Vibration Characteristics Investigation of Hard-Coated Mistuned Blisk With Multipackets by Lumped Parameter Model

The integrally bladed disk (called as a blisk) is a new-type structure in aeroengine, which incorporates the blade and disk integrally based on the three-dimensional printing technology compared with the traditional bladed disk [1]. The blisk is a one-piece unit that consists of several blades attached to a rotor, which can reduce the number of engine's components and parts, weight, aerodynamic losses, and eliminate each rotor blade's dovetail attachment roots [2]. However, it may induce the high-level vibration and be vulnerable to being subject to greater vibration stress, and even cause fatigue failures in the severe conditions due to the lack of contact friction generated by the assembly of the disk and blades for the blisk. Therefore, it is very vital to find out an external damping approach to effectively control and reduce the vibration response of the blisk to avoid the above circumstances.

Practically, the forced vibration of blisk has been studied in-depth over past decades. First, the dynamic responses were investigated effectively considering the influence of the fretting wear, friction ring dampers, vibration absorber, vibration prediction, and clearance joints in adjustable stator cascades on bladed disks [3–7]. In addition, some other researchers introduced dampers to study the modal properties and forced response characteristics by parallel piezoelectric shunt damping, transient vibration characteristics, multipackets, lacing wire damage, and identification of damage, respectively [8–15]. These scholars have achieved a lot of achievements, but they did not consider the special effect of hard coating on the vibration or the vibration reduction of blades by hard coating. Actually, the hard-coating is becoming an emerging damping technology, which can be regarded as the coating material constituted by the ceramic substrate, the metal substrate and their mixtures, and generally applied on the surface of substrate and can improve the performance of the anticorrosion and thermal resistance of the structure [16–17]. Correspondingly, the hard-coating was gradually used in the blisk to reduce the vibration response [18–20]. Besides, the vibration characteristics of the cylindrical shell with hard-coating were analyzed as well [21–23]. So, the hard-coating damping has a favorable positive effect on the vibration suppression. However, the common influence between the multipackets and the hard coating damping on the vibration characteristics for the blisk have not yet been studied in previous work. Therefore, it is very meaningful to conduct multipackets and hard-coating simultaneously and explore some useful regularities on the vibration characteristics for the blisk.

Generally, the blisk is designed as a cyclic symmetric structure and it has the identical physical and geometric properties in each sector, which is called the tuned blisk. However, some small differences generated by geometry scatters, assembling blades, blade wear, etc., exist in the blade and they break the cyclic symmetric property and lead to the localization behavior such as the modal and the forced response localization, which is called the mistuned blisk [24]. Some research on the mistuned blisk have been carried out. The modal and forced vibration characteristics of the mistuned blisk were studied, and the vibration responses were reduced by introducing the intentional mistuning [25–27]. Also, an approach to the subset simulation on the mistuned blisk was investigated [28]. Then, the reduced-order model (ROM) was established to investigate the structural dynamics of the mistuned blisk [29]. There is no doubt that mistuning will largely increase the risk of the resonant response and broaden the resonant frequency domain. Therefore, it is quite necessary and vital to analyze the vibration characteristics of the mistuned blisk.

In addition, an appropriate dynamic model needs to be established to study the vibration characteristics of the mistuned blisk. Commonly, there are three models, i.e., finite element method (FEM), continuous parameter model (CPM) and lumped parameter model (LMP). So far, kinds of FEMs, particularly, the ROMs effectively improve the efficiency and accuracy [30–33]. Besides, there is another analytical method that involves the CPM. The blade was simplified to the beam model in one-dimension [34], the plated or shell model in two-dimension [35], and the free and forced vibration of the blisk could be formulated by CPMs. Compared with another two models, the lumped parameter model (LPM) is simple and can be directly utilized to analyze the characteristics of the blisk and describe the dynamic behavior of the system. These advantages can be fully reflected, especially in the mistuned blisk. At present, many investigations are also concentrated on the LPM of the blisk. Beirow et al. [36] used a multibody model named “equivalent blisk model” to study the forced responses of the mistuned compressor blisk. Baek et al. [37] developed the LPM of the blisk with a ring damper to validate the rationality of the proposed ROMs. Yumer et al. [38] completed the neural networks identification by LPM. Besides, Zhang et al. [39] investigated the nonlinear and dual-parameter chaotic vibrations for the blisk by LPM. Therefore, it is proper and meaningful to utilize the LPM to explore the vibration characteristics of the mistuned blisk. However, these dynamic models are not considered the influence of multipackets on the blisk, which is of great significance for the vibration reduction design of blisk, so multipacket is not negligible in the actual project.

By summarizing recent researches, even if some people consider multipackets and hard-coating to study the blisk, but only studied tuned blisk not consider the mistuning, and this paper not only considers the multipackets and hard-coating, but also studies the mistuned blisk. Thus, a novel damping approach combining hard-coating and multipackets based on LPM is presented to improve the damping level and effectively reduce the resonant response for the mistuned blisk in this paper. In what follows, Sec. 2 establishes the LPM of the mistuned blisk without hard-coating and multipackets, the hard-coated mistuned blisk without multipackets and the mistuned blisk with hard-coated and multipackets, and gives corresponding dynamical equations. Then, the theoretical solution processes and the vibration response characteristics of the mistuned blisk are described. Section 3 carries out the comparisons of the solved results with the literatures, which validates the feasibility and correctness of the established models, and then investigates the natural frequency and the forced response characteristics for the mistuned blisk based on the Sec. 2. Section 4 givens several meaningful conclusions.

Generally, the suppression of the vibration response is a core problem for the design of the blisk. Depositing hard-coating on the blade surface or lacing wire to connect blades by shrouds to create a bladed packet is the common approach to reduce the blisk vibration. However, few scholars considered the effects of the hard-coating and the multipackets simultaneously on the vibration characteristics for the blisk up to now. Therefore, it is necessary to establish the dynamical model to explore the influences of the hard-coating and the multipackets on the mistuned blisk. The hard-coating is hard composite damping material that is used to coat on the surface of the blades. In this paper, NiCrAlCoY-YSZ (Changsha, Hunan Province, China) material is applied on the surface of the blade to absorb the energy released by the blisk, so as to reduce the vibration and deformation of the blades. (Note: however, this paper only did digital simulation and compared it with other people's research results, and did not do experiments to verify research results. It is not that the experiments are not important, but the next paper focuses on the test, and studies the vibration and multiple failure modes of the blisk by building the test platform. The dynamic model and vibration response of the blisk under various failure modes will be investigated by pasting vibration piezoelectric exciting elements to different parts of the blisk. Also, this paper only does linear vibration research on the blisk to facilitate the research, and the next paper will further in-depth studies nonlinear vibration and reliability on the blisk. Thirdly, this paper mainly conducts theoretical research and the data does not from the company, because the company data is confidential. In fact, the change trend of the simulated data is consistent with that of the real data, and the difference is that the amplitude size and frequency domain range are different, but it does not affect the guiding role of the actual work.) The multipackets group blades to form blade shroud during the analysis of blisk vibration with the purpose of reducing blade vibration and improving life span. For details, see the model in Fig. 1 in Sec. 2

Firstly, the LMP (it is actually a discrete model, in the study process, the disk is simplified as a concentrated mass block, each blade is simplified as a concentrated mass block, so the model has finite number of degrees-of-freedom (DOFs) of blisk without hard-coating and multipackets is given. The blisk is divided into 18 sectors and each sector includes the disk x_i and the blade x_i+18 (i = 1, 2, 3 … 18). The total number of DOFs is 36 in this model. In addition, the structure parameters of the blade and the disk can be simplified as the equivalent parameters, namely, the lumped mass m_b and m_d, the lumped stiffness k_b and k_d, the lumped damping c_b and c_d, are called as mass, stiffness, and damping of blades and disk, respectively. The symbol of disk and blade, e.g, m_d, k_d, c_d and m_b, k_b, c_b can be replaced by m₁, k₁, c₁ and m₂, k₂, c₂, respectively, to better express the dynamical equation of LPM. Then, the coupling stiffness between sector disk is expressed as k_cd by the spring. The LMP is shown in Fig. 1(a), which is called Style 1 to simplify the following description.

where, M, C, K are, respectively, the mass matrix, the damping matrix, and the stiffness matrix of the blisk; F is the external excitation force.

where F₀ represents the amplitude of the exciting force, ω represents the frequency of the exciting force, N represents the number of blades, and θ_j represents the phase Angle of the exciting force on the jth blade. Engine order (EO) represents the order of the harmonic excitation force.

However, the hard-coating and the multipackets have significant influence on the blisk. So, the hard-coated mistuned blisk without multipackets in Sec. 2.2 and the hard-coated mistuned blisk with multipackets in Sec. 2.3 are further investigated.

The coating has a good damping capacity on the blades, and the influences of the hard-coating depositing on the blades are investigated in this part based on the dynamical model in Fig. 1(b). The coating-covered area is the whole surface of the blades and does not involve the surface of the disk in the model. It is regarded that the parameter of the hard-coating in the LPM is sintered on the blade. Thus, the hard-coated LPM of blisk without multipackets is established in Fig. 1(b), and this model is also called style 1 (hard-coated mistuned blisk without multipackets).

Furthermore, the damping ratio is an indispensable condition and needs to be identified, thus, the coating damping is calculated by the modal loss factor of coating material in this paper, whereas the substrate damping is analyzed by the equivalent viscous damping [40]. Meanwhile, the hard-coating damping is combined with the real damping stiffness k_a to form the complex stiffness $ka*$ as shown in Fig. 1(b).

where $ηa$ represents the modal loss factor of coating damping, $ka$ is the real stiffness of coating.

The multipackets and the hard-coating are introduced simultaneously, and the influences of the two factors on vibration characteristics of the mistuned blisk are discussed. Generally, the multipackets consist of several bladed-packets formed by adjacent blades with shroud to increase the couple degree among blades. Different bladed-packets may have certain effects on the vibration performance of the mistuned blisk. So, the models of the hard-coated mistuned blisk with one-packet, two-packets, three-packets, six-packets, and nine-packets are, respectively, investigated.

The mistuned blisk is assumed as only one bladed-packet which consists of 18 shrouds, and the parameter of the shroud can be denoted by k_cb, which represents the coupling stiffness among blades as shown in Fig. 1(c). Similarly, this type of model can be called ‘Style 2’ with the same purpose as “style 1.”

The blisk is divided into two bladed-packets and each packet consists of nine blades by eight shrouds as depicted in Fig. 1(d) and this model is called “style 3”.

The blisk is divided into three bladed-packets and each packet is composed of six blades by five shrouds as shown in Fig. 1(e) and this model is called “style 4.”

The blisk is divided into six bladed-packets and each packet is constituted by three blades with two shrouds as shown in Fig. 1(f) and this model is called “style 5.”

The blisk is divided into nine bladed-packets and each packet consists of two blades by one shroud as shown in Fig. 1(g) and this model is called “style 6.”

The novelties of the paper are the LPM is established, this model is different from the traditional model. The traditional model only considers the hard-coating or only consider the multipackets, however, there are no literature reports that both multipackets and hard-coating are considered, which exactly multipackets and hard-coating both have important effect on the damping of the shock absorption for the blisk, particularly, for the mistuned blisk. This investigation is really considering practical engineering problems. Therefore, the novelties of this paper are in Secs. 2.2 and 2.3. The stiffness parameters in the model in Secs. 2.2 and 2.3 are obviously different from those in the model in Sec. 2.1, and the changes in the stiffness parameters have completely different meanings for practical engineering.

where $ωr2$ and ${φr}$ are, respectively, the rth eigenvalue and eigenvector.

Then, the mode superposition method is adopted to formulate the forced vibration response for the damped mistuned blisk, which can quickly transform the multi-DOF equations into the single-DOF equation and can easily calculate the response of the system.

where $ζr$ is the rth modal damping ratio.

Therefore, the modal characteristics and the forced vibration responses of the mistuned blisk can be calculated based on the models and formulas above.

According to Sec. 2, the dynamical models and the equations are investigated. Then these models are utilized to analyze the vibration characteristics of the blisk. However, the validation needs to be conducted first to ensure the feasibility and correctness of the proposed model and the method. Therefore, the mistuned blisk without and with hard-coating will be illustrated in this section, and the results will be compared with the literature [40]. First, the parameters are listed in Table 1. The blisk models before and after coating are shown in Fig. 2 

Considering the model of the blisk without multipackets in the literature [40], the model of style 1 is selected as the validated model to solve the vibration characteristics of the blisk. First, the natural frequencies are solved, and the results are listed in Table 2 and Fig. 3.

It can be observed from Table 2 that the error is 0% on the first, third, fifth the coated blisk, and the maximum error is 0.0043% in all deviations for the uncoated and the coated blisk and it satisfies requirements in engineering, which may be caused by the computational-precision of different computer configurations. And the good consistency of results between the presented models and the literature can be seen in Fig. 3. There are two apparent mode families in the blisk, that are the frequency orders from the first to the tenth in the first mode family and from the 11th to the 12th in the second mode family. The order of the 1st, 10th, 11th is single frequency due to the orders of the zero nodal diameter (also called the nodal circle). Besides, the orders from the 2nd to the 9th and the 12th are the repetition frequency, which lies in the first nodal diameter to the eighth nodal diameter, respectively. In reality, there are 36 frequency orders, and they are from the 1st to the 18th in first mode family and from the 19th to the 36th in second mode family for the blisk with 36 DOFs. However, only single frequencies and the first two orders in the second mode family are taken here to contrast with the literature's results, as shown in Fig. 3 and Table 2. Moreover, it should be noted that natural frequencies decrease with the addition of the coating, but the reduction degree is not obvious. Overall, the natural frequency of the blisk can be correctly solved utilizing the proposed model.

Furthermore, the blisk is assumed to be subject to the harmonic excitation with 1 g (1 gravity acceleration). The comparisons of the resonant frequencies and responses for the uncoated and the coated blisk are listed in Table 3. The FRFs are in Fig. 4.

According to Table 3, it can be easily found that the deviations of the resonant response for the uncoated and the coated blisk are, respectively, 0.621% and 0.755%, 0.265% and 0.158%, and the largest deviation is within 0.8%, which can be ignored. Also, the resonant frequencies hardly change in the frequency domain. Besides, it is seen that the deviations of the coated blisk are less than that of the uncoated, which is generated by the lower resonant responses for the coated blisk. Moreover, it is noted in Fig. 4 that only two resonant peaks occur in the frequency domain for the uncoated and the coated blisk, which are induced that the excitation frequency is equal or close to the natural frequency and the excitation order is the same as the nodal diameter. It is well known that the blades occur bending vibration when the frequency is 700–800 Hz in the low-order modal, however, it is transition stage from bending vibration to the torsional vibration for the blades when the frequency is 1100–1200 Hz in the high-order modal. As we know, the vibration in low-order is more dangerous. Therefore, the amplitude of the bending vibration of the blades in low-order at 700–800 Hz is larger than that of the transition stage from blade bending vibration to blade torsional vibration at 1100–1200 Hz, which is consistent with the practical engineering. However, once the torsional vibration is aroused, the destructiveness of torsional vibration is greater than that of bending vibration. Thus, the accident severity is greater for the blade in the high-order modal that of in the low-order modal. In addition, the response amplitudes have an obvious reduction for the coated blisk, which indicates that the coating has a great damping capacity to suppress the vibration.

According to the calculation of the natural frequencies and the forced responses of the uncoated and the coated blisk, it manifests that the results obtained by the proposed method are identical basically with the literature whether the hard-coating is considered or not, which validates that the presented dynamical models are correct and reasonable.

The developed models have been fully validated and the solved results have a good consistency with the related literature, which satisfies the corresponding requirement as well in Sec. 3.1. Next, the calculations are conducted based the LPMs to research the influence of the hard-coating and the multipackets on the vibration characteristics for the mistuned blisk, and some meaningful rules and interesting phenomena can be found in this section.

The natural frequencies of the tuned blisk without and with hard-coating, and the natural frequencies of the mistuned with hard-coating blisk are, respectively, formulated via models of styles 1 to 6. Some typical mode-orders and their change rates are shown in Tables 4–6 and the corresponding curves are plotted in Figs. 5–7.

The natural frequencies of the style 1 are formulated and the comparisons of the tuned and the mistuned are shown in Table 4 and Fig. 5.

It is observed from Table 4 and Fig. 5 that two mode families are divided clearly, and there are 16 pairs repetition frequencies from the 2nd to the 17th order and the 20th to the 35th order for the tuned blisk, and other orders belong to the 0 nodal diameter. However, the characteristics do not exist for the mistuned blisk, because the period-cyclic feature is destroyed by the appearance of mistuning. Besides, the change rates of the uncoated and the coated tuned blisk are negative increased, and the change rates are more than 2% in Fig. 5(b), and then the natural frequencies reduce with hard-coating, which means that the hard-coating can lower the system stiffness. And, the maximum change rates happen in the intermediate position for the first and second mode family, respectively, which indicates that the hard-coating can produce a greater damping capacity for style 1. Moreover, it is seen from Figs. 5(a) and 5(c) that the discrepancies of the natural frequencies between the tuned and the mistuned blisk of style 1 are almost identical, and the largest difference is less than 0.1%, which proves that the coating has a weak influence on the natural frequency of the mistuned blisk.

The natural frequencies of style 2 are formulated and their comparisons are shown in Table 5 and Fig. 6.

Similarly, the repetition frequencies also happen in the style 2. Though the style is surrounded by 18 shrouds, they can form a whole packet, so it also can be regarded as a cyclic symmetric structure and has similar features with style 1. Also, 16 pairs of repetition frequencies except the 1st, 18th, 19th, 36th order of the tuned blisk can be observed according to Figs. 6(a) and 6(b). Then, the change rates between the uncoated and the coated blisk remain negative increase, and the minimum value is more than 1%, and natural frequencies still reduce with the hard-coating. However, it should be noted that the maximum change rates appear at the beginning of the first mode family and the end of the second mode family, that are 2.3136% and 4.6233%, respectively. This variation trend may be caused by the effect of multipackets in the style 2 compared with the style 1. Furthermore, there are very small differences between the tuned and the mistuned blisk for the style 2, and the maximum value is less than 0.08%, which shows that the coating also has little influence on the natural frequencies for the style 2 of the mistuning blisk. And, other features of natural frequencies are similar to the style 1.

The natural frequencies of the style 3 are formulated and the comparisons of the tuned and the mistuned are shown in Table 6 and Fig. 7.

According to the above data, the obvious feature of the style 3 is that the repetition frequencies do not occur like styles 1 and 2 even if the system is tuned, i.e., the 2nd and the 3rd, 16th and 17th, 20th and 21th, 34th and 35th, which are caused by the multipackets do not consist of 18 shrouds and is not cyclic symmetric strictly. However, other characteristics of style 3 for the mistuned blisk are identical with the style 2, and the hard-coating still reduces the natural frequency.

Furthermore, the natural frequencies and their change rates from the style 4 to the style 6 for the tuned and the mistuned blisk are shown in Tables 7–9 and Figs. 8–10.

The similar regulations can be found that the hard-coating can decrease the natural frequency for the style 4 to the style 6, which indicates the coating has the same effect on the natural frequency for the blisk without or with multipackets. Besides, the coating hardly affects the natural frequencies of the mistuning blisk, which proves that the coated mistuning has a quite weak influence on the modal characteristics of the substrate. Moreover, it is seen from Tables 4–9 that the natural frequencies of each style are the same in the 1st and the 19th for the tuned blisk, which are 666.92 Hz and 651.49 Hz, respectively, which is caused by the influence of the uniform excitation under zero nodal diameter. Then, an interesting rule can be found that the division of the mode families for different styles is quite regular. There are two apparent mode families for the styles 1, 2, 3, and 4. However, two mode families are divided into three and two subfamilies again for the styles 5 and 6 due to the different amounts of shrouds in each bladed-packet for kinds of styles, and the situation can be reflected by the natural frequencies or the change rate of the uncoated and the coated in Figs. 9(a) and 9(b) and Figs. 10(a) and 10(b).

Next, the comparison of the natural frequencies for the mistuned blisk in Fig. 11 is discussed.

It is observed from Fig. 11 that the natural frequencies of the styles 3 to 6 are limited to the range of the styles 1 and 2, which indicates the maximum and the minimum stiffness take place styles 1 and 2, and the others always fluctuate between the styles 1 and 2, which reveals the stiffness of the mistuned blisk can be improved due to the component of the multipackets. Further, the coupling stiffness among blades is introduced into the blisk due to the bladed-packet, thus the system stiffness can be improved whatever the form of multipackets. Moreover, other styles have some rules as well except for styles 1 and 2. The order of natural frequencies is that style 3 > style 4 > style 5 > style 6, which indicates that the greater number of shrouds, the greater stiffness for the mistuned blisk. Thus, the stiffness can be changed by a variety of multipackets. Besides, the value is increased properly than the blisk without multipackets though the variation degree of stiffness for blisk may be different in kinds of styles.

Natural frequencies of the tuned and the mistuned blisk are investigated and some meaningful regulations are obtained in Sec. 3.2.1. Further, the forced responses for the mistuned blisk will be researched in this section to analyze the influence of the hard-coating and the multipackets on the vibration characteristics of the mistuned blisk. Generally, the amplitudes of the vibration responses are different from the variation of EO, so different EO excitations need to be analyzed. Nevertheless, the excitation of 0E and 1E can play a major role in the vibration response of the blisk and effectively reflect its behavior, thus the tuned and the mistuned blisk under 0E and 1E will be discussed in the following content. The comparison of the forced responses is shown in Fig. 12.

It can be seen in Fig. 12 that frequencies of two mode families are excited in the frequency domain. Besides, only two resonant peaks can be induced in the frequency domain for the tuned blisk of the styles 1 and 2 by the resonance principle under the condition of same frequency and nodal diameter. However, multiple resonant peaks are induced in frequency domain for the styles 3 to 6, which shows that the vibration behavior of the four styles is not strictly tuned due to the influence of many bladed-packets. In addition, it notes that the resonant responses are coincident for all styles under 0E, because the effect of the excitation under 0E on the forced vibration is identical. Then, it is observed clearly that the resonant peaks of other styles are less than that of style 1 under 1E in the first mode family, although the features have some little fluctuations in the second mode family, which shows the multipackets can play an effective role to suppress the vibration response for the uncoated blisk. Furthermore, the response amplitudes generated by the first mode family are more obvious than the responses in the second mode family, so the circumstances of the vibration responses in the first mode family can represent the level of danger vibration and will be discussed in the following content.

The resonant responses for the tuned and the mistuned blisk with hard-coating under 0E and 1E are analyzed, and the comparisons of these responses between other styles and style 1 are given in Figs. 13–18.