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
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
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<
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.