# Extraction of Features due to Breathing Crack from Vibration Response of Rotated Blades considering Tenon Connection and Shroud Contact

Authors: Faming Yang; Yongmin Yang; Haifeng Hu; Fengjiao Guan; Guoji Shen; Suiyu Chen; Zifang Bian
Journal: Shock and Vibration
 (2019)

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## Abstract

Cracks are common failures of aeroengine rotated blades. Online monitoring of rotated blades through their vibration to identify cracks early in various working conditions is significant for operational safety. Breathing crack is a practical form of early cracks and results in nonlinear vibration response. Tenon connection and shroud contact are common structures in aeroengine rotated blades, which can also lead blades to vibrate nonlinearly and seriously interfere online identification of early cracks. Thus, it is important to extract vibration features due to breathing crack considering these two structures. Firstly, a blade with tenon and shroud is simplified and a lumped parameter model of the bladed disk is built. Then, dry friction and coupling force on a blade are analyzed and dynamics equations of the lumped parameter model are established. Next, the stiffness of the blade trunk with a breathing crack is analyzed. Finally, the vibration response of blade trunks with the occurrence of breathing crack is analyzed in time and frequency domains by numerical simulation. Effective features due to breathing crack for online identification are extracted. 2x components of spectrums can be the criterion to judge whether breathing crack occurs. Besides, by comparing the changes in vibration amplitudes with 1x component peaks of spectrums, the cracked blade trunk can be distinguished. These findings can provide important theoretical guidance for online identification of early cracks in aeroengine rotated blades.


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## Body

## 1. Introduction

Aeroengine is a power source and a core component of aviation equipment. As one of the most fatigue and essential parts in aeroengine, rotated blades are important components for energy conversion. They suffer various loads including centrifugal force, airflow impact, and alternating forces transferred from other parts of the engine. Under these excitations, vibration happens and causes various faults in some fragile structures. As a common fault, crack is a key object of aeroengine blades monitoring. Once a crack occurs without detection, it may expand rapidly under complex forces and seriously affect the operational performance of the aeroengine, which may cause disasters. On the other hand, the occurrence of cracks can be reflected by changes in vibration characteristics. Thus, online monitoring of rotated blades through their vibration can be useful for cracks identification and significant for operational safety of aeroengines [1–3]. Effects of cracks on vibration characteristics of blades can be reflected by natural frequency, mode shape, vibration response, and so on. In fact, rotating speed of aeroengine blades is often changed. Even being controlled constant, rotating speed fluctuates due to various factors, such as uneven loads, airflow waves, and rotor imbalance [4, 5]. So it is difficult to measure the changes in natural frequency or mode shape online. But effects of cracks on vibration response of blades can be an important point for online identification. For assembled bladed disks, tenon connection and shroud contact are common structures. Blades are connected to mortises around disks by tenons and transfer loads to disk. Besides, when blades vibrate, adjacent shrouds rub against each other. To some extent, these two structures can reduce vibration. But they also induce nonlinear characteristics, which have a great impact on vibration response of rotated blades in aeroengine [6–9]. This seriously interferes online identification of early cracks. Therefore, it is essential to extract features due to early crack from vibration response of rotated blades considering tenon connection and shroud contact.Cracks can be generally divided into two types: open crack and breathing crack. Open crack is penetrating and remains open under various conditions. Nowadays, systematic research findings have been achieved about how open crack affects vibration characteristics of rotated blades and how to identify it online. Wu and Huang [10] made a research to study effects of an open crack on blades vibration characteristics considering various depths and locations under different rotating speeds. He et al. [11] proposed a new method to identify an open crack in turbine blades online under strong noise basing on genetic algorithm. However, early cracks are often nonpenetrating, and their contact surfaces can keep transforming with blades vibration. This kind of cracks can be modeled as breathing crack. Gudmundson [12] proved that damage on blades is oversimplified if open crack is used to model early cracks. However, periodic opening-closing behavior of breathing crack is more suitable to the real situation of early cracks.Effects of breathing crack on blades vibration characteristics have attracted much attention in recent years. Various models have been built to simulate opening-closing behavior of breathing crack when blades are in working conditions, such as square wave function, cosine function, and some models that alter with loads changing [13–15]. Both open crack and breathing crack can change natural frequencies and mode shapes of blades. But breathing crack can cause nonlinear vibration response on blades. It is a big difference from open crack, which can be guidance for online identification of early cracks. Xu et al. [16] modeled breathing crack with a cosine function to analyze a single-blade vibration response. Then a method called vibration power flow analysis was utilized to catch subharmonics in vibration response of the blade caused by breathing crack. The finding provides a good criterion for early cracks online identification on a single blade with simple structure. However, blades are often not isolated. Tenon connection and shroud contact can bring coupling effects between blades. Since they can greatly change vibration characteristics of blades and even cause nonlinear vibration response, it is far from enough to detect breathing crack only with nonlinear characteristics. Considering tenon connection and shroud contact, online identification of early cracks can be a big challenge.Tenon connection causes rotated blades to withstand nonlinear frictions. Laxalde and Thouverez [17] utilized nonlinear dry friction models on turbine blades and discussed the effects on blades vibration characteristics under different parameters. As nonlinear characteristics are induced, blades will present nonlinear vibration responses. Li et al. [18] established dry friction model for dovetail tenons and studied the effects on vibration response of blades under various rotating speeds, friction coefficients, and external force amplitudes. It is found that, in certain conditions, vibration responses of tenons include harmonics not only with the same frequency as excitation force but its 3x and 5x components. Apart from dry friction, coupling force between blades is also nonlinear due to inevitable gaps between tenons and mortises. Zeng et al. [19] analyzed dry friction and coupling force in consideration of gaps and established dynamic equations based on a lumped parameter model of mistuned bladed disk. Through a numerical simulation analysis, harmonics with multiple frequencies of excitation force were found in vibration response on tenons. Thus, it can be concluded that tenon connection causes nonlinear vibration response on blades.Shroud contact is usually utilized to reduce vibration on blades in aeroengines. Dry friction between contact interfaces of adjacent shrouds can subtract vibration energy. On the other hand, it can cause nonlinear vibration response on blades. Yang et al. [20] proposed a contact friction model and delimit boundary conditions of three states: sticking, slipping, and separation. Besides, nonlinear dry friction is also modeled when two interfaces contact. Ji et al. [21] measured vibration response of blade trunks with damping and pointed out that nonlinear characteristics of vibration response are due to dry friction force. Voldřich et al. [22] studied nonlinear vibration response on blades induced by friction force between shrouds under sticking, slipping, and separation states, respectively. Wu et al. [23] characterized dry friction between shrouds with a nonlinear model to study the effects on blades vibration characteristics under various normal loads and external force. The study shows that shroud contact can lead rotated blades to vibrate nonlinearly.To sum up, it can be concluded that tenon connection and shroud contact can both cause nonlinear vibration response on blades, which can seriously interfere online identification of early cracks. When these two structures exist, online identification of cracks based on vibration response of blades can be a big challenge. Thus, how breathing crack affects the vibration response of blades with tenon connection and shroud contact should be analyzed. And it is urgently needed to extract features which are effective for online identification due to breathing crack from vibration response. However, research on this area has rarely been reported so far.Effects of breathing crack on vibration responses of blades are investigated by numerical simulation considering tenon connection and shroud contact. Based on it, features due to breathing crack are extracted to provide theoretical guidance for online identification. Contents of this paper are organized as follows: In Section2, a real blade with tenon and shroud is simplified and a lumped parameter model of bladed disk is built. Then, nonlinear forces on blades are analyzed and dynamic equations of the lumped parameter model are established. In Section 3, a blade trunk is regarded as a cantilever beam. Stiffness model of the blade trunk containing a breathing crack is established. In Section 4, effects on the stiffness of blade trunk by a breathing crack are further analyzed. Vibration responses of blades are obtained by solving the dynamic equations. Then, effects of breathing crack on vibration responses of blade trunks are analyzed in time and frequency domains. Based on it, features due to breathing crack effective for online identification are extracted. Finally, conclusions are drawn in Section 5.


## 2. Forced Vibration Modeling and Analysis of Blades considering Tenon Connection and Shroud Contact

### 2.1. Establishment of Lumped Parameter Model Based on a Bladed Disk

A blade can be divided into tenon and blade trunk. Tenon is connected with mortise on the outer edge of the disk. Forces on a blade are transferred to disk by tenon connection. When rotating at high speed in aeroengine, blade trunks inevitably vibrate under complex forces including airflow impacts. Excessive vibration can lead to failure and damage, which is harmful to operational safety of aeroengines. To reduce vibration, shrouds are often processed on blade trunks. It can extend life of blades as well as improve safety and stability of aeroengines. A type of blade with tenon and shroud is shown in Figure1. It can be seen that geometry structure of a real blade is rather complicated. The blade trunk has pretwist angle and presetting angle. Even tenon and shroud have no regular shape.Figure 1
A certain type of blade with tenon and shroud.Since tenon is subjected to dry friction and pressure by disk in rotating direction, simplify it to a cuboid, which withstands dry friction on its bottom surfaces and pressure on its side surfaces. The bevels of shroud are its contact interfaces. So consider shroud as a flat plate with a sharp angle. Affected by external forces and shroud contact, blade trunk will present bending and torsional vibration. But only bending vibration is studied here. Thus, blade trunk can be considered as a beam with shroud. The simplified model is shown in Figure2.Figure 2
A simplified model of blade with tenon and shroud.Tenon connection and shroud contact can cause complex nonlinear forces on blades. Bottom surface of tenon is tightly attached to mortise and there exists a normal load. Disk drives blades to rotate by dry friction in the tangential direction between interfaces of tenon contact. Gaps are inevitable between side surfaces of tenon and mortise, so tenon may slip in mortise along their contact interfaces. In slipping state, the dry friction between tenon and mortise is relative to slipping direction. Besides, for disk is not an absolutely rigid body, it has a slight deformation under stress by a tenon and will impact adjacent ones. Therefore, coupling force exists between tenons. There is also dry friction force between adjacent shrouds, which is relative to the motion of their contact interfaces [8, 19, 24–26].Three sectors are selected to build a lumped parameter model of a bladed disk similar to that in [19], as shown in Figure 3. In this model, the disk is considered as a flexible body with no mass. Equivalent mass of each tenon and blade trunk is mi and mi+n, respectively, where i=1,2,…,n, and n is the total number of blades. What need to be declared is that shroud is included in the blade trunk by default. Equivalent stiffness of tenon and blade trunk is ki and ki+n, respectively. Equivalent damping of tenon and blade trunk is ci and ci+n, respectively. Taking the disk as a reference system, displacements of tenon and blade trunk are xi and xi+n, respectively. fli and fri are the coupling forces on tenon, and gi is the dry friction force on tenon. Fli and Fri are the dry friction forces on shroud. fi is the external airflow force on blade trunk.Figure 3
The lumped parameter model of a bladed disk.


### 2.2. Analysis of Nonlinear Forces on Blades

Tenon connection and shroud contact can cause nonlinear forces on rotated blades under working condition, which are analyzed in this section to lay the groundwork for subsequent research on vibration response.


#### 2.2.1. Analysis of Dry Friction Force and Coupling Forces on Tenon

Forces on tenons are shown in Figure4. ith sector is analyzed. dli is the initial gap size between mortise and the left side of tenon, while dri is the initial gap size between mortise and the right side of tenon. A tenon withstands dry friction force gi by the corresponding mortise and coupling forces fli and fri, respectively, by left and right adjacent ones.Figure 4
Diagram of forces on tenons.Up till now, two types of model can be utilized to simulate contact friction performance: macroslip friction and microslip friction. Macroslip friction model only considers slipping states of interfaces [27], while microslip friction model considers both slipping states and sticking states [28]. Since vibration response of blade trunk is the main object to analyze in Section 4, and in sticking states, tenon has little effect on it, and macroslip friction model is used to simulate dry friction between tenon and mortise here. The friction coefficient of interfaces is expressed as μt and the normal load as Nt. Then, dry friction force gi can be calculated as follows:(1)gi=μtNtsgnx˙i.In most studies about tenon connection, disks are considered as rigid bodies that cannot be deformed. In practice, disk has a slight deformation inevitably under loads by blades and affects other blades assembled on it, especially the adjacent ones. Coupling effects between adjacent blades by disk are essentially effects between their tenons. Coupling forces on adjacent tenons are related to gaps between their side surfaces and mortises. Firstly, coupling force on a tenon by its left adjacent one namedfli is discussed. According to [8], when xi−1−xi−dri−1+dli>0, the gaps between side surfaces of these two tenons and mortises are considered to be closed. At this time, coupling force is relatively larger and can be simulated as a linear elastic force. The coupling stiffness is set to be kg. To simplify expression, li=xi−1−xi−dri−1+dli is introduced. Then fli=kgli. While li≤0, the gaps between these two tenons and mortises are not closed. For the disk with a high stiffness, the coupling force is very small and can be neglected. To sum up, fli can be calculated as follows:(2)fli=kgli,li>0,0,li≤0.In the same way, defineri=xi−xi+1−dri+dli+1. Coupling force on a tenon by its right adjacent one fri can be calculated as follows:(3)fri=−kgri,ri>0,0,ri≤0.


#### 2.2.2. Analysis of Dry Friction Forces on Shroud

When blades are assembled on a disk, their shrouds are attached and there exists normal loads between their interfaces. During rotation, excessive vibration on blades is avoided by dry friction between contact interfaces, as shown in Figure5. In practice, directions of contact interfaces and rotation are at an angle. Therefore, directions of dry friction forces are not absolutely the same as that of rotation. But the stiffness of the blade trunk in rotating direction is much smaller than that in others, so the vibration is much larger. Furthermore, vibration in rotating direction is the main object of online monitoring. Considering these reasons, dry friction forces on shroud only in rotating direction are considered, or we can say that the component of actual dry friction force in rotating direction is considered.Figure 5
Diagram of shroud contact on a simplified blade.Each shroud keeps in touch with its adjacent ones. The dry friction model proposed by Santhosh et al. [26] is applied, as shown in Figure 6. Contact of adjacent shrouds is regarded as a platform damper with a linear spring, whose stiffness is ks. The friction coefficient is μs, and the normal load is Ns. yli and yri are displacements of left and right dampers, respectively. Dry friction forces Fli and Fri are relative to motion state of contact interfaces, which can be divided into two types: sticking and slipping. First, dry friction force by left adjacent shroud Fli is discussed. When ksxi+n−xi−1+n−yli<μsNs, the elastic force has not reached the lower limit of slipping friction. At this time, interfaces are in sticking state and Fli is equal to the elastic force. While ksxi+n−xi−1+n−yli≥μsNs, elastic force has exceeded the lower limit of slipping friction. At this time, interfaces are in slipping state and Fli is equal to the slipping friction force. To sum up, Fli can be calculated as follows:(4)Fli=ksxi+n−xi−1+n−yli,ksxi+n−xi−1+n−yli<μsNs,μsNssgny˙li,ksxi+n−xi−1+n−yli≥μsNs,where damper’s displacement yli and velocity y˙li can be defined as follows:(5)yli=xi+n−xi−1+n−μsNs/ks,x˙i+n−x˙i−1+n>0,0,x˙i+n−x˙i−1+n=0,xi+n−xi−1+n+μsNs/ks,x˙i+n−x˙i−1+n<0,y˙li=x˙i+n−x˙i−1+n,x˙i+n−x˙i−1+n≠0,0,x˙i+n−x˙i−1+n=0.Figure 6
Dry friction model on shroud.Similar toFli, dry friction force by right adjacent shroud Fri can be calculated as follows:(6)Fri=ksxi+n−xi+1+n−yri,ksxi+n−xi+1+n−yri<μsNs,μsNssgny˙ri,ksxi+n−xi+1+n−yri≥μsNs,where damper’s displacement yri and velocity y˙ri can be defined as follows:(7)yri=xi+n−xi+1+n−μsNs/ks,x˙i+n−x˙i+1+n>0,0,x˙i+n−x˙i+1+n=0,xi+n−xi+1+n+μsNs/ks,x˙i+n−x˙i+1+n<0,y˙ri=x˙i+n−x˙i+1+n,x˙i+n−x˙i+1+n≠0,0,x˙i+n−x˙i+1+n=0.


### 2.3. Modeling of External Airflow Forces and Establishment of Dynamic Equations

In working conditions, blade trunks are subjected to airflow forces, as shown in Figure7. Consider in one compressor stage, a set of vanes create an upstream flow disturbance with uneven air pressure around the bladed disk. Blade trunks circulate through high-pressure and low-pressure zones, and there exist external airflow forces. Take ith blade trunk as an example. At the moment, it enters a high-pressure zone, and the airflow force is opposite to rotating direction because of pressure difference on both sides of blade trunk. In the same way, while it gets out from a high-pressure zone, the airflow force is the same as the rotating direction. Since blade trunks rotate at a rather high speed, it takes a very short time for them to pass through a pressure zone. Thus, airflow forces can be considered to be gradient forces. Similar to [29], airflow force fi is simulated to a harmonic and cosine function which is utilized as follows:(8)fi=f0cosωexcitet+φi,where f0 is the amplitude of force, ωexcite is the angular velocity of fi, and ωexcite=2πDE0/60. D is the rotating speed of the bladed disk, E0 is the order of airflow force, which is equal to the number of transitions between high and low pressure in one circle around the bladed disk (also equal to the number of high-pressure zones or low-pressure zones), and φi is the phase angle and can be calculated as φi=2πE0i−1/n.Figure 7
Diagram of airflow excitation.According to the lumped parameter model shown in Figure3 and force analysis on blades, dynamic equations of a simplified bladed disk can be established as equation (9). By solving it, the vibration response of blades under external airflow forces can be obtained:(9)mix¨i+cix˙i+ci+nx˙i−x˙i+n+kixi+ki+nxi−xi+n=fli+fri−gi,mi+nx¨i+n+ci+nx˙i+n−x˙i+ki+nxi+n−xi=fi−Fli−Fri.


## 2.1. Establishment of Lumped Parameter Model Based on a Bladed Disk

A blade can be divided into tenon and blade trunk. Tenon is connected with mortise on the outer edge of the disk. Forces on a blade are transferred to disk by tenon connection. When rotating at high speed in aeroengine, blade trunks inevitably vibrate under complex forces including airflow impacts. Excessive vibration can lead to failure and damage, which is harmful to operational safety of aeroengines. To reduce vibration, shrouds are often processed on blade trunks. It can extend life of blades as well as improve safety and stability of aeroengines. A type of blade with tenon and shroud is shown in Figure1. It can be seen that geometry structure of a real blade is rather complicated. The blade trunk has pretwist angle and presetting angle. Even tenon and shroud have no regular shape.Figure 1
A certain type of blade with tenon and shroud.Since tenon is subjected to dry friction and pressure by disk in rotating direction, simplify it to a cuboid, which withstands dry friction on its bottom surfaces and pressure on its side surfaces. The bevels of shroud are its contact interfaces. So consider shroud as a flat plate with a sharp angle. Affected by external forces and shroud contact, blade trunk will present bending and torsional vibration. But only bending vibration is studied here. Thus, blade trunk can be considered as a beam with shroud. The simplified model is shown in Figure2.Figure 2
A simplified model of blade with tenon and shroud.Tenon connection and shroud contact can cause complex nonlinear forces on blades. Bottom surface of tenon is tightly attached to mortise and there exists a normal load. Disk drives blades to rotate by dry friction in the tangential direction between interfaces of tenon contact. Gaps are inevitable between side surfaces of tenon and mortise, so tenon may slip in mortise along their contact interfaces. In slipping state, the dry friction between tenon and mortise is relative to slipping direction. Besides, for disk is not an absolutely rigid body, it has a slight deformation under stress by a tenon and will impact adjacent ones. Therefore, coupling force exists between tenons. There is also dry friction force between adjacent shrouds, which is relative to the motion of their contact interfaces [8, 19, 24–26].Three sectors are selected to build a lumped parameter model of a bladed disk similar to that in [19], as shown in Figure 3. In this model, the disk is considered as a flexible body with no mass. Equivalent mass of each tenon and blade trunk is mi and mi+n, respectively, where i=1,2,…,n, and n is the total number of blades. What need to be declared is that shroud is included in the blade trunk by default. Equivalent stiffness of tenon and blade trunk is ki and ki+n, respectively. Equivalent damping of tenon and blade trunk is ci and ci+n, respectively. Taking the disk as a reference system, displacements of tenon and blade trunk are xi and xi+n, respectively. fli and fri are the coupling forces on tenon, and gi is the dry friction force on tenon. Fli and Fri are the dry friction forces on shroud. fi is the external airflow force on blade trunk.Figure 3
The lumped parameter model of a bladed disk.


## 2.2. Analysis of Nonlinear Forces on Blades

Tenon connection and shroud contact can cause nonlinear forces on rotated blades under working condition, which are analyzed in this section to lay the groundwork for subsequent research on vibration response.


### 2.2.1. Analysis of Dry Friction Force and Coupling Forces on Tenon

Forces on tenons are shown in Figure4. ith sector is analyzed. dli is the initial gap size between mortise and the left side of tenon, while dri is the initial gap size between mortise and the right side of tenon. A tenon withstands dry friction force gi by the corresponding mortise and coupling forces fli and fri, respectively, by left and right adjacent ones.Figure 4
Diagram of forces on tenons.Up till now, two types of model can be utilized to simulate contact friction performance: macroslip friction and microslip friction. Macroslip friction model only considers slipping states of interfaces [27], while microslip friction model considers both slipping states and sticking states [28]. Since vibration response of blade trunk is the main object to analyze in Section 4, and in sticking states, tenon has little effect on it, and macroslip friction model is used to simulate dry friction between tenon and mortise here. The friction coefficient of interfaces is expressed as μt and the normal load as Nt. Then, dry friction force gi can be calculated as follows:(1)gi=μtNtsgnx˙i.In most studies about tenon connection, disks are considered as rigid bodies that cannot be deformed. In practice, disk has a slight deformation inevitably under loads by blades and affects other blades assembled on it, especially the adjacent ones. Coupling effects between adjacent blades by disk are essentially effects between their tenons. Coupling forces on adjacent tenons are related to gaps between their side surfaces and mortises. Firstly, coupling force on a tenon by its left adjacent one namedfli is discussed. According to [8], when xi−1−xi−dri−1+dli>0, the gaps between side surfaces of these two tenons and mortises are considered to be closed. At this time, coupling force is relatively larger and can be simulated as a linear elastic force. The coupling stiffness is set to be kg. To simplify expression, li=xi−1−xi−dri−1+dli is introduced. Then fli=kgli. While li≤0, the gaps between these two tenons and mortises are not closed. For the disk with a high stiffness, the coupling force is very small and can be neglected. To sum up, fli can be calculated as follows:(2)fli=kgli,li>0,0,li≤0.In the same way, defineri=xi−xi+1−dri+dli+1. Coupling force on a tenon by its right adjacent one fri can be calculated as follows:(3)fri=−kgri,ri>0,0,ri≤0.


### 2.2.2. Analysis of Dry Friction Forces on Shroud

When blades are assembled on a disk, their shrouds are attached and there exists normal loads between their interfaces. During rotation, excessive vibration on blades is avoided by dry friction between contact interfaces, as shown in Figure5. In practice, directions of contact interfaces and rotation are at an angle. Therefore, directions of dry friction forces are not absolutely the same as that of rotation. But the stiffness of the blade trunk in rotating direction is much smaller than that in others, so the vibration is much larger. Furthermore, vibration in rotating direction is the main object of online monitoring. Considering these reasons, dry friction forces on shroud only in rotating direction are considered, or we can say that the component of actual dry friction force in rotating direction is considered.Figure 5
Diagram of shroud contact on a simplified blade.Each shroud keeps in touch with its adjacent ones. The dry friction model proposed by Santhosh et al. [26] is applied, as shown in Figure 6. Contact of adjacent shrouds is regarded as a platform damper with a linear spring, whose stiffness is ks. The friction coefficient is μs, and the normal load is Ns. yli and yri are displacements of left and right dampers, respectively. Dry friction forces Fli and Fri are relative to motion state of contact interfaces, which can be divided into two types: sticking and slipping. First, dry friction force by left adjacent shroud Fli is discussed. When ksxi+n−xi−1+n−yli<μsNs, the elastic force has not reached the lower limit of slipping friction. At this time, interfaces are in sticking state and Fli is equal to the elastic force. While ksxi+n−xi−1+n−yli≥μsNs, elastic force has exceeded the lower limit of slipping friction. At this time, interfaces are in slipping state and Fli is equal to the slipping friction force. To sum up, Fli can be calculated as follows:(4)Fli=ksxi+n−xi−1+n−yli,ksxi+n−xi−1+n−yli<μsNs,μsNssgny˙li,ksxi+n−xi−1+n−yli≥μsNs,where damper’s displacement yli and velocity y˙li can be defined as follows:(5)yli=xi+n−xi−1+n−μsNs/ks,x˙i+n−x˙i−1+n>0,0,x˙i+n−x˙i−1+n=0,xi+n−xi−1+n+μsNs/ks,x˙i+n−x˙i−1+n<0,y˙li=x˙i+n−x˙i−1+n,x˙i+n−x˙i−1+n≠0,0,x˙i+n−x˙i−1+n=0.Figure 6
Dry friction model on shroud.Similar toFli, dry friction force by right adjacent shroud Fri can be calculated as follows:(6)Fri=ksxi+n−xi+1+n−yri,ksxi+n−xi+1+n−yri<μsNs,μsNssgny˙ri,ksxi+n−xi+1+n−yri≥μsNs,where damper’s displacement yri and velocity y˙ri can be defined as follows:(7)yri=xi+n−xi+1+n−μsNs/ks,x˙i+n−x˙i+1+n>0,0,x˙i+n−x˙i+1+n=0,xi+n−xi+1+n+μsNs/ks,x˙i+n−x˙i+1+n<0,y˙ri=x˙i+n−x˙i+1+n,x˙i+n−x˙i+1+n≠0,0,x˙i+n−x˙i+1+n=0.


## 2.2.1. Analysis of Dry Friction Force and Coupling Forces on Tenon

Forces on tenons are shown in Figure4. ith sector is analyzed. dli is the initial gap size between mortise and the left side of tenon, while dri is the initial gap size between mortise and the right side of tenon. A tenon withstands dry friction force gi by the corresponding mortise and coupling forces fli and fri, respectively, by left and right adjacent ones.Figure 4
Diagram of forces on tenons.Up till now, two types of model can be utilized to simulate contact friction performance: macroslip friction and microslip friction. Macroslip friction model only considers slipping states of interfaces [27], while microslip friction model considers both slipping states and sticking states [28]. Since vibration response of blade trunk is the main object to analyze in Section 4, and in sticking states, tenon has little effect on it, and macroslip friction model is used to simulate dry friction between tenon and mortise here. The friction coefficient of interfaces is expressed as μt and the normal load as Nt. Then, dry friction force gi can be calculated as follows:(1)gi=μtNtsgnx˙i.In most studies about tenon connection, disks are considered as rigid bodies that cannot be deformed. In practice, disk has a slight deformation inevitably under loads by blades and affects other blades assembled on it, especially the adjacent ones. Coupling effects between adjacent blades by disk are essentially effects between their tenons. Coupling forces on adjacent tenons are related to gaps between their side surfaces and mortises. Firstly, coupling force on a tenon by its left adjacent one namedfli is discussed. According to [8], when xi−1−xi−dri−1+dli>0, the gaps between side surfaces of these two tenons and mortises are considered to be closed. At this time, coupling force is relatively larger and can be simulated as a linear elastic force. The coupling stiffness is set to be kg. To simplify expression, li=xi−1−xi−dri−1+dli is introduced. Then fli=kgli. While li≤0, the gaps between these two tenons and mortises are not closed. For the disk with a high stiffness, the coupling force is very small and can be neglected. To sum up, fli can be calculated as follows:(2)fli=kgli,li>0,0,li≤0.In the same way, defineri=xi−xi+1−dri+dli+1. Coupling force on a tenon by its right adjacent one fri can be calculated as follows:(3)fri=−kgri,ri>0,0,ri≤0.


## 2.2.2. Analysis of Dry Friction Forces on Shroud

When blades are assembled on a disk, their shrouds are attached and there exists normal loads between their interfaces. During rotation, excessive vibration on blades is avoided by dry friction between contact interfaces, as shown in Figure5. In practice, directions of contact interfaces and rotation are at an angle. Therefore, directions of dry friction forces are not absolutely the same as that of rotation. But the stiffness of the blade trunk in rotating direction is much smaller than that in others, so the vibration is much larger. Furthermore, vibration in rotating direction is the main object of online monitoring. Considering these reasons, dry friction forces on shroud only in rotating direction are considered, or we can say that the component of actual dry friction force in rotating direction is considered.Figure 5
Diagram of shroud contact on a simplified blade.Each shroud keeps in touch with its adjacent ones. The dry friction model proposed by Santhosh et al. [26] is applied, as shown in Figure 6. Contact of adjacent shrouds is regarded as a platform damper with a linear spring, whose stiffness is ks. The friction coefficient is μs, and the normal load is Ns. yli and yri are displacements of left and right dampers, respectively. Dry friction forces Fli and Fri are relative to motion state of contact interfaces, which can be divided into two types: sticking and slipping. First, dry friction force by left adjacent shroud Fli is discussed. When ksxi+n−xi−1+n−yli<μsNs, the elastic force has not reached the lower limit of slipping friction. At this time, interfaces are in sticking state and Fli is equal to the elastic force. While ksxi+n−xi−1+n−yli≥μsNs, elastic force has exceeded the lower limit of slipping friction. At this time, interfaces are in slipping state and Fli is equal to the slipping friction force. To sum up, Fli can be calculated as follows:(4)Fli=ksxi+n−xi−1+n−yli,ksxi+n−xi−1+n−yli<μsNs,μsNssgny˙li,ksxi+n−xi−1+n−yli≥μsNs,where damper’s displacement yli and velocity y˙li can be defined as follows:(5)yli=xi+n−xi−1+n−μsNs/ks,x˙i+n−x˙i−1+n>0,0,x˙i+n−x˙i−1+n=0,xi+n−xi−1+n+μsNs/ks,x˙i+n−x˙i−1+n<0,y˙li=x˙i+n−x˙i−1+n,x˙i+n−x˙i−1+n≠0,0,x˙i+n−x˙i−1+n=0.Figure 6
Dry friction model on shroud.Similar toFli, dry friction force by right adjacent shroud Fri can be calculated as follows:(6)Fri=ksxi+n−xi+1+n−yri,ksxi+n−xi+1+n−yri<μsNs,μsNssgny˙ri,ksxi+n−xi+1+n−yri≥μsNs,where damper’s displacement yri and velocity y˙ri can be defined as follows:(7)yri=xi+n−xi+1+n−μsNs/ks,x˙i+n−x˙i+1+n>0,0,x˙i+n−x˙i+1+n=0,xi+n−xi+1+n+μsNs/ks,x˙i+n−x˙i+1+n<0,y˙ri=x˙i+n−x˙i+1+n,x˙i+n−x˙i+1+n≠0,0,x˙i+n−x˙i+1+n=0.


## 2.3. Modeling of External Airflow Forces and Establishment of Dynamic Equations

In working conditions, blade trunks are subjected to airflow forces, as shown in Figure7. Consider in one compressor stage, a set of vanes create an upstream flow disturbance with uneven air pressure around the bladed disk. Blade trunks circulate through high-pressure and low-pressure zones, and there exist external airflow forces. Take ith blade trunk as an example. At the moment, it enters a high-pressure zone, and the airflow force is opposite to rotating direction because of pressure difference on both sides of blade trunk. In the same way, while it gets out from a high-pressure zone, the airflow force is the same as the rotating direction. Since blade trunks rotate at a rather high speed, it takes a very short time for them to pass through a pressure zone. Thus, airflow forces can be considered to be gradient forces. Similar to [29], airflow force fi is simulated to a harmonic and cosine function which is utilized as follows:(8)fi=f0cosωexcitet+φi,where f0 is the amplitude of force, ωexcite is the angular velocity of fi, and ωexcite=2πDE0/60. D is the rotating speed of the bladed disk, E0 is the order of airflow force, which is equal to the number of transitions between high and low pressure in one circle around the bladed disk (also equal to the number of high-pressure zones or low-pressure zones), and φi is the phase angle and can be calculated as φi=2πE0i−1/n.Figure 7
Diagram of airflow excitation.According to the lumped parameter model shown in Figure3 and force analysis on blades, dynamic equations of a simplified bladed disk can be established as equation (9). By solving it, the vibration response of blades under external airflow forces can be obtained:(9)mix¨i+cix˙i+ci+nx˙i−x˙i+n+kixi+ki+nxi−xi+n=fli+fri−gi,mi+nx¨i+n+ci+nx˙i+n−x˙i+ki+nxi+n−xi=fi−Fli−Fri.


## 3. Stiffness Modeling of Blade Trunk with a Breathing Crack

Since the tenon’s stiffness is much larger than the blade trunk’s and the movement range of tenon is rather small in practice, probability of cracks occurring in tenon is much lower than that in blade trunk. Thus, blade trunk with a breathing crack is focused, and the vibration responses of blade trunks are analyzed in the subsequent section.Breathing crack affects vibration characteristics of the blade mainly by changing its bending stiffness rather than mass or damping. Therefore, the stiffness of the blade trunk with a breathing crack is focused.Since shroud is thin and its proportion to blade trunk is low, its mass is relatively small and has little influence on bending stiffness of blade trunk, and it is omitted here. A blade trunk is simplified to a cantilever beam with one end fixed and one end free, as shown in Figure8. The blade trunk’s length is L, width is W, and thickness is H. There exists a breathing crack with a distance of yc from the blade tip and the depth is d. First-order vibration is considered. According to [14], equivalent mass and stiffness of an intact blade trunk at statics state can be calculated as follows:(10)mi+n=0.228m′L,kp=EIπ432L3.where m′ is the unit length mass of the blade trunk and can be calculated as m′=ρWH, ρ is the material density of the blade trunk, E is Young’s modulus, and I is the torque of the blade trunk around its central axis and can be calculated as I=WH3/12.Figure 8
The simplified model of the blade trunk with a breathing crack.According to [16], the blade trunk withstands the effects of centrifugal rigidity at high rotating speed, and its stiffness is increased. Equivalent stiffness of the rotated blade trunk kr can be calculated as follows:(11)kr=kp+4π2mi+nBn2,where B is the calibration factor and can be calculated as follows:(12)B=π302∫0LR+y∫0Ydz/dy2dydy∫0Lz2dy,where R is the radius of the disk and z is the modal function of a simplified blade trunk and can be calculated as follows:(13)z=sinλry−shλry+ζrcosλr−chλry,where ζr=cosβr+chβr/sinβr−shβr and λr=βr/L. Since only the first-order mode is studied, r=1 and βr=β1=1.875.Consider a nonpenetrating crack occurs on the right side ofith blade trunk, as shown in Figure 9. In working conditions, it transits between two states: open and closed. At the moment entering a high-pressure zone, the blade trunk bends opposite to rotating direction under the airflow force. For tensile stress on that side, the breathing crack is open. At this time, the stiffness of the blade trunk is equal to that with an open crack. According to [16], crack brings additional flexibility to the blade trunk. Additional flexibility Δc can be calculated as follows:(14)Δc=72yc2π1−ν2EWH4gα,where ν is Poisson’s ratio and gα can be calculated as follows:(15)gα=d219.60α8−40.69α7+47.04α6−32.99α5+20.30α4−9.98α3+4.60α2−1.05α1+0.63,where α is relative to the depth of the crack and thickness of the blade trunk and can be calculated as α=d/H. Flexibility of an intact blade trunk can be calculated as cr=1/kr. When breathing crack is fully open, the stiffness of the blade trunk can be calculated as follows:(16)kro=1Δc+cr.Figure 9
Diagram of breathing crack in open and closed states.While the blade trunk gets out of a high-pressure zone, it bends in the rotating direction under airflow force. The crack on it is closed under bearing stress, and stiffness of the blade trunk is equal to that when it is intact. Since breathing crack cyclically transits between open and closed states, stiffness of a rotated blade trunk is time-varying and coincides with transition of external airflow force. Thus, similar to [16], the stiffness of the blade trunk with a breathing crack can be simulated as follows:(17)kt=kro+12kr−kro1+cosωexcitet.Substitutekt into equation (8) and solving it, the vibration response of the blade trunk with a breathing crack can be obtained.


## 4. Numerical Analysis of Effects by Breathing Crack on Vibration Response of Blade Trunks

Numerical analysis is done based on the dynamics model of the bladed disk in Section2 and the stiffness model of blade trunk with a breathing crack in Section 3. Effects of breathing crack on vibration response are studied to extract features effective for online identification. Before the analysis, some parameters of the bladed disk have to be set, as shown in Table 1.Table 1
Parameters of the bladed disk.
Symbols
Properties
Value
L
Length of a blade trunk (mm)
95
H
Thickness of a blade trunk (mm)
2.5
W
Width of a blade trunk (mm)
45
R
Radius of a disk (mm)
200
N
Total number of blades
12
E
0
Order of airflow force
1
D
Rotating speed (rpm)
5000
f
0
Amplitude of airflow force (N)
30
E
Young’s modulus (Pa)
0.689
×
10
11
ν
Poisson’s ratio
0.33
ρ
Material density of a blade trunk (kg/m3)
2800
ξ
Damping ratio
0.01
m
t
Mass of a tenon (kg)
0.0272
c
t
Damping of a tenon (N⋅s/m)
0.1445
k
t
Stiffness of a tenon (N/m)
903480
k
g
Coupling stiffness of tenons (N/m)
361392
μ
t
Friction coefficient of tenon connection
0.5
N
t
Normal load on tenon bottom surface (N)
20
μ
s
Friction coefficient of shroud contact
0.5
N
s
Normal load on shroud contact interface (N)
10
k
s
Stiffness of shroud damping (N/m)
10000Due to minor differences in material, wear condition, and manufacturing, the physical property of each blade trunk cannot be exactly the same. According to [19], this phenomenon can be equivalent to stiffness mistuning. Stiffness mistuning factors are defined as δi=ki+n−kr/kr, and the stiffness of rotated blade trunks can be calculated as ki+n=kr1+δi. The stiffness mistuning factors are randomly generated satisfying the normal distribution with a mean of 0 and a standard deviation of 0.01. The factors corresponding to 12 blade trunks are shown in Table 2.Table 2
Stiffness mistuning factors of blade trunks.
δ
1
δ
2
δ
3
δ
4
δ
5
δ
6
δ
7
δ
8
δ
9
δ
10
δ
11
δ
12
0.0033
−0.0075
0.0137
−0.0171
−0.0010
−0.0024
0.0032
0.0031
−0.0086
−0.0003
−0.0016
0.0063


### 4.1. Effects on Stiffness of Blade Trunk by a Breathing Crack

It can be seen from the analysis in Section3 that breathing crack changes stiffness of a blade trunk by introducing additional flexibility. Stiffness after a breathing crack occurs, and its relationship to related parameters is analyzed by numerical simulation.As shown in Figure10, the stiffness of a blade trunk with an open crack is less than that of an intact one, though they are constant. But once a breathing crack occurs, stiffness fluctuates between that of the above two cases. It reaches the level of intact trunk at a few moments, when it can be concluded that the crack is in a closed state. While it drops to the level with an open crack, fully open state is affirmed. In other cases, the crack is partly open. With stiffness decreasing, blade trunk can vibrate with larger amplitude after a breathing crack occurs. Furthermore, breathing crack can change the stiffness of blade trunk nonlinearly, which may affect frequencies in vibration response spectrum.Figure 10
The stiffness graph of the blade trunk with time under different conditions.It is easy to find that the fluctuation range of stiffness depends on that with an open crack. And parametersyc and d are related to it, according to equations (14) and (15). It means that location and depth are the factors affecting the stiffness of the blade trunk with an open crack. As shown in Figure 11, the stiffness of the blade trunk decreases with larger yc and d, respectively, although the trends are nonlinear. Thus, it can be indicated that a breathing crack has a larger effect on vibration response of the blade trunk when it is deeper or closer to blade root.Figure 11
The stiffness graph of the blade trunk with (a)yc and (b) d when there is an open crack. (a) The depth is set as d=1.5mm. (b) The distance is set as yc=75mm.
(a)
(b)Effects on vibration responses of blade trunks by breathing crack are analyzed in time and frequency domains in subsequent sections to extract features.


### 4.2. Extraction of Features in Time Domain

It is uniformly assumed that the breathing crack occurs in 1st blade trunk. As shown in Figure12, the amplitude of the 1st blade trunk shows a steady upward trend with yc and d, respectively, which is consistent with the analysis in Section 4.1. But for most other blade trunks, amplitudes fluctuate in various forms without obvious regularity and do not show much correlation with yc and d. This indicates that tenon connection and shroud contact have effects on vibration responses. And these structures can bring great uncertainties to vibration amplitudes of blade trunks.Figure 12
Vibration amplitude of each blade trunk with (a) the distanceyc and (b) the depth d. (a) The depth is set as d=1.5mm. (b) The distance is set as yc=75mm.
(a)
(b)Next, comparison between vibration amplitudes of blade trunks is made. In order to provide a reference, vibration amplitudes of blade trunks with no crack occurring are shown in Figure13. It can be seen that there are some differences and one of them is much higher than others. This phenomenon is called localization of vibration response, which is due to mistuning [30]. It is worth noting that the 1st blade trunk has a relatively medium amplitude.Figure 13
Comparison between vibration amplitudes of blade trunks with no crack occurring.Because of coupling effects between blades due to tenon connection and shroud contact, once breathing crack occurs in one blade trunk, vibration responses of others can be affected. Comparison between vibration amplitudes of blade trunks under differentyc and d is shown in Figure 14. It can be seen that localization of vibration response still exists after breathing crack occurs but is changed with yc and d. The amplitude of the 1st blade trunk increases with yc and d, respectively, and even becomes the largest with a distance of more than 65 mm or depth of more than 1.3 mm. However, location of vibration response is a phenomenon randomly existing between blades in different bladed disks. Thus, whether breathing crack occurs cannot be judged by comparison between amplitudes especially when the distance is short or the depth is shallow. But what can be noticed in Figure 14 is that the amplitude of the 1st blade trunk changes more rapidly than others. So it is worth focusing on the changes in vibration amplitudes after the occurrence of breathing crack.Figure 14
Comparison between vibration amplitudes of blade trunks when a breathing crack occurs under (a) different distanceyc and (b) different depth d. (a) The depth is set as d=1.5mm. (b) The distance is set as yc=75mm.
(a)
(b)A comparison between the changes in amplitudes of blade trunks after the occurrence of a breathing crack under differentyc and d is shown in Figure 15. The change equals amplitude of the blade trunk after crack minus that before crack. It can be seen that the changes in the amplitude of the 1st blade trunk is larger than others and the differences are obvious, under various yc and d. From this feature, we can know breathing crack occurs and distinguished the cracked blade trunk. Therefore, change in vibration amplitude can be a reliable evidence for online identification of breathing crack.Figure 15
Comparison between the changes in vibration amplitudes of blade trunks after a breathing crack occurs under (a) different distanceyc and (b) different depth d. (a) The depth is set as d=1.5mm. (b) The distance is set as yc=75mm.
(a)
(b)


### 4.3. Extraction of Features in Frequency Domain

First, vibration response spectrum of each blade trunk without crack is obtained as a reference for subsequent analysis and shown in Figure16. It can be seen that when no crack occurs, not only 1x but 3x and 5x components appear. This is a reflection of nonlinear characteristics, which is due to nonlinear forces induced by tenon connection and shroud contact.Figure 16
Vibration response spectrum of each blade trunk without crack.Vibration response spectrum of each blade trunk after the occurrence of a breathing crack is shown in Figure17. It can be seen that, in addition to 3x and 5x components, 2x and even 4x components appear in vibration response spectrum of each blade trunk. This proves that the appearance of breathing crack can lead to nonlinear vibration response. Thus, significant changes by breathing crack on vibration response of blade trunks can be found by frequency spectrum analysis. Since 2x components are more obvious than 4x components generally, choose them as features to judge that breathing crack occurs.Figure 17
Vibration response spectrum of each blade trunk after a breathing crack occurs. Parameters of the breathing crack are set as follows:yc=45mm and d=1.5mm.However, after a breathing crack occurs in the 1st blade trunk, 2x component appears not only in the vibration response spectrum of the 1st blade trunk, but that of all blade trunks. This is because coupling effects exist between blades due to tenon connection and shroud contact. Changes in vibration response of one blade can affect others. When we need to identify in which blade the breathing crack occurs, relying on 2x component is not enough. Then an attempt is made to extract features for determining the cracked blade trunk based on peaks in spectrums.Since 1x, 2x, and 3x components are relatively obvious, comparison between 1x, 2x, and 3x component peaks is made under differentyc, as shown in Figure 18. It can be seen that, in most cases, peaks of the 1st blade trunk have no significant feature compared with other blade trunks. Therefore, the specific blade trunk where breathing crack occurs cannot be distinguished by comparison between peaks in spectrums.Figure 18
Comparison between (a) 1x, (b) 2x, and (c) 3x component peaks in vibration response spectrums of blade trunks under differentyc. The depth of the crack is set as d=1.5mm.
(a)
(b)
(c)Next, a comparison between changes in peaks after the occurrence of a breathing crack is made under differentyc, as shown in Figure 19. The change equals value of component peak after crack minus that before crack. It can be seen that after the occurrence of a breathing crack, change in 1x component peak of the 1st blade trunk is the largest, and the differences from others are obvious, under different yc. Therefore, change in the 1x component peak can be a guidance for determining the specific blade where breathing crack occurs, while in Figures 19(b) and 19(c), changes in 2x and 3x component peaks of the 1st blade trunk do not have an obvious difference comparing others, so they cannot be the features to determine the cracked blade.Figure 19
Comparison between changes in (a) 1x, (b) 2x, and (c) 3x component peaks in vibration response spectrums of blade trunks after a breathing crack occurs under differentyc. The depth of the crack is set as d=1.5mm.
(a)
(b)
(c)To further prove this, a comparison between changes in 1x component peaks of blade trunks is made under differentd, as shown in Figure 20. Similar to Figure 19(a), regardless of various d, a change in the 1x component peak of the 1st blade trunk is much larger than others, even when the crack is rather shallow. It is proved again that change in the 1x component peak is suitable as guidance for determining in which blade the breathing crack is located.Figure 20
Comparison between changes in 1x component peaks in vibration response spectrums of blade trunks after a breathing crack occurs under differentd. The distance of the crack from blade tip is set as yc=75mm.


## 4.1. Effects on Stiffness of Blade Trunk by a Breathing Crack

It can be seen from the analysis in Section3 that breathing crack changes stiffness of a blade trunk by introducing additional flexibility. Stiffness after a breathing crack occurs, and its relationship to related parameters is analyzed by numerical simulation.As shown in Figure10, the stiffness of a blade trunk with an open crack is less than that of an intact one, though they are constant. But once a breathing crack occurs, stiffness fluctuates between that of the above two cases. It reaches the level of intact trunk at a few moments, when it can be concluded that the crack is in a closed state. While it drops to the level with an open crack, fully open state is affirmed. In other cases, the crack is partly open. With stiffness decreasing, blade trunk can vibrate with larger amplitude after a breathing crack occurs. Furthermore, breathing crack can change the stiffness of blade trunk nonlinearly, which may affect frequencies in vibration response spectrum.Figure 10
The stiffness graph of the blade trunk with time under different conditions.It is easy to find that the fluctuation range of stiffness depends on that with an open crack. And parametersyc and d are related to it, according to equations (14) and (15). It means that location and depth are the factors affecting the stiffness of the blade trunk with an open crack. As shown in Figure 11, the stiffness of the blade trunk decreases with larger yc and d, respectively, although the trends are nonlinear. Thus, it can be indicated that a breathing crack has a larger effect on vibration response of the blade trunk when it is deeper or closer to blade root.Figure 11
The stiffness graph of the blade trunk with (a)yc and (b) d when there is an open crack. (a) The depth is set as d=1.5mm. (b) The distance is set as yc=75mm.
(a)
(b)Effects on vibration responses of blade trunks by breathing crack are analyzed in time and frequency domains in subsequent sections to extract features.


## 4.2. Extraction of Features in Time Domain

It is uniformly assumed that the breathing crack occurs in 1st blade trunk. As shown in Figure12, the amplitude of the 1st blade trunk shows a steady upward trend with yc and d, respectively, which is consistent with the analysis in Section 4.1. But for most other blade trunks, amplitudes fluctuate in various forms without obvious regularity and do not show much correlation with yc and d. This indicates that tenon connection and shroud contact have effects on vibration responses. And these structures can bring great uncertainties to vibration amplitudes of blade trunks.Figure 12
Vibration amplitude of each blade trunk with (a) the distanceyc and (b) the depth d. (a) The depth is set as d=1.5mm. (b) The distance is set as yc=75mm.
(a)
(b)Next, comparison between vibration amplitudes of blade trunks is made. In order to provide a reference, vibration amplitudes of blade trunks with no crack occurring are shown in Figure13. It can be seen that there are some differences and one of them is much higher than others. This phenomenon is called localization of vibration response, which is due to mistuning [30]. It is worth noting that the 1st blade trunk has a relatively medium amplitude.Figure 13
Comparison between vibration amplitudes of blade trunks with no crack occurring.Because of coupling effects between blades due to tenon connection and shroud contact, once breathing crack occurs in one blade trunk, vibration responses of others can be affected. Comparison between vibration amplitudes of blade trunks under differentyc and d is shown in Figure 14. It can be seen that localization of vibration response still exists after breathing crack occurs but is changed with yc and d. The amplitude of the 1st blade trunk increases with yc and d, respectively, and even becomes the largest with a distance of more than 65 mm or depth of more than 1.3 mm. However, location of vibration response is a phenomenon randomly existing between blades in different bladed disks. Thus, whether breathing crack occurs cannot be judged by comparison between amplitudes especially when the distance is short or the depth is shallow. But what can be noticed in Figure 14 is that the amplitude of the 1st blade trunk changes more rapidly than others. So it is worth focusing on the changes in vibration amplitudes after the occurrence of breathing crack.Figure 14
Comparison between vibration amplitudes of blade trunks when a breathing crack occurs under (a) different distanceyc and (b) different depth d. (a) The depth is set as d=1.5mm. (b) The distance is set as yc=75mm.
(a)
(b)A comparison between the changes in amplitudes of blade trunks after the occurrence of a breathing crack under differentyc and d is shown in Figure 15. The change equals amplitude of the blade trunk after crack minus that before crack. It can be seen that the changes in the amplitude of the 1st blade trunk is larger than others and the differences are obvious, under various yc and d. From this feature, we can know breathing crack occurs and distinguished the cracked blade trunk. Therefore, change in vibration amplitude can be a reliable evidence for online identification of breathing crack.Figure 15
Comparison between the changes in vibration amplitudes of blade trunks after a breathing crack occurs under (a) different distanceyc and (b) different depth d. (a) The depth is set as d=1.5mm. (b) The distance is set as yc=75mm.
(a)
(b)


## 4.3. Extraction of Features in Frequency Domain

First, vibration response spectrum of each blade trunk without crack is obtained as a reference for subsequent analysis and shown in Figure16. It can be seen that when no crack occurs, not only 1x but 3x and 5x components appear. This is a reflection of nonlinear characteristics, which is due to nonlinear forces induced by tenon connection and shroud contact.Figure 16
Vibration response spectrum of each blade trunk without crack.Vibration response spectrum of each blade trunk after the occurrence of a breathing crack is shown in Figure17. It can be seen that, in addition to 3x and 5x components, 2x and even 4x components appear in vibration response spectrum of each blade trunk. This proves that the appearance of breathing crack can lead to nonlinear vibration response. Thus, significant changes by breathing crack on vibration response of blade trunks can be found by frequency spectrum analysis. Since 2x components are more obvious than 4x components generally, choose them as features to judge that breathing crack occurs.Figure 17
Vibration response spectrum of each blade trunk after a breathing crack occurs. Parameters of the breathing crack are set as follows:yc=45mm and d=1.5mm.However, after a breathing crack occurs in the 1st blade trunk, 2x component appears not only in the vibration response spectrum of the 1st blade trunk, but that of all blade trunks. This is because coupling effects exist between blades due to tenon connection and shroud contact. Changes in vibration response of one blade can affect others. When we need to identify in which blade the breathing crack occurs, relying on 2x component is not enough. Then an attempt is made to extract features for determining the cracked blade trunk based on peaks in spectrums.Since 1x, 2x, and 3x components are relatively obvious, comparison between 1x, 2x, and 3x component peaks is made under differentyc, as shown in Figure 18. It can be seen that, in most cases, peaks of the 1st blade trunk have no significant feature compared with other blade trunks. Therefore, the specific blade trunk where breathing crack occurs cannot be distinguished by comparison between peaks in spectrums.Figure 18
Comparison between (a) 1x, (b) 2x, and (c) 3x component peaks in vibration response spectrums of blade trunks under differentyc. The depth of the crack is set as d=1.5mm.
(a)
(b)
(c)Next, a comparison between changes in peaks after the occurrence of a breathing crack is made under differentyc, as shown in Figure 19. The change equals value of component peak after crack minus that before crack. It can be seen that after the occurrence of a breathing crack, change in 1x component peak of the 1st blade trunk is the largest, and the differences from others are obvious, under different yc. Therefore, change in the 1x component peak can be a guidance for determining the specific blade where breathing crack occurs, while in Figures 19(b) and 19(c), changes in 2x and 3x component peaks of the 1st blade trunk do not have an obvious difference comparing others, so they cannot be the features to determine the cracked blade.Figure 19
Comparison between changes in (a) 1x, (b) 2x, and (c) 3x component peaks in vibration response spectrums of blade trunks after a breathing crack occurs under differentyc. The depth of the crack is set as d=1.5mm.
(a)
(b)
(c)To further prove this, a comparison between changes in 1x component peaks of blade trunks is made under differentd, as shown in Figure 20. Similar to Figure 19(a), regardless of various d, a change in the 1x component peak of the 1st blade trunk is much larger than others, even when the crack is rather shallow. It is proved again that change in the 1x component peak is suitable as guidance for determining in which blade the breathing crack is located.Figure 20
Comparison between changes in 1x component peaks in vibration response spectrums of blade trunks after a breathing crack occurs under differentd. The distance of the crack from blade tip is set as yc=75mm.


## 5. Conclusions

Cracks are common faults in aeroengine rotated blades, which can result in serious threats to operational safety. Therefore, online vibration monitoring on rotated blades to identify early cracks under various working conditions is significant. As practical form of early cracks, breathing crack can cause nonlinear vibration response on blades. However, as common structures of aeroengine blades, tenon connection and shroud contact can also lead the blades to vibrate nonlinearly, which can seriously interfere with online identification of early cracks. Thus, it is necessary to extract features due to breathing crack in consideration of tenon connection and shroud contact and to provide a theoretical guidance for online identification. An aeroengine rotated blade with tenon and shroud is simplified, and a lumped parameter model of bladed disk is built. Then nonlinear forces on rotated blades are analyzed, and dynamic equations of the lumped parameter model are established. Next, a stiffness model of blade trunk with a breathing crack is launched. Finally, effects on stiffness and vibration response of blade trunks by a breathing crack are studied, based on which effective features are extracted due to breathing crack in time and frequency domains, respectively. Conclusions can be drawn as follows: (i) Tenon connection and shroud contact cause coupling effects between blades, which makes the vibration responses of all blades change once a breathing crack occurs in one of them. Besides, because of the nonlinear characteristics induced by tenon connection and shroud contact, vibration response spectrums of blade trunks consist of 3x and 5x components. (ii) Once breathing crack occurs, vibration amplitude of the cracked blade trunk increases with the distance from blade tip and the depth. However, amplitudes of other trunks fluctuate in random forms because of uncertainties caused by tenon connection and shroud contact; (iii) due to mistuning, localization of vibration response exists. Because of initial differences between vibration amplitudes of blade trunks, whether breathing crack occurs cannot be judged by comparison between amplitudes. But the change in amplitude of the cracked blade trunk is much larger than others, which can be a stable evidence for crack identification. (iv) Breathing crack can cause 2x components in vibration response spectrums because of nonlinear change in the stiffness of the blade trunk. Besides, the specific blade where breathing crack occurs can be distinguished based on the change in the 1x component peak. The findings in this research provide a significant guidance for online identification of breathing crack considering tenon connection and shroud contact.However, it must be pointed out that this research is still far from practice. Firstly, the subject is a lumped parameter model of simplified bladed disk. But a real bladed disk has a much more complex structure and more factors to affect vibration responses of blades. Secondly, only bending vibrations under airflow excitation are studied in the numerical simulation. Thus, the model proposed should be improved and torsional vibrations of blade trunks should be studied in future. Other forces besides airflow excitation should also be considered. In addition, the features due to breathing crack extracted in this research should be verified in the online identification of early cracks in rotated blades under real working conditions of aeroengine.



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*Source: 8729620-2019-07-29.xml*