A finite element based stability test for equilibria of flexible structures in circular orbits

Stability Analysis of an Orbiting Plate with Finite Elements

“…Furthermore, if the Hessian of V * (u) is positive definite at the equilibrium, we may conclude that the system is stable. In [5] it is shown that the matrix of second derivatives A ij of V * can be interpreted as the sum of the system's tangent stiffness matrix and the tensor product of the nodal loadvector f c due to centrifugal forces scaled by a factor:…”

“…After projecting A ij onto the eigenspace of K ij the subdeterminants of the Hessian can be expressed analytically by the eigenvalues of K ij and by the projections of a onto the eigenvectors of K ij . From [5] one can further conclude that: (1) If the lowest eigenvalue of K ij is positive, the system is stable. (2) If the lowest two eigenvalues of K ij are negative, the system is unstable.…”

“…However, it is still a challenge, to apply this methods to flexible structures discretized by finite elements. Therefore, in [5] a process is described how the reduced energy momentum method can be integrated into a finite element analysis. Two examples of one-dimensional continua (such as the space elevator) are discussed in this paper.…”