Type 91 Torpedo - Theory: Aerial Torpedo Motion Equation

Theory: Aerial Torpedo Motion Equation

Rear Admiral Shoji Naruse explained in his class as follows.

The torpedo motion equation is the set of simultaneous ordinary differential equations, which is to model pitch motion of airborne aerial torpedo as follows.

Eq.1: Falling velocity of the torpedo equation

Eq.2: Horizontal vector velocity of the torpedo mass equation

Eq.3: Vertical vector velocity of the torpedo equation

Eq.4: Vertical vector acceleration of the torpedo mass equation

Eq.5: Angular velocity equation with respect to time

Eq.6: Angular velocity differential equation with respect to time

$\begin{array}{lcll} dx/dt &= &V_{X} &\cdots(Eq.1)\\ W/g \times \left(dV_{X}/dt \right) &= &- D \cos \varphi - L \sin \varphi &\cdots(Eq.2)\\ dz/dt &= &V_{Z} &\ldots (Eq.3)\\ W/g \times \left(dV_{Z}/dt\right) &= &D \sin \varphi - L \cos \varphi + W &\cdots(Eq.4)\\ d\theta/dt &= &\omega &\cdots(Eq.5)\\ I \times \left(d\omega/dt\right) &= &57.3M - bV\omega &\cdots(Eq.6) \end{array}$

where constant 57.3 of Eq.6 is the coefficient from 1 (radian) = 57.2958°

b V ω of Eq.6 is the damping moment, where b is defined as;

Since the angular moment of a torpedo here is the lifting movement as follows;

V : The velocity of the torpedo

V_H : The horizontal axis velocity of the torpedo

V_Z : The vertical axis velocity of the torpedo

φ : The moving vector angle of the torpedo in reference to horizontal axis

θ : The posture angle of the torpedo in reference to the horizontal axis

α : The internal angle between φ and θ, which is equal to the lift angle of tail fins of the torpedo

W : Weight of the torpedo

g : Acceleration of gravity, 9.8 m/sec2

I : Inertia coefficient with respect to lift moment at the gravity center of the torpedo

ω : Lifting angular velocity (radian)

D : Drag moment of force

L : Lift moment of force

M : Roll moment of force around the longitudial axis of the torpedo

ρ : Air density

S : Cross-section area of the torpedo

l : Total length of the torpedo

l_H : The length between the gravity center and the center of lift moment of tail fins of the torpedo

The value l_H is measured in wind-tunnel test, as drag moment of force coefficient differences between the torpedoes with and without box type attachment of wooden tail stabilizer plates;

where each coefficient is defined as;

C_nH = C_XH sin α + C_ZH cos α

C_tH = C_XH cos α + C_ZH sin α

C_X : Drag moment of force coefficient D / (1/2 ρ V2 S)

C_Z : Lift moment of force coefficient L / (1/2 ρ V2 S)

C_mg : Roll moment of force coefficient around the gravity center of the torpedo M / (1/2 ρ V2 l)

C_XH : Drag moment of force coefficient of tail stabilizer box of torpedo

C_ZH : Lift moment of force coefficient of tail stabilizer box of torpedo

C_mgH : Roll moment of force coefficient around the gravity center of tail stabilizer box of the torpedo

Solving Eq.1 through Eq.4 with respect to movement under certain boundary conditions, we could derive the set of equations t, X, Z in definite integral equation form.

where, λ = - tan φ, λ₀ = - tan φ₀, at time t = 0

Definite integrals can be numerically solved by Composite Simpson's rule in the ordinary differential equations field.

Lifting motion Eq.5 can be numerically solved by The common fourth-order Runge-Kutta method to get values of ω.

Lifting stability of Torpedo Eq.6 can be numerically solved by Exponential moving average method for exponential equations.

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