Transient Response Analysis of Damped Rotor Systems by the Normal Mode Method

Abstract

A method to determine the transient response of damped single or multi-shaft rotor systems is presented. The rotor systems are idealized as rotating concentrated masses connected by massless beams, discrete springs, and dampers. The springs may have piecewise constant springs rates to simulate the stiffening effect of parts coming in contact after displacement through an initial offset. Arbitrary forcing functions are allowed. The method employs an incremental formulation in which damping gyroscopic and nonlinear terms are treated as external loads which are lagged in time. The equations of motion are uncoupled by performing a normal mode expansion of the response solution in terms of the non-rotating, undamped eigenvectors and their associated eigenvalues; modes and natural frequencies are obtained from a standard Prohl analysis. An analytical solution is used for each step of the incremental analysis. This technique has been used to study the response of a number of rotor systems to the sudden application of a rotating imbalance load. The systems studied include a dual shaft model of a rig, a single-shaft case from the written literature and a large multi-line (multi-shaft) system. The transient analysis was run out to steady-state and close agreement obtained with results from an independent steady-state forced response analysis. Orthogonality relations between the mode shapes were observed to be critical to the quality of the results. It was observed that transient analysis of multi-line systems can be accurately predicted only if the higher frequency modes which are participating in the response are included in the normal mode solution.