Keywords

1 Introduction

In recent years, many modern design methods and corresponding sciences have appeared in the field of mechanical design [1,2,3,4]. At present, reliability design and optimization design have reached a certain level in theory and methodology, but it is impossible to exert the great potential of reliability design and optimization design no matter whether reliability design or optimization design is carried out unilaterally. On the one hand, reliability design is sometimes not equal to optimization design. Thus, reliability design of mechanical products does not guarantee their best performance or parameters. On the other hand, optimization design does not necessarily include reliability design. Likewise, mechanical products in the absence of considering reliability optimization design cannot guarantee that it will complete the specified functions within a specified condition and time. Besides, even failures and malfunctions may occur. [5,6,7,8].

The firearms automaton is mainly to transform the chemical energy of gunpowder into the mechanical kinetic energy of the automaton, to complete the action cycle of firing, uncocking, recoiling, cartridge case ejecting, counter-recoiling, feeding and locking, etc. Due to the existence of numerous design parameters in the mechanism of the firearms automaton, the simple reliability design method becomes impotent to determine multiple design parameters at the same time. Therefore, it is very important to carry out the research on optimal design of firearms automatic mechanism reliability. In order to make the firearms automaton not only to ensure the reliability requirements, but also to ensure the best performance and parameters, it is necessary to combine the reliability design with the optimal design. In this way can we give full play to the great potential of the reliability design and optimal design. while giving full play to the strengths of the two design methods. Therefore, the best reliability performance of firearms automaton can be achieved.

In this paper, a reliability optimization design method based on SQP for firearms automaton mechanism is proposed as shown in Fig. 1. The main steps are:

Fig. 1
A flowchart for firearms automaton. It includes complete the dynamic modeling of the firmarms automation, complete the mechanism reliability modeling, determine the design variables for reliability optimization, determine the objective function and constrains, solve the problem, and obtain the design.

A reliability optimization design method based on SQP for firearms automaton mechanism

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  1. (1)

    Complete the dynamic modeling of the firearms automaton;

  2. (2)

    Complete the mechanism reliability modeling based on the performance margin theory;

  3. (3)

    Determine the design variables for reliability optimization;

  4. (4)

    Determine the objective function and constraints of reliability optimization;

  5. (5)

    Solve the problem by SQP;

  6. (6)

    Obtain the optimized design variables.

2 Dynamics and Reliability Modeling of Firearms Automaton

The firearms automaton has the dynamic characteristics of variable mass and multiple motion phases and collisions, which needs to be described by discrete dynamics. Taking a gas operating automatic rifle as the research object, based on its automatic cycle diagram, the motion of the automaton can be represented as a point and line diagram shown in Fig. 2, in which the line segments denote the motion process, and the nodes denote the collision combination or collision separation.

Fig. 2
2 point-line diagrams for automaton motion. Top. The line has 7 segments for free travel, locking travel, counter recoil travel before and after feeding, and before hammer and cartridge pushing. Bottom. The line 6 segments for free and unlocking travels, and recoil travel before cartridge pressing.

The point-line diagram of the motion of the automatic mechanism

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Due to the differences in the mass of the moving parts corresponding to the different stages of motion of the automaton and the magnitude of the combined forces applied, the dynamical equations are divided into 13 stages based on the nodes, where each parameter of the next stage of motion is updated at the nodes. The equations for each stage can be written as follows:

$${\varvec{M}}\mathop x\limits^{ \cdot \cdot } = F$$
(1)

where x denotes the displacement of the moving parts, M denotes the vector of the mass of the corresponding moving parts in each stage, and F denotes the vector of the magnitude of the combined force applied in each stage.

$${\varvec{M}} = \left[ {\begin{array}{*{20}c} {m_{1} } \\ {m_{1} + \frac{{k_{tr1}^{2} }}{\eta }m_{3}^{\prime } + \frac{{k_{tr2}^{2} }}{\eta }m_{3} } \\ {m_{1} + m_{2} + m_{3} + m_{5} } \\ {m_{1} + m_{2} + m_{3} + m_{5} } \\ {m_{1} + m_{2} + m_{3} } \\ {m_{1} + m_{2} + m_{3} } \\ {m_{1} + m_{2} + m_{3} } \\ {m_{1} + m_{2} + m_{3} + m_{4} } \\ {m_{1} + m_{3} + m_{4} } \\ {m_{1} + m_{3} + m_{4} } \\ {m_{1} + \frac{{k_{tr1}^{2} }}{\eta }m_{3}^{\prime } + \frac{{k_{tr2}^{2} }}{\eta }m_{3} } \\ {m_{1} } \\ {m_{2} } \\ \end{array} } \right],{\varvec{F}} = \left[ {\begin{array}{*{20}c} {F_{1} - k_{1} x_{1} + \mu m_{1} g} \\ {F_{1} - k_{1} x_{1} + \mu \left( {m_{1} + m_{3} } \right)g + \frac{{k_{tr1} }}{\eta }F_{tr1} + \frac{{k_{tr2} }}{\eta }F_{tr2} } \\ {F_{1} - k_{1} x_{1} + F_{2} + k_{2} \left( {d_{0} - x_{1} } \right) + \mu \left( {m_{1} + m_{2} + m_{3} + m_{5} } \right)g} \\ {F_{1} - k_{1} x_{1} + F_{2} + k_{2} \left( {d_{0} - x_{1} } \right) + \mu \left[ {F_{0} - \left( {m_{1} + m_{2} + m_{3} + m_{5} } \right)g} \right]} \\ {F_{1} - k_{1} x_{1} + F_{2} + k_{2} \left( {d_{0} - x_{1} } \right) + \mu \left[ {F_{0} - \left( {m_{1} + m_{2} + m_{3} } \right)g} \right]} \\ {F_{1} - k_{1} x_{1} + F_{2} + k_{2} \left( {d_{0} - x_{1} } \right) + \mu \left( {m_{1} + m_{2} + m_{3} } \right)g} \\ {F_{1} - k_{1} x_{1} + F_{2} + k_{2} \left( {d_{0} - x_{1} } \right) - \mu \left( {m_{1} + m_{2} + m_{3} } \right)g} \\ {F_{1} - k_{1} x_{1} + F_{2} + k_{2} \left( {d_{0} - x_{1} } \right) - \mu \left[ {F_{0} - \left( {m_{1} + m_{2} + m_{3} + m_{4} } \right)g} \right]} \\ {F_{1} - k_{1} x_{1} + F_{2} + k_{2} \left( {d_{0} - x_{1} } \right) - \mu \left[ {F_{0} - \left( {m_{1} + m_{3} + m_{5} } \right)g} \right]} \\ {F_{1} - k_{1} x_{1} - \mu \left( {m_{1} + m_{3} + m_{4} } \right)g} \\ {F_{1} - k_{1} x_{1} - \mu \left( {m_{1} + m_{3} } \right)g - \frac{{k_{tr1} }}{\eta }F_{tr1} - \frac{{k_{tr2} }}{\eta }F_{tr2} } \\ {F_{1} - k_{1} x_{1} + \mu m_{1} g} \\ {F_{2} - k_{2} x_{2} + \mu m_{2} g} \\ \end{array} } \right]$$
(2)

Firearms automaton mechanism reliability modeling can be achieved by the performance margin equation [9]:

$${\varvec{M}} = G\left( {{\varvec{P}},{\varvec{P}}_{{{\varvec{th}}}} } \right) > 0$$
(3)

where \({\varvec{M}}\) is the performance margin vector, P is the performance parameter vector, and Pth is the critical value vector of the performance parameter. When the performance parameter vector does not exceed its critical value, i.e. M > 0, the system can work reliably.

The reliability model of the mechanism based on performance margin is:

$$R_{J} = P\left( {\tilde{\user2{M}} > 0} \right) = P\left( {\frac{{\varepsilon - \left| {P_{j}^{r} - P_{j} } \right|}}{\varepsilon }} \right) > 0$$
(4)

where Pj is the performance parameter of the mechanism in state J, \(P_{j}^{r}\) is the actual performance parameter of the mechanism, \(\delta\) is the performance parameter error, and \(\varepsilon\) is the maximum value of the performance parameter error.

3 SQP-Based Optimal Design Method for Mechanism Reliability

The optimal design of firearms automaton mechanism reliability studied in this paper is a nonlinear planning problem, so a sequential quadratic programming algorithm is used to solve it. The algorithm is very suitable for solving small and medium-sized smooth nonlinear problems containing constraints because its fast convergence, time-saving, and high accuracy [10, 11]. A firearms automaton is a system that consists of subsystems such as the case ejecting mechanism, feeding mechanism and locking mechanism. Therefore, the objective function of reliability optimization is generally the maximum system reliability, and the reliability of the subsystems is used as a constraint. The mathematical description of the optimization model is assumed as follows:

Objective function: \(\max \,R\left( X \right)\)

Constraints: \(G_{i} = 0,\,G_{i} \le 0\)

where \(X = \left[ {X_{1} ,X_{2} , \ldots ,X_{n} } \right]^{T}\) are the design variables. \(G\left( X \right) = \left[ {g_{1} \left( X \right),g_{2} \left( X \right), \ldots ,g_{m} \left( X \right)} \right]^{T}\) is the constraint function. By introducing Lagrange multipliers, the SQP optimization algorithm firstly transforms the above constrained optimization problem into an unconstrained optimization problem.

$$L\left( {X,\lambda } \right) = R\left( X \right) + \sum\limits_{{}}^{m} {\lambda_{i} g_{i} \left( X \right)}$$
(5)

where \(\lambda\) is the Lagrange multiplier.

The constraints are then linearized to obtain the subproblem of quadratic programming:

Objective function:

$$\min \frac{1}{2}d^{T} H_{k} d + \nabla R\left( {X_{k} } \right)^{T} d$$
(6)

Constraints:

$$\nabla g_{i} \left( X \right)^{T} d + g_{i} \left( X \right) = 0,i = 1,2, \ldots ,m_{e}$$
(7)
$$\nabla g_{i} \left( X \right)^{T} d + g_{i} \left( X \right) \le 0,i = m_{e + 1} ,m_{e + 2} , \ldots ,m$$
(8)

where d is the search direction of the global variable; \(\nabla\) denotes the gradient; and Hk is the positive definite proposed Newtonian approximation of the Hession matrix of the Lagrange function.

The SQP optimization algorithm implements nonlinear constrained planning precisely by solving quadratic programming subproblems with the quadratically approximated Lagrange function. The transformed objective function can be solved using any quadratic programming algorithm.

4 Results and Discussion

4.1 Design Variables of Firearms Automata

The three most common failures of firearms automaton are case jam, cartridge jam, and insufficient recoiling. And the corresponding mechanisms are the case-ejecting mechanism, feeding mechanism, and locking mechanism, respectively. In this study, the reliability of the firearms automaton is the product of the reliability of the above three mechanisms, and the design variables are the pressure coefficient of air chamber x1, friction coefficient of the guide x2, stiffness of the counter-recoiling spring x3, and stiffness of the hammer spring x4, and the main design variables and their values are shown in Table 1.

Table 1 Main design variables and their values
Full size table

4.2 Objective Function and Constraints of Reliability Optimization

The goal of the reliability optimization of the firearms automaton is to maximize the reliability of the system. Hence, the objective function can be expressed as:

$$\max R\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = R_{1} \cdot R_{2} \cdot R_{3}$$
(9)

Where \(R_{1}\), \(R_{2}\) and \(R_{3}\) represent the reliability of case-ejecting mechanism, feeding mechanism and locking mechanism respectively. For firearms in the design stage, different mechanisms need to be allocated a corresponding reliability level requirement, this paper takes the reliability level of each mechanism is not less than 0.998, thus obtaining the reliability optimization constraints are as follows:

$$R_{1} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = P\left( {v_{1} \ge 3.5} \right) \ge 0.998$$
(10)
$$R_{2} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = P\left( {v_{2} \ge 2.8} \right) \ge 0.998$$
(11)
$$R_{3} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = P\left( {v_{3} \ge 2} \right) \ge 0.998$$
(12)

Where v1, v2 and v3 represent the case ejecting velocity, cartridge feeding velocity and recoil in place velocity, respectively. Through the calculation of the dynamics model of the firearms automaton, the velocity of each characteristic point can be represented by the following surrogate model, respectively:

$$\begin{aligned} & v_{1} = - 13.4 + 27.2 \cdot x_{1} - 2.8 \cdot x_{2} + 0.02 \cdot x_{3} + 0.06 \cdot x_{4} \\ & + 1.3 \cdot x_{1} \cdot x_{2} + 0.01 \cdot x_{1} \cdot x_{3} - 0.04 \cdot x_{1} \cdot x_{4} - 0.01 \cdot x_{2} \cdot x_{3} \\ & + 0.03 \cdot x_{2} \cdot x_{4} - 9.4 \cdot x_{1} \cdot x_{1} - 0.23 \cdot x_{2} \cdot x_{2} - 0.0001 \cdot x_{3} \cdot x_{3} - 0.0002 \cdot x_{4} \cdot x_{4} \end{aligned}$$
(13)
$$\begin{aligned} & v_{2} = - 33 + 6.6 \cdot x_{1} + 0.6 \cdot x_{2} + 0.01 \cdot x_{3} + 0.02 \cdot x_{4} \\ & + 0.25 \cdot x_{1} \cdot x_{2} + 0.002 \cdot x_{1} \cdot x_{3} - 0.005 \cdot x_{1} \cdot x_{4} - 0.004 \cdot x_{2} \cdot x_{3} \\ & - 0.016 \cdot x_{2} \cdot x_{4} - 2 \cdot x_{1} \cdot x_{1} + 0.79 \cdot x_{2} \cdot x_{2} \end{aligned}$$
(14)
$$\begin{aligned} & v_{3} = - 16.4 + 30 \cdot x_{1} - 0.5 \cdot x_{2} + 0.037 \cdot x_{3} + 0.05 \cdot x_{4} \\ & + 1.19 \cdot x_{1} \cdot x_{2} + 0.012 \cdot x_{1} \cdot x_{3} - 0.017 \cdot x_{1} \cdot x_{4} - 0.013 \cdot x_{2} \cdot x_{3} \\ & - 0.003 \cdot x_{2} \cdot x_{4} - 11.5 \cdot x_{1} \cdot x_{1} + 0.19 \cdot x_{2} \cdot x_{2} \end{aligned}$$
(15)

4.3 Reliability Optimization Results

Keeping other conditions unchanged, when the objective function is set as the maximum reliability of case ejecting, the maximum reliability of cartridge feeding, the maximum reliability of sufficient recoil and the maximum reliability of system, the optimization results are shown in Table 2 respectively.

Table 2 Reliability optimization results obtained with different objective functions
Full size table

As can be seen from Table 2, it is advantageous to increase the pressure coefficient of air chamber appropriately to improve the reliability of each mechanism and system. A smaller friction coefficient (0.149) is advantageous to improve the reliability of case ejecting mechanism, feeding mechanism, and locking mechanism, but the ideal value of friction coefficient corresponding to the maximum system reliability is 0.259.

Increasing the stiffness of counter-recoil spring and the hammer spring is advantageous to feeding and the recoiling, but it would increase the resistance in the recoil process, so it will reduce the reliability of case ejecting mechanism. From the point of view of the maximum reliability of the system, the ideal value of the stiffness of the counter-recoil spring and the hammer spring are 230 N/m and 80 N/m, respectively.

The optimized design variables are close to the values of the actual existing products, and the maximum relative error is 1.3%, which proves that the optimization method given in this paper is effective.

5 Conclusion

In this paper, a mechanism reliability optimization design method of firearms automaton is proposed. The dynamics and reliability model of a gas operating firearm automaton are established. And the reliability optimization design of firearms automaton is completed based on the SQP optimization algorithm, and the main conclusions obtained are as follows:

  1. (1)

    The discrete dynamics and SQP algorithm provide an effective solution to deal with the reliability optimization problem of mechanism with multi-phase and multi-collision;

  2. (2)

    The design variables after reliability optimization is close to the value of existing products, which proves the rationality and effectiveness of the mechanism reliability optimization design method of firearms automatic machine given in this paper;

  3. (3)

    The comparison of the reliability optimization results before and after shows that for the firearms automaton, the pressure coefficient of the air chamber is a larger-the-better parameter to a certain extent, while there is an ideal value in the range of friction coefficient and the spring force parameters.