Approach for stability analysis of a cracked rotor with time-varying stiffness

An approach for dynamic stability analysis of a cracked rotor system with transverse crack is addressed here. The timevarying area moments of inertia of the cracked section are employed in formulating the time-periodic finite element stiffness matrix which yields a linear time-periodic system. The harmonic balance method (HB) is used in solving the finite element (FE) equations of motion for studying the dynamic stability of the system. The sign of the determinant of the scaled coefficient matrix resulting from applying the HB solution to the cracked rotor system is found to be a reliable approach for identifying the major unstable regions of the system in the parameter plane obtained by plotting the shaft speeds of rotation vs. the crack depths. Specifically, the negative values of the determinant of this scaled coefficient matrix identify the unstable regions of the cracked system. This approach is applied here to the parametrically excited Mathieu's equation, two degree-of-freedom gyroscopic system, and then to the FE model of the cracked rotor system. The results of applying this approach are verified using the Floquet's theory. Compared with the theory, the sign of the determinant of the scaled coefficient matrix is found here to be an efficient tool for identifying the unstable regions of linear parametrically excited systems, especially the large scale dynamic systems where this approach requires considerably less computational time than the Floquet's theory.