Gas actuator interior ballistic analytical calculation method considering leakage

Abstract

The present application provides a kind of gas actuator interior ballistic analytical calculation method considering leakage, steps are as follows: step 1: gas actuator interior ballistic mathematical model establishment: step 2: gas actuator pressure characteristic analytical solution calculation: step 3: gas actuator work characteristic analytical solution calculation.The gas actuator interior ballistic state parameter calculation method provided by the present application can be used for the evaluation and optimization of the overall design stage of gas actuator, and improves the design efficiency.By analytical solution, the influence of gunpowder gas leakage on the output characteristics of gas actuator is corrected, and it is no longer constrained by the mass conservation assumption, thereby improving the accuracy of interior ballistic prediction results.Gas actuator output characteristics are directly calculated by analytical formula, without using numerical method to integrate and solve the entire working process, which significantly reduces the calculation time.

Description

Technical Field

[0001] This invention relates to a method for calculating the internal ballistics of a gas actuator, particularly an analytical calculation method for the internal ballistics of a gas expansion process under leakage conditions, belonging to the fields of weapons industry and pyrotechnics. Background Technology

[0002] Pyrotechnic devices are a class of instruments that utilize high-pressure gases generated by gunpowder combustion to perform predetermined functions. They possess advantages such as simple structure, low energy input, rapid action, and high reliability, playing a crucial role in spaceflight missions. Common pyrotechnic devices include gas actuators, pyrotechnic pin pullers, and release nuts. Among these, gas actuators can be used to perform tasks such as satellite release, interstage separation, and payload separation. The operation of gas actuators involves phenomena such as gunpowder combustion, mass flow, energy conversion, and pushrod movement, forming a complex internal ballistics problem.

[0003] To achieve accurate calculation of the internal ballistic state parameters of a gas actuator and overcome the slow calculation speed of existing methods, this invention establishes a mathematical model of the internal ballistics of a single-chamber gas actuator, considering gas leakage, based on the first law of thermodynamics. The higher-order non-homogeneous differential equations of the internal ballistics are decoupled based on a few approximations, yielding an analytical solution for the gas pressure of the gas actuator. By integrating the pressure analytical solution with the expansion ratio as the integration variable, and combining the few approximations, analytical solutions for the output energy and load separation velocity of the gas actuator are obtained. Summary of the Invention

[0004] The technical problem solved by this invention is: in order to solve the problem that the internal ballistic process of pyrotechnic devices is complex and the parameters in the mathematical model are severely coupled, making it impossible to obtain the analytical relationship between the output characteristics and the design parameters, this invention provides an analytical calculation method for the internal ballistic state parameters of gas actuators based on high-order small quantity approximation and considering the gas leakage phenomenon.

[0005] The technical solution of this invention is as follows: Based on the establishment of a ballistic mathematical model of a gas actuator, the analytical solution of the differential equation is achieved using a small number of approximate assumptions, including the following steps:

[0006] Step 1: Establishment of the internal ballistic mathematical model of the gas actuator.

[0007] This paper introduces the structure and working principle of the gas-fired actuator, and simplifies the working process appropriately. A geometrical law of combustion is used to mathematically model the gunpowder combustion process, and the Abelian equation of state is used to describe the relationships between the various states of the gunpowder gas. Based on the first law of thermodynamics, energy conservation equations for the various state quantities within the gas-fired actuator are established. By differentiating these energy conservation equations, the following ballistic mathematical model of the gas-fired actuator is obtained:

[0008]

[0009] Where: e is the thickness of the gunpowder after burning; a is the empirical combustion constant; n is the burning rate pressure exponent; r is the combustion rate; P is the gunpowder gas pressure; ψ is the combustion ratio; e1 is the gunpowder thickness; l is the length of the gunpowder particle; m e The mass of the propellant gas leaked into the external environment; c is the leakage flow coefficient; m g ω is the mass of the propellant gas in the chamber; V is the initial mass of the propellant; f is the free-moving volume of the propellant gas; b is the propellant force; ρ is the density of the propellant; γ is the adiabatic index; S is the piston area of ​​the push rod; x is the push rod displacement. For secondary work coefficient; m l For load quality.

[0010] Step 2: Analytical solution calculation of pressure characteristics of gas actuator

[0011] Based on the internal ballistic mathematical model of the gas actuator, focusing on the changes in internal ballistic state variables during the adiabatic expansion stage of the gas actuator, the required set of differential equations is obtained as follows:

[0012]

[0013] To solve the system of differential equations, we assume that the push rod is stationary at the initial moment of this stage, neglect the effect of leaked gas on the free volume of the gas actuator chamber, and rewrite the differential equations in the form of derivatives with respect to displacement:

[0014]

[0015] In the formula: x a V is the equivalent displacement of the initial free motion volume of the propellant gases during the adiabatic expansion stage. a / S;V a This represents the initial free-moving volume of the propellant combustion gases during the adiabatic expansion phase.

[0016] The velocity of the push rod is approximated using the Clausius equation:

[0017]

[0018] In the formula: x e The equivalent displacement is x. e =x a +x;P a This represents the initial pressure during the adiabatic expansion phase.

[0019] Substituting the approximate solution for velocity into the differential equation for pressure yields the analytical solution for pressure:

[0020]

[0021] In the formula: λ is the leakage index; χ is the expansion ratio, and its value is L is the leakage correction factor; V0 is the initial volume of the gas chamber.

[0022] Step 3: Analytical calculation of the working characteristics of the gas actuator

[0023] Based on the analytical solution of the calculated pressure characteristics of the gas actuator, the work done by the expansion of the propellant gas is obtained by integrating the product of the pressure and the piston area over the push rod displacement:

[0024]

[0025] In the formula: W represents the work done by the push rod; x f The maximum displacement of the push rod is generally referred to as the push rod stroke.

[0026] To simplify the integration process, the push rod displacement is normalized, and the integral is solved using the integration by parts method. Approximations are made for parts that cannot be solved analytically, ultimately yielding an analytical calculation method for the work done by the expansion of the propellant gas.

[0027]

[0028] Dividing the work done by the initial energy of the gunpowder yields the work ratio calculation method:

[0029]

[0030] In the formula: Π represents the work ratio; U a The initial internal energy of the gunpowder gas is [value missing].

[0031] Dividing the work done by the load mass and taking the square root yields the analytical formula for calculating the load speed:

[0032]

[0033] In the formula: v x Π is the velocity of the push rod when its displacement is x. x Let x be the work ratio when the push rod is displaced by x.

[0034] Compared with existing calculation methods, the beneficial effects of this invention are as follows:

[0035] 1) A new method for calculating the internal ballistic state parameters of a gas actuator is proposed, which can be used for evaluation and optimization in the overall design stage of the gas actuator, thereby improving design efficiency.

[0036] 2) By using analytical solutions, the influence of gunpowder gas leakage on the output characteristics of the gas actuator is corrected, and it is no longer constrained by the mass conservation assumption, thereby improving the accuracy of the internal ballistic prediction results.

[0037] 3) The output characteristics of the gas actuator can be directly calculated using analytical formulas, without the need to use numerical methods to integrate and solve the entire working process, which significantly reduces the calculation time. Attached Figure Description

[0038] Figure 1 Flowchart for ballistic analysis calculation of an actuator considering leaking gas.

[0039] Figure 2 This is an isometric view of a gas actuator.

[0040] Figure 3 This is a cross-sectional view of a gas actuator.

[0041] Figure 4 The pressure diagram is used for analytical solution comparison.

[0042] Figure 5 The pressure relative error diagram is used for comparison of analytical solutions.

[0043] Figure 6 The work ratio diagram is used to compare the analytical solutions.

[0044] Figure 7 The work ratio relative error diagram is used to compare the analytical solutions.

[0045] Figure 8 The velocity plot is used to compare the analytical solution.

[0046] Figure 9 The velocity relative error diagram is used to compare the analytical solutions.

[0047] The labels in the diagram are explained as follows:

[0048] 1-Gas actuator;

[0049] 101-Igniter, 102-Gunpowder, 103-Powder chamber, 104-Sealing ring, 105-Actuator sleeve, 106-Push rod, 107-Shear pin. Detailed Implementation

[0050] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are not intended to limit the present invention.

[0051] Step 1: Establishment of the ballistic mathematical model within the gas actuator

[0052] like Figure 1 , Figure 2The gas actuator 1 includes an igniter 101, gunpowder 102, a chamber 103, a sealing ring 104, an actuator sleeve 105, a push rod 106, and a shear pin 107; wherein, the cavity between the actuator sleeve 105 and the push rod 106 is a gas chamber. During operation, the igniter 101 ignites the gunpowder 102 to generate high-pressure gas, which drives the push rod 106 to perform work. The working process of the gas actuator 1 is divided into three stages:

[0053] 1) Constant volume combustion stage: The gunpowder 102 is ignited and begins to burn, producing high-pressure gas. The pressure in the gas chamber increases, but the shear pin does not fail and the push rod remains stationary.

[0054] 2) Expansion and combustion stage: The pressure in the gas chamber exceeds the shear strength of shear pin 107, shear pin 107 fails, push rod 106 starts to move, and at this time some gunpowder is still burning;

[0055] 3) Adiabatic expansion stage: The gunpowder 102 is completely burned, and the adiabatic expansion of the gas in the gas chamber drives the push rod 106 to do work.

[0056] To facilitate the establishment of a mathematical model for the gas actuator, the working process of the gas actuator needs to be simplified. Therefore, based on classical internal ballistics theory, the following assumptions are made:

[0057] 1) The gunpowder particles start burning simultaneously, and the combustion process conforms to the geometrical law of combustion;

[0058] 2) The parameters of the propellant gas are the same throughout the gas chamber, and the flow and pressure distribution of the gas are not considered.

[0059] 3) The heat loss and kinetic energy of the leaked gas during the operation of the gas actuator are ignored;

[0060] 4) Under the action of gas, the push rod begins to move in one dimension axially the instant the shear pin stress reaches the shear strength.

[0061] When the physicochemical properties and combustion temperature of gunpowder are constant, the combustion rate r of gunpowder is only a function of pressure, and is generally described using an exponential function:

[0062]

[0063] In the formula: e is the thickness of the gunpowder burnout; a is the empirical combustion constant; n is the combustion rate pressure index; r is the combustion rate; and P is the gunpowder gas pressure.

[0064] The mass ratio of gunpowder burned, i.e., the combustion ratio ψ, can be expressed using the shape geometry parameters of the gunpowder:

[0065]

[0066] In the formula: e1 is the gunpowder thickness; l is the length of the gunpowder particle. Differentiating formula (12) with respect to time t yields:

[0067]

[0068] The Abelian equation of state is used to describe the propellant gas:

[0069] P(V c -m g b) = m g RT (14)

[0070] In the formula: P is the pressure of the propellant gas; V c m is the volume of the air chamber. g For the quality of the propellant gas in the gas chamber;

[0071] b is the residual volume of the gunpowder gas; R is the universal gas constant; T is the temperature of the gunpowder gas.

[0072] To simplify the equation, the free-moving volume V of the propellant combustion gases is defined as:

[0073]

[0074] In the formula: V0 is the initial volume of the gas chamber; S is the area of ​​the push rod piston; x is the push rod displacement; ω is the initial mass of the gunpowder; ρ is the density of the gunpowder. Then formula (14) can be written as:

[0075] PV = m g RT (16)

[0076] Since the combustion process of gunpowder is very short, it can be assumed that the gas does no work during this stage. Therefore, the gas temperature remains constant at the combustion temperature T1. In internal ballistics, the propellant force f is conventionally defined as:

[0077] f = RT1 (17)

[0078] The work done by the propellant gas during its high-temperature expansion phase conforms to the first law of thermodynamics:

[0079] E+U+U e +W=E0 (18)

[0080] In the formula: E is the energy of unburned gunpowder; U is the internal energy of the remaining propellant gas in the gas chamber; U e The internal energy of the propellant gases leaking into the external environment is represented by W; the work done by the propellant gases on the surrounding environment is represented by E0; and the initial propellant energy is represented by E0. The calculation methods for each element are as follows:

[0081]

[0082] In the formula: γ is the adiabatic index; m eThe mass of the propellant gas leaked into the external environment; the internal energy U of the leaked gas. e Using the gunpowder force f as an approximation facilitates subsequent calculations and can correct for the effects of heat loss.

[0083] Differentiating equation (18) with respect to time t yields:

[0084]

[0085] In the formula: dx is the differential of the push rod position, satisfying the following constraints:

[0086]

[0087] In the formula: For secondary work coefficient; m l For load mass. Substitute equation (16) into From:

[0088]

[0089] In the formula:

[0090]

[0091] The rate of change in the mass of gunpowder gas for:

[0092]

[0093] Generally, the leakage rate of propellant gas is directly proportional to the pressure:

[0094]

[0095] In the formula: c is the leakage flow coefficient. Therefore, equation (20) can be reorganized into the following form:

[0096]

[0097] The differential equations for the changes in various physical quantities during the operation of the gas actuator are summarized as follows:

[0098]

[0099] Step 2: Analytical solution calculation of pressure characteristics of gas actuator

[0100] Since the constant-volume combustion stage and expansion combustion stage of the gas actuator last for milliseconds, accounting for less than 5% of the total working time, and the displacement of the push rod is limited during these two stages, contributing less than 1% to the output energy of the actuator, they can be ignored. Therefore, this invention focuses on the adiabatic expansion stage. When the gas actuator enters the adiabatic expansion stage, since the gunpowder has been completely burned, equation (27) only has two differential equations remaining:

[0101]

[0102] It is not difficult to see that equations (28) and (29) together form a high-order non-homogeneous ordinary differential equation system, which has no elementary function solution. Therefore, it is necessary to simplify and decouple the differential equation system.

[0103] Assume the push rod is stationary at the initial moment of this stage. Considering the very small volume of the leaked propellant gas, and that the magnitude of the Pb term is less than 5% of the f term, making it a high-order small quantity, equation (28) is simplified to:

[0104]

[0105] In the formula: V a Let be the initial free-moving volume of the propellant gas during the adiabatic expansion stage. Note that equation (30) represents the derivative with respect to time, but the terminal value of time t is difficult to determine, making it inconvenient to calculate the analytical solution. Therefore, it is rewritten as a derivative with respect to displacement x:

[0106]

[0107] In the formula: x a V is the equivalent displacement of the initial free motion volume of the propellant gases during the adiabatic expansion stage. a / S.

[0108] Since the effect of gas leakage on pressure is much smaller than the effect of push rod displacement, the pressure P in equation (29) is approximated by the adiabatic expansion process under no-leakage conditions, i.e., the Clausius equation:

[0109]

[0110] In the formula: P * The pressure is an approximation; P a This represents the initial pressure during the adiabatic expansion stage. After approximation, the original high-order ordinary differential equations are decoupled into two low-order ordinary differential equations:

[0111]

[0112] First, solve the displacement differential equation and define the equivalent displacement x. e With respect to the speed v of the push rod:

[0113]

[0114] After the replacement, the following is true:

[0115]

[0116] Solving the differential equation, we get:

[0117]

[0118] Equation (36) is an approximate solution for the velocity v of the push rod. Next, we only need to solve for the pressure P and the equivalent displacement x. e The resulting first-order differential equation with constant coefficients. Substituting equation (36) into equation (31), we get:

[0119]

[0120] After sorting, we get:

[0121]

[0122] Integrating both sides and substituting the initial conditions, we get:

[0123]

[0124] It can be observed that the analytical solution for pressure is quite similar to the Clausius equation, but its form is overly complex. To simplify it, the expansion ratio χ and the leakage index λ are defined as follows:

[0125]

[0126] Therefore, equation (39) can be written as:

[0127]

[0128] Equation (41) differs from the Clausius equation by only one coefficient term, and the leakage correction coefficient L can be defined as:

[0129]

[0130] Then equation (41) can be written as:

[0131]

[0132] In practical engineering, it is difficult to measure the initial pressure P during the adiabatic expansion stage. a and the initial free-moving volume of the combustion gas V a Considering the rapid combustion speed of gunpowder, the energy loss of the propellant gases in the first two stages is small compared to the total internal energy. Therefore, the following approximation method is adopted:

[0133]

[0134] Accordingly, the leakage index λ becomes:

[0135]

[0136] The analytical solution for the pressure is obtained as follows:

[0137]

[0138] The simulation conditions are shown in the table below:

[0139]

[0140] The results are as follows Figure 3 , Figure 4 As shown, compared with the traditional Clausius equation calculation method, the pressure analytical solution calculation method proposed in this invention has further improved accuracy.

[0141] Step 3: Analytical calculation of the working characteristics of the gas actuator

[0142] Based on the analytical solution of the pressure during the adiabatic expansion stage, the work characteristics of the gas actuator during this stage can be further analyzed. Integrating the product of pressure P and piston area S over the push rod displacement x yields the work W done by the expansion of the propellant gas:

[0143]

[0144] In the formula: x f The maximum displacement of the push rod is generally referred to as the push rod stroke.

[0145] Define intermediate variable z:

[0146] z=χγ-1 (48)

[0147] Substituting the integral in equation (47), we get:

[0148]

[0149] Integrating by parts, we obtain:

[0150]

[0151] By the mean value theorem for integrals:

[0152]

[0153] In the formula: z a , z b All are in the domain [1, z] f Within the domain:

[0154]

[0155] at the same time:

[0156]

[0157] Therefore, the following approximation is made:

[0158]

[0159] Substituting equation (54) into equation (50), we get:

[0160]

[0161] Therefore, the total work W done by the propellant gas during the adiabatic expansion stage is:

[0162]

[0163] At the same time, the initial internal energy U of the gunpowder gas at this stage a for:

[0164]

[0165] Considering the secondary work coefficient for correction, the work ratio Π is defined as follows:

[0166]

[0167] Using the initial gunpowder energy E0 to calculate the initial internal energy U of the gunpowder gas during the adiabatic expansion phase... a Approximating the speed at which the load separates (v) when the gas actuator finishes working, calculate the speed at which the load separates. f :

[0168]

[0169] More generally, the analytical solution for the push rod velocity can be obtained through equation (59):

[0170]

[0171] In the formula: v x Π is the velocity of the push rod when its displacement is x. x Let x be the work ratio when the push rod is displaced by x.

[0172] Simulation results are as follows Figures 5-8 As shown in the figure, the time consumption of each calculation method is compared in the following table:

[0173]

[0174] The analytical calculation method for work characteristics proposed in this invention significantly reduces the time consumption compared to traditional numerical integration methods, and has higher calculation accuracy compared to calculation formulas that do not consider the influence of gas leakage.

Claims

1. A method for analytical ballistic calculation of a gas actuator considering leakage, characterized in that, The steps include the following: Step 1: Establishment of the ballistic mathematical model within the gas actuator: The geometric combustion law is used to mathematically model the gunpowder combustion process of the gas actuator, the Abelian equation of state is used to describe the relationship between the states of the gunpowder gas, and the energy conservation equations of the state quantities in the gas actuator are established according to the first law of thermodynamics. Step 2: Analytical solution calculation of the pressure characteristics of the gas actuator: Based on the internal ballistic mathematical model of the gas actuator, the changes in the internal ballistic state variables during the adiabatic expansion stage of the gas actuator are analyzed. Step 3: Analytical calculation of the working characteristics of the gas actuator: Based on the analytical solution of the calculated pressure characteristics of the gas actuator, the work done by the expansion of the propellant gas is obtained by integrating the product of the pressure and the piston area over the displacement of the push rod. In step 2, the required system of differential equations is obtained as follows: To solve the system of differential equations, we assume that the push rod is stationary at the initial moment of this stage, neglect the effect of leaked gas on the free volume of the gas actuator chamber, and rewrite the differential equations in the form of derivatives with respect to displacement: In the formula: x a V is the equivalent displacement of the initial free motion volume of the propellant gases during the adiabatic expansion stage. a / S;V a This is the product of the initial free motion of the propellant gases during the adiabatic expansion stage; The velocity of the push rod is approximated using the Clausius equation: In the formula: x e The equivalent displacement is x. e =x a +x;P a This represents the initial pressure during the adiabatic expansion phase. Substituting the approximate solution for velocity into the differential equation of pressure, we obtain the analytical solution for pressure: In the formula: λ is the leakage index; χ is the expansion ratio, and its value is L is the leakage correction factor; V0 is the initial volume of the gas chamber.

2. The method for analyzing the ballistics of a gas actuator considering leakage, as described in claim 1, is characterized in that: In step 1, by differentiating the energy conservation equation, the following mathematical model of the internal ballistics of the gas actuator is obtained: Where: e is the thickness of the gunpowder after burning; a is the empirical combustion constant; n is the burning rate pressure exponent; r is the combustion rate; P is the gunpowder gas pressure; ψ is the combustion ratio; e1 is the gunpowder thickness; l is the length of the gunpowder particle; m e The mass of the propellant gas leaked into the external environment; c is the leakage flow coefficient; m g ω is the mass of the propellant gas in the chamber; V is the initial mass of the propellant; f is the free-moving volume of the propellant gas; b is the propellant force; ρ is the density of the propellant; γ is the adiabatic index; S is the piston area of ​​the push rod; x is the push rod displacement. For secondary work coefficient; m l For load quality.

3. The method for analyzing the ballistics of a gas actuator considering leakage, as described in claim 1, is characterized in that: In step 3, specifically: In the formula: W represents the work done by the push rod; x f The maximum displacement of the push rod is called the push rod stroke.

4. The method for analyzing the ballistics of a gas actuator considering leakage, as described in claim 3, is characterized in that: To simplify the integration process, the push rod displacement is normalized, and the integral is solved using the integration by parts method. Approximations are made for parts that cannot be solved analytically, ultimately yielding an analytical calculation method for the work done by the expansion of the propellant gas.

5. The method for analyzing the ballistics of a gas actuator considering leakage, as described in claim 4, is characterized in that: The work done is divided by the initial energy of the gunpowder to obtain the work ratio calculation method: In the formula: Π represents the work ratio; U a The initial internal energy of the gunpowder gas is [value missing].

6. The ballistic analysis calculation method for a gas actuator considering leakage, as described in claim 5, is characterized in that: Divide the work done by the load mass and take the square root to obtain the analytical formula for calculating the load speed: In the formula: v x Π is the velocity of the push rod when its displacement is x. x Let x be the work ratio when the push rod is displaced by x.

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