# Nonproportional Intentionally Mistuned Turbine Blisk Design with Improved Component Modal Synthesis

Authors: Bin Bai; Qi Yang; Guang Wei Zhu; Qi Liang Wu; Xin ye Li
Journal: Shock and Vibration
 (2021)

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## Abstract

An improved component modal synthesis-based nonproportional mistuning method (ICMS-NPMM) is proposed to investigate mistuned turbine blisks (MTBs) since the high-fidelity finite element models (HFEMs) involve large number of computations, which leads to low calculation efficiency. To reduce degrees of freedom and suppress the flutter of MTB, it is divided into mistuned blade structure and tuned disk structure, and the intentional mistuning is considered. Furthermore, the mistuned parameters, nonproportional mistuning, and complex loads are also considered. Firstly, the basic theory of ICMS-NPMM is investigated; secondly, the model of MTB is established via ICMS-NPMM; finally, the intentionally mistuned design of modal shape amplitudes (MSAs) is investigated via ICMS-NPMM. The results indicate that the calculation efficiency is enhanced via ICMS-NPMM relative to that of via HFEM. In addition, the sensitivity and the flutter are decreased; meanwhile, the amplitude fluctuations of MSAs are distinctly decreased and become comparatively smooth. This investigation provides an important guidance for the vibration characteristic study of complex mechanical structures in engineering practice.


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## Body

## 1. Introduction

The turbine blisks work in complex environment and bear complicated loads such as centrifugal force and thermal shock. Mistuning induces vibration and reduces the service life of turbine blisks. Besides, vibration causes the reduction of reliability and durability for turbine blisks. Moreover, the mistuning of turbine blisks is difficult to be identified and predicted. It is necessary to perform the design of mistuned turbine blisks (MTBs) to ensure the performance of aeroengine. The mistuned blisks have been researched such as vibration performance, reliability, and optimization. [1–3].In general, the performance of mistuned blisks is affiliated to the frequency and output. For instance, the influence of mistuning on the blade-disk coupling was investigated [4]. The frequency, critical speeds, and Campbell diagram were studied [5]. Based on the investigation of the frequency, lots of scholars studied the response, and the forced and transient responses were conducted by establishing the sector finite element model (FEM) or lumped mass model, while these models were rarely applied to engineering [6–8].As for the above studies, it is necessary to build the model of blisks firstly. Usually, the accuracy of blisks is related with the model. The lumped mass model was constructed to study the blisks [9], but the model is simplified severely and its precision is not high. Thus, the continuous parameter model is investigated and its precision is higher than the lumped mass model. The beams or shell was used to analyze the natural frequency or vibration response for a rotational shaft-disk-blade system or a compressor blade [5, 10–12]. This model can guide significance, but it is still few applied in engineering. Accordingly, the FEM [13] was developed. However, the computational efficiency of high-fidelity FEM (HFEM) is low; then, an improved method named the reduced-order model (ROM) is developed, and its efficiency enhanced markedly. Numerous researchers have studied the ROM from different perspectives to improve the calculation efficiency by reducing the degrees of freedom (DOFs) of the model. A way, each bladed disk sector regarded as a substructure, was presented to address the ROM [14]. The ROM was sublimed and devoted to small geometric and stiffness mistuning, etc. [15–17]. The above investigations reveal that the ROM can increase the calculation efficiency observably. Therefore, this method is extensively utilized in engineering practice. In this work, the ROM is improved to discuss the modal shape amplitudes of the blisks.However, the above research studies are on the blisks with passive mistuning. Virtually, the blisks of aeroengine often generate vibration or flutter during flight, which seriously affects the service life. Thus, the investigation of flutter for mistuned blisks is significant to engineering practice. To suppress the flutter of bladed disk, the intentionally mistuned design is researched. Intrinsically, the intentionally mistuned design is a robustness control and design, which got a rapid development since the 1990s. The robustness of blisks was analyzed [9, 18, 19]. The robustness of turbocharged components was investigated [20]. Jarrett et al. [21] introduced the nonintrusive polynomial chaos into the design process for a gas turbine aeroengine to facilitate a rapid robustness analysis. Saikia and Sahu [22] studied the robustness of frequency response function of a gas turbine power plant. Mohammadi and Montazeri-Gh [23] investigated the robust performance for an industrial two-shaft gas turbine. Meanwhile, numerous researchers researched the wind turbines as well. For instance, the high extremum out-of-plane loads on the outlier robustness for wind turbines were discussed [24]. Some other scholars also studied the robustness of wind turbines, such as optimization efficiency and multiobjective robust optimization [25–27]. However, the robustness design of blades [28–31] had been focused since the beginning of the 21st century. The intentionally mistuned design of blisks is performed in the recent years. For example, the intentional mistuning was introduced to reduce the forced response of blade [32–35]. In addition, some other scholars conducted the reliability and optimization of blades [36–38].However, the above investigations on intentionally mistuned blisks only focus on the displacement modal shape amplitudes, the localization factor, or only considering blade elasticity modulus as mistuned parameter; furthermore, the above approaches have unacceptable computational efficiency. To cater the actual engineering, more parameters such as blade thickness and density besides its elasticity modulus are regarded as mistuned parameters. The stress amplitudes (SAs) and strain energy amplitudes (SEAs) are regarded as the research objectives as well as displacement amplitudes (DAs), concurrently focusing the centrifugal force and thermal load. In this case, a methodology named the improved component modal synthesis-based nonproportional mistuning method (ICMS-NPMM) is proposed to investigate the modal shape amplitudes of intentionally MTB involving DAs, SAs, and SEAs.In the other sections, Section2 proposes the methodology of ICMS-NPMM, including tuned disk ROM, mistuned blade ROM, improved ROM, and the synthesis of blade and disk substructures, which lays a foundation for dynamic characteristic analysis of MTB, Section 3 establishes the MTB model which supports the modal shape amplitude analysis, Section 4 investigates the intentionally mistuned design of the modal shape amplitudes for MTB, including DAs, SAs, and SEAs, and Section 5 gives conclusions.


## 2. Improved Component Modal Synthesis and Reduced-Order Model

For the complex mechanical structure such as MTB, the computational efficiency is very low for HFEM. To deal with this problem, the ROM is adopted to investigate the modal shape amplitudes of MTB. Usually, blade stiffness mistuning which is a constant parameter was considered in classical component modal synthesis (CCMS). This method is named the CCMS-based proportional mistuning method (CCMS-PMM). However, the change of the modal and response is different proportion in practical engineering. Actually, mistuning parameters are constantly changing during the operation of blisks. This method is termed as the CCMS-based nonproportional mistuning method (CCMS-NPMM), which only considers the elastic modulus of blade as nonproportional mistuned parameter [39]. However, different mistuned blades have different modes and responses. Therefore, not only nonproportional stiffness mistuning but also nonproportional density and thickness mistuning are considered for different blades, which is named the improved component modal synthesis-based nonproportional mistuning method (ICMS-NPMM). And the modal shape amplitudes of MTB are analyzed by ICMS-NPMM.Due to the influences of manufacturing error, material dispersion, installation error, and wear on MTB, the mistuning is inevitable. According to the working condition, the blade is taken as mistuned structure and disk is taken as tuned structure in this work.


### 2.1. Tuned Reduced-Order Model of Disk

For the substructure of tuned disks, the displacement is denoted as(1)x=xΘxΠ=ΦΘΨΘΦΠΨΠpφpψ,where Φ is the truncated set of normal modes; Ψ is the complete set of attachment modes; pφ and pψ are, respectively, the modal coordinates of Φs and Ψs; Θ is the internal DOFs of disk; and Π is the interfacial DOFs between blades and disk.By coordinate transformation, the mass and stiffness matrices of ROM for the disk are written as(2)μ=IΦTMSΨΨTMΦΨTMΨ,κ=ΛΦTKΨΨTKΦΨΠ,where M and K are the mass and stiffness matrices of the tuned disk.


### 2.2. Mistuned Reduced-Order Model of Blade

For the mistuned blade, the displacement is described as(3)x′=Ipψ′=pψ′,where I is the unit matrix and pψ′ is the modal coordinates.The mass and stiffness matrices are described as(4)μ′=ITM′I=M′,κ′=ITK′I=K′,where M′ and K′ are, respectively, the mass and stiffness matrices in physical coordinates and μ′ and κ′ are, respectively, the mass and stiffness matrices of mistuned blade ROM.


### 2.3. Substructure Improvement

Only the stiffness is considered as a constant mistuned parameter on the CCMS-PMM, i.e., the elastic modulus is expressed as(5)En=E01+δnE,where E0 is the nominal elastic modulus of the tuned turbine blisks; En is the nth bladed elastic modulus of MTB; and δnE is the mistuned parameter of the elastic modulus for the nth blade, which is a constant.However, theδnE is changing with the operation of MTB on the CCMS-NPMM. To cater the practical engineering, the nonproportional mistuning of blade thickness and mass is considered as well as that of the blade elastic modulus. And this methodology is termed as ICMS-NPMM.When the mistuning of theith blade thickness is written as(6)Δdi=d0⋅Aint.sin2π⋅NhNbi−1,d0 is the blade thickness of disk, Aint is the mistuned amplitude, Nb is the number of blades, and Nh is the number of harmonics.Theith blade thickness of MTB is defined as(7)di=d0+Δdi.The mistuning of theith blade stiffness is(8)ΔEi=E0⋅Aint⋅sin2π⋅NhNbi−1.Theith blade elasticity modulus of MTB is defined as(9)Ei=E0+ΔEi.When the mistuning of theith blade density is(10)Δρi=ρ0⋅Aintsin2π⋅NhNbi−1,ρ0 is the blade density of tuned blisks.Theith blade density of MTB is defined as(11)ρi=ρ0+Δρi.Three mistuned parameters are considered, which caters the real engineering.


### 2.4. Substructure Synthesis

To combine the disk and blades ROM, the coordinated condition is given, which is expressed as(12)xΠ=x′,fΠ=f′.According to equations (1)–(3) and equation (12), the coordinated condition is rewritten as(13)xΠ=ΦΠpφ+ΨΠpΨ=pΨ=x′.The mass and stiffness matrices of MTB and their modal coordinates are expressed as(14)μsys=μ+ΦΠTM′ΦΠΦΠTM′ΨΠΨΠTM′ΦΠSΨΠTM′ΨΠ=I+ΦΠTM′ΦΠΦTMΨ+ΦΠTM′ΨΠΨTMΦ+ΨΠTM′ΦΠΨTMΨ+ΨΠTM′ΨΠ=I+ΦΠTM′ΦΠΛ−1ΦΠT+ΦΠTM′ΨΠΦΠΛ−1+ΨΠTM′ΦΠΨTMΨ+ΨΠTM′ΨΠ,(15)κsys=κ+ΦΠTK′ΦΠΦΠTK′ΨΠΨΠTK′ΦΠΨΠTK′ΨΠ=Λ+ΦΠTK′ΦΠΦTKΨ+ΦΠTK′ΨΠΨTKΦ+ΨΠTK′ΦΠΨΠ+ΨΠTK′ΨΠ=Λ+ΦΠTK′ΦΠΦΠT+ΦΠTK′ΨΠΦΠ+ΨΠTK′ΦΠΨΠ+ΨΠTK′ΨΠ,(16)psyn=pφpψ,where psyn is comprehensive modal coordinate.The intrinsic frequencies of tuned blisks are still dense after a small mistuning (0%–5%), and the mistuned intrinsic frequencies can be described by a tuned intrinsic subset. Thus, theΨ and pΨ in equations (14) and (15) can be ignored for the above ROM, and the ROM of MTB is rewritten as(17)μsys=I+ΦΠTM′ΦΠ,(18)κsys=Λ+ΦΠTK′ΦΠ,(19)psyn=pφ,where M′=Bdiagn−1,…,NMn′; K′=Bdiagn−1,…,NKn′; Bdiag. is block diagonal matrix; and N is the number of blades.If only the normal modal is used to describe blade vibration, the boundary displacement between blades and disk cannot be obtained; therefore, the additional boundary modal must be considered. Assume that the additional boundary modal is defined as(20)ΨCBI,where ΨCB is the internal DOFs of blades.The relationship of mass matrixMCB and stiffness matrix KCB are, respectively, described as(21)JmΨCB=ΨCBITMiiCBMibCBMibCBMbbCBΨCBI,JkΨCB=ΨCBITKiiCBKibCBKibCBKbbCBΨCBI,where i and b are, respectively, the internal and boundary DOFs of blades.The relationship of the minimum boundary modalΨCB,m and ΨCB,k is, respectively, obtained via transforming of MCB and KCB based on the first-order differential of Jm and Jk, which are expressed as(22)MiiCBΨCB,m+MibCB=0,KiiCBΨCB,m+KibCB=0.In fact, the boundary modal is the smallest which contributes kinetic and potential energy to the blades. The relationship of normal modal and boundary modal corresponding to the mass and stiffness mistuning is, respectively, expressed as(23)ΦΠ,nm=ΦCBΨCB,m0Ιqφ,nCB,mqΨ,nCB,ΦΠ,nk=ΦCBΨCB,k0Ιqφ,nCB,kqΨ,nCB,where qφ,nCB,m, qΨ,nCB, and qφ,nCB,k are, respectively, the nth bladed normal and boundary modals.Therefore, the relationship of all blades can be expressed as(24)ΦΠm=I⊗UCB,mqCB,m,(25)ΦΠk=I⊗UCB,kqCB,k,where(26)UCB,m=ΦCBΨCB,m0Ι,UCB,k=ΦCBΨCB,k0Ι,qCB,m=⋮qφ,nCB,mqΨ,nCB,m⋮,qCB,k=⋮qφ,nCB,kqΨ,nCB⋮,where ⊗ is Kronecker product.Equations (24) and (25) are substituted into equations (17) and (18), and the mass and stiffness matrices of ROM are, respectively, rewritten as(27)μsyn=I+qCB,mTI⊗UCB,mTM′I⊗UCB,mqCB,m=I+Bdiagn=1,…,NqnCB,mTUCB,mTMn′UCB,mqnCB,m=I+Bdiagn=1,…,NqnCB,mTμφφ,n′μφΨ,n′μφΨ,n′TμΨΨ,n′qCB,m,(28)κsyn=Λ+qCB,kTI⊗UCB,kTM′I⊗UCB,kqCB,k=Λ+Bdiagn=1,…,NqnCB,kTUCB,kTMn′UCB,kqnCB,k=Λ+Bdiagn=1,…,NqnCB,kTκφφ,n′κφΨ,n′κφΨ,n′TκΨΨ,n′qnCB,k,where(29)μφφ,n′=ΦCBTMii,n′ΦCB+μφΨ,n′=ΦCBTMii,n′ΨCB,m+Mib,n′,μΨΨ,n′=ΨCB,mTMii,n′ΨCB,m+Mib,n′+Mib,n′TΨCB,m+Mbb,n′,κφφ,n′=ΦCBTKii,n′ΦCB+κφΨ,n′=ΦCBTKii,n′ΨCB,k+Kib,n′,κΨΨ,n′=ΨCB,kTKii,n′ΨCB,k+Kib,n′+Kib,n′TΨCB,k+Kbb,n′.Equations (27) and (28) can be used in various mistuned blisks, but mistuned value associated with boundary modal needs to be known; however, it is almost impossible to achieve. Therefore, assuming that the boundary displacement of the normal modal is smaller, or there is no mistuning of boundary unit, i.e., the kinetic and potential energy of boundary displacement is ignored. In this case, the mass and stiffness matrices of ROM for MTB can be expressed as(30)μsyn≈I+Bdiagn=1,…,Nqφ,nCB,mTμφφ,n′qφ,nCB,m,κsyn≈Λ+Bdiagn=1,…,Nqφ,nCB,kTκφφ,n′qφ,nCB,k.This ROM only has blade normal modal, and the kinetic and potential energy of boundary displacement is ignored. This model can be used to analyze the maximum modal shape amplitudes of MTB.


## 2.1. Tuned Reduced-Order Model of Disk

For the substructure of tuned disks, the displacement is denoted as(1)x=xΘxΠ=ΦΘΨΘΦΠΨΠpφpψ,where Φ is the truncated set of normal modes; Ψ is the complete set of attachment modes; pφ and pψ are, respectively, the modal coordinates of Φs and Ψs; Θ is the internal DOFs of disk; and Π is the interfacial DOFs between blades and disk.By coordinate transformation, the mass and stiffness matrices of ROM for the disk are written as(2)μ=IΦTMSΨΨTMΦΨTMΨ,κ=ΛΦTKΨΨTKΦΨΠ,where M and K are the mass and stiffness matrices of the tuned disk.


## 2.2. Mistuned Reduced-Order Model of Blade

For the mistuned blade, the displacement is described as(3)x′=Ipψ′=pψ′,where I is the unit matrix and pψ′ is the modal coordinates.The mass and stiffness matrices are described as(4)μ′=ITM′I=M′,κ′=ITK′I=K′,where M′ and K′ are, respectively, the mass and stiffness matrices in physical coordinates and μ′ and κ′ are, respectively, the mass and stiffness matrices of mistuned blade ROM.


## 2.3. Substructure Improvement

Only the stiffness is considered as a constant mistuned parameter on the CCMS-PMM, i.e., the elastic modulus is expressed as(5)En=E01+δnE,where E0 is the nominal elastic modulus of the tuned turbine blisks; En is the nth bladed elastic modulus of MTB; and δnE is the mistuned parameter of the elastic modulus for the nth blade, which is a constant.However, theδnE is changing with the operation of MTB on the CCMS-NPMM. To cater the practical engineering, the nonproportional mistuning of blade thickness and mass is considered as well as that of the blade elastic modulus. And this methodology is termed as ICMS-NPMM.When the mistuning of theith blade thickness is written as(6)Δdi=d0⋅Aint.sin2π⋅NhNbi−1,d0 is the blade thickness of disk, Aint is the mistuned amplitude, Nb is the number of blades, and Nh is the number of harmonics.Theith blade thickness of MTB is defined as(7)di=d0+Δdi.The mistuning of theith blade stiffness is(8)ΔEi=E0⋅Aint⋅sin2π⋅NhNbi−1.Theith blade elasticity modulus of MTB is defined as(9)Ei=E0+ΔEi.When the mistuning of theith blade density is(10)Δρi=ρ0⋅Aintsin2π⋅NhNbi−1,ρ0 is the blade density of tuned blisks.Theith blade density of MTB is defined as(11)ρi=ρ0+Δρi.Three mistuned parameters are considered, which caters the real engineering.


## 2.4. Substructure Synthesis

To combine the disk and blades ROM, the coordinated condition is given, which is expressed as(12)xΠ=x′,fΠ=f′.According to equations (1)–(3) and equation (12), the coordinated condition is rewritten as(13)xΠ=ΦΠpφ+ΨΠpΨ=pΨ=x′.The mass and stiffness matrices of MTB and their modal coordinates are expressed as(14)μsys=μ+ΦΠTM′ΦΠΦΠTM′ΨΠΨΠTM′ΦΠSΨΠTM′ΨΠ=I+ΦΠTM′ΦΠΦTMΨ+ΦΠTM′ΨΠΨTMΦ+ΨΠTM′ΦΠΨTMΨ+ΨΠTM′ΨΠ=I+ΦΠTM′ΦΠΛ−1ΦΠT+ΦΠTM′ΨΠΦΠΛ−1+ΨΠTM′ΦΠΨTMΨ+ΨΠTM′ΨΠ,(15)κsys=κ+ΦΠTK′ΦΠΦΠTK′ΨΠΨΠTK′ΦΠΨΠTK′ΨΠ=Λ+ΦΠTK′ΦΠΦTKΨ+ΦΠTK′ΨΠΨTKΦ+ΨΠTK′ΦΠΨΠ+ΨΠTK′ΨΠ=Λ+ΦΠTK′ΦΠΦΠT+ΦΠTK′ΨΠΦΠ+ΨΠTK′ΦΠΨΠ+ΨΠTK′ΨΠ,(16)psyn=pφpψ,where psyn is comprehensive modal coordinate.The intrinsic frequencies of tuned blisks are still dense after a small mistuning (0%–5%), and the mistuned intrinsic frequencies can be described by a tuned intrinsic subset. Thus, theΨ and pΨ in equations (14) and (15) can be ignored for the above ROM, and the ROM of MTB is rewritten as(17)μsys=I+ΦΠTM′ΦΠ,(18)κsys=Λ+ΦΠTK′ΦΠ,(19)psyn=pφ,where M′=Bdiagn−1,…,NMn′; K′=Bdiagn−1,…,NKn′; Bdiag. is block diagonal matrix; and N is the number of blades.If only the normal modal is used to describe blade vibration, the boundary displacement between blades and disk cannot be obtained; therefore, the additional boundary modal must be considered. Assume that the additional boundary modal is defined as(20)ΨCBI,where ΨCB is the internal DOFs of blades.The relationship of mass matrixMCB and stiffness matrix KCB are, respectively, described as(21)JmΨCB=ΨCBITMiiCBMibCBMibCBMbbCBΨCBI,JkΨCB=ΨCBITKiiCBKibCBKibCBKbbCBΨCBI,where i and b are, respectively, the internal and boundary DOFs of blades.The relationship of the minimum boundary modalΨCB,m and ΨCB,k is, respectively, obtained via transforming of MCB and KCB based on the first-order differential of Jm and Jk, which are expressed as(22)MiiCBΨCB,m+MibCB=0,KiiCBΨCB,m+KibCB=0.In fact, the boundary modal is the smallest which contributes kinetic and potential energy to the blades. The relationship of normal modal and boundary modal corresponding to the mass and stiffness mistuning is, respectively, expressed as(23)ΦΠ,nm=ΦCBΨCB,m0Ιqφ,nCB,mqΨ,nCB,ΦΠ,nk=ΦCBΨCB,k0Ιqφ,nCB,kqΨ,nCB,where qφ,nCB,m, qΨ,nCB, and qφ,nCB,k are, respectively, the nth bladed normal and boundary modals.Therefore, the relationship of all blades can be expressed as(24)ΦΠm=I⊗UCB,mqCB,m,(25)ΦΠk=I⊗UCB,kqCB,k,where(26)UCB,m=ΦCBΨCB,m0Ι,UCB,k=ΦCBΨCB,k0Ι,qCB,m=⋮qφ,nCB,mqΨ,nCB,m⋮,qCB,k=⋮qφ,nCB,kqΨ,nCB⋮,where ⊗ is Kronecker product.Equations (24) and (25) are substituted into equations (17) and (18), and the mass and stiffness matrices of ROM are, respectively, rewritten as(27)μsyn=I+qCB,mTI⊗UCB,mTM′I⊗UCB,mqCB,m=I+Bdiagn=1,…,NqnCB,mTUCB,mTMn′UCB,mqnCB,m=I+Bdiagn=1,…,NqnCB,mTμφφ,n′μφΨ,n′μφΨ,n′TμΨΨ,n′qCB,m,(28)κsyn=Λ+qCB,kTI⊗UCB,kTM′I⊗UCB,kqCB,k=Λ+Bdiagn=1,…,NqnCB,kTUCB,kTMn′UCB,kqnCB,k=Λ+Bdiagn=1,…,NqnCB,kTκφφ,n′κφΨ,n′κφΨ,n′TκΨΨ,n′qnCB,k,where(29)μφφ,n′=ΦCBTMii,n′ΦCB+μφΨ,n′=ΦCBTMii,n′ΨCB,m+Mib,n′,μΨΨ,n′=ΨCB,mTMii,n′ΨCB,m+Mib,n′+Mib,n′TΨCB,m+Mbb,n′,κφφ,n′=ΦCBTKii,n′ΦCB+κφΨ,n′=ΦCBTKii,n′ΨCB,k+Kib,n′,κΨΨ,n′=ΨCB,kTKii,n′ΨCB,k+Kib,n′+Kib,n′TΨCB,k+Kbb,n′.Equations (27) and (28) can be used in various mistuned blisks, but mistuned value associated with boundary modal needs to be known; however, it is almost impossible to achieve. Therefore, assuming that the boundary displacement of the normal modal is smaller, or there is no mistuning of boundary unit, i.e., the kinetic and potential energy of boundary displacement is ignored. In this case, the mass and stiffness matrices of ROM for MTB can be expressed as(30)μsyn≈I+Bdiagn=1,…,Nqφ,nCB,mTμφφ,n′qφ,nCB,m,κsyn≈Λ+Bdiagn=1,…,Nqφ,nCB,kTκφφ,n′qφ,nCB,k.This ROM only has blade normal modal, and the kinetic and potential energy of boundary displacement is ignored. This model can be used to analyze the maximum modal shape amplitudes of MTB.


## 3. Modeling

MTB modeling is the foundation for investigating the DAs, SAs, and SEAs. Thus, the model parameters need to be given at first, and the main parameters of MTB are shown in Table1. The variations of blade stiffness, density, and thickness are shown in Figures 1∼3.Table 1
Parameters of MTB.
DiskBladesMTBρd%×103 (kg/m3)8.560denb%×103 (kg/m3)8.249w1120ed%×1011 (Pa)1.841eb%×1011 (Pa)1.7823t1100prd0.3143prb0.3181h3.0αd%×10−5 (°C−1)1.216αb%×10−5 (°C−1)1.268——kd%W (m.°C)-127.21kb%W (m.°C)−129.72——Figure 1
Blade thickness.Figure 2
Blade elasticity modulus.Figure 3
Blade density.The interfaces between blades and disk as well as the interfaces between disk hole and disk are the key joints. Therefore, only the interface DOFs are calculated, which is a huge saving in terms of computational cost. The master DOFs of the MTB are shown in Figure4.Figure 4
Master DOFs of the MTB. (a) Interfaces of master DOFs. (b) Master DOFs.
(a)(b)Only the disk is regarded as superelement [40]; however, the reduced-order FEM of MTB is established via ICMS-NPMM in Figure 5.Figure 5
Reduced-order FEM of MTB. (a) Disk cohesion and blade substructure. (b) Blade cohesion and disk substructure.To verify the DOFs are reduced by ISCMS, the elements are calculated as shown in Table2.Table 2
The elements of MTB.
——HFEMClassical SCMSRatio of decline (%)ISCMSRatio of decline (%)Element nos.110251917391184.224299896.11It can be seen from Table2 and Figure 5 that compared with classical SCMS and HFEM, ratios of decline are, respectively, 84.22% and 96.11%; thus, the DOFs of MTB are distinctly reduced so that the computational time is obviously shorted relative to CCMS and HFEM.


## 4. Maximum Modal Shape Amplitude Analysis

The MTB works in severe environment, and it is very sensitive to mistuning which can trigger vibration localization even high fatigue stress and fracture. Therefore, it is very necessary to investigate the modal shape under centrifugal force and thermal load. The difference from previous studies (localization factor as research object [40])is that the modal shape amplitudes are regarded as the research object so as to make sure the variation tendency of most dangerous position and reasonably prevent MTB from destroying by mistuning. This method is simple and has high computational efficiency. To comprehensively study vibration characteristics of MTB, the DAs, SAs, and SEAs are investigated, respectively.


### 4.1. Verification

The investigation indicates that the one vibration period of blisks is about 23-24-order frequencies, and the blisks are mainly affected by the low-order modes. Meanwhile, in order to improve the calculation efficiency, the first two vibration period modes are studied. Thus, the first 38 intrinsic frequencies of MTB withw = 1120 rad·s–1, t = 1100°C are discussed by ICMS-NPMM in Figure 6 to verify the effectiveness and rationality of ICMS-NPMM. The first 10, 20, 30, and 38 intrinsic frequencies are, respectively, calculated by ICMS-NPMM, CCMS-NPMM, and HFEM in Figure 7 and Table 3.Figure 6
Intrinsic frequency analysis.Figure 7
Computational time of different order frequencies.Table 3
Computational time and time saving ratio of three methods.
Firstn ordersHFEMtCCMS-NPMMt γ (%)ICMS-NPMMt γ (%)1011.4510.597.5110.2510.482018.6715.8914.8914.4722.493028.7921.7524.4518.3936.123856.7840.2429.1328.8749.15Note:t is the computational time of different order frequencies; γ is the time saving ratio for CCMS-NPMM and ICMS-NPMM relative to HFEM.Figure6 indicates that the errors of inherent frequencies for MTB are 0.004%∼0.102% and 0.006%∼0.159% by ICMS-NPMM and CCMS-NPMM relative to that of HFEM. The inherent frequencies acquired by ICMS-NPMM and CCMS-NPMM are practical unanimity with that of HFEM. Thus, the accuracy of SCMS-NPMM and CCMS-NPMM is comparable to that of HFEM. Meanwhile, the computational time and saving ratio are given in Table 3 and Figure 7. The computational time of MTB is, respectively, reduced by 7.51%∼29.13% and 10.48%∼49.15% with CCMS-NPMM and ICMS-NPMM relative to HFEM. It is observed that the larger of the orders, the higher of the time saving ratio. Thus, the calculative time of ICMS-NPMM is lower than that of CCMS-NPMM, which indicates that the ICMS-NPMM is superior to CCMS-NPMM. Also, it can be seen from Table 3 that the time-saving rate is higher with the rise of the frequency order and the main reason is that the energy of the blade disk cannot be uniformly transmitted out due to the existence of mistuning. The accumulated energy increases with the improvement of order so as to the calculation time becomes slower and slower for the HFEM. However, ICMS-NPMM proposed solved the problem.


### 4.2. Modal Shape Amplitudes

The MTB is subjected to common action of centrifugal force and thermal load; therefore, it is necessary to investigate the intrinsic shape amplitudes. To study the common effect of centrifugal force and thermal load on the modal shape amplitudes, the DAs, SAs, and SEAs of the tuned and mistuned blisks are researched under interaction of centrifugal force and thermal load. The first 38 order DAs, SAs, and SEAs, are respectively, calculated using ICMS-NPMM, as shown in Figures8(a), 9(a), and 10(a); the deviations of DAs, SAs, and SEAs are shown in Figures 8(b), 9(b), and 10(b); the increments of DAs, SAs, and SEAs and their comparison are shown in Figures 8(c), 9(c), and 10(c). The explanation of Figures 8–10 is shown in Table 4.Figure 8
Displacement amplitudes with centrifugal force and thermal load. (a) Das. (b) Deviation of Das. (c) Variable quantity of DAs.
(a)(b)(c)Figure 9
Stress amplitudes with centrifugal force and thermal load. (a) SAs. (b) Deviation of SAs. (c) Variable quantity of SAs.
(a)(b)(c)Figure 10
Strain energy amplitudes with centrifugal force and thermal load. (a) SEAs. (b) Deviation of SEAs. (c) Variable quantity of SEAs.
(a)(b)(c)Table 4
The explanation of Figures8–10.
Figures8(a)/9(a)/10(a)Figures8(b)/9(b)/10(b)Figures8(c)/9(c)/10(c)Tuned blisksMistuned blisksDeviations of mistuned blisks relative to tuned blisksTuned blisksMistuned blisksy1:w = 1120 rad. s−1y4:w = 1120 rad. s−1y1:w = 1120 rad. s−1y1:w = 1120 rad·s−1, t = 780°C relative to w = 1120 rad·s−1y2:w = 1120 rad·s−1, t = 780°C relative to w = 1120 rad·s−1y2:t = 780°Cy5:t = 780°Cy2:w = 1250 rad. s−1y3:w = 1120 rad·s−1, t = 780°C relative to w = 1120 rad·s−1y4:w = 1120 rad·s−1, t = 1100°C relative to w = 1120 rad·s−1y3:w = 1120 rad·s−1, t = 780°Cy6:w = 1120 rad·s−1, t = 780°Cy3:t = 780°Cy5:t = 1100°C relative to t = 780°Cy6:t = 1100°C relative to t = 780°Cy7:t = 1100°Cy9:t = 1100°Cy4:t = 1100°Cy7:w = 1120 rad·s−1, t = 1100°C relative to w = 1120 rad·s−1, t = 780°Cy8:w = 1120 rad·s−1, t = 1100°C relative to w = 1120 rad·s−1, t = 780°Cy8:w = 1120 rad·s−1, t = 1100°Cy10:w = 1120 rad·s−1, t = 1100°Cy5:w = 1120 rad·s−1, t = 780°C——y11:w = 1250 rad. s−1y13:w = 1250 rad. s−1y6:w = 1120 rad·s−1, t = 1100°C——y12:w = 1250 rad·s−1, t = 780°Cy14:w = 1250 rad·s−1, t = 780°Cy7:w = 1250 rad·s−1, t = 780°C——As seen in Figures8∼10, the variation tendencies of DAs, SAs, and SEAs under the combined action with speeds and temperatures are analogical to those of only speeds or temperatures considered; however, there are some differences.Figure8 indicates that the DAs are closer to those of only speeds considered with the combined action of speeds and temperatures, and the research indicates that DAs are more effected by speed than temperature. Besides, the change trend of DAs is analogical to those of only speeds or temperatures taken into account and corresponding values are reduced. Also, the mistuning is more effected than the speed when the speed and temperature simultaneously act in MTB.Figure9 manifests that the instability of SAs is observed, the danger rises obviously, and the destroy of MTB increases from the 24th order when the temperatures and speeds are both considered. The main reason is the fluctuation obviously increases from the 24th order model. In addition, the variable quantities of SAs arise in low-order and high-order frequencies when the temperature is changeless and the speed raises, or the speed is changeless and temperature rises, which intensifies destruction of MTB.Figure10 shows that the SEAs are sensitive to the temperature in high-order frequency and the MTB is more likely failure from the 24th order model. Thus, the influence of temperature on the SEAs will strengthen when the temperature enhances to 1100°C. In addition, the deviations of SEAs increase sharply in the first vibration period. Distinctly, the destructive of MTB increases when the rotational speed is invariant and temperature is enhanced to a certain degree. Meanwhile, the SEAs in the 9th order frequency occur larger fluctuation when the temperature is changeless but speed is varied. Therefore, the SEAs are impacted by speed in low-order frequency. This investigation indicates that the effect of temperature on SEAs in high-order frequency is larger. The increment of SEAs is a bit more complicated, and the fluctuation is observed in low-order frequency.


### 4.3. Intentionally Mistuned Design

Section4.2 manifests the localization of the modal shape amplitude caused by mistuning remarkably increases and exacerbates MTB compared with the TTB. This can induce flutter which can cause catastrophic damage to MTB, especially, to blades. In fact, the aeroengine will fluctuate or vibrate seriously due to the disturbance of air flow during the flight of aircraft, and the installation error or wear will result passive mistuning, which will induce cracks failure or fatigue fracture for blades. The intentionally mistuned design can reduce passive mistuning. To suppress the flutter, the intentionally mistuned design is investigated for MTB, this design can improve the safety and prolong its service life.To decline the flutter of mistuned blisks, it is vital to find the most influential factors on the structural characteristics and the best combination. The mainly influence factors are shown in Figure11 by sensitivity analysis [41]. However, the severe mistuning can cause intensive vibration of blades. The experimental study indicates that the density and elastic modulus of each three blades are the same, which can reduce the passive mistuning. Thus, the elastic modulus and density of blades can be reasonably designed to improve the robustness and reduce the flutter of MTB as well as decline its fluctuation, which are shown in Figure 12.Figure 11
Sensitivity of the modal shape amplitudes for MTB. (a) Sensitivity of DAs. (b) Sensitivity of SAs. (c) Sensitivity of SEAs.
(a)(b)(c)Figure 12
Intentionally mistuned design. (a) Density of blades. (b) Elastic modulus of blades.
(a)(b)Then, the DAs, SAs, and SEAs of intentional MTB are investigated via ICMS-NPMM, and the results are drawn in Figures13∼15.Figure 13
Displacement amplitudes of intentionally MTB. (a) DAs of tuning, passive, and intentional mistuning. (b) DAs of intentional mistuning. (c) Deviation of DAs. (d) Increment of DAs.
(a)(b)(c)(d)Figure 14
Stress amplitudes of intentionally MTB. (a) SAs of tuning, passive, and intentional mistuning. (b) SAs of intentional mistuning. (c) Deviation of SAs. (d) Increment of SAs.
(a)(b)(c)(d)Figure 15
Strain energy amplitudes of intentional MTB. (a) SEAs of tuning, passive, and intentional mistuning. (b) SEAs of intentional mistuning. (c) Deviation of SEAs. (d) Increment of SEAs.
(a)(b)(c)(d)Figures13(a) ∼ 15(a) manifest that the fluctuations of DAs, SAs, and SEAs become smooth and steady in the first vibration period and distinctly decline in the second vibration period for intentionally MTB compared with passive MTB, which means the stability and safety enhances of blisks; meanwhile, it is observed that their variation tendencies of intentional mistuning are similar to those of tuning and passive mistuning, but the values are less than those of passive mistuning, which are significantly reduced. Figures 13(b) ∼ 15(b) indicate that the DAs, SAs, and SEAs are greatly affected by rotational speed in low-order frequency while by temperature in high-order frequency. Although the values significantly increase, they are stable dominated by blade vibration, which means the reliability of bladed disk enhances. Figures 13(c) ∼ 15(c) show that deviations of DAs, SAs, and SEAs in intentional mistuning relative to those of in passive mistuning in centrifugal force and thermal load are closer to those of only centrifugal force considered in the first vibration period and closer to those of only thermal load considered in the second vibration period. Figures 13(d) ∼ 15(d) reveal that the increments of DAs SAs and SEAs in intentional mistuning relative to in passive mistuning have changed much in the second vibration period than in the first vibration period, the main reason is that the change in passive mistuning is larger than in intentional mistuning.It is seen from Figures13–15 that the fluctuation of modal shape amplitudes is reduced and becomes smooth and steady of intentionally MTB; that is, the response is less sensitive to the variables and the robustness is improved.


## 4.1. Verification

The investigation indicates that the one vibration period of blisks is about 23-24-order frequencies, and the blisks are mainly affected by the low-order modes. Meanwhile, in order to improve the calculation efficiency, the first two vibration period modes are studied. Thus, the first 38 intrinsic frequencies of MTB withw = 1120 rad·s–1, t = 1100°C are discussed by ICMS-NPMM in Figure 6 to verify the effectiveness and rationality of ICMS-NPMM. The first 10, 20, 30, and 38 intrinsic frequencies are, respectively, calculated by ICMS-NPMM, CCMS-NPMM, and HFEM in Figure 7 and Table 3.Figure 6
Intrinsic frequency analysis.Figure 7
Computational time of different order frequencies.Table 3
Computational time and time saving ratio of three methods.
Firstn ordersHFEMtCCMS-NPMMt γ (%)ICMS-NPMMt γ (%)1011.4510.597.5110.2510.482018.6715.8914.8914.4722.493028.7921.7524.4518.3936.123856.7840.2429.1328.8749.15Note:t is the computational time of different order frequencies; γ is the time saving ratio for CCMS-NPMM and ICMS-NPMM relative to HFEM.Figure6 indicates that the errors of inherent frequencies for MTB are 0.004%∼0.102% and 0.006%∼0.159% by ICMS-NPMM and CCMS-NPMM relative to that of HFEM. The inherent frequencies acquired by ICMS-NPMM and CCMS-NPMM are practical unanimity with that of HFEM. Thus, the accuracy of SCMS-NPMM and CCMS-NPMM is comparable to that of HFEM. Meanwhile, the computational time and saving ratio are given in Table 3 and Figure 7. The computational time of MTB is, respectively, reduced by 7.51%∼29.13% and 10.48%∼49.15% with CCMS-NPMM and ICMS-NPMM relative to HFEM. It is observed that the larger of the orders, the higher of the time saving ratio. Thus, the calculative time of ICMS-NPMM is lower than that of CCMS-NPMM, which indicates that the ICMS-NPMM is superior to CCMS-NPMM. Also, it can be seen from Table 3 that the time-saving rate is higher with the rise of the frequency order and the main reason is that the energy of the blade disk cannot be uniformly transmitted out due to the existence of mistuning. The accumulated energy increases with the improvement of order so as to the calculation time becomes slower and slower for the HFEM. However, ICMS-NPMM proposed solved the problem.


## 4.2. Modal Shape Amplitudes

The MTB is subjected to common action of centrifugal force and thermal load; therefore, it is necessary to investigate the intrinsic shape amplitudes. To study the common effect of centrifugal force and thermal load on the modal shape amplitudes, the DAs, SAs, and SEAs of the tuned and mistuned blisks are researched under interaction of centrifugal force and thermal load. The first 38 order DAs, SAs, and SEAs, are respectively, calculated using ICMS-NPMM, as shown in Figures8(a), 9(a), and 10(a); the deviations of DAs, SAs, and SEAs are shown in Figures 8(b), 9(b), and 10(b); the increments of DAs, SAs, and SEAs and their comparison are shown in Figures 8(c), 9(c), and 10(c). The explanation of Figures 8–10 is shown in Table 4.Figure 8
Displacement amplitudes with centrifugal force and thermal load. (a) Das. (b) Deviation of Das. (c) Variable quantity of DAs.
(a)(b)(c)Figure 9
Stress amplitudes with centrifugal force and thermal load. (a) SAs. (b) Deviation of SAs. (c) Variable quantity of SAs.
(a)(b)(c)Figure 10
Strain energy amplitudes with centrifugal force and thermal load. (a) SEAs. (b) Deviation of SEAs. (c) Variable quantity of SEAs.
(a)(b)(c)Table 4
The explanation of Figures8–10.
Figures8(a)/9(a)/10(a)Figures8(b)/9(b)/10(b)Figures8(c)/9(c)/10(c)Tuned blisksMistuned blisksDeviations of mistuned blisks relative to tuned blisksTuned blisksMistuned blisksy1:w = 1120 rad. s−1y4:w = 1120 rad. s−1y1:w = 1120 rad. s−1y1:w = 1120 rad·s−1, t = 780°C relative to w = 1120 rad·s−1y2:w = 1120 rad·s−1, t = 780°C relative to w = 1120 rad·s−1y2:t = 780°Cy5:t = 780°Cy2:w = 1250 rad. s−1y3:w = 1120 rad·s−1, t = 780°C relative to w = 1120 rad·s−1y4:w = 1120 rad·s−1, t = 1100°C relative to w = 1120 rad·s−1y3:w = 1120 rad·s−1, t = 780°Cy6:w = 1120 rad·s−1, t = 780°Cy3:t = 780°Cy5:t = 1100°C relative to t = 780°Cy6:t = 1100°C relative to t = 780°Cy7:t = 1100°Cy9:t = 1100°Cy4:t = 1100°Cy7:w = 1120 rad·s−1, t = 1100°C relative to w = 1120 rad·s−1, t = 780°Cy8:w = 1120 rad·s−1, t = 1100°C relative to w = 1120 rad·s−1, t = 780°Cy8:w = 1120 rad·s−1, t = 1100°Cy10:w = 1120 rad·s−1, t = 1100°Cy5:w = 1120 rad·s−1, t = 780°C——y11:w = 1250 rad. s−1y13:w = 1250 rad. s−1y6:w = 1120 rad·s−1, t = 1100°C——y12:w = 1250 rad·s−1, t = 780°Cy14:w = 1250 rad·s−1, t = 780°Cy7:w = 1250 rad·s−1, t = 780°C——As seen in Figures8∼10, the variation tendencies of DAs, SAs, and SEAs under the combined action with speeds and temperatures are analogical to those of only speeds or temperatures considered; however, there are some differences.Figure8 indicates that the DAs are closer to those of only speeds considered with the combined action of speeds and temperatures, and the research indicates that DAs are more effected by speed than temperature. Besides, the change trend of DAs is analogical to those of only speeds or temperatures taken into account and corresponding values are reduced. Also, the mistuning is more effected than the speed when the speed and temperature simultaneously act in MTB.Figure9 manifests that the instability of SAs is observed, the danger rises obviously, and the destroy of MTB increases from the 24th order when the temperatures and speeds are both considered. The main reason is the fluctuation obviously increases from the 24th order model. In addition, the variable quantities of SAs arise in low-order and high-order frequencies when the temperature is changeless and the speed raises, or the speed is changeless and temperature rises, which intensifies destruction of MTB.Figure10 shows that the SEAs are sensitive to the temperature in high-order frequency and the MTB is more likely failure from the 24th order model. Thus, the influence of temperature on the SEAs will strengthen when the temperature enhances to 1100°C. In addition, the deviations of SEAs increase sharply in the first vibration period. Distinctly, the destructive of MTB increases when the rotational speed is invariant and temperature is enhanced to a certain degree. Meanwhile, the SEAs in the 9th order frequency occur larger fluctuation when the temperature is changeless but speed is varied. Therefore, the SEAs are impacted by speed in low-order frequency. This investigation indicates that the effect of temperature on SEAs in high-order frequency is larger. The increment of SEAs is a bit more complicated, and the fluctuation is observed in low-order frequency.


## 4.3. Intentionally Mistuned Design

Section4.2 manifests the localization of the modal shape amplitude caused by mistuning remarkably increases and exacerbates MTB compared with the TTB. This can induce flutter which can cause catastrophic damage to MTB, especially, to blades. In fact, the aeroengine will fluctuate or vibrate seriously due to the disturbance of air flow during the flight of aircraft, and the installation error or wear will result passive mistuning, which will induce cracks failure or fatigue fracture for blades. The intentionally mistuned design can reduce passive mistuning. To suppress the flutter, the intentionally mistuned design is investigated for MTB, this design can improve the safety and prolong its service life.To decline the flutter of mistuned blisks, it is vital to find the most influential factors on the structural characteristics and the best combination. The mainly influence factors are shown in Figure11 by sensitivity analysis [41]. However, the severe mistuning can cause intensive vibration of blades. The experimental study indicates that the density and elastic modulus of each three blades are the same, which can reduce the passive mistuning. Thus, the elastic modulus and density of blades can be reasonably designed to improve the robustness and reduce the flutter of MTB as well as decline its fluctuation, which are shown in Figure 12.Figure 11
Sensitivity of the modal shape amplitudes for MTB. (a) Sensitivity of DAs. (b) Sensitivity of SAs. (c) Sensitivity of SEAs.
(a)(b)(c)Figure 12
Intentionally mistuned design. (a) Density of blades. (b) Elastic modulus of blades.
(a)(b)Then, the DAs, SAs, and SEAs of intentional MTB are investigated via ICMS-NPMM, and the results are drawn in Figures13∼15.Figure 13
Displacement amplitudes of intentionally MTB. (a) DAs of tuning, passive, and intentional mistuning. (b) DAs of intentional mistuning. (c) Deviation of DAs. (d) Increment of DAs.
(a)(b)(c)(d)Figure 14
Stress amplitudes of intentionally MTB. (a) SAs of tuning, passive, and intentional mistuning. (b) SAs of intentional mistuning. (c) Deviation of SAs. (d) Increment of SAs.
(a)(b)(c)(d)Figure 15
Strain energy amplitudes of intentional MTB. (a) SEAs of tuning, passive, and intentional mistuning. (b) SEAs of intentional mistuning. (c) Deviation of SEAs. (d) Increment of SEAs.
(a)(b)(c)(d)Figures13(a) ∼ 15(a) manifest that the fluctuations of DAs, SAs, and SEAs become smooth and steady in the first vibration period and distinctly decline in the second vibration period for intentionally MTB compared with passive MTB, which means the stability and safety enhances of blisks; meanwhile, it is observed that their variation tendencies of intentional mistuning are similar to those of tuning and passive mistuning, but the values are less than those of passive mistuning, which are significantly reduced. Figures 13(b) ∼ 15(b) indicate that the DAs, SAs, and SEAs are greatly affected by rotational speed in low-order frequency while by temperature in high-order frequency. Although the values significantly increase, they are stable dominated by blade vibration, which means the reliability of bladed disk enhances. Figures 13(c) ∼ 15(c) show that deviations of DAs, SAs, and SEAs in intentional mistuning relative to those of in passive mistuning in centrifugal force and thermal load are closer to those of only centrifugal force considered in the first vibration period and closer to those of only thermal load considered in the second vibration period. Figures 13(d) ∼ 15(d) reveal that the increments of DAs SAs and SEAs in intentional mistuning relative to in passive mistuning have changed much in the second vibration period than in the first vibration period, the main reason is that the change in passive mistuning is larger than in intentional mistuning.It is seen from Figures13–15 that the fluctuation of modal shape amplitudes is reduced and becomes smooth and steady of intentionally MTB; that is, the response is less sensitive to the variables and the robustness is improved.


## 5. Conclusions

An improved component modal synthesis-based nonproportional mistuning method (ICMS-NPMM) is proposed, which considers elasticity modulus, thickness, and density of blades, as well as centrifugal force and thermal load. Based on this theory, the reduced-order FEM of turbine blisks is established and its DOFs are obviously reduced. The investigation can obtain the following conclusions:(1)
The intentional mistuning is adopted, and MTB is decomposed into mistuned blade and tuned disk. Meanwhile, the substructures are improved and synthesized. The reduced-order FEM is established by ICMS-NPMM, which can be used to analyze small or large mistuning of turbine blisks.(2)
The DAs, SAs, and SEAs of first 38 order frequencies of the passive MTB are investigated using ICMS-NPMM and analyzed the influence of mistuning, centrifugal force, and thermal load on them. This investigation indicates that the modal shape amplitudes in low-order frequency are caused by disk or blade-disk coupling bending vibration, and they in high-order frequency are caused by blade torsional vibration or blade-disk coupling torsional vibration, which shows that the destructive of torsional vibration is more severe and the mistuning is more sensitive to the vibration governed by blades.(3)
Based on the investigation of the modal shape amplitudes in passive mistuning, the intentionally mistuned design is performed to improve the robustness and decrease the destructiveness of flutter for MTB. The modal shape amplitudes are reduced significantly by intentionally mistuned design, and the change is smooth and steady. Furthermore, the amplitude of fluctuation governed by blades decreases distinctly and become comparatively smooth and steady. Particularly, the modal shape amplitudes are more stable and less fluctuant, and the sensitivity of output response to input variables is reduced when the intentional MTB enters into the second vibration period.



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*Source: 6658694-2021-01-30.xml*