Method for Determining the Time Parameter

Nauka i Obrazovanie

K. P. Baslyk¹  N. N. Generalov¹  B. G. Kuleshov¹ 

[1] Bauman Moscow State Technical University;

关键词: launch vehicle;    active trajectory leg;    program of flight;    the program of attack angle;    project and ballistic parameters;    equations of motion of the launch vehicle;    multiplier method;    modified Lagrange function;    Newton method;   

DOI  :  

来源: DOAJ

【 摘 要 】

This article proposes a method for calculating one of the characteristics that represents the flight program of the first stage of ballistic rocket i.e. time parameter of the program of attack angle.

In simulation of placing the payload for the first stage, a program of flight is used which consists of three segments, namely a vertical climb of the rocket, a segment of programmed reversal by attack angle, and a segment of gravitational reversal with zero angle of attack.

The programed reversal by attack angle is simulated as a rapidly decreasing and increasing function. This function depends on the attack angle amplitude, time and time parameter.

If the projected and ballistic parameters and the amplitude of attack angle were determined this coefficient is calculated based the constraint that the rocket velocity is equal to 0.8 from the sound velocity (0,264 km/sec) when the angle of attack becomes equal to zero. Such constraint is transformed to the nonlinear equation, which can be solved using a Newton method.

The attack angle amplitude value is unknown for the design analysis. Exceeding some maximum admissible value for this parameter may lead to excessive trajectory collapsing (foreshortening), which can be identified as an arising negative trajectory angle.

Consequently, therefore it is necessary to compute the maximum value of the attack angle amplitude with the following constraints: a trajectory angle is positive during the entire first stage flight and the rocket velocity is equal to 0,264 km/sec by the end of program of angle attack. The problem can be formulated as a task of the nonlinear programming, minimization of the modified Lagrange function, which is solved using the multipliers method.

If multipliers and penalty parameter are constant the optimization problem without constraints takes place. Using the determined coordinate descent method allows solving the problem of modified Lagrange function of unconstrained minimization with fixed values of multipliers and parameter of penalty. The solving problem of the optimization without constraints is one step of the optimization problem with constraints.

The proposed method was realized as a computational Pascal-language program. There is the multiple call of Runge-Kutta method procedure for integration of motion equations. To reduce operation-use time, normalization of motion equations is used, and the 4-th order RungeKutta method time step accuracy control is also applied.

The program test results were compared with the solution, which was obtained using the existing software for the one stage missile design.

The general results obtained in this paper are following: numerical determination method of the attack angle time parameter and maximum allowed value of the attack angle amplitude as functions of the projected and ballistic parameters; software implementation of this method. Deviation of angle trajectory value at the end of active trajectory leg as a function of the error of the time parameter was obtained as well. This paper can be used in the courses of learning such as “Introduction to rocketry” and “Launch vehicle design”.

【 授权许可】

Unknown