Calculation of trajectory of decoy for torpedoes, which are decreased speed by parachute by double deployment method

Nội dung text: Calculation of trajectory of decoy for torpedoes, which are decreased speed by parachute by double deployment method

1. Mechanics & Mechanical engineering CALCULATION OF TRAJECTORY OF DECOY FOR TORPEDOES, WHICH ARE DECREASED SPEED BY PARACHUTE BY DOUBLE DEPLOYMENT METHOD Lai Thanh Tuan*, Uong Sy Quyen Abstract: This article presents a theoretical model to calculate and solve the problem of the motion of decoy for torpedoes, which arrange in the missile. The missile is decreased speed by parachute with double deployment method. The trajectory of the missile is devided into stages, on each stage different forces act on the missile. At each stage, the motion of the missile is determined by a system of differential equations. The system of differential equations is solved for each stage of the trajectory. The results of the calculation allow determining the appropriate time to activate the drogue and main parachutes to ensure the required speed of the missile when it hits the water and ensures the minimum flight time to increase the efficiency of missiles. Keywords: Decoy for torpedoes; Parachute by double deployment method; Missile’s trajectory. 1. INTRODUCTION Recently, with the aim of gradually mastering the design and technology of manufacturing weapons for the Navy's forces, one of the research directions, which increase the survivability of naval ships is design, manufacturing decoy for torpedo for ships. To ensure the safety and effective operation of the equipment inside the missile, it is required to reduce the speed of the missile before hitting the water. The use of a dual deployment parachute is an effective method, which allows ensuring the drop speed of the missile as required before going to the water. Calculating the movement trajectory of the decoy for the torpedo allows determining the right time to activate the reducer of the missile. On the basis of the designed parachute, it will determine the parameters of the missile before hitting the water to evaluate the operability of structures, which arrangements inside the missile. 2. STRUCTURE OF DECOY FOR TORPEDOES, WHICH ARE DECREASED SPEED BY PARACHUTE AND WORKING MECHANISM OF THE MISSILE Decoy of the torpedo, which is decreased speed by parachute by the dual deployment method includes the: head block, in which arrange warhead, payload block, and bulkhead; the middle block, in which arrange the drogue parachute and the trigger of drogue parachute; the electronics block with bulkheads and extended block, in which arrange the main parachute, the trigger of the main parachute, the engine, and fins (Fig. 1). Figure 1. Arrangement of compartments in the decoy of a torpedo which are decreased speed by parachute. 1 - The head block; 2 - The middle block; 3 - The electronics block; 4 - The extended block. 70 L. T. Tuan, U. S. Quyen, “Calculation of trajectory of by double deployment method.”
2. Research The dual deployment is done as follows: At the highest point of the missile's trajectory, the drogue parachute will be deployed. At this point, the vertical velocity of the missile is almost zero, so it is safe for both the missile and the parachute. After that, the missile will start to fall quickly and the drift is small, and even though it is small so it is less affected by horizontal winds. When the missile falls to a predetermined height, the main parachute will be deployed. Because the main parachute has a large area, it will quickly reduce the speed of the missile before hitting the water and the drift is small. As a result, the accuracy of projectile placement is guaranteed. 3. CALCULATION OF THE MOVEMENT OF THE DECOY FOR TORPEDOES, WHICH ARE DECREASED SPEED BY PARACHUTE WITH DUAL DEVELOPMENT METHOD 3.1. Phases of missile's movement The trajectory of the missile, which is decreased speed by parachute with dual development method consists of four stages. In each stage the forces act on the missile are shown as follows (Fig. 2): Figure 2. Forces act on missile when it is in motion. R - Aerodynamic drag; Q - Gravity; P - Engine thrust. Stage 1 (the active phase of the missile's trajectory): The projectile moves under the action of engine thrust P; aerodynamic drag R and gravity Q. Stage 2 (the passive phase of missile's trajectory when the reducer by parachute has not been activated): The missile moves like a bullet under the action of aerodynamic drag R and gravity Q. Stage 3 (the moving phase with the drogue parachute): Forces, which act on the missile on the projectile include aerodynamic drag R; gravity Q, and parachute drag force Fdp. Stage 4 (last stage of the trajectory, the missile moves with the main parachute): Forces, which act on the missile on the projectile include gravity Q and the parachute drag Fdc (the aerodynamic drag is ignored due to the velocity of the missile has been signficantly reduced by the main parachute and their value is much smaller than the parachute drag, which is created by the reducer). In the general case, the above forces (except gravity) are not located at the center of mass of the missile. Therefore, when leaving the position of the force to the center of mass, they are equivalent to a force and a corresponding moment. Journal of Military Science and Technology, Special Issue, No.75A, 11 - 2021 71
3. Mechanics & Mechanical engineering 3.2. Determine the forces acting on the missile on the trajectory 3.2.1. Determine the thrust of the engine P The engine thrust of the missile is determined by the general expression [1]: P m Ua S a p a p k (1) in which, m - The mass flow rate; Ua - The gas flow rate at output cross-section of the nozzle; Sa - The outlet cross-sectional area of the nozzle; pa - The gas pressure at outlet section of the nozzle; pk - The atmospheric pressure. In the calculation of the missile's external ballistic, the engine thrust is usually written through the effective ejection speed Ue: gS p Pm U ; U U By* 1 ; B* a 0c (2) e e e0  Where: Ue0 - The effective ejection speed at height y = 0;  - The weight of the propellant; g - The gravity acceleration. As for the solid propellant, which is usually used in missiles nowadays: Ue0 = 1800 ÷ 2100 m/s. 3.2.2. Determine the aerodynamic drag R The aerodynamic drag R consists of three components, which arrange along the axes of the speed coordinate system (orthogonal coordinate system whose origin coincides with the center of mass O: the frontal drag Rx , the lifting drag Ry , and the lateral drag Rz (Fig. 3). With the hypothesis that the missile is symmetrical about the longitudinal axis, then Rz = 0 and the components Rx and Ry are determined as follows [1]: Figure 3. The speed coordinate system and components of aerodynamic drag. 22 v v v v 2 Rx SC x , SC x0 1 K (3) 22 aa 22 v v v v Ry SC y , SC y (4) 22 aa in which S - The maximum cross-sectional area of the missile (Miden cross-section); Cx0  - The aerodynamic coefficient of front drag Cx corresponding to the angle  = 0; Cy - 72 L. T. Tuan, U. S. Quyen, “Calculation of trajectory of by double deployment method.”
4. Research The aerodynamic coefficient of lifting drag Cy corresponding to the angle  = 1; K - The experimental coefficient. 3.2.3. Determine the gravity Q Gravitational acceleration at height y is approximated by the following expression: y ggy 0 12 (5) R In which, the gravitational acceleration changes both in value and direction with latitude and altitude, but this change is insignificant, so when solving the problem of unguided missile's motion, we consider that the vector of gravitational acceleration is vertical and it is constant in direction and value, usually g = 9.81 m/s2. 3.2.4. Determine the parachute drag force Fd ( Fdc , Fdp ) The parachute drag force Fd is calculated according to the formula [3]: v2 FCS (6) ddd 2 in which Cd - The drag coefficient of the parachute; ρ - The air density; Sd - The drag area of the parachute; v - The relative velocity of the missile to the environment. The drag coefficient of a parachute is normally determined by experiment or theoretical calculation. According to data of the drag coefficient of some shapes [4], we see that the round parachute has a fairly large drag coefficient, so it provides high efficiency for deceleration and it is easy to manufacture, and at the same time, when operating the parachute is less prone to drift. Therefore, we have chosen a round parachute to design a deceleration parachute for the missile. 3.3. The establishment a system of differential equations of motion 3.3.1. The initial hypothesis To simplify the process of calculation while ensuring accuracy, we use some hypotheses. Firstly, the values of meteorological factors for the calculation are taken according to Soviet artillery standards. Second, the weight of the missile and propellant are taken exactly as the values in the design. The characteristic quantities of a jet engine correspond to the standard temperature of the propellant (+15 ºC). In addition, we use the standard motion conditions: ignore the asymmetry of the missile with respect to the longitudinal axis; the angle of attack is zero (δ = 0) during the motion; the value and direction of the gravitational acceleration are constant; ignore the curvature of the Earth's surface, the Coriolis force due to the ejection of gases and due to the rotation of the Earth. 3.3.2. The system of differential equations of motion of the center of mass of the warhead On the basis of the phases of the motion of the missile, the system of differential equations of the missile's motion on each stage is expressed as follows: On the first stage (the active phase): Journal of Military Science and Technology, Special Issue, No.75A, 11 - 2021 73
5. Mechanics & Mechanical engineering UC vy F ve0 g  sin ; Q  11    0 (7) g  cos ;y v sin ; x v cos . v Over the entire trajectory during crosswinds: UCFe 0Fdp dc v  y F v   g sin  ; Q 1  1  ( M m ) ( M m ) 0 dp dc (8) g  cos  ;y v sin  ; x v cos  V . v g in which Fdp, Fdc - The drag force of the main parachute and drogue parachute; Vg - The wind speed; α, β, η, γ - The control coefficients; π(y) - The pressure function by altitude; F(vτ) - The function to determine the front drag Rx of the aerodynamic force. 3.3.3. The method of solving system of differential equations of motion of missile In order to solve the problem of calculating the trajectory of the missile's motion with the above purposes, we solve the system of differential equations on each stage in turn according to the following steps: In the first step, we solve the system of equations (7) with the hypotheses that the missile moves without using a reducer by parachute. The purpose of this step is to find out the parameters of the missile's ejection at the time of activation of the reducer by parachute (determine the parameters of the missile at the highest point of the trajectory). In the next step, we solve the system of equations to the end of the third stage to find out the law of variation of velocity and the moment with the smallest velocity, which is suitable to choose as the deployment time of the main parachute. In the last step, we solve the system of equations to the end of stage 4 to find the missile's range, the flight time, the drift and check with the requirements set out when designing the structure. 4. SIMULATION RESULTS AND DISCUSSIONS 4.1. The input parameters The input parameters for calculating the trajectory of the missile are taken from the results of solving the interior ballistic in monograph [2] and the technical requirements of missile: the weight of the missile: q =15 KG; coefficient by the shape of the missile: i58 = 1.05; caliber: d = 0.12 m; muzzle velocity: v0 = 31.5 m/s; horizontal wind speed: Vg = 5 m/s; launch angle: 45º; effective injection speed of jet engine: Ue = 1930 m/s; the working time of engine: tđc = 0.32 s; the weight of the propellant of the engine: ω = 9.2 KG; the required speed when hitting the water of the missile: Vn = 5 m/s; the safe speed of missile when deploying main parachute: Vat ≤ 100 m/s; the weight of warhead: M =10 KG; the weight of reducer by parachute (expected before design): m = 3 KG; the weight of parachute and cables: md = 0.5 KG. 4.2. The results of calculation of the missile's trajectory In the first and second stages, considering that the missile moves without a reducer by parachute, based on the graph of the trajectory of the missile (Fig. 4), we determined the 74 L. T. Tuan, U. S. Quyen, “Calculation of trajectory of by double deployment method.”
6. Research time, in which the missile reaches the highest point is t = 10.36 s. This time is selected as the time to activate the main parachute. In the third stage, when using the drogue parachute, the missile's velocity gradually decreases to a certain value, then begins to increase. This is explained that when the missile is subjected to an additional force, which is proportional to the velocity and this force does not overcome the gravity, so the missile continues to be accelerated by gravity after reaching the minimum speed. At the final stage of the trajectory, because the decoy for torpedoes requires a small time for parachute deployment, we choose the time, in which the main parachute deploys is t = 24.15 s. This time is close to the stopping time at the third step. At this moment the missile has a velocity of 48 m/s (Fig. 5). 500 150 450 400 350 100 300 250 y(m) v(m/s) 200 50 150 100 50 0 0 0 200 400 600 800 1000 1200 1400 1600 1800 0 5 10 15 20 25 30 35 x(m) t(s) Figure 4. The trajectory of the missile Figure 5. The law of variation of the without activating the reducer by parachute. missile's velocity on the trajectory. 5. CONCLUSIONS The results of the problem show that: the speed of the missile before hitting the water is 5 m/s; the flight time on the trajectory: 31.53 s; the missile's range: 1650 m; the speed at the highest moment when opening the drogue parachute: 109.76 m/s. These parameters are guaranteed to satisfy the technical requirements set out for the decoy of torpedoes. By calculation of the trajectory of decoy for the torpedoes, which are decreased speed by parachute by double deployment method, we have determined the time to activate the drogue parachute and the time to activate the main parachute to ensure the required speed before hitting the water. At the same time, the results of the calculation also ensure the flight time of the missile is as small as possible to increase the effectiveness of the missile. REFERENCES [1]. Nguyen Van Tho, Nguyen Dinh Sai, “Exterior ballistics”, Military Technical Academy, 379 p., 2003. [2]. Nghiem Xuan Trinh, Nguyen Quang Luong, Nguyen Trung Hieu, Ngo Van Quang, “Intorior ballistics”, Military Technical Academy, 318 p., 2015. [3]. Zhukov V.P., Rasskazov A.V. License of invention “High-speed rocket braking device” on www.FindPatent.ru. [4]. Sighard F. Hoerner, “Aerodynamic Drag”, Practical data on aerodynamic drag evaluated and presented by Sighard F. Hoerner, 1951. Journal of Military Science and Technology, Special Issue, No.75A, 11 - 2021 75
7. Mechanics & Mechanical engineering TÓM TẮT TÍNH TOÁN QUỸ ĐẠO CHUYỂN ĐỘNG CỦA ĐẠN MỒI BẪY NGƯ LÔI ĐƯỢC GIẢM TỐC BẰNG DÙ THEO PHƯƠNG PHÁP TRIỂN KHAI KÉP Bài báo trình bày mô hình tính toán và giải bài toán chuyển động của đạn mồi bẫy ngư lôi giảm tốc bằng dù theo phương pháp triển khai kép. Quỹ đạo chuyển động của đạn được chia thành các giai đoạn với các lực tác dụng thành phần khác nhau. Trên mỗi giai đoạn, chuyển động của đạn được xác định bởi hệ phương trình vi phân. Kết quả tính toán cho phép xác định thời điểm thích hợp kích hoạt dù phụ, kích hoạt dù chính nhằm đảm bảo yêu cầu về tốc độ của đạn khi chạm nước và thời gian bay của đạn nhỏ nhất nhằm tăng tính hiệu quả của đạn. Từ khóa: Decoy for torpedoes; Parachute by double deployment method; Missile’s trajectory. Received 16th August 2021 Revised 10th October 2021 Accepted 11th November 2021 Author afilication: Faculty of Weapons, Le Quy Don Technical University. *Corresponding author: thanhtuan711@gmail.com. 76 L. T. Tuan, U. S. Quyen, “Calculation of trajectory of by double deployment method.”