Longitudinal movement with Preset Pitch Angle#

block — Block appearance in REPEAT VISION#

Consider the longitudinal controlled motion in the starting coordinate system, which is denoted 0xyz to reduce indexing. Suppose that the roll and yaw control ensures the flight of the missile in a vertical plane passing through the launch site and target. It can be assumed that the forces applied to the rocket operate in the same plane - in the firing plane, and the flight path is a flat curve (see Fig. 1 below).

longitudinalMovementP — Figure 1. Diagram of force action on projectile aircraft moving during starting engine operation#

Components of forces acting on the projectile aircraft along axes τ and n

Force	Tangential component (projection on the τ axis)	Normal component (projection on the n axis)
Thrust force	P\(\cdot\) cos(\(\alpha - \xi \))	-P\(\cdot\) sin(\(\alpha - \xi \))
Drag	-X	-
Lifting force	-	-Y
Gravity	-m\(\cdot g \cdot sin \theta \)	m\(\cdot g \cdot cos \theta \)
Loss of steering wheel traction	-X\(_{p1} \cdot cos(ϑ - \theta)\)	X\(_{p1} \cdot sin(ϑ - \theta)\)
Steering force	-Y\(_{p1} \cdot cos(ϑ - \theta)\)	Y\(_{p1} \cdot sin(ϑ - \theta)\)

Let us write the equations of motion in projections on the tangent and the normal (1):

(1)#\[\begin{split} \begin{cases} m \cdot \dot{\nu}=P \cdot cos (\alpha - \xi ) -X - m\cdot g \cdot sin\Theta - X_{p_1}\cdot cos(\theta -\Theta) - Y_{p_1}\cdot sin(\theta -\Theta)\\-m \cdot \nu \cdot \dot{\Theta}=-P \cdot sin (\alpha - \xi ) -Y + m\cdot g \cdot cos\Theta - X_{p_1}\cdot sin(\theta -\Theta) - Y_{p_1}\cdot cos(\theta -\Theta)\end{cases} \end{split}\]

The rotational equation will be (2):

(2)#\[J_z \cdot \dot{\theta}=M_z+M_{pz}\]

where \(J_z\) is the equatorial moment of inertia of the projectile aircraft relative to 0Z1 axis, which in this case coincides with Oz axis.

It is necessary to add two obvious kinematic equations to the three equations written(3) and (4):

(3)#\[\dot{x}=\nu \cdot cos \Theta\]

(4)#\[\dot{y}=\nu \cdot sin \Theta\]

Basic diagram of aircraft pitch control (refer to Fig. 2 below):

During operation, the program block generates an electrical signal corresponding to the program pitch angle value at a given time ϑpr(t). At the same time, the sensing element measures the actual pitch angle ϑ and transmits the corresponding signal to the comparing unit, which, after comparing the signals proportional to the program and actual pitch angles, outputs a mismatch signal. Amplified and converted in the control signal generating unit, it is transmitted to a steering drive which rotates the controls accordingly. In this case, the direction of movement of the rocket will change so as to parry the resulting deviation from the program trajectory. Pitch mismatch angle can be approximately related to angle of rotation of control elements by means of first summand of control equation (5):

(5)#\[\delta_z=k_{0ϑ}\cdot (ϑ-ϑ_{np})\]

where \(k_{0ϑ}\) - static coefficient of the control system.

Then (6)and (7):

(6)#\[M_z+M_{pz}=0\]

(7)#\[\frac{\rho \cdot \nu^2}{2}\cdot S \cdot lm^{\alpha}_z \cdot \alpha_{\delta} + Y^{\delta}_{p_1} \cdot \delta_z \cdot l_p=0\]

Hence (8), (9), (10) and (11):

(8)#\[\alpha_{\delta}=- \frac{2\cdot Y^{\delta}_{p1}\cdot l_p \cdot \delta_z}{\rho \cdot \nu^2 \cdot S \cdot lm^{\alpha}_z}=-\epsilon_ϑ \cdot \delta_z\]

(9)#\[\alpha_{\delta}=-\epsilon_ϑ \cdot k_{0ϑ}\cdot (ϑ-ϑ_{np})\]

(10)#\[ϑ=\theta + \alpha_{\delta}\]

(11)#\[\alpha_{\delta}=\frac{\epsilon_ϑ \cdot k_{0ϑ}\cdot(ϑ_{np}- \theta)}{1+\epsilon_ϑ \cdot k_{0ϑ}}\]

For small angle of attack take (12):

(12)#\[sin\alpha=\alpha_{\delta}; cos\alpha=1; Y=Y^{\alpha}\cdot \alpha_{\delta}\]

As a result, the system of equations describing the longitudinal controlled flight of the AD is as follows (13), (14)и (15):

(13)#\[\dot{\nu}=\frac{P-X}{m}-g \cdot sin \theta\]

(14)#\[\dot{\theta}=\frac{\epsilon_ϑ \cdot k_{0ϑ}\cdot(P+Y^{\alpha})\cdot (ϑ_{np}- \theta)}{m \cdot \nu \cdot (1+\epsilon_ϑ \cdot k_{0ϑ})}- \frac{g \cdot cos \theta}{\nu}\]

(15)#\[ϑ=\theta + \alpha_{\delta}=ϑ_{np}(t); \dot{y}=\nu \cdot sin \theta; \dot{x}=\nu \cdot cos \theta\]

1. Properties#

Property	Default Value	Description	Units of measurement
Distance from the center of pressure of the controls to the center of mass of the AD	0.45	-	m
AD surface area required to determine the moment Mz along Z axis	0	-	m\(^2\)
Initial speed of AD	55	-	m/s
Initial value of the trajectory inclination angle	0	-	rad
X0	0	-	-
Y0	0	-	-

2. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
P	Input	Traction force vector in the related coordinate system	N	Automation
X	Input	Frontal resistance of forces in the half-speed coordinate system	N	Automation
Y_alpha (Y\(_\alpha\))	Input	Lifting force in half-speed coordinate system	N	Automation
Mz_alpha (M\(_{z \alpha}\))	Input	Moment projection on the Z1 axis in the related coordinate system	N\(\cdot\)m	Automation
Y\(_{p1}\)	Input	Sum of forces generated by controls in a related coordinate system	N	Automation
m	Input	Mass	kg	Automation
Q	Input	Velocity head	N/\(m^2\)	Automation
Tang\(_{prog}\)	Input	Program angle value	rad	Automation
K\(_{0tan}\)	Input	Control system factor	-	Automation
\(v\)	Output	AD speed module	m/s	Automation
theta (\(\theta\))	Output	Flight-path angle	rad	Automation
X\(_z\)	Output	Coordinates of longitudinal motion of the AD along the X axis	m	Automation
Y\(_z\)	Output	Coordinates of longitudinal motion of the AD along the Y axis	m	Automation
alpha (\(\alpha\))	Output	Attack angle	rad	Automation

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