An efficient method for modal analysis of multistage cyclic structures with gyroscopic, stress stiffening, and spin softening effects

An efficient method is developed in this work for the modal analysis of multistage cyclic structures considering gyroscopic, stress stiffening, and spin softening effects. The eigen-solution analysis for gyroscopic systems is redefined as the eigen-solution analysis of a Hamiltonian matrix within a state-space framework, and an improved adjoint symplectic subspace iteration method is employed to determine the eigenvalues and eigenvectors of the Hamiltonian matrix. The algorithmic cost of the improved adjoint symplectic subspace iteration method is reduced by exploiting the matrix properties of the structure. Specifically, we demonstrate that for the Hamiltonian matrix corresponding to the gyroscopic system, the eigenvectors associated with a pair of conjugate eigenvalues exhibit a symmetry: their real and imaginary parts are invariant under mutual exchange. This property enables the required dimension of the iterative subspace to be halved. Subsequently, recognizing that the stress stiffening matrix is numerically much smaller than the stiffness matrix, a tailored preconditioner is designed to ensure the rapid convergence of the preconditioned conjugate gradient method with minimal iterations. Furthermore, the Guyan reduction and group theory are utilized to further reduce the computational cost of solving the linear algebraic equations by exploiting the cyclic symmetry of each stage. The proposed method achieves an accuracy comparable to full-order modal analysis while demonstrating superior computational efficiency. The computational accuracy and efficiency of the proposed method are validated through three case studies.