Full Movement of AD

Block appearance in REPEAT VISION

 The movement of missiles will be considered in 0XeYeZe coordinate system rigidly connected to the Earth. Moreover, to simplify the task, the influence of the Earth will be taken into account by the introduction of a constant acceleration in magnitude and direction based on gravity, neglecting Coriolis acceleration; the curvature of the Earth will also not be taken into account. When writing the right-hand sides of the equations, the components of the thrust force, gravity, aerodynamic and control forces will be taken into account.

 A system of equations describing the motion of the rocket center of mass in the half-speed coordinate system 0XYZ (1):

(1)\[\begin{split} \begin{cases} \dot{\nu} =\frac{1}{m}\cdot (P_{x_1}-\sum X_{p}-X+Q_x) \\ \nu \cdot\dot{\theta} =\frac{1}{m}\cdot (P_{y^*}-\sum Y^*_{p}+Y^*+Q_y^*) \\ \nu \cdot\dot{\Psi}\cdot cos\theta =-\frac{1}{m}\cdot (P_{z^*}+\sum Z^*_{p}+Z^*) \end{cases} \end{split}\]

where:

 \(P_x, P_{y^*}, P_{z^*}\) are projections of thrust force on half-speed coordinate axes;

 \(Q_x, Q_{y^*}\) are projections of gravity on half-speed coordinate axes;

 \(X_p, Y_{p^*}, Z_{p^*}\) is a projection of aerodynamic control forces on half-speed coordinate axes;

 \(X, Y^*, Z^*\) is a projection of aerodynamic forces on half-speed coordinate axes.

 The equations of rotational motion of rockets and aircraft are usually written in projections on the associated coordinate axes (2), (3) и (4):

(2)\[J_{x_1}\cdot \dot\omega_{x_1}+(J_{z_1}-J_{y_1})\cdot \omega_{y_1} \cdot \omega_{z_1}=\sum M_{x_1}+\sum M_{px_1}\]

(3)\[J_{y_1}\cdot \dot\omega_{y_1}+(J_{x_1}-J_{z_1})\cdot \omega_{x_1} \cdot \omega_{z_1}=\sum M_{y_1}+\sum M_{py_1}\]

(4)\[J_{z_1}\cdot \dot\omega_{z_1}+(J_{y_1}-J_{x_1})\cdot \omega_{x_1} \cdot \omega_{y_1}=\sum M_{z_1}+\sum M_{pz_1}\]

where:

 \(\sum M_{x_1}, \sum M_{y_1}, \sum M_{z_1}\) is the sum of the projections of the moments of external forces and thrust force to the associated coordinate axes (excluding control forces);

 \(\sum M_{px_1}, \sum M_{py_1}, \sum M_{pz_1}\) are the sums of the projections of control forces moments on the associated coordinate axes;

 \(J_{x_1}, J_{y_1}, J_{z_1}\) are the moments of inertia along the main axes of AD.

  To establish links between velocity derivative and angular rate \(w_{x_1}, w_{y_1}, w_{z_1}\), use (5):

(5)\[\begin{split} \begin{cases} \omega_{x_1} =\dot{\psi}\cdot sin\upsilon + \dot{\gamma} \\ \omega_{y_1} =\psi\cdot cos\upsilon \cdot sin\gamma + \upsilon \cdot sin \gamma\\ \omega_{z_1} =-\dot{\psi}\cdot cos\upsilon \cdot sin \gamma + \upsilon \cdot cos\gamma \end{cases} \end{split}\]

  When determining the values of aerodynamic forces in the process of solving the spatial problem of missile motion, it is necessary to know the values of α, β and \(γ_s\) angles.

  By defining the directional cosines of the sequential transition from the related axes to the wind axes, from the wind axes to the half-speed axes and from the half-speed axes to the earth’s axes and equating them to the directional cosines of the direct transition from the related axes to the earth’s axes, the following relations are obtained between (6), (7) and (8) angles:

(6)\[sin\Theta=cos\alpha \cdot cos\beta \cdot sin\upsilon-(sin\alpha \cdot cos\beta \cdot cos\gamma+sin\beta\cdot sin\gamma)\cdot cos\upsilon\]

(7)\[sin\Psi\cdot cos\Theta=cos\alpha \cdot cos\beta \cdot sin\psi \cdot cos\upsilon-sin\alpha \cdot cos\beta \cdot (cos\psi \cdot sin\gamma +sin\psi \cdot sin\upsilon \cdot cos\gamma) - sin\beta \cdot(cos\psi \cdot cos\gamma -sin\psi \cdot sin\upsilon \cdot sin\gamma)\]

(8)\[cos\upsilon \cdot sin \gamma=sin\gamma_c \cdot cos\beta \cdot cos\theta-sin\beta \cdot sin\theta\]

 Hence (9):

(9)\[\gamma_c=arcsin \cdot \frac{cos\upsilon\cdot sin\gamma +sin\beta \cdot sin\theta}{cos\beta \cdot cos\theta}\]

  If expressions for projections of the velocity vector of the rocket center of mass on the axis of the Earth’s coordinate system are used, (10), (11) and (12) are obtained:

(10)\[\frac{dx_з}{dt}=\nu \cdot cos\theta \cdot cos\Psi\]

(11)\[\frac{dy_з}{dt}=\nu \cdot sin\theta \]

(12)\[\frac{dz_з}{dt}=-\nu \cdot cos\theta \cdot sin\Psi\]

1.  Properties

Property	Default Value	Description	Units of measurement
Initial angular speed relative to the connected axis X1 \(\omega_{X0}\)	0	-	rad/s
Initial angular speed relative to the connected axis Y1 \(\omega_{Y0}\)	0	-	rad/s
Initial angular speed relative to the connected axis Z1 \(\omega_{Z0}\)	0	-	rad/s
Initial speed \(v_0\)	200	-	m/s
Initial angle of trajectory inclination \( heta_0\)	0	-	rad
Initial angle of trajectory rotation \(\Psi_0\)	0	-	rad
Initial yaw angle \(\psi_0\)	0	-	rad
Initial pitch angle tang0	0	-	rad
Initial roll angle \(\gamma_0\)	0	-	rad
Initial coordinate X\(_0\)	0	-	m
Initial coordinate Y\(_0\)	3000	-	m
Initial coordinate Z\(_0\)	0	-	m

2.  Inputs and outputs

Parameters	Input/Output	Description	Units of measurement	Port type
{Px,Py,Pz}	Input	Engine thrust vector in the half-speed coordinate system	N	Automation
{Rx,Ry,Rz}	Input	Vector of aerodynamic forces in half-speed coordinate system	N	Automation
g	Input	-	m/s\(^2\)	Automation
mass	Input	-	kg	Automation
{Ix,Iy,Iz}	Input	AD inertia moments	kg\(\cdot m^2\)	Automation
{Mx,My,Mz}	Input	Vector of aerodynamic moments in a related coordinate system	N\(\cdot\)m	Automation
Velocity	Output	AD speed module	m/s	Automation
theta (\(\theta\))	Output	Flight-path angle	rad	Automation
omegaX (\(\omega_X\))	Output	Angular velocity in the X axis	rad/s	Automation
omegaY (\(\omega_Y\))	Output	Angular velocity in the Y axis	rad/s	Automation
omegaZ (\(\omega_Z\))	Output	Angular velocity in the Z axis	rad/s	Automation
PSI (\(\Psi\))	Output	Trajectory rotation angle	rad	Automation
psi (\(\psi\))	Output	Yaw angle	rad	Automation
tang (\(\upsilon\))	Output	Pitch angle	rad	Automation
gamma (\(\gamma\))	Output	Roll angle	rad	Automation
alpha (\(\alpha\))	Output	Attack angle	rad	Automation
beta (\(\beta\))	Output	Gliding angle	rad	Automation
gammaC (\(\gamma_C\))	Output	Roll angle (angle between planes 0X1Y1 and 0XY*)	rad	Automation
X	Output	Coordinate on X axis of earth coordinate system	m	Automation
Y	Output	Coordinate on Y axis of earth coordinate system	m	Automation
Z	Output	Coordinate on Z axis of earth coordinate system	m	Automation

In order to see the connection capabilities of the block, you need to go to page Description of block ports