Simplified lateral movement#

block — Block appearance in REPEAT VISION#

When dividing complex movement of missiles into longitudinal and lateral, it is assumed that in the first approximation the longitudinal movement does not depend on lateral one. Thus, only those forces and moments that operate in the firing plane are taken into account in the equations for determining the characteristics of longitudinal movement. When determining the characteristics of the lateral movement of a missile, it cannot be considered independent of the longitudinal one, therefore, all forces and moments, which in some degree can cause the missile to deviate from the firing plane, are included in the equations.

System of equations describing lateral motion of rocket center of mass in half-speed coordinate system 0XYZ (1), (2), (3), (4), (5) and (6):

(1)#\[\dot{\Psi}=-\frac{1}{m\cdot \nu \cdot cos\theta}\cdot(-(P_{x_1}-\sum X_{p_1})\cdot cos\gamma_c \cdot sin\beta + (P_{y_1}+\sum Y_{p_1})\cdot sin\gamma_c+(P_{z_1}+\sum Z_{p_1})\cdot cos\beta +Z*)\]

(2)#\[J_{y_1}\cdot \dot{\omega_{y_1}}=\sum M_{y_1}+ \sum M_{py_1}\]

(3)#\[\dot{\psi}=\frac{1}{cos\nu}\cdot \omega_{y_1}\]

(4)#\[sin\psi=sin\Psi \cdot cos\beta \cdot cos\gamma_c + cos\Psi \cdot (sin\beta \cdot cos\theta + sin\gamma_c \cdot sin\theta \cdot cos\beta)\]

(5)#\[sin\gamma_c=tg\beta \cdot tg\theta\]

(6)#\[\dot{z_3}=-\nu \cdot cos\theta \cdot sin\Psi\]

Equations describing only lateral motion cannot be solved independently, without considering the basic values determined by longitudinal motion. For example, in a system, the first equation is basic in determining lateral motion parameters, but it is solved only if m(t), v(t) and 0(t) are known. Aerodynamic forces and laterally acting moments cannot also be determined unless speed and altitude are known.

At large trajectory rotation angles, it is necessary to check the limits of applicability of the formula determining Yc.

A significant simplification of the system can be obtained if an artificial technique is applied and the pitch and roll angles of the trajectory are assumed to be equal to 0 in the equations defining the angular parameters of the rocket motion. This assumption is equivalent to the idea that the rocket flies in a horizontal position, its longitudinal axis makes an angle p with the velocity vector of the center of mass, and the angle \(\gamma_c\)=0.

&Emsp;To obtain results close to the actual results, in the equations defining the conditional horizontal trajectory of the missile’s center of mass, its velocity should be taken equal to \( u \cdot cos heta\), the multipliers of which are determined by solving the system of equations describing the longitudinal motion of the missile’s (projectile’s) center of mass. With the assumptions made, a simpler system of differential equations (7), (8), (9), (10) and (11) follows from the system of differential equations above:

(7)#\[\dot{\Psi}=-\frac{1}{m\cdot \nu \cdot cos\theta}\cdot(-(P_{x_1}-\sum X_{p_1})\cdot sin\beta + (P_{z_1}+\sum Z_{p_1})\cdot cos\beta +Z*)\]

(8)#\[J_{y_1}\cdot \dot{\omega_{y_1}}=\sum M_{y_1}+ \sum M_{py_1}\]

(9)#\[\dot{\psi}=\omega_{y_1}\]

(10)#\[sin\psi=sin\Psi \cdot cos\beta + cos\Psi \cdot sin\beta =sin(\Psi + \beta)\]

(11)#\[\dot{z_3}=-\nu \cdot cos\theta \cdot sin\Psi\]

The penultimate equation can be replaced by the sum of angles (12):

(12)#\[\psi=\Psi + \beta\]

1. Properties#

Property	Default Value	Description	Units of measurement
Initial value of the trajectory rotation angle	-13/57.3	-	rad
Yaw angle initial value	0	-	rad
Initial value of angular velocity of AD rotation	0	-	rad/s
Initial values for lateral displacement	0	-	m

2. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
\(v\)	Input	AD speed module	m/s	Automation
Theta (\(\theta\))	Input	Flight-path angle	rad	Automation
P\(_y\)	Input	AD engine thrust in projection on the half-speed axis 0Z*	N	Automation
Z	Input	Aerodynamic force on axis 0Z*	N	Automation
MR\(_{y}\)	Input	Projection of the moment in a related coordinate system on axis Y	N\(\cdot\)m	Automation
J\(_y\)	Input	AD inertia moments	kg\(\cdot m^2\)	Automation
m	Input	AD mass	kg	Automation
PSI \(\Psi\)	Output	Trajectory rotation angle	rad	Automation
W\(_{y1}\)	Output	Angular speed of AD rotation relative to connected axes	rad/s	Automation
psi \(\psi\)	Output	Yaw angle	rad	Automation
betta \(\beta\)	Output	Gliding angle	rad	Automation
Z\(_{z}\)	Output	Lateral offset	m	Automation

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