Shock and Vibration

Volume 2016, Article ID 4656032, 9 pages

http://dx.doi.org/10.1155/2016/4656032

## Vibration Characteristics of a Mistuned Bladed Disk considering the Effect of Coriolis Forces

State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China

Received 27 March 2016; Revised 12 July 2016; Accepted 14 July 2016

Academic Editor: Jörg Wallaschek

Copyright © 2016 Xuanen Kan and Bo Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To investigate the influence of Coriolis force on vibration characteristics of mistuned bladed disk, a bladed disk with 22 blades is employed and the effects of different rotational speeds and excitation engine orders on the maximum forced response are discussed considering the effects of Coriolis forces. The results show that if there are frequency veering regions, the largest split of double natural frequencies of each modal family considering the effects of Coriolis forces appears at frequency veering region. In addition, the amplitude magnification factor considering the Coriolis effects is increased by 1.02% compared to the system without considering the Coriolis effects as the rotating speed is 3000 rpm, while the amplitude magnification factor is increased by 2.76% as the rotating speed is 10000 rpm. The results indicate that the amplitude magnification factor may be moderately enhanced with the increasing of rotating speed. Moreover, the position of the maximum forced response of bladed disk may shift from one blade to another with the increasing of the rotational speed, when the effects of Coriolis forces are considered.

#### 1. Introduction

Generally, when the vibration characteristics of bladed disk are designed and analyzed, the bladed disk is assumed to be tuned [1–4]. However, practical experience shows that there are always deviations of blade-to-blade caused by manufacturing tolerances and wears during operation. These small deviations are often called mistuning of blades. Mistuning of bladed disk may cause vibration localization which may accelerate high cycle fatigue [5]. Wei and Pierre [6, 7] developed perturbation methods to investigate the vibration localization and pointed out that a small mistuning may result in strong vibration localization for weakly coupled systems. Yoo et al. [8] established a simplified pendulum model to research the influence of stiffness coupling, damping parameters on the vibration localization. Chiu and Huang [9] investigated the influence of mistuning caused by blade’s stagger angle on the stability of mistuned bladed disk. Bladh et al. [10] researched the relationship between mistuning strength and amplitude magnification factor. Their results indicated that forced response amplitude and stress of mistuned bladed disk are related to mistuning strength and coupling strength.

In the previous works, the effects of Coriolis force are usually negligible. However, blades of jet engine system become thinner and geometrical shapes become more and more complicated [11]; in particular, the rotational speed becomes higher and higher. The model without considering the effects of Coriolis force cannot accurately describe the vibration characteristics of mistuned bladed disks. Therefore, Huang and Kuang [12] used Galerkin’s method to research the mode localization of bladed disk considering the effects of Coriolis forces. Their results pointed out that rotating speed has a significant effect on the mode localization of mistuned bladed disk. Nikolic et al. [13] established an experiment, for the first time, to validate the split of double natural frequencies considering the effect of Coriolis forces and used a lumped parameters mass-spring model to investigate the forced response localization. The Coriolis forces of blades are related to the geometry of bladed disk. Therefore, Xin and Wang [14] used realistic bladed disk to investigate the effect of Coriolis forces on vibration characteristics and pointed out that the localization of mode shapes was significantly influenced by the Coriolis effects.

On the other hand, if there are frequency veering regions in curves of natural frequencies versus nodal diameters, the degree of double natural frequencies split of every modal family is not thoroughly discussed in previous work. It is important for avoiding resonance of bladed disk. In addition, the effects of Coriolis forces are affected by rotational speed, but the different rotational speeds on the maximum forced response with and without considering the Coriolis effects are not thoroughly investigated in previous works.

In this paper, if there are frequency veering regions, the sensitivity of degree of double natural frequencies split of every modal family to the Coriolis effects is discussed. In addition, the effects of different rotational speeds and excitation engine orders on the maximum forced response considering the Coriolis effects are investigated.

The remaining parts of this study are organized as follows. In Section 2, the theory of rotating bladed disk with considering the effects of Coriolis force is presented. In Section 3, if there are frequency veering regions, the sensitivity of degree of double natural frequencies split of every modal family to the Coriolis effects is discussed. In Section 4, the effects of different rotational speeds and excitation engine orders on the maximum forced response considering the Coriolis effects are investigated. In Section 5, main conclusions are summarized.

#### 2. Theory of Rotating Blade Disk System with Considering the Coriolis Effects

The general differential equation of motions of forced response for rotating bladed disk is described aswhere is mass matrix; is Coriolis matrix; is damping matrix; , where is stiffness matrixes, is stress stiffening matrix (it is described in detail in Appendix), and is spin softening matrix; si displacement response vector of the system; is vector of external forces. The Coriolis matrix is generated:where is the shape function matrix and the rotational matrix:An engine order forcing is introduced:where is the force amplitude; is the engine order; is the rotational speed; is the time; is the circumferential position.

Generally, the amplitude magnification factor is used to describe the change of forced response between mistuned and tuned bladed disk, and it is a critical parameter for evaluating high cycle fatigue of bladed disk. Quantitatively, amplitude magnification factor is the ratio between the maximum forced response of mistuned bladed disk and tuned bladed disk [15]: where and are the maximum forced response of mistuned and tuned bladed disk, respectively.

#### 3. Sensitivity of Degree of Double Natural Frequencies Split of Every Modal Family to the Coriolis Effects

The effects of Coriolis forces of bladed disk are influenced by the geometrical shapes of the blade, so a realistic bladed disk of jet engine is employed in this paper. The bladed disk contains 22 blades with 330659 eight-node solid elements. The mesh of finite element of the bladed disk is shown in Figure 1. The working rotational speed is 16100 rpm. Young’s modulus is 117.2 GPa, mass density is 4539.5 Kg/m^{3}, Poisson’s ratio is 0.3, and the structural damping is 0.002. Finite element method is used in the simulation.