Nonlinear cyclic reduction for the analysis of mistuned cyclic systems

Abstract

Predicting the vibratory response of bladed-disks in turbomachinery is of the utmost importance to design a reliable and optimized engine. Yet, such simulations are challenging mainly due to the large size of the finite-element model used to describe the system, the existence of random mistuning coming from manufacture tolerances, and the nonlinear effects arising from the different components and their coupling. As a consequence, very few reduced-order models handling nonlinear mistuned systems have yet been proposed and the existing ones try to find the best compromise between a computationally efficient simulation and a correct description of the nonlinear phenomena. In this paper, a novel approach is proposed to tackle this challenge. Its key ideas rely on the substructuring concept and the cyclic properties of the structure. A reduction basis composed of cyclic complex nonlinear normal modes is created on each sector of the system leading to a compact nonlinear superelement per sector. The system is then assembled and a synthesis simulation is performed. Due to the main concepts employed, the method is referred to as the Substructuring method based on Nonlinear Cyclic Reduction (SNCR). It is validated on a finite element model of a tuned, a randomly mistuned and an intentionally mistuned bladed-disks. It will be shown to be fast and accurate despite strong nonlinearities. Due to its flexibility and efficiency, the method is expected to have a wide range of possible applications within the community.

Impact of Mistuned Underplatform Dampers on the Nonlinear Vibration of Bladed Disks

“…The SNCR [24] employs a substructuring approach [32] in which the full structure is broken down into 𝑁 substructures (the 𝑁 sectors). Hence, for each different kind of sector (two in this case), the associated cyclic nonlinear normal modes (CNNMs) must be computed.…”

“…In practise, this procedure is not used because it requires large memory space to store the mass, damping, and stiffness matrices, and the computation time would be excessively long, mainly due to the computation of the nonlinear forces for all sectors. In this paper, we employ a recent nonlinear ROM methodology called the Substructuring method based on Nonlinear Cyclic Reduction (SNCR) [24] to compute the frequency forced responses of systems with different intentional mistuning patterns. This approach is briefly explained in Section 4.1, and then tested and analyzed for cyclic symmetric structures in Section 4.2.…”

“…Different strategies have been proposed [20][21][22][23] to tackle this problem. Recently, the authors have developed a new approach called the Substructuring method based on Nonlinear Cyclic Reduction (SNCR) [24] which employs a substructuring approach with cyclic complex nonlinear normal modes. This paper will use several theoretical and numerical strategies [13,14,24] with the following original objective: optimizing an intentional mistuning pattern of UPDs to minimize the maximal displacement of the structure over two distinct modes.…”

Application of the reproducing kernel method for solving linear Volterra integral equations with variable coefficients

“…Some natural phenomena can be modeled by linear and non-linear systems of integral and differential equations, which are commonly used in fields such as biology, chemistry, and physics [1][2][3][4][5][6][7][8][9][10][11]. Many numerical methods, such as collocation boundary value methods, discontinuous Galerkin approximations, Euler matrix method, spectral element method, Chebyshev wavelets approach, and Radial basis Functions, have been provided to solve linear and nonlinear Volterra integral equations [12][13][14][15][16][17].…”

Dynamic characteristics of vibration localization of mistuned bladed disk due to shroud and blade damages

“…[6][7][8][9][10] The mistuning will lead to more complicated vibration localization. [11][12][13][14] The mistuning may cause vibration localization leading to premature high-cycle fatigue. 15 A perturbation method is proposed to study mode localization and vibration localization.…”