The eigenvalue problem for a thin linearly elastic shell, of thickness 2c, clamped along its lateral surface is considered. Under the geometric assumption on the middle surface of the shell that the space of inextensi...The eigenvalue problem for a thin linearly elastic shell, of thickness 2c, clamped along its lateral surface is considered. Under the geometric assumption on the middle surface of the shell that the space of inextensional displacements is non-trivial, the authors obtain, as e - 0, the eigenvalue problem for the two-dimensional "flexural shell" model if the dimension of the space is infinite. If the space is finite dimensional, the limits of the eigenvalues could belong to the spectra of both flexural and membrane shells. The method consists of rescaling the variables and studying the problem over a fixed domain. The principal difficulty lies in obtaining suitable a priori estimates for the scaled eigenvalues.展开更多
The authors consider the exact controllability of the vibrations of a thin shallow shell, of thickness 2ε with controls imposed on the lateral surface and at the top and bottom of the shell. Apart from proving the ex...The authors consider the exact controllability of the vibrations of a thin shallow shell, of thickness 2ε with controls imposed on the lateral surface and at the top and bottom of the shell. Apart from proving the existence of exact controls, it is shown that the solutions of the three dimensional exact controllability problems converge, as the thickhess of the shell goes to zero, to the solution of an exact controllability problem in two dimensions.展开更多
文摘The eigenvalue problem for a thin linearly elastic shell, of thickness 2c, clamped along its lateral surface is considered. Under the geometric assumption on the middle surface of the shell that the space of inextensional displacements is non-trivial, the authors obtain, as e - 0, the eigenvalue problem for the two-dimensional "flexural shell" model if the dimension of the space is infinite. If the space is finite dimensional, the limits of the eigenvalues could belong to the spectra of both flexural and membrane shells. The method consists of rescaling the variables and studying the problem over a fixed domain. The principal difficulty lies in obtaining suitable a priori estimates for the scaled eigenvalues.
文摘The authors consider the exact controllability of the vibrations of a thin shallow shell, of thickness 2ε with controls imposed on the lateral surface and at the top and bottom of the shell. Apart from proving the existence of exact controls, it is shown that the solutions of the three dimensional exact controllability problems converge, as the thickhess of the shell goes to zero, to the solution of an exact controllability problem in two dimensions.
