Mathematical modeling of temperature fields in antenna reflectors of space vehicles

Abstract

The main factor that determines the distortion of the shape of the reflectors in outer space is the temperature deformation due to the uneven distribution of heat fluxes in the structural elements. Therefore, it is important to develop models and methods for calculating temperature fields in reflectors with uneven distribution of heat fluxes on the surface. The use of such methods will reduce the number of expensive field experiments. The paper first constructs a mathematical model for calculating temperature fields in a parabolic reflex antenna, in the form of a paraboloid of rotation rotating at a constant angular velocity, taking into account the finite velocity of heat propagation as a boundary value problem of mathematical physics for the hyperbolic equation of thermal conductivity. that the thermophysical properties of the body are constant. At the initial moment of time, the body temperature is constant, and on the outer surface of the body are known values of heat flux which are continuous coordinate functions. To solve the obtained boundary value problem, a new integral transformation for a two-dimensional finite space was constructed. The formula of inverse transformation is given. Eigenvalues and eigenfunctions for the kernel of integral transformation are found using finite element methods and Galorkin. The division of the region into simplex elements was made. Thus the problem of finding eigenvalues and eigenfunctions was reduced to the algebraic problem of finding eigenvalues and eigenfunctions. After applying the constructed new integral transformation to the obtained boundary value problem, we obtained the Cauchy problem, the solution of which was found analytically. The obtained solution of the boundary value problem is twice continuously differentiated by spatial coordinates and once in time. The solution of the boundary value problem found can be used to modulate the temperature fields that occur in a parabolic reflex antenna. The paper first constructs a mathematical model for calculating temperature fields in a paraboloid rotating at a constant angular velocity, taking into account the finite velocity of heat propagation as a boundary value problem of mathematical physics for the hyperbolic equation of thermal conductivity with Neumann boundary conditions. Using the developed integral transformation, the temperature fields in the paraboloid in the form of convergent series by Fourier functions were found. The solution of the generalized boundary value problem of heat exchange of the paraboloid of rotation can be used to modulate the temperature fields that occur in the antenna reflectors of spacecraft.