**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.

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

The method of

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

where _{} - amplitude of fluctuations of a angle of attack.

_{} , (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)

_{} . (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<

Fig. 4

Fig. 5

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

Speed
of a landing: _{}

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

The entry conditions:

_{} - angle of entry in an atmosphere,

_{} - initial speed,

_{} - initial height,

_{} - initial angular speeds.