Abstract
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.
1. Introduction
Rocket-towed systems [1–5], using rockets as power source, belong to a certain type of equipment applied in aerospace engineering fields. After launching, the payload is towed by the rocket out of its container and delivered to the target territory. Engineers are capable of achieving remote control and rapid deployment with these systems. In this paper, we focus our energy on the study of a rocket-towed net system (RTNS) which is one of the systems as stated above. The payload part of the system is several units of flexible net which are arranged in the container beforehand.
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 [1, 2] and shaped charge array deploying devices [3] were the main research targets in previous research. Gu et al. [1] presented a dynamic model of line throwing rocket with flight motion based on Kane's method; the kinematics description of the system and the forces acting on the system were both taken into consideration. Lu et al. [2] also applied Kane's method in developing a lumped mass model of rescue rope which was divided into several discrete finite segments. Djerassi et al. [5] studied the deployment of a cable from two moving platforms, they proposed a model of masses, and the cable was regarded as a collection of links. Mankala and Agrawal [6] studied the dynamic behavior of a tether-net/gripper system under impact. The tether was modeled as a continuum and simulation was finished by using the ordinary differential equation (ODE) solver in MATLAB.
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 [7–14] and ocean engineering fields [15–22].
Williams and Trivailo [7, 8] studied the dynamics of a circling aircraft-towed cable system. A lumped mass model of the cable was built for the circularly towed configuration; practical towing solutions that achieve small motion of the towed body were obtained by using constrained numerical optimization. Meanwhile, they studied the transitional dynamics as the aircraft changes from straight flight to circular flight by assuming no tension in the cable. Williams lead his team [9–12] in investigating the dynamic deployment of aircraft-towed cable dropping systems. They proposed a zero-tension cable model by using lumped masses connected via rigid links and modified it by introducing the linear elastic tension hypothesis. Then aerodynamic force acting upon the discrete links was considered when simulating the flight attitude of the cable and optimal control schemes were presented through a parameterization study on essential control variables. Trivailo and Kojima [13] developed the comprehensive nonlinear models of following net capturing systems for debris removal. Dynamics of the net systems and their interaction with the capturing objects were studied. Sharf et al. [14] presented a concept for a net closing mechanism; they finished the design and testing of a debris containment system for use in a tether-net approach to space debris removal.
Buckham et al. [15–18] adopted a mass-spring model to reveal the dynamic behavior of the cable in a towed underwater vehicle system by taking into account the cable viscosity. Furthermore, they introduced the flexural rigidity of the cable into the previous proposed model following Galerkin Principle. Bending moment acting on the cable segment was transformed to concentrated force imposed on the discrete mass in order to increase the model accuracy in describing the attitude of the cable under low-tension situations. Takagi T et al. [19–22] developed a net-configuration and load-analysis system under ocean current effect based on a mass-spring model.
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.
2. Multibody Model
2.1. Deployment Process of RTNS
As shown in Figure 1, RTNS is a quite complex set of devices. It consists of a rocket, a net part, a launching platform (including a tractor and a net container), and connecting devices (including a wire rope and a retaining rope) between them.
During the working process of RTNS, the launching platform and the net container are almost stationary. Therefore, as shown in Figure 2, the major research targets of RTNS are the net, the rocket, and the connecting devices.
As shown in Figure 2, the flexible fabric net is made of six net units and two transition sections. Each unit contains 4 longitudinal mesh-belts and 24 transverse mesh-belts. Two transition sections refer to the front and back mesh-belts which connect the six units to the wire rope and the retaining rope. The mesh-belts are woven from nylon. In every longitudinal-transverse intersection, there is a screw which connects the belts in orthogonal directions. In order to keep the net stretching out and tight longitudinally, seven equidistant transverse supporting bars are set at the front, back, and middle of the net. The bars link six net units together. Two ends of the net are connected to the rocket and the fixing device by the wire rope and the retaining rope, respectively.
The launching-flying-landing deployment process of RTNS can be divided into four phases as follows.(1)The rocket is launched from the platform and starts pulling the wire rope connecting behind it.(2)The net begins being towed out of the container unit by unit until the rocket engine finishes working.(3)RTNS continues flying forward with inertia until the retaining rope starts functioning.(4)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 3 indicates a certain moment of RTNS in flight.
2.2. Two-Dimensional Assumptions
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.(1)Transverse translational motion is ignored.(2)There is no aerodynamic force acting upon the system during its flight.(3)The thrust eccentric effect on the rocket is neglected; the ballistic curve sticks to one same coordinate plane.(4)Velocity and force states in the net are highly consistent transversely. Directions of the vectors are always parallel with the rocket trajectory plane.
2.3. Multibody Model of RTNS
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 [23, 24]. Under the circumstance of this coordinate system, the absolute coordinates of the position and attitude are adopted to constitute the dynamic equations. Advantages of this approach can be summed up as follows.(1)Equations built by absolute coordinates possess a clarity of physical meanings.(2)The values of mass matrix elements remain constant, which is vital to decouple equations during the course of computer simulation procedure.(3)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 4. With x-axis and y-axis along the horizontal and vertical directions, respectively, we define the first supporting bar’s mapping point on the ground before launching as the origin of coordinates.
The angle between rocket axis and horizontal direction along with the absolute position coordinates of the rocket centroid and lumped masses is marked in Figure 4. When numbering the subscripts of coordinates, the lumped mass located at the bottom center of the rocket is specified as and then the subscripts of the following masses are made in turn starting from 1 to N. To sum up, the whole flexible part of the wire rope, the retaining rope, and the net is divided into N segments. The total number of generalized coordinates expressed in (1) is 2N+5:
3. Dynamic Equations
3.1. Derivation of Equations
To a kinetic multibody system containing n generalized coordinatesThe universal dynamic equation in a variational form can be expressed aswhere and are the generalized inertia force and generalized force of the generalized coordinate (denoted as ), respectively. Supposing the mutual independence between all n generalized coordinates, due to the fact that the variation of generalized displacement is arbitrary, (3) only holds under the circumstance that the coefficients of are identically equal to 0. We can obtain the dynamic equations composed of n formulae:
However, almost all kinds of engineering equipment, including RTNS, contain constraints. S mutual independent constraints existing in the system can be expressed as
Consequently, the n generalized coordinates are no longer completely independent and dynamic equations of the system could not be set up directly from (4). To cope with this situation, the Lagrange multiplier method is adopted in our study [25]. Following the skeleton of this methodology, constraint functions are converted to penalty functions and embedded into the variational principle. The variational form of (5) derived by variational calculation is
S Lagrange multipliers indicated by are put to use in establishing the penalty functions. Then we introduce the penalty functions into the universal dynamic equation ( (3)):
As is shown in (7), with an applicable group of , s coefficients of are equal to 0, keeping the rest n-s generalized coordinates mutually independent. In order to maintain (7) tenable, values of the other (n-s) coefficients are prescribed as 0:
Equation (8) is well known as the Lagrange equation of the first kind.
With the existence of n generalized coordinates and s Lagrange multipliers, there are (n+s) unknown quantities of the kinetic multibody system. Combining (5) and (8), the closed multibody dynamic equations are
In the following part of this subsection, the absolute coordinate expression of (9) in the Cartesian coordinate system shown in Figure 4 is fulfilled in preparation for subsequent computational simulation. As can be indicated from Section 2.3 above, only one fixed-end constraint exists in the RTNS multibody model; the constraint equations can be expressed as follows:where is the distance from the rocket centroid to bottom center. The Jacobi matrix of (10) is expressed as
According to (10), there are two Lagrange multipliers denoted as (). The multibody dynamic equations of RTNS arewhere is the mass matrix; is the generalized force matrix. It contains the moment of the resultant external force acting on the rocket and the resultant external forces imposed on lumped masses and the rocket:
Apart from gravity , each lumped mass is also affected by the force,which is signified as , of its adjacent force element. Specifically, is defined as the elastic force imposed from the force element on the dumped mass. Meanwhile, the rocket is boosted by its engine thrust . It is worth noting that the moment and the force imposed from the wire rope on the rocket, namely, the restriction caused by the fixed-end constraint as mentioned earlier, do not correlate with . For this reason, in (10), we have . The generalized force acting on the rocket is
The generalized force acting on each lumped mass is
Equation (16) works on all lumped masses with the exception of the first and last individuals, which refer to the two masses coinciding with the front end of wire rope and the back end of retaining rope, respectively. Both of them receive only one tension force in a single direction:
Equation (12) is a typical example of differential-algebraic equations (DAEs) which are frequently engaged in the research field of multibody dynamics. With both second order ODE and algebraic equation as its constituents, DAEs could not be solved directly. It is essential to transform (12) into a set of authentic second order ODEs. In the interest of this issue, we adopt the contracted method to eliminate the Lagrange multipliers and all nonindependent generalized coordinates [26].
Constraint relations between five generalized coordinates, including , , , , and , are given in (10). Taking this into account, we can choose any two coordinates of them as nonindependent coordinates and represent the two with other independent ones. In our case , referred to as the joint of the wire rope and the rocket, are selected.
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 and the Jacobi matrix are expressed by nonindependent and independent block matrices:where subscripts and denote the nonindependent and independent matrix elements, respectively, and
By derivation of (10), the velocity constraint equations and the acceleration constraint equations are derived and transformed in block matrix forms:
where subscript indicates the time partial derivatives; the term in (26) represents the Jacobian matrix of to . Here, we define as
Equation (26) indicates the relation between the independent accelerations and the nonindependent accelerations . Substituting (23) in (26), we have
Therefore, the nonindependent accelerations () are eliminated:where
Substituting (29) in the second order ODEs of (12), we havewhere and in block matrix forms are
By left-multiplying (32) by , the Lagrange multipliers are eliminated:
Up to now, (12) is converted into (35) from DAEs to second order ODEs. Equation (35), containing 2N+3 equations, is the contracted multibody dynamic equations of RTNS. It is noteworthy that among all the matrix elements in only the ones of the 3×3 block in the top left corner are nonzero. Thus is a large-scale sparse matrix. Meanwhile, is a diagonal matrix. As a result, the mass matrix of (35), namely, , possesses a comparatively low nonlinear degree, which is a significant property in computational programming and simulation.
3.2. Generalized Force
3.2.1. Rocket Thrust
In Section 2.2, we assume no thrust eccentric effect on the rocket. Therefore, the thrust operates along with the axis of the rocket. Through the time-history curve of the rocket thrust recorded in the engineering manual, we havewhere is the rocket thrust matrix and
3.2.2. Tension Force
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 bar segment is . After extension, the length turns intowhere and are the position vectors of the (i-1 and the lumped masses, respectively, when . Particularly, when , is the position vector of the rocket centroid.
Based on the discussion in Section 2.3, when . We assume that the tension and the strain exhibit a linear elastic response in all flexible parts of RTNS. According to the engineering strain theory, can be expressed as
where is the real-time length of the force element, is the origin length of the force element, is the tensile moduli, and is the cross-sectional area of all four mesh-belts in the segment. Both and contain three different values corresponding to the wire rope, the restrain rope, and the net, respectively. The elastic force vector imposed from the force element on the dumped mass is
Obviously, the one from the force element on the (i-1 dumped mass is due to Newton’s third law.
In (39), the flexible parts of RTNS, including the wire rope, the restrain rope, and the net, are considered to be linear elastic materials. However, since the mesh belts are the main components of the net, the woven fabric structure in the belts should not be overlooked. Ordinarily, the constitutive relations of woven fabric materials are quite different from those of other engineering materials [27–30]: on the one hand, the inner structure of the belts remains slack under natural conditions; as a result of this feature, there is a pretightening stage during the initial phase of deformation; on the other hand, the stress-strain curve varies in the process of continuous loading and unloading due to the elastic hysteresis of woven fabrics.
As is shown in Figure 5, an experiment study on mechanical performance of the mesh belts is accomplished under the support of a material test system (MTS).
(a)
(b)
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 6 shows the stress-strain curve of the sample under three cycles. It is clearly that the unloading-stage stress is much lower than the loading-stage one, which is a remarkable manifestation of elastic hysteresis phenomena. Energy loss occurs in every load-unload cycle.
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 (39) and elastic hysteresis of woven fabrics while modifying the tension model of mesh-belts. A hysteresis coefficient is proposed to introduce the elastic hysteresis into the linear elastic model; the tension force expressed in (39) can be modified aswhere is based on the average energy loss ratio in all load-unload cycles. The tensile modulus is modified to when the mesh belts are contracting.
Similarly, since the retaining rope is composed of high strength fabric material, the tensile modulus is also modified by adding a coefficient . In this work, ,where and denote the hysteresis coefficients of the net and the retaining rope, respectively.
3.2.3. Gravity
Gravity in the Cartesian coordinate system is expressed as follows:
3.3. Construction of the Initial State
Initial values of generalized speeds () and generalized coordinates () ought to be given while solving (12). Initial generalized speeds are 0 since the initial time () refers to the moment in which the engine starts. As for , constructing the initial state under the actual operating conditions needs to be conducted. As is shown in Figure 7, the net is arranged in its container and stays at a static draped state before launching. Transverse supporting bars in the net are hung on the slide rails on both sides of the container isometrically.
(a)
(b)
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.
3.3.1. Initial Positions
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 8, the supporting bars reach an equilibrium status by gravity, the friction forces, and the bracing forces. The mesh-belts are kept free-draped under gravity and the tension force.
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 (12).
Taking one single net unit as the research object, we make as the position coordinates of the mass in the unit. Two formulae representing static equilibrium conditions are introduced into (12):
Equation (43) is the static equilibrium equation of the net unit. It can be solved by the Newton method.
Similarly, the initial positions of all lumped masses in RTNS (shown in Figure 9) are acquired by applying the above approach.
3.3.2. Initial Constraint Principle
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 (10).
The initial constraints imposed on the supporting bars can be expressed aswhere is one part of in (1), representing the generalized coordinates of seven supporting bars.
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 ((43) in Section 3.3.1). So we do not need to set up initial constraints for other parts in RTNS. According to the initial constraint principle, when conducting integral operation of the multibody dynamic equations ((43) in Section 3.1), we check in every time step to decide whether to exert the initial constraint forces over the supporting bars.
4. Simulation
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 3.3. All calculation parameters applied in numerical examples are shown in Table 1.
As stated above in Section 3.1, the mass matrix of the dynamic equations possesses a low nonlinear degree. By taking into consideration both low nonlinear degree in (35) and relatively short duration of RTNS deployment process, one-step integral methods are able to ensure the precision of simulation results. In this work’s solution, the classical fourth-order Runge-Kutta method is applied to solve the ODEs presented in (35). A step size is chosen under the consideration of both precision and time cost.
Figure 10 shows the simulation result of the launching-flying-landing deployment process, where the green line denotes the rocket, the black line denotes the wire rope, the pink lines signify the front and back transition sections, the red line signifies six net units, and the yellow line denotes the retaining rope.
(a) t=0s
(b) t=0.4s
(c) t=0.8s
(d) t=1.2s
(e) t=1.6s
(f) t=2.0s
(g) t=2.4s
(h) t=2.51s
5. Flight Test
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 11 shows the RTNS prototype and its installation before launching.
(a)
(b)
(c)
The test parameters are made, consistent with the calculation parameters in Table 1, to ensure the comparability between the simulation and the flight test. The layout of the test site is shown in Figure 12.
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 13 shows the whole deployment process.
(a) t=0s
(b) t=0.4s
(c) t=0.8s
(d) t=1.2s
(e) t=1.6s
(f) t=2.0s
(g) t=2.4s
(h) t=2.51s
Ballistic curves of all six runs are indicated in Figure 14. Meanwhile, test results including flight time, ballistic peak point, and shooting range (denoted as , , and ) are listed in Table 2, which are three essential factors to assess how well the RTNS prototype meets its engineering aims. As can be seen in Figures 13 and 14, the rocket is launched at the top of the platform; the net is pulled out and successfully deployed to the target territory at a fully expanded state in every shot. The consistency of essential engineering factors is acceptable. Experimental fluctuations of flight time, ballistic peak point, and shooting range between all runs are 0.16s, 0.86m, and 0.61m, respectively. It can be concluded that the prototype functions well in meeting the engineering aims in the deploying process.
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 14, ballistic curves of different runs begin to diverge right after the launching and present fluctuations during the first half stage. In the second half period, the curves become steady gradually along with the rocket approaching the peak point and then differ again before the rocket landing.
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.
6. Discussion
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 15, ballistic curves of simulation and six flight tests are manifested. As can be seen, the ballistic simulation matches the test ballistics well. Based on three essential engineering factors shown in Table 2, the error ranges in flight time, ballistic peak point, and shooting range between simulation and Run#1~#6 are -0.02~+0.11s, -0.57m~+0.29m, and -3.9~-3.3m (-0.7%~+4.1%, -5.0%~+2.75%, and -9.8%~-8.4% relatively). The small error ranges indicate the valuable accuracy of our model in predicting the engineering factors of RTNS.
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 ((41)). Though the constitutive relations of the mesh belts and the retaining rope can be approximately simulated by the modified tension model, it is found that the simulation stress value at the low strain stage is exhibited remarkably higher than reality. Thereby, the simulation stiffness of the retaining rope becomes larger than its real value. The unusual large stiffness magnifies its buffer action on the rocket and causes the eccentric reverse in the numerical example.
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 16–19. Furthermore, Tables 3–6 reveal the statistical correlations between simulation results and fight tests. Correlation coefficients between the curves shown in Figures 16–19 are listed in these tables right after each figure correspondingly.
(a) Run #1
(b) Run #2
(c) Run #3
(d) Run #4
(e) Run #5
(f) Run #6
(a) Run #1
(b) Run #2
(c) Run #3
(d) Run #4
(e) Run #5
(f) Run #6
(a) Run #1
(b) Run #2
(c) Run #3
(d) Run #4
(e) Run #5
(f) Run #6
(a) Run #1
(b) Run #2
(c) Run #3
(d) Run #4
(e) Run #5
(f) Run #6
As can be observed in Figures 16–19, four parameters indicating dynamic characteristics of RTNS all undergo a constant oscillation course during the whole deploying process. Therefore, when assessing the performance of the model, the RTNS operation is divided into four particular phases based upon different levels of agreements on the time-variation curves between simulation and tests. Both accuracies in predicting average behaviors and catching oscillations of the measured parameters are taken into consideration phase by phase.
With several characteristic times (as , , , and marked in Figures 16–19) defined in four phases, connections between the oscillations in simulation and real test phenomena are studied. Moreover, listed out as statistical sustains, Table 7 shows key engineering elements emerging in flight tests and simulation to help in assessing the oscillation-catching accuracy of the model, and Tables 8–11 provide average values of the measured parameters in different phases to help in assessing the accuracy of the model in predicting average behaviors.
Phase 1 is characterized from to , where represents the ignition time when the system starts working and denotes the moment when first of the six net units is fully pulled out of the container. Compared with the follow-up phases, time-variation curves of four dynamic parameters in Phase 1 go through a high-frequency and large-amplitude oscillation course. The rocket experiences a constant intense disturbance imposed by the pulled-out ending (exit in front of the container) of the net part when flying forward and towing the net units. The stress value of the disturbance is proportional to the mass inertia of the stationary net ending which is being pulled out of the container. That is to say, sudden changes in longitudinal linear density of the pulled-out ending lead to the loading oscillations on the rocket. As can be inferred in Table 1 and Figure 3, the rocket successively pulls out the wire rope, the transition section, and the first net unit; therefore the longitudinal linear density of the pulled-out ending increases two times in Phase 1 (first from 0.507 kg/m up to 1.112 kg/m and then from 1.112 kg/m up to 3.168 kg/m). As a result, two sharp growths of the wipe rope tension loading on the rocket occur and cause two major large-amplitude oscillations of the curves just around the time shown in Figures 16–19. In Table 7, the moment fluctuates from 0.197s to 0.218s in six test runs.
After observing the test photographs taken around , it is found out that the rocket exactly starts to pull out the supporting bar of the first net unit at this moment, which leads to the second jump of the pulled-out ending’s linear density. All four measured parameters go through the biggest oscillation and valuable engineering elements such as the maximum wire rope tension, the maximum resultant velocity, the maximum longitudinal velocity and the maximum horizontal velocity of the rocket centroid (denoted as , and in Table 7) appear around . It is also worth mentioning that since the flying length of the net is relatively short in Phase 1 the waving behavior of flexible net belts is not remarkable enough to dominate the dynamic process. The rocket flies under an intense dynamic loading condition due to sudden changes in longitudinal linear density of the pulled-out ending, which is the reason why time-variation curves in Figures 16–19 differ little in Phase 1.
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 8–11). Meanwhile, simulation curves exhibit the similar high-frequency and large-amplitude oscillations and perform well in reproducing the overall changing trends and frequency of the oscillations. As can be seen in Figures 16–19, the model is able to predict the occurring time of the maximum wire rope tension () and three other key engineering elements (, , and ) as approximate engineering estimates. The error ranges are -0.024~-0.003s, -5.94~-1.1m/s, -9.14~-4.12m/s, and +2.77~+4.71m/s, respectively. Figure 20 indicates the wire rope tension curve in simulation. The changing frequency of tension value stays at an extremely high level. It can be inferred that the model manages to predict the two sharp growths of the wire rope tension emerging around in Phase 1. Maximum instantaneous value of the tension reaches over 10KN, which is about ten times the value of the rocket thrust. With such a high frequency and large amplitudes in its oscillations, the wire rope tension becomes a fundamental element for the flight stability since it determines the moment imposed on the rocket.
Phase 2 is characterized from to , where represents the moment in which the rocket stops working. The net is pulled out of the container unit by unit and flies at a waving status. Longitudinal linear density of the pulled-out ending no longer jumps due to the more even density of the net. Consequently, due to the waving behavior of flexible net belts, the constant stress waves spread through the net and the wire rope and become the second major disturbance on the attitude and velocity of the rocket except for its own thrust. Since the stress waves are much slighter than the strong impact process in Phase 1, the amplitudes of the oscillations in Figures 16–19 all undergo a big drop and the dynamic parameters fluctuate around steady average values in Phase 2.
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 8–11).
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 [23], there is still room for improvement of the modified tension model especially in characterizing the stress in net belts during the unloading cycle. As can be inferred in Figure 6, after three loading-unloading cycles, when the net belt contracts below 1/3 of the maximum strain in the third cycle, the corresponding stress falls down to 1/10 of the maximum stress. However, the modified tension model still remains a relatively higher stress under the unloading circumstance. The eccentric higher restoring force of the model that occurred in the contracting stage of net belts is the main factor causing the larger amplitudes of the oscillations in simulation.
Phase 3 is characterized from to , where represents the time when the retaining rope starts functioning as a buffer. The rocket stops working and the whole system enters into a free flight phase during this period. Since the rocket thrust is no longer imposed as a strong driving force, the oscillations of the dynamic parameters begin to decay. The aerodynamic force is no more ignorable and becomes the major factor in maintaining the constant waving action of the system. Accordingly, the amplitudes of the oscillations in all test phenomena continue to decrease.
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 8–11).
Phase 4 is characterized from to the end of the flight when the rocket lands on the ground. As stated above, the unusual large stiffness of the tension model in this work magnifies the buffer action of the retaining rope and causes the eccentric ballistic reverse in the numerical example. So a fully description distortion of the model appears in this phase. Comparisons between flight tests and simulation in Phase 4 are no more necessary and are not included in Tables 8–11.
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 , , , ,, , and .
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 16–19 is fulfilled to assess the agreements between simulation and flight tests from a statistical point of view. Firstly, the first six columns in Tables 3–6 indicate the correlations between experimental data of different runs. Almost all correlation coefficients between measured values of the same parameter in six runs stay at the scope of 0.8~0.9, which proves a high consistency in test phenomena. Secondly, divergences between the correlation coefficients in the last two columns also separate the ending phase of the ballistic simulation, which has a clear disagreement with the tests due to deficiencies existing in the tension model, from Phase 1 to 3 of RTNS movement. Comparing simulation results with six runs in Phases 1~3, correlation coefficients between resultant velocities of the rocket centroid, longitudinal velocities of the rocker centroid, horizontal velocities of the rocket centroid, and rocket pitching angles fluctuate at 0.6975~0.7415, 0.7170~0.7824, 0.7542~0.7722, and 0.6008~0.6160, respectively. Therefore, significant correlations are found between test data and simulation results, which proves that the model statistically matches the flight tests well.
7. Conclusions and Future Work
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 (, , , and ) defined in four phases, connections between the oscillations in simulation and real test phenomena are studied.
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 8–11), which indicates the capability of the model in predicting the overall kinetic energy of the system. Furthermore, the model achieves the goals on catching the high-frequency and large-amplitude oscillations in the intense dynamic loading phase (0~0.5s as Phase 1). Several key engineering elements (including the maximum resultant velocity, the maximum longitudinal velocity and the maximum horizontal velocity of the rocket centroid, the maximum wire rope tension, and its occurring time) emerging in this phase are successfully predicted by the simulation results with small fluctuations between the tests (shown in Table 7), which signifies that the multibody model proposed in this work functions well as a qualified theoretical guidance for experimental design and achieves in predicting essential engineering factors during the RTNS deploying process as an approximate engineering reference. The multibody model appears to be a referential tool with potential applications for the future modifying of RTNS.
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.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare no conflicts of interest.
Authors’ Contributions
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.
Acknowledgments
This research was funded by the National Natural Science Foundation of China, Grant number 3020020121137.