Shock and Vibration

Volume 2017, Article ID 1785759, 18 pages

https://doi.org/10.1155/2017/1785759

## Effects of Crack on Vibration Characteristics of Mistuned Rotated Blades

^{1}Science and Technology on Integrated Logistics Support Laboratory, National University of Defense Technology, Changsha, Hunan 410073, China^{2}State Key Laboratory for Manufacturing and Systems Engineering, Xi’an Jiaotong University, Xi’an, Shanxi 710049, China

Correspondence should be addressed to Zhongsheng Chen; nc.ude.tdun@gnehsgnohznehc

Received 19 October 2016; Accepted 16 January 2017; Published 15 February 2017

Academic Editor: Vadim V. Silberschmidt

Copyright © 2017 Hailong Xu et al. 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

Rotated blades are key mechanical components in turbine and high cycle fatigues often induce blade cracks. Meanwhile, mistuning is inevitable in rotated blades, which often makes it much difficult to detect cracks. In order to solve this problem, it is important and necessary to study effects of crack on vibration characteristics of mistuned rotated blades (MRBs). Firstly, a lumped-parameter model is established based on coupled multiple blades, where mistuned stiffness with normal distribution is introduced. Next, a breathing crack model is adopted and eigenvalue analysis is used in coupled lumped-parameter model. Then, numerical analysis is done and effects of depths and positions of a crack on natural frequency, vibration amplitude, and vibration localization parameters are studied. The results show that a crack causes natural frequency decease and vibration amplitude increase of cracked blade. Bifurcations will occur due to a breathing crack. Furthermore, based on natural frequencies and vibration amplitudes, variational factors are defined to detect a crack in MRBs, which are validated by numerical simulations. Thus, the proposed method provides theoretical guidance for crack detection in MRBs.

#### 1. Introduction

Rotated blades are one of the main parts of turbine, which play important roles of energy conversion and often are called the heart of turbines. Low pressure turbine stages have long slender blades which undergo high bending deformation caused by natural frequencies in lower operating range. In order to limit vibratory deformations, these blades of low pressure stages are often stiffened with lacing wire connections. During working, rotated blades are often exposed to severe environments including high-speed rotating aerodynamic force, large centrifugal forces, and vibration transmission from other parts. In addition, some of them are affected by thermal stresses. Thus, rotated blades usually cause different kinds of faults under these extreme operation environments. In particular, blade cracks are one class of dangerous faults. If cracks cannot be detected as soon as possible, they can expand dramatically under the complex incentive conditions and thus lead to disastrous consequences [1–3]. Ideally, rotated blades generally are considered to be cyclic symmetry. In this case, a microcrack can cause vibration localization in tuned rotated blades [4, 5], so detection of microcracks became very easy in tuned rotated blades because they are sensitive to microcracks. However, manufacturing tolerances, using abrasion and material properties, can result in little differences between rotated blades. This phenomenon is usually called mistuning [6, 7]. Vibration localization appears in MRBs and leads to high vibration stress concentrate in minority blades of MRBs, which can induce fatigue cracks in MRBs. When cracks appear in MRBs, they inevitably change mistuned characteristics and vibration characteristics of MRBs. So crack detection in MRBs is difficult to be reached, which is a necessary and important thing for operational safety of MRBs.

Up to now, many methods have been studied to detect and identify cracks in rotated blades. Vibration monitoring is a common and effective method for damage identification which requires us to fully understand the effects of cracks on vibration characteristics of rotated blades. So vibration characteristics of rotated blades and effects of crack on vibration characteristics of rotated blades are hot research issues. Many scholars have done lots of researches. Kim and Stubbs [8] presented a practical methodology to nondestructively localize cracks and estimate the sizes of the cracks in beam-type structures using changes in frequencies. Piovan and Sampaio [9] used continuous parameters model to analyze rotated beams with functionally graded properties and quantificationally studied vibration characteristics of the blade based on centrifugal force. The common characteristic of the above researches is that an isolated blade model is constructed and analyzed. So the model is simple and can get vibration characteristics of a rotated blade quickly. But rotated blades are interconnected with each other by lacing wire, and vibration characteristics of MRBs are different from a single inevitable blade. So it is inappropriate to use isolated blade model to analyze vibration characteristics of MRBs.

Some scholars further propel related researches. They used coupled multiple-blades dynamic model to analyze vibration characteristics of rotated blades. Prohl [10] was the first to analyze blade group dynamics using lumped-parameter modeling of blades with lacing wire attached to the blade tip. Here, lacing wires were considered as spring elements and forced vibration was computed using Holzer Method. This analytical method computes intricately for blades with varying cross section and twist along the length. Finite element model is well suited for such complex problems and it was used by Bajaj [11] to determine the natural frequencies of packeted blade in coupled bending-torsion modes. Lim et al. [12] have carried out an extensive modal analysis based on energy integral for the multipacket blade system considering disk flexibility, angular speed, and shroud flexibility. The above researched works in this segment are done based on ideal blades. In other words, blades have been considered to be identical, both geometrically and structurally. Such ideal type of blades is called tuned blades. The tuned condition preserves symmetry in the mathematical model. However, in practice, the cyclic symmetry of blades is inevitably destroyed by manufacturing tolerance, using abrasion and properties of materials. So the analytical results have a little difference compared with actual rotated blades.

Next, some related researches considered mistuning of blades and used mistuned model to study vibration behaviors of rotated blades. Chandrashaker and Adhikari [13] proposed the Modal Assurance Criterion Localization Factor which may help designers to easily obtain crucial information, such as the degree of localization for all the vibration modes in the system. Wang et al. [14] analyzed resonance characteristics of blisk under random mistuning. And localization of low order bending mode was most greatly impacted by random mistuning. Hai et al. [15] studied the effects of different mistuning factor on the natural frequency and vibration characteristics of blisk. It was found that quality and stiffness of blades were more sensitive to mistuning than damping of blades. It is mature that rotated blades are analyzed based on mistuning. In the operation of MRBs, vibration induces high cycle fatigue and then leads to cracks. It is a difficult and important problem of cracks affecting vibration characteristics of MRBs and related researches are studied rarely. But some scholars study effects of crack on vibration characteristics of tuned rotated blades. Huang [4, 5] studied the effect of a number of blades and distribution of cracks on vibration localization in a cracked pretwisted blade system. Hou [16] presented a study of the mechanisms of cracking-induced mistuning in bladed disks. An analytical model was formulated and then analyzed to understand the relationship between the blade dynamic response and the crack length.

In the aforementioned researches, vibration characteristics of MRBs and effects of crack on tuned rotated blades are studied by some scholars. However, practical experience shows that mistuning is inherent feature of rotated blades and crack can change vibration characteristics of MRBs by change mistuning of MRBs. To the best of our knowledge, rare works have been reported in this area. In this paper, coupled multiple-blades model is built based on inherent mistuning in rotated blades and then used to analyze the effects of crack on vibration characteristics of MRBs. Moreover, analytical results are used to study crack detection in MRBs. The left contents are organized as follows: In Section 2, a simplified model is formulated and then analyzed to understand vibration characteristics of MRBs with a breathing crack. In addition, a local parameter and a variational factor are introduced to quantificationally describe vibration characteristics of MRBs. In Section 3, modal characteristics of MRBs are analyzed by numerical analysis. Next, the effects of depths and positions of a crack on natural frequency, vibration amplitude, and localization parameters of vibration response (LPVRs) of MRBs are studied. Based on natural frequencies and vibration amplitudes, variational factors are used to detect a crack in MRBs. Then, nonlinear behaviors of LPVRs are explained by bifurcation diagram of vibration displacement. Finally, conclusions are marked in Section 4.

#### 2. Modeling and Analysis of MRBs with a Breathing Crack

The 3D model of MRBs is shown in Figure 1. Rotated blades are fixed on central disk, which are long slender blades and undergo high bending deformation caused by natural frequencies in lower operating range. These blades go through transient resonant condition at nozzle passing frequency [17]. In order to limit vibratory deformations, these blades are often stiffened with lacing wire connections. In addition, micromistuning occurs between each blade due to manufacturing tolerances, using abrasion and material properties. Mistuning can be changed as fatigue cracks appeared in MRBs during operational life. These can change vibration characteristics of MRBs. Therefore, crack detection in MRBs can be realized probably through changes of vibration characteristics. So model of MRBs is built first and then is used to study the effects of crack on vibration characteristics of MRBs. For the sake of simplicity, the disk is assumed to be rigid and a blade in Figure 1 is simplified to cantilever beam in Figure 2, which is conducive to establish and solve follow-up model. Length, width, and height of the blade in Figure 2 are denoted as , , and , respectively. When a crack appears in a blade, the distance between the crack and the blade tip is denoted as , and the depth of the crack is denoted as . Then, we make an investigation into modal characteristics of tuned blades and MRBs using a lumped-parameter model approach similar to [16]. The simplified lumped-parameter model is shown in Figure 3. In this model, every blade is represented by a single-degree-of-freedom (SDOF) lumped-parameter model with the equivalent stiffness and the equivalent damping . The equivalent mass is concentrated at the blade tip. Lacing wire between two adjacent blades adds stiffness to blades; hence, it is modeled as a massless spring and stiffness is . is the serial number of a blade and the total number of blades is , so . Lumped-parameter model is adopted here mainly to facilitate parametric study, particularly investigating the effect of lessened stiffness in a blade due to a crack. The objective is primarily to study changes of vibration characteristics in MRBs for different possibilities of cracks, and then expect to instruct crack detection in MRBs.