Mission YES2: dynamics of movement in an atmosphere

Zabolotnov Yu., Lyubimov V., Prokofiev A.

 

Systems of coordinates


Where - centre of mass of capsule;

- speed of the centre of mass of capsule;

- classical corners Euler L.;

- spatial corner of attack;

- the main connected system of coordinates;

- system of coordinates connected to a vertical plane , taking place through a vectors of gravitational acceleration and speed ;

differs from system of coordinates turn around of a vector of speed on a corner of a roll ;

differs from system of coordinates turn around of an axis on a corner of attack .

The equations of movement of capsule in inertial system of coordinates

 

The equations of movement centre of mass

, . (1)

The equations of rotary movement of capsule

,

, (2)

,

, , , (3)

Where - mass of capsule;

- aerodynamic force; - gravitational force;

, , - axial moments of inertia of capsule;

, , - components of angular speeds;

, , - components of the aerodynamic moment;

, , - individual vectors of the main connected system of coordinates .

The equations of movement (1), (3) are projected on an axis of inertial system of coordinates.

Inertial system of coordinates : - the geometrical centre of the Earth; - plane of equator, the axis is directed on north; the axis is directed to a point of a spring equinox.

 


The accepted assumptions in model

 

1. The gravitational acceleration corresponds to factor of compression of the Earth , radius of equator , - distance from the centre of the Earth up to its surface, , , - coordinates of the centre of mass in inertial system.

2.   The standard atmosphere NASA.

3.  The gravitational moment is not taken into account.

4. The atmosphere rotates together with the Earth with angular speed .

 

Account of aerodynamic forces and moments for symmetric capsule

 

The aerodynamic forces and moments are set in system of coordinates .

Calculation of aerodynamic forces

, , , (4)

, ,

where - factors of aerodynamic force in the main connected system of coordinates , - high-speed pressure, - density of an atmosphere, - characteristic area.

The factors are set as function of a corner of attack and Mach number : .

Calculation of the aerodynamic moments

, , ,

, , (5)

,

where - factors of aerodynamic moment in the system of coordinates ,

- characteristic size,

- factors of aerodynamic moment concerning the centre of mass of capsule and the nose of capsule,

- coordinate determining situation of the centre of mass rather nose of capsule.

The factor are set as function of a corner of attack and Mach number : .

The situation of a point of action of aerodynamic force rather nose is defined by the formula

. (6)

For spherical capsule

, ,

, (7)

where - coordinate determining situation of the centre of sphere,

- diameter of sphere,

- factor of aerodynamic force of sphere.

 

 

 

 



The approached calculation of factors of forces and moments for capsule YES2 by a method of Newton

 

 

The method of Newton is applicable for numbers more than five.

The form of capsule is represented as set of two forms: a segment and truncated cone. And these forms are interfaced smoothly.

For a spherical segment the factors of forces are calculated under the following formulas

At

, (8)

,

where - corner at top of a cone.

At

(9),

where , , .

The similar formulas for the truncated cone look like.

At

, (10)

.

At

(11) ,

The factors of aerodynamic forces for a cone with spherical nose turn out through factors of forces of a segment and truncated cone as follows

, , (12)

where , - radius spherical nose, - radius of a ground part of capsule.

Factor of the restoring aerodynamic moment rather nose of capsule is calculated under the formula

, (13)

where , - length of capsule, - length of the truncated cone, - size determining a situation of the centre of reduction of aerodynamic forces for a truncated cone; .

 


Static stability of movement of capsule

 

Condition of static stability of movement

at . (14)

 

Dependence for capsule YES2


Fig. 3

Parameters of capsule: , , , .

 

Dynamic stability of movement of capsule

 

Condition of dynamic stability of movement

, (15)

where - amplitude of fluctuations of a angle of attack.

 

The approached differential equation for

, (16)

where , (17)

, , ,

- factor of lift force of capsule,

- factor of viscous friction in a plane of a spatial corner of attack.

As the differential equation (17) has the decision

 

. (18)

Condition of dynamic stability of movement

. (19)

On the top site of re-entry (height of flight H=70 -100 km)

 

. (20)

The analytical decision for H=70 -100 km

 

, (21)

where and - initial meanings of amplitude and frequency of fluctuations (H=100 km),

- frequency of flat fluctuations of capsule, .

 

 

 

 

 

Influence of lift force on dynamic stability at H<70 km

 

Dependence for capsule YES2


Fig. 4

 

Dependence for capsule YES2


Fig. 5

 


Change of parameters of a trajectory at re-entry capsule YES2

 

Dependence of height of flight (km) on time (s)

 


Fig. 6

 

Dependence of a high-speed pressure () on time (s)


Fig. 7

 

 

 

Dependence of speed () on time (s)

 

Speed of a landing:

 

 


Fig. 8

 

Dependence of a angle of an inclination of a trajectory (degr) on time (s)

 


Fig. 9

 

 

 

Dependence of parameter () on time (s)

 


Fig. 10

 

Dependence of a angle of attack (degr) on time (s) at

 


 

Fig. 11

 

 

Dependence of a angle of attack (degr) on time (s) at

 


Fig. 12

Dependence of a thermal flow () on time (s)


Fig. 13

Action of aerodynamic forces at dynamic stability of capsule YES2

 

 

Fig. 14

 

Action of aerodynamic forces at dynamic instability of capsule YES2

 

 


 

Fig. 15

 

 

 

Initial data

 

Parameters of capsule: , , , , .

 

The entry conditions:

 

- angle of entry in an atmosphere,

- initial speed,

- initial height,

- initial angular speeds.