Dynamic Response and Parameter Analysis of Electromagnetic Railguns under Time Varying Moving Loads

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Introduction
Electromagnetic (EM) launch technology is a revolution in the launch mode which uses EM energy to propel objects to high speed or even ultrahigh speed [1].EM railguns are the most important application of the EM launch technology, which have many advantages such as high speed, long range, and strong power [2].As one of the newest powerful weapons, EM railguns have attracted great attention in military feld in recent years, especially in America, Russia, and China [3][4][5].
During the launch process, the armature moves along the rails of EM railguns under the action of the ampere force generated by the heavy current.Meanwhile, the rail is subjected to moving loads which will result in its nonlinear lateral vibration.Rail vibration can afect the contact status and shot accuracy, even causing damage between the armature and the rail [6].In order to ensure the launch accuracy and reliability of EM railguns, it is necessary to analyze the vibration response of the rail.
A typical barrel of EM railguns is mainly composed of two rails, elastic insulation layers, and the containment structure with fastening parts.When the out containment is assumed to be a rigid boundary and the insulation layer is an elastic support, the rail can be seen as a beam on an elastic base [7].Terefore, many works established a beam model sitting on an elastic foundation to analyze the vibration of the rail.Tzeng [8] simplifed the rail of EM railguns as a Timoshenko beam founded on an elastic base.Te infuences of design parameters and material characteristics on the critical velocity of the rail were analyzed.Johnson and Moon [9] adopted a fnite element code to calculate the dynamic defection of the rail which was simplifed as a Timoshenko beam.Simulation results showed that the contact pressure between the armature and the rail changes when the speed of the armature reaches the critical velocity of the rail.In addition, Nechitailo and Lewis [10,11] modeled an Euler-Bernoulli beam subjected to uniform pressure to estimate the critical velocity of the rail.Numerical tests using the fnite element analysis were used to illustrate the existence of the group resonance phenomenon.Daneshjoo et al. [12] built an Euler-Bernoulli beam model of the rail to study its dynamic behavior.Results showed that the resulting maximum defection varies with physical and mechanical parameters of the rail.However, these results are mainly applicable to the analysis of the critical velocity and resonance in a hypervelocity launcher, always assuming constant pressure on the rail as well as constant speed of the armature.
To analyze the dynamic response of the rail, many approaches have been presented.Yang et al. [13] studied the contact between the armature and the rail during launch by a fnite element model with LS-DYNA code.Tree types of C-shaped armatures were designed to match rails with curvatures.Reck et al. [14] also used the LS-DYNA code to simulate a projectile launch with a moving armature.Assuming the pressure acting on the rail is uniform, the dynamic deformations of the rail and the dynamic stress states were studied.Chen et al. [15] applied the numerical analysis methods to obtain the dynamic response of the rail.Te actions of the EM repulsion and the thermal expansion pressure of the armature were considered.Tian et al. [16] simplifed the rail as a beam on an elastic foundation.Te dynamic response of the beam was solved by using twodimensional Fourier integral transformation.Yin et al. [17] calculated the critical velocity of a fexural wave in the composite housing of the railgun barrel using the 3D transient fnite element.Te dynamic responses and the development of damage for the railgun barrel were analyzed.Zhang et al. [18] also treated the rail as an Euler-Bernoulli beam and proposed a nonlinear fnite element model to study the dynamic characteristics of the rail.In the work, the repulsive force and the contact pressure acting on the rail were considered.
As we know, the dynamics of beams under moving loads has been widely studied for their extensive applications in engineering, such as railroads, bridges, and transport pipelines.Te research procedures on beams can provide helpful ideas for solving the dynamic equation of the rail under time varying moving loads in this paper.Based on the references we have found, there are two types of moving loads on beams which are mainly studied.One is a single moving load, and the other is a harmonic moving load.On the aspect of a single moving load, for instance, AlSaleh et al. [19] investigated the dynamic response of Euler-Bernoulli beams under a traversing moving load based on Green's functions combined with a decomposition technique.Te load was assumed to be moving with diferent values of constant velocity.Akbas et al. [20] analyzed the vibration of a simply supported porous microbeam made of functionally graded materials subjected to a moving load.Te governing equations were obtained by the Lagrange procedure and were solved by Ritz and Newmark average acceleration methods.And then, they also studied the dynamics of carbon nanotube-reinforced composite beams under a moving load using the same methods [21].Guo et al. [22] studied the forced vibration response of thick microplates under a moving load with acceleration in speed on its upper surface.Te governing equations for simply supported microplate were solved by developing a state space method in conjunction with a set of mathematical series.Bozyigit [23] proposed an analytical method based on a combination of transfer matrix formulations and modal superposition to obtain the forced vibration of damaged beams subjected to a moving concentrated load.Kumar et al. [24] derived a simple closed-form expression for free vibration response, and the dynamic behavior of simply supported uniform beams subjected to a single moving point load was analyzed.Moreover, Esen [25] investigated the forced vibration of microbeams under a moving point load with constant velocity using the Newmark-β method and the fnite element method.For the case of a harmonic moving load, Eyvazian et al. [26] investigated the dynamic analysis of a composite cylindrical nanoshell on an elastic foundation subjected to a moving harmonic load.Te equations of motion were derived based on frst-order shear deformation theory and the nonlocal strain gradient theory.Kim and Cho [27] investigated the vibration and buckling of an infnite shear beam-column resting on an elastic foundation.Te response of the beam under moving harmonic loads was obtained using a Fourier transform.Yang et al. [28] established the governing equation of a simply supported thinwalled beam under a harmonic moving load, and closed-form solutions were presented for the lateral, vertical, and torsional vibration of the beam.Chen et al. [29] studied the responses of an infnite beam resting on a tensionless viscoelastic foundation under a harmonic point force moving with a constant speed.To solve the governing equation, the infnite beam is replaced by a fnite long beam expanding the beam defection into a harmonic series.In addition, in the study of dynamic behaviors of the beam supported on a viscoelastic foundation.Bozyigit et al. [30] developed a comprehensive method to solve the motion equations by combing the Adomian decomposition method and the diferential transform method.
In terms of EM railguns, the existing research works provide a good foundation for the vibration analysis and structural design of rails.However, most results are applicable to constant pressure or constant speed of armature motion.In fact, since the exciting current is changing with time, the armature undergoes a variable acceleration motion along the rail.Te dynamic response of beams is signifcantly afected by varying the moving velocity of loads [19].In this paper, in order to study the dynamic behavior of EM railguns deeply, more consideration is given on the nonuniform motion of the armature and the moving loads on rail.Tree kinds of forces acting on the rail are involved, which are time varying and nonperiod, more complex than the constant and periodic moving loads.Te dynamic governing equation of an Euler-Bernoulli beam under time varying moving loads is established.Due to its high nonlinearity, an analytical method combined with numerical integration is presented to get solutions of the governing equation.Te lateral vibration response of the rail in the process of launching is achieved.Furthermore, the infuences of parameters on the vibration amplitude of the rail are analyzed.Te results obtained can provide a meaningful tool for the vibration analysis of EM railguns.Shock and Vibration and the containment structure which includes the stifening steel plates and steel bolts.During launch, the current fows in from one rail through the armature and then out from the other rail, as shown in Figure 2. Te current generates a strong magnetic feld between the two rails, and so the armature moves along the rails propelled by the ampere force.In addition, when the armature is sliding, friction exists between the armature and the rail.Te rail has a rectangular cross section and has a large slenderness ratio.During launch, the rail mainly bears lateral loads perpendicular to the axial.Assuming that the cross section is always perpendicular to the axis when the lateral vibration of the beam occurs, the rail can be simplifed as an Euler-Bernoulli beam with an elastic insulation support wrapped in a rigid constraint, as shown in Figure 3.In addition, the insulation layer is thin and the outer containment is rigid.Under these conditions, the elastic support behavior is similar to the spring model.Terefore, the elastic foundation of the rail is modeled as a Winkler foundation model ignoring the shear defection.Te Winkler model has a simple form and is mathematically easy to handle, and it is used popularly in the analysis of EM railguns in most previous studies [9][10][11][12][13][14][15][16].

Dynamic Modeling of the Rail
To improve the stability and life of the barrel of EM railguns, the insulation should use materials with sufcient strength and stifness to bear the forces transmitted from the rail.Te greater the stifness of the insulation gets, the smaller the efect of its damping is.Terefore, most studies ignore the damping in modeling for simplifcation [6][7][8][9][10][11][12][13][14][15][16][17][18].
Te governing equation of the beam subjected to the time varying moving load is established as follows [7]: where ω(x, t) is the lateral vibration amplitude of the rail, E is the modulus of rail material, I is the moment of inertia of the cross section, m is the mass per unit length of rail, k is the elastic foundation coefcient, and q(x, t) represents the time varying moving load.When the armature moves along the rail at ultrahigh speed, the rail sufers moving loads.In many studies, to obtain the critical velocity of rail, only the EM repulsive force on rail is considered [7][8][9][10][11][12].With further researches, Cao et al. [31] established the governing equation of an Euler-Bernoulli beam under moving loads to investigate the dynamic response of the rail.In the study, two forces including the repulsive force between rails and the thermal pressure of armature on rail were considered.Wu et al. [32] also simplifed the rail to an Euler-Bernoulli beam supported by an elastic base to analyze the vibration response of the rail.In the work, the repulsive force and the contact pressure acting on the rail were considered.Based on the above works, we believe that there are three kinds of forces acting on the rail actually; i.e., the EM repulsive force between rails, the contact pressure between the armature and the rail due to factors such as preloading for tight ftting, and the thermal expansion pressure of the armature induced by the Joule heating efect, respectively.Terefore, the time varying moving loads on rail are composed of three parts as follows: where q 1 (t), q 2 (t), and q 3 (t) are the EM repulsive force, the contact pressure, and the thermal expansion pressure, respectively, H is a Heaviside step function, u(t) is the displacement of the armature, and l is the contact length between the armature and the rail.

EM Repulsive Force.
Te EM feld coordinate of the rail is established as shown in Figure 4. Considering the skin efect of the current and ignoring the distribution of current in the width direction of the rail, the magnetic induction intensity at any point between two rails can be obtained.
According to the Biot-Savart law and reference [33], the magnetic induction intensity B generated by rail 1 at the infnitesimal dy′ of rail 2 is as follows: B � where μ 0 is the vacuum permeability, h is the height of the rail, s is the distance between the inner walls of two rails, and i(t) is the current.Terefore, the EM repulsive force q 1 between rail 1 and rail 2 is derived as follows:

Contact Pressure.
In order to avoid the occurrence of transition and ensure steady contact between the armature and the rail, the Anker law is usually used to judge whether the contact pressure provided by the tight ft can meet the launch demand.However, when the armature moves along the rail at a superhigh speed, the contact pressure will be impacted by the initial interference between the armature and the rail and the tension of the armature under the action of current as well as the wear of the armature [34].So, the calculation of the contact pressure q 2 is complex to some extent.Te contact pressure can be approximately calculated by the following equation [31]: where L ′ is the inductance gradient.Te calculation method of L ′ is detailed in reference [35].

Termal Expansion Pressure.
When the current passes through the armature, the armature will inevitably generate a large amount of heat inside it because of the Joule heating efect.Te Joule heat will cause thermal expansion of the armature and therefore produce a thermal expansion pressure on the rail.According to references [31,36], assuming that the current inside the armature distributes uniformly, the temperature distribution of the armature along its height direction is computed by the following equation: where h a is the height of the armature, α F is the heat transfer coefcient of armature material, T f is the temperature of the medium adjacent to the armature, λ T is the thermal conductivity coefcient of armature material, σ d is the conductivity of armature material, and P(t) � 0.86(J 2 (t)/σ d ) is the power of the Joule heat with the current density J(t) � (i(t)/ l•h a ).Given an extremely short acting time, the change of the temperature can be ignored.Te average temperature along the height direction of the armature is taken as the basis for calculation.Based on equation ( 6), ignoring the medium temperature, the thermal strain of the armature is obtained as follows: where α T is the linear expansion coefcient of armature material.Terefore, assuming that the armature expands uniformly and the force acting on the rail is even, the thermal expansion pressure of the armature on rail is derived as follows: where E d is the elastic modulus of armature material.

Movement of the Armature
As shown in Figure 4, according to the Biot-Savart law, the magnetic induction intensity B 1 generated by rail 1 at the infnitesimal dz is as follows: Similarly, the magnetic induction intensity B 2 generated by rail 2 at the infnitesimal dz is as follows: Tus, the Ampere force on the armature is calculated by the following equation: In addition, there is friction between the armature and the rail.Ignoring the static friction at the initial startup stage, the thrust force of the armature in the sliding stage is as follows: where F f is the friction force, μ is the coefcient of sliding friction, and F N � (q 2 + q 3 )hl is the normal pressure.Terefore, according to the Newton's second law, the velocity and displacement of the armature during launch can be derived as follows: where M and u 0 are the mass and the initial position of the armature, respectively.

Solving Method
Here, the method of separation of variables is adopted to solve the diferential equation of the rail.Te solution of equation ( 1) is assumed as follows [19]: where θ(x) and ϕ(t) represent the vibration mode and the vibration rule, respectively.Te homogeneous vibration equation of equation ( 1) is as follows: Substituting equation ( 14) into equation (15), there is Equation ( 16) can also be expressed as follows: In equation ( 17), x and t are independent of each other.So, both sides of the equation must be a constant at the same time, and the constant should be nonnegative.Assuming the constant is c 2 , equation ( 17) is transformed into the following two independent equations: with Now, solutions of equation ( 18) can be expressed as follows: Introducing hyperbolic functions, i.e., cosh x � (e x + e − x )/2 and sinh x � (e x − e − x )/2 into equation (20), the solutions of equation ( 18) are rewritten as follows: where a 1 , a 2 , a 3 , and a 4 are constant coefcients which can be determined by the boundary conditions and the initial conditions, c 1 and c 2 , are constant coefcients.
According to Appendix A, the constant coefcients a 1 , a 2 , a 3 , and a 4 can be obtained.Substituting equation (A.7) into equation (21), the ith-order vibration mode of the rail θ i (x) is derived as follows: Te function θ i (x) is an orthogonal function which satisfes the following formula: Next, in order to obtain ϕ(t) in equation ( 14) under the time varying moving load q(x, t), the Lagrange equation of the rail is established as follows [37]: where T and U are the kinetic energy and the total potential energy of the beam, respectively.Also, Q(t) is a generalized force, which is defned as follows: Shock and Vibration According to Appendix B, the kinetic energy T and the total potential energy U of the beam can be obtained.So, based on equations (B.3) and (B.7), the following equations are obtained: Substituting equations ( 26)-( 28) into equation ( 24), the Lagrange equation about the generalized coordinator ϕ i (t) is simplifed as follows: According to the Duhamel integral, the general solution of ϕ i (t) in equation ( 29) can be expressed as follows: Te initial conditions of ϕ i (t) at zero time are as follows: Substituting equation ( 31) into (30), there is where According to the properties of the Heaviside step function, the generalized force Q i (t) can be further expressed by the following equation: 6 Shock and Vibration So, substituting equation (34) into equation ( 32), the ithorder vibration rule ϕ i (t) can be achieved as follows: where

Shock and Vibration
Ten, the vibration rule ϕ(t) can be calculated by performing numerical integration on equation (36).Based on the procedures presented above, the solution of equation ( 1) is obtained at last.

Results and Analysis
5.1.Physical Parameters.Te detailed parameters of the armature and the rail under the laboratory conditions are shown in Table 1.Te rail is made of copper, and the armature is made of aluminum.Te exciting current curve is shown in Figure 5. Te peak value of the current is about 400 KA at the time of 0.6 ms.Heavy pulse current with a certain pulse width can make the armature accelerate instantaneously and speed up to an extremely high level [1]. 1, the repulsive force, the contact pressure, and the thermal expansion pressure acting on the rail during launch are calculated as shown in Figures 6-8, respectively.It can be seen from Figure 6 that the repulsive force on rail always exists at the rear position of the armature as it is moving along the barrel.While in Figures 7 and 8, the contact pressure and the thermal expansion pressure only exist at the contact position between the armature and the rail.With the varying of the current, these forces change with time and reach their peaks within approximately 0.6 ms.Te maximum values of the repulsive force, the contact pressure, and the thermal expansion pressure are 1.71 × 10 6 N/m 2 , 4.08 × 10 6 N/m 2 , and 2.24 × 10 8 N/m 2 , respectively.It is illustrated that the thermal expansion pressure is much bigger than the repulsive force and the contact pressure.So, it is necessary to consider the thermal expansion pressure in the vibration analysis of the rail.

Movement of the Armature.
When the armature is moving along the rails, the contact between the armature and the rail is seen as a sliding electrical contact.Te friction coefcient of the sliding electrical contact is from 0.04 to 0.09 generally [38].In this paper, the sliding friction coefcient is selected as 0.05.Te ampere force and the friction force acting on the armature are obtained as shown in Figure 9.It can be seen that both the ampere force and the friction force are time varying.When the current reaches its peak, the forces reach their maximum values, too.Te friction force is approximately 16% of the ampere force.According to the forces analysis in Section 5.2, the friction force is mainly generated by the thermal expansion force.Terefore, the armature undergoes a variable acceleration motion, whose velocity and displacement during launch are illustrated in Figure 10.It can be seen that the acceleration of the armature increases rapidly in the frst half of the motion and then decreases gradually because the thrust reduces.If the initial position of the armature u 0 is assumed to be zero, the whole launch time is about 1.87 ms and the muzzle velocity of the armature is up to 1072.0 m/s.

Vibration Response of the Rail.
Based on the solving procedure presented in Section 4, the frst six vibration modes of the rail are obtained as shown in Figure 11.It can     Shock and Vibration be seen that as the mode order increases, the natural frequency increases.Using high-order modes to calculate the vibration response of the rail can improve the calculation accuracy, but the calculation cost is high.According to the vibration mechanics theory, the vibration of a beam is mainly afected by low-order modes.So, the frst three order modes are used for the calculation of vibration amplitudes [39].Te vibration response of the rail under time varying moving loads is shown in Figure 12.As the armature moves along the rail, the dynamic response of the rail at diferent times and positions can be obtained from the graph.It can be seen from Figure 12 that the rail vibration is low frequency defection wave.Tis tendency agrees with the results obtained in reference [14], approximately.When the defection is positive, it means that the railgun caliber is constricted [14].In Figure 12, the vibration amplitude at the middle of the rail is obviously larger than that at other locations.When the armature runs for 1.1 ms and reaches the position of 0.983 m of the rail, the maximum vibration amplitude (MVA) of the rail is about 1.79 × 10 − 4 m.Te MVAs at diferent positions of the rail are diferent during the period of launch.From a local perspective, the MVAs at the positions of 0.8 m, 1.0 m, and 1.2 m of the rail are 1.72 × 10 − 4 m, 1.78 × 10 − 4 m, and 1.33 × 10 − 4 m, respectively, as given in Figure 13.

Elastic Foundation Coefcient.
Te elastic foundation coefcient of the supporting structure is an important parameter, which will afect the lateral vibration amplitude of the rail.But the armature motion will not be afected because the exciting current and the geometric structure parameters of the rail are unchanged.Now, the elastic foundation coefcient is set to be 7 GPa, 10 GPa, and 15 GPa, respectively.Te lateral vibration amplitudes at the position of 1.0 m are shown in Figure 14.It can be seen that the MVA of the rail decreases with the increasing of the elastic foundation stifness.When the elastic foundation coefcients are 7 GPa, 10 GPa, and 15 GPa, the MVAs are 2.55 × 10 − 4 m, 1.78 × 10 − 4 m, and Shock and Vibration 1.17 × 10 − 4 m, respectively.Terefore, to reduce the vibration amplitude of the rail, the elastic foundation stifness should be enhanced as much as possible by changing the foundation material or dimensions.

Rail Width.
When the rail width is varied, the magnetic induction intensity and the inductance gradient as well as the moment of inertia of the rail will change correspondingly.Here, the rail widths of 8 mm, 10 mm, and 12 mm are selected for calculation.Te moment of inertia, the inductance gradient, and the magnetic induction intensity under diferent widths are calculated, as listed in Table 2.It can be seen that when the rail width increases, the moment of inertia increases, and the bending stifness of the rail is enhanced.However, the inductance gradient and the magnetic induction intensity decrease slightly, and it means that the forces acting on the rail will get smaller.
Te velocity and the displacement of the armature under diferent rail widths are given in Figures 15 and 16.It can be seen that the running speed of the armature decreases with the increasing of the rail width.When the rail widths are 8 mm, 10 mm, and 12 mm, the muzzle velocities of the armature are 1085.1 m/s, 1072.0 m/s, and 1056.9 m/s, respectively.Correspondingly, the running times of the armature on the 2-meter long rail are 1.81 ms, 1.87 ms, and 1.93 ms, respectively.
Ten, the lateral vibration amplitudes at 1.0 m of the rail are shown in Figure 17.It can be seen that the vibration amplitude of the rail decreases with the increasing of the width of the rail.When the rail widths are 8 mm, 10 mm, and 12 mm, the MVAs are 1.89 × 10 − 4 m, 1.78 × 10 − 4 m, and 1.62 × 10 − 4 m, respectively.

Rail Height.
Here, the rail heights of 18 mm, 20 mm, and 22 mm are selected for calculation.Te moment of inertia, the inductance gradient, and the magnetic induction intensity under diferent heights are calculated as listed in Table 3.It can be seen that when the rail height increases, the moment of inertia increases, while the inductance gradient and the magnetic induction intensity decrease.
Te velocity and the displacement of the armature under diferent rail heights are given in Figures 18 and 19.It is illustrated that the running speed of the armature decreases with the increasing of the rail height.When the rail heights are 18 mm, 20 mm, and 22 mm, the muzzle velocities of the armature are 1079.2m/s, 1072.0 m/s, and 1065.9 m/s, respectively.Correspondingly, the running times of the armature on the 2-meter long rail are 1.85 ms, 1.87 ms, and 1.90 ms.
Ten, the lateral vibration amplitudes at 1.0 m of the rail are shown in Figure 20.It can be seen that the lateral vibration amplitude of the rail decreases with the increasing of the rail height.When the rail heights are 18 mm, 20 mm, and 22 mm, the MVAs are 2.19 × 10 − 4 m, 1.78 × 10 − 4 m, and 1.46 × 10 − 4 m, respectively.Now, in order to compare the efects of diferent factors on the muzzle velocity of the armature and the MVA of the rail, the relative reduction rates are listed in Table 4. Te relative reduction rate R i is computed by the following equation: where D 1 is the frst data and D i (i � 2, . .., m) represents the other data.
It can be seen from Table 4 that the changing of parameters has a little efect on the muzzle velocity of the armature, but an obvious efect on the MVA of the rail.Relatively speaking, the infuences of the rail height and the elasticity foundation stifness are greater than that of the rail width.Among them, the relative reduction rate of the MVA reaches 54.2% when the foundation elastic stifness increases to 15 GPa, and the relative reduction rate of the MVA reaches 33.3% when the rail height increases to 22 mm.Shock and Vibration

Conclusions
In this paper, an analytical method combined with numerical integration is proposed to investigate the lateral vibration of rails under time varying moving loads.Te dynamic response of the simplifed Euler-Bernoulli beam is obtained.Te main results are as follows: (1) A dynamic analysis of an EM rail under time varying moving load is carried out based on the Winkler model by using the method of separation of variables and the Lagrange procedure.An intuitive 3D graph refecting the change of rail vibration with launch time and rail position is achieved.It can be seen that the lateral vibration pattern mainly appears as    Shock and Vibration deviation from the axis of the railgun bore.Te vibration amplitude at the middle position of the rail is much bigger than that at other locations.Te maximum vibration amplitude can reach 1.79 × 10 − 4 m.Tis provides an idea for the structure design of the rail, i.e., an irregular section rail with high stifness in its middle part is needed to reduce the vibration amplitude of the rail.(2) Te time for the armature to pass through a 2-meter long rail is approximately 1.8 ms.Te muzzle velocity of the armature is about 1043 m/s.During the whole launch time, the time varying forces reach their peaks when the current is biggest at 0.6 ms.Te thermal expansion pressure induced by the Joule heating efect at the contact area between the armature and the rail is much bigger than the repulsive force and the contact pressure.It is about a hundred times that of the repulsive force.So, it is the main cause of friction and wear between the armature and the rail.(3) Te vibration response is infuenced by the dimensions of the rail and the brace stifness obviously.
Increasing the width and the height of the rail and the elastic foundation stifness can reduce the vibration magnitude.Relatively speaking, the relative reduction rates of the MVAs obtained by increasing the foundation stifness and the rail height are bigger than that obtained by increasing the rail width.Te relative reduction rate reaches 54.2% by increasing the foundation stifness from 7 GPa to 15 GPa and 33.3% by increasing the rail height from 18 mm to 22 mm.It should be noted that although the relative reduction rate is signifcantly higher, it is difcult to enhance the foundation stifness to such a high level.Terefore, it is advisable to increase the rail height as a priority for vibration reduction.In addition, it is good to see that the variation of parameters has slight efects on the muzzle velocity of the armature.
Next, further works will be conducted on the forces and dynamic responses of irregular rails (with concave or convex cross-section) of EM railguns, as well as the variable cross section rails like the trapezoidal rail.Te aim is to study the efects of rail beams with special shapes on the vibration reduction of EM railguns.Moreover, it is necessary to investigate the infuence of the rail vibration on the shot accuracy of EM railguns in the future.where δ i � (i− 0.5)π/L, (i � 1, 2... n) and n represents the number of modes.

B. Details of Solving the Kinetic Energy and the Total Potential Energy of the Beam
Te kinetic energy of the beam T is expressed as follows: T where M i represents the generalized mass of the beam, which is expressed as follows: In addition, the total potential energy is determined by the strain energy of the rail and the strain energy of the elastic foundation.
Te strain energy of the rail U s is computed by the following equation: Tus, the total potential is achieved as follows:

2. 1 .
Mathematical Model.Te schematic diagram of the barrel of an EM railgun is shown in Figure1, which is mainly composed of two rails, an armature, elastic insulation layers, 2

Figure 1 :Figure 2 :Figure 3 :
Figure 1: Schematic diagram of the barrel of an EM railgun.

Figure 4 :
Figure 4: Coordinate system of the EM feld.

Figure 5 :
Figure 5: Curve of the exciting current.

Figure 6 :
Figure 6: Repulsive force on the rail.

Figure 11 :
Figure 11: First six vibration modes of the rail.

Figure 15 :Figure 16 :
Figure 15: Velocities of the armature under diferent widths of the rail.

Figure 18 :
Figure 18: Velocities of the armature under diferent heights of the rail.

Figure 17 :
Figure 17: Efects of rail widths on the lateral vibration of the rail.

Figure 19 :
Figure 19: Displacements of the armature under diferent heights of the rail.

Figure 20 :
Figure 20: Efects of rail heights on lateral vibration of the rail.

Table 1 :
Parameters of the armature and the rail.

Table 2 :
Mechanical parameters under diferent rail widths.

Table 3 :
Mechanical parameters under diferent rail heights.

Table 4 :
Summary of results for comparison.
Combining equation (B.2) with equation (23), the kinetic energy of the beam T can be obtained as follows: Also, the strain energy of the elastic foundation U b is computed by the following equation: i  j ϕ i (t)ϕ j (t)  i  j ϕ i (t)ϕ j (t)