacement response $\Delta w$ is period-one vibration when rail width is relatively large (above 8 mm), and it becomes complicated when rail width is less than 8 mm. The obvious bifurcations occur when the rail width is equal to 3.29 mm, 3.43 mm, 4.61 mm, 4.80 mm, 5.04 mm or 6.38 mm. It shows that changes of the rail width have significant effects on dynamic performance of the rail gun.

Figure 2.

Bifurcation diagram with rail width.

Figure 3.

Poincare map of the displacement response for the rail.

Figure 3 gives the displacement response $\Delta w$ when the rail width is equal to 3.29 mm, 3.43 mm, 4.61 mm and 5.04 mm. Figures 2 and 3 show:

At $w=$ 3.29 mm, jumping dynamic displacement $\Delta w$ occurs in the bifurcation diagram, which corresponds to a ring shape of Poincare map. Here, two Lyapunov Exponent values are negative (the maximum Lyapunov Exponent value is $-$ 0.87). These show that the quasi-periodic vibration occurs in the drive system for the rail width.

At $w=$ 3.43 mm or 4.61 mm, the jumping dynamic displacement $\Delta w$ occurs in the bifurcation diagram as well, however, it corresponds to a non ring shape of Poincare map. Here, one Lyapunov Exponent value is negative and another is positive (the maximum Lyapunov Exponent values are 0.07 and 0.06, respectively) [17]. These show that the chaotic vibration occurs in the drive system for the rail width.

At $w=$ 5.04 mm, jumping dynamic displacement $\Delta w$ occurs again in the bifurcation diagram, which corresponds to a ring shape of Poincare map as well. Here, two Lyapunov Exponent values are negative (the maximum Lyapunov Exponent value is $-$ 0.51). So, the quasi-periodic vibration occurs again in the drive system for the rail width.

Figure 4 shows the bifurcation diagram of the displacement displacement $\Delta w$ with rail thickness $h$ varying from 15 mm to 25 mm (here, $w=$ 5 mm, $b=$ 10 mm, $I_{0}=$ 25 kA and $I=$ 125 kA). Figure 4 shows that the displacement response $\Delta w$ is period-one vibration for ordinary rail thickness, and it becomes complicated when rail thickness is equal to 15.85 mm, 18.95 mm, 21.51 mm or 24.73 mm. Figure 5 gives Poincare map of the displacement response $\Delta w$ for the rail thickness. Here, ring shape of Poincare map for the displacement response $\Delta w$ occurs for the rail thickness. Here, all of the Lyapunov Exponent values are negative (the maximum Lyapunov Exponent value is $-$ 0.71, $-$ 0.88, $-$ 0.57 and $-$ 0.55, respectively). These mean that quasi-periodic vibration occurs for the rail thickness.

Figure 4.

Bifurcation diagram with rail thickness.

Figure 5.

Poincare map of the displacement response for several rail thickness.

Figure 6 shows the bifurcation diagram of the dynamic displacement $\Delta w$ with rail distance $b$ varying from 5 mm to 25 mm (here, $w=$ 5 mm, $h=$ 20 mm, $I_{0}=$ 25 kA and $I=$ 125 kA). Figure 6 shows that the displacement response $\Delta w$ is period-one vibration for ordinary rail distance, and it becomes complicated when rail distance is equal to 5.69 mm, 7.59 mm, 9.57 mm, 12.22 mm, 15.10 mm or 18.46 mm. Figure 7 gives Poincare map of the displacement response $\Delta w$ for the rail distance 5.69 mm, 7.59 mm, 9.57 mm, 12.22 mm, 15.10 mm.

Figure 6.

Bifurcation diagram with rail distance $b$ .

Figure 7.

Poincare map of the displacement response for several rail distances.

At $b=$ 5.69 mm, jumping dynamic displacement $\Delta w$ occurs in the bifurcation diagram, which corresponds to a non ring shape of Poincare map. Here, one Lyapunov Exponent value is negative and another is positive (the maximum Lyapunov Exponent value is 0.05). These show that the chaotic vibration occurs in the railgun system.

At $b=$ 7.59 mm or 12.22 mm, another jumping dynamic displacement $\Delta w$ occurs in the bifurcation diagram, however, it corresponds to a ring shape of Poincare map. Here, two Lyapunov Exponent values are negative (the maximum Lyapunov Exponent value is $-$ 0.30 and $-$ 0.31, respectively). These show that the quasi-periodic vibration occurs in the drive system for the distance 7.59 mm or 12.22 mm.

At $b=$ 15.10 mm, jumping dynamic displacement $\Delta w$ occurs again in the bifurcation diagram, which corresponds to a non ring shape of Poincare map as well. Here, one Lyapunov Exponent value is negative and another is positive (the maximum Lyapunov Exponent value is 0.13). So, the chaotic vibration occurs in the railgun system for the distance $b=$ 15.10 mm.

Figure 8 shows the bifurcation diagram of the dynamic displacement $\Delta w$ with rail current $I$ varying from 100 kA to 150 kA (here, $w=$ 5 mm, $h=$ 20 mm, $I_{0}=$ 25 kA and $b=$ 10 mm). Figure 8 shows that the displacement response $\Delta w$ is period-one vibration for ordinary rail current, and it becomes complicated when rail current is equal to 103.09 kA, 113.82 kA, 125.10 kA or 137.23 kA. Figure 9 gives Poincare map of the displacement response $\Delta w$ for the rail currents 102.94 kA, 103.09 kA, 113.82 kA, and 125.10 kA.

Figure 8.

Bifurcation diagram with rail currents.

Figure 9.

Poincare map of the displacement response for several rail currents.

At $I=$ 102.94 kA, the dynamic displacement $\Delta w$ is near to its first jumping peak in the bifurcation diagram, which corresponds to a ring shape of Poincare map. Here, Lyapunov Exponent values are negative (the maximum Lyapunov Exponent value is $-$ 0.52). These show that the quasi-periodic vibration occurs in the drive system for the rail current ( $I=$ 102.94 kA).

At $I=$ 103.09 kA, the first jumping peak of the dynamic displacement $\Delta w$ occurs in the bifurcation diagram, it corresponds to a non ring shape of Poincare map. Here, one Lyapunov Exponent value is negative and another is positive (the maximum Lyapunov Exponent value is 0.75). These show that the chaotic vibration occurs in the drive system for the rail current ( $I=$ 103.09 kA).

At $I=$ 113.82 kA, the second jumping peak of the dynamic displacement $\Delta w$ occurs in the bifurcation diagram, however, it corresponds to a ring shape of Poincare map. Here, two Lyapunov Exponent values are negative (the maximum Lyapunov Exponent value is $-$ 0.51). These show that the quasi-periodic vibration occurs in the drive system for the rail current ( $I=$ 113.82 kA).

At $I=$ 125.10 kA, the third jumping peak of the dynamic displacement $\Delta w$ occurs in the bifurcation diagram, and it corresponds to a non ring shape of Poincare map. Here, one Lyapunov Exponent value is negative and another is positive (the maximum Lyapunov Exponent value is 1.22). These show that the chaotic vibration occurs in the drive system for the rail current ( $I=$ 125.10 kA).

4. Conclusions

In this paper, the nonlinear electromagnetic force equation of the rail for the railgun is given. Based on the equation, the nonlinear electromechanical coupled dynamic equations for the rail are proposed. Using the equations, the nonlinear dynamic performance of the rail-gun is investigated. Results show:

When the current parameter is large (larger than 100 kA), quasi-periodic vibration and chaotic vibration could occur. The current in worm coils has the most important effect on the nonlinear vibration of the railgun system.

When the rail thickness is small (less than 25 mm), quasi-periodic vibration and chaotic vibration could occur. When the rail width is small (less than 8 mm), quasi-periodic vibration and chaotic vibration could occur as well. The rail width and rail thickness have important effect on the nonlinear vibration of the railgun system.

The rail distance has also effect on the nonlinear vibration of the railgun system. At some rail distance, quasi-periodic vibration and chaotic vibration could occur as well.

To obtain good dynamic performance of the railgun system, the current parameter in rails, rail width, rail thickness and rail distance should be selected properly.

Footnotes

Acknowledgments

This project is supported by the Doctoral Research Program Foundation of Education Ministry of China (Priority Development Areas, no. 20131333130002).

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