Blade－Loss－Caused Rubbing Dynamic Characteristics of Rotor－Bladed Disk－Casing System

0 Introduction

Aero－engine usually operates in harsh working conditions such as high temperature，high pressure and high speed［1－2］.Accordingly，rotor，bladed disk，casing and bearing，as the key components of aero－engine，often suffer from the combined alternating effects of centrifugal，aerodynamic and thermal loads［3－4］.Therefore，fatigue crack is easily initiated under these circumstances.According to the existing literatures，blade crack is studied by many scholars as a common problem［5－8］.Once the crack generates，then propagates rapidly and fractures finally［9］.Released rotating blade can further perforate the casing as well as other airplane components and endangers the flight safety［10－11］.Therefore，blade containment test must be conducted during engine development［9］.However，due to high experimental cost and less useful datum，numerical simulation is mostly adopted to carry out blade containment analysis in many cases［10－12］.Sun et al.［13］simplified the dual－rotor as a lumped mass model，and casing is established by mixed beam－solid elements.Besides，the coupling effects between rotors and casing are considered through spring elements.Then dynamics of the system due to blade loss event is discussed.Considering the effects of blade loss.Sinha［14］derived the equations of motion of unsymmetrical rotor－bladed disk system using analytical methods，and discussed the rubbing characteristics between blade and rigid casing.Based on LS－DYNA software，Carney et al.［12］and Jain et al.［15］simulated the interaction process between blade fragment，rotor and casing as well as the motion trajectory of local components.Heidari et al.［16］adopted implicit－explicitimplicit analysis process to discuss the complicated rubbing dynamics of whole aero－engine based on MD Nastran software.He et al.［17－19］combined numerical simulation with experiments to carry out the research on blade containment，and the results obtained from LS－DYNA agreed well with those obtained from experiments.

According to the literatures listed above，blade containment and blade loss－induced rubbing dynamics between unbalanced rotor and casing are the two main topics concerned by many scholars.As for the research on blade containment，rotor and bearing supports are usually removed from the system.As for blade loss－induced rubbing dynamics between unbalanced rotor and casing，most researchers simulate blade loss effect only through introducing unbalanced force but ignoring blade loss－induced asymmetrical inertia and mass variation of the system，moreover，material property of the components is often linear elastic and casing flexibility is paid less attention.Therefore，this paper mainly combines the blade containment with blade loss－induced rubbing dynamics，and establishes a single rotor－bladed disk－casing system using commercial software ANSYS/LS－DYNA.Then the complicated rubbing characteristics between blade fragment，casing and unbalanced rotor system are simulated.In addition，vibration responses of disk mass center，changing rules of rubbing force and contact states among system components are also analyzed in detail.Finally，some conclusions are made.

1 Finite Element Model of Rotor－

Bladed Disk－Casing System

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The finite element model of rotor－bladed disk－casing system is shown in Fig.1.In Fig.1，rotor，bladed disk and casing are meshed by solid element Solid164，and bearing supports are simulated by spring/damping element Combin165.Furthermore，gap between blade tip and casing is defined as 0.8mm.Table 1lists the material parameters of corresponding components.After finite element discretization，the equations of motion of the system can be written as follows

In order to consider the effect of initial stress caused by the operational speed on the vibration response of the system，an implicit－explicit sequence solution is adopted，and corresponding flow chart is shown in Fig.3.

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Fig.1 Finite element model of rotor－bladed disk－casing system

Table 1 Parameter settings for rotor－disk－blades system with spring－damping supports

Component Density ρ/（kg·m－3）Young′s modulus E/MPa Poisson′s ratioυ Tangent modulus G/MPa Yield strength σs/MPa Stiffness kxi，kyi/（N·m－1）（i＝1，2）Damping cxi，cyi/（N·s·m－1（i＝1，2））Rotor 7 850 2.066×105 0.3 Disk 7 850 2.066×105 0.3 Blade 4 370 1.25×105 0.3 927 825 Casing 4 370 1.25×105 0.3 927 825 Spring－damper kx1＝2.7×108 ky1＝4.89×108 kx2＝9.4×107 ky2＝7.67×108 cx1＝3.84×105 cy1＝6.08×105 cx2＝1.33×105 cy2＝1.89×106

Fig.2 Local modes of rotor－bladed disk－casing system

Fig.2shows the local modes of rotor－bladed disk－casing system .Figs.2（a，b）represent the the one nodal diameter vibration of bladed disk coupling with the lateral vibration of shaft.Figs.2（c，d）show the first order bending vibration of blade.Figs.2 （e，f）show the first order bending vibration of blade coupling with the lateral vibration of disk－shaft.

Rayleigh damping is adopted in the system when conducting transient dynamic analysis.Corresponding formula is written as follows

（1）Contact between broken blade，blade component and casing is considerably nonlinear.Multiple frequencies even fractional frequency can exist in the rubbing responses caused by blade loss.Under low operational speed，free vibration in the rubbing responses mainly gives priority to one nodal diameter vibration of bladed disk coupling with the lateral vibration of the shaft as well as the first order bending vibration of blade.With the increasing operational speed，the first order bending vibration of blade in the rubbing responses gradually becomes indistinct，but the first order bending vibration of blade coupling with the lateral vibration of the shaft and disk becomes dominant.

where M，Cand Kare the mass matrix，damping matrix and structural stiffness matrix of the system；G（ω）and K（ω）the gyroscopic and stiffness matrices corresponding to operational speedω（ω＝πn/30，n represents the operational speed and its unit is′rev/min′），respectively；Rand Rc the external load and contact force，respectively；， and uthe acceleration，velocity and displacement of the system，respectively.

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Fig.3 Implicit－explicit analysis flow chart of the rotorbladed disk－casing system

2 Dynamic Characteristics of System Under Blade Loss Events

In this section，dynamic characteristics of the system with blade loss are discussed under n＝5 000，10 000and 15 000rev/min.It′s worth noting that the broken blade will not be released from the disk until t＝5/fr（fr＝n/60）.

The finite element model of rotor－bladed disk－casing is established using ANSYS/LS－DYNA.Then the effects of blade loss under different operational speeds on the rubbing dynamic characteristics of the system are analyzed in detail.Some conclusions can be made as follows：

The time－history curve of rubbing force is shown in Fig.5.Fig.6shows the contact states between components at local moments under n＝5 000rev/min.Figs.5（b—d）are the enlarged views of rubbing region A （see Fig.5（a））.Broken blade firstly contacts with the casing （see Fig.5（b）and Fig.6（b）），then contacts with both blade components and casing （see Fig.5（c）and Fig.6（c）），and finally contacts with the casing again（see Fig.5（d），Figs.6（d，e））.Furthermore，contact status at other local moments is also added into Fig.6.It′s worth noting that no component is damaged and local plastic deformation exists under n＝5 000rev/min（Fig.6（f））.

Fig.4 Vibration responses of disk mass center and bearing dynamic anti－force under n＝5 000rev/min

Fig.5 Time－history curve of rubbing force under n＝5 000rev/min

Fig.6 Contact states between components at local moments under n＝5 000rev/min

This work was supported by the National Natural Science Foundation of China（No.11772089），the Fundamental Research Funds for the Central Universities （Nos.N160312001and N160313004），and the Research Project of State Key Laboratory of Mechanical System and Vibration（No.MSV201707）.

Fig.7 Vibration responses of disk mass center and bearing dynamic anti－force under n＝10 000rev/min

Fig.8 Time－history curve of rubbing force under n＝10 000rev/min

Fig.9 Contact states between components at local moments under n＝10 000rev/min

Figs.10—12show the rubbing dynamic behaviors of rotor－bladed disk－casing system under n＝15 000rev/min.Compared with vibration responses under n＝5 000，10 000rev/min，the distinct difference is the frequency component in the spectrum （see Fig.10）.Only natural frequency fn19can be observed.Moreover，there also exist some differences in the time－history curve of rubbing force relative to the previous two cases（see Fig.11）.Rubbing force in Fig.11（b）doesn′t show multiple－pulse variation relative to that in Fig.5（b）and Fig.8（b）.In Fig.11（d），the rubbing force obtained from blade component always exists even if blade component has no contact with casing.This is mainly because permanent plastic deformation exists in the blade component leading to the appearance of self－contact in the blade.Fig.12gives the contact states of the system components at local moments.Fig.12（a）represents that broken blade just contacts with casing.Fig.12（b）shows the contact between broken blade，blade component and casing.In Fig.12（c），casing is cut off by broken blade.Fig.12（d）shows the final contact states when termination time is arrived.

Fig.10 Vibration responses of disk mass center and bearing dynamic anti－force under n＝15 000rev/min

Fig.11 Time－history curve of rubbing force under n＝15 000rev/min

Fig.12 Contact states between components at local moments under n＝15 000rev/min

3 Conclusions

Vibration responses of disk mass center under n ＝ 5 000rev/min are shown in Fig.4.Figs.4（a，b）show that an instantaneous impact exists when t＝5/frbecause of sudden unbalance caused by blade loss.Fig.4（c）depicts the motion trajectory of disk mass center before and after blade loss.Transient and steady vibration responses after blade loss are processed by Fourier transform （see Figs.4（d，e））.The spectrums reveal that multiple frequencies such as 2fr，3fr and 3.7fraccompanying with natural frequencies such as fn1and fn3occur.This can be attributed to rubbing nonlinearity between components.Fig.4（f）gives the poincarémap of motion trajectory of disk mass center in the last 5/fr.Period－1orbit can be observed clearly.

whereω1＝2πf1，ω2＝2πf2，ξ1＝ξ2＝0.02.［f1，f2］andξ1，ξ2are the concerned low－frequency range covering the first 20order natural frequencies（see Fig.2）and damping ratios.In this paper，f1and f2equal 300Hz and 500Hz，respectively.

（2）During rubbing process，three distinct rubbing phases can be observed under the given operational speeds in this paper.The first phase is the contact between broken blade and casing，the second phase is the mixed contact between broken blade，blade component and casing，and the rubbing mechanical behaviors in the third phase depend on the operational speed，i.e.，contact occurs between broken blade and casing under low operational speed but between blade component and casing under high operational speed.Moreover，self－contact in the blade may appear especially at high operational speed.

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Acknowledgements

Rubbing dynamic characteristics of rotorbladed disk－casing system under n＝10 000rev/min are shown in Figs.7—9.Compared with vibration responses under n＝5 000rev/min，response amplitude after blade loss is much larger than that before blade loss （see Figs.7（a—c））.Besides，spectrums in Figs.7（d，e）show that both multiple and fractional frequencies occur.Moreover，unlike fn1being excited under n ＝5 000rev/min（see Figs.2and 4），fn19is excited here（see Figs.2，7）.These phenomena indicate that rubbing becomes more severe.Poincarémap in Fig.7（f）shows that the orbit of disk mass center also presents period－1motion.Fig.8shows that both the contact time and amplitude of rubbing force are larger than those under n＝5 000 rev/min，which further validates the severity of rubbing.It′s worth noting that the reason for period－1motion of disk mass center is that there is no nonlinearity in the system during steady vibration process after blade loss（see Fig.8，when t≥0.098s，the rubbing force is 0N）.In Fig.8（b），an interesting phenomenon is that blade component（see Fig.1（c））also slightly contacts with casing when t＝0.031 11sand 0.031 26s.The reason causing this phenomenon is that when broken blade contacts with casing，local vibration of casing exists and the gap between blade component and casing then varies（see Figs.9（a，b）），which leads to the occurrence of rubbing between casing and blade component.When t＝0.033 55s，there exists a peak in the curve representing the rubbing force of casing（see Fig.8（c））.This phenomenon can be explained as follows：On the one hand，the broken blade makes centrifugal movement under the effect of centrifugal force and persistently impacts the casing，and the rubbing force can reach the maximum at certain moment，such as t ＝0.033 55sin this paper.On the other hand，assuming that the total energy is definited when rubbing between blade component and broken blade occurs，the rubbing force can be reduced because of large flexural deformation existing in the blade contacting with the broken blade（see Fig.9（c））.Fig.8（d）demonstrates that the rubbing force obtained from both blade component and casing are nearly the same except at the moment t＝0.055 78s.This also indicates that rubbing occurs between blade component and casing（see Figs.9（d，e））.As for the occurrence of catastrophe point at t＝0.055 78s，it is because blade component contact with not only casing but also itself.In addition，Figs.9（a，f）represent the initial and final contact states between components.

In this paper，blade loss is released from the disk and simulated by elements deletion（see Fig.1（b））.Because of solid elements lacking rotational freedoms，in order to make the rotor－bladed disk system rotate about the z－axis （see Fig.1（a）），the regions being 5mm away from left and right shaft ends are defined as rigid bodies（see Fig.1（d）），and only ux，uyand rot z are unconstrained.Considering that casing is a kind of thinshell structure and may be destroyed by rotating broken blade，the failure stress of casing is defined as 902MPa.Moreover，since the contact between components cannot be predicted in advance，the eroding－single－surface contact type is adopted in the system，and this can be helpful to simulate complicated contact behaviors between components.

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