# 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

Fig. 3

# ,                                   (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

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)

## Dependence of speed () on time (s)

Speed of a landing:

# 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

# Parameters of capsule: , , , ,  .

The entry conditions:

- angle of entry in an atmosphere,

-  initial speed,

- initial height,

- initial angular speeds.