12.3.1 State Vector

For the launch dynamics model of the MLRS shown in Figure 12.2, noting the sign convention, according to the MSTMM, the state vectors of the connection point of the MLRS are defined as follows:

(12.1)

(12.2)

(12.3)

(12.4)

(12.5)

(12.6)

The form of , , , , , , , and is similar to , the subscripts denote the sequence number of bodies and joints at the connection point, and i is the sequence number of the launch tube.

12.3.2 Transfer Matrix and the Transfer Equation of a Component in the MLRS

12.3.2.1 From Ground to Wheels (axes)

According to the transfer equation of a spatial spring, the transfer equation from contact points between the wheels and the ground to the wheel (axle) points can be obtained:

(12.7)

is the transfer matrix from the ground to the wheels (axes):

(12.8)

where is the transfer matrix of the spatial spring.

12.3.2.2 Wheels (axes)

According to the transfer equation of a lumped mass with spatial vibration, the transfer equation of the wheels (axes) from every axis point to point can be obtained:

(12.9)

is the transfer matrix of the wheels (axes):

(12.10)

where is the transfer matrix of a lumped mass with spatial vibration.

12.3.2.3 From Wheels (axes) to Vehicle

According to the geometric and mechanic relation between the wheel (axes) points and the vehicle points , the transfer equations from the wheel (axes) points to the vehicle points can be obtained:

(12.11)

(12.12)

where

(12.13)

(12.14)

(12.15)

(a_1,2, a_2,2, a_3,2), (a_1,3, a_2,3, a_3,3), (a_1,4, a_2,4, a_3,4), (a_1,5, a_2,5, a_3,5), and (a_1,6, a_2,6, a_3,6) are the coordinates of points (19, 14), (19, 15), (19, 16), (19, 17), and (19, 18) at the coordinate system whose origin is fixed on the first input end of body 19.

12.3.2.4 Vehicle

A vehicle is a rigid body with six inputs and one output. According to the transfer equation of a rigid body, the transfer equation from the vehicle points to (19, 20) can be obtained:

(12.16)

where is the transfer matrix of a rigid body (vehicle) with six input ends and one output end (see Equation (3.25)).

12.3.2.5 Traversing Mechanism and Elevating Mechanism

The actions of traversing and elevating mechanisms are equivalent to the actions of rotary springs and springs. According to the transfer matrices of spatial springs and rotary springs, the transfer equations from the traversing mechanism point (19, 20) to point (21, 20) and from the elevating mechanism point (21, 22) to point (23, 22) can be obtained as, respectively,

(12.17)

(12.18)

where and are the corresponding transfer matrices of the elastic joints, respectively (see Equation (3.36)), and and are the corresponding coordinate transformation matrices with angles of azimuth α and departure angle θ, respectively.

12.3.2.6 Gyration Part

A gyration part is a rigid body with one input end and one output end. According to the transfer equation of a rigid body, the transfer equation of the gyration part can be obtained:

(12.19)

where is the transfer matrix of a rigid body (gyration part) with one input end and one output end (see Equation (3.19)).

12.3.2.7 Pitching Part

A pitching part is a rigid body with one input end and two output ends. According to the transfer equation of a rigid body, the transfer equation of the pitching part can be obtained:

(12.20)

where is the transfer matrix of a rigid body (pitching part) with one input end and two output ends.

12.3.2.8 The ith Launch Tube Trail (to rear bearing)

Similar to the gyration part, the transfer equation of the ith launch tube rear end point to point can be obtained:

(12.21)

where is the transfer matrix of a rigid body (the ith launch tube trail) with one input end and one output end.

12.3.2.9 Transfer Matrix of a Connection Point from the Pitching Part to the ith Launch Tube

The ith launch tube is connected to the pitching part through two points and , forming a loop between the pitching part and the launch tube. Using the geometric relation and the force equilibrium between point and point , we obtain

(12.22)

Equation (12.22) can be written as

(12.23)

where

(12.24)

Similarly, using the geometric relation and the force equilibrium between point and point , we obtain

(12.25)

where

(12.26)

The displacement (including angular displacement) relation between points and is

(12.27)

where

(12.28)

The displacement (including angular displacement) relation between points and is

(12.29)

namely

(12.30)

where

(12.31)

(12.32)

12.3.2.10 The ith Launch Tube

The ith launch tube in which the moving rocket is located is regarded as a transversely vibrating elastic beam, and its longitudinal and torsional deformation are neglected. The transfer equations from point to point and from point to point at the ith launch tube are, respectively,

(12.33)

(12.34)

where and are the transfer matrices of transversely vibrating elastic beams without considering longitudinal and torsional deformation (see Equations (11.41) and (11.42)).

12.4.1 Transfer Equations from Ground (0, 1‐6) to Pitching Part

According to the transfer equations (12.7), (12.9), (12.11), (12.16), (12.17), (12.19), (12.18) and (12.20), the transfer equation from the ground to the pitching part can be obtained:

(12.35)

where

(12.36)

12.4.2 Transfer Equation from Ground to Vehicle

According to the transfer equations (12.7), (12.9), (12.11) and (12.12), the transfer equation from point to point can be obtained:

(12.37)

where

(12.38)

12.4.3 Transfer Equation from Launch Tube Trail to Launch Tube Muzzle

According to the transfer equations (12.21), (12.23), (12.33), (12.25) and (12.34), the transfer equation from point to point can be obtained:

(12.39)

where

(12.40)

12.4.4 Transfer Equation from Point to Point

According to the transfer equations (12.21), (12.23), (12.33) and (12.29), we obtain

(12.41)

where

(12.42)

12.4.5 Transfer Equation from Point to Point

According to the transfer equcations (12.21) and (12.27), we obtain

(12.43)

According to Equations (12.39) and (12.43), we obtain

(12.44)

(12.45)

Let

(12.46)

then

(12.47)

Substituting this into Equation (12.35) yields

(12.48)

Substituting this into Equation (12.41), it follows that

(12.49)

12.4.6 Overall Transfer Equation and Transfer Matrix of MLRS

According to Equations (12.37), (12.48) and (12.49), the overall transfer equation of MLRS can be obtained:

(12.50)

where

(12.51)

(12.52)

is composed of state vectors at the boundary points of the system and is the overall transfer matrix of the MLRS, a 30 × 60 matrix.

12.5.1 Characteristic Equation of the System

It is can be seen from Equation (12.51) that is composed of the state vectors at the boundary points of the system, including ground points , launch tube trail point and launch tube muzzle . Half of the elements of the state vector at the boundary points are determined by the boundary condition. There are 36 elements in total in , including the displacement and force in three directions at six contact points between six wheels and the ground, where 18 elements respresenting displacement are zero. Getting rid of these zeros, can be written as

(12.53)

The state vectors and of point and point include linear displacement, angular displacement, force and moment in three directions of corresponding points, and within the 12 elements involved in the state vector of each point, six of them (forces and displacements) are zeros. Getting rid of these zeros, we obtain

(12.54)

(12.55)

Getting rid of these 30 zeros from , a new state vector can be obtained:

(12.56)

Getting rid of columns 1, 2, 3, 7, 8, 9, 13, 14, 15, 19, 20, 21, 25, 26, 27, 31, 32, 33, 43, 44, 45, 46, 47, 48, 55, 56, 57, 58, 59, 60 in , a square matrix with 30 orders can be obtained. Then Equation (12.50) is written as

(12.57)

or

(12.58)

It can be seen from the derivation process that both and are only relative to the total parameters of the system and the natural frequency . When the total parameters of the system are determined, the determinant of matrix corresponding to the natural frequency ω_k is zero and Equation (12.58) must have a nontrivial solution, namely

(12.59)

Equation (12.59) is the eigenequation of the MLRS. Solving Equation (12.59), the eigenfrequency of the MLRS can be obtained. After the eigenfrequency of the MLRS is obtained, under the defined normalized condition (such as letting the maximal absolute value of the 30 elements of to be 1), solving Equation (12.58), and corresponding to ω_k can be obtained, namely the state vectors , and corresponding to ω_k can be obtained. All state vectors (such as , , Z_19,20, Z_21,20, Z_21,22, , , , , , and so on) at every connection joint and any point at the beam corresponding to ω_k can be obtained through the transfer equation of each element.

12.5.2 Vibration Characteristics of the System

Substituting the corresponding parameters into the characteristic equation and solving it, the natural frequency of the MLRS in any case can be obtained. The state vectors corresponding to natural frequency at any point of the MLRS can be obtained through the transfer equation of the system.

The quantitative relation between the vibration characteristics of the MLRS and the total parameters of the system is established. By changing the total parameters of the system, the vibration characteristics can be modified easily. The order of the transfer matrix and the eigenequation of the MLRS all are very low, which ensures the high‐speed computation of the vibration characteristics.

12.6.1 Body Dynamics Equations of the System

The body dynamics equation of single body element is

(12.60)

The body dynamics equation of every element is rearranged and assembled according to their sequence number, and so the body dynamics equation of the MLRS can be obtained:

(12.61)

where

(12.62)

is the column matrix of system displacement coordinates, and are augmented operators, is the damping augmented operator, is a column matrix of external force, and are the sequence numbers of the body element.

For proportional damping we have

(12.63)

where α and β are constant.

12.6.2 Orthogonality of Augmented Eigenvectors of the System

Let

(12.64)

where k is mode rank, q(t) is the generalized coordinate, and V is the augmented eigenvector of the MLRS.

The parameter matrices , , , and of the body elements and the augmented eigenvector of the system are introduced in the following.

12.6.2.1 Six Wheels (Element )

For six wheels, from Equation (5.7) we obtain

(12.65)

where m_j is the mass of the wheel and [f_xC,j, f_yC,j, f_zC,j]^T is the column matrix of external force acting on the wheel.

For convenience, this is denoted as

(12.66)

12.6.2.2 Vehicle, Gyration Parts, Pitching Parts and Launch Tube Trail (Element )

For the vehicle, gyration parts, pitching parts and launch tube trail, we obtain

(12.67)

where m_j is the mass of rigid body j, C is the mass center, is the moment of inertia matrix about the first input end I₁, [m_xC,j, m_yC,j, m_zC,j]^T is the column matrix of external moment coordinates acting on the rigid body, [f_xC,j, f_yC,j, f_zC,j]^T is the column matrix of external force acting on the rigid body, I_n is the nth input point, and O_l is the lth output end.

12.6.2.3 Launch Tube (Element )

For the launch tube, if the lognitudinal and torsional deformations are neglected, we obtain

(12.68)

where x_j are the coordinates of the point on elastic beam j, l_j is the length of beam j, is the mass per unit length of beam j, EI is bending stiffness, m_j is the total mass of beam j, m_j=l_j and J_j,x is the moment of inertia with respect to the axis of beam j.

So we obtain

(12.69)

The augmented eigenvector of the MLRS is

(12.70)

where

(12.71)

From the orthogonality of the augmented eigenvectors of the multibody system established in this book, we obtain

(12.72)

where is the natural frequency of the system and M_p is the p th rank mode mass.

It is then possible to solve the dynamic responses of the MLRS once the following problems have been solved: calculating eigenvalues, constucting and verifying the orthogonality of the augmented eigenvectors, and establishing the launch dynamics equation of the MLRS. Some ways are offered here to calculate and analyze dynamics performance and the quantitative relation between dynamics performance and the total parameters of system, and the basis for decreasing rocket consumption in the MLRS test and improving the weapon performance is provided. As the multi‐uninterrupted firing weapon, the initial conditions of each firing and its vibration characteristics of corresponding system are different, and attention should be given to this when analyzing the dynamics response.

12.6.3 Dynamics Response of the System

Substituting Equations (12.64) and (12.63) into Equation (12.61), we obtain

(12.74)

Finding the inner products of and the terms on both sides of Equation (12.74), we obtain

(12.75)

So can be obtained by substituting the column matrix of external force f and step‐by‐step integration of the above equation, using the Equation (12.64), gives the exact analysis of the dynamics response and high‐speed computation of the MRFS for the MLRS. This is an exact analysis because the difficult problem of orthogonality of eigenvectors of the MRFS for the MLRS is solved by the MSTMM, and it is a high‐speed computation because the problem of computation of eigenvalues and the dynamics response of the MRFS for the MLRS is also solved by the MSTMM, which ensures that the order of the involved matrix is low all the time, which cannot be achieved by other methods.

12.6.4 Initial Value of Each Firing for the System

For the MLRS, the condition of the rocket launcher is changed with each rocket being fired, that is, when t is changed from to , the relative positions of point , point , point , point , and point are also changed. The mass, relative position of mass center, and inertial matrix of body 23 are changed, as are the eigenfrequency and eigenvector of the system. In this case, it is wrong to regard the numerical value of the response at time as that of time directly. It is necessary to convert the response at time into the initial value of the dynamics response relative to the new balance position, and to deduce the system generalized coordinates in the new mode of the system from the initial value. Superscript i shows the squence number of the rocket fired, superscript + means after firing, while superscript – means before firing. Supposing the initial value of the system is known before firing the th rocket, that is to say, the initial value at time is known. How to determine the initial values of and corresponding to the balance position when the system contains rockets is as follows. The detailed calculation method can be seen in reference [1].

The weight of the rocket being fired is considered in the launch dynamics equations, and its action affecting the rocket launcher is embodied by contact force. The initial conditions of the dynamics response of the MLRS are displacement and velocity when one rocket is fired, and the eigenvectoer, balance position and dynamics response relative to the balance position of the system excluding the fired rocket. Although the balance positions of the system in every firing are different, they are all static; the difference in the orientation angles of two neighboring balance positions caused by the weight of each rocket is small. So for two neighboring balance positions, the projections and of system velocity in the corresponding coordinate system can be regarded as being the same, where I is a point fixed on body 7,……, 12, 19, 21 or 23. In interval , the corresponding launch tube and rocket can be regarded as parts of body 23, so linear the displacement and angular displacement of every point at bodies , and are determined by the rigid body motion of body 23.

12.6.4.1 Initial Value of the First Rocket Being Fired

When the first rocket is fired, the other 39 rockets are contained in the system and the initial value relative to the balance position is

(12.76)

where is the displacement of point I at the moment of t relative to the balance position of the system when the launch tube contains rockets, represents the difference between the displacement relative to the balance position of the system when the launch tube contains rockets and that when launch tube contains rockets.

When the first rocket is fired, the system initial value relative to the balance position of the system with a full load is

(12.77)

12.6.4.2 Initial Value of the Second Rocket Being Fired

When the second rocket is fired, the other 38 rockets are contained in the system. The mode shape of the system, the balance position and the dynamics response are all relative to the balance position of the system with 38 rockets, namely

(12.78)

When the second rocket is fired, the system initial value relative to the balance position of the full‐load system is

(12.79)

12.6.4.3 Initial Value of the ith Rocket Being Fired

When the ith rocket is fired, there are rockets contained in the system. The mode shape of the system, the balance position and the response are all relative to the balance position of the system containing rockets, namely

(12.80)

When the ith rocket is fired, the initial value of the system relative to the balance position of the full‐load system is

(12.81)

where

(12.82)

The initial values of each rocket are determined as shown above, including the values relative to the corresponding balance position and the balance position of the full‐load system. These values are used to calculate the dynamics response of the system at every period. Finally, these dynamics reponses of the system represented in their local equilibrium positions will be converted to the corresponding values represented in the initial (full‐load) equilibrium position.

12.6.5 Solving the Dynamics Response of the System

After the first rocket is fired, from the initial conditions we obtain

(12.83)

Solving Equations (12.74) and (12.75) and taking notice of Equation (12.83), q^k(t), in interval , the dynamic response of the system, the rocket motion in the launch tube and the initial disturbance can be obtained.

According to this calculation, , , displacement and velocity at time can also be obtained. Using Equation (12.78), displacement and velocity relative to the balance position of the system at time (38 rockets are contained in the system) can be obtained. Similar to determining and , and after the firing of the second rocket can be obtained. Using the augmented eigenvector V^k corresponding to interval , using Equations (12.73) and (12.75), q^k(t), in interval , the dynamics response of the system, the parameters of rocket motion in the launch tube, and the initial disturbance can be obtained. Proceeding with this, the dynamics response of the system during the whole firing process can be obtained as follows:

(12.84)

12.7.1 Coordinate Systems

The o₁xyz coordinate system (translational coordinate system): The origin o₁ is attached to the geometric center of the rocket projectile at the initial position of the rocket, the x axis is directed toward the launch tube muzzle and along the tangent of the launch tube axis, the y axis is directed along the vertical plane and perpendicular to the x axis, and the z axis completes the triad, so the direction of every coordinate axis is then fixed. The system is mostly used to determine the basis of the orientation of the rocket, as shown in Figure 12.3.

The o_ox_oy_oz_o coordinate system (rocket launcher coordinate system): The origin o_o is the crosspoint of the cross‐section of the rocket at the mass center and launch tube axis. The x_o axis is directed toward the launch tube muzzle and along the tangent of the launch tube, the y_o axis is directed along the vertical plane and perpendicular to the x_o axis, and the z_o axis completes the triad. The system is used to determine the basis of the orientation of the rocket relative to the launch tube.

The o₁ξηζ coordinate system (rocket projectile coordinate system): The origin o₁ is attached to the geometric center of rocket projectile, the ξ axis is directed toward the head of the rocket and along the geometric longitudinal axis of the rocket, the η axis is directed along the vertical plane and perpendicular to the ξ axis, and the ζ axis completes the triad. As shown in Figure 12.4, the system is used to determine the orientation of the rocket axis.

The o₁x₂y₂z₂ coordinate system (velocity coordinate system): The origin o₁ is attached to the geometric center of the rocket projectile, the x₂ axis is directed toward the velocity of the geometric center of the rocket, the y₂ axis is directed along the vertical plane and perpendicular to the x₂ axis, and the z₂ axis completes the triad. As shown in Figure 12.5, the system is used to determine the translation of the rocket.

12.7.2 Forces Analysis of the Launching Process

Forces on the MLRS include weight, the mechanic interactive forces among the components and between the MLRS launcher and the ground, the projectile‐launcher action force, the thrust of the engine, the impact force of the jet flow and so on. Mechanic interactive forces among components and between the MLRS launcher and the ground may be equivalent to the corresponding spring and damping forces of elastic joints with damping, which are all internal forces of the system. In solving the launch dynamics of the MLRS using the MSTMM, the supporting forces of the ground are included in the system dynamic model as internal forces, and these internal forces are all considered in the state vector and need not be expressed with intricate external forces. These are also features of solving the launch dynamics of the MLRS by the MSTMM.

The static balance position of the system is regarded as the basis of system displacement, where the rockets fired and being fired are not included in the system. In this way, the function of system weight (including all components and the rocket not fired) in dynamics equations is offset by that of the static deformation of system components, so in the chosen coordinate system the action related to the weight of the system and the static deformation need not be considered, which greatly simplifies the forces analyzed and the dynamics solved for the MLRS. The large impact force on the rocket launcher [265] is made by the rocket exhaust, which causes vibration of the rocket launcher. The initial disturbance of the subsequently fired rocket is affected, and the firing dispersion of the MLRS is also affected. The forces between the rocket and the launch tube are a pair of acting forces and counterforces. The kinetic equation of the rocket launcher and the vibration equation of the rocket launcher couple together, via the interaction force between bourrelet and launch tube, as well as the interaction force between the guide button and the guide trough. These forces include the normal collision force and the tangential friction force between the bourrelet and the launch tube, and between the guide button and the guide trough. That the weight of the fired rocket affects the rocket launcher is demonstrated by the contact force between the rocket and the launcher. The thrust of the engine and the impingement force of the rocket exhaust on the firing equipment can be calculated theoretically or measured by testing.

The forces acting on the rocket in the launch tube include weight, the thrust of the engine, air resistance, as well as the force between the launch tube and the guide button, and the contact force between the bourrelet and the launch tube. The expressions are given in reference [1].

12.7.3 The Launch Dynamics Equation of the Rocket

As launch dynamics is very complex, the launch dynamics of a multibody system such as an MLRS with high temperature, high pressure, short time and violent change cannot be solved using the usual kinds of research methods. In the research field and the expansive literature of launch dynamics at home and abroad, the model of launch dynamics has had to be simplified again and again, so the differences between the model and the reality are so great that the resulting level of error is large. In launch dynamics research an assumption is always made by the researcher that the rear bourrelet center moves along the bore axis as this simplifies the research problem. In 1984, the general differential equation of motion of a projectile in a bore was developed by Marting, an American expert in weapon dynamics [266]. To overcome the shortcomings of the Marting’s equation, a uniform launch dynamics equation of a rocket suitable for all kinds of motion has been established by the authors. Not only are the factors considered more general and closer to reality than those of Marting’s model, but also the form of the differential equation of motion is more concise than that of Marting’s equation. In the model of launch dynamics, the rocket weight, thrust, thrust tmalalignment angle, linear thrust malalignment, rocket‐tube clearance, contact force between bourrelet and launch tube, mass eccentricity, and dynamic unbalance are all considered. The translational and rotational equations are established in the rocket launcher coordinate system o_ox_oy_oz_o and the rocket axis system o₁ξηζ, respectively [1].

The general dynamics equations of an asymmetric rocket in the launch tube are

(12.73)

where

, and are the longitudinal, vertical and sidewise accelerations, respectively, of the mass center of the rocket projectile in the rocket launcher coordinate system o_ox_oy_oz_o, , and are the projections in the rocket launcher coordinate system o_ox_oy_oz_o of acceleration of the mass center of the rocket launch tube with respect to the ground coordinate system, F_p is thrust, μ is the frictional coefficient between the guide button and the launch tube, m is the mass of the rocket, θ₁ is the departure angle of the rocket launcher, γ is the roll angle of the rocket, γ₀ is the initial orientation angle of the guide slot, g is the acceleration due to gravity, a_p is the axial acceleration at the origin point o_o of the rocket launcher coordinate system o_ox_oy_oz_o for launching the tube, and are the vertical and sidewise components of the contact force between the rear bourrelet and the launch tube, respectively, and and are the vertical and sidewise components of the contact force between the front bourrelet and the launch tube, respectively. and are the vertical and sidewise components of the bracket angle between the projectile spin axis and the tangent of the launch tube axis, and are the projections of thrust malalignment angle β_p in axes η and ζ of the rocket axis coordinate system o₁ξηζ, respectively, α is the twist angle of the guide flute, r_b is the neutral circular radius of the guide button, and are the vertical and sideways components of the bracket angle between the tangent of the launch tube axes and the aiming line, respectively, A is the transverse moment of inertia of the rocket, C is the polar moment of inertia, and are the projections of dynamic unbalance angle in axes η and ζ of the rocket axis coordinate system o₁ξηζ, respectively, and are the projections of mass eccentricity in axes η and ζ of the rocket axis coordinate system o₁ξηζ, respectively, L_η and L_ζ are the projections of thrust malalignment in axes η and ζ of the rocket axis coordinate system o₁ξηζ, respectively, l_R is the distance between the mass center of the rocket and the rear bourrelet, and l₁ is the distance between the mass center of the rocket and the front bourrelet.

12.8.1 Composition, Flow and Functions of the Simulation System

The launch dynamics simulation system of the MLRS based on the MSTMM is as follows:

1. General control module: This module is used to mobilize all function modules, simulate the launch dynamics, carry out the statistical analysis and output the results.
2. Random number module: This module produces the random number for other the function modules according to the statistical characteristics of the random parameter, which are prescribed by a specified distribution.
3. Weapon vibration characteristics module: This module is used to simulate the eigenfrequency, eigenvector, damping ratio and so on.
4. Interior ballistics module: This module is used to simulate the thrust and interior ballistics with the random number obtained in (2) as one part of the basic parameter.
5. Impingement force of the rocket exhaust module: This module is used to simulate the impingement force of the rocket exhaust.
6. Weapon dynamic characteristics of the firing process module: This module is used to simulate projectile and barrel motion, projectile–barrel interaction, and the initial disturbance of the rocket according to the data obtained in (2), (3), (4) and (5).
7. Exterior ballistics module: This module is used to simulate the ballistics data of the rocket projectile and hitting point scattering according to the data obtained in (2), (4) and (6).
8. Statistical analysis module: This module is used to deal with the simulation results statistically.
9. Correlation analysis module: This module is used to analyze correlation among all kinds of factors, the dynamic performance of the MLRS and the firing accuracy according to the data obtained in (2), (6) and (7), to offer the primary influence factors of dynamic performance of the MLRS and the firing accuracy, and to provide the basis for the general parameter dynamic design of the MLRS.
10. Result output module: This module is used to output simulation results in the form of character, curve and figure etc.

The simulation flow of the launch dynamics of the MLRS based on the MSTMM is shown in Figure 12.6.

The functions of the launch dynamics simulation system based on the MSTMM are:

1. Simulation of the vibration characteristics of the MLRS and its variance caused by the variance of the total rocket number in the launcher.
2. Simulation of the dynamic response of the MLRS (including displacement, velocity, acceleration, angular displacement, angular velocity, angular acceleration etc.).
3. Simulation of the initial disturbance of the rocket (including initial swing angle, initial swing velocity, muzzle rotation angle, rotation angular velocity, displacement, velocity, and other motion parameters).
4. Simulation of exterior ballistics and firing precision of the MLRS (including exterior ballistics data, ballistics parameters of burning phase end, flight time, points of fall, firing accuracy, firing dispersion, etc.).

12.8.2 Simulation Results of the Dynamics Response of the System

The firing sequence and the loading position of a MLRS with 40 launch tubes are shown in Figure 12.7. The loading position of every rocket is denoted with a dual (n₁, n₂), where n₁ shows the row number with 1, 2, 3 and 4 from up to down, while n₂ shows column number with 1, 2… 10 from left to right. In the figure, each circle represents a launch tube, the nearside number inside the circle represents the firing sequence of a 40‐rocket salvo, and the dextral number inside the circle represents the loading position of the rockets. For example, the loading position of the 26th rocket in the firing sequence is represented by (4, 6).

Using the launch dynamics simulation system of the MLRS based on the MSTMM, the launch dynamic simulation results of a MLRS with 40 launch tubes and rockets in the process of firing are obtained as follows.

12.8.2.1 Simulation Results of the Launch Dynamics of a MLRS with Single Firing

The numerical simulation of an MLRS with 40 launch tubes single firing was carried out. Some of the simulation results of the launch tube vibration during single tube firing are shown in Figures 12.8–12.17. The time histories of the motion and the impact forces of the rocket are illustrated in Figures 12.18–12.27.

12.8.2.2 Simulation Results of the Launch Dynamics of a MLRS with a 40‐rocket Salvo

The numerical simulation for an MLRS with a 40‐rocket salvo was carried out. Some of the numerical simulation results are shown in Figures 12.28–12.43.

12.8.3 Simulation Result of the Initial Disturbance of the Rocket

The initial disturbance of an MLRS with 40 launch tubes is simulated by the launch dynamics simulation system of the MLRS based on the MSTMM. Some of the numerical simulation results are shown in Table 12.1.

	Swing angle		Swing angular velocity		Yaw angle
	φa/rad	φ2/rad			ψ1/rad	ψ2/rad
Mean value	0.866	−2.03 × 10−5	−0.102	−5.07 × 10−3	−4.19 × 10−3	−9.45 × 10−4
Mean square	4.96 × 10−5	5.61 × 10−5	3.03 × 10−3	2.40 × 10−3	6.27 × 10−5	6.04 × 10−5

12.8.4 Simulation and Test Validation of the Vibration Characteristics of the System

The simulation results and the mode test results of the first 14 eigenfrequencies of an MLRS with 40 launch tubes under the condition of full‐load and non‐load are shown in Tables 12.2 and 12.3, respectively. The comparison indicates that they have good agreement, and the simulation results are validated by a modal test.

12.8.5 Simulation and Test Validation of the Vibration of the System

The simulation results and test results for the vehicle response of an MLRS with a salvo of 40 launch tubes are shown in Figure 12.44 (departure angle is 50°, azimuth angle is 45°) and the simulation results and test results of the vehicle response when firing 16 rounds with the B scheme are shown in Figure 12.45. It is obvious that the simulation and test results have good agreement. The simulation results are validated by the tests.

12.8.6 Simulation and Test Verifying of Firing Dispersion

The simulation and test results for firing dispersion at a certain range of an MLRS with 40 launch tubes are shown in Table 12.4 and show good agreement. The simulation results are also validated by the tests.

12.9.1 Principle of Decreasing Rocket Consumption in MLRS Tests

12.9.1.2 Statistics Principle of Decreasing Rocket Consumption in the MLRS Test

The impact coordinates are random variables and are usually considered to have a normal distribution. The distance coordinate of the impact point is given by x and the transverse coordinate by z. Both these show normal distribution and are independent from each other, so the situation can be explained by discussing only distance x. The probability p_E(n) is called the confidence level of the estimated value located in interval . For different DOFs (where n is the number of test rounds), the probability is p_E(n) and the value is found from the computation where is the confidence level of estimated value located in interval when testing infinite rounds and denotes the confidence level of replacing the estimated value of infinite rounds by estimated value of n rounds. The confidence levels of the estimation of the mean and its variation tendency along with the number of test rounds are shown in Table 12.5 and Figure 12.46.

n	3	4	5	6	7	8	9	10	40
pE(n)	0.429	0.450	0.461	0.469	0.474	0.477	0.480	0.482	0.496	0.50
	0.858	0.900	0.922	0.938	0.948	0.954	0.960	0.964	0.992	1

It can be seen from Table 12.5 and Figure 12.46 that when the number of test rounds bigger than seven, the confidence level increases very slowly and at the same time the confidence level reaches then exceeds 94.8% of that of the infinite rounds test. Therefore, for the same batch of ammunition with steady performance and single‐round firing with a launch tube, only seven rounds are needed to describe the influence of random factors of the rocket itself on the mean value of the coordinates of the points of fall.

When estimating firing dispersion, we usually suppose that it is a random variable with normal distribution. E is the system firing dispersion and its estimated value is Ê, where and . The probabilities (confidence level) of the estimated value of the standard deviation located in interval are shown in Table 12.6 and Figure 12.47 for . This is also the probability (confidence level) of the estimated value Ê of the dispersion located in interval . The probability (confidence level) of the estimated value Ê located in interval is therefore 91.7% for seven test rounds with a launch tube. Table 12.6 and Figure 12.47 show that when the number of test rounds is more than seven, confidence level increases slowly as number of test rounds increases.

n	3	4	5	6	7	8	9	10	40
Confidence level	0.682	0.779	0.842	0.886	0.917	0.938	0.954	0.966	0.999	1

Taking both of these factors into account, firing seven rounds is the routine method for the firing dispersion test of a weapon with a single launch tube for the same batch of ammunition with steady performance, the basis being that the corresponding confidence level of the estimated value of the firing dispersion can reach 91.7%. For an MLRS with 40 launch tubes, if the number of test rounds is bigger than seven in one group the confidence level of the estimated value of the firing dispersion should reach 91.7%. If the influence of factors such as coupling between the rocket and the gun on firing dispersion is considered, it is necessary to optimize the loading position, firing sequence and firing interval when searching the test scheme, substituting the nonfull‐charge loading test for the full‐charge loading test.

12.9.2 Programming Technology of Substituting Nonfull Loading for Full Loading

A difficult engineering problem that urgently needs to be solved is how to optimize and design a system that includes random and discrete variables at the same time because the existing theories and methods of optimization design cannot do this. Because of the low efficiency of optimization design for discrete variables and the sample effect of mathematical statistics which results in huge computational work, it is difficult to determine the optimal solution for the global area. Compared with the case of continuous variables, the optimization design of discrete variables has many different characteristics, such as the discontinuity of the variable, the discontinuity of the feasible area space (feasible area becomes feasible assemblage), the undifferentiability of the function, the nonapplicability of the Kuhn–Tucker condition and so on. These features make the optimization design for discrete variables different to that for continuous variables. Not only is the selection method for the chosen design variable different, but the judgement condition of optimization is also different. To address this problem, a totally new method for optimization design, random integer programming, has been developed by the authors. In this method, the continuity, discrete, randomicity and hypothesis tests of design variables are considered, with the aim of first ascertaining the necessary solution space and then solving the optimal solution. Thus, the difficult problem, namely that optimization design for discrete variables based on statistics cannot be used in large engineering due to its low computation efficiency, is overcome. The engineering optimization problem of decreasing rocket consumption in the firing dispersion test of an MLRS is also solved using this method. This method has been verified and further details can be found in reference [1].

The general approach to decreasing rocket consumption in the firing dispersion test of an MLRS is to take the MLRS system as research background, synthetically apply the new theory and technology of launch dynamics based on the MSTMM, adjust the loading position, firing sequence and firing interval of the nonfull‐loading scheme, and search the nonfull‐loading scheme whose firing dispersion is same as that of the original 40‐rocket full‐loading scheme. In this way rocket consumption in the MLRS test can be freatly decreased as long as the test quality is assured.

The detailed procedure to decrease rocket consumption in the firing dispersion test of the MLRS is to apply the newest study results for the launch dynamics based on the MSTMM, along launch scheme → initial disturbance of rocket → dispersion of the MLRS, taking the theory of launch dynamics of the MLRS as the foundation, the launch dynamics simulation system as the core, the random integer programming [1] as a tool, using the same firing dispersions for two systems as the constraint condition, taking the minimum rocket consumption as the target, by mean of adjusting loading position, firing sequence and firing interval, through plenty of computation and optimization, getting a new test scheme, and inspecting whether the estimated values of firing dispersion for the two schemes are distinctly different with statistical methods. Thus, the new test scheme substituting full loading with non‐full loading for dispersion test of MLRS can be found. The programmed technology for decreasing rocket consumption in the MLRS test by substituting full loading with nonfull loading is shown in Figure 12.48. The steps involved are as follows:

1. Select the applicable launch dynamics model of the MLRS from the library of models and determine the statistical property (such as mean and standard deviation) of the correlated parameters according to measured parameters.
2. Using the launch dynamic simulation system of the MLRS based on the MSTMM, carry out simulations for the firing dispersion of the MLRS.
3. Optimize the test scheme, substituting full loading with nonfull loading for dispersion of the MLRS using random integer programming.

12.9.3 Optimization of a Nonfull‐load Firing Scheme and Test Verifying

When determining a nonfull‐loading MLRS system A₁ to simulate the firing dispersion of a full‐loading salvo, random integer programming is used to optimize the firing dispersion test scheme of the MLRS. The factors that influence the dispersion of the MLRS are divided into two types, one is decided by the inherent mechanics characteristics of the rocket‐launcher system and the other is decided by various random factors. The main influence of the inherent mechanics characteristics of a rocket‐launcher system on firing dispersion is the change in the initial disturbance of the rocket due to the difference in rocket‐launcher vibration produced by the mass and stiffness of the rocket launcher, firing sequence, interval of the rocket, rocket exhaust and so on.

The main influence of various random factors is the change in rocket ballistics caused by the randomness of parameters such as rocket‐tube clearance, dynamic unbalance, mass eccentricity, thrust malalignment and firing interval. For the same batch of rockets, with steady performance and the same rocket launcher, the statistical properties of various random parameters are steady, their influence on firing dispersion is rather slight, but the variance of the inherent mechanics characteristics of the MLRS (including loading position, firing sequence, and so on) is main factor that influences the variance of the firing dispersion. The influence of these factors on firing dispersion must be fully considered when determining a test scheme. The firing dispersion E_A of the MLRS is a multivariable function that has the following form:

(12.85)

where is a constant structure parameter of the system which influences firing dispersion, is the rocket loading position vector (), x_i is an integer variable equal to 0 or 1, and is the mean of firing interval t.

Let , , , and . The random part of the firing interval is considered in , where is a random vector that influences the firing dispersion (such as mass of rocket, thrust, thrust malalignment, dynamic unbalance, mass eccentricity, propellant mass, the random part of firing interval, or weather condition). Because the influence of random vector and constant structure parameter on the firing dispersion E_A of full‐load salvo system A₀ and nonfull‐load system A₁ is steady, and are the main factors causing a difference in the firing dispersion between the two systems. and are the main design variables when determining the nonfull‐loading system, and the influence of various random factors on system firing dispersion and the discontinuity of must be considered at the same time. The following discrete variable mathematical programming problem based on statistics is constructed:

(12.86)

where y is the target function, x_i is the loading position function of the rocket for the MLRS with 40 launch tubes, , where x_i is an integer equal to 0 or 1, x_i = 0 means there is no loading rocket in the ith position of the launch tube and x_i = 1 means a rocket is loaded in the ith launch tube. is a hypothesis test of the firing dispersion of the full‐load salvo system A₀ and the target system A₁.

On the foundation of the launch dynamics of the MLRS based on the MSTMM, using the method of random integer programming [1], the nonfull‐load system A₁ is obtained with an estimated value of firing dispersion that is not distinctly different from that of the full‐load salvo system A₀, so the firing test scheme substituting full‐load salvo with nonfull‐load is obtained:

1. number of rounds in each group: 7
2. firing interval: 0.5 ± 0.05 s
3. loading position and firing sequence: 1 (2,6), 2 (3,6), 3 (4,5), 4 (1,7), 5 (2,4), 6 (2,7), 7 (3,4).

In Figure 12.49 each circle represents a launch tube, the number on the left of each circle represents the original firing sequence in a 40‐round salvo, and the number on the right of each circle represents the new firing sequence in the seven‐round string firing scheme. The serial numbers of the launch tube corresponding to the seven‐round string firing scheme are 6, 22, 28, 10, 16, 14, and 32.

According to the static measure results for the rocket (including rocket weight, mass eccentricity, eccentricity position, bourrelet diameter and so on), using the established simulation system, the firing dispersions of the seven‐round string firing scheme and the original 40‐round salvo are numerically simulated. The variance trends for firing dispersion along with the number of firing groups for the seven‐round string firing scheme and the original 40‐round salvo scheme are shown in Figure 12.50, and the critical values of the F test is shown at the same time. It can be seen from the predicted results that the variation tendencies of the firing dispersions of the two schemes share the same pattern as the number of the test groups increases, and the firing dispersions always meet the requirement of the F test, which shows that the two schemes have the same firing dispersion. In order to verify the correctness of the seven‐round string firing scheme, special dispersion contrast tests for both the seven‐round string firing scheme and the 40‐round salvo scheme are completed. The test results prove that the seven‐round string firing scheme is correct and the test technology for the firing dispersion of the MLRS substituting full‐charge loading with nonfull‐charge loading is effective. The simulation and test results for the firing dispersion of the seven‐round string firing scheme and the 40‐round salvo are shown in Table 12.7 and are in good agreement. The F test shows that there are no distinct differences between the test results for the seven‐round string firing scheme and the 40‐round salvo when , so the system firing dispersions of the two launch schemes are the same, which validates the seven‐round optimization test scheme.

The following conclusions can be drawn from the test results:

1. The simulation results are validated by the test, and the launch dynamics theory of the MLRS based on the MSTMM is also validated.
2. The test results for the firing dispersion of the seven‐round firing scheme and the 40‐round salvo are same, so the seven‐round optimization test scheme is validated, which shows that the technology for decreasing rocket consumption in the MLRS test is right and that the rocket consumption in the firing dispersion test of the MLRS is decreased by 82.5% compared to the common method.

The difficult problem of decreasing rocket consumption in the MLRS test was solved by the test technology. Under the precondition of ensuring test quality, the test budget greatly decreases when the firing dispersion test of the MLRS of the full‐loading salvo is substituted with the nonfull‐loading scheme, and such a substitution has good operability and can be easily applied in engineering practice.

12.10.1 Technique for Improving the Firing Dispersion of the System

A lot of research for improving the firing dispersion of the MLRS has carried out by rocket experts all over the world. The concept of passive control [270–272] for improving firing dispersion was first put forward by John E. Cochran. The basic idea is that the ballistic difference caused by the initial disturbance of the rocket could be counteracted by the ballistic difference caused by thrust malalignment, dynamics unbalance and mass eccentricity by a thorough study of the amplitude frequency and phase frequency response function of the launch equipment. The main purpose of passive control is to improve the firing dispersion, mainly by reducing the angular velocity dispersion of the deflection angle at the end of the power period. No outside energy source is needed, the control is realized only by using the feedback of rocket launch system itself and the control sources are rocket’s own random factors. Although passive control looks tempting in theory, the model is very simple, which is very important in reality because it lacks a way of establishing the quantitative relation between the total parameters of the MLRS and its dynamic performance. This means it is difficult to reach the goal of reducing initial disturbance through adjusting the total parameters of the MLRS, and so far there has been no practical application of this result. It is, however, an important way of using simple control to improve the firing accuracy of the MLRS, for example a rocket with simple control has been used in the Russian PC30 CMEPЧ MLRS and the US M270 MLRS, and is being developed in Italy, Britain and other countries to improve the firing accuracy of the MLRS. The cost of a rocket with simple control is several times higher than that of an ordinary rocket. The search continues around the world for a new method of improving the firing accuracy of the MLRS.

The firing technology of an MLRS with high precision has been established by the authors using the theory and method of MSTMM launch dynamics. The main features of this technology are as follows:

1. Through optimizing the parameters of the rocket‐launcher system (such as firing sequence and firing interval) the initial disturbance can be reduced.
2. Using the concept of equivalent initial disturbance, through the design of the parameters of the rocket‐launcher system, the initial disturbance of the rocket counteracts the equivalent initial disturbance caused by its own deficiency (mass eccentricity, dynamics unbalance, thrust malalignment and other factors). In addition, the ballistic difference caused by the initial disturbance of the rocket is counteracted by the ballistic difference caused by its own deficiency.
3. The integration soft/hard module of launch dynamics is installed on a rocket launcher to control the initial disturbance of the rocket by controlling the motion of the launch tube of the MLRS. This greatly reduces the cost of improving the firing precision of noncontrol MLRSs compared to using simple control. This method and its application in improving the firing dispersion of an MLRS by reducing the initial disturbance due to optimizing the firing sequence and firing interval are discussed in this section.

The MLRS is a kind of string‐firing weapon, and its firing sequence and firing interval have a very big influence on its dynamic performance. It has been shown that different firing sequences and different total system parameters will lead to different vibration characteristics and different initial conditions for each rocket fired, resulting in different initial disturbance of the rocket. The matching relation between the firing frequency and the natural frequency also largely influences dynamic performance, and a reasonable firing interval is very important to ensure and improve firing dispersion. Therefore, the goal of improving the firing dispersion of the MLRS through reasonable adjustment of the firing sequence and firing interval can be achieved.

12.10.2 Random Combinatorial Optimization for Firing Dispersion

How can we optimize the firing sequence and firing interval to ensure the highest firing dispersion of the MLRS? For an MLRS with n launch tubes, there are n ! kinds of firing sequences. To obtain the optimum solution for the whole area, all kinds of firing sequences must be scanned one by one. If a group of calculations is taken as one scan, the total computation time is about 1.2 quadrillion years using a PIV‐3.2G computer, which is not feasible. In addition, the theory and algorithm of global optimization is not as advanced as that of local optimization, so the analysis condition similar to the condition “gradient is zero and the second derivative matrix is orthogonality” of the local minima point has not yet been achieved, that is to say, it there is not an affirmative method of judging whether or not a local minima point is a global minima point. Even the filling function method, a kind of global optimization algorithm proposed in recent years, can only solve an optimization problem whose target function is analytical expression of continuous variables and whose constructed filling function is an analytical function about target function and optimization variables, which is useless for an optimization problem about discrete variables. Because it is difficult to find an optimal solution of global area, many approximation algorithms have been constructed, including the greed method, the local scanning method and the incomplete enumeration method. These methods behave better in seeking preferable solutions. Based on these methods, focused on the essence of the problem, the arithmetic of random combinatorial optimization is constructed, in which a random combination is adopted in each optimization and every combination is forbidden to repeat the former. Thus, the local optimal solution can be searched and the firing dispersion of the original system is improved to meet engineering requirements.

In order to find the local optimal solution, a constraint condition for the firing sequence is first decided according to the research results of MLRS launch dynamics. The firing sequency (from far to near) is symmetrical about the center of the launcher, and the rocket nearest to the center of the launcher is fired at last. Next, a random firing sequence is produced, which must satisfy the constraint conditions, and the firing dispersion is simulated corresponding to the firing sequence combination and estimated using the maximum entropy method. Lastly, the hypothesis test is applied to judge whether or not the firing dispersion is higher than that of original firing sequence. After many random combinatorial optimizations, the firing sequence corresponding to a high firing dispersion can be obtained. In fact, the key to improve the firing dispersion via the optimization of the firing sequence is that the firing dispersion of the MLRS corresponding to different firing sequences can be simulated exactly, and the optimization of the firing sequence is merely a combinatorial optimization problem of mathematics. The flowchart for random combinatorial optimization is shown in Figure 12.51.

12.10.3 High Firing Dispersion Scheme and its Test Verification

Taking the launch dynamics simulation system of the MLRS to be a platform based on the MSTMM, the influences of various factors on firing dispersion, such as rocket, launcher, propellant and environment, are all taken into consideration, and the firing sequence and firing interval of the optimization firing scheme are obtained using random combinatorial optimization. The simulation results for the corresponding relation between firing interval and firing dispersion for an MLRS are shown in Figure 12.52, from which it can be seen that a reasonable firing interval for the weapon system is 1.0 s.

Many test results have indicated that firing dispersion is improved more than four times by the optimization scheme relative to the original firing sequence and firing interval.