Longitudinal Movement

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.

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

(1)\[\begin{split} \begin{cases} \dot{\nu} =\frac{1}{m}\cdot ((P_{x_1}-\sum X_{p_1})\cdot cos\alpha - (P_{y_1}+\sum Y_{p_1})\cdot sin\alpha - X - Q \cdot sin\theta) \\\dot{\theta} =\frac{1}{m\cdot \nu}\cdot ((P_{x_1}-\sum X_{p_1})\cdot sin\alpha + (P_{y_1}+\sum Y_{p_1})\cdot cos\alpha + Y^* - Q \cdot cos\theta) \\J_{z_1} \cdot \dot{\omega_{z_1}}=\sum M_{z_1}+\sum M_{pz_1} \\ \dot{\upsilon}=\omega_{z_1} \\ sin \upsilon=sin\theta \cdot cos\alpha+cos\theta \cdot sin\alpha=sin(\theta + \alpha) \\ \dot{x_3}=\nu \cdot cos \theta \\ \dot{y_3}=\nu \cdot sin \theta \\ r=\sqrt{x^2_3+y^2_3} \\ m=m_0-\int^t_0|\dot{m}|dt\end{cases} \end{split}\]

 In some cases it is more convenient to replace the trigonometric equation in the system linking the pitch, roll and attack angles with equation (2):

(2)\[\upsilon=\theta+ \alpha\]

 The resulting system is closed and can be solved.

1.  Properties

Property	Default Value	Description	Units of measurement
Initial speed of AD	150	-	m/s
Initial value of the trajectory inclination angle	0	-	rad
Pitch angle initial value	0.17452006980802794	-	rad
Initial value of angular speeds of AD rotation	0	-	rad/s
Initial X coordinates	0	-	m
Initial Y coordinates	3000	-	m

2.  Inputs and outputs

Parameters	Input/Output	Description	Units of measurement	Port type
p*	Input	Thrust vector in the half-speed coordinate system	N	Automation
R*	Input	Vector of aerodynamic forces in half-speed coordinate system	N	Automation
MR\(_{z1}\)	Input	Vector of aerodynamic moments in a related coordinate system	N\(\cdot\)m	Automation
J\(_z\)	Input	AD inertia moments	kg\(\cdot m^2\)	Automation
m	Input	AD mass	kg	Automation
\(v\)	Output	AD speed module	m/s	Automation
Theta (\(\theta\))	Output	Flight-path angle	rad	Automation
X\(_z\)	Output	Coordinates on X axis of earth coordinate system	m	Automation
Y\(_z\)	Output	Coordinates on Y axis of earth coordinate system	m	Automation
W\(_{z1}\)	Output	Angular speed of FD rotation relative to related axes	rad/s	Automation
Tang (\(\upsilon\))	Output	Pitch angle	rad	Automation
Alpha (\(\alpha\))	Output	Attack angle	rad	Automation

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