Rocket-towed systems are commonly applied in specific aerospace engineering fields. In this work, we concentrate on the study of a rocket-towed net system (RTNS). Based on the lumped mass method, the multibody dynamic model of RTNS is established. The dynamic equations are derived by the Cartesian coordinate method and the condensational method is utilized to obtain the corresponding second order ordinary differential equations (ODEs). Considering the elastic hysteresis of woven fabrics, a tension model of mesh-belts is proposed. Through simulation in MATLAB, the numerical deploying process of RTNS is acquired. Furthermore, a prototype is designed and flight tests are conducted in a shooting range. Ballistic curves and four essential dynamic parameters are studied by using comparative analysis between simulation results and test data. The simulation acquires a good accuracy in describing average behaviors of the measured dynamic parameters with acceptable error rates in the main part of the flight and manages to catch the oscillations in the intense dynamic loading phase. Meanwhile, the model functions well as a theoretical guidance for experimental design and achieves in predicting essential engineering factors during the RTNS deploying process as an approximate engineering reference.
Rocket-towed systems [
When designing the engineering factors, a dynamic model of the deployment process is urgently needed to be developed. The dynamic process of RTNS mainly involves motions of the rocket, the net, and the connecting devices between parts. In exterior ballistics phase, the rocket is affected by engine thrust, aerodynamic force, gravity, and other factors. Meanwhile, its movement is also coupled with the flexible net and the connecting devices. Combining the rigid-body swing motion of the rocket and the flexible vibration of the net, it is appropriate to apply multibody dynamics theories in RTNS modeling.
Rocket-towed systems have been seldom studied before. Rocket-towed systems such as rocket-propelled remote rope erection devices [
Throughout the studying history of line/cable system dynamics, the continuum method and the multibody theory are two mainstream modeling approaches adopted in theoretical research and engineering applications. Although the continuum method is considered to be more accurate, the multibody theory represented by the lumped mass method, an emerging methodology in the research field of dynamic characteristics of flexible cable/net, is more compatible to be applied to complex multibody dynamics and computer programming. Many other scholars continued applying and developing lumped mass method in aerospace engineering [
Williams and Trivailo [
Buckham et al. [
In this paper, a lump-mass multibody model of RTNS is established by the Cartesian coordinate method in the case of a two-dimensional assumption. This model is involved in the elasticity of the wire rope, flexible mesh-belts, and retaining rope. Due to the complex operating conditions, we finish the calculation of the initial RTNS position when the net is arranged in its container before launching. By taking into account the elastic hysteresis of woven fabrics, we proposed a modified tension model of mesh-belts. Furthermore, a prototype of RTNS is designed and built. Flight tests are conducted in a shooting range. Ballistic curves and four essential dynamic parameters are studied by using comparative analysis between simulation results and test data. The simulation acquires a good accuracy in describing average behaviors of the measured parameters with acceptable error rates in the main part of the flight and manages to catch the oscillations in the intense dynamic loading phase.
As shown in Figure
Rocket-towed net system: 1, rocket; 2, wire rope; 3, transition section; 4, net; 5, supporting bar; 6, net container; 7, launching platform.
During the working process of RTNS, the launching platform and the net container are almost stationary. Therefore, as shown in Figure
Net, rocket, and connecting devices: 1, rocket; 2, wire rope; 3, front transition section; 4, supporting bars; 5, mesh belts; 6, back transition section; 7, retaining rope.
As shown in Figure
The launching-flying-landing deployment process of RTNS can be divided into four phases as follows. The rocket is launched from the platform and starts pulling the wire rope connecting behind it. The net begins being towed out of the container unit by unit until the rocket engine finishes working. RTNS continues flying forward with inertia until the retaining rope starts functioning. RTNS lands on the ground. Under the influence of the retaining rope and seven supporting bars, the net is deployed on the target territory in a fully stretched state.
Figure
RTNS in flight: 1, rocket; 2, wire rope; 3, supporting bars; 4, mesh belts; 5, transition section; 6, retaining rope; 7, net container; 8, fixing point.
The dynamic behaviors of RTNS can be divided into three simultaneous types: the longitudinal deploying motion, the transverse translational motion of the net, and its rotation around the center longitudinal axis. Obviously, the large-scale displacement caused by the longitudinal deployment motion is much larger than the transverse one. And effect resulting from the latter two motions on the longitudinal motion is very little. Without regard to transverse windage, we can also neglect the rotation and consider no transverse disturbance during the rocket ballistics phase due to the symmetry of RTNS. Furthermore, as a result of lacking rigid support along the longitudinal direction, the longitudinal extension is expected to be much more considerable than the transverse one when we discuss the deformation in all parts of the system.
Given the above that we have discussed, the dynamic features in longitudinal direction is the fundamental issue in this paper. Hereon, we consider the launching-flying-landing deployment process of RTNS as a two-dimensional subject and introduce the following assumptions as follows. Transverse translational motion is ignored. There is no aerodynamic force acting upon the system during its flight. The thrust eccentric effect on the rocket is neglected; the ballistic curve sticks to one same coordinate plane. Velocity and force states in the net are highly consistent transversely. Directions of the vectors are always parallel with the rocket trajectory plane.
Based on the assumptions stated above, it becomes viable to acquire a practical multibody model of RTNS. Firstly, the wire rope, the retaining rope, and the net are handled as an integral flexible body. A series of straight elastic bar segments associated end to end by undamped hinges are applied to substitute for the above body. The positions of transverse supporting bars coincide with the ends of segments. Then we divide the mass of each segment evenly into both ends of it and remove all undamped hinges out of this model.
At this point, the wire rope, the retaining rope, and the net have been modeled into a string of lumped masses linked by massless elastic force elements between them. These force elements describe the longitudinal elasticity of the flexible body. According to the longitudinal mechanical properties of the system, value of each force element which equals the tension between lumped masses is calculated based on the tensile moduli of different materials. Meanwhile, each force element reacts along with the direction of its corresponding bar segment. In particular, when dealing with force elements which belong to the net part, the cross-sectional area of the net is taken as the total areas of all four mesh-belts in its relevant position. Considering that the mesh-belt has almost no resist compression strength, the tension force is assumed to be zero when the mesh-belt is compressed shorter than its original length.
On the basis of RTNS configuration, one end of the retaining rope is connected to the net; the other end is mounted to the fixing device. Correspondingly, the last lumped mass of the model remains stationary to the ground during the whole working process. The first lumped mass in the string coincides with the joint of the wire rope and the rocket; as a result, a fixed-end constraint is added in the multibody model.
The rocket is regarded as a rigid body operating in a two-dimensional plane. Displacement of the rocket is described by its centroid coordinate and the angle between its axis and horizontal direction is chosen to reveal the rotation mechanism.
Up to present, RTNS has been built into a discrete mechanical model composed of a string of lumped masses and the rigid rocket. The dynamic characteristics of them are the crucial topic in analyzing the deployment process. Moreover, the rocket engine thrust, gravity, and tension forces between lumped masses are generalized forces imposed on the RTNS model.
To derive the dynamic equations, the Cartesian coordinate method is often applied in multibody dynamics [ Equations built by absolute coordinates possess a clarity of physical meanings. The values of mass matrix elements remain constant, which is vital to decouple equations during the course of computer simulation procedure. This method is more suitable for multibody systems within few kinematic constraints.
In our case, there is only one fixed-end constraint existing between the first lumped mass and the bottom center of the rocket. Therefore, we select the Cartesian coordinate method to investigate the model.
On the basis of a Cartesian coordinate system, the inertial reference frames of RTNS are set and shown in Figure
Inertial reference frames of RTNS.
The angle
To a kinetic multibody system containing
However, almost all kinds of engineering equipment, including RTNS, contain constraints.
Consequently, the
As is shown in (
Equation (
With the existence of
In the following part of this subsection, the absolute coordinate expression of (
According to (
Apart from gravity
The generalized force acting on each lumped mass is
Equation (
Equation (
Constraint relations between five generalized coordinates, including
Large-scale matrices in multibody dynamics are frequently rewritten into block matrix forms. Matrices in this form are more concise and clear, making it more feasible for researchers to disclose relations between independent and nonindependent coordinates. Therefore, the generalized coordinate matrix
By derivation of (
where subscript
Equation (
Therefore, the nonindependent accelerations (
Substituting (
By left-multiplying (
Up to now, (
In Section
The force elements proposed in the lumped mass multibody model reveal the tensile capacity of RTNS in the longitudinal direction. For instance, the original length of the
Based on the discussion in Section
where
Obviously, the one from the
In (
As is shown in Figure
Material test of the mesh belts: (a) material test system (MTS); (b) a sample under test.
First, we load the testing force up to 3000N and unload it down to 100N with a constant rate; then we repeat the loading-unloading cycle several times to finish the experiment. Figure
Stress-strain curve of the sample under three cycles; the maximum loads for three cycles are 3000N, 5000N, and 7000N.
In view of the experimental data, we find that the stress-strain relationship of the mesh belts is quite complicated and engineering strain is not suitable in our case. In this work, consideration is given to both compatibility of (
Similarly, since the retaining rope is composed of high strength fabric material, the tensile modulus is also modified by adding a coefficient
Gravity in the Cartesian coordinate system is expressed as follows:
Initial values of generalized speeds (
Net arrangement before launching: (a) 3D schematic representation; (b) 2D cross-section. Blue parts denote the supporting bars, red parts denote the net units, and black parts denote the container.
Study on the initial state is divided into two steps: first, the initial positions of all lumped masses in RTNS are acquired by solving out the static draped state; after that, initial constraints are presented in order to keep the system stationary before launching.
Since the net part is made up of the mesh belts and the supporting bars, we consider them as one integral object and carry out a force analysis of the net at the static draped state. As is shown in Figure
Force analysis of the net.
The net is separated into six units by seven supporting bars. Constructions and stress states in all units are identical before launching. So it is reasonable to solve the draped state of one certain unit and get access to all positions of the lumped masses by coordinate translations. Apart from the lumped masses coinciding with the supporting bars, the position coordinates of other lumped masses still satisfy (
Taking one single net unit as the research object, we make
Equation (
Similarly, the initial positions of all lumped masses in RTNS (shown in Figure
Initial position of RTNS.
During the computational simulation, initial constraints ought to be imposed on the lumped mass multibody model. On the basis of the actual deployment process, an initial constraint principle is defined as follows: every lumped mass is fixed by initial constraints and remains stationary until the x-component of its resultant force turns to a positive value. It is worth noting that the resultant force mentioned above does not include the inner constraint force caused by (
The initial constraints imposed on the supporting bars can be expressed as
After adding the initial constraints on the supporting bars, the static draped states of the wire rope, the net, and the retaining rope can be settled due to the static equilibrium equation ((
Based on the lumped mass multibody model proposed above, the dynamic deployment process of RTNS is manifested through self-programmed codes in MATLAB. Calculation parameters, such as the tensile moduli and the longitudinal linear density of the wire rope, the net, and the retaining rope, are determined in accordance with the actual engineering design and experimental data of material tests. The initial state is constructed by the method proposed in Section
Calculation parameters.
Parameter | Unit | Value | Parameter | Unit | Value |
---|---|---|---|---|---|
Rocket mass | kg | 4 | Rocket moment of inertia | Nm2 | 0.0832 |
Transition section linear density | kg/m | 1.112 | Rocket thrust | N | 1176 |
Propellant burning time | s | 1.8 | Rocket launching angle | ° | 15 |
Wire rope cross-sectional area | mm2 | 584.6 | Wire rope tensile modulus | GPa | 208 |
Wire rope linear density | kg/m | 0.507 | Net longitudinal linear density | kg/m | 3.168 |
Mesh belt longitudinal tensile modulus | GPa | 3.57 | Equivalent net longitudinal cross-sectional area | mm2 | 382.5 |
Retaining rope longitudinal linear density | kg/m | 0.48 | Retaining rope longitudinal tensile modulus | GPa | 3.57 |
Retaining rope cross-sectional area | mm2 | 420 | Supporting bar mass | kg | 1.21 |
As stated above in Section
Figure
Simulation results of the launching-flying-landing deployment process.
t=0s
t=0.4s
t=0.8s
t=1.2s
t=1.6s
t=2.0s
t=2.4s
t=2.51s
For the sake of validating the accuracy of the lumped mass multibody model, a prototype of RTNS is developed and successfully deployed in a flight test at a shooting range.
Figure
RTNS prototype: (a) its installation before launching; (b) rocket; (c) net.
The test parameters are made, consistent with the calculation parameters in Table
Layout of the test site: 1, fixing device; 2, net container; 3, rocket; 4, surveyor's pole; 5, high-speed camera.
Flight tests for the same prototype are conducted following the identical experiment layout and repeated for six times. All six runs of the RTNS prototype are recorded by a Phantom high-speed camera. Figure
RTNS test.
t=0s
t=0.4s
t=0.8s
t=1.2s
t=1.6s
t=2.0s
t=2.4s
t=2.51s
Ballistic curves of all six runs are indicated in Figure
Essential engineering factors of all runs.
Run #1 | Run #2 | Run #3 | Run #4 | Run #5 | Run #6 | |
---|---|---|---|---|---|---|
| 2.80s | 2.73s | 2.68s | 2.84s | 2.71s | 2.75s |
| 11.41m | 11.18m | 10.76m | 11.02m | 10.55m | 11.10m |
| 39.80m | 39.49m | 39.35m | 39.96m | 39.88m | 39.76m |
Ballistic curves.
However, an experimental fluctuation analysis is not negligible. Out flied tests of a complex device such as RTNS are usually intervened by external working condition, resulting in the fluctuation between the experiment data. As can be seen in Figure
The rockets operated in the flight tests are all expendable one-time products using solid propellant engines. Subtle flaws occur in the microstructure of the propellant after a relatively long period of storage in a seaside warehouse. Consequently, small changes in chemical properties of the propellant are inevitable, which leads to distinctions in burning rate and burning surface between different rockets during the preliminary stage of propellant combustion. Therefore, there are minor differences existing between the output thrusts of different rockets in the initial trajectory. As a result, ballistic curves present fluctuations during the first half stage.
Aerodynamic force is considered to be another element which causes the fluctuation in the ending period. After the engine stops working at t=1.8s, the rocket starts to descend towards the ground; aerodynamic force is no more ignorable and turns into the main disturbance source affecting the flight stability and attitude of the rocket. Consequently, a landing point 0.61m fluctuation occurs under different wind scale circumstances.
Several essential factors in the RTNS deployment process are discussed. Ballistic curve, pitching angle, centroid velocity of the rocket, and the wire rope tension are investigated through comparisons between numerical simulation results and flight test data.
In Figure
Comparison of ballistic curves between flight tests and simulation.
However, an eccentric reverse occurs at the terminal ballistic simulation, resulting in the -3.9m~-3.3m landing-point error range between the simulation and the tests. This abnormal phenomenon is mainly caused by deficiencies existing in the tension model ((
Moreover, the temporal synchronization of the model in describing the RTNS deploying process is evaluated and considered qualified. The average relative errors in real time flight distances (longitudinal X values at the same moment) between Run#1~#6 and simulation are 8.2%, 5.4%, 5.7%, 5.9%, 7.1%, and 12.7%, respectively.
The agreements on the attitude and velocity of the rocket between simulation and all six flight tests are studied based on a time correlation analysis. Time-variation curves of four essential dynamic parameters (including the resultant velocity of the rocket centroid, the longitudinal velocity of the rocker centroid, the horizontal velocity of the rocket centroid, and the rocket pitching angle) are shown in Figures
Correlation coefficients between resultant velocities of the rocket centroid in tests and simulation.
Run #1 | Run #2 | Run #3 | Run #4 | Run #5 | Run #6 | Simulation: Phases (1~3) | Simulation: Phase 4 | |
---|---|---|---|---|---|---|---|---|
Run #1 | 1.0000 | 0.8391 | 0.8293 | 0.8293 | 0.8464 | 0.8362 | 0.7051 | 0.2212 |
Run #2 | 0.8391 | 1.0000 | 0.8463 | 0.8254 | 0.8370 | 0.8696 | 0.6975 | 0.2187 |
Run #3 | 0.8293 | 0.8463 | 1.0000 | 0.8687 | 0.8848 | 0.8532 | 0.7415 | 0.1666 |
Run #4 | 0.8293 | 0.8254 | 0.8687 | 1.0000 | 0.8491 | 0.9010 | 0.7059 | 0.2360 |
Run #5 | 0.8464 | 0.8370 | 0.8848 | 0.8491 | 1.0000 | 0.8302 | 0.7002 | 0.2171 |
Run #6 | 0.8362 | 0.8696 | 0.8532 | 0.9010 | 0.8302 | 1.0000 | 0.7156 | 0.2554 |
Correlation coefficients between longitudinal velocities of the rocket centroid in tests and simulation.
Run #1 | Run #2 | Run #3 | Run #4 | Run #5 | Run #6 | Simulation: Phases (1~3) | Simulation: Phase 4 | |
---|---|---|---|---|---|---|---|---|
Run #1 | 1.0000 | 0.8428 | 0.8511 | 0.8496 | 0.8569 | 0.8384 | 0.7438 | 0.4409 |
Run #2 | 0.8428 | 1.0000 | 0.8516 | 0.8373 | 0.8481 | 0.8717 | 0.7170 | 0.4093 |
Run #3 | 0.8511 | 0.8516 | 1.0000 | 0.8554 | 0.8588 | 0.8519 | 0.7824 | 0.4996 |
Run #4 | 0.8476 | 0.8373 | 0.8554 | 1.0000 | 0.8601 | 0.8277 | 0.7410 | 0.4799 |
Run #5 | 0.8569 | 0.8481 | 0.8588 | 0.8601 | 1.0000 | 0.8409 | 0.7231 | 0.4440 |
Run #6 | 0.8384 | 0.8717 | 0.8519 | 0.8277 | 0.8409 | 1.0000 | 0.7283 | 0.4647 |
Correlation coefficients between horizontal velocities of the rocket centroid in tests and simulation.
Run #1 | Run #2 | Run #3 | Run #4 | Run #5 | Run #6 | Simulation: Phases (1~3) | Simulation: Phase 4 | |
---|---|---|---|---|---|---|---|---|
Run #1 | 1.0000 | 0.8938 | 0.8803 | 0.8904 | 0.8907 | 0.8932 | 0.7624 | -0.2266 |
Run #2 | 0.8938 | 1.0000 | 0.8785 | 0.8878 | 0.8898 | 0.8916 | 0.7609 | -0.2555 |
Run #3 | 0.8803 | 0.8785 | 1.0000 | 0.8730 | 0.8774 | 0.8764 | 0.7722 | -0.2279 |
Run #4 | 0.8904 | 0.8878 | 0.8730 | 1.0000 | 0.8881 | 0.8891 | 0.7542 | -0.2676 |
Run #5 | 0.8907 | 0.8898 | 0.8774 | 0.8881 | 1.0000 | 0.8911 | 0.7581 | -0.2191 |
Run #6 | 0.8932 | 0.8916 | 0.8764 | 0.8891 | 0.8911 | 1.0000 | 0.7577 | -0.2553 |
Correlation coefficients between rocket pitching angles in tests and simulation.
Run #1 | Run #2 | Run #3 | Run #4 | Run #5 | Run #6 | Simulation: Phases (1~2) | Simulation: Phases (3~4) | |
---|---|---|---|---|---|---|---|---|
Run #1 | 1.0000 | 0.8640 | 0.8418 | 0.8460 | 0.8392 | 0.8244 | 0.6096 | 0.2072 |
Run #2 | 0.8640 | 1.0000 | 0.8510 | 0.8515 | 0.8464 | 0.8432 | 0.6008 | 0.2364 |
Run #3 | 0.8418 | 0.8510 | 1.0000 | 0.8764 | 0.8736 | 0.8774 | 0.6160 | 0.1922 |
Run #4 | 0.8460 | 0.8515 | 0.8764 | 1.0000 | 0.8471 | 0.8395 | 0.6130 | 0.2353 |
Run #5 | 0.8392 | 0.8464 | 0.8736 | 0.8471 | 1.0000 | 0.8591 | 0.6024 | 0.2286 |
Run #6 | 0.8244 | 0.8432 | 0.8774 | 0.8395 | 0.8591 | 1.0000 | 0.6685 | 0.2523 |
Comparisons of resultant velocities of the rocket centroid between flight tests and simulation.
Run #1
Run #2
Run #3
Run #4
Run #5
Run #6
Comparisons of longitudinal velocities of the rocket centroid between flight tests and simulation.
Run #1
Run #2
Run #3
Run #4
Run #5
Run #6
Comparisons of horizontal velocities of the rocket centroid between flight tests and simulation.
Run #1
Run #2
Run #3
Run #4
Run #5
Run #6
Comparisons of rocket pitching angles between flight tests and simulation.
Run #1
Run #2
Run #3
Run #4
Run #5
Run #6
As can be observed in Figures
With several characteristic times (as
Comparisons of key engineering elements between flight tests and simulation.
Run #1 | Run #2 | Run #3 | Run #4 | Run #5 | Run #6 | Simulation | Error range | |
---|---|---|---|---|---|---|---|---|
| 0.212 | 0.214 | 0.197 | 0.210 | 0.208 | 0.218 | 0.194 | -0.024~-0.003 |
| 36.87 | 39.04 | 38.99 | 36.20 | 37.99 | 41.04 | 35.10 | -5.94~-1.1 |
| 34.22 | 36.17 | 35.55 | 33.78 | 36.07 | 38.80 | 29.66 | -9.14~-4.12 |
| 15.20 | 14.69 | 16.00 | 15.64 | 14.77 | 14.06 | 18.77 | +2.77~+4.71 |
Average values of the resultant velocity (m/s) in different phases.
Run #1 | Run #2 | Run #3 | Run #4 | Run #5 | Run #6 | Simulation | Error rate | |
---|---|---|---|---|---|---|---|---|
Phase 1 | 21.54 | 21.99 | 22.09 | 21.57 | 21.58 | 21.98 | 19.78 | -10.5%~-8.2% |
Phase 2 | 16.13 | 16.15 | 15.96 | 16.27 | 16.23 | 16.20 | 16.63 | +2.2%~+4.2% |
Phase 3 | 14.09 | 13.94 | 14.14 | 13.91 | 14.12 | 14.03 | 13.95 | -1.3%~+0.3% |
Average values of the longitudinal velocity (m/s) in different phases.
Run #1 | Run #2 | Run #3 | Run #4 | Run #5 | Run #6 | Simulation | Error rate | |
---|---|---|---|---|---|---|---|---|
Phase 1 | 19.35 | 19.86 | 19.90 | 19.44 | 19.47 | 19.86 | 17.03 | -14.4%~-12.0% |
Phase 2 | 15.80 | 15.81 | 15.55 | 15.89 | 15.79 | 15.84 | 15.91 | +0.1%~+2.3% |
Phase 3 | 10.57 | 10.35 | 10.50 | 10.34 | 10.56 | 10.48 | 9.81 | -8.6%~-6.4% |
Average values of the horizontal velocity (m/s) in different phases.
Run #1 | Run #2 | Run #3 | Run #4 | Run #5 | Run #6 | Simulation | Error rate | |
---|---|---|---|---|---|---|---|---|
Phase 1 | 8.78 | 8.73 | 8.81 | 8.63 | 8.80 | 8.72 | 9.73 | +10.4%~+12.8% |
Phase 2 | 2.83 | 2.86 | 2.85 | 2.93 | 2.90 | 2.87 | 3.24 | +10.6%~+14.4% |
Phase 3 | -8.80 | -8.87 | -8.88 | -8.84 | -8.84 | -8.78 | -8.66 | +1.4%~+2.5% |
Average values of the rocket pitching angle (°) in different phases.
Run #1 | Run #2 | Run #3 | Run #4 | Run #5 | Run #6 | Simulation | Error rate | |
---|---|---|---|---|---|---|---|---|
Phase 1 | 30.66 | 30.62 | 29.94 | 31.07 | 31.02 | 30.73 | 24.59 | -20.9%~-17.9% |
Phase 2 | 28.82 | 28.51 | 28.98 | 28.66 | 28.69 | 29.22 | 28.29 | -3.2%~-0.8% |
Phase 3 | 19.64 | 18.48 | 19.50 | 19.68 | 20.00 | 19.15 | 13.97 | -30.2%~-24.4% |
Phase 1 is characterized from
After observing the test photographs taken around
In Phase 1, it can be concluded that the simulation results acquire good accuracies in both predicting average behaviors and catching oscillations of the measured parameters. Compared with all six runs, the error rates of average values in four essential dynamic parameters are -10.5%~-8.2%, -14.4%~-12.0%, +10.4%~+12.8%, and -20.9%~-17.9% (seen in Tables
Wire rope tension in simulation.
Phase 2 is characterized from
Compared with test results in Phase 2, the model curves exhibit medium-amplitude oscillations around similar average values, which means that the model is still able to predict the overall kinetic energy of the system in high precision. The error rates of average values in four essential dynamic parameters are +2.2%~+4.2%, +0.1%~+2.3%, +10.6%~+14.4%, and -3.2%~-0.8% (seen in Tables
However, the model fails to describe the changing trends of the oscillations and the amplitudes of the oscillations are larger than test runs. These disagreements are also caused by deficiencies of the tension model mentioned above. Although the elastic hysteresis of woven fabrics in net belts is considered in this work by introducing a hysteresis coefficient when correcting the linear elastic model applied in previous research [
Phase 3 is characterized from
The model is simplified under several assumptions; aerodynamic force is ignored when establishing the model. Instead of a constantly damping swing-motion appearing in the flight test, the rocket in simulation reaches at a stable gliding state without any oscillation right after its extinction. The model curves can no more exhibit oscillations. However, it still functions well in predicting the overall kinetic energy of the system with acceptable deviation of average values in four essential dynamic parameters. The corresponding error ranges are -1.3%~+0.3%, -8.6%~-6.4%, +1.4%~+2.5%, and -30.2%~-24.4% (seen in Tables
Phase 4 is characterized from
Through the phase-by-phase oscillation analysis stated above, the agreement between simulation results of the model and flight tests is fully evaluated. Firstly, during the main part (0~2.54s as Phases 1~3) of RTNS movement, the simulation results acquire a good accuracy in describing average behaviors of the measured parameters with acceptable error rates, which indicates the capability of the model in predicting the overall kinetic energy of the system. Furthermore, the model performs well in catching the high-frequency and large-amplitude oscillations in the intense dynamic loading phase (0~0.5s as Phase 1). Combining the discussion on oscillation analysis with the ballistic curve comparison, it can be concluded that the model succeeds in predicting several key engineering elements including
Taken altogether, the multibody model established in this work functions well as a qualified theoretical guidance for experimental design and achieves the goals on predicting essential engineering factors during the RTNS deploying process as an approximate engineering reference. It appears to be a referential model with potential applications for the future modifying of RTNS. As an example, for future RTNS prototypes designed with different distributions of longitudinal linear density, the model is able to predict almost the exact time points when the sharp growths of the wire rope tension emerge in Phase 1 for each prototype. The numerical intense-loading times will be a valuable guidance for active control scheme on the rocket. Moreover, the numerical maximum wire rope tension is also an instruction for strength check of the supporting bars.
Furthermore, a time correlation analysis based on correlation coefficients between time-variation curves in Figures
In present study, a lumped mass multibody model of RTNS is established by the Cartesian coordinate method. After modifying the model by introducing the elastic hysteresis of woven fabrics and setting up the initial state, computer codes are self-programmed and numerical simulations are accomplished in MATLAB. Furthermore, we design a RTNS prototype and conduct six flight tests in a shooting range. With a good consistency of the essential engineering factors, the prototype manages to function well and meet the engineering aims in the deploying process. Inconsistent thrust curves between different rockets in the initial trajectory and aerodynamic force in the ending phase are two main factors causing the fluctuation between test data of six runs according to the experimental fluctuation analysis.
Comparison of ballistic curves is finished. The ballistic simulation matches the test ballistics well with small error ranges (all within 10%) in flight time, ballistic peak point, and shooting range.
In order to carry out a further investigation on the numerical performance of the multibody model, four essential dynamic parameters including the rocket pitching angle, the resultant velocity, the horizontal velocity, and the longitudinal velocity of the rocket centroid are studied by using comparative analysis between simulation results and test data. The RTNS operation is divided into four particular phases based upon different levels of agreements on the time-variation curves between simulation and tests. A phase-by-phase oscillation analysis is fulfilled while both accuracies in predicting average behaviors and catching oscillations of the measured parameters are taken into consideration. Meanwhile, with several characteristic times (
During the main part (0~2.54s as Phases 1~3) of the flight, the simulation results acquire a good accuracy in describing average behaviors of the measured parameters with acceptable error rates (within 15% mostly in Tables
Furthermore, a time correlation analysis based on correlation coefficients between simulation and test curves is conducted. Significant correlations are found between test data and simulation results, which proves that the model statistically matches the flight tests well.
However, there are few deficiencies existing in the multibody model which lead to the simulation errors. The model curves fail to exhibit oscillations in Phase 3 due to its neglection of aerodynamic force. The eccentric higher restoring force in the model that occurred at the contracting stage of net remains unsolved, which is the main cause for larger numerical amplitudes of the oscillations in Phase 2 and the full description distortion in Phase 4. These issues should be considered in future research to modify the multibody model.
The data used to support the findings of this study are included within the article.
The authors declare no conflicts of interest.
Methodology was done by Qiao Zhou and Feng Han; numerical validation was made by Qiao Zhou and Feng Han; flight test was made by Qiao Zhou and Fang Chen; data accuracy was done by Qiao Zhou and Fang Chen; writing—original draft preparation— was achieved by Qiao Zhou; writing—review and editing—was achieved by Qiao Zhou, Fang Chen, and Feng Han; project administration was done by Feng Han; funding acquisition was got by Feng Han and Fang Chen.
This research was funded by the National Natural Science Foundation of China, Grant number 3020020121137.