Optimization Research of Firing Interval for Multiple Launch Rocket Systems

Abstract:

Optimization research of firing interval is applied to improve firing density level of MLRS. Firstly, the finite element model of MLRS is built to obtain the dynamic response of the launch guider muzzle and the rocket initial disturbance is calculated. Secondly, the equation of exterior ballistic is solved using Monte Carlo simulation and Runge-Kutta method and with mathematical statistic method the estimate of dispersion is obtained. Thirdly, modal analysis for MLRS is carried out and the new firing interval is proposed combined with the dynamic response obtained above. Finally, substitute the new firing interval into the model of MLRS, the results show that the new firing interval shorten the firing time by 21.4% and improve the firing density effectively without affecting the dispersion of the MLRS.

• Periodical: Applied Mechanics and Materials (Volumes 411-414)
• Pages: 3046-3051
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.411-414.3046
• Online since: September 2013
• Authors: Jian Lin Zhong*, Da We Ma, Jian Guo Hu
• Keywords: Dispersion, Dynamic Simulation, Firing Density, Firing Interval, MLRS, Modal Analysis

References

[2] to [6], based on transfer matrix method of multi-body system and launch dynamics theory, the influence of firing order and firing interval on the weapon system's performance of MLRS is researched, in reference [7] to [9], the rigid -flexible-coupling model of MLRS is built, a technical scheme of two guide tube simultaneous firing is proposed for improving the firing density and dispersion. The transport case of the MLRS which researched in this paper is small, so the rockets tend to collide in middle air if the technical scheme of two guide tube simultaneous firing is applied, on the premise that the dispersion won't be affected, shortening the firing interval reasonably is the selected way to improve the firing density in this paper. The MLRS is selected as the research object in this paper, the dynamic model is built and the initial disturbance is calculated. With the use of Monte Carlo simulation, n groups of random numbers are generated by the qualified random number algorithm. Substitute n groups of random numbers and the initial disturbance into the exterior ballistic equation, n groups of landing site location are calculated and the estimate of dispersion is obtained with Mathematical Statistic Method. The modal characteristic of MLRS is obtained using Lanczos algorithm and the new firing interval is proposed combined with the dynamic response of the launch guider muzzle, substitute the new firing interval into the model of MLRS, the new dispersion meets the design indices, the firing time is shortened and firing density is improved.

[2] dynamic Dynamic model and initial disturbance 2. 1 launch dynamic model of MLRS The initial disturbance is caused by the launching system. To provide parameters for the calculation of rocket initial disturbance, the dynamic model of MLRS is built and the vibration response of rocket launch tube is obtained. The dynamic model includes transport case, reeling frame, upper carriage, elevating mechanism, tire, cabin and so on. According to the real structure of the MLRS, the finite-element model of the weapon system is built with shell element and beam element, the tire and the hydraulic support leg are simulated by the nonlinear connecting unit while the cabin is built as display body . Replace the fully loaded ammunition with partial loaded ammunition to reduce the amount of the computation, the original firing interval is 0. 7s, the dynamic model of MLRS is shown in figure 1. Fig. 1 dynamic Dynamic model of MLRS Simulation calculation is done with Abaqus/explicit module, then the pitch and yaw angular rate , angular displacement, velocity of the launch guider muzzle in the bomb-axis coordinate system are obtained. Limited by space, the pitch and yaw angular rate and velocity of the 8th launch guider are shown from Fig. 2 to Fig. 5 . Fig. 2 pitch Pitch angular rate curve Fig3 yaw Yaw angular rate curve Fig. 4 pitch Pitch velocity curve Fig. 5 yaw Yaw velocity curve 2. 2 calculation Calculation of initial disturbance To calculate the loading firing dispersion of MLRS, the rocket initial disturbance should be taken into consideration, transfer the initial disturbance of MLRS obtained from simulation of dynamic model to the rocket initial disturbance. Suppose that the rocket is rigid, regardless of the gaps between projectiles and guns and micro-bendor of the launch tube, the motion modeling of the rocket in the semi-captive period is shown in Fig. 6. Fig 6 motion Motion modeling in semi-captive period The initial disturbance angular rate caused by the vibration of the MLRS can be written as (1) where, is angular rate of launch guider muzzle when the mid-bourrelet deorbit, is angular displacement of launch guider muzzle when the mid-bourrelet deorbit, is angular displacement of launch guider muzzle when last bourrelet deorbit, is the velocity of launch guider muzzle when the mid-bourrelet deorbit, is the velocity of launch guider muzzle when the low bourrelet deorbit, is the distance between the center of mass and the last bourrelet, is the distance between the mid-bourrelet and the low bourrelet, is the equatorial radius of gyration, is the velocity of rocket when the mid-bourrelet deorbit, is the velocity of rocket when the last bourrelet deorbit. Substitute the data obtained from the calculation of the dynamic model to equation (1), we get initial disturbance of the 8 rockets, the results are shown in Table 1. Tab. 1 Results of initial disturbance firing order pitch angular rate(rad/s) yaw angular rate(rad/s).

[1] -0. 00560606 -0. 00845675.

[2] -0. 00476794 -0. 0322077.

[3] 0. 00638398 0. 00807136.

[4] 0. 0042376 0. 0108227.

[5] 0. 00549997 0. 00243357.

[6] 0. 00383613 0. 00938455.

[7] 0. 00674465 -0. 00988541.

[8] 0. 00595192 0. 00979516.

[3] Simulation of Firing Dispersion Firing dispersion includes vertical target dispersion, air burst dispersion and ground dispersion. The ground dispersion is studied in this paper, firstly, based on the equation of exterior ballistic in reference, the pseudo random sequence is generated with the Monte Carlo method, secondly, secondly, substitute the pseudo random sequence and initial disturbance into exterior ballistic equation, landing site location is calculated, finally, repeat the steps above n times until calculation accuracy meet the demand of Monte Carlo method. Based on the idea above, the simulation system is built with Matlab, calculate the exterior ballistic with Runge-Kutta method. This method is actually based on Tailor Progression. （2） where, is the remainder. The error is in proportional to h if the forth order derivative is used. The formula to calculate the increasing function in the Runge-Kutta method is as follows （3） where Calculate the derivative, in equation (2) and express them with the first order derivative , and operates ,, etc., substitute them into equation (3). Then let the increasing function in the equation(2) and (3) equals with each other. The appropriate value of the undetermined coefficient,, are selected to meet the demand that all coefficient in ,, , equal with each other. Then substitute these values into equation (3), we get the Runge-Kutta increasing function formula: （4） Calculate derivative of the first and last pointsand middle points , then the average value of them are calculated, use the value as gradient of the secant line between the first and last points to calculate the value of the increment. n groups of landing site location , , is obtained after calculation, the dispersion , is obtained with math- ematical statistic method, the simulation results and the design indices of loading firing dispersion is shown in table2. Tab. 2 Simulation results and the design indices of loading firing dispersion loading firing dispersion Simulation value 1/207 1/133 design indices 1/190 1/120.

[4] optimization Optimization of firing interval If the firing frequency and the natural frequency are close to each other, the resonance phenomenon will occur, so the modal analysis model of MLRS is built, the top 5 inherent frequencies are obtained. Limited to space, the top 2 inherent frequencies are shown in Fig7 and Fig8, the frequencies and period of top 5 inherent are shown in Tab3. Fig. 7 first First inherent modal response Fig. 8 second Second inherent modal response Tab. 3 frequencies Frequencies and period of top 5 inherent Inherent Frequency/Hz Period/s.

DOI: 10.1109/ivelec.2002.999301

[1] 2. 901 0. 345.

[2] 3. 846 0. 26.

[3] 6. 347 0. 158.

[4] 6. 84 0. 146.

[5] 8. 737 0. 114 The original firing interval of MLRS is 0. 7s, the flight time inside the launch guider is 0. 2s, the MLRS start to level off when the rockets depart from the launch guider for about 0. 2s based on Fig2 to Fig5, so we can fire the next rocket, take natural frequency and the dynamic response into consideration, the firing interval is shortened to 0. 55s.

[5] dispersion simulation with new firing interval Substitute the new firing interval into the model of MLRS, simulation and calculation are carried according to the second section, limited by space, the pitch and yaw angular rate of the 8th launch guider are shown in Fig. 9 and Fig10. The initial disturbance and the dispersion with the new firing interval are shown in table4 and table5. From the values in table4 and table5, we get that the new firing interval shorten the firing time and the firing density is improved effectively. Fig. 9 pitch Pitch angular rate curve Fig10 yaw Yaw angular rate curve Tab. 4 Results of initial disturbance with the new firing interval firing order pitch angular rate(rad/s) yaw angular rate(rad/s).

DOI: 10.7554/elife.38169.011

[1] -0. 005613068 -0. 008474847.

[2] -0. 004721432 -0. 032283066.

[3] 0. 006399665 0. 008090247.

[4] 0. 004328708 0. 010820059.

[5] 0. 005518945 0. 002432976.

[6] 0. 003849365 0. 009416927.

[7] 0. 006767919 -0. 009919515.

[8] 0. 005972454 0. 009828953 Tab. 5 Simulation results and the design indices of loading firing dispersion the new firing interval Dispersion Original simulation value 1/205 1/132 New simulation value 1/201 1/128 Design indices 1/198 1/125.

[6] Conclusions In this paper, the launch dynamics model of MLRS is built with finite element method, the dynamics response of the launch guider muzzle is obtained, the initial disturbance is calculated based on the motion modeling in semi-captive period, then we get the disper- sion of the original firing interval, take the dynamics response and modal characteristic into consideration, the new firing interval is proposed, the simulation of dynamics model with the new firing shows that the new firing interval do not affect the dispersion while the firing time is shortened by 21. 4%, the firing density is improved, the research method can provide technical support for the improvement of firing density of MLRS. References.

DOI: 10.1115/imece2022-95035

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