An efficient and modular modeling for launch dynamics of tubed rockets on a moving launcher

Qin-bo Zhou, Xiao-ting Rui, Guo-ping Wang, Jian-shu Zhang

Institute of Launch Dynamics, Nanjing University of Science and Technology, Nanjing, Jiangsu Province, 210094, PR China

Keywords:Launch dynamics Marching fire Transfer matrix method for multibody systems Multiple launch rocket system

ABSTRACT This paper develops a modular modeling and efficient formulation of launch dynamics with marching fire (LDMF) using a mixed formulation of the transfer matrix method for multibody systems (MSTMM)and Newton-Euler formulation.Taking a ground-borne multiple launch rocket systems(MLRS),the focus is on the launching subsystem comprising the rocket, flexible tube, and tube tail. The launching subsystem is treated as a coupled rigid-flexible multibody system,where the rocket and tube tail are treated as rigid bodies while the flexible tube as a beam with large motion. Firstly,the tube and tube tail can be elegantly handled by the MSTMM,a computationally efficient order-N formulation.Then,the equation of motion of the in-bore rocket with relative kinematics w.r.t. the tube using the Newton-Euler method is derived. Finally, the rocket, tube, and tube tail dynamics are coupled, yielding the equation of motion of the launching subsystem that can be regarded as a building block and further integrated with other subsystems.The deduced dynamics equation of the launching subsystem is not limited to ground-borne MLRS but also fits for tanks, self-propelled artilleries, and other air-borne and naval-borne weapons undergoing large motion. Numerical simulation results of LDMF are given and partially verified by the experiment.

1. Introduction

Launch dynamics has been developed continuously since the 1960s and is mainly dedicated to launcher and projectile/rocket motion as well as subjected forces during the acceleration phase.Attention has been paid on the in-bore motion of a projectile[1-4]and then gradually transitted onto a complex multibody system composed of a projectile and other main parts of the launcher[5-14].Meanwhile,the focus is drawn on from stock-still fire[5-8]to marching fire [9-14]. Therein, the dynamics approach used covers the Newton-Euler method, Lagrangian method, Kane method, transfer matrix method for multibody systems (MSTMM),and the use of commercial software,e.g.,ANSYS,ABAQUS,ADAMS,RecurDyn, Amesim, etc. For stock-still fire, the launcher does not experience large motion;thus,can be elegantly handled by theory of linear structural mechanics or linear multibody system theory.However,only theory of the general multibody system can be used for marching fire,which results in a totally different formulation.

Among many of the dynamics methods,the MSTMM[15]shows its superiority for its features that it always avoids establishing the global dynamics equation and keeps low order of matrices involved independently from system degree of freedom (DOF), thus results in fast computational speed.In the MSTMM,a multibody system is broken up into elements, including bodies and hinges. On the element level,each element's dynamics properties are summarized in a transfer matrix that relates the state vector of its input end with that of its output end. Then, on the system level, these element transfer matrices are considered as building blocks and are assembled by simple matrix operations into the system's overall transfer equation that can be solved for system dynamics.The first version of MSTMM [16] is dedicated to linear systems. For rigidflexible coupling where beam-like elements are involved,MSTMM can even deliver an exact solution in the frequency domain, thus has been successfully applied to launch dynamics of linear multibody systems with stock-still fire [5]. To consider general multibody system dynamics, recently, Rui et al. [17] present a new version of MSTMM by switching the kinematic variables summarized in the state vectors from a position level to acceleration level. This results in different element transfer matrices and algorithms while maintaining the inherent merits of MSTMM.Further,it is shown that this acceleration-based version is an order-N strategy which is comparable to the recursive method.This acceleration-based version can be applied to launch dynamics of marching fire(LDMF).Thus,in the following we refer MSTMM to this new version.

Weapons like rocket launchers request high maneuverability.Notably, with the development of artillery reconnaissance technology, the multiple launch rocket system (MLRS) has a high potential to be exposed during the launching process.Marching fire is a significant routine to improve mobility, where the involved launch dynamics is the key complexity. However, the traditional ground-borne MLRS is not capable of marching fire. Under this background, the concept of marching fire of a MLRS is proposed by some scholars. Also, air-borne and naval-borne MLRS, e.g.,helicopter-borne rockets, anti-aircraft missiles, anti-submarine deep bombs, jamming rockets, etc., are typical examples of MLRS where marching fire is involved. However, there are three significant problems involved in state of the art:most reported models for launch dynamics with marching fire (LDMF) are (i) either oversimplified, which can not reflect the real situation; (ii) or too complicated, which use finite element method and is computational costly;furthermore,(iii)research on launch dynamics of the MLRS is mainly devoted to stock-still fire and few on marching fire.Feng et al.[11]study the LDMF of rocket system mounted on an allterrain vehicle considering the vehicle-launcher-road coupling using ADAMS.Dziopa et al.[10]establish the equation of motion of a naval-borne anti-aircraft missile system considered as a rigid multibody system composed of the launcher,missile,and gyro with 11 DOFs,where the large overall motion of the launcher is involved.However, their dynamics model does not consider the flexiblity of the launcher. Meanwhile, the missile has only two DOFs w.r.t. the launcher,i.e.,longitudinal translation and spin.Thus,the transverse movement of the missile w.r.t. to the launcher can not be considered,which has a great contribution to the initial disturbance of the missile. A notable work is done by Cochran et al. [9] where the equation for launch dynamics of a flexible rocket launcher using the Newton-Euler method is established without the restriction of the motion of the laucher. However, since the classical dynamics method leads to a global equation of motion with large dimensions,a simulation of a launch process of 0.25s in the physical world leads to a CPU time of 15 min, which is not acceptable for optimization and design purpose.

The development of an efficient formulation for LDMF of a MLRS is not a trivial task. It requires deep knowledge of mechanics,mathematics,numerics,and the mechanical structure of the MLRS.To tackle the abovementioned three challenges,this paper develops an efficient,modular,and novel formulation by combining MSTMM with the Newton-Euler method for launch dynamics of a rocket/projectile from a moving base.A ground-borne MLRS is taken as an demonstrating example in this paper, where its engineering prototype is shown in Fig. 1 (a), and the cooresponding dynamics model is shown in Fig. 1 (b). The wheels are treated as lumped masses with 3 DOFs. Vehicle chassis, azimuth and elevation platforms are treated as rigid bodies with spatial motion. Only the launching tube and the cooresponding in-bore rocket are modeled seperately while other tubes and rockets are integrated into the elevation platform. The launching tube is devided into threes segments by the connections with the elevation platform.The tube tail numbered 1 is treated as a rigid body with spatial motion,while the other two segments are combined as a single element numbered 2 and treated as an Euler-Bernoulli beam with large overall motion.Elements 1 and 2 are rigidly welded together. The firing in-bore rocket numbered 3 is treated as a rigid body with spatial motion.The connection and contact between the wheel and the ground,and those between the wheel and chassis are modeled as springand-damper-like elastic hinges (or alternatively called force elements). The connection between the chassis and the azimuth platform, and that between the azimuth and elevation platforms are modeled as revolute hinges. The connections between the launching tube and the elevation platform are modeled as elastic hinges.Due to the use of elastic hinges,equations of motion of the bodies connected by the elastic hinge can be decoupled by treating the reaction forces caused by the elastic hinges as“known external forces” since the system positions and velocities are known. Then,the whole system can be decoupled on the acceleration level into three subsystems:the marching subsystem comprising six wheels;the orientating subsystem consisting of the chassis, the azimuth,and elevation platforms; and the launching subsystem comprising the tube tail, the flexible tube, and the firing in-bore rocket.

Typically, there are two crucial common features involved in LDMF for various weapons such as tanks,air-borne and naval-borne rocket launchers:(i)large overall motion of the launching base,and(ii)highly coupled dynamics between the rocket(or projectile)and the tube due to contact,constraints and impact.Notably,both two features are reflected in the dynamics of the launching subsystem.Also, the launching subsystem is a rather general topology for different weapons where the launch is involved.For instance,for a tank[12]or a self-proprelled artillery[6],elements 1 thru 3 in Fig.1(b) stand for gun breech, gun barrel, and projectile, respectively.Therefore,in the following,the focus is only paid on the dynamics of the launching subsystem. For dynamics of the orientating and marching subsystems, see Ref. [13]. The rest of the paper is organized as follows. First, in section 2, on the element level, the transfer equation of beam 2 is briefly deduced to describe the basic concept and formalism of the MSTMM.Next,in the main part of the paper in Section 3,on element level,the dynamics equation of the firing in-bore rocket 3 treated as a rigid body is deduced, where it distincts from the classical rigid body dynamics for the relative kinematic description w.r.t. the beam 2. Then, in the main part of the paper in Section 4, the dynamics of elements 1, 2 and 3 are assembled yielding the coupled equation of motion of the launching subsystem on system level. Finally, in Section 5, numerical simulation for the LDMF of the MLRS is given. Part of the simulation results are validated by experiment. Two notations are used throughout the paper: (i) vector notation using an arrow hatted over a letter, which is especially useful for formular derivations regarding time derivative of a vector; and (ii) matrix notation using bold letters for computation,which can be regarded as projections of vectors into a specific coordinate system.

2. Dynamics of an Euler-Bernoulli beam with large spatial motion using MSTMM

Since deformation of the tube has a vital influence on the initial disturbance of the launching rocket and thus the impact dispersion[5], element 2 shown in Fig.1 (b) is treated as an Euler-Bernoulli beam in the floating frame of reference formulation (FFRF), where the motion of a free beam is decomposed into a nonlinearly described overall motion of the reference frame KR(see Fig.2),and a linear elastic deformation w.r.t. KR. Notice that the tube tail 1 and the flexible tube 2 in Fig. 1 (b) form a chain system, which is elegantly and efficiently dealt with MSTMM in this paper. The transfer direction is firstly defined from the tail to the tip of the launching subsystem,as shown in Fig.1 (b). Then for an element j(e.g.,beam 2 with spatial motion),the connecting point I into which the transfer direction goes is defined as the input point.Meanwhile,the connecting point O from which the transfer direction comes out is defined as the output point.With the free body diagram of beam 2 shown in Fig. 2, its transfer equation can be deduced. In the sequel, first, how a beam's dynamics is classically treated in the FFRF is briefly given following [18]. This also lays a foundation for the relative kinematic description of the rocket 3 w.r.t.to the beam 2 illustrated in Section 3.Then,the transfer equation of the beam is obtained by rewriting the equation of motion via relating O quantities with I quantities,as we follow Rui's method[17]where details can be found in Ref. [19].

Fig.1. A ground-borne MLRS: (a) an engineering prototype and (b) its dynamics model.

Fig.2. Free body diagram of an Euler-Bernoulli beam with spatial motion described in FFRF.

2.1. Kinematics of an Euler-Bernoulli beam

As shown in Fig.2,for a fixed point P located on the neutral axis of the Euler-Bernoulli beam, the position of P in the global inertial frame KIcan be written as

The transformation matrix of a coordinate system KPwhich is rigidly attached to a cross section of the beam with origin P and with x-axis tangent to the neutral axis, defined in KIis approximated by

Then,the first-and second-time derivatives of Eq.(4)lead to the absolute angular velocity and acceleration of KPwritten as

respectively, where ΦP（RRRP）?R3×nqand ΨP（RRRP）?R3×nqare two space-dependent global shape functions of the translational and rotational deformation field, which can be provided by model order reduction methods, e.g., the Craig-Bampton modes [20];Rq（t）?Rnq×1is the corresponding elastic coordinates; nqis the dimension of the elastic coordinates. Furthermore, the second equality in Eqs. (7)and(8)holds because the material coordinates of P in KPis

2.2. Equation of motion of an Euler-Bernoulli beam

where M is the time-variant symmetric mass matrix; hais the generalized force including interial force term, elastic force term,volume force term, and surface force term exclusive of reaction forces; LTPand LRPwith P?｛I,O｝ are Jacobian matrices for translation and rotation defined in Eqs. (13) and (16).

2.3. Transfer equation of an Euler-Bernoulli beam

We are now in the position to derive the transfer equation of the Euler-Bernoulli beam.This is achieved by rewritting the kinemaics on acceleration level (see Eqs. (12) and (15)) and the equation of motion (17) by relating the O quantities with I quantities. First,according to Eq.(17),the generalized acceleration of the beam can be expressed as

Further, Eqs. (19)-(22) can be arranged into matrix form as

3. Dynamics of an in-bore rocket

With kinematics of the beam in the context of FFRF discussed in Section 2.1, the rocket's relative kinematics w.r.t. the tube is firstly discussed in this section. Then, based on the model type II of the projectile/rocket established in Ref. [3], the equation of motion of the firing in-bore rocket treated as a rigid body with spatial motion is established. Finally, constraint force and constraint equation regarding the connection between the guided slot of the tube and the directional knob on the rocket are analyzed and given.

3.1. Kinematic description of an in-bore rocket treated as a rigid body

In classical launch dynamics [21], the in-bore rocket/projectile kinematics is usually described using relative coordinates w.r.t the tube.This greatly facilitates the contact force analysis between the rocket and the tube.Fig.3 shows the rocket numbered 3 as a rigid body moving along the flexible tube numbered 2 represented by its neutral axis. Herein, two significant Cartesian coordinate systems are established.The rocket's body-fixed frame｛O1; e→Bα｝,α =｛x,y,z｝,simply noted as KB, is located at the center O1of the rear adaptor,and x-axis coincides with the rocket's symmetry axis pointing to the front. Further, to identify the position and orientation of KB, a transitional coordinate system called the sliding frame｛P′; e→Sα｝,α =｛x,y,z｝, simply noted as KS, is defined. The origin P′is a moving point and the perpendicular foot is obtained by making a perpendicular to the neutral axis over the point O1.Meanwhile,the three axes e→Sαcoincide with those of KPdefined in Section 2.1, which is rigidly attached to the beam cross-section, and the origin P is the implication point fixed on the neutral axis cooresponding to P′.

Now, as shown in Fig. 3, the position of O1w.r.t. the reference frame KRcan be uniquely identified using coordinates

where xRis the material coordinate of the implication point P in KR,ySand zSare the y- and z-coordinates of O1defined in KS, respectively. According to the definition of KS, it is clear that the x-coordinate of O1in KSis zero.Further,the orientation of KBw.r.t.KScan be described using the three body-fixed 3-2-1 Euler angles as

Therefore, the generalized coordinates and velocites of the rocket can be selected as

respectively, whereSωSBis the angular velocity of KBw.r.t. KSdescribed in KS.Further,the relationship between y and ˙x,as well as ˙y and ¨x is obtained as

Fig. 3. Relative kinematic description of an in-bore rocket w.r.t. the tube neutral axis.

where the Jacobian matrices H and ˙H can be found in the Appendix.

3.2. Kinematics of the sliding frame

Since we study the in-bore rocket's motion using relative kinematics, it is necessary to study the sliding frame KS’s motion before describing the motion of the body-fixed frame KB. Subsequently,we analyze the kinematics of KSon position,velocity,and acceleration level, respectively. According to the definition, KScoincides with an instant KP. This leads to identical position and orientation defined in KIas

But since P′is sliding along the neutral axis, its cooresponding implication point P has a time-varying material coordinate xR（t）.Then,for any kinematic quantity related to P′uniformly denoted as B,B is not only time-dependent but also space-dependent and thus can be written as a function of xRand t as B（xR（t）,t）.Further,the first and second time derivative of B can be otained as

respectively, where vP′:=dxR/dt and aP′:=d2xR/dt2are the relative velocity and acceleration that P′slides along the neutral axis.

Then, via Eq. (31), the absolute angular velocity of KScan be obtained using the angular velocity superposition theorem as

Seen from Figs. 2 and 3, the absolute velocity of P′can be obtained with Eq. (31) as

Finally, with Eq. (32), the time derivative of Eq. (35) w.r.t. KIleads to the absolute acceleration of P′as

3.3. Kinematics of the in-bore rocket

With the previously discussed kinematics of KS, the kinematics of KBfixed on the rocket can be studied by considering the relative motion between KBand KS. For the rotational motion, the transformation matrix that defines the orientation of KBin KIcan be expressed as

Further, the angular velocity of KBcan be obtained using the angular velocity superposition theorem as

Consequently, considering Eqs. (29), (34) and (36), the matrix form of Eq. (47) yields

Eventually, the absolute time derivative of Eq. (47) leads to the acceleration of O1in a vector form

and a matrix form in terms of ˙η and ˙y considering Eqs. (29), (38)and (40) as

resulting in a matrix form in KIas

3.4. Equation of motion of an in-bore rocket

Then Eqs. (52) and (54) are the equation of motion that describes the in-bore rocket with six DOFs.

3.5. Constraint force analysis

During launching, a rocket is mainly subjected to gravity f→C,engine thrust f→pΣ,contact force f→B,i(i=1（1）3 corresponds to the rear,middle and front adapters)between the adapters and the tube wall,and the constraint force f→Ebetween the guided slot and the directional knob.Herein only constraint force is analyzed,which is necessary to discuss coupled dynamics in Section 4. For other forces,see Ref. [13].

The rocket and the tube are assembled by plugging the directional knob(see Fig.3)on the rocket's rear adaptor into the guided slot carved on the tube wall.As shown in Fig.4(a),the guided slot can be depicted in a plane by cutting the tube along its symmetry axis and then unfolding it.Therein, x represents the distance from the origin o of the slot to the contact point E along the symmetry axis;α is the angle between the tangent of the slot and symmetry axis of the tube, and is usually a prescribed function of x. Then, as shown in Fig.4(a),the contact force f→Eacting on the rocket due to the contact pair,slot and knob,can be decomposed into a constraint force N normal to the slot and a frictional force along the tangent of the slot.Further,they are described along and perpendicular to the tube axis as

Fig. 4. Schematic for constraint force analysis: (a) guided slot depicted in an unfolded tube, (b) orientation of the directional knob in KB, and (c) forces acting on the directional knob.

respectively, where μ is the Coulomb coefficient of friction.

As shown in Fig.4(b),the contact point E on the knob can also be viewed from the rear of the rocket, where there is an angle γbbetween the knob and the y-axis of KB. Then, when the unfolded plane in Fig.4(a)is folded back into a pipe,the contact force lies in the tangent plane of the cylindrical tube over E,which can be easily described in KSas shown in Fig. 4 (c) and can be written as

where ex=[1 0 0]Tand ez= [0 0 1]T. Considering Eq. (30)the above equation can be rewritten in terms of y and ˙y as

4. Coupled dynamics of the launching subsystem

In Sections 2 and 3, on element level, the dynamics of the flexible tube 2 and tube tail 1 are handled by MSTMM.Meanwhile,the in-bore rocket 3 is treated as a rigid body and handled by the classical Newtown-Euler method with relative kinematic description sliding along the beam. In the sequel, we show how these dynamics of elements using different methods are assembled and solved for the system dynamics.

4.1. The rewritten equations of elements considering constraint force

Since the elements 2 and 3 shown in Fig. 1 (b) are coupled through the unknown constraint force N(see Eq.(57)),N has to be explicitly written in their transfer equation or equation of motion to solve the system dynamics.

4.1.1. Transfer equations rewritten for the beam

4.2. Coupled equation of motion of the launching subsystem

The equation of motion of the launching subsystem can be obtained in the following three steps.In the first step,elements 1 and 2 in Fig.1(b)are coupled via a fixed hinge(or rigid weld)and forms a chain system that can be elegantly handled by MSTMM. Due to virtual cutting,the input point I of element 2 and output point O of element 1 are the same. This yields

which essentially describes the identical relationship between adjacent state vectors on element j and j+1 (j=1 in our case).Notice that all quantities summarized in the state vector are described in the global inerial frame KI. This avoids further coordinates transformation to hold Eq. (73).

In the second step,the rocket 3 is coupled with the chain formed by elements 1 and 2. This is achieved by substituting Eq. (75) into Eqs. (67) and (68) so that ˙η vanishes.This results in

In the last step,Eqs.(74),(76)and(77) and constraint equation(61)are assembled into matrix form yielding the coupled equation of motion of the launching subsystem as

It is worth mentioning that Eq.(78)is a general expression that is not only fit for ground-borne, air-borne, and naval-borne rocket launchers, but also barrel weapons. In the latter case, the rear adaptor of a rocket becomes the bearing band of a projectile.Then,the constraint force results from interactions between the rifling and the bearing band. For a smoothbore gun tube, one only has to eliminate the constraint equation corresponding to the last row in Eq.(78)and the constraint force N together with the last column in Uall. This will further lead to the decoupled dynamics of projectile and barrel on the acceleration level.

4.3. Solution procedure of the subsystem dynamics

The independent generalized coordinates xalland velocity yallof the entire launching subsystem are chosen as a combination of coordinates and velocities of flexible tube 2 and rocket 3 as

5. Numerical simulation and experimental verification

With the established equation of motion (80) of the launching subsystem, its co-simulation with other subsystems,including the orientating and marching subsystems shown in Fig. 1 (b), is achieved.Noting that elastic hinges connect different subsystem,data are exchanged among subsystems only in position and velocity level. This modular modeling owns major merit that the development of the dynamics of this complex MLRS can be concurrent and distributed, that is, divided between different teams and/or external suppliers, each in their own domain and each with their own tools and expertise.Further,simulation of the rocket's exterior ballistics is integrated, resulting in a joint simulation environment of the LDMF of the MLRS covering the whole process of launch and free flight,see Ref.[13].The joint simulation environment is coded in C++ with objected-oriented programming techniques. Moreover, stochastic factors can be considered using the Monte Carlo method.These stochastic factors cover aspects of meteorology,e.g.,a gust of wind,and manufacturing errors of the rockets,e.g.,mass,mass eccentricity, dynamic imbalance, the mass of propellants,magnitude of engine thrust, the eccentricity of the thrust, and clearance between tube walls and rocket adaptors.

5.1. Simulation results

Theretofore, simulation results related to the launching subsystem are given and discussed only,as is our focus.To exhibit the large motion of the launcher, firstly, under different translational marching velocities, the dynamic response of the tube and the inbore rocket are simulated, respectively. Then, we simulate four rockets launched in the process that the launcher experiences large rotation. For confidential concerns, parameters related to the dynamics are not given.

5.1.1. Dynamic response of the tube tip w.r.t. different marching velocities

We consider the case that the MLRS is marching along an A-level road with the same determined profile for wheels on both sides of the vehicle. While marching, a single rocket is fired from the launcher. For disturbance rejection purpose, the desired pitch and yaw of the elevation platform are set and controlled by two stablizers(modeled in the orientating subsystem in parrell with the two revolute hinges shown in Fig.1(b),see Refs.[13,14])to be 19?8°and 0?0°w.r.t. the xz- and xy-plane (see Fig.1 (b)) of KI, respectively.Figs.5 and 6 show the time response of the output point O and its corresponding fixed frame KOon the flexible tube 2 under marching velocities of v=0 km/h, v=15 km/h, and v= 30 km/h,respectively. The cooresponding time response of the in-bore rocket is depicted in Figs. 7-9.

Seen from Fig. 5, for v=0 km/h, the time response of the tip attenuates promptly with the usage of stablizers after the in-bore rocket moves out from the tube (t>0?15 s), however for v= 15 km/h and v=30 km/h, still oscillates back and forth due to the constant excitations from the road. Moreover, the amplitude of oscillation increase with the increasing marching speed accompanied by higher frequency components. This is because higher marching velocity means larger amplitude and higher frequency of road excitations due to the stochastic road profile. Together with the vibration characteristics of the MLRS,the road excitation leads to the increased amplitude and higher frequency components.Besides,in Fig.6,there are three obvious jumps of the tip response when the rocket moves in the tube for t<0?15 s.As shown in Fig.9,these three jumps correspond to three collisions between the tube wall and the rocket adaptors.Thus,such collision is the major cause for the sudden change of the tip velocity and angular velocity.

5.1.2. Dynamic response of the in-bore rocket w.r.t. different marching velocities

From Figs. 7 (a), (b), and 8 (c), one can conclude that the marching of the MLRS does not have a noticeable influence on the translation of the center point O1of the rear adaptor, thus no evident effect on the spin of the rocket due to constraint.However,according to Figs. 7 (c) and 8 (a), (b), marching has a significant impact on the pitch and yaw of the in-bore rocket. Notably, significant distinctions of angular movement are found among different marching velocites in the half-binding period (approximately when t>0?1 s)that both the front and mid adaptors moves out from the tube but with rear adaptor inside. Further, since the rocket is less constrained in the half-binding period, higher marching velocity leads to larger pitch and yaw movements resulting from the larger relative angular motion between the inbore rocket and the tube caused by larger road excitations.This will significantly affect the initial disturbance of the rockets thus their impact dispersion. According to Fig. 9, there are three evident jumps in contact forces among the rocket adaptors and the tube wall. They correspond to: (i) the impact among the rear adaptor,front adaptor,and the tube wall when the rocket is unlocked from the launcher at the initial stage;(ii)impact among the rear adaptor,mid adaptor, and the tube wall after the front adaptor moves out from the tube; (iii) impact between the rear adaptor and the tube wall after the mid adaptor moves out.

Fig. 5. Time histories of dynamics of tube tip on position level: (a) y-coordinate and (b) z-coordinate in KI; orientation angles about (c) y-axis and (d) z-axis of KI.

Fig. 6. Time histories of dynamics of tube tip on velocity level: velocities along (a) y-axis and (b) z-axis of KI; time derivatives of angles about (c) y-axis and (d) z-axis of KI.

5.1.3. External ballistics of rockets launched while orientating part undergoing large rotation

5.2. Experimental validation

Fig. 7. Dynamics of in-bore rocket on position level: (a) longitudinal displacement xR w.r.t. time; phase graphs of (b) transverse displacement of O1 and (c) pitch-and-yaw angle w.r.t. the tube.

The run time of the in-bore rocket counts from the rocket's ignition to the seperation of rear adaptor from the tube tip.Together with the muzzle velocity, they are key indices for the launch dynamics of rockets. As shown in Fig.1 (a), the stock-still firing experiment is conducted where the high-speed camera records the key time instants,and the millimeter-wave radar records the muzzle velocity vM.These key time instants include tF,tM,tRat which the front, mid, and rear adaptors leave from the tube tip,respectively.All time instants are conunted from the ignition that is also captured by the camera. The camera and the radar are synchronized. Two salvos of rockets are shot, with each salvo of 18 rockets launched in a determined sequence. In comparison, numerical simulation is implemented with 10 salvos of rockets considering the random factors mentioned in the beginning of Section 5. Both results are shown in Table 1, which shows good agreement.

Fig. 8. Dynamics of in-bore rocket on velocity level: time derivatives of (a) vertical transverse angle , (b) horizontal transverse angle , and (c) spinning angle γ w.r.t.the tube.

6. Conclusion

Fig.9. Time histories of norm of resultant contact force between:(a)the rear-adaptor and tube wall, (b) the mid-adaptor and tube wall, and (c) the front-adaptor and tube wall.

In this paper,a mixed formulation of MSTMM with the Newton-Euler method is proposed for an efficient and module modeling of LDMF of a MLRS. Attention is focused on the launching subsystem that is quite general topology for various weapons where the launch is involved. The established equation of motion for the launching subsystem is not restricted to stock-still fire only but is also suitable for marching fire or fire from a general moving base.The launching subsystem can then be regarded as a module plugged into the whole system also composed of orientating and marching subsystems, resulting in the overall system dynamics to which the design parameters such as clearance between tube and rocket, firing interval, firing sequence, etc., are related. Part of the simulation results are validated by experiment. Further, the developed numerical simulation environment can be used for a weapon system's analysis and design of the entire ballistics.

Fig.10. Exterior ballistics of four rockets launched in a sequence while adjusting the azimuth and elevation platforms.

Table 1 Key time instants of in-bore rockets and their average muzzle velocity by computation and experiment.

With the established simulation environment, future research may include: (i) analysis of the influence of firing interval and sequence to the impact dispersion of a moving MLRS; (ii) disturbance rejection and trajectory tracking control of two stabilizers to reduce the impact dispersion of a moving MLRS.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors thank Ms.Lilin Gu,a PhD candidate in the Institute of Launch Dynamics,Nanjing University of Science and Technology for providing the experimental data. The research is financially supported by the National Natural Science Foundation of China(No.11972193).

Appendix

Herein, we discuss the relationship between the generalized coordinate x and velocity y of the in-bore rocket. The angular velocitySωSBcan be described in term of ˙δBusing the Euler's equations of kinematics as