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Article

Parameterized Design and Dynamic Analysis of a Reusable Launch Vehicle Landing System with Semi-Active Control

1
Key Laboratory of Exploration Mechanism of the Deep Space Planet Surface, Ministry of Industry and Information Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 211100, China
2
Field Engineering College, Army Engineering University of PLA, Nanjing 210001, China
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(9), 1572; https://doi.org/10.3390/sym12091572
Submission received: 17 August 2020 / Revised: 15 September 2020 / Accepted: 17 September 2020 / Published: 22 September 2020
(This article belongs to the Special Issue Multibody Systems with Flexible Elements)

Abstract

:
Reusable launch vehicles (RLVs) are a solution for effective and economic transportation in future aerospace exploration. However, RLVs are limited to being used under simple landing conditions (small landing velocity and angle) due to their poor adaptability and the high rocket acceleration of current landing systems. In this paper, an adaptive RLV landing system with semi-active control is proposed. The proposed landing system can adjust the damping forces of primary strut dampers through semi-actively controlled currents in accordance with practical landing conditions. A landing dynamic model of the proposed landing system is built. According to the dynamic model, an light and effective RLV landing system is parametrically designed based on the response surface methodology. Dynamic simulations validate the proposed landing system under landing conditions including the highest rocket acceleration and the greatest damper compressions. The simulation results show that the proposed landing system with semi-active control has better landing performance than current landing systems that use passive liquid or liquid–honeycomb dampers. Additionally, the flexibility and friction of the structure are discussed in the simulations. Compared to rigid models, flexible models decrease rocket acceleration by 51% and 54% at the touch down moments under these two landing conditions, respectively. The friction increases rocket acceleration by less than 1%. However, both flexibility and friction have little influence on the distance between the rocket and ground, or the compression strokes of the dampers.

1. Introduction

As one of the most important technologies for aerospace exploration, advances in launch vehicles have greatly promoted aerospace developments [1]. Reusable launch vehicles (RLVs) can achieve fast and cheap launches by dividing the launch costs into several launch missions. Since the 1950s, many countries have focused on developing RLVs. From American X-series spacecraft to the Falcon-series rockets of SpaceX, RLVs have always been a hot topic in aerospace technology [2].
The landing system is a critical subsystem of RLVs, the malfunction of which can cause recycle failure [3]. The design of RLV landing systems is quite difficult, because it requires high reusability, effective impact absorption, reliability, and heat resistance [4]. The Delta Clipper proposed by the McDonnell-Douglas Corporation was the first RLV to use a vertical soft-landing system. The Delta Clipper was to be a single-stage-to-orbit vehicle that took off vertically and landed vertically [5]. However, the project was aborted due to lack of funding, and it has not been used in practical engineering. The New Shepard proposed by the Blue Origin Corporation completed a sub-orbital experiment and landed vertically via retractable landing legs. However, it was only just able to reach the Kármán Line (100 km a.s.l.) [6]. The Falcon-series rockets proposed by SpaceX were the first to realize the application of vertical landing in aerospace missions. The landing gears of the Falcon-series rockets use four sets of landing gears with liquid dampers that include primary struts, auxiliary struts, and locking mechanisms [7,8,9]. The Ariane Group, the French Space Agency, and the Deutsches Zentrum für Luft- und Raumfahrt have also proposed a new low-cost reusable rocket project called Callisto. Callisto uses four landing legs to absorb the impact energy, which is similar to the Falcon-series rocket [10,11]. However, it is still in the design stage. Yue [12,13,14,15] and Lei [16] proposed a vertical landing system with novel oleo–honeycomb dampers and conducted many landing experiments. The oleo–honeycomb dampers were able to improve the landing performance of the RLV under dangerous conditions. However, it was necessary to replace the aluminum honeycomb after every landing. In summary, the current landing systems employed by RLVs use passive liquid or liquid–honeycomb dampers to absorb impact energy. These two kinds of passive dampers have complex structures, as well as greater mass and rocket accelerations. Additionally, they are not able to adjust damping forces in order to meet practical landing conditions, which require a low landing velocity and angle, making the recycling of rockets more difficult. There is limited research on controllable landing systems in RLVs.
In this paper, an RLV landing system with semi-active control is proposed. Its dynamic landing model, control approach, and parameterized design are introduced. Furthermore, the proposed landing system is validated by means of multiple rigid bodies and multiple coupled flexible–rigid dynamic simulations. The influence of structural flexibility and friction during the RLV landings is also discussed in the dynamic simulations. Section 2 introduces the overall scheme, working principles, and control approach of the proposed RLV landing system. Section 3 introduces the landing dynamic model and parameterized design of the RLV landing system. Section 4 validates the proposed landing system and discusses the influence of structural flexibility and friction on the RLV landing performance with reference to the simulations.

2. Working Principles of the RLV Landing System

2.1. Overall Scheme of the RLV Landing System with MRF Dampers

Figure 1 shows the overall scheme of the RLV landing system with magnetorheological fluid (MRF) dampers, which consists of a rocket and four sets of landing gears. Each set of landing gear includes a primary strut (including a damper and deployment), an auxiliary strut, and a pad. The primary strut damper is full of MRF, whose profile is shown in Figure 2. The primary strut damper includes a master cylinder (the inner diameter is D), a piston (compose of a magnetic core and a group of coils, the length is L), a piston rod (the diameter is d), a nitrogen accumulator, and a gap between the piston and cylinder (the width is h). The magnetic properties of the primary strut materials are shown in Table 1. The MRF-132DG produced by the Lord Company is used as an example in this paper. Its main specifications are shown in Table 2.
There are N turns coils on the magnetic core of the piston. The length of the magnetic flux density lines along the coils is L1, the length of the magnetic flux density lines in the gap is h, and the length of the arched magnetic flux density lines out the cylinder is L2. Based on the Maxwell equation and Ampere circuit rule,
L 1 B 1 · d l + 2 h B 2 · d l + L 2 B 3 · d l = μ m N I
where B1 is the magnetic flux density in the magnetic core area, B2 is the magnetic flux density in the gap, B3 is the magnetic flux density out the cylinder, and μm is the magnetic constant. The magnetic flux density B is in T, the magnetic field intensity H is in A/m, the electric current I is in A. Due to the magnetic field intensity in the magnetic core area and gap is much larger than that out the cylinder, the magnetic field intensity in the gap is
H 2 = B 2 μ m = N I H 1 L 1 2 h
The relationship between yield stress and magnetic field intensity of MRF-132DG [17,18] is
τ = 2.717 × 10 5 × 0.32 1.5239 t a n h ( 6.33 × 10 3 · H 2 )
During the landing, the viscosity and plasticity of the MRF change quickly under the magnetic fields produced by energized coils. When the piston moves and pushes the MRF to flow through the gap between the cylinder and piston, the coils energize and produce magnetic fields. The MRF becomes semi-solid from the liquid in milliseconds, and its yield stress is controlled by different magnetic fields to absorb the impact energy. After landing, MRF will return to the liquid state without the magnetic fields [19].

2.2. Working Principles of the RLV Landing System

When the rocket approaches the recycle-platform, four sets of landing gears deploy simultaneously and prepare for the landing impact. After the sensors of pads touch the recycle-platform and the RLV entries landing state, four primary strut dampers absorb the impact energy by their compressions and extensions. Their damping forces of primary strut dampers consist of the controllable parts and uncontrollable parts. The uncontrollable parts are determined by the viscosity and velocity of the MRF, and the air-spring forces of accumulators. The controllable parts are related to the yield stress of MRF, which are controlled according to the acceleration, jerk, pitch angle, and roll angle of the rocket [20].

2.3. Control Approach of the RLV Landing System

Due to the landing process is quite short, which requires a fast and robust control approach. Fuzzy control is suitable for complex systems and can decrease the response time significantly. Furthermore, the nonlinear characteristics of fuzzy control can increase the system robustness [21,22,23,24].
During the landing, the acceleration, jerk, pitch angle, and roll angle of the rocket are set as inputs. These four inputs are from rocket sensors to the control system of damping forces. Meanwhile, the yield stresses of four primary struts controlled by currents are set as outputs. The currents can control the damping forces of every primary strut, respectively. These four outputs are from the control system of damping forces and act on four primary strut dampers.
The highest acceleration of the RLV should be smaller than 2 g to protect the precise electronic instruments. Its jerk is set as [−2amax, 2amax]. The pitch angle and roll angle are set as [–3, 3], due to landing angles of the current RLVs are from −3° to 3° [2]. The output yield stresses of MRF in four primary strut dampers are set as [0, 100% Maximum] to adapt to different landing conditions. Considering the control accuracy and efficiency, the acceleration a is divided into four equal fuzzy sets (Z, S, M, B). The jerk da is divided into two equal fuzzy sets (Z, B). Both the pitch angle alpha and roll angle beta are divided into three equal fuzzy sets (N, Z, P). The output yield stresses are divided into seven equal fuzzy sets (Z, S, SM, M, SB, MB, B). The membership affiliations between physical parameters and fuzzy sets for the inputs are shown in Figure 3. The membership affiliations between physical parameters and fuzzy sets for the outputs are shown in Figure 4.
The control principles are (a) While the acceleration is increasing and less than the setting value, controllable damping forces are small. (b) While the acceleration is increasing and larger than the setting value, controllable damping forces are zero. (c) While the acceleration is decreasing and larger than the setting value, controllable damping forces are zero. (d) While the acceleration is decreasing and less than the setting value, controllable damping forces are big. Moreover, the output damping forces are also determined by the pitch angle and roll angle of the rocket. Detailed fuzzy control rules for inputs and outputs are shown in the Appendix A.

3. Landing Dynamic Analysis and Parameterized Design of RLV Landing System

3.1. Landing Dynamic Analysis of the RLV Landing System

Based on the working principles and control approach of the proposed RLV landing system, its landing dynamic model is required to design the detailed structures. The RLV is composed of an elastic part and four non-elastic parts, as shown in Figure 5. The elastic part includes the rocket, four primary strut deployments, and cylinders of four primary strut dampers. The non-elastic parts include piston rods of four primary strut dampers, four auxiliary struts, and four pads [25]. Due to the RLV being symmetric, a quarter landing dynamic model of the RLV is built, as shown in Figure 6.
The coordinate system is at the center of the bottom surface of the rocket. The revolute joint between the primary strut and rocket is A (xA, yA). The sphere joint between the primary strut and auxiliary strut is B (xB, yB). The projection of the revolute joint between the auxiliary strut and rocket is C (xC, yC). The horizontal distance xC between the origin and C is R. The angle between the primary strut and ground is α. The angle between the auxiliary strut and ground is θ. The vertical distance yA between the origin and A is H1. The mass center of the elastic part is P1 (0, H1 + H2). Due to auxiliary struts occupy most mass of non-elastic parts, whose center can be simplified as the mass center of non-elastic parts. It is 0.5(xB + xC,yB + yC), which is shown as
P 2   ( 1 2 H 2 tan α tan θ + R , 1 2 ( 3 H 1 + H 2 + R 2 tan α + H 2 R tan α tan α tan θ ) )
The landing dynamic models of elastic parts are
{ m p 1 x · · p 1 = F p cos α F a cos θ m p 1 y · · p 1 = F p sin α + F a sin θ m p 1 g
where Fp is the damping force of the primary strut, Fa is the damping force of the auxiliary strut. The landing dynamic models of non-elastic parts are
{ m p 2 x · · p 2 = F p cos α + F a cos θ μ F n m p 2 y · · p 2 = F n F p sin α F a sin θ m p 2 g
where μ is the friction coefficient. Fn is the contact force between the pad and ground, shown as follows [26]
F n = { 0 ,   q > q 0 k ( q 0 q ) e c q · s t e p ( q , q 0 d , 1 , q 0 , 0 ) ,   q q 0
where q is the distance criterion of the impact function, q0 is the trigger distance of the impact function. k is the stiffness, e is the contact force exponent, c is the contact damping, and d is the penetration depth.
According to the cross-section diagram of the primary strut damper in Figure 2, the damping force of the primary strut damper Fp is
F p = F c + F u + F Ni = 3 π D 2 L τ 4 h + 3 η L [ π ( D 2 d 2 ) ] 2 4 π D h 3 v + 0.5 ρ K entry A p 4 A gap 2 v 2 + 0.5 ρ K exit A p 4 A gap 2 v 2 + A n P 0 ( V 0 V ) 1.1
where Fu is the uncontrollable damping force of the MRF damper. Fc is the controllable damping force of the MRF damper. FNi is the air-spring force caused by the accumulator [27,28,29]. Fentry is local resistance caused by the abrupt enlargement, and Fexit is local resistance caused by the abrupt contraction. ρ is the density of MRF. Kentry is the local resistance coefficient of the entry, and Kexit is the local resistance coefficient of the exit. v is the piston velocity. Ap is the piston area. Agap is the gap area between the master cylinder and piston, and An is the cross-section area of the master cylinder. P0 is the initial pressure of the accumulator. V0 is the initial volume of the accumulator, and V is the volume of the accumulator during the landing.

3.2. Parameterized Design of the RLV Landing System

According to the proposed landing dynamic model, H1, α, and θ determine the buffer effects of Fa and Fp, and the efficiency and performance of landing systems [30]. Hence, a parameterized design of the RLV landing system is proposed according to these three parameters to get an effective landing system. The lower and upper limits of these three parameters are given in Table 3. The rocket acceleration, compression strokes of dampers, and the distance between the rocket and ground are the most important indexes for the design of a landing system [31]. A large rocket acceleration will damage structures and instruments [32]. Large compressions of primary strut dampers will cause the rocket to incline or tip over. The distance between the rocket and the ground should be large enough for a safe landing [33]. The mass is also an important index for spacecraft, a lighter landing system means a lower launch cost. Hence, these four design targets of the landing system are selected as responses, as shown in Table 4. The parameterized design principle based on the response surface methodology (RSM) is shown as follows
{ Minimize { D } = Min { R 1 R 2 ( R P 1 R 3 ) R 4 4 } s . t . { 24 θ 30 42 α 53 1200 H 1 1800
where RP1 is the initial distance between the mass center of the rocket and the ground.
{D} is the desirability function, which shows the desirable ranges for each response Ri. The function combines these four responses in a non-dimensional way. Its design goal is the smallest rocket acceleration, compression stroke, the mass of a set of landing gear, and the largest distance between the rocket and ground.
The RSM builds an approximate model between the codes (design parameters) and responses (design targets) via function fitting. The RSM assumes every code is an n-dimensional vector xEn, which is the independent variable of its response function y. Their relationship is y = f(x). Based on lots of simulation data, an approximate function of the response y ˜ is obtained by the undetermined coefficient method. Considering the efficiency and accuracy, a quadratic function with cross terms is used, which is shown as follows
y ˜ = a 0 + j = 1 n a j x j + i = 1 n j = 1 n a i j x i x j
where a0 is the undetermined coefficient of the constant term, aj is the undetermined coefficient of the one-degree term, and aij is the undetermined coefficient of the quadratic term. y ˜ is close to y by keeping their sum of error squares smallest via the least square principle [36].
According to the ranges of the three parameters (factors) in Table 3, landing dynamic simulations are carried to get corresponding four design targets (responses) under different parameter combinations. Their results are the RSM sampling, as shown in the appendix. Based on the RSM sampling and fit function in Equation (10), accurately fitted functions between codes and responses are obtained by the undetermined coefficient method, as shown in Table 5. These code coefficients of functions show the influences of codes on responses [37]. The influences of three codes and their extended codes on R1 is C > B > A > BC > AB > A2 > C2 > AC > B2. The influences of three codes and their extended codes on R2 is B2 > A2 > A > AC > C2 > BC > C > B > AB. The influences of three codes and their extended codes on R3 is A > C > AC > C2 > B2 > A2> AB > BC > B. The influences of three codes and their extended codes on R4 is C > A > AC > A2, B2, C2 > BC > AB > B.
Based on these four functions, the predicted values versus actual values of four responses are shown in Figure 7, Figure 8, Figure 9 and Figure 10. The points above or below the line indicate that they are over or under prediction. The data points of plots are randomly scattered along the 45° oblique line, which suggests that these four functions are accurate. These fitted functions can provide powerful support for the following parameterized design.
Combining the design principle in Equation (10) and the fitted functions in Table 5, the final design result based on RSM is shown in Table 6.

4. Landing Dynamic Simulations

Based on the design parameters in Section 3, a dynamic model of the RLV with semi-active control is built in MSC Adams to validate the proposed landing system, as shown in Figure 11. The rocket diameter is 2250 mm. The entire RLV weighs 5200 kg. Its center of mass is located at (0, 6017 mm, 0). The coordinate system is at the center of the bottom surface of the rocket, as shown in Figure 6. The highest rocket acceleration and greatest damper compression conditions are selected as examples because they are two of the most important design parameters of landing gears. These two conditions are shown in Figure 12. Their motion parameters are shown in Table 7.
Furthermore, the influences of structural flexibility and friction on landing performance are discussed in dynamic simulations. The end centers of the primary strut deployments and damper cylinders are fixed in their modal analysis. Their 20 order models are calculated in MSC Patran to obtain flexible primary struts. The flexible structures are imported into MSC Adams to conduct multiple coupled flexible–rigid dynamic simulations. Structural flexibility and friction will influence rocket acceleration, energy absorption, and compression strokes [38]. Different combinations of rigid structures, flexible structures, and frictions are simulated to analyze the proposed RLV landing system more accurately.

4.1. Highest Rocket Acceleration Condition

Under the highest rocket acceleration condition, four sets of landing gears touch the ground at the same time. The accelerations and the distances between the rocket and the ground are shown in Figure 13 and Figure 14, respectively. L1 is taken as an example, whose damping forces and damper compression strokes are shown in Figure 15 and Figure 16, respectively.
Figure 13 shows that all rocket accelerations of these four situations possess the same tendency. At about 0.003 s, four pads touch the ground, and peaks appear vertically. Subsequently, the rocket accelerations decrease vertically and remain at about 4 m/s2. During the landing, the controllable damping forces Fc of four primary strut dampers belong to Z and S. Their uncontrollable damping forces Fu slowly decrease versus time due to compression velocity decrease. The highest rocket acceleration of the entire rigid model is 25.95 m/s2. The highest rocket acceleration of the entire rigid model with friction is 27.57 m/s2, which is the largest in these four situations. The highest rocket acceleration of the model with flexible primary struts is 12.73 m/s2, which is the smallest in these four situations. Additionally, structural flexibility causes fluctuations in acceleration and damping force. Adding friction to the flexible model, the highest rocket acceleration increases to 13.13 m/s2, and the fluctuations of the rocket acceleration and damping force also increase. The highest rocket accelerations for these two flexible situations decrease by about 51% at the touch down moment. At the same time, the damping force peaks of L1 decrease by about 5%, because the flexible structures absorb parts of the impact energy. However, after the instantaneous contact, rocket accelerations and damping forces of these four situations are close to each other.
The highest rocket acceleration of current landing systems with passive liquid dampers is 37.2 m/s2 under the highest acceleration landing condition [2,17]. Compared to this, the highest rocket acceleration of the proposed landing system with semi-active control decrease about 30.2%. By controlling the damping forces of the four primary strut dampers, the RLV has much lower rocket accelerations and impact forces, which can protect the structures and instruments better during rocket recycle. As shown in Figure 14 and Figure 16, the distance between the rocket and the ground and the compression strokes of L1 are close to each other in these four situations. In conclusion, friction has little influence on landing performance. However, structural flexibility has a strong influence on rocket acceleration and the damping forces of primary struts.

4.2. Greatest Damper Compressions Condition

Under the greatest damper compressions condition, L3 touches the ground first. Second, L2 and L4 touch the ground together. Finally, L1 touches the ground. In brief, it is a kind of 1–2–1 landing condition. The rocket accelerations and the distances between the rocket and the ground are shown in Figure 17 and Figure 18. Because L3 touches the ground first, L3 is taken as an example. The damping forces and compression strokes of L3 are shown in Figure 19 and Figure 20.
At about 0.005 s, L3 touches the ground, and rocket accelerations and damping forces increase vertically. At about 0.085 s, L2 and L4 touch the ground at the same time. The rocket accelerations increase vertically again. At about 0.167 s, L1 touches the ground, and the rocket accelerations increase vertically for a third time. From 0.167 s to 0.310 s, the controllable damping force Fc of L1 belongs to Z, and the controllable damping forces Fc of L2, L3, and L4 belong to S. Their uncontrollable damping forces Fu decrease versus time slowly due to the decrease of compression velocities. The air-spring forces FNi increase because damper compressions increase. Hence, their resultant forces remain basically stable. Additionally, the pitch angle gradually decreases to 0 due to the horizontal velocity and control of the damping forces. The rocket accelerations increase slightly during this time. After 0.310 s, four controllable damping forces Fc belong to Z together, and the rocket accelerations have a small vertical decrease.
Under the greatest damper compressions condition, the compression strokes of the proposed landing system with semi-active control are close to those of current landing systems. However, the highest rocket acceleration of current landing systems with passive liquid dampers is about 22.5 m/s2 [2]. The highest rocket acceleration of the proposed landing system with a rigid model is 5.37 m/s2, which is a decrease of about 76.1%. Additionally, the highest rocket accelerations also decrease by about 54% at the three touch down moments in these two flexible situations. At the same time, damping forces decrease by about 11% in these two flexible situations. Except for the three touch down moments, the rocket accelerations and damping forces of flexible situations are a little higher than those of rigid situations. Structural flexibility also causes an approximately 0.005 s delay, and fluctuations of rocket accelerations and damping forces. The compression strokes of primary strut dampers and distances between the rocket and ground are also close in these four situations.
In conclusion, these two typical landing conditions prove that the proposed landing system has better landing performance than current landing systems with passive liquid or liquid–honeycomb dampers. On one hand, structural flexibility decreases rocket acceleration and damping force. On the other hand, friction increases rocket acceleration and damping force a little. Both flexibility and friction have little influence on the compression strokes of the primary strut dampers and the distances between the rocket and the ground. Structural flexibility should be considered in the design of RLV landing systems.

5. Conclusions

A landing system for reusable launch vehicles with semi-active control was proposed in this paper. Its control approach and landing dynamic model were built. According to the dynamic model, an effective and light landing system was parametrically designed based on the response surface methodology. The parameterized design achieved the best-desired design targets under limited ranges of design parameters, guiding the design by fitted functions between design parameters and targets. The parameterized design provided a fast and high-efficiency approach to designing a landing system. Dynamic landing simulations validated the proposed landing system under landing conditions with the highest rocket acceleration and greatest damper compressions. The simulation results proved that the proposed landing system with semi-active control has better landing performance than currently available landing systems that use passive liquid or liquid–honeycomb dampers. Additionally, the simulation results show that structure flexibilities decrease rocket accelerations by about 50% at the touch down moments. At the same time, they also decrease the damping forces of the primary strut dampers by 5% and 11% at the touch down moments under two typical conditions. However, the friction has little influence on landing performance.

Author Contributions

Conceptualization, C.W. and J.C.; Methodology, C.W. and S.J.; Data curation, C.W. and H.C.; Software, C.W.; Validation, C.W.; Formal analysis, C.W.; Investigation, C.W. and J.C.; Resources, C.W. and J.C.; Writing—original draft preparation, C.W.; Writing—review and editing, H.C.; Supervision, J.C. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by [National Natural Science Foundation of China] grant number [51675264], [National Natural Science Foundation of China] grant number [52075242], [National Natural Science Foundation of China] grant number ]11902157], [Basic Research Program (Natural Science Foundation) of Jiangsu Province] grant number [BK20180417], and [Talents Startup Foundation of Nanjing University of Aeronautics and Astronautics] grant number [1011-YAH20114]. And The APC was funded by [National Natural Science Foundation of China] grant number [51675264].

Conflicts of Interest

The authors declare no conflict of interest.

Notation

The following symbols are used in this paper:
αAngle between the primary strut and ground
θAngle between the auxiliary strut and ground
μFriction coefficient
τMaximum yield stress of MRF
ρDensity of MRF
ηViscosity of MRF
a0undetermined coefficient of the constant term
ajthe undetermined coefficient of one-degree term
aijthe undetermined coefficient of the quadratic term.
cContact damping of the impact function
dPenetration depth of the impact function
eContact force exponent of the impact function
kStiffness of the impact function
qDistance function of the impact function
q0Trigger distance of the impact function
vPiston velocity
ApPiston area
AgapThe gap area between the master cylinder and piston
AnCross-section area of the master cylinder
DDiameter of piston
DDiameter of piston rod
FaForce of auxiliary strut acting at point C
FcControllable damping force of MRF damper
FpForce of primary strut acting at point A
FuUncontrollable damping force of MRF damper
FNiAir-spring force caused by the accumulator
KentryLocal resistance coefficient of the entry
KentryLocal resistance coefficient of the exit
H1Vertical distance between the origin and point A
H2Vertical distance between the mass center P1 of elastic parts and point A
LLength of coils
RHorizontal distance between the origin of rocket coordinate system and point C
R1Highest rocket acceleration
R2Greatest compression stroke
R3Distance between rocket and ground
R4Mass of a set of landing gear
RP1Initial distance between the mass center of the rocket and the ground
VVolume of accumulator
V0Initial volume of accumulator

Appendix A

Table A1. Fuzzy control rules for inputs and outputs [39].
Table A1. Fuzzy control rules for inputs and outputs [39].
InputsOutputs
AccelerationJerkalphabetaτ1τ2τ3τ4
BAllAllAllZZZZ
ZZZZBBBB
ZBZZMBMBMBMB
SZZZSBSBSBSB
SBZZMMMM
MZZZSMSMSMSM
MBZZSSSS
ZZPZSSBMBSB
ZBPZSMSBM
ZZNZMBSBSSB
ZBNZSBMSM
ZZZPSBSSBMB
ZBZPMSMSB
ZZZNSBMBSBS
ZBZNMSBMS
ZZNPMBSBSBMB
ZBNPSBMMSB
ZZPNSBMBMBSB
ZBPZMSBSBM
ZZNNSBMBMBSB
ZBNNSBSBMM
ZZPPSBSBMBMB
ZBPPMMSBSB
SZPZZMSBM
SBPZZSMMSM
SZNZSBMZM
SBNZMSMZSM
SZZPMZMSB
SBZPSMZSMM
SZZNMSBMZ
SBZNSMMSMZ
SZPPMMSBSB
SBPPSMSMMM
SZNNSBSBMM
SBNNMMSMSM
SZPNMSBSBM
SBPNSMMMSM
SZNPSBMMSB
SBNPMSMSMM
MZPZZSMMSM
MBPZZSSMS
MZNZSMZMSM
MBNZSZSMS
MZZNSMMSMZ
MBZNSSMSZ
MZZPSMZSMM
MBZNSSMSZ
MZNNMMSMSM
MBNNSMSMSS
MZNPMSMSMM
MBNPSMSSSM
MZPNSMMMSM
MBPNSSMSMS
MZPPSMSMMM
MBPPSSSMSM
Table A2. RSM sampling.
Table A2. RSM sampling.
RunFactor 1: A(θ/°)Factor 2: B(α/°)Factor 3: C(H1/mm)Response 1: R1 (amax/m/s2)Response 2: R2 (Strokes/mm)Response 3: R3 (Distance/mm)Response 4: R4 (Mass/kg)
12742180023.349579.57872240108.446
22442150017.791364.47431112.891.6878
33042150033.193113.3552113.5109.83
42742120020.514582.25041448.689.412
53053150030.023946.4953902.99979.913
62747.5150061.879221.4722112185.2949
72747.5150061.879221.4722112185.2949
82447.5180069.025518.54791028.887.5495
92447.5120021.664556.0716894.95485.2949
102747.5150061.879221.4722112185.2949
112753180029.546440.7394873.65282.5555
123047.5120044.333170.82671013.282.452
133047.5180036.578562.36151923.298.0229
142753120027.705343.8536602.33672.0231
152453150031.397395.6979375.02175.1617
162747.5150061.879221.4722112185.2949
172747.5150061.879221.4722112185.2949

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Figure 1. The overall scheme of the RLV landing system with MRF dampers.
Figure 1. The overall scheme of the RLV landing system with MRF dampers.
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Figure 2. The cross-section diagram of the primary strut damper.
Figure 2. The cross-section diagram of the primary strut damper.
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Figure 3. Membership Affiliations of inputs: (a) Membership affiliation of a; (b) Membership affiliation of da; (c) Membership affiliations of alpha and beta.
Figure 3. Membership Affiliations of inputs: (a) Membership affiliation of a; (b) Membership affiliation of da; (c) Membership affiliations of alpha and beta.
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Figure 4. Membership Affiliations of outputs.
Figure 4. Membership Affiliations of outputs.
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Figure 5. Elastic and non-elastic parts of the RLV.
Figure 5. Elastic and non-elastic parts of the RLV.
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Figure 6. Quarter landing dynamic model of the RLV landing system.
Figure 6. Quarter landing dynamic model of the RLV landing system.
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Figure 7. Predicted versus actual values of the highest rocket acceleration.
Figure 7. Predicted versus actual values of the highest rocket acceleration.
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Figure 8. Predicted versus actual values of damper greatest compression stroke.
Figure 8. Predicted versus actual values of damper greatest compression stroke.
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Figure 9. Predicted versus actual values of the distance between rocket and ground.
Figure 9. Predicted versus actual values of the distance between rocket and ground.
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Figure 10. Predicted versus actual values of mass.
Figure 10. Predicted versus actual values of mass.
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Figure 11. Top view of the RLV.
Figure 11. Top view of the RLV.
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Figure 12. Two typical landing conditions of the RLV.
Figure 12. Two typical landing conditions of the RLV.
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Figure 13. Rocket accelerations of the RLV under the highest acceleration landing condition.
Figure 13. Rocket accelerations of the RLV under the highest acceleration landing condition.
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Figure 14. Distance between rocket and ground under the highest acceleration landing condition.
Figure 14. Distance between rocket and ground under the highest acceleration landing condition.
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Figure 15. Damping forces of L1 of the RLV under the highest acceleration landing condition.
Figure 15. Damping forces of L1 of the RLV under the highest acceleration landing condition.
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Figure 16. Damper compression strokes of L1 of the RLV under the highest acceleration landing condition.
Figure 16. Damper compression strokes of L1 of the RLV under the highest acceleration landing condition.
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Figure 17. Accelerations of RLV under the greatest compressions landing condition.
Figure 17. Accelerations of RLV under the greatest compressions landing condition.
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Figure 18. Distance between rocket and ground under the greatest compressions landing condition.
Figure 18. Distance between rocket and ground under the greatest compressions landing condition.
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Figure 19. Force of primary strut 1 of RLV under the greatest compressions landing condition.
Figure 19. Force of primary strut 1 of RLV under the greatest compressions landing condition.
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Figure 20. Strokes of primary strut 1 of RLV under the greatest compressions landing condition.
Figure 20. Strokes of primary strut 1 of RLV under the greatest compressions landing condition.
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Table 1. The magnetic properties of primary strut materials.
Table 1. The magnetic properties of primary strut materials.
StructureMaterialRelative PermeabilityConductivity (S/m)Relative Permittivity
CylindersAluminum13.774 × 1071.000022202
Magnetic coreAISI10105006.452 × 1061
CoilsCopper15.998 × 1070.9999935542
Gas in the accumulatorNitrogen101
Table 2. Main specifications of MRF-132DG.
Table 2. Main specifications of MRF-132DG.
ParametersValue
Viscosity0.112 Pa s
Density2.95 g/cm3
Solid content by Weight20.98%
Maximum Yield Stress48 kPa
Operating Temperature−40∼+130 °C
Table 3. Lower and upper boundaries of parameters [34,35].
Table 3. Lower and upper boundaries of parameters [34,35].
CodesDesign ParametersLower LimitsUpper Limits
AThe angle between the auxiliary strut and ground (θ)24°30°
BAngle between the primary strut and ground (α)42°53°
CThe vertical distance between point A and C (H1)1200 mm1800 mm
Table 4. Design target parameters.
Table 4. Design target parameters.
ResponsesDesign TargetsGoal
R1Highest rocket acceleration (m/s2) Minimize
R2Greatest compression stroke (mm) Minimize
R3Distance between rocket and ground (mm) Maximum
R4Mass of a set of landing gear (kg)Minimize
Table 5. Fitted functions between codes and responses.
Table 5. Fitted functions between codes and responses.
ResponseFitted Functions
R1R1 = +61.879 + 0.531A + 2.978B + 5.535C − 4.194AB − 13.779AC − 0.248BC − 8.078A2 − 25.670B2 − 10.901C2
R2R2 = +21.472 + 7.281A − 14.109B − 6.472C − 24.521AB + 7.265AC − 0.111BC + 24.440A2 + 34.093B2 + 6.040C2
R3R3 = +1121.000 + 317.666A − 520.112B + 263.320C − 118.180AB + 194.039AC − 130.021BC − 35.514A2 + 40.594B2 + 129.553C2
R4R4 = +87.578 + 3.815A − 11.215B + 5.924 C − 3.3477AB + 3.329AC − 2.125BC
Table 6. Final design parameters and targets.
Table 6. Final design parameters and targets.
Parameters (Design Parameters)Responses (Design Targets)
A (θ)B (α)C (H1)Highest rocket accelerationGreatest compression strokeDistance between rocket and groundMass
29.58°52.42°1800 mm17.79 m/s247.31 mm1286.82 mm85.12 Kg
Table 7. Parameters of two typical landing conditions.
Table 7. Parameters of two typical landing conditions.
Landing ConditionVertical VelocityHorizontal VelocityPitch Angle
Highest acceleration−2 m/s1 m/s
Greatest compression−2 m/s1 m/s

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Wang, C.; Chen, J.; Jia, S.; Chen, H. Parameterized Design and Dynamic Analysis of a Reusable Launch Vehicle Landing System with Semi-Active Control. Symmetry 2020, 12, 1572. https://doi.org/10.3390/sym12091572

AMA Style

Wang C, Chen J, Jia S, Chen H. Parameterized Design and Dynamic Analysis of a Reusable Launch Vehicle Landing System with Semi-Active Control. Symmetry. 2020; 12(9):1572. https://doi.org/10.3390/sym12091572

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Wang, Chen, Jinbao Chen, Shan Jia, and Heng Chen. 2020. "Parameterized Design and Dynamic Analysis of a Reusable Launch Vehicle Landing System with Semi-Active Control" Symmetry 12, no. 9: 1572. https://doi.org/10.3390/sym12091572

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