Abstract
A numerical algorithm to calculate the periodic response, stability and bifurcations of a periodically excited non-conservative, Multi-Degree of Freedom (MDOF) system with strong local non-linearities is presented. First, the given large order system is reduced using a fixed-interface component mode synthesis procedure (CMS) in which the degrees of freedom associated with non-linear elements are retained in the physical co-ordinates while all others, whose number far exceeds the number of non-linear DOF, are transformed to modal co-ordinates and reduced using real mode CMS. A shooting and continuation method is then applied to the reduced system to solve for the periodic response. Floquet stability theory is used to calculate stability and bifurcations of the periodic response. The algorithm is applied to study the response to imbalance, stability, and bifurcations of a 24 DOF flexible rotor supported on journal bearings. The results indicate that the proposed algorithm, though approximate, can yield very accurate information about dynamic behavior of large order non-linear systems, even with few numbers of retained component modes. The algorithm, which imposes less demand on computer time and memory, is believed to be of considerable potential in analyzing a variety of practical problems.
Highlights
The dynamic analysis of multiple shaft rotor-bearing systems can require the assembly and solution of large order sets of ordinary differential equations of motion
Very often the non-linear equations of motion are linearized in the neighborhood of an operating point and the order of the resulting linear system of equations is reduced using modal transformations and the reduced model is solved for system response
A few analysts have dealt with non-linear rotordynamics for large scale systems and even a smaller number of them have dealt with steady state response and stability of such systems under forced excitations
Summary
The dynamic analysis of multiple shaft rotor-bearing systems can require the assembly and solution of large order sets of ordinary differential equations of motion. Fluid-film bearings produce forces which are highly non-linear functions of displacements when operating at high eccentricities (heavily loaded) In such a case, the linearization procedure masks potentially dangerous bifurcations such as subcritical Hopf bifurcations that are predicted by the original non-linear equations of motion. Very often stability information is predicted using eigenanalysis of the unforced system even when the system is under forced excitation due to imbalance or any other source of excitation Such a procedure may be valid only for weakly non-linear systems and more general procedures based on modern dynamical systems theory are required to handle strongly non-linear systems. It is the objective of this work to develop numerical procedures to calculate the steady state periodic response, its stability, and bifurcation for a large order nonlinear system under forced excitation
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