## 3 Theoretical Model  
Depending on the hypothesis of the Bernoulli-Euler beam theory [7; 17; 18; 19; 20; 21; 22], the tube vibration function with moving force and moving mass was solved [10; 23; 24; 25]. Therefore, two theories are used, and the first is the ballistic theory, the second is the unbalanced rotor theory. The vertical deformation of the barrel axis is represented by \(y_{1}\), and the longitudinal deformation of the barrel axis is represented by \(y_{2}\), as shown Figure 5, The vibration equation of the barrel considering the weight and acceleration of the projectile is:  
\[\frac{\partial^{2}}{\partial x^{2}}[EI(x)\frac{\partial^{2}y_{1}(x,t)}{ \partial x^{2}}]+c_{1}\frac{\partial y_{1}(x,t)}{\partial t}+\rho(x)A\frac{ \partial^{2}y_{1}(x,t)}{\partial t^{2}}=-(mS+F_{2})\delta(x-\xi) \tag{9}\]  
\[\frac{\partial^{2}}{\partial x^{2}}[EI(x)\frac{\partial^{2}y_{2}(x,t)}{ \partial x^{2}}]+c_{2}\frac{\partial y_{2}(x,t)}{\partial t}+\rho(x)A\frac{ \partial^{2}y_{2}(x,t)}{\partial t^{2}}=(mg+F_{1})\delta(x-\xi) \tag{10}\]  
\[S=\frac{\partial^{2}y_{1}}{\partial t^{2}}+2v\frac{\partial^{2}y_{1}}{ \partial x\partial t}+v^{2}\frac{\partial^{2}y_{1}}{\partial x^{2}} \tag{11}\]  
\[\zeta=\int_{0}^{t}v(t)dt \tag{12}\]  
\[y=\sqrt{y_{1}^{2}+y_{2}^{2}} \tag{13}\]  
where: \(E\) is the elastic modulus of the material; \(I(x)\) is the moment of inertia of the cross-section of a cage-like structure; \(c_{1},c_{2}\) is the viscous damping coefficient; \(y_{1}\) is the vertical deformation of the axis; \(y_{2}\) is the longitudinal deformation; \(y\) is the resultant displacement. \(\rho(x)\) is the material density; \(A\) is the cross-sectional area of the barrel; \(I(x)\) and \(A\) are constants under the simplification of uniform section; \(m\) is the mass of the projectile; \(F\) is the axial force of barrel from muzzle to tail, \(v(t)\) is the moving speed of the projectile; \(g\) is the acceleration of gravity; \(\xi\) is the travel of the bullet from interior ballistic calculation; \(S\) is the additional term of an external force acting on the barrel after considering the inertial effect of the bullet, and \(\delta(\ldots)\) is the Dirac function. Similar to as Figure 9, the calculation formula of \(I(x)\) is as follows:  
\[I(x)=nm_{sg}(\frac{D}{2})^{2}+I_{zg} \tag{14}\]  
where: \(I_{zg}\) is the moment of the barrel connector and 6 barrels, \(m_{sg}\) is the mass of the bullet, \(n\) is the number of the bullet. After calculation, its internal ballistic time is only 1.2 ms. The practical firing rate of M134 is 2000 rpm, that is, \(n=1\);  
\[I(x)=m_{sg}(\frac{D}{2})^{2}+I_{zg} \tag{15}\]  
The bullet goes through an unbalanced force, under the action of the unbalanced force, the rotor will produce vibration. At high speed, even a small mass eccentricity will produce a larger centrifugal force. The data are obtained from their modeling, and some are obtained from relevant literature [13; 14; 15; 17]. The gun has an rpm from 300 to 6000, and the time interval is 200 ms to 10 ms. By the time the second bullet was started, the first bullet was already out of the gun. There is no interaction between the first and second bullet. In addition, two bullets do not appear in the barrels at the same time. The Gatling gun rotation rate is at Table 2.  
According to the size of each part, the solid is drawn by 3D software, and then imported into the finite element software, which is divided by hexahedral mesh, and the calculated model is obtained. During the rotation of the barrel group continuously driven by the motor, the bullet moves along the barrel from back to front, which will cause the vibration caused by the additional mass. In addition, it is also a recognition that the bullet causes the additional vibration of the barrel assembly due to the unbalanced mass. Therefore, it is better that the lighter mass of the bullet and which will not affect the barrel assembly.