A method for finding repeated roots in transcendental eigenvalue problems

The eigenvalues of a gyroscopic system depend on a parameter, and the system may become unstable when eigenvalues collide. For continuous systems, the eigenvalue problem is represented by a matrix with transcendental elements. A method is developed to determine the value of the parameter that causes eigenvalues to collide. The method is implemented using a certain formula for the determination of the derivative of a matrix determinant without the use of symbolic expansion of the determinant. This formula simplifies the computational task considerably. Applications of the method are demonstrated by problems involving buckling by circulatory forces and oscillations of a gyroscopic system.