Gas actuator inner trajectory analytical calculation method considering leakage

Abstract

The invention provides a gas actuator inner trajectory analytical calculation method considering leakage. The method comprises the following steps: step 1, establishing a gas actuator inner trajectory mathematical model; step 2, calculating a gas actuator pressure intensity characteristic analytical solution; and step 3, calculating a gas actuator acting characteristic analytical solution. The method for calculating the ballistic state parameters in the fuel gas actuator can be used for evaluation and optimization of the overall design stage of the fuel gas actuator, and design efficiency is improved. Through an analytical solution, the influence of gunpowder gas leakage on the output characteristics of the gas actuator is corrected, and the influence is not restrained by a mass conservation hypothesis any more, so that the precision of an inner trajectory prediction result is improved. The output characteristics of the gas actuator are directly calculated through an analytical formula, integral solution is not required to be carried out on the whole working process by using a numerical method, and the calculation time consumption is remarkably reduced.

Description

Technical Field

[0001] The invention relates to a method for calculating the internal ballistics of a gas actuator, in particular to an analytical calculation method for the internal ballistics of a gas expansion process under leakage conditions, and belongs to the field of weapons industry and pyrotechnics. Background Art

[0002] Pyrotechnic devices are devices that use high-pressure gas generated by the combustion of gunpowder to complete the intended design function. They have the advantages of simple structure, small input energy, rapid action, and high reliability, and play a vital role in space flight missions. Common pyrotechnic devices include gas actuators, pyrotechnic pin pullers, and separation nuts. Gas actuators can be used to complete tasks such as satellite release, interstage separation, and payload separation. The working process of gas actuators involves phenomena such as gunpowder combustion, material flow, energy conversion, and push rod movement, forming a complex internal ballistics problem.

[0003] In order to achieve accurate calculation of the internal ballistic state parameters of the gas actuator and overcome the problem of slow calculation speed of the existing methods, the present invention aims at a single-chamber gas actuator and establishes a mathematical model of the internal ballistics of the gas actuator taking into account gas leakage according to the first law of thermodynamics. Based on the small quantity approximation assumption, the internal ballistic high-order non-homogeneous differential equations are decoupled to obtain the analytical solution of the gas pressure of the gas actuator. The analytical solution of pressure is integrated with the expansion ratio as the integral variable, and the analytical solution of the output energy and load separation speed of the gas actuator is obtained in combination with the small quantity approximation assumption. Summary of the invention

[0004] The technical problem solved by the present invention is: in order to solve the problem that the internal ballistic process of the pyrotechnic device is complex in form, the parameters in the mathematical model are severely coupled, and the analytical relationship between the output characteristics and the design parameters cannot be obtained, the present invention provides a method for analytical calculation of the internal ballistic state parameters of the gas actuator based on high-order small quantity approximation and taking into account the gas leakage phenomenon.

[0005] The technical solution of the present invention is: on the basis of establishing a mathematical model of the internal ballistics of the gas actuator, using a small amount approximate assumption to realize the analytical solution of the differential equation, including the following steps:

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

[0007] The structure and working principle of the gas actuator are introduced, and the working process of the gas actuator is appropriately simplified. The geometric combustion law is used to mathematically model the gunpowder combustion process, and the Abel state equation is used to describe the relationship between the various states of gunpowder gas. According to the first law of thermodynamics, the energy conservation equation of each state quantity in the gas actuator is established. By differentiating the energy conservation equation, the internal ballistic mathematical model of the gas actuator can be obtained as follows:

[0008]

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

[0010] Step 2: Calculation of the analytical solution of the gas actuator pressure characteristics

[0011] Based on the mathematical model of the internal ballistics of the gas actuator, the required differential equations are obtained mainly for the changes in the internal ballistic state quantity of the gas actuator during the adiabatic expansion stage:

[0012]

[0013] In order to solve the differential equations, it is assumed that the push rod is stationary at the initial moment of this stage, the effect of leaked gas on the free volume of the gas actuator chamber is ignored, and the differential equations are rewritten in the form of derivatives with respect to displacement:

[0014]

[0015] Where: x a is the equivalent displacement of the initial free motion volume of the gunpowder gas during the adiabatic expansion stage, and its value is V a / S;V a is the initial free movement volume of gunpowder gas during the adiabatic expansion stage.

[0016] Use the Clausius equation to approximate the push rod's velocity:

[0017]

[0018] Where: x e is the equivalent displacement, whose value is x e =x a +x;P a is the initial pressure during the adiabatic expansion phase.

[0019] Substituting the velocity approximate solution into the pressure differential equation, we can obtain the analytical solution for pressure:

[0020]

[0021] Where: λ is the leakage index; χ is the expansion ratio, and its value is L is the leakage correction coefficient; V0 is the initial volume of the air chamber.

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

[0023] According to the calculated analytical solution of the gas actuator pressure characteristic, the product of pressure and piston area is integrated with the push rod displacement to obtain the work done by the expansion of gunpowder gas:

[0024]

[0025] Where: W is the work done by the push rod; x f It is the maximum displacement of the push rod, generally referred to as the push rod stroke.

[0026] In order to simplify the integration process, the push rod displacement is normalized, the integral is solved using the method of integration by parts, and the part that cannot be solved analytically is approximated. Finally, the analytical calculation method for the work done by the expansion of gunpowder gas is obtained:

[0027]

[0028] Dividing the work value by the initial energy of the gunpowder gives the work ratio calculation method:

[0029]

[0030] Where: Π is the work ratio; U a is the initial internal energy of the gunpowder gas, and its value is

[0031] Dividing the work value by the load mass and taking the square root gives the analytical formula for the load velocity:

[0032]

[0033] Where: v x is the push rod speed when the push rod is displaced x; x is the work ratio when the push rod is displaced by x.

[0034] Compared with the existing calculation method, the beneficial effects of the present invention are:

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

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

[0037] 3) The output characteristics of the gas actuator are directly calculated by analytical formulas, without the need to use numerical methods to integrate and solve the entire working process, which significantly reduces the calculation time. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 This is the flow chart of analytical calculation of internal ballistics of gas actuator considering leakage.

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

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

[0041] Figure 4 Pressure diagram for comparison of analytical solutions.

[0042] Figure 5 This is the relative error diagram of pressure compared with the analytical solution.

[0043] Figure 6 Work ratio diagram for comparison of analytical solutions.

[0044] Figure 7 Relative error diagram of work ratio compared with analytical solution.

[0045] Figure 8 Velocity diagram for analytical solution comparison.

[0046] Fig. 9 Velocity relative error diagram for comparison with analytical solution.

[0047] The numbers in the figure 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 DESCRIPTION

[0050] The technical solution of the present invention will be clearly and completely described below in conjunction with the accompanying drawings of the present invention. Obviously, the described embodiments are not limitations of the present invention.

[0051] Step 1: Establishment of the mathematical model of gas actuator internal ballistics

[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 an air chamber. When working, the igniter 101 ignites the gunpowder 102 to generate high-pressure gas to push the push rod 106 to do work externally. The working process of the gas actuator 1 is divided into three stages:

[0053] 1) Constant volume combustion stage: the gunpowder 102 ignites and starts to burn to produce high-pressure gas, the pressure in the gas chamber increases, but the shear pin does not fail, and the push rod is in a stationary state;

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

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

[0056] In order to facilitate the establishment of the mathematical model of the gas actuator, it is necessary to simplify the working process of the gas actuator. Therefore, the following assumptions are made based on the classical interior ballistic theory:

[0057] 1) The gunpowder particles begin to burn at the same time, and the combustion process conforms to the geometric combustion law;

[0058] 2) The parameters of the gunpowder gas at all locations in the gas chamber are equal, and the flow and pressure distribution of the gas are not considered;

[0059] 3) Ignore the heat loss during the working process of the gas actuator and the kinetic energy of the leaked gas;

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

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

[0062]

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

[0064] The mass ratio of gunpowder burned, that is, the combustion ratio ψ, can be expressed using the shape and geometric parameters of gunpowder:

[0065]

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

[0067]

[0068] The Abel equation of state is used to describe the gunpowder gas:

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

[0070] Where: P is the pressure of gunpowder gas; V c is the volume of the air chamber; m g The quality of the gunpowder gas in the gas chamber;

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

[0072] In order to make the equation more concise, the free movement volume V of gunpowder gas is defined as:

[0073]

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

[0075] PV=m g RT (16)

[0076] Since the duration of the gunpowder combustion process is very short, it can be considered that the gas does not do work externally during this stage, so the gas temperature remains unchanged at the combustion temperature T1. The gunpowder force f is usually defined as:

[0077] f=RT1 (17)

[0078] The high-temperature expansion stage of gunpowder gas does work externally in accordance with the first law of thermodynamics:

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

[0080] Where: E is the energy of the unburned gunpowder; U is the internal energy of the remaining gunpowder gas in the gas chamber; U e is the internal energy of the gunpowder gas leaked into the external environment; W is the work done by the gunpowder gas to the outside world; E0 is the initial gunpowder energy. The calculation methods for each item are as follows:

[0081]

[0082] Where: γ is the adiabatic index; m eis the mass of the explosive gas leaked into the external environment; the internal energy of the leaked gas U e The use of gunpowder force f for approximate calculations facilitates subsequent calculations and can correct for the effects of heat loss.

[0083] The derivative of formula (18) with respect to time t is:

[0084]

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

[0086]

[0087] Where: is the secondary work coefficient; m l is the load mass. Substituting equation (16) into In the result:

[0088]

[0089] Where:

[0090]

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

[0092]

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

[0094]

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

[0096]

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

[0098]

[0099] Step 2: Calculation of the analytical solution of the gas actuator pressure characteristics

[0100] Since the constant volume combustion stage and the 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 in these two stages is limited, the contribution to the output energy of the actuator is less than 1%, which can be ignored. Therefore, the present invention studies the adiabatic expansion stage. When the gas actuator enters the adiabatic expansion stage, since the gunpowder has been completely burned, there are only two differential equations left in formula (27):

[0101]

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

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

[0104]

[0105] Where: V a is the initial free motion volume of the gunpowder gas in the adiabatic expansion stage. Note that equation (30) is represented by the time derivative, and the terminal value of time t is difficult to determine, which is not convenient for the calculation of the analytical solution. Therefore, it is rewritten as the derivative form with respect to the displacement x:

[0106]

[0107] Where: x a is the equivalent displacement of the initial free motion volume of the gunpowder gas during the adiabatic expansion stage, and its value is V a / S.

[0108] Since the effect of gas leakage on pressure is much smaller than the effect of push rod displacement, the pressure P is approximately calculated in equation (29) according to the adiabatic expansion process in the absence of leakage, that is, the Clausius equation:

[0109]

[0110] Where: P * is the pressure after approximate processing; P a is the initial pressure in the adiabatic expansion stage. After approximate processing, 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 And the push rod movement speed v:

[0113]

[0114] After replacement:

[0115]

[0116] Solving the differential equation yields:

[0117]

[0118] Equation (36) is the approximate solution for the push rod motion velocity v. Next, we only need to solve the pressure intensity P and the equivalent displacement x e The first-order constant coefficient differential equation is formed. Substituting equation (36) into equation (31) yields:

[0119]

[0120] After sorting:

[0121]

[0122] Integrating both sides and substituting the initial conditions yields:

[0123]

[0124] It can be found that the analytical solution of pressure is similar to the Clausius equation, but the form is too complicated. In order to simplify it, the expansion ratio χ and the leakage index λ are defined as:

[0125]

[0126] So formula (39) can be written as:

[0127]

[0128] It is found that 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 formula (41) can be written as:

[0131]

[0132] In engineering practice, it is difficult to measure the initial pressure P during the adiabatic expansion stage. a and the initial gas free motion volume V a Considering that the gunpowder burns very quickly, the loss of internal energy of the gunpowder gas in the first two stages is small compared to the total internal energy, so the following method is adopted for approximation:

[0133]

[0134] Correspondingly, the leakage index λ becomes:

[0135]

[0136] The analytical solution of pressure is as follows:

[0137]

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

[0139]

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

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

[0142] Based on the analytical solution of the pressure in the adiabatic expansion stage, the work characteristics of the gas actuator in this stage can be further analyzed. Integrating the product of the pressure P and the piston area S with respect to the push rod displacement x can yield the work W done by the expansion of the gunpowder gas:

[0143]

[0144] Where: x f It is the maximum displacement of the push rod, generally referred to as the push rod stroke.

[0145] Define the intermediate variable z:

[0146] z=χγ-1 (48)

[0147] Substituting the integral in equation (47) yields:

[0148]

[0149] Integrating by parts gives:

[0150]

[0151] According to the integral mean value theorem:

[0152]

[0153] Where: z a , z b Both are in the domain [1,z f ]. Considering that within the domain:

[0154]

[0155] at the same time:

[0156]

[0157] Therefore, the following approximation is made:

[0158]

[0159] Substituting formula (54) into formula (50), we obtain:

[0160]

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

[0162]

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

[0164]

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

[0166]

[0167] Using the initial gunpowder energy E0 to calculate the initial internal energy U of the gunpowder gas in the adiabatic expansion stage a By approximation, calculate the load separation speed v at the end of the gas actuator operation f :

[0168]

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

[0170]

[0171] Where: v x is the push rod speed when the push rod is displaced x; x is the work ratio when the push rod is displaced by x.

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

[0173]

[0174] The analytical calculation method for work characteristics proposed in the present invention consumes significantly less time than a traditional numerical integration method, and has higher calculation accuracy than a calculation formula that does not consider the influence of gas leakage.

Claims

1. A method for calculating the internal ballistics of a gas actuator taking leakage into consideration, characterized in that: The steps include: Step 1: Establishment of the mathematical model of gas actuator internal ballistics: The geometric combustion law is used to mathematically model the gunpowder combustion process of the gas actuator, the Abel state equation is used to describe the relationship between the various states of gunpowder and gas, and the energy conservation equation of each state quantity in the gas actuator is established according to the first law of thermodynamics; Step 2: Calculation of analytical solution of gas actuator pressure characteristics: Based on the mathematical model of internal ballistics of gas actuator, the changes of internal ballistic state quantity of gas actuator in adiabatic expansion stage are analyzed. Step 3: Calculation of analytical solution of gas actuator work characteristics: According to the calculated analytical solution of the gas actuator pressure characteristic, the product of pressure and piston area is integrated with the push rod displacement to obtain the work done by the expansion of gunpowder gas.

2. The method for calculating the internal ballistics of a gas actuator taking leakage into consideration according to claim 1, characterized in that: In step 1, by differentiating the energy conservation equation, the mathematical model of the gas actuator interior ballistics is obtained as follows: Where: e is the thickness of the gunpowder burned; a is the empirical combustion constant; n is the burning rate pressure index; r is the burning rate, P is the gunpowder gas pressure; ψ is the combustion ratio; e1 is the gunpowder thickness; l is the gunpowder particle length; m e is the mass of the propellant gas leaked into the external environment; c is the leakage flow coefficient; m g is the mass of gunpowder gas in the gas chamber; ω is the initial mass of gunpowder; V is the free movement volume of gunpowder gas; f is the gunpowder force; b is the residual capacity of gunpowder gas; ρ is the gunpowder density; γ is the adiabatic index; S is the push rod piston area; x is the push rod displacement; is the secondary work coefficient; m l is the load mass.

3. The method for calculating the internal ballistics of a gas actuator taking leakage into consideration according to claim 1 or 2, characterized in that: In step 2, the required differential equations are analyzed as follows:

4. The method for calculating the internal ballistics of a gas actuator taking leakage into consideration according to claim 3, characterized in that: In order to solve the differential equations, it is assumed that the push rod is in a stationary state at the initial moment of this stage, the influence of the leaked gas on the free volume of the gas actuator chamber is ignored, and the differential equation is rewritten in the form of a derivative with respect to displacement: Where: x a is the equivalent displacement of the initial free motion volume of the gunpowder gas during the adiabatic expansion stage, and its value is V a / S;V a is the initial free motion product of the gunpowder gas during the adiabatic expansion stage.

5. The method for calculating the internal ballistics of a gas actuator taking leakage into consideration according to claim 4, characterized in that: Use the Clausius equation to approximate the push rod's velocity: Where: x e is the equivalent displacement, whose value is x e =x a +x;P a is the initial pressure during the adiabatic expansion stage.

6. The method for calculating the internal ballistics of a gas actuator taking leakage into consideration according to claim 5, characterized in that: Substituting the velocity approximate solution into the pressure differential equation to obtain the analytical solution for pressure: Where: λ is the leakage index; χ is the expansion ratio, and its value is L is the leakage correction coefficient; V0 is the initial volume of the air chamber.

7. The method for calculating the internal ballistics of a gas actuator taking leakage into consideration according to claim 1, characterized in that: In step 3, specifically: Where: W is the work done by the push rod; x f The maximum displacement of the push rod is called the push rod stroke.

8. The method for calculating the internal ballistics of a gas actuator taking leakage into consideration according to claim 7, characterized in that: In order to simplify the integration process, the push rod displacement is normalized, the integral is solved using the method of integration by parts, and the part that cannot be solved analytically is approximated. Finally, the analytical calculation method for the work done by the expansion of gunpowder gas is obtained:

9. The method for calculating the internal ballistics of a gas actuator taking leakage into consideration according to claim 8, characterized in that: Divide the work value by the initial energy of the gunpowder to get the work ratio calculation method: Where: Π is the work ratio; U a is the initial internal energy of the gunpowder gas, and its value is 10. The method for calculating the internal ballistics of a gas actuator taking leakage into consideration according to claim 9, characterized in that: Divide the work value by the load mass and take the square root to get the analytical calculation formula for the load speed: Where: v x is the push rod speed when the push rod is displaced x; x is the work ratio when the push rod is displaced by x.

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