2.1 Calculation model of floating machine gun
The model of the two degree of freedom floating machine gun is shown in Fig. 1. The BCG and the inner receiver are connected by the recoil spring. The BCG cycles in a circular motion in the inner receiver and has an independent degree of freedom. The BCG is designed with a locking mechanism and a bullet feeding mechanism. The two are connected with the BCG by a cam curve groove and are equivalent to the BCG by mass. The inner receiver and the barrel are rigidly connected to form a floating body. The floating body and the outer receiver are connected by a bidirection buffer floating spring. The floating body moves in a circular motion in the outer receiver, with another independent degree of freedom. The above components together constitute a two degree of freedom mechanical system with impulse interaction and offset. \({O}_{1}\)、\({O}_{2}\)、\({O}_{3}\) is the marked point on the outer receiver, \({I}_{1}\)、\({I}_{2}\)、\({I}_{3}\) is the marking point of inner receiver,\({I}_{1}\)、\({O}_{2}\) is the displacement zero of the BCG and the floating body respectively, \({O}_{1}\)、\({O}_{3}\) is the extreme recoil point and the extreme return point of the floating body respectively. The movement of the BCG from \({I}_{3}\) to \({I}_{1}\),and floating body from \({O}_{3}\), o\({O}_{1}\) is called recoil༌which is the positive directi, n of displacement, and the movement of the BCG from \({I}_{1}\) to \({I}_{3}\)༌and floating body from\({O}_{1}\) to \({O}_{3}\) is cal, ed return༌which is the negative direction of dis, lacement. The black, red, green and yellow arrows in Fig. 1 represent the internal ballistic force, the piston chamber force to the floating body, the piston chamber force to the BCG and the feeding resistance respectively. The piston chamber force to the floating body and the BCG are equal in magnitude and opposite in direction.
The motion cycle of the floating machine gun is shown in Fig. 2. Figure 2 (a) shows the initial state, the BCG starts to recoil under the action of the recoil spring, affected by the feeding resistance; In Fig. 2 (b), the BCG moves to \({I}_{3}\) and collide with the floating body. In Fig. 2 (c), after the collision,t, e BCG pushes the the floating body which is at \({O}_{2}\) ༌and, forces it to start return and fire bullet. Then the floating body recoils under the action of the internal ballistic force and piston chamber force༌ th, BCG recoils under the piston chamber force. The recoil impulse of the floating body partially offsets the impulse of the internal ballistic force.In Fig. 2 (d), the BCG decelerates under the action of the recoil spring, and the floating body decelerates under the action of the floating spring. In Fig. 2 (E), the BCG recoils to \({I}_{1}\)༌colli, e with and push the floating body to recoil further. As shown in Fig. 2 (f), the floating body and the BCG recoil to the maximum recoil position respectively. After that, the BCG starts return under the action of the recoil spring, and the floating body decelerates under the action of the bidirection buffer floating spring. Finally, the two return to the initial state shown in Fig. 2 (a). The dynamic simulation model based on ADAMS is shown in Fig. 3. Model parameters are shown in Table 1.
Table 1
The parameters fo the floating machine gun model
Name
|
Energy storage(J)
|
Name
|
Mass(kg)
|
Name
|
Length(mm)
|
Bidirectional buffer floating spring
|
4.6
|
BCG
|
1.33
|
Extreme recoil displacement
|
19.1
|
Recoil spring
|
8
|
Floating body
|
5.6
|
Extreme return displacement
|
19.1
|
2.2 Determination of main load
The forces applied in the floating machine gun model include interior ballistic force, piston chamber force and feeding resistance. The internal ballistic pressure can be expressed by the following formula[9].
$$\left\{\begin{array}{c}\psi =\chi Z(1+\lambda Z+\mu {Z}^{2})\\ \frac{dZ}{dt}=\frac{P}{{I}_{k}}\\ Spdt=\phi mdv\\ Sp(l+{l}_{\psi })=f\omega \psi +\frac{\theta }{2}\phi m{v}^{2}\\ v=\frac{dl}{d{t}_{i}}\\ {l}_{\psi }={l}_{0}(1-\frac{△}{\delta }-△(\alpha -\frac{1}{\delta })\psi )\end{array}\right.$$
\({t}_{i}\) 、\(l\)、\(p\)、\(v\)、\(\psi\)、\(Z\)、\(S\)、\({W}_{0}\)、\(\omega\)、\(\delta\) 、\(m\)、\({l}_{0}\)、\(△\)、\({I}_{k}\)、\(\alpha\)、\(f\)、\(\phi\)、\({p}_{0}\) denotes the interior ballistic time, bullet displacement, the pressure of gas in the barrel, velocity of bullet, the mass percentage of burned gunpowder, relative burned thickness of gunpowder, area of the barrel, barrel chamber volume, gunpowder quantity, gunpowder density, mass of bullet, the initial equivalent volume of the barrel, charge density of gunpowder, total impulse of gunpowder gas pressure, residual volume of gunpowder gas, gunpowder force, coefficient, extrusion pressure respectively. \(\chi\)、\(\lambda\)、\(\mu\) denotes the characteristic quantity of powder shape.
The after effect period of interior ballistic pressure can be expressed by the following formula.
$$\left\{\begin{array}{c}{p}_{a}={p}_{k}{e}^{-\frac{{t}_{h}}{b}}\\ b=\frac{(\beta -0.5)\omega {v}_{0}}{S({p}_{k}-{p}_{e})}\\ \beta =\frac{1110}{{v}_{0}}\end{array}\right.$$
\({p}_{a}\) 、\({v}_{0}\)、\({P}_{k}\)、\({P}_{e}\)、\({t}_{h}\)、\(\beta\) denotes the after effect period pressure, muzzle velocity, the average pressure in the barrel at the moment when the bullet flies out of the barrel, 1.8 times atmospheric pressure, time calculated from the moment when the bullet flies out of the barrel, after effect coefficient.
The piston chamber pressure is calculated according to Bravin's formula
$$\left\{\begin{array}{c}{P}_{s}={P}_{d}{e}^{-\frac{{t}^{{\prime }}}{c}}(1-{e}^{-a\frac{{t}^{{\prime }}}{c}})\\ {i}_{0}=\frac{{P}_{d}+{P}_{k}}{2}{t}_{dk}+\frac{\beta -0.5}{S}\omega {v}_{0}\\ b=\frac{{i}_{0}}{{p}_{d}}\end{array}\right.$$
\({P}_{s}\) 、\({P}_{d}\)、\(t{\prime }\)、\(\text{a}\)、\(\text{c}\)、\({t}_{dk}\) denotes the piston chamber pressure, average pressure in the barrel at the moment when the bullet passes through the gas hole, time from the moment when the bullet passes through the gas hole, structural coefficient, time constant, the movement time of bullet from the gas hole to the muzzle.
The internal ballistic pressure and piston chamber pressure are shown in Fig. 4. The internal ballistic force and piston chamber force can be expressed by the following formula.\(\)\(\left\{\begin{array}{c}{F}_{i}=\left\{\begin{array}{c}pS, 0\le t\le t{\prime }\\ {p}_{a}S, t\ge t{\prime }\end{array}\right.\\ {F}_{ch}={P}_{s}{S}_{s}\end{array}\right.\)
\({F}_{i}\) 、\({F}_{ch}\)、\({S}_{s}\) denotes the internal ballistic force, piston chamber force, area of the piston chamber respectively. The rigid flexible coupling model of the belt is established according to MPC-BRE2 method[10][11], and the feeding resistance is measured and shown in Fig. 5.