# A Reduced-Order Model for the Vibration Analysis of Mistuned Blade-Disc-Shaft Assembly
## 1 Introduction  
Blade-disc-shaft assemblies are the key components of engines and compressors and take an important position in the design of these machines [1]. As the basis for design, developing accurate dynamic models for such assemblies and gaining deep insight into their vibration characteristics are of great significance. For this reason, the vibration issues of blades and rotor systems attracted enormous attention from researchers and engineers, and a huge number of investigations were published in the past few decades [1; 2; 3; 4]. Nevertheless, most of the available investigations concentrated either on the vibration of the blade and disc by neglecting the flexibility of the shaft or on the dynamics of rotor systems by assuming the blade and disc to be rigid and tuned. Some researchers considered the mutual interactions between bladed discs and shafts and developed several dynamic models for blade-disc-shaft assemblies. [PERSON] et al. [5] presented a simplified analytical model for rotating a flexible blade-rigid disc-flexible cantilevered shaft system. [PERSON] et al. [6] addressed an equivalent modeling technique for a blade-shaft system by dividing the system into a shaft plus additional equivalent mass-spring models of blades. [PERSON] et al. [7] presented a method for dynamic analysis of flexible blade-disc-shaft systems by means of the finite element method and a modal synthesis approach. [PERSON] et al. [8] developed an analytical approach for the coupled shaft torsion and blade bending vibration analysis using the weighted residuals method. [PERSON] et al. [9] proposed an analytical substructure synthesis method for analyzing the coupled free vibrations between shaft and blades. [PERSON] [10; 11] presented efficient models for coupled shaft-torsional and blade-bending vibration analysis using Lagrange's equation and the assumed modes method. [PERSON] al. [12] developed a dynamic model for rotor-blade systems by representing blades as Timoshenko beams and shafts as multiple lumped mass points connected by massless springs. [PERSON] et al. [13] developed an analytical model for the coupling vibration analysis of a blade-disc-shaft system. In these investigations, the mistuning in blades and discs, which is an inevitable issue due to manufacturing tolerances, material deviations, and in-service wear, tended to be neglected. Regarding this issue, some researchers took mistuning into account. [PERSON] et al. [14] presented a simple analytical model for non-rotating flexible blade-rigid disc-flexible shaft assemblies by representing blade mistuning as frequency deviations. [PERSON] [15] developed a simplistic model for describing the in-plain whirling behavior of a mistuned bladder rotor using complex coordinate transformation and modulation. [PERSON] et al. [16] established a dynamic model for a shaft-disc-blade system considering the mistuning in blade length and stagger angle. These investigations provided effective ways for the modeling of blade-disc-shaft assemblies. Nevertheless, the models developed in these investigations were mainly analytical or lumped with, more or less, some simplifications and assumptions. Although such models have the advantages of ease of use, they have limitations in modeling and analysis of complex rotor assemblies, such as those of engines and compressors in practical industrial application. Three-dimensional (3D) finite element models could overcome these limitations by discretizing the complex assemblies as solid elements, where the detailed geometry can be accurately captured, and the flexibility of all components can be taken into account. Such models were widely employed in the modeling and analysis of bladed discs [17; 18; 19; 20] and complex rotor systems [21; 22; 23; 24]. Nevertheless, the huge number of degrees of freedom (DOFs) in 3D finite element models hinders the computational efficiency in analysis, especially for systems containing random factors, such as mistuning.