Deformation Calculating of Electromagnetic Launcher's Rail Subjected to Sinusoidal Magnetic Pressure

Abstract

The electromagnetic launcher's rail can be modeled as a beam on elastic foundation with simply supported beam by moving load. In this paper, Euler beam theory is applied to build the mechanical model, and the analytical solution of the equation subjected to sinusoidal magnetic pressure is derived in detail, which has successfully avoided the errors which are caused by using the uniform pressure to approximately replace the variable force. Numerical analysis of the influences brought from the elastic coefficient, the damping coefficient, the mass of rail, and the load's velocity on the deformation of beam by the MATLAB software show that the elastic coefficient and the load's velocity have quite obvious effect on the deformation of the beam while the damping coefficient and the mass of rail have not obvious effect on the deformation of the beam.

1. Introduction

As a new concept weapon, the electromagnetic railgun has adopted the electromagnetic launching method, not only broken through the limitation of projectile velocity by the traditional chemical energy, but also greatly surpassed the rate of cannon projectile. Hence, it has a huge application potential in the military field. Since the 1980s, especially in the recent ten years, with the development of new technology and new material, the research of launcher, launching weight, projectile velocity, and high efficiency power source in the electromagnetic railgun has reached a series of achievements. The Su Rense.Livermore Country Laboratory and The Lowes.Alamos Country Laboratory, once have cooperated to accelerate a projectile weighed 2.2 g to a supervelocity of 10 km/s. Fluid Physics Institute of the Chinese Engineering Academy had built the first electromagnetic rail launcher, which can accelerate the projectile weighted 0.34 g to 16.8 km/s [1]. While the velocity of the conventional cannon is only 2 km/s, which is so close to the limitation of physics that the range is not possible to be farther. On the contrary, the thrust of the electromagnetic railgun is ten times bigger than that of the traditional launcher. The projectile can be accelerated to several kilometers or even to dozens of kilometers in one second, for it possesses huge kinetic energy which greatly enhance the range and power of the weapon [2–4].

Tzeng and sun [5] used the elastic foundation beam to build mechanical model of the electromagnetic railgun and deduced the solve process of governing equation. Yuwei [6] analyzed theoretical built model and simulation analysis for the work of the process of electromagnetic railgun. Sheng [7] adopted the Fourier transform to study displacement field due to a moving load on Euler beam resting on an elastic half-space.

However, the damping force to the response of beam is ignored in the above researches. There is no doubt that the calculation of this kind of situation has the defects of analysis and calculation of mechanical. As a high-tech and high-precision electromagnetic launcher, accurate theoretical analysis and calculation in engineering are require. But until now, no researcher has given any exact analytical solution; thus, further analytic solution of the equation subjected to variable pressure is of great significance. In fact, it is of theoretical value to research the theoretical analytic solution of various disciplines, including the analytical solution of practical engineering problems. On the one hand, its mechanical picture can be completely stated; on the other hand, it can be used as a standard solution to widely produce variety of numerical solution.

In this paper, regarding the rail as simply supported beam on the elastic foundation and considering the damping force, a mechanic model which is under the effect of moving load is proposed. Moreover, making use of variable method and the Lagrange equation considering the damping force, the analytical solution [8, 9] of the governing equation subjected to nonlinear function pressures is derived and the influences brought from the elastic coefficient, the damping coefficient, the mass of rail, and the load’s velocity on the response of beam are analyzed.

2. Mechanical Model

Figure 1 shows a schematic of an electromagnetic railgun composed of power source, rail, armature, and projectile. When the electric current of armature goes through the rail, it forms a strong magnetic field in the area of their encirclement. With the reaction by the magnetic field and the electric current, it emerges powerful electromagnetic force, which pushes the armature and projectile to do the accelerating motion along the rail till the projectile is launched out of the rail.

Figure 1 

The general diagram of the railgun.

Figure 2 is the mechanical model of the railgun, simply supported beam partially subject to nonlinear load in a time-varying region sitting on the elastic foundation. Considering the effect of the beam by the damping force and basing on the Euler beam theory, we obtain the governing equation of elastic foundation beam by moving load which is a transient fourth-order differential equation as follows [10, 11]:𝑚𝜕²𝑤𝜕𝑡²𝜕+𝐸𝐼⁴𝑤𝜕𝑥⁴+𝑘𝑤+𝑐𝜕𝑤()𝜕𝑡=𝑝𝑥,𝑡, (2.1) where 𝑤 is the deflection, 𝑚=𝜌𝐵ℎ is the mass per unit length, 𝜌 is the density of rail material, 𝐵 and ℎ are, respectively, the width and thickness of the rail, 𝐸𝐼 is the bending stiffness of beam, 𝑘 is the elastic constant, and 𝑐 is the damping coefficient. We have the function 𝑝(𝑥,𝑡)=𝑞(𝑥)[1−𝐻(𝑥−𝑣𝑡)] In (2.1), represents the magnetic pressure front traveling along the rail with velocity 𝑣 represented by a Heaviside step function 𝐻(𝑥−𝑣𝑡) [12], and 𝑞(𝑥)=𝑞(sin𝑎𝑥+𝑏).

Figure 2 

The rail is modeled as a beam on elastic foundation.

3. Solution of the Homogeneous Equation

The homogeneous equation is a fourth-order partial differential equation, and in order to change it into the ordinary differential equation, we solve it by the method of variable separation.

The solution of the homogeneous equation of (2.1) can be expressed as follows:𝑤(𝑥,𝑡)=𝜙(𝑡)𝜃(𝑥). (3.1) Substituting (3.1) into the homogeneous equation (2.1):𝑚𝜕²𝜙𝜕𝑡²𝜃+𝐸𝐼𝜙𝜃⁽⁴⁾+𝑘𝜙𝜃+𝑐𝜃𝜕𝜙𝜕𝑡=0. (3.2) That can be expressed as follows:−𝜕²𝜙/𝜕𝑡²−𝑐𝜙𝜕𝜙/𝜕𝑡=𝑚𝜙𝐸𝐼𝜃𝑐𝑚𝜃⁽⁴⁾+𝑘𝑐𝑚. (3.3) From (3.3), let−𝜕²𝜙/𝜕𝑡²−𝑐𝜙𝜕𝜙/𝜕𝑡𝑚𝜙=𝜆², (3.4)𝐸𝐼𝜃𝑐𝑚𝜃⁽⁴⁾+𝑘𝑐𝑚=𝜆². (3.5)

That is,𝜃⁽⁴⁾−𝛽⁴𝜃=0, (3.6) where 𝛽⁴=𝜆²−𝑘𝑐𝑚𝑐𝑚𝐸𝐼. (3.7)

Solution of (3.4) can be expressed as follows:𝜙(𝑡)=𝐴𝑒^{((−𝑐+𝑅)/2𝑚)𝑡}+𝐵𝑒^{((−𝑐−𝑅)/2𝑚)𝑡}, (3.8) where𝑅²=𝑐²−4𝑐𝑚²𝜆²>0. (3.9)

Based on the boundary condition of the simple beam,𝜃(0)𝜕=0,₂𝜃(𝑥)𝜕𝑥²_𝑥=0𝜃(𝐿)𝜕=0,=0,₂𝜃(𝑥)𝜕𝑥²_𝑥=𝐿=0. (3.10)

Solution of (3.6) can be expressed as follows [13]:𝜃_𝑖(𝑥)=2𝑚𝐿sin𝑛𝜋𝐿𝑥. (3.11)

Accordingly:𝑤_𝑖(𝑥,𝑡)=𝜃_𝑖(𝑥)𝜙_𝑖(𝑡)=2𝑚𝐿sin𝑛𝜋𝐿𝑥×𝐴_𝑖𝑒^{((−𝑐+𝑅)/2𝑚)𝑡}+𝐵_𝑖𝑒^{((−𝑐−𝑅)/2𝑚)𝑡}. (3.12)

In terms of the orthogonality of 𝜃_𝑖(𝑥) [14], we obtain^𝐿₀𝜃_𝑖𝜃_𝑗𝑑𝑥=0,𝑎𝑡𝑖≠𝑗,1,𝑎𝑡𝑖=𝑗. (3.13)

Hence, deformation 𝑤(𝑥,𝑡) of the beam can be expressed by the linear combination of 𝜃_𝑖(𝑥)𝑤(𝑥,𝑡)=_𝑖𝜃_𝑖(𝑥)𝜙_𝑖(𝑡)=_𝑖2𝑚𝐿sin𝑛𝜋𝐿𝑥×𝐴_𝑖𝑒^{((−𝑐+𝑅)/2𝑚)𝑡}+𝐵_𝑖𝑒^{((−𝑐−𝑅)/2𝑚)𝑡}, (3.14)

where constants 𝐴_𝑖, 𝐵_𝑖 are determined by the initial conditions.

4. Analytical Solution of Governing Equation

The analytical solution of (2.1) can be obtained by the Lagrange equation including the damping force, where 𝑇 is the kinetic energy of the beam, 𝑈 is the total stain energy, and 𝐺 is the dissipation function [15]. We have 𝑑𝑑𝑡𝜕𝑇𝜕𝜙_𝑖−/𝜕𝑡𝜕𝑇𝜕𝜙_𝑖+𝜕𝑈𝜕𝜙_𝑖−𝜕𝐺𝜕𝜙_𝑖/𝜕𝑡=𝑄_𝑖. (4.1)

The kinetic energy of the beam 𝑇 can be expressed as follows [16]:1𝑇=2_𝑖𝑀_𝑖𝜕𝜙_𝑖𝜕𝑡², (4.2) where 𝑀_𝑖=∫^𝐿₀𝑚𝜃²_𝑖(𝑥)𝑑𝑥 represents the general mass of the beam.

The total strain energy of the beam 𝑈 is consisted of the strain energy 𝑈_𝑏 of the beam and the strain energy 𝑈_𝑓 of the foundation𝑈_𝑏=12^𝐿₀𝜕𝐸𝐼²𝑤(𝑥,𝑡)𝜕𝑥²²=1𝑑𝑥2_𝑖𝑗𝜙_𝑖𝜙_𝑗^𝐿₀𝜕𝐸𝐼²𝜃_𝑖𝜕𝑥²𝜕²𝜃_𝑗𝜕𝑥²=1𝑑𝑥2𝐸𝐼𝑚_𝑖𝛽⁴_𝑖𝑀_𝑖𝜙²_𝑖,𝑈_𝑓=12^𝐿₀𝑘𝑤²𝑘𝑑𝑥=2^𝐿_0𝑖𝜃_𝑖𝜑_𝑖²=𝑘𝑑𝑥2_𝑖𝑗𝜙_𝑖𝜙_𝑗^𝐿₀𝜃_𝑖𝜃_𝑗=𝑘𝑑𝑥𝑀2𝑚^𝑖_𝑖𝜙²_𝑖. (4.3) The total strain energy 𝑈 is obtained as𝑈=𝑈_𝑏+𝑈_𝑓=12𝐸𝐼𝑚_𝑖𝛽⁴_𝑖𝑀_𝑖𝜙²_𝑖+𝑘𝑀2𝑚^𝑖_𝑖𝜙²_𝑖=12_𝑖𝑀_𝑖𝜙²_𝑖𝐸𝐼𝑚𝛽⁴_𝑖+𝑘𝑚=12_𝑖𝑐𝑀_𝑖𝜙²_𝑖𝜆²_𝑖. (4.4)

The dissipation function 𝐺 can be expressed as1𝐺=−2^𝐿₀𝑐𝜕𝑤𝜕𝑡²1𝑑𝑥=−2_𝑖𝑗𝜕𝜙_𝑖𝜕𝑡𝜕𝜙_𝑗𝜕𝑡^𝐿₀𝑐𝜃_𝑖𝜃_𝑗𝑐𝑑𝑥=−2𝑚_𝑖𝑀_𝑖𝜕𝜙_𝑖𝜕𝑡². (4.5)

The virtual work done by the magnetic pressure 𝑝(𝑥,𝑡)=𝑞(𝑥)[1−𝐻(𝑥−𝑣𝑡)] in a virtual displacement 𝛿𝜑_𝑖 can be expressed as follows:𝜕𝑊=^𝐿₀𝑝(𝑥,𝑡)𝛿𝑤_𝑖𝑑𝑥=_𝑖𝜕𝜙_𝑖^𝐿₀𝑝(𝑥,𝑡)𝜃_𝑖(𝑥)𝑑𝑥=_𝑖𝜕𝜙_𝑖𝑄_𝑖, (4.6)

where we define 𝑄_𝑖 as the generalized force 𝑄_𝑖=^𝐿₀𝑝(𝑥,𝑡)𝜃_𝑖(𝑥)𝑑𝑥=₀^𝑣𝑡𝑞(𝑥)𝜃_𝑖(𝑥)=𝑑𝑥2𝑚𝐿₀^𝑣𝑡𝑞(sin𝑎𝑥+𝑏)sin𝑛𝜋𝑥𝐿=()𝑑𝑥𝑞𝐿𝑎𝐿+𝑛𝜋(𝑎𝐿−𝑛𝜋)1()2𝑚𝐿𝑎𝐿+𝑛𝜋sin𝑣𝑡𝑎𝐿−𝑛𝜋𝐿−(𝑎𝐿−𝑛𝜋)sin𝑣𝑡𝑎𝐿+𝑛𝜋𝐿−2𝐿𝑚𝑏𝑞𝑛𝜋cos𝑛𝜋𝑣𝑡𝐿−1. (4.7)

Substituting 𝑇, 𝑈, 𝐺, 𝑄_𝑖 into the Lagrange equation including the damping force, we obtain an ordinary differential equation:𝜕²𝜙_𝑖𝜕𝑡²+𝑐𝑚𝜕𝜙𝜕𝑡+𝑐𝜆²_𝑖𝜙_𝑖=𝑄_𝑖(𝑡)𝑀_𝑖=𝐹(𝑡), (4.8) where𝐹(𝑡)=𝑞𝐿(𝑎𝐿+𝑛𝜋)𝑀_𝑖(𝑎𝐿−𝑛𝜋)1)2𝑚𝐿(𝑎𝐿+𝑛𝜋sin𝑣𝑡𝑎𝐿−𝑛𝜋𝐿−(𝑎𝐿−𝑛𝜋)sin𝑣𝑡𝑎𝐿+𝑛𝜋𝐿−2𝐿𝑚𝑏𝑞𝑛𝜋𝑀_𝑖cos𝑛𝜋𝑣𝑡𝐿−1. (4.9)

The general solution of (4.8) is 𝜙_𝑖(𝑡)=𝑚𝑅𝜕𝜙_𝑖(0)+𝜕𝑡𝑐+𝑅𝜙2𝑚^𝑖(0)𝑒^{((−𝑐+𝑅)/2𝑚)𝑡}+𝑚𝑅−𝜕𝜙_𝑖(0)+𝜕𝑡−𝑐+𝑅𝜙2𝑚^𝑖(0)𝑒^{((−𝑐−𝑅)/2𝑚)𝑡}+𝑚𝑅^𝑡₀𝐹(𝜉)𝑒^{((−𝑐+𝑅)/2𝑚)(𝑡−𝜉)}−𝑒^{((−𝑐−𝑅)/2𝑚)(𝑡−𝜉)}𝑑𝜉. (4.10)

So, the general solution of (2.1) can be expressed as follows:𝑤(𝑥,𝑡)=_𝑖𝜃_𝑖(𝑥)𝜙_𝑖(𝑡)=_𝑖2𝑚𝐿sin𝑛𝜋𝐿𝑥×𝑚𝑅𝜕𝜙_𝑖(0)+𝜕𝑡𝑐+𝑅𝜙2𝑚^𝑖(0)𝑒^{((−𝑐+𝑅)/2𝑚)𝑡}+𝑚𝑅−𝜕𝜙_𝑖(0)+𝜕𝑡−𝑐+𝑅𝜙2𝑚^𝑖(0)𝑒^{((−𝑐−𝑅)/2𝑚)𝑡}+𝑚𝑅^𝑡₀𝐹(𝜉)𝑒^{((−𝑐+𝑅)/2𝑚)(𝑡−𝜉)}−𝑒^{((−𝑐−𝑅)/2𝑚)(𝑡−𝜉)}𝑑𝜉. (4.11) The initial conditions are𝜙(0)(𝑡)=0,𝜕𝜙𝜕𝑡_𝑡=0=0. (4.12)

So,𝜙_𝑖(𝑡)=𝑚𝑅₀^𝑣𝜉𝐹(𝜉)𝑒^{((−𝑐+𝑅)/2𝑚)(𝑡−𝜉)}−𝑒^{((−𝑐−𝑅)/2𝑚)(𝑡−𝜉)}𝑑𝜉. (4.13)

According, substituting the solution of (4.13) into (4.11), we can get the solution 𝑤(𝑥,𝑡) of (2.1). The moment and the shear force of the beam in the rail can be further derived from 𝑤(𝑥,𝑡), which provides basis for the overall investigation of the dynamic behavior of the electromagnetic railgun.

5. Numerical Analysis

Since there are differences among materials of electromagnetic rail launcher, the damping force and the rate of moving load will possibly bring influence to the response of the rail [17]. Thus, it is necessary to consider the elastic coefficient, the damping coefficient, the mass of rail, and the load’s velocity to compare the response of the rail.

A known Material is modulus of rail material 𝐸=120 GPa, the elastic constant 𝑘=2.532×10¹⁰ N/m², the density of rail material 𝜌=8700 kg/m³, the width of rail 𝐵=3×10⁻² m, the thickness of rail ℎ=1×10⁻² m, the length of rail 𝐿=2 m, and the magnetic load collection degree 𝑞(𝑥)=110sin𝑥 MPa [18].

Figure 3 shows the deformation of the beam by the elastic coefficient. Along with the elastic coefficient (𝑘) increasing, the curve of time deformation is a decreasing trend. Under the calculating conditions given by this paper, for the rail of which 𝑘 equals to 2.532×10¹⁰ N/m², the deformation (𝑤) of the beam is 1×10⁻³ m when the armature moves to the moment 𝑡=1×10⁻³. While for the rail of which 𝑘 equals to 5.064×10¹⁰ N/m³, the deformation (𝑤) of the beam is 0.45×10⁻³ m at the same moment, and we can see that the former is 122% bigger than the latter.

Figure 3 

Deformation curve by differences elastic coefficient.

Figure 4 shows the deformation of the beam by the damping coefficient. Along with the damping coefficient (𝑐) increasing, the curve of time deformation is a slowly decreasing trend.

Figure 4 

Deformation curve by differences damping coefficient.

Figure 5 shows the deformation of the beam by the mass of rail. Compared with copper and aluminum rail, the curve of time deformation has no significant changes.

Figure 5 

Deformation curve by differences mass of rail.

Figure 6 shows the deformation of the beam by the load’s velocity. Along with the load’s velocity (𝑣) increasing, the curve of time deformation is an increasing trend. Under the calculating conditions given by this paper, for the rail of which 𝑣 equals to 1000 m/s, the deformation (𝑤) of the beam is 1.5×10⁻³m when the armature moves to the moment 𝑡=1.6×10⁻³ s. While for the rail of which 𝑣 equals to 1200 m/s, the deformation (𝑤) of the beam is 2.05×10⁻³ m at the same moment, and we can see that the former is 27% smaller than the latter.

Figure 6 

Deformation curve by differences velocity of the load.

6. Conclusions

(1) Taking the rail as a simply support beam on the elastic foundation and considering the damping force, a mechanical model for the electromagnetic railgun is built. “We do not have general solution to nonlinear problems and some particular solutions are as few as treasures in history” [19]. In this paper, making use of variable method and the Lagrange equation including the damping force, the general solution of the homogeneous part, and the analytical solution of the governing equation subjected to sinusoidal pressures is derived which enriched and developed the theory of elastic mechanics with the hope to lay the foundation for solving the difficulty problem of electromagnetic rail subjected to arbitrary distribution function pressures.

(2) The deformation of beam which is influenced by the elastic coefficient, the damping coefficient, the mass of rail, and the load’s velocity is analyzed by the MATLAB software. When the elastic coefficient is larger, the deformation of beam is smaller; when the load’s velocity is larger, the deformation of beam is larger, the damping coefficient and the mass of rail have not obvious effect on the deformation of the beam.

Acknowledgment

The project support by the National Nature Science Foundation of China under (Grant no. 50875230).