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Features of rotational motion of a spacecraft descending in the Martian atmosphere

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ISSN 0010-9525, Cosmic Research, 2007, Vol. 45, No. 4, pp. 331–338. © Pleiades Publishing, Ltd., 2007.Original Russian Text © V.S. Aslanov, A.S. Ledkov, 2007, published in Kosmicheskie Issledovaniya, 2007, Vol. 45, No. 4, pp. 351–357.
331
1. FORMULATION OF THE PROBLEMIt is agreed that one of main causes, resulting inanomalous behavior of spacecraft at atmospheric entry,is the parametric resonance [1, 2]. It arises in the pres-ence of a small mass-inertial and aerodynamic asym-metry, when the motion relative to the center of massdepends on two angular variables: the spatial angle of attack and the angle of spinning. When the frequency of oscillation of the angle of attack and the average angu-lar velocity of spinning become multiple to the ratio of prime integers under an effect of disturbances, the res-onance arises. The resonance, as a phenomenon of con-siderable change in the amplitude of oscillations, canalso arise in the absence of asymmetry, when themotion depends on a single angular variable, the spatialangle of attack, while the aerodynamic restoringmoment coefﬁcient
m
α
(
α
)
vanishes at three points onthe segment [0,
π
]. In this case, on the phase portrait
=
one can observe three regions separated by aseparatrix [3]. Under an effect of disturbances, such asdynamic pressure variation at spacecraft descent in theatmosphere, the phase trajectory can intersect the sepa-ratrix, thus transferring from one region to another,which is accompanied by a jump change of the oscilla-tion amplitude and represents a resonance [4]. Fordescent in the rareﬁed Martian atmosphere the blunt-shaped bodies of small elongation are used, which pro-vides for effective drag. Such bodies, depending ontheir mass conﬁguration, can possess, along with twobalancing positions of the angle of attack:
α
* = 0,
π
,also the third equilibrium position:
α
*
∈ (
0,
π)
.Figure 1 presents the segmental-conic body and depen-dencies of a restoring aerodynamic moment coefﬁcient
m
α
(
α
)
on the spatial angle of attack for various posi-tions of the center of mass measured from the body’s
α
˙
α
˙
α
() nose (
=
x
T
/
l
, where
l
is the characteristic size of abody), found using the Newton’s shock theory.To approximate the restoring moment coefﬁcient wemake use of a bi-harmonic dependence of the form
(1.1)
For the considered class of spacecraft the position
α
= 0is stable; therefore, the derivative of the restoringmoment coefﬁcient with respect to the angle of attack at this point is negativeor
(1.2)
And if there exists an intermediate balancing positionon the interval (0,
π
), thenwhich holds true, if
(1.3)
It is obvious that inequalities (1.2) and (1.3) are validsimultaneously at
b
< 0. Note that the dependencies
m
α
(
α
)
presented in Fig. 1 satisfy conditions (1.2) and(1.3).The problem is stated to demonstrate the possibilityof appearance of resonances for axi-symmetric bodiesintended for entering the Martian atmosphere, to ﬁndthe motion stability conditions, to obtain the averagedequations of disturbed motion, and to construct the pro-cedure for calculating the upper and lower estimates of motion parameters with the use of the averaged equa-tions.
x
T
m
α
α()
a
α
b
2
α
.sin+sin=
a
α
2
b
2
α
cos+cos
()
a
0=
0,
<
2
b
–
a
.
<
m
α
α()
a
α
sin
b
2
α
sin+==
α
a
2
b
α
cos+
()
sin0,=2
ba
.
>
Features of Rotational Motion of a Spacecraft Descending in the Martian Atmosphere
V. S. Aslanov and A. S. Ledkov
Korolev Samara State Aerospace University, Samara, Russia
Received February 1, 2006
Abstract
—Angular motion at atmospheric entry is studied in the paper for a spacecraft with a bi-harmonicmoment characteristic. Special attention is given to the case when the spacecraft possesses two stable balancedpositions, and, hence, it can oscillate in dense atmospheric layers in the ranges of small or large angles of attack.The averaged equations of spacecraft motion are derived, which allow one to increase the speed of calculationsby several orders of magnitude. A real example is presented, which concerns a spacecraft specially designed fordescending in the Martian atmosphere.PACS: 45.40.Gj
DOI:
10.1134/S0010952507040065
332
COSMIC RESEARCH
Vol. 45
No. 4
2007
ASLANOV, LEDKOV
2. EQUATIONS OF MOTION AND THE SYSTEM’S PHASE PORTRAITWe write the equations of three-dimensional motionof an axi-symmetric body at descent in the atmospherein the following form [2]:
(2.1)
where
z
= (
R
,
G
,
V
,
θ
,
H
)
is the vector of slowly varyingparameters;
α
is the spatial angle of attack,
ε
is a smallparameter,
R
and
G
are, to an accuracy of a multiplier,the projections of the angular momentum vector ontothe longitudinal axis and onto the velocity direction,respectively;
V
is the spacecraft motion velocity,
θ
isthe trajectory inclination angle,
H
is the ﬂight altitude,
g
is the acceleration of gravity,
c
x
α
(
α
)
is the drag forcecoefﬁcient,
q
=
ρ
V
2
/2
is the dynamic pressure,
ρ
is thedensity of the atmosphere,
S
is the middle cross sectionarea,
m
is the spacecraft mass,
M
α
=
m
α
qSL
/
I
is therestoring moment to an accuracy of a multiplier (
I
is the
α
˙˙
GR
α
cos–
()
RG
α
cos–
()α
3
sin----------------------------------------------------------------+–
M
α
α
z
,()
–
m
z
z
()α
˙,=
R
˙–
ε
m
x
z
()
R
εΦ
R
z
()
,==
G
˙–
ε
m
y
z
()
Gm
x
z
()
m
y
z
()–
[]
R
α
cos+
{}
==
εΦ
G
α
z
,()
,
V
˙–
c
x
α
α
()
qS m
------
g
θ
sin–
εΦ
V
α
z
,()
,==
θ
˙–
θ
cos
V
------------
gV
2
R
P
H
+-----------------–
εΦ
θ
α
z
,()
,==
H
˙
V
θ
sin
εΦ
H
α
z
,()
,==transverse moment of inertia of the body, and
L
is itscharacteristic size),
R
P
is the planet’s radius;
ε
m
x
(
z
),
ε
m
y
(
z
)
, and
ε
m
z
(
z
)
are the projections of a small damp-ing moment onto the axes of the right-handed coordi-nate system
Oxyz
chosen in such a manner that the
Ox
axis is directed along the spacecraft’s axis of symmetry,the
Oy
axis lies in the plane formed by
Ox
and velocityvector
V
.It should be noted that the right-hand sides of theequations of system (2.1) can also be written in a morecomplicated form, such as that in [1, 2]. However, of principal signiﬁcance is here the circumstance that theright-hand sides are functions of only one “fast” vari-able, the spatial angle of attack
α
. We present the sys-tem (2.1) in a short-cut form:
(2.2)
Disturbed system (2.2) for
ε
= 0 is reduced to theundisturbed system with a single degree of freedom.The evolution of motion parameters proceeds under aneffect of disturbances arising due to small dampingmoments and dynamic pressure variability. Now weshould ﬁnd the relationship between the presence of three balancing positions of a bi-harmonic characteris-tic (1.1), under conditions (1.2) and (1.3), and the exist-ence of stable and unstable equilibrium positions on thephase portrait of an undisturbed system obtained from(2.2) for
ε
= 0:
(2.3)
α
˙˙
F
α()
+–
ε
m
z
z
()α
˙,
z
˙
εΦ
z
α
z
,()
.==
α
˙˙
F
α()
+0,=
0 . 2 2 0 . 3 8
0.3250.130.41
306090120150180
α
,
deg
0–0.500.51.01.52.0
m
α
x
Ú
= 0.3
x
Ú
= 0.5
x
Ú
= 0.7
Fig. 1.
The coefﬁcient of the aerodynamic restoring moment versecs the spatial augle of attack at varicus positions of the center of mass.
COSMIC RESEARCH
Vol. 45
No. 4
2007
FEATURES OF ROTATIONAL MOTION OF A SPACECRAFT DESCENDING333
where
(2.4)
Equation (2.3) has the integral of energy
(2.5)(2.6)
where
W
g
(
α
) = (
G
2
+
R
2
– 2
GR
cos
α
)/[2
sin
2
α
],
W
r
(
α
) =
A
cos
α
+
B
cos
2
α
.There is one-to-one correspondence between thevalues of variable
u
=
cos
α
on the segment [–l, +l] andthe values of angle
α
on the segment [0,
π
]. With regardto replacement
u
=
cos
α
integral of energy (2.5) can bewritten as
(2.7)
where
W
g
(
u
) = (
G
2
+
R
2
– 2
GRu
)/[2(1 –
u
2
)],
W
r
(
u
) =
Au
+
Bu
2
.Now let us present (2.7) as follows:
–
f
(
u
)
= 0,where
(2.8)
The character of a phase portrait of the systemdescribed by equation (2.7) is determined by the formof potential function
W
(
u
)
. In particular, the numberand position of extreme points of this function deter-mine the number and type of singular points. The stablepoint of the center type corresponds to the minimum,and the unstable point of the saddle type to the maxi-mum. The behavior of function
W
(
u
) =
W
g
(
u
) +
W
r
(
u
)
for various combinations of
R
,
G
,
A
,
and
B
parameterswas studied in [2]; we present here only the results of this study. The potential function
W
(
u
)
has no inﬂectionpoints on the (–1, 1) interval, provided that
(2.9)
since its second derivative
W
''(
u
) = (
u
) +
(
u
)
with respect to variable
u
is non-negative. This impliesthat there is no saddle singular point on the phase por-trait. According to (2.9), quantity
B
*
is always negative.For
R
=
G
= 0 function
(
u
)
degenerates; therefore,
B
*
= 0, and condition (2.9) assumes the form of
B
≥
0
.The saddle point will also be absent at a relatively smallvalue of coefﬁcient
b
as compared to
a
(the motionclose to the Lagrangian case). Really, if
(2.10)
F
α
()
G R
α
cos–
()
R G
α
cos–
()α
3
sin----------------------------------------------------------------=–
A
α
B
2
α
,sin–sin
A aSL I
----------
q
,
B bSL I
----------
q
.==
α
˙
2
/2
W
α()
+
E
,=
W
α()
F
α()α
d
∫
W
g
α()
W
r
α()
,+==
u
˙
2
/21
u
2
–
()[]
W
g
u
()
W
r
u
()
++
E
,=
u
˙
2
f u
()
21
u
2
–
()
E Au
–
Bu
2
–
()
=+2
GRu G
2
–
R
2
.–
B
–0.5
W
g
''
u
()()
–1
u
1
≤≤
min
[]
B
*,
≡≥
W
g
''
W
r
''
W
g
''
b
0.5
a
,
≤
then function
(
u
)
has one and the same sign through-out the interval, and, hence, the derivative
W
'(
u
) =(
u
) + (
u
)
vanishes at a single point, and function
W
(
u
)
has a single extremum (minimum). Condition(2.10) contradicts condition (1.3), and, if condition(2.9) is not met, function
W
(
u
)
can have two minimaand one maximum on the (–1, 1) interval, which corre-sponds to the presence of an unstable singular point of the saddle type on the phase portrait. It is obvious thatthe aforementioned situation arises upon satisfying thecondition
(2.11)
where
u
*
1
and
u
*
2
are the roots of the equation
W
''(
u
)
= 0.When condition (2.11) is met, the phase plane isdivided by a separatrix into three regions: the outerregion
A
0
and two inner regions
A
1
and
A
2
. If
E
>
W
*
,where
W
∗
is the value of
W
(
u
)
at a saddle point
u
=
u
∗
,then the motion proceeds in the outer region
A
0
(Fig. 2).In the opposite case (
E
<
W
*
) the motion can occur inany of innner regions
A
1
or
A
2
, depending on the initialconditions. The equality
E
=
W
∗
corresponds to themotion along the separatrix.
W
r
''
W
g
'
W
r
'
W
'
u
*
1
()
W
'
u
*
2
()
0,
<
uuuu
*
u
*
f
(
u
)
u. A
1
A
2
A
0
u
*
W
(
u
)
Fig. 2.
Phase portrait.
334
COSMIC RESEARCH
Vol. 45
No. 4
2007
ASLANOV, LEDKOV
3. STABILITY OF PERTURBED MOTIONUnder the action of perturbations, arising duringspacecraft descent in the atmosphere, the phase trajec-tory, while remaining in one of the regions, eithermoves apart from a separatrix or approaches it. In theﬁrst case the trajectory is “immersed” deeper into agiven region, and in the second case it is “pushed out”from it. Accordingly, we will refer to regions
A
0
,
A
1
,
and
A
2
as stable or unstable. The motion can start either inouter region
A
0
or in any of inner regions
A
1
and
A
2
. If the region in which the motion has begun is unstable,the phase trajectory intersects the separatrix in someﬁnite time. Obviously, two situations can take place atthe separatrix intersection instant: 1) two regions areunstable and one is stable, and 2) on the contrary, oneregion is unstable and two are stable. In the ﬁrst case themotion continues in the stable region only, and in thesecond case the further behavior of a trajectory dependson the current phase of the angle of attack. If the phaseis not determined, to fall into any region is of a randomcharacter. The author of [4] proposes to use, for choos-ing the continuation of motion, the concept of probabil-ity of “capture” into each region. This probability isdetermined on the basis of calculating the areas of regions encompassed by a separatrix. Analytical ﬁnd-ing of these areas is reduced to calculation of improperintegrals.In order to estimate the stability of the regions it isnot necessarily to calculate their areas. Under the actionof small perturbations the average value of the totalenergy slowly changes, as well as the value of poten-tial energy , calculated at the saddle point
u
=
For determining the stability it is sufﬁcient to make useof time derivatives of mentioned functions [2]. Theinner region (
A
1
or
A
2
) is stable, if the following condi-tion is satisﬁed near the separatrix:(3.1)For the outer region
A
0
the stability condition is as fol-lows:(3.2)The value of function (2.8) at the saddle point
u
=
is equal to:(3.3)In the neighborhood of the separatrix
(
z
) –
W
(,
z
) =
O
(
ε
), (
z
) =
O
(
ε
)
, and the differentiation of function(3.3) with respect to time, to an accuracy of quantitiesof the order of
ε
2
, gives the following result:(3.4)
E W
*
u
*.
E
˙
z
()
W
˙
u
*
z
,
().
<
E
˙
z
()
W
˙
u
*
z
,
().
>
u
*
f
*
f u
*
z
,()≡
21
u
*
2
–
()
E z
()
W u
*
z
,()
–
[]
.=
E u
*
u
˙*
f
˙*21
u
*
2
–
()
E
˙
z
()
W
˙
u
*
z
,()
–
[]
.=It follows from (3.4) that the conditions
< 0
and > 0 correspond, respectively, to conditions (3.1)and (3.2) [3]. Indeed, if in the inner region (
A
1
or
A
2
) thevalue of polynomial
f
(
u
)
at point decreases, then thisregion is stable. In the opposite case the region is unsta-ble, and the phase trajectory will not fall into it at anyinitial conditions. Similarly, the outer region
A
0
will bestable or unstable with increasing or decreasing
f
∗
,respectively.It follows from energy integral (2.5) that the totalenergy is equal to potential one
E
=
W
(
α
m
)
, calculatedfor the amplitude value of the angle of attack
α =
α
m
(for = 0). It is obvious that(3.5)where
α
m
and
z
correspond to the averaged equations.We suppose that the averaged equations of motion cor-responding to system (2.2) are obtained. Let us calcu-late the derivatives and in virtue of theaveraged equations:andNow we introduce the criterion which determines sta-bility of the perturbed motion in the separatrix neigh-borhood:(3.6)and then, ﬁnally, we can write the stability conditions:for the inner regions
A
1
and
A
2
(3.1)
Λ
< 0(3.7)
and for outer region
A
0
(3.2)
Λ
> 0.(3.8)
4. AVERAGED EQUATIONS AND MODELING OF THE PERTURBED MOTIONThe stability criterion
Λ
is a function of the ampli-tude value of the angle of attack; therefore, it is expedi-ent to write the averaged system for this angle directly.In addition, the numerical modeling of the perturbedmotion is convenient to be performed with the use of
f
˙*
f
˙*
u
*
α
˙
E z
()
W
α
m
z
,()
,=
E
˙
z
()
W
˙
α
*
z
,
()
E
˙
z
()
∂
W
∂α
--------
αα
m
=
α
˙
m
∂
W
∂
z
--------
αα
m
=
z
˙+==
F
α
m
z
,()α
˙
m
∂
W
∂
z
--------
αα
m
=
z
˙+
W
˙
α
*
z
,
()
∂
W
∂
z
--------
αα
*=
z
˙.=
Λ
F
α
m
z
,()α
˙
m
∂
W
∂
z
--------
α
*
α
m
z
˙,+
≡
COSMIC RESEARCH
Vol. 45
No. 4
2007
FEATURES OF ROTATIONAL MOTION OF A SPACECRAFT DESCENDING335
the averaged equations. For the known solution of unperturbed system (2.3) [2], we write, by means of V.M. Volosov’s method [5] for perturbed system (2.1),the averaged equations, having chosen the maximumangle of attack as an amplitude
α
m
:
(4.1)
Here
α
˙
m
2
ε
TF
α
m
()
-------------------
m
z
I
1
R
2
m
x
G
2
m
y
+
()
I
2
+---=–2
GRm
y
I
3
R
2
m
x
m
y
–
()
I
4
–
A
q
I
5
B
q
I
6
+
()Φ
q
+–
G R
α
m
()
cos–
α
m
()
2
sin-----------------------------------
m
x
m
y
–
()
RI
5
–
ε
F
α
m
()
---------------
R G
α
m
()
cos–
α
m
()
2
sin-----------------------------------
m
x
RG R
α
m
()
cos–
α
m
()
2
sin-----------------------------------
m
y
G
++---
A
q
α
m
()
cos
B
q
α
m
()
2
cos+
()Φ
q
,
R
˙
ε
m
x
R
,
G
˙
ε
m
x
G
2
T
---
m
x
m
y
–
()
RI
5
+,==
V
˙–2
T
---
c
x
α
α
()
α
d
2
E W
α()
–
()
------------------------------------
α
min
α
m
∫
qS m
------
g
θ
sin–==
εΦ
V
α
m
z
,
(),
θ
˙–
θ
cos
V
------------
g V
2
R
P
H
+-----------------–
εΦ
θ
z
()
,==
H
˙
V
θ
sin
εΦ
H
z
()
,==
εΦ
q
α
m
z
,()
q
˙
d dt
-----
ρ
V
2
/2
()
===
ερ
V
Φ
V
α
m
z
,()ρ
H
Φ
H
z
()
V
2
/2+
[]
,
A
q
dAdq
-------,
B
q
dBdq
-------.==
I
1
α
˙
α
,
I
2
d
α
min
α
m
∫
α
d
α
˙
α
2
sin-----------------,
α
min
α
m
∫
==
(4.2)
The integrals
I
i
can be reduced to complete normalelliptic Legendre integrals of the ﬁrst, second, and thirdkind [6]. For this purpose, depending on the form andposition of roots of the polynomial
f
(
u
)
(see Table), oneshould make use of one of changes of variables [2]. If there exist four real roots, it is necessary to use thechange
(4.3)
and if there are two real and two complex-conjugateroots, then
(4.4)
The following designations are introduced in formulas(4.2) and (4.3):
u
1
=
cos
α
max
,
u
2
=
cos
α
min
,
u
3
,
u
4
,
and
u
34
±
i
v
are the roots of the polynomial
f
(
u
);
ξ
=
cos
χ
1
/
cos
χ
2
, = (
u
1
–
u
34
)/
v
, = (
u
2
–
u
34
)/
v
.When calculating integrals (4.2) it is convenient tomake use of the following expressionwhich is obtained from (2.8), (4.3), and (4.4). The val-ues of coefﬁcients
k
,
β
, and period
T
are determineddepending on the type of roots:—four roots are real: change (4.3)—two roots are real and two ones are complex-con- jugate: change (4.4)
I
3
α
cos
α
d
α
˙
α
2
sin-------------------,
I
4
α
min
α
m
∫
α
2
cos
α
d
α
˙
α
2
sin---------------------,
α
min
α
m
∫
==
I
5
α
cos
α
d
α
˙-------------------,
I
6
α
min
α
m
∫
α
2
cos
α
d
α
˙---------------------.
α
min
α
m
∫
==
α
cos
uu
1
u
2
u
3
–
()
u
3
u
1
u
2
–
()γ
2
cos+
u
2
u
3
–
()
u
1
u
2
–
()γ
2
cos+-------------------------------------------------------------------------,==
α
cos
uu
2
u
1
ξ
+
()
u
2
u
1
ξ
–
()γ
cos–1
ξ
+
()
1
ξ
–
()γ
cos–-------------------------------------------------------------------.==
χ
1
tan
χ
2
tan
d
αα
˙-------–
d
γ β
1
k
2
γ
2
sin–----------------------------------,=
k u
1
u
2
–
()
u
3
u
4
–
()
u
1
u
3
–
()
u
2
u
4
–
()
------------------------------------------,=
β
–12---
B u
1
u
3
–
()
u
2
u
4
–
()
,
T
2
K k
()β
---------------;==
k
12---1
u
1
u
34
–
()
u
2
u
34
–
()
v
2
+
u
1
u
34
–
()
2
v
2
+
()
u
2
u
34
–
()
2
v
2
+
()
-----------------------------------------------------------------------------------------–
,=

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