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
, , , (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
For spherical capsule
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
where - corner at top of a cone.
where , , .
The similar formulas for the truncated cone look like.
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
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
where - amplitude of fluctuations of a angle of attack.
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
On the top site of re-entry (height of flight H=70 -100 km)
The analytical decision for H=70 -100 km
where and - initial meanings of amplitude and frequency of fluctuations (H=100 km),
- frequency of flat fluctuations of capsule, .
of lift force on dynamic stability at H<
Change of parameters of a trajectory at re-entry capsule YES2
Speed of a landing:
Dependence of a thermal flow () on time (s)
Action of aerodynamic forces at dynamic stability of capsule YES2
Action of aerodynamic forces at dynamic instability of capsule YES2
The entry conditions:
- angle of entry in an atmosphere,
- initial speed,
- initial height,
- initial angular speeds.