International audienceReal applications in structural mechanics where the dynamic behavior is linear are rare. Usually, structures are made of components assembled together by means of joints whose behavior maybe highly nonlinear. Depending on the amount of excitation, joints can dramatically change the dynamic behavior of the whole system, and the modeling of this type of constraint is therefore crucial for a correct prediction of the amount of vibration. The solution of the nonlinear equilibrium equations by means of the Harmonic Balance Method (HBM) is widely accepted as an effective approach to calculate the steady-state forced response in the frequency domain, in spite of Direct Time Integration (DTI). The state-of-the-art contact element used to model the friction forces at the joint interfaces is a node-to-node contact element, where the local contact compliance is modeled by means of linear springs and Coulomb's law is used to govern the friction phenomena. In the literature, when the HBM is applied to vibrating systems with joint interfaces and the state-of-the-art contact model is used, an uncoupled approach is mostly employed: the static governing equations are solved in advance to compute the pre-stress effects and then the dynamic governing equations are solved to predict the vibration amplitude of the system. As a result, the HBM steady-state solution may lead to a poor correlation with the DTI solution, where static and dynamic loads are accounted for simultaneously. In this paper, the HBM performances are investigated by comparing the uncoupled approach to a fully coupled static/dynamic approach. In order to highlight the main differences between the two approaches, a lumped parameter system, characterized by a single friction contact, is considered in order to show the different levels of accuracy that the proposed approaches can provide for different configurations

Friction damping is one of the most exploited systems of passive control of the vibration of mechanical systems. In order to mitigate vibration of turbine blades, friction dampers are commonly included in the bladed disk design. A common type of blade-to-blade friction dampers are the so-called underplatform dampers (UPDs); these are metal devices placed under the blade platforms and held in contact with them by the centrifugal force acting during rotation. The effectiveness of UPDs to dissipate energy by friction and reduce vibration amplitude depends mostly on the damper geometry and material and on the static loads pressing the damper against the blade platforms. The common procedure used to estimate the static loads acting on UPDs consists in decoupling the static and the dynamic balance of the damper. A preliminary static analysis of the contact is performed in order to compute the static pressure distribution over the damper/blade interfaces, assuming that it does not change when vibration occurs. In this paper a coupled approach is proposed. The static and the dynamic displacements of blade and UPD are coupled together during the forced response calculation. Both the primary structure (the bladed disk) and the secondary structure (the damper) are modeled by finite elements and linked together by contact elements, allowing for stick, slip and lift off states, placed between each pair of contact nodes, by using a refined version of the state-of-the-art friction contact model. In order to model accurately the blade/damper contact with a large number of contact nodes without increasing proportionally the size of the set of nonlinear equations to be solved, damper and blade dynamics are modeled by linear superposition of a truncated series of normal modes. The proposed method is applied to a bladed disk under cyclic symmetric boundary conditions in order to show the capabilities of the method compared to the classical decoupled approaches.

Reduced order models (ROMs) of turbine bladed disks (blisks) are essential to quickly yet accurately characterize vibration characteristics and effectively design for high cycle fatigue. Modeling blisks with contacting shrouds at adjacent blades is especially challenging due to friction damping and localized nonlinearities at the contact interfaces which can lead to complex stick–slip behavior. While well-known techniques such as the harmonic balance method (HBM) and Craig–Bampton component mode synthesis (CB-CMS) have generally been employed to generate ROMs in the past, they do not reduce degrees-of-freedom (DoFs) at the interfaces themselves. In this paper, we propose a novel method to obtain a set of reduction basis functions for the contact interface DoFs as well as the remaining DoFs called “adaptive microslip projection” (AMP). The method is based on analyzing a set of linear systems with specifically chosen boundary conditions on the contact interface. Simulated responses of full order baseline models and the novel ROMs under various conditions are studied. Results obtained from the ROMs compare very favorably with the baseline model. This study addresses the case of a shrouded blisk in microslip close to stick. The AMP procedure may be possibly applied to other systems with Coulomb friction contacts, but its accuracy and effectiveness will need to be evaluated separately.