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Dynamic Analysis of Rockets Launcher

This paper deals with dynamic analysis of rockets launcher on a moving vehicle. Mechanical model of a launch rail system with a rocket is represented as a set of solid bodies and deformable elements with a damped elastic support. These launch rail systems enable to fire rockets of different weights and geometric configurations. Unlike them, today ́s multiple rocket launcher systems are located in containers. Movement of this type of mechanical system is described with three defined coordinates: movement of a rocket on a deformable ramp, curvature of the top of a deformable ramp and an angle of rotation of the ramp around its longitudinal axis. Mathematical model of the whole system, including rockets launcher and vehicle, represents a system of five nonlinear, nonhomogeneous differential equations of second order. This system is transformed to a system of 10 differential equations of the first order, suitable for numerical solving through the software package MATLAB. Rocket launcher designers can find the obtained simulation and experimental results, represented in a graphical form which is very useful during the development of a new and modification of existing constructions.


INTRODUCTION
The rocket launcher belongs to a group of artillery weapons intended for attacks against enemy personnel, as well as individual fortified points deep in enemy territory. Therefore, military theorists classify it within a group of artillery weapons aimed at supporting own infantry. A rocket launcher is characterized with high firepower and maneuverability, thanks to the possibility of moving with the rockets on the launch pad. Rockets used on this type of launcher are unguided rockets. In order to achieve the necessary effect, rockets must be launched at an exactly determined angle of elevation and azimuth. As a vital part of a launcher, the launch pad has a task of holding and guiding a rocket at a desired elevation. Launch pad has to ensure with its length that it guides a rocket until a vertical component of the rocket's engine thrust vector balances the force of the rocket's weight. Dynamic parameters of a launch pad are quite broadly considered in literature. However, launch pads are considered as a solid body which does not fully correspond to real conditions in exploitation [1]. Also, in [1] author does not take elasticity of the ramp support and elasticity of the launcher support into consideration, which is quite significant having in mind that a launcher can work on the basis of different categories.
In [2] authors stated that the launcher models used in most of the previous studies had a few degrees of freedom that could not possibly capture many important motion characteristics. They used a flexible multibody model of a missile launcher system. The fixed, azimuth and elevation platforms and the launch pad were modeled as flexible bodies. The joint flexibility effects were also included to the model by spring-damper elements.
A case of a physical and mathematical model of a launcher of unguided rockets with an elastic support and solid launch pad was considered [3]. The mathematical model was classified as a coupled system with timevarying coefficients, determined, variable in time, dissipative and bounded, with five degrees of freedom.
In order to study the coupling effect of the equipment and the launching site during the erecting process of the vehicle missile system, the flexible multi-body dynamic model is established in ADAMS [4]. The coupling calculation model of the weapon system and the launching site is established in the co-simulation of the ADAMS and MATLAB/Simulink.
In [5] authors considered the stability of launching devices during firing. They assumed that the launching device and the moving rocket form a complex oscillating system of rigid bodies bound by elastic elements (the vehicle chassis, the tilting platform and the rockets in the containers). The developed mathematical model of this oscillating system is used to determine the oscillations and their effects on the initial conditions of the rocket path.
Taking into consideration that during the firing, the rocket-launching device system oscillates and these oscillations may have a negative influence on the unguided rocket firing precision, it is necessary to evaluate the exact influence of the disturbance factors on the rocket movement upon the launcher. In [6] authors present the equations of the rocket movement upon the launcher under the disturbance factors action, taking into consideration that the rocket is the integrated part of the rocket-launching device system. In [7] authors deal with the parameters important for defining the mechanism for elasticity elimination in the vehicle launching device. The presented basic dynamic analysis of rocket launcher is of crucial importance for discovering the influential parameters necessary for calculating the required power of the mechanism.
In [8] authors present an analysis and design of active vibration control algorithm for Multiple Launch Rocket System (MLRS). It has interesting advantages as follows: 1) Active vibration control is more competent, more pliable, and more realizable in engineering compared to passive vibration control; 2) The guidance and control package will not be destroyed. In this research, transfer matrix method of multibody system is used to build the controlled dynamic model of MLRS.
In [9] authors studied the dynamics of a vehicular missile system and determined initial disturbances of a missile. The multi-rigid-flexible launch dynamics model of a vehicular missile system was developed by using transfer matrix method for multibody system and launch dynamics theory.
Poor dispersion characteristics of rockets due to the vibration of Multiple Launch Rocket System (MLRS) have always restricted the MLRS development for several decades. Vibration control is a key technique to improve the dispersion characteristics of rockets. To approach this problem, a computed torque controller integrated with a radial basis function neural network is proposed to achieve the high-precision vibration control for MLRS [10].
In [11] authors considered the oriented control as a key technique to improve the dispersion characteristics of rockets. They proposed an orienting control method for launcher of the MLRS in a salvo firing. Because the MLRS is a typical nonlinear system, the major difficulty in designing the orienting controller lies in the nonlinearity. To deal with the nonlinearity, the concept of computed torque control is introduced.
The efficient vibration control of complex multibody systems is an important issue in engineering design and has drawn substantial attention in the field of multibody system dynamics and control engineering. In [12] authors presented an active vibration control design method for the dynamic modeling and vibration control design of multibody systems by combining the transfer matrix method for multibody systems and modal space control.
In [13] authors presented the control modeling and analysis of the prototype of Multiple Launch Rocket System (MLRS) using Matlab Simulink. The mathematical model was determined using physical law and theoretical foundation of the turning mechanism control system and the elevating mechanism control system.
In [14] authors considered a comprehensive multibody system dynamics model of a Multiple Launch Rocket System (MLRS), and its simulation and experimental studies. The new version of transfer matrix method of multibody system and the launch dynamics theory are used in deriving the equations of motion coupled with rockets and barrels. The missile launching system is a typical complex mechanical system, which is becoming increasingly larger in scale and more complicated in structure. Designing these types of weapons often requires repeated complex and time-consuming model design, calculation, and analysis of the dynamics of the system [20]. In [15] authors put forward a new method for bidirectional-driving parameter between CAD and CAE, by changing the geometric model parameters, which directly changes the dynamics model.
The transfer matrix method for multibody systems, namely the "Rui method", is a new method for studying multibody system dynamics, which avoids the global dynamics equations of the system, keeps high computational speed, and allows highly formalized programming. Some aspects regarding the transfer matrix method for multibody systems are reviewed systematically in [16]. This paper sets and analyses a mechanicalmathematical model taking into consideration influential parameters in the rocket's launch phase. It also analyses the deformation of a launch pad, rocket's speed, influence of stiffness of the ramp support on a mechanical model with elastic support. Then, some numerical examples of a specific construction were worked out and the conclusions drawn were of interest for improving the existing or a new construction in the future.
Mechanical model of a launcher while launching a rocket ( Fig. 1) was greatly based on mechanical models of analogue objects: auto-cranes and telescopic crane shafts [17]. The model consists of solid bodies and deformable elements with elastic connections. The lower part of the launcher (Fig. 1, position 1), consists of a chassis, cabin, power train aggregates and supports (tires and stabilizers). Chassis with aggregates and its corresponding elements will be considered as a solid plate supported on corresponding springs. This approximation is allowed, especially in structurally reinforced chassis. Stiffness of these springs corresponds to reduced stiffness of supports and substrate.

MECHANICAL-MATHEMATICAL MODEL
The upper part of the launcher (Fig. 1, position 2) including a mount, radial-axial bearing, mechanisms for raising and rotating a ramp, is considered as a solid body supported on the chassis. The launch pad (Fig. 1, position 3) represents a beam bracket (a smaller beam bracket is neglected) that is elastic and deformable in the vertical and horizontal axis (

Generalized Coordinates
Movement of the presented mechanical system is defined with the following generalized coordinates (Fig. 1, Fig. 2, Fig. 3

Coordinate Systems
Movement of such complex mechanical system that is considered here, was interpreted in [17], in detail. In this case, the movement is observed with respect to the following coordinate systems Fig. 1, Fig. 3): 0 0 0 x y z , 1 1 1 x y z , 2 2 2 x y z , When determining kinetic and potential energy, the curve of oscillation Potential energy of the adopted mechanical system is equal to the sum of potential energy of supports and base, centre of mass of the chassis and potential energy of deformation of the launching ramp For supporting contour presented in Fig ; Potential energy of the launch pad [19] is:

Figure 5 Supporting contour and configuration of stabilizers
Therefore, total potential energy of the launching pad is: where the constants are respectively: The expression for potential energy of the mechanical system has the final form of: 2  2  11 11  11  12  12  12   2  2  21  21  21  22  22  22   0  2  3  11 22 33

Coordinate Systems
According to the defined expression, generalized forces are: Generalized forces corresponding to coordinates  and u are: where p F -thrust force (Fig. 6 It can be considered that other generalized forces are approximately equal to zero, i.e. : 0

Ordinates and Velocities of Characteristic Points
Determining the radius vectors (especially ordinates) and velocity of characteristic points of such complex mechanical systems with a classical method is a very complex procedure and practically unachievable. Therefore, the paper uses a matrix method based on multiple coordinate systems, primarily set in [17] and then applied in [18].
Assuming that the position vector of any point of the system with respect to the fixed coordinate system 0 n  , k r -vector between centres of the adjacent coordinate systems Thus, the radius vectors of the centre of gravity of the mount (L), centre of gravity of the launch pad (S) and the centre of gravity of the rocket (R), respectively, are: 10 10 1  20   10  10 1  20 2  30 3   10  10 1  20 2  30 3 , (12) Corresponding matrices of cosine of angles are:

r T r T r r T r T r T r r T r T r T r
Determining velocities by this process [17], that is increasingly used in robot dynamics, is completely formalized and comes down to derivatives of matrices, position vectors and matrix multiplications. Velocity of an arbitrary point in the system is:  If velocities of the points C, L, R and S are written in a summarized form i.e. matrix form, it would simplify differential equations on the movement of a mechanical system.
-Velocity of mass centre of the chassis, point C, is: Technical Gazette 28, 2(2021), 530-539 -Velocity of mass centre of the mount, point L: 11 -Velocity of mass centre of the rocket (point R) and velocity of mass centre of the shaft (point S) are:

Mathematical Model-Differential Equations of the Motion of the Mechanical System
In order to derive differential equations of the motion, the Lagrange's equations of the second kind are used: Obtained mathematical model represents a system of five nonlinear, nonhomogeneous differential equations of second order. The given system of differential equations is similar to Lyapunov's system and it is solved by separating small parameters in the form of a row (Selmic, 1979). However, the aforementioned system is solved faster and simpler by using discrete numerical methods, which are quite efficiently done on computers. Runge-Kutta method is recommended for solving these systems of differential equations. Due to this, systems of differential equations should be written in the form of a matrix: Software package MATLAB enables solving nonlinear differential equations. Due to the application of MATLAB it was necessary to write differential equations of motion in the form of a matrix, as follows: where the following is obtained: Expressions for accelerations and velocities prepared in such a way can be solved by the software through the previous replacement of variables in the following way: 1  2  3  4  5  6   7  8  9  1 0   ,  ,  ,  ,  ,  , , , , Thus the system of differential equations has the following form:     , , , , , , , , , , , , , and it is transformed to a system of 10 differential equations of the first order, suitable for numerical solving through the software package MATLAB. A numerical analysis of this problem was performed for a concrete construction.

RESULTS ANALYSIS
The numerical analysis was performed in the case when the hydro-cylinder is replaced with resistance force linearly proportionate to the first degree of velocity of point B (Fig. 2), i.e. the force in hydro-cylinder is: Graphic overview of characteristic parameters is given in Fig. 9 to Fig. 12. Based on the input data and obtained diagrams it can be concluded that: 1) displacement of mass centre of the chassis occurs in the limits of up to 0,009 m (Fig. 9), 2) rotation around x -axis does not exist because all masses are on that axis (Fig. 10), 3) rotation around y -axis exists to thrust force and momentum during movement of the rocket on the launch pad, and is performed to harmonic function (Fig. 11), 4) deflection of the launch pad is the biggest at the end of the launch pad and it is of a nonlinear character and amounts to 0,03 m (Fig. 12).
From the attached diagrams, especially Fig. 12 it is visible that the hydro-cylinder needs to be approximated only by damping high stiffness. Curve of deflection authentically follows the curve obtained by measuring. An analysis of influential parameters was performed for the adopted mechanical model. Graphic overview of characteristic parameters is given in Fig. 13 to Fig. 16. Based on the variations of the given parameters it can be concluded that: 1) by reducing the values of coefficients of viscous friction of a hydro-cylinder, deflection of the launch pad's top increases (Fig. 13), 2) depending on the required velocity of deflection of the launch top of the ramp, it is possible to pick the value of the coefficient of viscous friction of a hydro-cylinder (Fig. 14), 3) deflection of the launch pad reduces with reduction of rocket's mass almost linearly (Fig.15, Fig. 16).

MEASURING AND STATISTICAL PROCESSING OF RESULTS
Measurements were performed on a self-propelled launcher while launching a rocket. Single and burst fire launches were observed. A point was observed, firstly the left and second time the right (Fig. 2) part of the launch pad and its motion in the vertical plane (deflection of the top). In case when the measuring point was right part of the top of launch pad, three shots were measured (Fig. 18). In case when the measuring point was left part of the top of the ramp, four shots were measured (Fig. 19). In case when three rockets were fired at once, two measurements were taken (Fig. 20). During the aforementioned measurements, the launch pad was directed at the material axis of the system with a direction x 0 -axis, i.e. angle of azimuth was 0   (Fig. 3). Three measurements (  the top of the ramp was in function of the time that the rocket moved on the ramp. When performing experiments according to this one in Fig. 17, the following equipment was used: 1) Camera 1: stationary camera; 2) Camera 2: manual tracking of the rocket (from hand); 3) radar for measuring speed; 4) observation equipment. character and their change can be characterized as irregular harmonic change of small amplitudes. Deflection curve of the top of the pad obtained by numerical interpretation of an adopted mechanicalmathematical model (curve 5) is relatively agreed to the curves obtained measurements (curves 1, 2, 3, 4). Clearly expressed oscillatory character of curve 5, relative to the large oscillatory amplitudes is the consequence of elasticity of the pad and support in hydro-cylinder which are taken in the considered model.

CONCLUSION
Influence of each construction and exploitation parameter on the behaviour of the rocket system can be determined by mathematical modelling and simulation on a computer in the design or modification phase. This is of great importance because it rationalizes the entire project. Since a solution of the mathematical model can be used when analysing the influence of concrete construction parameters on certain properties of the rocket system, from that part of the research, the following conclusions can be made: (1) It is known that angular velocities around characteristic axis (transversal and longitudinal axis) have an important influence on the position of hits. The smallest spread on the target is obtained by positioning the launch pad in a position that causes the smallest values of the mentioned angular velocities. Simulation can find parameters of a most favourable position of the system, considering accuracy and precision. (2) Declination angle of a launch pad has a significant influence on the flow of the oscillatory process of a rocket system. It influences the deflection but in a smaller degree than stiffness of the support. This paper gives a comprehensive approach to the problem of oscillatory motion of the rocket system. Basic assumption of the dynamic model is that, after a rocket is launched from the launch ramp, rocket system performs muffled oscillations. Disturbances during rocket's start represent initial conditions for oscillations. Rocket system requirements regarding weight and range require more complete calculation methods for the entire construction, especially a more precise determination of dynamic loads. The method presented in this paper enables determining dynamic loads of mechanical stabilizers (supports) taking into consideration elasticity of the base. It is possible to determine dynamic loads of the launch pad, depending on parameters of the support and stiffness of the base.
Abbreviations and symbols: Oz is directed vertically downward, 1 1 1 x y z -Mobile coordinate system connected to the chassis with the centre O 1 in the point C (chassis' centre of mass). 1 Ox axis coincides with the longitudinal and 1 Oy axis with the transversal central axis of inertia in an equilibrium and 1 Oz axis is directed vertically vertically downwards, 2 2 2 x y z -Mobile coordinate system adopted for a mount and a launch pad with the centre in point O 2 which intersects the longitudinal axis of the mount and horizontal plane which passes through the joint of the launching ramp, 3 3 3 x y z -Mobile coordinate system with the centre O 3 in the joint of the launch pad, directed axis 3 x at the top of the launch pad, 3 y perpendicular to the launch pad and 3 z axis is directed perpendicularly to the 3