Abstract
In modern turbo machines such as aircraft jet engines, structural contacts between the casing and bladed disk may occur through a variety of mechanisms: coincidence of vibration modes, thermal deformation of the casing, rotor imbalance due to design uncertainties to name a few. These nonlinear interactions may result in severe damage to both structures and it is important to understand the physical circumstances under which they occur. In this study, we focus on a modal coincidence during which the vibrations of each structure take the form of a k-nodal diameter traveling wave characteristic of axi-symmetric geometries. A realistic two-dimensional model of the casing and bladed disk is introduced in order to predict the occurrence of this very specific interaction phenomenon versus the rotation speed of the engine. The equations of motion are solved using an explicit time integration scheme in conjunction with the Lagrange multiplier method where friction is accounted for. This model is validated from the comparison with an analytical solution. The numerical results show that the structures may experience different kinds of behaviors (namely damped, sustained and divergent motions) mainly depending on the rotational velocity of the bladed disk.
Highlights
In rotordynamics, nonlinear coupling forces between the rotating and surrounding stationary parts can result in unexpected significant displacements and subsequent high stresses leading to structural failure
Starting from simple considerations on structures that are rotationally periodic or exhibit axi-symmetry, it is proved that k-nodal diameter rotating modes that separately propagate in the bladed disk and casing, may interact through direct contact due to the small tip clearance
Taking into account physical considerations on the direction of the contact and friction forces between the two structures - forward in the casing and backward in the bladed disk due to the chosen direction of rotation, and invoking the action/reaction principle of Newton’s third law, only one of these equations can be considered as dangerous, ωc = kΩct − ωbd
Summary
Nonlinear coupling forces between the rotating and surrounding stationary parts can result in unexpected significant displacements and subsequent high stresses leading to structural failure. The related works usually analyze the vibrations of a rotating shaft with a non-uniform cross-section supported by journal bearings where different levels of nonlinearity are considered : oil-film pressure field implicating nonlinear hydrodynamic equations [2, 3], direct rub and friction forces, viscous damping forces, nonconstant angular velocity to name a few These studies mainly necessitate small models with a few coupled nonlinear second-order differential equations suitable for the investigation of a real shaft behavior. Starting from simple considerations on structures that are rotationally periodic or exhibit axi-symmetry, it is proved that k-nodal diameter rotating modes (see Appendix A for discussion) that separately propagate in the bladed disk and casing, may interact through direct contact due to the small tip clearance.
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