Analytical solution to approximate equations of the launch vehicle motion under the gust action for the dynamic loading calculation

Introduction. The launch vehicle (LV) in flight and the dynamic components of loads from the impact of a trapezoidal wind gust are considered. It is proposed to determine the dynamic components of the force factors using analytical solutions for the structure points accelerations. The work objective is to create a technique for selecting the duration of the standard gust, under the influence of which maximum loads are provided in the sections of the LV structure.Materials and Methods. The launch vehicle is presented as an uneven beam. The description of its vibrations is reduced to a system of independent ordinary differential equations that determine the motion of an equivalent system of oscillators. The equation of oscillator vibrations under the action of a trapezoidal pulse load is solved by the overlay method, and it is reduced to the calculation of the Duhamel integral. It is proposed to get the parameters of an equivalent system of oscillators based on the results of the calculation of dynamic characteristics for a finite element LV model in the Nastran program.Results. Analytical relations for the LV structure point accelerations under the action of a trapezoidal wind gust are given. For the beam model, test calculations of accelerations were carried out according to the technique proposed in this paper. These data are compared to the results of finite element modeling. With the help of analytical solutions, dependences are constructed that determine the nature of the change in the magnitude of the bending moment for different sections of the launch vehicle when the duration of the wind gust varies.Discussion and Conclusions. The presented technique provides building an equivalent dynamic model of systems with a large number of degrees of freedom on the example of a LV and obtaining analytical solutions for accelerations of points of a mechanical system under trapezoidal external action. These solutions are applicable for the study of dynamic loads. The analysis results enable to select the duration of the wind gust, at which maximum loads are reached in the sections of the LV structure. Calculations based on the analytical solutions are very economical in terms of time spent. They can be used in design calculations for preliminary assessment of loading.