Determination of the optimal positions of the supports of a homogeneous plate under non-stationary action

Abstract

In this work, a study of micrometeoroid protection (MMP) is carried out. It is a thinwalled shell designed to protect the equipment on the nose of the spacecraft from micrometeorites. However, an equally dangerous equipment is posed by an astronaut who can accidentally affect the shell during repair work and damage the equipment. To counteract this, the shell is specially supported by supports from the inside. To develop a method for calculating the number of supports and their position, a simplified version of the problem is considered. The interaction of a plate hinged along the edges and having additional supports over the area with a special load in the form of a non-stationary Delta function is studied. The KirchhoffLowe plate was chosen as the model of the plate. The origin is placed at the top left corner of the plate. It is required to determine the optimal location of additional supports, based on the fact that the maximum deflection should not allow the maximum permissible value. The deflection function is defined as the sum of the convolutions of the influence functions with the corresponding external load and reactions in the additional supports. To determine the value of the influence function, all functions included in the expressions for the motion of the plate are expanded into Fourier series in such a way that the boundary conditions at the edges of the plate are satisfied and the Laplace transform in time is applied. Further, in the equation of motion of the plate, both the external load and the convolution of the Delta function with reactions in additional supports and in time are taken into account. The values of the reactions in the supports are determined from the boundary conditions, based on the fact that the displacements of the fixed points are equal to zero. After that, from the obtained equation of normal displacements, the coordinates of the location of the supports around the applied external load are determined, so that the condition of not exceeding the specified displacement value is satisfied. The normal displacements are determined, the inverse Laplace transform is performed in time, and the sum of the series is found.

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