This contribution addresses the vibratory analysis of unilateral-contact induced structural interactions between a bladed impeller and its surrounding rigid casing. Such assemblies can be found in helicopter or small aircraft engines for instance and the interactions of interest shall arise due to the always tighter operating clearances between the rotating and stationary components. The investigation is conducted by extending to cyclically symmetric structures an in-house time-marching based tool dedicated to unilateral contact occurrences in turbomachines. The main components of the considered impeller together with the associated assumptions and modeling principles considered in this work are detailed. Typical dynamical features of cyclically symmetric structures, such as the aliasing effect and frequency clustering are explored in this nonlinear framework by means of thorough frequency-domain analyses and harmonic trackings of the numerically predicted impeller displacements. Additional contact maps highlight the existence of critical rotational velocities at which displacements potentially reach high amplitudes due to the synchronization of the bladed assembly vibratory pattern with the shape of the rigid casing. The proposed numerical investigations are also compared to a simpler and (almost) empirical criterion: it is suggested, based on nonlinear numerical simulations with a linear reduced order model of the impeller and a rigid casing, that this criterion may miss important critical velocities emanating from the unfavorable combination of aliasing and contact-induced higher harmonics in the vibratory response of the impeller. Overall, this work suggests a way to enhance guidelines to improve the design of impellers in the context of nonlinear and nonsmooth dynamics.
Presented is an approach for finding periodic responses of structural systems subject to unilateral contact conditions. No other non-linear terms, e.g. large displacements or strains, hyper-elasticity, plasticity, etc. are considered. The excitation period due to various forcing conditions-from harmonic external or contact forcing due to a moving contact interface-is discretized in time, such that the quantities of interest-displacement, velocity, acceleration as well as contact force-can be approximated through time-domain schemes such as backward difference, Galerkin, and Fourier. The solution is assumed to exist and is defined on a circle with circumference T to directly enforce its periodicity. The strategy for approximating time derivative terms within the discretized period, i.e. velocity and acceleration, is hence circulant in nature. This results in a global circulant algebraic system of equations with inequalities that can be translated into a unique linear complementarity problem (LCP). The LCP is then solved by dedicated and established methods such as Lemke's algorithm. This allows for the computation of approximate periodic solutions exactly satisfying unilateral contact constraints on a discrete time set. The implementation efficiency and accuracy are discussed in comparison to classical time marching techniques for initial value problems combined with a Lagrange multiplier contact treatment. The LCP algorithm is validated using a simple academic problem. The extension to large-scale systems is made possible through the implementation of a Craig-Bampton type modal component synthesis. The method shows applicability to industrial rotor/casing contact set-ups as shown by studying a compressor blade. A good agreement to time marching simulations is found, suggesting a viable alternative to time marching or Fourier-based harmonic balance simulations.
The phenomenology of rotor-casing setups experiencing contact interactions is still poorly understood, particularly when complex geometries such as centrifugal compressors are involved. Although interaction phenomena have been witnessed and recorded during industrial experiments, the physical understanding of what occurs during these interactions is limited. The usual design approach is to consider possible modal interaction points in a linear framework and move these outside of normal operating conditions by means of minor geometric changes. Based on this linear approach, no information on the severity of these interactions is available to the designer. Besides, a possible interaction point appearing in the linear framework may not produce any harmful interactions, thereby increasing design restrictions. Based on an in-house numerical strategy previously presented, contact interactions for a flexible centrifugal compressor from a helicopter engine and rigid casing setup are investigated. By imposing a small deformation on the casing geometry, blade/casing contact is initiated and subsequent interactions feature complex phenomena that are analyzed. In comparison to previous interaction simulations involving axial compressors, a higher degree of complexity of the numerical simulations stems from a strong curvature of the blade and very significant blade-disk coupling. This coupling presents itself towards the trailing edge where compressor mode shapes indicate a significant component normal to the casing surface. Accordingly, these modes may lead to large amplitude contact forces. Time simulation results are confronted with experimental observations, and the consistency of the behavior of the numerical model with respect to industrial observations is underlined. A frequency domain post-processing of the results reveals specific engine order interactions and frequency spectra are plotted in order to interpret the phenomenon of interest. Such methodology will enable designers to more efficiently discriminate potential critical interaction speeds as compared to the classical linear frequency approach.
The “Tie-Dye” (TD) method is a well-known preliminary flutter design method for subsonic low pressure turbine (LPT) blades. In this paper, a study of 2D mode shape sensitivity using the TD-method for supersonic exit Mach numbers is presented. Using a harmonic balance CFD method, TD maps displaying the critical reduced frequency for a range of pitching axis locations were created. The TD method was run on two geometrically different blades. Subsonically, the characteristic appearance does not change much over airfoil types. An even lesser amount of morphing can be observed between the different profiles in the supersonic range, than for the subsonic cases. Pure bending modes show a high sensitiviy to the actual bending direction. Therefore the single critical reduced frequency value criteria does not hold up for all cases. The method is applicable for supersonic exit flows, and is even more predictable and universal than for the subsonic cases.
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