Numerical continuation of the interaction of flutter and forced response

The simultaneous presence of flutter instability and forced response is a common situation in the operation of modern turbomachinery. The interaction of these two effects is typically of nonlinear character due to the friction effects at the contact interfaces. Previous works have found that flutter and forced response do not appear combined in the system response; either flutter dominates (asynchronous response), or the vibration is locked to the forcing frequency (synchronous response). Therefore, the simple linear superposition of the vibration levels from flutter and forced response alone is not accurate, and it can lead to a considerable overestimation of the final vibration amplitude. These studies relied on the numerical integration of lumped systems over a very large number of elastic cycles to reach a converged solution. Due to the presence of very long time scales, there is always some level of uncertainty on whether a true final state has been achieved. To avoid this problem, in this work, a numerical continuation procedure is applied to compute the different solutions that set in, their traveling wave content, and their stability. Bifurcation diagrams showing the different solutions present in the system are obtained, and used to identify the flutter or forced response dominated regions, depending on the forcing engine order and frequency.