Nonproportional Intentionally Mistuned Turbine Blisk Design with Improved Component Modal Synthesis

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.


Introduction
e 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. e mistuned blisks have been researched such as vibration performance, reliability, and optimization. [1][2][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]. e 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][7][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. e lumped mass model was constructed to study the blisks [9], but the model is simplified severely and its precision is not high. us, the continuous parameter model is investigated and its precision is higher than the lumped mass model. e 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][11][12]. is model can guide significance, but it is still few applied in engineering. Accordingly, the FEM [13] was developed. However, the computational efficiency of highfidelity 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]. e ROM was sublimed and devoted to small geometric and stiffness mistuning, etc. [15][16][17]. e above investigations reveal that the ROM can increase the calculation efficiency observably. erefore, 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.
us, 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. e robustness of blisks was analyzed [9,18,19]. e 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 outof-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][26][27]. However, the robustness design of blades [28][29][30][31] had been focused since the beginning of the 21st century. e 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][33][34][35]. In addition, some other scholars conducted the reliability and optimization of blades [36][37][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. e 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, Section 2 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.

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). is 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. is method is termed as the CCMSbased 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. erefore, 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.

Tuned Reduced-Order Model of Disk.
For the substructure of tuned disks, the displacement is denoted as where Φ is the truncated set of normal modes; Ψ is the complete set of attachment modes; p φ and p ψ are, respectively, the modal coordinates of Φ s and Ψ s ; 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 Shock and Vibration us, the Ψ and p Ψ in equations (14) and (15) can be ignored for the above ROM, and the ROM of MTB is rewritten as where 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 where Ψ CB is the internal DOFs of blades. e relationship of mass matrix M CB and stiffness matrix K CB are, respectively, described as where i and b are, respectively, the internal and boundary DOFs of blades. e relationship of the minimum boundary modal Ψ CB,m and Ψ CB,k is, respectively, obtained via transforming of M CB and K CB based on the first-order differential of J m and J k , which are expressed as In fact, the boundary modal is the smallest which contributes kinetic and potential energy to the blades. e relationship of normal modal and boundary modal corresponding to the mass and stiffness mistuning is, respectively, expressed as where q CB,m φ,n , q CB Ψ,n , and q CB,k φ,n are, respectively, the nth bladed normal and boundary modals. erefore, the relationship of all blades can be expressed as where 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 where 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. erefore, 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 is ROM only has blade normal modal, and the kinetic and potential energy of boundary displacement is ignored.
is model can be used to analyze the maximum modal shape amplitudes of MTB.

Modeling
MTB modeling is the foundation for investigating the DAs, SAs, and SEAs. us, the model parameters need to be given at first, and the main parameters of MTB are shown in Table 1.
e variations of blade stiffness, density, and thickness are shown in Figures 1∼3. e interfaces between blades and disk as well as the interfaces between disk hole and disk are the key joints. erefore, only the interface DOFs are calculated, which is a huge saving in terms of computational cost. e master DOFs of the MTB are shown in Figure 4.
Only the disk is regarded as superelement [40]; however, the reduced-order FEM of MTB is established via ICMS-NPMM in Figure 5.
To verify the DOFs are reduced by ISCMS, the elements are calculated as shown in Table 2.
It can be seen from Table 2 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.

Maximum Modal Shape Amplitude Analysis
e MTB works in severe environment, and it is very sensitive to mistuning which can trigger vibration localization even high fatigue stress and fracture. erefore, it is very necessary to investigate the modal shape under centrifugal force and thermal load. e 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. is method is simple and has high computational efficiency. To comprehensively study vibration characteristics of MTB, the DAs, SAs, and SEAs are investigated, respectively.

Verification.
e 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. us, the first 38 intrinsic frequencies of MTB with w � 1120 rad·s -1 , t � 1100°C are discussed by ICMS-NPMM in Figure 6 to verify the effectiveness and rationality of ICMS-NPMM. e first 10, 20, 30, and 38 intrinsic frequencies are, respectively, 29.72 --     Figure 7 and Table 3. Figure 6 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. e inherent frequencies acquired by ICMS-NPMM and CCMS-NPMM are practical unanimity with that of HFEM. us, 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. e 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. us, 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. e 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.

Modal Shape Amplitudes.
e 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. e first 38 order DAs, SAs, and SEAs, are respectively, calculated using ICMS-NPMM, as shown in Figures 8(a) Table 4.
As seen in Figures 8∼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. Figure 8 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. Figure 9 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. e 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. Figure 10 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. us, 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. erefore, the SEAs are impacted by speed in low-order frequency. is investigation indicates that the effect of temperature on SEAs in high-order frequency is larger. e increment of SEAs is a bit more complicated, and the fluctuation is observed in low-order frequency.   Shock and Vibration      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. e 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.

Shock and Vibration
To decline the flutter of mistuned blisks, it is vital to find the most influential factors on the structural characteristics and the best combination. e mainly influence factors are shown in Figure 11 by sensitivity analysis [41]. However, the severe mistuning can cause intensive vibration of blades. e experimental study indicates that the density and elastic modulus of each three blades are the same, which can reduce Shock and Vibration the passive mistuning. us, 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. en, the DAs, SAs, and SEAs of intentional MTB are investigated via ICMS-NPMM, and the results are drawn in Figures 13∼15.  Figures 13(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    y1: w � 1120 rad·s −1 , t � 780°C relative to w � 1120 rad·s −1 y2: w � 1120 rad·s −1 , t � 780°C relative to w � 1120 rad·s −1 y2: t � 780°C y5: t � 780°C y2: w � 1250 rad. s −1 y3: w � 1120 rad·s −1 , t � 780°C relative to w � 1120 rad·s −1 y4: w � 1120 rad·s −1 , t � 1100°C relative to w � 1120 rad·s −1  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 Figures 13-15 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.

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. e investigation can obtain the following conclusions: (1) e intentional mistuning is adopted, and MTB is decomposed into mistuned blade and tuned disk. Meanwhile, the substructures are improved and synthesized. e reduced-order FEM is established by ICMS-NPMM, which can be used to analyze small or large mistuning of turbine blisks. (2) e 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. is 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. e 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.

Data Availability
e data used to support the findings of this study are currently under embargo while the research findings are commercialized. Requests for data, (6/12 months) after publication of this article, will be considered by the corresponding author.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.