A general geometrical theory of turbine blade underplatform asymmetric dampers

This paper presents a general geometric theory of in-plane motion of underplatform dampers as a function of blade neck deflection and the resulting platform kinematics. In-plane treatment is compatible with Mode 1 blade vibration, in which the rigid-body platform rotation associated with pure neck bending largely predominates over other rotations. The method applies when the amplitude of platform vibrational rotation reaches at least (or is greater than) the minimum value required to have, at both ends of the forward and reverse half-cycles, the onset of full slip in the last contact element still in stick.In this context, the first objective of this paper is to present a general geometric theory for damper kinematics and damper-platform forces, for both the In-Phase and Out-of-Phase blade vibrations and is then developed in detail for the In-Phase case. The concept of equivalent virtual parallel geometry is introduced, which transforms a case with nonnegligible inclination between adjacent blades into one, more tractable, valid for a very large (ideally infinite) number of blades. The equations are developed for the In-Phase case according to a formulation whose structure is particularly suited to highlighting the properties of the system, thus facilitating both its solution and interpretation.The second objective is to employ the equations in a piecewise linear algorithm to pre-compute a cycle as a sequence of a few finite linear sections in the number, per half-cycle, of Jenkins-type contact elements, so as to avoid both the time-step integration of the damper cycle used in the HBM-AFT approach and its repetition at each iteration of the solving system. The concept of a force Base-Cycle is introduced for the contact forces, unique for a given damper shape and its set of contact parameters, along with a moment Base-Cycle for the platformdamper system, an in-plane moment on the platform, generated by the contact forces, function of platform rotation.It is shown that such Base Cycles uniquely characterize a damper shape so that, once produced for a convenient Reference Case, they can be scaled linearly without recalculation according to size, radial force, platform proportions, and proportional variation of contact stiffness.To better illustrate the procedure, a practical example is developed in parallel with the theory, concluding with indications on how the moment Base-Cycle can be used in a reduced version of the Platform Centered Reduction, proposed in recent papers, meant to be a utility tool to support the initial damper design phase.