COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Coupled Thermal-Dynamic Stability Analysis of Large Scale Space
Structures by FEM
Wei Li *, Zhihai Xiang, Mingde Xue
Department of Engineering Mechanics, Tsinghua University, Beijing, 100084 China
Email: [email protected]
Abstract Thermally induced vibrations of flexible spacecraft appendages, which are subjected to incident solar heat
flux during the orbital eclipse transitions, have been typical failure of modern spacecrafts since 1960’s. When the
thermally-induced vibration is unstable, it is called thermal flutter, which is due to the coupling effect between
structural deformations and the incident normal solar heat flux. Owing to the existence of the coupling effect and the
transient heat conduction problem involving heat transfer by radiation, it is really a highly nonlinear problem. In this
paper, the thermally induced nonlinear vibration of practical thin-walled large-scale space structures subjected to
suddenly applied thermal loading is analyzed. By using the finite element method, the governing equations of the
coupled system are established, which contain the temperature field and dynamic response accurately for actual
complex space structures and heating conditions. In the coupled thermal-dynamic system, the stability property of
the system depends on only the heat incident angle vectorν and damping ratio ζ . Finally the criterion of thermal
flutter is investigated under the frame of nonlinear vibration theories.
Key words:
coupled, thermal flutter, nonlinear vibration, stability, FEM
INTRODUCTION
Thermally induced vibrations typically occur on spacecrafts’ flexible appendages with very low frequency due to
sudden heating changes during the orbital eclipse transitions. Suddenly heating an appendage may cause temperature
changes and resultant time-dependent bending moments which causes the structure motion. These vibrations may
degrade the system operations, and in some conditions, self-excited vibrations of increasing amplitude may occur. The
latter phenomenon is known as thermal flutter, which is due to the coupling effect between structural deformations and
the thermal loading. Since the 1960’s, several spacecraft have experienced thermally induced vibrations, such as the
UGO-IV satellite [1], the UARS satellite [3] etc. Among them, the most notable is the Hubble Space Telescope (HST)
[2]. Therefore, the thermal flutter and stability problems have attracted some researcher’s attentions.
Boley [4, 5] was the first to include inertia effects in calculating the thermally induced response of a thin cantilever
beam subjected to suddenly incident heat flux. Boley [6], Seibert and Rice [7], Manolis and Beskos [8], analyzed the
beams’ thermally induced vibrations problem theoretically, but none of them included the highly nonlinear term of
heat transfer by radiation. Namburu and Tamma [9] used a generalized transform method based on the finite element
method in the dynamic analysis, and they studied the transient response of an Euler-Bernoulli beam under rapid
heating, but thermal analysis for solid beam structure is carried out by using brick elements in order to account for
temperature variation through thickness. Thornton and Kim [10] adopted an analytical approach to obtain the
thermally induced response of a rolled-up solar array of the HST by using simplified beam model. Both uncoupled and
coupled thermal-structural analyses were presented. Additionally, the stability criterion of the dynamic response for
the beam model is established. Xue and Ding proposed a Fourier-FEM method [11] for transient thermal analysis of
thin-walled tube structures with closed cross-section considering heat transfer by radiation. Both the thermal axial
forces due to the average temperature and the thermal bending moments due to the temperature gradient in the
cross-section can be obtained by the special tube elements.
In this paper, the thermally induced nonlinear vibration of practical thin-walled large-scale space structures (LSSS)
subjected to suddenly thermal loading is analyzed. For the coupled thermal-dynamic system of a given structure, there
⎯ 434 ⎯
are two variable parameters, namely the heat incident angle vectorν and damping ratios ζ . The stability property of
the system depends on these two parameters. So the influence of the two parameters is studied and the phenomenon of
the thermally-induced Hopf bifurcation is introduced. Finally the criterion of thermal flutter is investigated under the
frame of nonlinear vibration theories.
THE COUPLED TRANCIENT THERMAL ANALYSIS
Usually, the LSSS are composed of thin-walled beams. To conduct the thermally-induced vibration analysis of this
type of structures, it is necessary to calculate the temperature perturbations on the cross-section of beams so that it is
possible to consider transverse thermal bending moments. In this paper, special tube elements [11] are adopted for this
purpose, which are also adopted in Ref. [12].
According to Ref. [11, 12], the coupled heat conduction equation by FEM can be expressed as:
CT& (t ) + K T T (t ) = Q (t , a ) ≈ Q (t , 0) + Q′(t , 0)a = Q (t ) + Ba
(1)
where temperature vector T (t ) = ( (T 0 )T , (T 1 )T ) ; T 0 is the nodal average temperature vector corresponding to the
T
axial thermal expansion and is discretized along the axial direction of the beam. T 1 is corresponding to the transverse
thermal bending and contains two sets of amplitudes of perturbation temperatures at the nodes. C is the thermal
capacity matrix; K T is the heat conduction matrix, which contains the contribution of heat radiation. a is nodal
displacement vector, Q is the heat load vector, which is the function of time, t, and a due to considering the influence of
deformation on heat flux absorbed by the structure. Q (t ) is the heat load vector in spite of structure deformation, B is
the coupling matrix, which bridges the temperature field and the displacement field, see Ref. [12].
THE COUPLED THERMAL-STRUCTURAL DYNAMIC ANALYSIS
The dynamic FEM equation of structures is:
Ma&& + Da& + Ka = PT (t ) + Pf
(2)
where M is the mass matrix; D is the damping matrix, K is the stiffness matrix, PT(t) is the nodal thermal load vector
and Pf is the nodal external load vector which can be considered separately. With the obtained temperature field, the
effective nodal temperature load PT (t ) can be calculated easily. For linear elastic materials, the following relation
holds: PT =WT, where W is the matrix relating to the cross-section, material properties and element interpolating
functions of the tube. Using the modal superposition method a = Φ U and adopting Rayleigh damping [12], Eq. (2)
can be changed into:
U&& + DsU& + ΩsU = RT + R f
(3)
where Φ is the matrix of the first I order of modals, U is the vector of the generalized displacement,
, ωi is the ith natural frequency; and ζ i is
the ith damping ratio for simplifying ζ1=ζ2 =ζ . Here, WI is in I×3J dimension and T is in 3J×1 dimension. Therefore,
the coupled thermal-structural dynamic equations are obtained as:
CT& (t ) + KT T (t ) = Q (t ) + BΦ U = Q (t ) + B1U
(4)
U&& + DS U& + ΩS U = W I T + R f
(5)
The set of coupled thermal-dynamics FEM equations can be solved in a step-by-step manner in the domain of time. In
each time step Δt, T(t+Δt) is firstly solved with Eq. (4) from T(t) and U(t) at the end of last step by using the Wilson-θ
method and Newton-Raphson iteration, then U(t+Δt) is secondly solved with Eq. (5) from T (t+Δt) and U(t) by the
Newmark method. The details can be seen in Ref. [12].
THE STABILITY ANALYSIS
According to Eq. (4) and Eq. (5), the simultaneous equations of the coupled thermal-structural system can be expressed
as:
⎯ 435 ⎯
⎡C
⎢0
⎢
⎢⎣ 0
− B1
0 0 ⎤ ⎛ T ⎞ ⎡ KT
d⎜ ⎟
I 0 ⎥⎥ ⎜ U ⎟ + ⎢⎢ 0
dt
0 I ⎥⎦ ⎝⎜ U& ⎠⎟ ⎣⎢ −WI
0
ΩS
0 ⎤⎛T ⎞ ⎛ Q
⎜ ⎟ ⎜
− I ⎥⎥ ⎜ U ⎟ = ⎜ 0
DS ⎦⎥ ⎝⎜ U& ⎠⎟ ⎜⎝ R f
⎞
⎟
⎟
⎟
⎠
(6)
Where B1 and Q are the coefficient matrices relating to the heat incident angle vectorν ; DS is the damping
matrix relating to the damping ratioζ ; C, KT and Ωs are all coefficient matrix relating to the material parameters of the
structure. Therefore, for a given structure, there are only two varying parameters in this system, namely the heat
incident angle vectorν and the damping ratio ζ .
Eq. (6) is in 3J + 2I dimension(J is total number of nodes and I is the number of the orders of truncating modals). The
(
system’s state variables are defined as H = T T
UT
T
U& T ) , so the corresponding state space contains the average
temperature, the amplitudes of perturbation temperature, the generalized displacement and the generalized velocity.
Because KT contains radiation terms, Eq. (6) is nonlinear. This is a stability problem of a nonlinear vibration system.
(
Supposing Eq.(6) has a particular solution, which is denoted as H ps = TpsT , U Tps , U& Tps
⎡C
⎢0
⎢
⎢⎣ 0
0 0⎤
⎡ KT
I 0 ⎥⎥ H& ps + ⎢⎢ 0
⎢⎣ −WI
0 I ⎥⎦
− B1
0
ΩS
)
T
, and it satisfies:
⎛Q⎞
0⎤
⎜
⎟
− I ⎥⎥ H ps = ⎜ 0 ⎟
⎜ Rf ⎟
DS ⎥⎦
⎝ ⎠
(7)
This solution represents an unperturbed stable state (or steady state), which could be an equilibrium state or a periodic
movement. If the initial condition H (t0 ) diverges from H ps (t0 ) , the system will undergo a perturbed movement.
(
In order to analyze the stability of the system, a perturbation vector Z = H − H ps = X T , Y T , Y& T
)
T
is introduced
here. X , Y , Y& are the perturbations of temperature, displacement and velocity, respectively. The perturbation
equation can be obtained by subtracting Eq. (7) from Eq. (6):
⎡C
⎢0
⎢
⎢⎣ 0
0 0 ⎤ ⎛ X ⎞ ⎛ − K T (T ) T + K T (Tps ) Tps + B1Y ⎞
⎟
d⎜ ⎟ ⎜
I 0 ⎥⎥ ⎜ Y ⎟ = ⎜
Y&
⎟
dt ⎜ & ⎟ ⎜
⎟
&
0 I ⎥⎦ ⎝ Y ⎠ ⎜
WI X − ΩSY − DSY
⎟
⎝
⎠
(8)
Then the stability of Eq. (6) is equivalent to the stability analysis of Eq. (8) around the zero solution, which corresponds
to H = H ps . Eq. (8) can be rewritten as:
⎛ X ⎞ ⎡C
d⎜ ⎟ ⎢
Y = 0
dt ⎜⎜ & ⎟⎟ ⎢
⎝ Y ⎠ ⎣⎢ 0
−1
0 0 ⎤ ⎛ − K T (T ) T + K T (Tps ) Tps + B1Y ⎞
⎜
⎟
I 0 ⎥⎥ ⎜
Y&
⎟
⎜
⎟
0 I ⎦⎥ ⎜
WI X − ΩsY − DsY&
⎟
⎝
⎠
(9)
The system only has two parameters: ν and ζ , then a new vector F is defined as:
⎡C
F ( Z,ν , ζ ) = ⎢⎢ 0
⎣⎢ 0
−1
0 0 ⎤ ⎛ − K T (T ) T + K T (Tps ) Tps + B1Y ⎞
⎜
⎟
I 0 ⎥⎥ ⎜
Y&
⎟
⎜
⎟
0 I ⎥⎦ ⎜
WI X − ΩsY − DsY&
⎟
⎝
⎠
(10)
KT(T)T is a highly nonlinear item about T, so it is expanded as Taylor's series around steady temperature Tps :
K T (T )T = K T (Tps )Tps + J (Tps ) (T − Tps ) + o ( (T − Tps ) 2 )
= K T (Tps )Tps + J (Tps ) X + o ( X 2 )
⎯ 436 ⎯
(11)
where J (Tps ) =
d ⎡
K (T )T ⎤⎦ . Substituting Eq. (11) into Eq. (10), we get:
dT ⎣ T
T
ps
⎡C
F ( Z,ν , ζ ) = ⎢⎢ 0
⎢⎣ 0
2
0 ⎤ ⎛ X ⎞ ⎛ −o ( X ) ⎞
⎟
⎜ ⎟ ⎜
0 ⎟
I ⎥⎥ ⎜ Y ⎟ + ⎜
⎜
⎟
Ds ⎥⎦ ⎜⎝ Y& ⎟⎠ ⎜
0 ⎟
⎝
⎠
−1
0 0 ⎤ ⎡ − J (Tps ) B1
0
I 0 ⎥⎥ ⎢⎢ 0
0 I ⎥⎦ ⎢⎣ WI
-Ωs
(12)
According to Eq. (9-12), the system state equation is obtained finally as follows:
Z& = L (ν , ζ ) Z + o( Z 2 )
(13)
Because Eq. (9) contains singular point of hyperbolic type, according to the Hartman’s theory [13], the stability of Eq.
(13) can be discussed by taking only the linear part:
Z& = L (ν , ζ ) Z
(14)
The stability property is determined by the eigenvalues of matrix L. Among these eigenvalues, if there is at least one
eigenvalue with positive real part, the system is unstable; if the real parts of all eigenvalues are negative, the system is
overdamped; if there is a pair of conjugate eigenvalues with zero real parts and the real parts of the rest of eigenvalues
are negative, Hopf bifurcation [14] happens.
(
Since B1 is dependent on the incident vectorν = ν 1 ,ν 2 , 1 − (ν 12 +ν 22 )
) , and D
T
S
is dependent on damping ratio
ζ, the stability property of the system depends on only the combination of ν1, ν2 and damping ratios ζ.
THE NUMERICAL EXAMPLES
1. The analyses of the HST solar array Fig. 1 is the model of the HST solar array presented in reference [10]. The
solar heat flux lies in the x-z plane and has an incident angle φ with respect to the z axial (see Fig. 1). Therefore, the
three components of the heat incident angle vectorν areν 1 = − sin φ , ν 2 = 0, ν 3 = cos φ , respectively. In Ref. [10]
the spreader bar was treated as a rigid beam with mass Ms to transfer the forces between the booms and the solar
blanket was modeled as an inextensible membrane with density ρsb and zero rigidity, which has only transverse
deflection in z direction. The thermal analyses were only conducted on two booms. Each boom is subjected to an axial
compressive force of 14.75 N. Table 1 lists the material properties of the solar array, which is also shown in Ref. [10].
solar blanket
φ
right boom
z
z
A
spread bar
left boom
S0
R
P
x
deflection
A
L
h
y
A-A
Boom of the HST solar array
Figure 1: Model of the HST solar array
Table 1 The material properties of the HST (from Ref. [10])
αs
αΤ
σ
(1/K)
(W/m2•K4)
0.5 1.692×10-5
5.67×10-8
ε
ρb
c
k
Ε
ρ sb
(J/kg•K) (kg/m3) (W/m•K) (GPa) (kg/m2)
0.13
502
7010
16.61
193.0
1.589
Ms
(kg)
1.734
The comparisons of structural dynamic responses between the present and the analytical results in Ref. [10] are shown
in Fig. 2, where w is the boom deflection.
⎯ 437 ⎯
10
0
ζ = 0.0001
φ = 80°
Present
Ref.[10]
-3
1
ζ=0.0001
Unstable
Ref.[10]
Boom1
Boom2
Boom3
Boom4
0.1
1
w/R
-2
αΤ T B sinφcr / R
-1
-4
0.01
Stable
-5
1E-3
-6
-7
0
100
200
300
400
1E-4
1E-4
1E-3
0.01
t (s)
Figure 2: Comparison of the coupled analysis
for the unstable beam deflection
(Τ)
1/ω1τ1
0.1
1
10
Figure 3: Comparison of the stability boundary
for the HST solar array
It observes that the phases of the two results are almost the same, and the discrepancy is less than 2﹪ because of the
numerical error.
The stability property of the system is determined by the eigenvalues of matrix L, which is the function of damping
ratio ζ and incident angleφ. Among all conjugate eigenvalues of matrix L, supposing the one with the maximum real
part is λ (φ , ζ ) = α (φ , ζ ) ± i β (φ , ζ ) , the stability of the system is determined by the sign of α. Fixing the damping
ratio, series of eigenvalue analyses can be conducted by varying the incident angle φ. When α = 0 , the corresponding
incident angle can be regarded as the critical incident angle and denoted as φcr . In this way, if ζ = 0.0001 , the
critical incident angle of the HST solar array model is 2.76°, which is almost on the stability boundary plotted in Ref.
[10] (see Fig. 3). To verify the present method, the critical incident angles of other three beams differing on the
cross-section dimensions (in Table 2) are calculated and the corresponding stability states are compared with the
prediction of Ref. [10] (see Fig. 3).
Table 2 Different boom configurations for the HST solar array model and the corresponding results when ζ = 0.0001
*
Boom
R (mm)
h (mm)
τ 1(T ) (s)
ω1 (1/s)
τ 1(T )ω1
B
T 1 (K)
φcr ( o )
1
17.02
0.3379
58.47
0.1953×2π
71.75
3.087
16.59
16.37
2*
10.92
0.2350
23.76
0.0970×2π
14.48
4.018
9.955
2.760
3
9.525
0.1811
18.09
0.0671×2π
7.627
5.904
9.850
0.882
4
9.000
0.1500
16.07
0.0446×2π
4.503
11.85
10.60
0.241
The boom of the real HST solar array
Generally speaking, the discrepancies between the present solution and the solution in Ref. [10] are very small. This
confirms the validity of the methods proposed in this paper.
2. The thermal flutter analyses of satellite antenna A satellite antenna (see Fig. 4) is used to further illustrate the
capabilities of the methods proposed in this paper. The FEM model contains 748 elements and 382 nodes.
The dimension, element type and material of each component are listed in Table 3. Table 4 shows the mechanical and
thermal properties of each kind of material. In this table, E is the Young’s modulus; G is the shear modulus; ρ is the
material density; αT is the coefficient of thermal expansion; k is the thermal conductivity; c is the specific heat; αs is the
absorptivity and ε is the emissivity.
In order to express the heat incident angle vectorν , two parameters, φ1 and φ2 , are introduced as follows:
uuur
Supposing OP is the heat incident vector ν, P′ and P′′ are the projections of P in the X-Z plane and the X-Y plane,
uuuur
respectively (see Fig. 4). φ1 is the included angle between OP ′ and X axial, and φ2 is the included angle between
uuuur
OP ′′ and X axial. The three components of ν can be calculated by φ1 and φ2 as following:
⎯ 438 ⎯
View 1
Z
Z
S0
3.8m
P
Y
12.25m
Y
X
3.0m
ν
View 2
φ1
Z
A
P′
P ′′
φ2
Y
X
O
X
O
Figure 4: Model of the satellite antenna
Table 3 The element types, dimensions and materials of the components of the solar array
Component
Element type
Dimension (mm)
Material
Spar boom
Thin-walled
circular tube
Outside radius: 30
Thickness: 5
Carbon fiber
reinforced composite
Ring beam
Outside radius: 7.5
Thickness: 0.5
Thin-walled
circular tube
Frame
Vertical beam
Outside radius: 15
Thickness: 1
Thin-walled
circular tube
Reflective network
Carbon fiber
reinforced composite
Outside radius: 5
Thickness: 0.1
Stainless steel
Table 4 The material properties of the components of satellite antenna
E
G
(Gpa) (Gpa)
Carbon fiber
56.70 20.20
reinforced composite
Stainless steel
ν1 =
cos φ2
1 + tan 2 φ1 cos 2 φ2
190.0 73.08
, ν2 =
ρ
(kg/m3)
αT
k
(W/m•K)
c
(J/kg•K)
αs
ε
(1/K)
1600
1.4E-6
11.56
924.0
0.91
0.82
7930
16.1E-6
158.4
502.0
0.5
0.13
sin φ2
1 + tan 2 φ1 cos 2 φ2
, ν3 =
tan φ1 cos φ2
1 + tan 2 φ1 cos 2 φ2
(15)
Therefore, the stability of the system is dependent on the three parameters: φ1 , φ2 and ζ , whose influence on the
stability is discussed as follows.
1) The influence of damping ratio on the stability of the system To evaluate the influence of the damping ratio, the
incident angle φ1 is fixed as 20°, φ2 is fixed as 0° and only the damping ratio ζ is changeable. The maximum real
part of the eigenvalues of L is zero when damping ratio is 1.12×10-4 as shown in Fig. 5. Therefore, the critical
damping ratio is ζ 0 = 1.12 × 10-4 when φ1 =20° and φ2 =0°.
Supposing the initial condition is T0 = 290K and U& = U = 0 , and setting ζ = ζ 0 , φ1 = 20o , φ2 = 0o , the history of
displacement at node A (see Fig. 4) in Z direction shown in Fig. 6 is calculated by the present method and the
corresponding locus is plotted in Fig. 7. It observes that constant amplitude vibrations appear in the Fig. 6, and an
⎯ 439 ⎯
w(mm)
0
-5
4
α×10
-5
3
φ1=20°
φ2=0°
2
Response
Quasi-static response
-10
-5
ζ×10
1
-15
0
0
5
10
15
20
-1
-20
-2
t (s)
-25
-3
0
Figure 5: The maximum real part α of eigenvalues
of L versus the damping ratio ζ.
400
800
1200
1600
2000
Figure 6: The displacement in Z direction at node A
( ζ = ζ 0 , φ1 = 20o , φ2 = 0o )
Figure 7: The locus of node A in Z direction ( ζ = ζ 0 , φ1 = 20o , φ2 = 0o )
isolated close orbit exits in the phase plane from Fig. 7. As the time goes on, the path, which starts from a point outside
the close orbit, will approach this close orbit spirally. The close orbit is called limit loop and the thermally induced
Hopf bifurcation clearly occurs.
With the same initial condition and incident angle φ1 , φ2 , when ζ is less than ζ 0 , the system is unstable (see Fig. 8);
while if ζ > ζ 0 , the thermally-induced vibration will fade out (see Fig. 9).
0
w(mm)
-5
Response
Quasi-static response
-10
-15
-20
t (s)
-25
0
Figure 8: The displacement in Z direction
at node A ( ζ = 1 × 10
−5
200
400
600
800 1000 1200 1400 1600 1800 2000
Figure 9: The displacement in Z direction
at node A ( ζ = 1 × 10 −2 )
)
2) The influence of incident angle on the stability of the system In this example, ζ is fixed as 0.5 × 10−4 and φ1 is
fixed as 20o , only the other incident angle φ2 is variable. The maximum real part of eigenvalues of L is zero
when φ1 = −61.28o or φ1 = 61.50o as shown in Fig. 10. It means that both the two angles are critical incident angle
denoted as φcr1 and φcr 2 , respectively.
⎯ 440 ⎯
- 5
α (×10
2
)
w(mm)
0
The maximum real part
1
-5
ζ= 5×10
-5
0
-180 -150 -120 -90 -60 -30
φ 1= 0 °
Response
Quasi-static response φ2=-61.28°
φ2°
0
30
60
90 120 150 180
-10
-1
-15
-2
-20
-3
t (s)
0
-4
Figure 10: The maximum real part α of eigenvalues
of L versus the incident angle φ2
400
800
1200
1600
2000
Figure 11: The displacement in Z direction at node A
( ζ = 5 × 10 −5 , φ1 = 0o , φ2 = φcr1 )
Supposing the initial condition is T0 = 290 K and U& = U = 0 , and setting ζ = 5 × 10 −5 , φ1 = 0o , φ2 = φcr1 , the
history of displacement in Z direction at node A (see Fig. 4) is calculated and shown in Fig. 11. The corresponding
locus is plotted in Fig. 12. As expected, the thermally induced Hopf bifurcation is clearly observed. The same
phenomenon can be observed as well under the same initial conditions when φ1 = 0o and φ2 = φcr 2 .
Figure 12: The locus of node A in Z direction ( ζ = ζ 0 , φ1 = 0o , φ2 = φcr1 ).
3) The combined influence of damping ratio and incident angle The critical damping ratio is calculated for different
incident angles based on the presented method. The ζ − φ1 − φ2 parameters’ space is divided by a critical surface
into a stable zone and an unstable zone as shown in Fig. 13. Fig. 13 shows that a stable vibration will occur in a
structure with the damping and incident angular parameters located in the stable zone under arbitrary initial
condition, whereas, a thermal flutter will induced in a structure with the parameters located in the unstable zone.
Thermal flutter will never occur in the given satellite antenna with damping ratio more than 0.00012 in spite of
incident angle of sudden heat flux.
Stable
Critical surface
ζ
φ2
Unstable
φ1
Figure 13: The critical damping ratio ζ versus the incident angle φ1 , φ2 .
⎯ 441 ⎯
CONCLUSIONS
Based on the FEM, a computational method is developed in this paper to analyze the thermally-induced vibration of
complex LSSS. The perturbation temperature on the cross section of thin-walled beams as well as the coupling effect
of structural deformations on the incident normal heat flux is taken into account by the present method. Further more,
the criterion of thermal flutter is established under the frame of nonlinear vibration theories. All the examples illustrate
the capabilities of the proposed methods for the analysis of practical LSSS.
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