Study on the spatial motion model of underwater projectile

This paper studies the three-dimensional motion modeling for a super cavitating projectile. 6-degrees of freedom equation of the motion was constructed by defining the forces and moments acting on the supercavitating body while moving underwater.

Research STUDY ON THE SPATIAL MOTION MODEL OF UNDERWATER PROJECTILE Nguyen Huu Thang*, Nguyen Hai Minh, Dao Van Doan Abstract: This paper studies the three-dimensional motion modeling for a super cavitating projectile 6-degrees of freedom equation of the motion was constructed by defining the forces and moments acting on the supercavitating body while moving underwater The impact force of the projectile tail with the cavity wall is determined by the super-cavity size calculations when considering the effect of the reduction of the super-cavity size The calculation results obtained the integrated motion of the super cavitating projectile in the water environment, which was the basis for the study of scattering for the underwater projectile Keywords: Underwater projectile; Super-cavity; Supercavitating projectile; Planing force; Cavity model INTRODUCTION In literature, the dynamics of a supercavitating body is more complex than that of moving body in a flow regime that does not separate in the air or in water The complexity causes are the cavity instability surrounding the body and the discrete impact between the body tail and the cavity wall At each moment, the force is determined from the relative position and relative motion of the body compared with the cavity and the cavity shape, formed in the first time Experimental studies in [6] have shown that the stability of motion of supercavitating body can be achieved with all velocities ranging from 50-1450 (m/s) The analysis showed that the four different mechanisms of motion stabilization sequentially act at motion velocity increase (Fig.1) Figure Schemes of motion of supercavitating models Two - cavity flow scheme (Fig.1a), V  50 (m/s): In this case the hydrodynamic drag center is placed behind the mass center, and stabilizing moment of the force Y2 acts to the model It means that the classic condition of stability is fulfilled Sliding along the internal surface of the cavity (Fig 1b), V  70-200 (m/s): To compensate the buoyancy losses the body's tail part is planing along the lower internal cavity surface In this case, the motion may be stable as a whole Impact interaction with cavity boundaries (Fig 1c), V  300-900 (m/s): Presence of initial perturbations of the model attack angle and the angular velocity causes to the impact of the model tail part against the internal boundary of the cavity The mathematical modeling showed that after this impact the model can perform steady Journal of Military Science and Technology, Special Issue, No.60A, May 2019 117 Mechanics & Mechanical engineering or damped oscillations accompanied by periodic impacts of the tail part alternately against the upper and lower cavity walls Then the motion can remain stable as a whole Aerodynamic interaction with vapor-splash in the cavity (Fig 1d), V  1000-1450 (m/s): At very high velocities the aerodynamic and splash forces of interaction with vapor filling the cavity and splashes near the cavity boundaries have a considerable effect on the body motion Since the clearance between the body surface and the cavity boundary usually is small compared to the cavity radius, they apply the known methods of near-wall aerodynamics to estimate the arising forces The result of longitudinal inertial motion model (2D) was obtained on the basis of the assumption the cavity has an axial symmetrical ellipse [1, 2, 3] Therefore, this model does not mention the horizontal plane motion caused by the original disturbance factors In this paper, we establish the spatial motion model (3D) of an inertial motion projectile considering the gravity effect, the reduction of the super-cavity size, the impact force and the friction between the tail and the cavity wall, the effect of movement around the projectile center to the motion trajectory of the center MODELING OF SUPERCAVITATING PROJECTILE 2.1 The equations of 3D motion of supercavitating projectile According to the general order, the mathematical model describes the motion of the supercavitating projectile (SC-projectile), including the equations of the solids of six degrees of freedom, equations for calculating the shape and size of the cavity, the equations determine the relationship to calculate the interaction forces between them Figure shows a scheme of the 6-DOF SC-projectile motion The body coordinate system O1X1Y1Z1 and the flow coordinate system O1XVYVZV are shown (Fig 2a), An origin of both the coordinate systems is placed at the projectile mass center O1 O1XV of the flow coordinate system is directed along the velocity vector V of the projectile mass center The axis O1X1 of the body coordinate system is directed along the longitudinal body axis The axes O1Y1, O1Z1 together with O1X1 axis form the positive triangle Also, we will use the fixed coordinate system OXYZ and the semi-body coordinate system O1XgYgZg (Fig 2b) The direction of the semi-body coordinate system axes coincides with the direction of axes of the fixed coordinate system Oxyz at each time instant a) b) Figure Scheme of forces acting onto SC- projectile and the coordinate systems 118 N H Thang, N H Minh, D V Doan, “Study on the spatial … underwater projectile.” Research Writes a set of equations of the 6-DOF motion of a solid body in projections on the axes of the body coordinate system O1X1Y1Z1, which are the principal axis of inertia of the body:   dVx1  m  dt   yVz1   zVy1    Fx1       dVy1 m  dt   zVx1   xVz1    Fy1    dV   m z1   V   V x y y x     Fz1   dt   d x I x   y z  I z  I y    M x1  dt  d y   z  x  I x  I z    M y1 I y dt   I d  z     I  I   M  z1 x y y x  z dt  d  ( y cos    z sin  )   dt cos   d   y sin    z cos   dt  d   x  tang  ( y cos    z sin  )  dt   dx  V cos(   ) cos(   )  dt  dy   V sin(   )  dt  dz  V cos(   ) sin(   )  dt (1) Where:Vx1 ,Vy1 ,Vz1 - is the projection of the velocity vector of the body mass center in the  body coordinate system;    x , y , z  - is the angular velocity vector relative to the body mass center; m- is the body mass; I x , I y , I z - are the moments of inertia relatively to the axes O1X1, O1Y1, O1Z1, respectively;  F ,  F ,  F ,  M , M ,  M x1 y1 z1 x1 y1 z1 - are the projections of the composite force vector and the main moment on the same axis;  , ,  - are the yaw, pitch, and roll angle, respectively;  ,  - are the angle of attack and sliding angle, respectively, they are present in the relationship between the velocity coordinate system O1XvYvZv and the body coordinate system O1X1Y1Z1 (figure 2a) In this case, the following relations are valid: Vx1  V cos  cos  ; V  V sin  Vy1  V sin  cos  ; z1 We accept the following assumptions for the formulation of the general problem of the SC-Projectile dynamics: Journal of Military Science and Technology, Special Issue, No.60A, May 2019 119 Mechanics & Mechanical engineering The SC- projectile is a slender body of revolution and symmetrical through the x-axis, in this case, Iy = Iz; The disk-shaped projectile nose with Dn diameter is the Cavitator shape; The mass m, the mass center position xc, and the vehicle moments of inertia Ix, Iz not vary during motion The Components of force acting on the projectile consist: the projections of the   gravity force mg , the hydrodynamic force on the cavitator Fn   Fnx1 , Fny1 , Fnz1  and the force of interaction between the projectile tail and the cavity wall  Fp   Fpx1 , Fpy1 , Fpz1  :  F  F  F  mg sin  ;  F  F  F  mgcos cos ;  F  F  F  mgcos sin  x1 nx1 px1 y1 ny1 py1 z1 nz1 pz1 2.2 Cavity model The cavity is a major component of the SC-projectile system The behavior of the cavity bubble around the projectile affects the body immersion The cavitator continuously creates a cavity while the projectile is moving (Fig 3a) The cavity shape model is taken from a solution presented in Lovinovich’s work (1972) The shape and size of the cavity depending on the motion parameters of the projectile To determine the shape and size of the cavity, an approximate method is given in the theory of G.V Logvinovich [10]: Divide the entire cavity into finite sections; each section is cylindrical, limited by the cross-section perpendicular to the cavity axis (Fig 3b) Each part of the cavity is formed from the moment the water environment begins to separate from the surface of the Cavitator center Due to the effect of inertial force, the cavity expands in a direction perpendicular to the orbit of the Cavitator After the cavity reaches its maximum size, it starts to contraction and disappears due to hydrostatic pressure pressing on the outer surface of the cavity The important parameter representing the characteristics of the cavity is the cavitation number:   2( p  pc ) / ( V ) (2) where: p - is the ambient pressure the cavity; pc - is the pressure inside the cavity; V - projectile velocity;  - density of water Let the projectile starts a motion with the arbitrary angle, hydrostatic pressure varies with depth The hydrostatic pressure around of the cavity at each moment is determined by the expression: p (t )  patm   g  H  S (t ) sin  (t ) cos (t)  (3) where: patm - atmospheric pressure at the surface; H0 - is initial depth; 120 N H Thang, N H Minh, D V Doan, “Study on the spatial … underwater projectile.” Research S (t ) - is the trajectory of the cavitator center formed at the time instant t; sin  (t ) and cos (t) - is the deflection of the cavitator center trajectory line at each time for the axis O1Xg a) b) Figure Scheme of symmetrical axis and sections of cavity According to G.V Logvinovich, the cavity sections expand and shift independently of each other, radius of cavity section at the distance x from cavitator is determined [10]: Rc (x)  Rn K 0,82 (1   ) (4)  and the cavity contraction rate R c is: Rc   20 (1   ) 0,5 4,5 (23/17) (0,82 ) V (1  ) K1 19  1  1,92 K2 (  3) (5)  Here, K1, K2 are two functions of  , that defined to represent the cavity model: 0,5 40   4,5  17  2l 1,92 1 K1  ( )  ; K  1  1  K   dn    1    The projectile's nose (cavitator) is a fundamental part of the projectile; it creates a cavity bubble around the body and generates force components on projectile’s nose The forces acting on a disk-type cavitator in steady flow are well understood To a very high accuracy, it may be assumed that the force vector acts through the center of the disk and normal to its wetted face The drag, lift and side force are then given by [8]: Journal of Military Science and Technology, Special Issue, No.60A, May 2019 121 Mechanics & Mechanical engineering Fnx1  Fny1   V 2 Dn2 C D (1+ )cos n cos n V 2 Dn2 C D (1+ ) sin  n cos n (6) (7) V 2 Dn2 C D (1+ ) sin  n cos n (8) Where:  n    z xc / Vx1 - is the cavitator attack angle;  n     y xc / Vx1 - is Fny1  the cavitator sliding angle, dn- is the disk cavitator diameter; C D =0.82- is the drag coefficient of the cavitator at an attack angle of zero and a Cavitation number of zero 2.3 Planing forces model The interactive force ( Fplane ) on the wet surface of the projectile tail affects the projectile speed and the stability of motion So that, the stability of the SCprojectile depends on the moments associated with the nose and the aft The contact of the projectile tail with the cavity wall creates the planing force [8], that interaction force is nonlinear The shape and size of the cavity and the relative position between the bullet and the cavity will determine immersion The interactive force included planing force ( Fp ) and friction force ( Ff ) acting at the transom of the sliding section is shown in Fig.4 Figure Scheme of the interaction force between the projectile tail with a cavity wall 122 N H Thang, N H Minh, D V Doan, “Study on the spatial … underwater projectile.” Research In planing force modeling methods, it is assumed that the planing force depends entirely on the vertical velocity Hassan has given the following set of equations for the planing force and moment for cylindrical body planing on a curved surface as [11]:   R  R 2   R  h  c b b Fp  V ( R ) sin  p cos  p 1        h  Rc  Rb    Rb  2h  2 c   Rb  h  h2 M p  V ( Rc2 ) cos  p     h  Rc  Rb  Rb  2h  (9) (10) where: Rb , Rc - is the radius of the body and the cavity at the immersion position, respectively; lp, l - are the wet length of the projectile tail and the total length of the projectile, respectively;  p - is the immersion angle; h - is the immersion depth Since the planing force is a non-linear function, an appropriate method is to calculate the planing force along the plane of sliding planing (not necessarily along the Y or Z directions) by determining the immersion angle along that direction This force can then be that separated into the Y and Z directions (see Fig 4) Since the Projectile's tail is in contact with the liquid at high speed, so the friction force is generated [11]: Ff  V S p C p cos  p (11) p R3   1  uc2  arctan(uc  us )   S p  4R (us  ) arcsin us  us  us2     tan  2 tan   2  p p  p  Rc  R  h ; C p  0.031  ul p     k  ; uc  p h  p h ; us  R p  p - is the distance between centers of cross-section of cavity and projectile at the transom The planing force in the body frame are obtained through the projections of the plane of immersion into the X-Y and X-Z: Fplane  Fpx1    Ff      Fpy1  Fp cos p       Fpz1    Fp sin  p  (12) Because the shape of the cavity at the tail is closely related to the cavitator angle of attack  n and cavitator sliding angle  n , in the steady-state regime, by assuming Journal of Military Science and Technology, Special Issue, No.60A, May 2019 123 Mechanics & Mechanical engineering instantaneous cavity formation and cylindrical cavities, the total cavitator angle of attack expressed in polar coordinate system as:  t   n2   n2 , it determines the n ) defines the angle between the plane of n immersion and the vertical symmetry axis of the cavity cross section at immersion location seen in Figure From the size of the cavity and the relative position of the projectile to the cavity (Figure 4), it is possible to determine when the bullet tail impacts the cavity wall through the immersion depth The parameter h in equation (9) is expressed as: planing direction  p  arctan( 0 When ( Rc  R)   t l h (13)   t l  (R c  R) Othewise ( Rc  R)   t l and the immersion angle at body tail with considering the effect of contraction rate of the cavity determined as: When ( Rc  R)   t l   t l  (R c  R) Othewise ( Rc  R)   t l 0 p   (14) SIMULATION RESULTS AND DISCUSSION The differential equation system (1) combines the formula for calculating the cavity size (4), (5), the impact condition (13) and the immersion angle (14), solved by the numerical integration method, using the algorithm Runge Kuta math- Applying calculations for underwater projectile parameters in table We received results in the variation of change of dynamics parameters and the motion trajectory on figure 5-8 Table Initial parameters of the simulation Parameters Projectile mass (m) Projectile length (l) Cavitator diameter (dn) Body diameter (d) Distance from tail to mass center of projectile (lp) Shot depth (h) Density of water (  ) Moment of inertia about X-axis (Ix) Moment of inertia about Y-axis (Iy) Moment of inertia about Z-axis (Iz) Initial center velocity in the X- direction (V0x) Initial center velocity in the Y- direction (V0y) Initial center velocity in the Z- direction (V0z) 124 Value 13,3.10-3 48.10-3 1,2.10-3 5,56.10-3 14,31.10-3 0,8 1000 39.10-9 2,1.10-6 2,1.10-6 350 5 Unit (kg) (m) (m) (m) (m) (m) (kg/m3) (kg.m2) (kg.m2) (kg.m2) (m/s) (m/s) (m/s) N H Thang, N H Minh, D V Doan, “Study on the spatial … underwater projectile.” Research a) b) Figure Time evolution of angular velocity  y (a) and angular velocity z (b) Under the gravity effect, the lift force at the nose of the projectile and the condition of the loss of Archimedes force, the projectile moves around the center in the cavity with rules angular velocity in figure Due to the movement of projectile around the center, the oscillates angular velocity around the value “0” When the impact of the projectile tail with the cavity wall happens, the moment of the impact force serves as the stable moment of the projectile Thereby creating the projectile movement is stable through three phases: the movement does not involve the force with the cavity wall, the movement involves the impact with the cavity wall and the movement slides on the “wet road” After that, when the projectile totally soaks, a monotonous increase of the angle of attack leads to the projectile instability on the flight path a) b) Figure Time evolution of angle of attack  (a) and sliding angle  (b) Due to movement around the center of mass, the angle of attack and sliding angle begins to oscillate around the value "0" (Figure 6) Under the influence of stable moment, the bullets move stably through three stages: motion does not affect Journal of Military Science and Technology, Special Issue, No.60A, May 2019 125 Mechanics & Mechanical engineering on the wall of super-cavity, the movement has an effect on the wall of super-cavity and sliding motion is on "wet road" When the bullets are completely wetted, angle of attack and of sliding angle increase monotonically That leads to unstable ammunition on the trajectory The variation of the projectile center velocity in the X, Y and Z directions is shown in Figure and Figure 8a Accordingly, the value of the X- directional speed decreases monotonically, the speed value in the Y and Z directions fluctuates around the value "0" and increases monotonically at the stage of instability on the flight path b a) Figure Time evolution of velocity of mass center in the X- direction (a) and Y- direction (b) The trajectory of the projectile mass center in three-dimensional space is shown in Figure 8b, including the value of the range (X), the height (Y) and the drift (Z) a b Figure The velocity in the Z- direction (a) and the trajectory of the mass center in space (b) 126 N H Thang, N H Minh, D V Doan, “Study on the spatial … underwater projectile.” Research CONCLUSION Based on the study of the dynamics model of the Supercavitating projectile taking into account the effect of super cavity size change and the motion around the center during the stable motion phase, we obtain the results as given: Under the gravity effect, the lift force at the nose of the projectile and the condition of the loss of Archimedes force, resulting in the projectile may be unstable on the flight path Moment of the impact force serves as the stable moment for the projectile Thereby, it helps the projectile movement is stable through three phases: the movement does not involve the force with the cavity wall, the movement involves the impact with the cavity wall and the movement sliding on the cavity surface This conclusion is consistent with the results of empirical analysis [6] The modeling allows the study to simultaneously moving in the vertical and horizontal planes of the underwater projectile, which was the basis for the study of scattering for the underwater projectile REFERENCES [1] Đào Văn Đoan, Nguyễn Hải Minh, Nguyễn Hữu Thắng, “Nghiên cứu ảnh hưởng hiệu ứng siêu khoang đến tính ổn định chuyển động quỹ đạo đạn súng bắn nước”, Tạp chí Khoa học Kỹ thuật (Học viện KTQS), Số 187 (12-2017), tr.21-30 [2] Đào Văn Đoan, Nguyễn Văn Hưng, “Xây dựng mơ hình tính tốn thuật phóng ngồi đạn bắn nước”, Tạp chí Khoa học Kỹ thuật (Học viện KTQS), Số 163 (8-2014), tr.114-122 [3] Nguyen Thai Dung, Nguyen Duc Thuyen, “Study on impact forces of the underwater cavity projectile”, Journal of Science and Technology, Viet Nam Academy of Science and Technology, Vol 54, No 54 (6), 2016, pp 797-807 [4] Nguyễn Đông Anh, “Động lực học hệ vật rắn”, NXB Xây dựng, Hà Nội (2000) [5] Rand R, Pratap R, Ramani D, Cipolla J, Kirschner I, “Impact dynamics of a supercavitating underwater projectile”, In: Proceedings of the 1997 ASME Design Engineering Technical Conferences, 16th Biennial Conference on Mechanical Vibration and Noise, Sacramento 1997 [6] Zhao Xinhua, Sun Yao, “Advances in supercavitating vehicle control technology”, College of Automation, College of Civil Engineering, Harbin Engineering University, Harbin, 150001, China [7] Dzielski J, Kurdila A , “A benchmark control problem forsupercavitating vehicles and an initial investigation of solutions”, J Vib Control 9, 2003, pp 791–804 [8] Kirschner IN, Rosenthal BJ, Uhlman JS, “Simplified dynamical systems analysis of supercavitating high-speed bodies”, Cav03-OS-7-005 In: proceedings of the fifth international sym-posium on Cavitation (CAV2003), 2003, Osaka, Japan [9] Fine NE, Kinnas S, “A boundary element method for the analysis of the flow around 3-D cavitating hydrofoils”, J Ship Res 37, 1993, pp 213–224 Journal of Military Science and Technology, Special Issue, No.60A, May 2019 127 Mechanics & Mechanical engineering [10] G.V Logvinovich, “Hydrodynamics of free boundary flows, IPST, Jerusalem (1972) [11] Seong Sik Ahn, Massimo Ruzzene, “Optimal design of supercavitating vehicles based on trimmed flight performance), In: 8th Biennial ASME Conference on Engineering Systems Design and Analysis, July 4-7, 2006, Torino, Italy TÓM TẮT NGHIÊN CỨU MƠ HÌNH CHUYỂN ĐỘNG KHƠNG GIAN CỦA ĐẠN BẮN DƯỚI NƯỚC Bài báo nghiên cứu mơ hình chuyển động không gian đạn bắn nước với hiệu ứng siêu khoang Hệ phương trình chuyển động với bậc tự xây dựng thông qua xác định lực mô men tương tác lên đạn chuyển động môi trường nước Lực tương tác đuôi đạn với thành khoang xác định tính tốn hình dạng - kích thước khoang có xem xét đến ảnh hưởng thu nhỏ kích thước Kết tính tốn thu chuyển động tổng hợp đầu đạn môi trường nước, làm sở để xây dựng mơ hình tản mát điểm chạm đạn bắn nước Từ khóa: Đạn nước; Siêu khoang; Đạn siêu khoang; Lực trượt; Mô hình khoang Received 4th March 2019 Revised 8th May 2019 Accepted 15th May 2019 Author affiliations: Military Technical Academy *Corresponding author: nguyenhuuthang260806@gmail.com 128 N H Thang, N H Minh, D V Doan, “Study on the spatial … underwater projectile.” ... the center MODELING OF SUPERCAVITATING PROJECTILE 2.1 The equations of 3D motion of supercavitating projectile According to the general order, the mathematical model describes the motion of the. .. and the motion around the center during the stable motion phase, we obtain the results as given: Under the gravity effect, the lift force at the nose of the projectile and the condition of the. .. not mention the horizontal plane motion caused by the original disturbance factors In this paper, we establish the spatial motion model (3D) of an inertial motion projectile considering the gravity