New techniques are presented for generating reduced-order models of the vibration of mistuned bladed disks from parent finite element models. A novel component-based modeling framework is developed by partitioning the system into a tuned bladed disk component and virtual blade mistuning components. The mistuning components are defined by the differences between the mistuned and tuned blade mass and stiffness matrices. The mistuned-system model is assembled with a component mode synthesis technique, using a basis of tuned-system normal modes and attachment modes. The formulation developed is general and can be applied to any mistuned bladed disk, including those with large geometric mistuning (e.g., severe blade damage). In the case of small (i.e., blade frequency) mistuning, a compact reduced-order model is derived by neglecting the attachment modes. For this component mode mistuning model, the blade mistuning is projected first onto the component modes of a tuned, cantilevered blade, and then projected again onto the tuned-system normal modes via modal participation factors. In this manner, several natural frequencies of each mistuned blade can be used to capture systematically the effects of the complex physical sources of mistuning. A numerical validation of the developed methods is performed for both large and small mistuning cases using a finite element model of an industrial rotor. Nomenclature c = aerodynamic coupling damping matrix projected onto tuned-system normal modes E = Young's modulus F = real Fourier matrix f = excitation force vector j = 1 p K = stiffness matrix in physical coordinates M = mass matrix in physical coordinates N = no. of blades N h = no. of the retained tuned-system normal modes corresponding to harmonic h p = modal coordinates q,q h = set of tuned-cantilevered-blade mode participation factors for the blade motion in the retained tunedsystem modes in physical coordinates and in cyclic coordinates U = set of the retained tuned-cantilevered-blade normal and boundary modes v, v = mistuned-cantilevered-blade normal mode participation factor for a tuned-cantilevered-blade normal mode and set of the factors x = physical coordinates = structural damping coefficient = nondimensional mistuning parameter = reduced stiffness matrix or stiffness projection to the retained component modes , = eigenvalue and diagonal matrix of eigenvalues of the retained component normal modes = reduced mass matrix or mass projection to the retained component modes , = set of the retained component normal modes in physical coordinates and in cyclic coordinates = component interface modes ! = frequency Subscripts b, i = finite element degrees of freedom (DOF) of the cantilevered-blade boundary and interior H = maximum harmonic number h = harmonic number n = blade number o = tuned blade R = set of the retained cantilevered-blade normal mode numbers r = rth cantilevered-blade normal mode = finite element DOF of the blades = finite element DOF of the disk = generalized coordinates for the retained component normal m...

The forced response of a mistuned bladed disk can be significantly amplified compared to that of a tuned bladed disk. Various reduced-order models have been studied to predict the response of mistuned bladed disks. Most of these models have been tested only for simple mistuning cases that may not be realistic for actual systems. In this paper, a new approach to generate a general reduced-order model for a mistuned system is presented. From this general formulation, a compact reduced-order model for small blade mistuning is also derived in which mistuning is projected to tuned-system normal modes using modal participation factors of cantilevered-blade component modes. The presented mistuning projection method can estimate the effects of complicated mistuning easily from measurable modal mistuning values of mistuned blades.

In this paper, a new reduced-order modeling technique is presented for bladed disks that feature large, geometric deviations from a nominal design. Various finite-element-based reduced-order models (ROMs) have been proposed in the literature for bladed disks with small blade-to-blade differences, called mistuning. Many of these techniques rely on the fact that mistuned-system normal modes can be effectively represented using a linear combination of the normal modes of the nominal (tuned) system. However, when the mistuning or geometric deviation is large, the number of tuned-system normal modes required to describe the mistuned-system normal modes increases dramatically. In this work, a method for large mistuning is formulated by employing a mode-acceleration method with static mode compensation. By accounting for the effects of mistuning as though they were produced by external forces, a set of basis vectors is established using a combination of tuned-system normal modes compensated by static modes. The obtained basis vectors approximately span the space of the mistuned-system modes without requiring a much more expensive modal analysis of the mistuned system, and they provide much better convergence than tuned-system normal modes. Furthermore, in order to extend the method to higher frequency ranges, quasi-static modes, in which inertia effects are included, are employed in place of static modes in the modeacceleration formulation. It is seen that ROMs based on the new technique are extremely compact, yet they accurately capture the vibration response of bladed disks subject to geometric mistuning or design changes.

The forced vibration response of bladed disks can increase dramatically due to blade mistuning, which can cause major durability and reliability problems in turbine engines. To predict the mistuned forced response efficiently, several reduced-order modeling techniques have been developed. However, for mistuned bladed disks, increases in blade amplitude levels do not always correlate well with increases in blade stress levels. The stress levels may be computed by postprocessing the reduced-order model results with finite element analysis, but this is cumbersome and expensive. In this work, three indicators that can be calculated directly from reduced-order models are proposed as a way to estimate blade stress levels in a straightforward, systematic, and inexpensive manner. It is shown that these indicators can be used to predict stress values with good accuracy relative to finite element results, even for a case in which the displacement and stress levels show different frequency response trends.

Intentional mistuning is the deliberate incorporation of blade-to-blade parameter variations in the nominal design of a bladed disk. Previous studies have shown that this is a promising strategy for mitigating the damaging effects of unintended, random mistuning. In this paper, the mechanisms of intentional mistuning are studied by investigating the relation between blade response and vibration energy flow in lumped parameter models. Based on key observations from the energy flow analysis, a few design guidelines are proposed that drastically reduce the design space for intentional mistuning patterns. Thus, an optimization may be performed on the reduced design space or skipped altogether, yielding dramatic reductions in computational costs. The guidelines are validated by extensive Monte Carlo simulations for the lumped parameter models as well as for a finite-element-based reduced-order model of an industrial rotor. It is shown that the reduced design space includes optimal or near-optimal intentional mistuning patterns.