Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 1, pp. 69-90, Warsaw 2009 INFLUENCE OF ATMOSPHERIC TURBULENCE ON BOMB RELEASE Grzegorz Kowaleczko Andrzej Żyluk Airforce Institute of Technology, Warsaw, Poland e-mail: g.kowaleczko@chello.pl Amodel of the bombreleasedynamics is presented in thepaper, andhow it is influenced by awind field. The appliedwayof describing this field is discussedwith account taken of its stochastic nature. Exemplary results of a numerical simulation of bombing are submitted and concluded. Key words: atmospheric turbulence, bombing 1. Introduction Any bomb release should be carried out in such a way that the target is hit with the maximum possible degree of accuracy. Release conditions can vary significantly. Different usually are: air speed at bomb release, bomb release altitude and the angle of bomb release. Depending on these parameters, a different point on the Earth’s surface is reached. Wind is another factor that can influence the bomb trajectory. If the wind exhibits a constant speed and direction, then the situation is simple. However, each time the influence of atmospheric turbulence on the bomb release can be different because of its stochastic nature. Therefore, even if initial conditions of bomb release are the same, bomb trajectories are different. The aim of this study is to estimate the influence of the stochastic wind field on flight of a small training bomb.A series of numerical simulations were carried outwith a six-degrees-of-freedommodel ofmotion applied. Themodel describes the bomb gliding in the three-dimensional space. To determine the wind field, Shinozuka’smethod has been applied. Thismethod is usually used to model stochastic processes. 70 G. Kowaleczko, A. Żyluk 2. Mathematical description of bomb motion 2.1. Assumptions for a physical model Toanalyse thebombflightdynamics, the following assumptionsweremade that allow formulation of a mathematical description of motion: 1. The bomb is a rigid body of constantmass, constantmoments of inertia and a constant position of the centre of mass. 2. Thebombhas two symmetryplanes.These are the Oxz and Oxy planes (Fig.1) that areplanesof geometric,mass, andaerodynamic symmetries. 2.2. Coordinates systems To formulate amathematical model of the bomb, the following orthogonal coordinate systems were used: Oxyz – bomb-fixed systemwith its origin at the bombcentre ofmass Oxayaza – air-trajectory reference frame Oxgygzg – Earth-referenced system with its origin at the bomb centre of mass. These systems are related to each other bymeans of the following angles: — Oxyz and Oxgygzg systems: with the angle of yaw Ψ, the angle of pitch Θ and the angle of roll Φ — Oxyz and Oxayaza systems: with the angle of sideslip β and the angle of attack α. Subsequent turns by angles Ψ, Θ and Φ about the coordinate axes can result in finding the matrix of transition from the Oxgygzg system to the Oxyz system    x y z    =Ls/g    xg yg zg    (2.1) where the Ls/g matrix is Ls/g = (2.2) =    cosΨ cosΘ sinΨ cosΘ −sinΘ cosΨ sinΘsinΦ− sinΨ cosΦ sinΨ sinΘsinΦ+cosΨ cosΦ cosΘsinΦ cosΨ sinΘcosΦ+sinΨ sinΦ sinΨ sinΘcosΦ− cosΨ sinΦ cosΘcosΦ    Influence of atmospheric turbulence on bomb release 71 Further turns by angles β and α result in finding thematrix of transition from the Oxayaza system to the Oxyz system    x y z    =Ls/a    xa ya za    (2.3) where the Ls/g matrix has elements Ls/a =    cosαcosβ −cosαsinβ −sinα sinβ cosβ 0 sinαcosβ −sinαsinβ cosα    (2.4) Fig. 1. Coordinate systems with angles of transition 2.3. Equations of motion of the bomb 2.3.1. A general form of the equations of motion Since tunnel measurements of aerodynamic forces are usually taken in the air-trajectory reference frame Oxayaza, equations of equilibrium of forces will be determined in this system. However, equations of equilibrium of moments will be determined in the bomb-fixed coordinate system Oxyz, because the tensor of moments of inertia is independent of time in this system. • The vector equation of motion of the bomb centre of mass takes the following form d(mV ) dt = ∂(mV ) ∂t +Ω× (mV )=F (2.5) 72 G. Kowaleczko, A. Żyluk and can be rewritten as three scalar equations in any rectangular moving system of coordinates m(U̇ +QW −RV )=X m(V̇ +RU−PW)=Y (2.6) m(Ẇ +PV −QU)=Z where m – mass of a bomb V – velocity vector with components V = [U,V,W ]⊤ in the moving system of coordinates Ω – vector of angular velocity of themoving system against the inertial reference frame with components Ω = [P,Q,R]⊤ in the moving system of coordinates F – resultant vector of forces acting on the bomb with components [X,Y,Z]⊤ in the moving system of coordinates. In the air-trajectory reference frame Oxayaza, the velocity vector has only one component Ua = V (which should not be mistaken for the second com- ponent of the vector V according to the designations above). Equations (2.6) take the following form mV̇ =Xa mRaV =Ya −mQaV =Za (2.7) Assuming that both the angular velocity of the bomb-fixed system Oxyz against the inertial reference frame Ωs and the velocity of the system Oxyz against the Oxayaza system are known, the vector of the angular velocity of the Oxayaza system against the inertial reference frame can be determined Ωa =Ωs+Ωs/a =Ωs+ β̇− α̇ (2.8) In the Oxyz frame, the Ωs vector has the following components: Ωs = [P,Q,R] ⊤; in the Oxayaza coordinate system, the β̇ vector has the following components: β̇= [0,0, β̇]⊤; whereas in the Oxyz frame, the α̇ vec- tor has the components: α̇ = [0, α̇,0]⊤. Taking the foregoing into account, and using transition matrix (2.4), on the basis of (2.8), we receive Pa =P cosαcosβ+(Q− α̇)sinβ+Rsinαcosβ Qa =−P cosαsinβ+(Q− α̇)cosβ−Rsinαsinβ (2.9) Ra =−P sinα+Rcosα+ β̇ Influence of atmospheric turbulence on bomb release 73 Having applied (2.9) to equations (2.7), after some transformations, the following system of equations is arrived at V̇ = 1 m Xa β̇= 1 mV Ya+P sinα−Rcosα (2.10) α̇= 1 cosβ [ Za mV +Qcosβ− (P cosα+Rsinα)sinβ ] • The vector equation of moments of force equilibrium has the following form d(K) dt = ∂(K) ∂t +Ω×K =M (2.11) where M is the resultant moment of forces acting on the bomb with compo- nents M = [L,M,N]⊤ in the moving system of coordinates. The vector of the bomb angular momentum is K = IΩ (2.12) where the inertia tensor (moments and products of inertia) I is determined as I=    Ix −Ixy −Ixz −Iyx Iy −Iyz −Izx −Izx Iz    (2.13) As said above, equations (2.11) will be written down in the bomb-fixed system Oxyz. The stability ofmass characteristics of the bomb in this system makes all derivatives of components of the moment-of-inertia tensor against time equal to zero. It means that ∂K ∂t = ∂(IΩs) ∂t = ∂I ∂t Ωs+ I ∂Ωs ∂t = I ∂Ωs ∂t (2.14) What is arrived at after some transformations, on the basis of (2.11) and with (2.14) applied, is a system of three scalar equations that describe angular motion of the bomb in the moving bomb-fixed system of coordinates Oxyz. It takes the following form IxṖ − Iyz(Q2−R2)− Izx(Ṙ+PQ)− Ixy(Q̇−RP)− (Iy − Iz)QR=L IyQ̇− Izx(R2−P2)− Ixy(Ṗ +QR)− Iyz(Ṙ−PQ)− (Iz − Ix)RP =M (2.15) IzṘ− Ixy(P2−Q2)− Iyz(Q̇+RP)− Izx(Ṗ −QR)− (Ix− Iy)PQ=N 74 G. Kowaleczko, A. Żyluk Since the Oxz and Oxy planes are the bomb planes of symmetry, the following dependences occur Ixy = Iyx = Izy = Iyz =0 (2.16) Therefore, system of equations (2.15) is reduced to IxṖ − (Iy − Iz)QR=L IyQ̇− (Iz − Ix)RP =M (2.17) IzṘ− (Ix− Iy)PQ=N Finally, after some elementary transformations, system (2.17) takes the form Ṗ = 1 Ix [L+(Iy − Iz)QR] Q̇= 1 Iy [M+(Iz − Ix)RP ] (2.18) Ṙ= 1 IxIz [L+(Iy − Iz)QR] Complementary to systems (2.10) and (2.18) are kinematic relations that permit determination of the rates of changes of the angles Ψ,Θ and Φ, if the angular velocities P,Q,R are known Φ̇=P +(RcosΦ+QsinΦ)tanΘ Θ̇=QcosΦ−RsinΦ (2.19) Ψ̇ = 1 cosΘ (RcosΦ+QsinΦ) Furthermore,with relationships (2.1) and (2.3) applied, the velocity vector of the bomb centre of mass in the Oxgygzg system can be determined    Ug Vg Wg    =    ẋg ẏg żg    =L−1 s/g Ls/a    V 0 0    (2.20) Particular components are as follows: ẋg =V [cosαcosβ cosΘcosΨ+sinβ(sinΦsinΘcosΨ−cosΦsinΨ)+ +sinαcosβ(cosΦsinΘcosΨ+sinΦsinΨ)] ẏg =V [cosαcosβcosΘsinΨ+sinβ(sinΦsinΘsinΨ+cosΦcosΨ)+ +sinαcosβ(cosΦsinΘsinΨ+sinΦcosΨ)] (2.21) żg =V [−cosαcosβ sinΘ+sinβ sinΦcosΘ+sinαcosβ cosΦcosΘ] Influence of atmospheric turbulence on bomb release 75 Equations (2.10), (2.18), (2.19), and(2.21) composea systemof12ordinary differential equations that describe spatial motion of the bomb treated as a rigid body. It can be written down in the following form dX dt =F(t,X,S) (2.22) where X – twelve-element vector of the bomb flight parameters, X = [V,α,β,P,Q,R,Φ,Θ,Ψ,xg,yg,zg] ⊤ V – velocity of the bomb (absolute value of the bomb velocity vector) α – angle of attack β – angle of sideslip P,Q,R – angular velocities of rolling, pitching and yawing in the Oxyz system of coordinates Θ,Φ,Ψ – bomb angles of pitch, roll and yaw. 2.3.2. General expressions to describe forces and moments acting on the bomb Forces acting on the bomb The right-hand side of equation (2.5) represents forces that act on the bomb F =Q+R (2.23) According to designations in equations (2.7), there are the following compo- nents Xa =Qxa +Rxa Ya =Qya +Rya (2.24) Za =Qza +Rza Particular components in expression (2.24) are determined below. They are as follows: • the bombweight Q that has only one component Q= [0,0,mg]⊤ in the Oxgygzg system. Using dependences (2.1) and (2.3), one can calculate components of the vector Q in the Oxayaza system    Qxa Qya Qza    =L−1 s/a Ls/g    0 0 mg    (2.25) 76 G. Kowaleczko, A. Żyluk We get Qxa =mg(−cosαcosβ sinΘ+sinβcosΘsinΦ+sinαcosβcosΘcosΦ) Qya =mg(cosαsinβ sinΘ+cosβcosΘsinΦ− sinαsinβcosΘcosΦ) Qza =mg(sinαsinΘ+cosαcosΘcosΦ) (2.26) • the aerodynamic force R that has the following components in the Oxayaza system Rxa =−Pxa =−Cxa ρV 2 2 S Rya =Pya =−Cya ρV 2 2 S (2.27) Rza =−Pza =−Cza ρV 2 2 S where Cxa,Cya,Cza – coefficients of aerodynamicdrag, side and lift forces S – cross-sectional area of the bomb ρ – air density. Moments of forces acting on the bomb The right-hand side of system of equations (2.17) contains the vector M = [L,M,N]⊤ that is a resultant vector of moments of forces acting on the bomb. Since equations (2.18) are determined within the system of the bomb principal axes of inertia with its origin at the bomb centre of mass, the aerodynamic moments are the only moments acting on the bomb. Therefore, particular components are as follows L=Cl ρV 2 2 Sd M =Cm ρV 2 2 Sd N =Cn ρV 2 2 Sd (2.28) where Cl,Cm,Cn – coefficients of rolling, pitching and yawing moments, respectively d – diameter of the bomb. 2.4. Aerodynamic forces and moments acting on the bomb Aerodynamic forces and moments acting on the bomb, described with expressions (2.27) and (2.28), are determined according to their aerodyna- mic coefficients. These coefficients depend on many factors such as the bomb Influence of atmospheric turbulence on bomb release 77 shape, angle of attack, angle of sideslip,Mach number, Reynolds number, and angular velocities. There are no general methods for finding these characteri- stics for any spatial position of the bomb.Therefore, variousmethods are used depending on the problem discussed, availability of source data on the bomb, and research apparatus. Themostwidely usedmethod iswind-tunnel testing.Both real objects and models thereof can be tested. While conducting the model testing, account should be taken of the so-called similarity criteria. These should be met to provide highly reliable results. Another way of determining the aerodynamic coefficients is to applymethods of numerical fluidmechanics. Thesemethods, based on equations that describe continuous-media flows, allow for numerical calculations of basic aerodynamic characteristics of different kinds of objects. There are alsomethods of the so-called identification: characteristics are found in effect of several flight tests. Various parameters are measured during these tests to enable identification of the bomb aerodynamic characteristics. In the case of both aerodynamic forces and moments, it is assumed that the total aerodynamic coefficient is the sumof the static component and com- ponents effected by the bomb non-zero angular velocities. The superposition principle understood in this way can be applied in the following general form Ca =Castatic(α,β)+C p aP+C q aQ+C r aR+C pq a PQ+C pr a PR+C qr a QR (2.29) For individual aerodynamic coefficients, some components above are equal to zero or are so small that can be omitted. A detailed description of the way of determining all aerodynamic coeffi- cients can be found in Lebedew (1973) andOstalawskij (1957). Static aerody- namic characteristics obtained from the tunnel testing have been used in these calculations. They are shown in Figs.2, 3, and 4. Fig. 2. Aerodynamic drag coefficient Cxa(α) 78 G. Kowaleczko, A. Żyluk Fig. 3. Aerodynamic lift and side force coefficients Cza(α), (Cya(β)) Fig. 4. Coefficient of aerodynamic pitching (yawing)moment Cm(α), (Cn(β)) Dynamic derivatives have been found using dependences taken fromOsta- lawskij (1957). The most important are derivatives that determine damping moments. They are as follows Cqm =−C αH zaH SHL 2 H Sd (2.30) where C αH zaH – derivative of aerodynamic lift coefficient as related to the angle of attack, C αH zaH = ∂CzaH/∂αH SH – bomb ’tail-plane’ area LH – distance between control surfaces of the bomband its centre of mass. Because of the symmetry, these two are equal Crn =C q m (2.31) Influence of atmospheric turbulence on bomb release 79 While determining the aerodynamic drag coefficient, account was taken of the effect of compressibility of the air on the change in value of this coefficient at the zero angle of attack. It was assumed that this coefficient remains constant up to the Mach number reaching its critical value. Above this critical value, the aerodynamic drag coefficient increases to reach themaximumvalue for the Mach number 1.1. The calculations resulted in the following Cxa0 = { 0.11 for Ma