=Paper=
{{Paper
|id=Vol-1490/paper23
|storemode=property
|title=Numerical simulation of the resonance effect at re-entry of a rigid body with low inertial and aerodynamic asymmetries into the atmosphere
|pdfUrl=https://ceur-ws.org/Vol-1490/paper23.pdf
|volume=Vol-1490
}}
==Numerical simulation of the resonance effect at re-entry of a rigid body with low inertial and aerodynamic asymmetries into the atmosphere==
https://ceur-ws.org/Vol-1490/paper23.pdf
Mathematical Modeling
Numerical simulation of the resonance effect at Re-entry
of a rigid body with low inertial and aerodynamic
asymmetries into the atmosphere
Lyubimov V.V.
Samara State Aerospace University
Abstract. We consider a motion relative to centre of mass of a rigid body with
low inertial and aerodynamic asymmetries at re-entry into the atmosphere.
Numerical and mathematical simulation provide a means to study the resonance
effect that results in changes of the direction of the rigid body rotation. It is
shown that the realization of the effect in question is observed when selecting
the initial conditions of the integration and defined values of the asymmetry
parameters.
Keywords: resonance effect, numerical simulation, asymmetry, rigid body,
atmosphere
Citation: Lyubimov V.V. Numerical simulation of the resonance effect at Re-
entry of a rigid body with low inertial and aerodynamic asymmetries into the
atmosphere. Proceedings of Information Technology and Nanotechnology
(ITNT-2015), CEUR Workshop Proceedings, 2015; 1490: 198-210. DOI:
10.18287/1613-0073-2015-1490-198-210
1. Introductions
It is known [1]-[2] that the presence of the small asymmetry can lead to realization
of the resonance that is observed in re-entry of a spacecraft viewed as a rigid body
(RB) into the atmosphere. A continuous resonance leads to violation of the
technological limitations on the RB’s angle of attack or angular velocity. Besides the
resonance itself, some secondary resonance effects can lead to violation of the
technological limitations. The mentioned effects were found by Sadov Y.A. in
dynamical systems with slow and fast variables [3]. The essence of these effects is: in
the area close to the resonance, characteristic evolutions of slow variables of the
system are observed, caused by this resonance. From the mathematical point of view,
the secondary resonance effects are explained with presence of the resonant frequency
mistuning in the denominators of averaging method’s highest approximations that are
received in the non-resonant case. Secondary resonance effects were studied
particularly in the spherical rotation of the heavy asymmetrical rigid body [4].
Applying to the task of re-entry of an asymmetrical RB in the atmosphere, the
mentioned resonance effects were studied in detail in the work [5]. In particular, it has
been revealed that the secondary resonance effect itself can lead to the realization of
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Mathematical Modeling Lyubimov V.V. Numerical simulation of the…
the RB’s strong spin-up that was received earlier in the work [1]. In the following,
using of traditional mathematical analysis methods allowed to extend the
classification of the secondary resonance effects applying to the task of atmospheric
re-entry of RB [6]. Other features of the task of atmospheric re-entry that affect the
behaviour of the descent craft, are random character of the initial conditions on the
separatrix (that lead or don’t lead to the transfer into the resonance area) [7], and also
the possibility of losing part of data on the craft’s movement [8]. In the work [9], it is
shown that the negative consequences of the resonance effects in respect to the task of
atmospheric re-entry of the spacecraft with low mass and aerodynamic asymmetries
can be solved with introducing the control over the amount of asymmetry. In the
process of numerical simulation of movement of the RB with low mass and
aerodynamic asymmetries, a secondary resonance effect has been considered, leading
to changes in the direction of the craft’s rotation [10]. Numerical simulation and
analytical study of the similar resonance effect in the task of atmospheric re-entry of
RB with low aerodynamic and inertial asymmetries are of practical interest.
2. Mathematical models
Initial non-linear equation of the asymmetrical RB’s movement relating to the
centre of mass in the atmosphere appear, for example, in the work [11]. However, the
significant non-linearity of these equations and instability of frequencies of the system
significantly complicate the application of such studies for detection and detailed
study of secondary resonance effects. It is known that application of the integral
manifold method [12] allow to decrease the order of differential equation system. For
instance, in the work [13] it is shown that the application of one of the variants of
integral manifold method makes it possible to receive from initial non-linear system
of asymmetrical RB’s movement relating to the centre of mass an equation system
that is approximated to the non-linear. With allowance only for aerodynamic and
inertial asymmetries, this system takes the following form:
m sin 2
x 2
cos 2( 3 ) , (1)
I x 1,2
2m a 1,2 sin 1,2
2
sin 2
x cos(2 23 )
Fa 2a
2m Aa 4mzп 2a SL dq
cos( 1 ) 3 , (2)
Fa IFa dt
x 1,2 cos . (3)
Here, ε is a small parameter that characterizes the value of RB’s small inertial and
aerodynamic asymmetries, and the slowness of changing the dynamic pressure
q ( dq / dt O( ) ), x is the RB’s angular velocity relating to axis with least moment
3
of inertia; α is the space angle of attack; mzп qSLctg / I ; S and L are
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characteristic area and size of the RB; I x and I= I y I z are the moments of RB’s
inertia relating to axes of XYZ body-fixed coordinate system; I x I x / I ,
п / 2; φ is the aerodynamic roll angle; m A and m are generalized
parameters of aerodynamic in inertial asymmetries, m A m m ,
1
A 2 A 2
2
Ix
1,2 x ,
2
m1A
1 I x x 31,2 * 2 (mф C z)tg, sin m A / m A ,
y x 1 1
2a mzп
m2A
1 I x x 31,2 * 2 (mф C y)tg, cos m A / m A , m A m A / 2 ,
z x 1 2
2a mzп
m ( I yz )2 (I )2 , sin 23 I / m , cos 23 I yz / m , mфy , mzф -
aerodynamic shape asymmetry coefficients, I ( I y I z ) / 2, I ( I z I y ) / 2,
I I / I , I yz I yz / I - inertial (dynamic) asymmetry, mzп - coefficient of
2
restoring moment, a I x 2x / 4 2 , x 1,2 - resonant frequency mistuning,
Fa (x , , q) is the known function of the slow variables. Equation system (1)-(3) is
solved together with three differential equations that describe change in
characteristics of the RB’s centre of mass movement [1]: airspeed, flight-path
inclination angle and flight altitude. Mentioned three variables are considered as slow.
Equation system (1)-(3) takes into account the possibility of implementing the
main resonance: x 1,2 0 . When passing through the main resonance, significant
perturbations of movement parameters (compared to multiple resonances of higher
orders) are observed [14]. By solving the equation x 1,2 0 , we find the
resonance values of the angular velocity:
rx . (4)
I
1 x
I
The sign in the expression (4) matches the sign of angular velocity x . The
equation system (1)-(3) contains in its right side the dependency on the phase of rapid
movement θ, which complicates the analysis of non-resonant evolutions at secondary
resonance effects. For further simplification of this system, we will use averaging
method in non-resonant case [15].
System (1)-(3) applies to the class of systems with single rapid phase and
several slow variables. It has a standard form for application of averaging method:
u U (u, , ), (5)
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(u). (6)
Here u x , is the vector of slow variables, (u) x 1,2 , U (u, , ) is
the vector function of right side of equations (1) and (2). Equations for centre of mass
movement parameters do not depend on , so they are not taken into account in
process of averaging.
System (1)-(3) is averaged on the non-resonant areas of movement under the rapid
phase . Resulted averaging of equations for slow variables have the following form:
du o
A1 (u o ) 2 A2 (u o ) 3 A3 (u o ) ... , (7)
dt
where the functions Ai , i=1,2,3,... are determined through standard averaging
method [15].
After averaging the system (1)-(3) in non-resonant case we obtain: A1 0 , A2 0
A3 0 . Hence, the evolution of slow variables x , , caused by the secondary
resonance effects, is determined by the members of the third approximation of
averaging method. The equations for slow variables x , , being averaged on non-
resonant areas of movement and taking into account first three approaches of
averaging method, are
d x A
m g 2 g3 A
A
m g3 A g
3 ( m g ) ( m g3 2 )
dt 3
3
2
A
m cos(21 22 ) ,
2
3(m g3 )2 g 2
( g ) (8)
2
4 8
Averaged equations (8) and (9) contain generalized parameters of asymmetry m A ,
m , 1 , 2 in numerator. The parameters m A , m define the value of RB’s
asymmetry, and parameters 1 , 2 characterize the location of asymmetry on the
rigid body. From the equations (8)-(9) it appears the main resonance =0 contributes
to the realization of secondary resonance effects, for all the members of mentioned
averaged equations contain frequency mistuning x 1,2 in their denominators. In
particular, while approaching the resonance in the boundaries of non-resonant area, a
decrease of x 1,2 value occurs, which leads to an increase of speed in averaged
values of variables x , . Such behaviour of slow variables is typical for secondary
resonance effects.
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A
d 3 m g3 g1 A A 2
( m g ) m g
3
3 3
dt 2
A 2
m g3 A g g A
(m g3) 1 12 (m g3 )
3
A
m g2 g3 A A
( m g 3 ) 2m g 3
3 x
A A
(m )2 g3 g1 g3 (m ) g3
2 2
g
7 g2 4 2
2
2 4
x
(m )2 g32 2 2 g1
A 2
m cos(21 22 )
g4 .
3
2 g1 (9)
2 4
2
8
2a 1,2 sin 1,2
2
sin 2 1,2
2
sin 2 2a 2
Here g1 ( x ) , g2 , g3 ,
Fa 2a Ix Fa
4mzп 2 SL dq
g4 .
IFa dt
To analyse the secondary resonance effects, we also have to obtain the equation for
the second derivative of the averaged angular velocity:
d 2 x x d x x d x d
. (10)
dt 2
x dt dt dt
3. Numeric simulation with different initial conditions
The numerical integration of the equation system (1)-(3) (with allowance of the
equation for derivative dq/dt) is implemented via four-staged Runge-Kutta method
with adaptive stepsize. The secondary resonance effect is considered, when the
angular velocity x (t ) , changing from initial positive value, envelopes the x (t )
r
ascending curve and reaches significant values with further keeping its sign in the
area of descending curve x (t ) . At certain decrease of initial conditions of
r
integration or decrease of asymmetry parameters, the situation of angular velocity
behaviour changes dramatically. If in the area of ascending curve x (t ) the angular
r
velocity x (t ) also increases, in the area of descending curve x (t ) , the realization
r
of continuous resonance occurs that can be followed with transition to the negative
range of values x (t ) . Let us assume that the time of integration is 300 s. We
consider the influence of various initial condition of integration on the realization of
the aforementioned resonance effects. Here, two cases shall be pointed.
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Case 1. The resonance effect followed by change of rotation direction is observed
in various values of initial angular velocity of the body. Let us assume that the
condition x (0) x (0) has been met. In the first case, generalized parameters of
r
asymmetry take the following values: m A 0.011 , m 0.04 , 1 2 0 . Initial
values of the angle of attack: (0) 5 grad, and initial value of aerodynamic roll
angle is * (0) 0 .
In the process of numerical integration of equations (1)-(3), a narrow interval of
1
positive values for initial angular velocity x (0) [17, 23]s , has been found. With
boundary values of x (0) within this interval, the resonance effect in question is
observed. Let us have a more detailed review of these numerical results.
1
1. We assume that the initial value of angular velocity x (0) 23s . In the Fig.1,
the upper curve describes the increase of angular velocity x (t ) , and the curve with
maximum is associated with the change of resonance values of angular velocity
rx (t ) . At that, in the area of ascending branch of resonance curve rx (t ) , a
monotonic non-resonance increase of x (t ) occurs. Further, the withdrawal from
resonance values is observed, and angular velocity x (t ) is stabilized. From the Fig.1
it appears that the craft acquires the terminal positive angular velocity which exceeds
1
the initial velocity x (0) more than twice. Thus, with x (0) 23s , a slow increase
of spacecraft angular velocity positive values is observed. The angle of attack also
increases slowly from the initial value of 5 grad, but does not reach the value in 90
angle degree, typical for realization of main resonance. The direction of RB’s rotation
remains unchanged and positive. The increase of angular velocity x (t ) in the area of
ascending branch of rx (t ) curve is explained by positivity of the derivate (8). The
change of signature of the x (t ) curve’s convexity happens due to flip of the second
derivate’s sign (10). The increase of the x (t ) fluctuation area after passing the
maximum of the resonance curve can be explained with the following: in the right
side of the equation (8) there are terms directly proportional to sinα. After 220
seconds of flight, the angle of attack takes values from 40 to 50 grades, which leads to
a certain perturbation on the angle of attack in the right side of the equation (8).
1
Fig. 1. – Change of angular velocity at x (0) 23s
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1
2. Let the initial angular velocity be x (0) 20s . In this case value of the initial
1
angular velocity is within the x (0) 17 s range. Also, from the Fig.2 it appears
that the angular velocity x (t ) in the area of ascending curve x (t ) changes
r
similarly to the previously considered variant. But in the area of the descending
resonance curve, angular velocity x (t ) passes a bit lower than on the Fig.1. Also,
after passing the maximum of the resonance curve, the fluctuation amplitude x (t )
increases significantly, which also promotes the angular velocity to reach its
resonance values x (t ) . In its turn, an increase of the resonance curve fluctuation
r
area is observed. The expansion of the angular velocity fluctuation area occurs due to
increase of angle of attack that reaches 75 grads on the 200 th second. It should be
noticed that the continuous resonance does not occur nevertheless.
1
3. Let the initial angular velocity be x (0) 17 s . From the Fig.3 it appears that
in the variant under consideration (similarly to the first one), a growth of non-
resonance RB spin-up in the area of ascending resonance curve is realized.
1
Fig. 2. – Change of angular velocity at x (0) 20s
Yet, the spin-up is slightly smaller than in variant 2, so the increase of amplitude
x (t ) in the area of the curve maximum leads to meeting the resonance values x (t )
r
. Further, the main resonance is realized, and the angle of attack quickly reaches 90
grades. After the resonance realization, angular velocity returns to the non-resonance
movement in the negative range of its values.
Comparing the results of the Fig.1 and Fig.3, it may be noticed that with values of
1
the initial angular velocity that are the boundaries of the x (0) [17, 23]s interval,
in the process of evolution x (t ) , opposite directions of the RB rotating motion are
reached.
Case 2. Resonance effect followed by change of the rotation direction is also
observed in various values of the initial angle of attack (0) . In the second case,
generalized parameters of asymmetry take the following values: m A 0.011 ,
m 0.002 , 1 2 0 . Initial value of the angular velocity: x (0) 25s 1 , and
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initial value for the aerodynamic roll angle is (0) 0 . In the process of numerical
integration of the equations (1)-(3), we found a narrow range of values for initial
angle of attack (0) [3; 5.7] grad, with boundary values of which, the studied
resonance effect is observed. The results obtained in the second case are shown on the
Fig. 4-6.
4. Let the initial value for the angle of attack be (0) 3 grad. On the Fig.4, in the
area of ascending branch of the resonance curve, a monotonic non-resonant increase
x (t ) occurs, similar to variant 1 shown on Fig.1. Further, a withdrawal of x (t )
from resonance values x (t ) is also observed, and the angular velocity of the body is
r
1
stabilized at x (0) 47s . Yet, different from variant 1, the angular speed changes
in monotonic way throughout the whole integration interval. It occurs due to fact that
the angle of attack at the slow increase from initial value of 5 grad reaches the value
of 38 grad in the end of integration interval. As a result, no perturbations of angular
velocity is observed taking place with angles of attack higher than 50 grades and
more. The direction of the craft’s rotation also remains positive throughout the whole
integration interval. The increase of the angle velocity x (t ) in the area of ascending
branch of the curve x (t ) is also explained by positivity of the derivative (8), and the
r
change of the character of the x (t ) curve’s convexity occurs due to flip of the
second derivative’s sign (10).
1
Fig. 3. – Change of angular velocity at x (0) 17s
Fig. 4. – Change of angular velocity in the second case at α(0)=3 grad
5. Let us assume that the initial value for the angle of attack is α(0)=4.7 grad. Here,
the value for the initial angle of attack is within the section of (0) [3;5.7] grad. At
that, the angular velocity x (t ) in the area of ascending curve x (t ) changes
r
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similarly to variant 4. Yet in the area of descending resonance curve, angular velocity
x (t ) passes slightly lower than on the Fig.4. Compared to variant 2, in this variant
the angular speed perturbation in the area of descending resonance curve is not
significant. This is due to fact that the angle of attack in the considered area takes
value of about 40 grades. Comparing to variant 4, a slightly increase takes place in the
fluctuation are of the resonance curve. Also, in the area of maximum x (t ) on the
r
curve x (t ) , another inflexion point arrives. As a result, on the curve x (t ) there are
two inflexion points, where the mentioned curve changes the character of the
convexity twice. From the Fig.5 it appears that in the end of the integration interval,
angular velocity x (t ) doesn’t reach its resonance values x (t ) .
r
Fig. 5. – Change of angular velocity in the second case at α(0)=4.7 grad
6. Let the initial angle of attack be α(0)=5.7 grad. From the Fig.6 it appears that in
the considered variant, similarly to variants 4 and 5, a growths of the non-resonant RB
spin-up occurs in the area of the ascending resonance curve. Still, the mentioned spin-
up is slightly lower than in the variant 5. As a result, the expansion of range of the
resonance values leads angular velocity x (t ) to reaching the resonance values
rx (t ) Further, main resonance is realized, and angle of attack reaches 90 grades.
After leaving the continuous resonance, RB’s angular velocity x (t ) takes non-
resonance values in the negative range. Comparing the results presented on Fig.4 and
Fig.6, a conclusion can be made that with the initial values for angle of attack that are
the boundaries of section (0) [3; 5.7] grad, in the process of the evolution the
angular velocity x (t ) reaches opposite values of the rotating motion.
4. Numerical simulation at different values of asymmetry parameters
We perform a numerical simulation that allow us to study the issue of the influence
of different values of asymmetry parameters on the realization of the considered
resonance effect. We conduct an individual analysis of influence of inertial and
aerodynamic parameter values on the considered resonance effect. Hence, two cases
are found.
Case 3. Realization of the resonance effect with changing the rotation direction
occurs at different values of the generalized parameter of m inertial asymmetry. In
the process of the RB rotating motion numerical simulation, a range of small values
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was found for m [0.012, 0..040] parameter, for which three types of body’s
angular velocity evolution, similar to the first case, were revealed. Let us give them
more detailed overview. At values of the parameters m 0.040 , m A 0.011 ,
1
1 2 0 and initial condition of integration x (0) 23s , (0) 5 grad,
(0) 0 we obtain the result shown in Fig.1. When integrating the system (1)-(3)
with asymmetry parameter m 0.012 (all other parameters and initial conditions
are the same as in variant 1), we obtain the result shown in Fig.7. This result confirms
that with parameter value m 0.012 angular velocity changes the rotation direction.
Fig. 6. – Change of angular velocity in the second case at α(0)=5.7 grad
In numerical simulation with parameter m 0.013 we obtain result that conforms
the result on Fig.2 in the qualitative manner.
Case 4. Also, the realization of the resonance effect with change of the rotation
direction occurs with different values of m A aerodynamic asymmetry generalized
parameter. In numerical simulation of RB rotating motion, a narrow range of small
values for parameter m [0.008, 0.011] was found. With three values from this
A
range we obtain three types of spacecraft’s angular speed evolution which are similar
to the first case. Let us give them more detailed overview. The initial variant
conforms the case shown on the Fig.1. Indeed, the parameters of the craft and initial
Fig. 7. – Change of angular velocity at m 0.012
data are identic to the variant 1: m∆ =0.040, mᴬ=0.011, θ1 =θ2 =0, ωx (0)=23c -1 ,
(0) 5 grad, φ(0)=0. If the aerodynamic asymmetry parameter is considered equal
to m A 0.008 (leaving other parameters and initial integration conditions
unchanged), we obtain the result of integration presented on the Fig.8. This result is
approximate to the results shown in Fig.3 and Fig.7. Here, the angular velocity x (t )
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also transits to non-resonant negative range of its values after the realization of the
continuous resonance and withdrawal from it. Numerical simulation of angular speed
x (t ) at the parameter value m A 0.0082 allow us to obtain a result identical to the
result of simulation shown on Fig.2.
Fig. 8. – Change of angular velocity at m A 0.008
5. Conclusion
In this work, application of numerical simulation together with analytical research
allows us to study in detail the new resonance effect observed in the process of
movement in relation to centre of mass of the rigid body with low inertial and
aerodynamic asymmetries at re-entry into the atmosphere. An unusual feature of the
studied resonance effect is the following: while changing in the boundaries of the
ascending branch of the resonance curve, angular velocity x (t ) in absolutely most
cases “glides” with delay of relatively resonant curve x (t ) without reaching the
r
resonant values. At the transition to the area of the descending part of the curve x (t )
r
, two typical cases were peculiar to angular velocity x (t ) behaviour:
angular velocity kept increasing the positive values (practically by the linear law)
with further gradual non-linear transition to some large constant value;
angular velocity reached resonant value, resonance realization occurred with
further withdrawal from it and transition to non-resonant negative range of values.
In numerical simulation, narrow intervals of initial integration conditions x (0) ,
(0) and a small interval of asymmetry of m , m A parameters values were found.
Taking into account the values on the boundaries of intervals in the process of
numerical integration led us to realization of the two described typical cases. While
accounting intermediate values from aforementioned intervals in the process of
numerical simulations, we found a number of cases where an increase of amplitudes
of x (t ) and x (t ) oscillations was observed after passing the maximum by the
r
resonance curve. At that, change of the x (t ) curve convexity character occurred.
The conducted numerical simulation and results of analytical research allow us to
draw the following conclusions:
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1. The values of initial integration conditions x (0) , (0) and values of asymmetry
parameters m , m A , 1 2 0 together have an effect on realization of the
considered resonance event.
2. One may talk of a common pattern of influence by the values x (0) and m , m A
on the realization of the found resonance effect that takes place in case when
mentioned initial conditions and asymmetry parameters are chosen from the
intervals found in this study.
3. The comparison of the numerical results obtained in process of integration of the
approximated system (1)-(3) with the results of numerical integration of initial
non-linear system has shown good qualitative agreement with these results.
4. The further study of the considered dynamical event of x (t ) sign flip is of
practical interest, for instance in the process of movement of a RB with low inertial
and aerodynamic asymmetries in the atmosphere.
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