Flexural vibration of multistep rotating Timoshenko shafts using hybrid modeling and optimization techniques

In this paper, the flexural vibration of uniform and non-uniform rotating shafts based on the Timoshenko beam theory is investigated. Considering shear deformation and gyroscopic effects to build an exact framework of a solution, the fourth-order differential equation of vibration is solved by an analytical method. The overall transfer matrix of the system formed by a series of flexible distributed in addition to lumped elements is achieved. The results obtained by the distributed lumped modeling technique (DLMT) are verified with other techniques. The damped frequencies obtained by the hybrid modeling method for a multistep gas turbine rotor system using the Euler-Bernoulli and Timoshenko beam theories are compared with the results of the transfer matrix method (TMM). It is shown that the presented method provides highly accurate results, while with no compromise, it can be simply and effectively employed for complex systems. It is also shown how the Hooke and Jeeves and the ant colony optimization (ACO) besides the direct enumeration method can be applied to obtain the damped natural frequencies of complicated vibrational systems such as the gas turbine rotors. The comparison between the frequency and damping ratio values obtained by the optimization and direct enumeration methods shows less than 1% error for different cases.