Mistuning identification and model updating of a blisk with piezoelectric excitation components

Abstract

Integrally bladed disks (blisks) are extensively used in military and commercial aircraft engines. There are inevitable deviations between blades called mistuning, leading to the vibration localization of blisks, which potentially results in high cycle fatigue (HCF). Furthermore, the dynamic performance is sensitive to mistuning patterns. Hence, to predict the vibration response of mistuned blisks accurately, the mistuning pattern needs to be identified experimentally and verified under traveling wave excitation (TWE). Piezoelectric TWE is a promising testing technique for blisk dynamics in a non-rotating state, offering advantages such as high excitation force, wide bandwidth, and a simple setup. However, piezoelectric materials are generally arranged on blades forming an electromechanically coupled structure, which challenges the existing mistuning identification methods. In this paper, an integral mistuning identification and model updating method with traveling wave excitation verification is developed. The detuning strategy is used to split natural frequencies of blades, whereupon the response is measured blade by blade. Measured response functions of isolated blades can be retrieved by data reconstruction. Then, the mistuning patterns of elastic modulus and damping ratio are identified by Kirchhoff-plate theory and half-power bandwidth method, respectively. Experimental studies are carried out to validate the method. A dummy blisk with piezoelectric patches is designed and the dynamic properties are calculated. Blade-by-blade measurement and TWE test are conducted. Results show that natural frequencies of the updated model are in excellent agreement with the measured ones. Under TWE, the deviations of natural frequencies for all blades are less than 0.25%. The response deviation of the updated model is decreased by 89.4% on average compared with the original model and the average response deviation of the updated model is 7.4%.

Keywords

blisk mistuning identification detuning strategy traveling wave excitation piezoelectric

1. Introduction

Blisks are extensively used in military and commercial aircraft engines. Compared with conventional blade-assembled disks, they have higher rigidity, lower air leak, and provide high compressor efficiency, thrust-to-weight ratio, and operation reliability for aero-engines (Fu et al., 2017). Nevertheless, just as the bladed disks suffer from, the vibration problems caused by the mistuning still exist in blisks and are even severer, especially for high-order modes. Due to manufacturing tolerances, material inhomogeneity, field wear, etc., there are always deviations between the blades, called mistuning.

Even though mistuning is typically small, it can result in vibration localization, that is, the stress level of one or several blades is significantly increased and cause HCF. Mistuned blisks exhibit two distinct characteristics. First of all, mode shapes may become spatially localized, called mode localization. The concept of localization was first proposed by Anderson (1958). As the acoustical analogs of condensed-matter problems, Hodges (1982) demonstrated the localization phenomenon in near periodic structures and found that the spatial extent of the modes depends on the ratio of disorder to coupling. The research of Ewins (1973) on bladed disks showed that the mistuned assemblies have mode shapes with several diametral components leading to the irregularity of mode shapes. When the localization occurs, the vibrational energy is confined to a few, or even a single blade. Hence, the majority of blades are of low response and the response of the rest greatly exceeds the tuned response, namely, forced response amplitude magnification which is the second characteristic. Random mistuning can cause forced response to be significantly greater than in a tuned case (Whitehead 1966; Dye and Henry 1969; Wei and Pierre 1990; Liao et al., 2010). Regarding the mistuning strength, there exists a critical mistuning strength yielding a maximum resonant response (MacBain and Whaley 1984; Ottarsson and Pierre 1995). This remarkable effect provides an idea that the mistuning pattern, that is, the distribution of blade deviations, can be designed intentionally so that a further increase in mistuning actually results in lower forced response (El-Bayoumy and Srinivasan 1975; Ewins and Imregun 1984; Yao et al., 2011; Tan et al., 2019; Castanier and Pierre 1997, 2002; Brewer et al., 1999).

To predict the vibration response of mistuned blisks accurately, the mistuning pattern needs to be identified, whereupon the simulation model can be corrected. Regarding bladed disks, the mistuning can be identified by the measurements on the individual blades. However, this strategy is impractical for blisks since blades cannot be removed for individual measurements. The mistuning identification for blisks is usually achieved through global measurements on the whole structure followed by various inverse form of reduced order modeling (ROM) techniques (Feiner and Griffin 2004; Judge et al., 2001). The modal test for blisks in a stationary state was conducted to obtain frequency response functions (FRFs) serving as inputs for mistuning identification. This experiment is often cumbersome and technically difficult since blisks are lightly damped structures with numerous closely spaced natural frequencies (Zhou et al., 2022).

The mistuning identification technique based on blisk detuning facilitates the identification experiment by generating blade-dominated mode skillfully. The detuning strategy has been proposed to “isolate” blades of blisk, thus the blade-by-blade measurement can be employed to identify mistuning (Beirow et al., 2009; Hönisch et al., 2011; Maywald et al., 2017; Lupini et al., 2021; Zhou et al., 2022; Cimpuieru et al., 2023). The detuning mass is attached on the blades except the one under test to isolate individual blade natural frequencies. This single blade under test is excited independently and the “blade-alone” frequencies are measured. Nevertheless, a single blade cannot be isolated completely from a disk due to the inter-blade coupling effect. Lupini et al. (2021) presented a procedure for the calculation of coefficients representing the effect that neighboring blades’ mistuning has on the frequency measurement of other blades. Zhou et al. (2022) presented a novel method to quantify the residual inter-blade coupling coefficient. They showed that the influence coefficients are insensitive to the minor perturbations of the blade mistuning pattern; therefore, the influence coefficients can be computed delicately by simulating the blade detuning test with a known mistuning pattern. Recently, Cimpuieru et al. (2023) developed the calculation of residual inter-blade coupling coefficients. The cyclic modeling error is avoided by considering a Taylor approximation of the eigenvalue of the blisk with one isolated blade and expressing it as a function of the detuning mass and the blisk mistuning.

As can be seen, experimental testing and validation are indispensable for the mistuning identification. First of all, a blade level test needs to be conducted for the identification of the mistuning pattern. Although the mistuning also exists between the sectors of disk, it can be dismissed for the fact that the amplitude magnification is sensitive to the mistuning of blades rather than disk (Yang and Griffin 2001; Lim et al., 2007; She et al., 2021). Secondly, a blisk level test needs to be conducted further to verify the accuracy of the identification results. The experimental simulation of engine order excitation is of great importance. Rotating blisks at realistic speeds requires complex and expensive test fixtures (Jones and Cross 2003). As an alternative, the rotating excitation, that is, traveling wave excitation (TWE), can be applied on the stationary blisk. Several TWE systems are proposed using piezoelectric, acoustic, or magnetic actuators applied to stationary bladed disks (Kruse and Pierre 1997; Judge et al., 2001; Jones and Cross 2003; Gibert et al., 2010; Gillaugh et al., 2019). The acoustic actuator does not physically disturb the system while it is limited by the quite low maximum achievable amplitude and multiple mode excitation cannot be easily obtained. Compared with the magnetic actuator, the range of excitation frequencies of the piezoelectric actuator is much wider. Therefore, the piezoelectric TWE system exhibiting high force amplitude, wide ranges of excitation frequencies, and traveling wave characteristics is preferred for the experimental simulation of engine order excitation, especially for the dangerous modes with high order. Nevertheless, the intrusion of piezoelectric excitation components makes the mistuning identification more complex. In addition, the coatings as vibration reduction application scenarios have also been considered in the mistuning identification (Xu et al. 2019, 2020, 2021a, 2021b, 2022; Yan et al., 2022).

However, the effectiveness of the mistuning identification method is verified through modal testing most of the time. For the mistuned blisk, the modes are no longer pure nodal diameter modes and are less likely to have a large response to any particular engine order excitation (Castanier and Pierre 2002). The forced response is a combination of various modes of mistuned blisk. Therefore, the modal analysis cannot replace the response analysis. The verification through forced response experiment is more persuasive than the modal testing. Nevertheless, the verification under engine order excitation of the mistuning identification is completely absent. To achieve the engine order excitation, the piezoelectric TWE system as rotating excitation is preferred for the experimental simulation. Although the piezoelectric TWE system will increase the complexity of the system, the piezo patches can be used adequately in the mistuning identification. It is able to generate the sinusoidal sweeping excitation on each single blade with high signal to noise ratio and wide range of frequency. Therefore, the primary focus of this paper is to identify the mistuning and update the model of a blisk with piezoelectric excitation components. A dummy blisk with piezo patches is designed, manufactured, and assembled. An experimental system is set up capable of the blade-by-blade test and the TWE test, and the mistuning identification scheme is presented. The FRFs of blades are retrieved by data reconstruction. The damping ratios of blades are also identified based on the half-power bandwidth method. The responses of the mistuned blisk under TWE are measured to verify the scheme.

The highlights of this manuscript are as follows:

- A mistuning identification and model updating method with traveling wave excitation verification is developed for the first time.

- The mistuning identification is based on forced response integrated with data reconstruction and half-power bandwidth.

- A dummy blisk with piezo patches and related piezoelectric actuation system are designed, whereupon experimental studies are performed to validate the method.

With the identified and fully verified mistuning pattern, the FE model can be updated and the vibration response of mistuned blisks can be predicted more accurately. Then, the security and service life of blisks are able to be evaluated. Moreover, it provides an experimental guide to the retuning or a certain intentional mistuning of blisk for vibration reduction. For example, when a tuning or a specific mistuning pattern is expected to be achieved on a blisk by some structural modifications, the identified mistuning pattern of the blisk provides a basis for modification of blades. Once the blades are modified, the mistuning identification method can be used again to check if the expected pattern is achieved.

In the remainder of this paper, the methodology based on the detuning strategy and the modeling of the blisk with piezoelectric excitation components will be firstly described (Section 2). Then, a dummy blisk is designed based on the simulation model of the blisk with piezoelectric patches. The dynamic properties of the simulation model are calculated. The experimental system is set up and the experimental scheme is introduced (Section 3). Subsequently, the results of the mistuning identification are shown and the model is updated accordingly. The outcomes of the mistuning identification and the model updating for the mistuned blisk are verified by the blade-by-blade test and the TWE test (Section 4). Finally, we draw conclusions and outline further perspectives (Section 5).

2. Problem formulation

2.1. Modeling of the blisk with piezoelectric system

The tuned blisk is cyclically symmetric. The equation of motion for the discrete piezoelectric blisk is given by (Wu et al., 2023)

M_{a} {\ddot{x}}_{a} + C_{a} {\dot{x}}_{a} + K_{a} x_{a} = w_{a},

(1)

where the vector of DOFs x_a is the concatenation of the displacements {u_i} and voltage {V_i}, i denotes the ith sector. Accordingly, the vector w_a consists of the excitation force {f_i} and the electric charge {Q_i}.

\begin{array}{l} x_{a} = {[{u_{1}}^{T}, \dots, {u_{i}}^{T}, \dots, {u_{N}}^{T}, V_{1}, \dots, V_{i}, \dots, V_{N}]}^{T}, \\ w_{a} = {[{f_{1}}^{T}, \dots, {f_{i}}^{T}, \dots, {f_{N}}^{T}, Q_{1}, \dots, Q_{i}, \dots, Q_{N}]}^{T}, \end{array}

(2)

N denotes the number of sectors. The coefficient matrices have the forms:

\begin{array}{l} K_{a} = [\begin{array}{l} K_{ma} & K_{mpa} \\ {K_{mpa}}^{T} & K_{ea} \end{array}] = \\ [\begin{array}{l} Bcirc (K_{s}, K_{c}, 0, {K_{c}}^{T}) & Bdiag (K_{mp}, \dots, K_{mp}) \\ Bdiag ({K_{mp}}^{T}, \dots, {K_{mp}}^{T}) & Bdiag (- C_{p}, \dots, - C_{p}) \end{array}], \\ M_{a} = [\begin{array}{l} M_{ma} & 0 \\ 0 & 0 \end{array}] = [\begin{array}{l} Bcirc (M_{s}, M_{c}, 0, {M_{c}}^{T}) & 0 \\ 0 & 0 \end{array}], \\ C_{a} = [\begin{array}{l} C_{ma} & 0 \\ 0 & 0 \end{array}] = [\begin{array}{l} Birc (C_{s}, C_{c}, 0, {C_{c}}^{T}) & 0 \\ 0 & 0 \end{array}], \end{array}

(3)

where Bcirc(⋅) denotes the block circulant operation and Bdiag denotes the block diagonal operation. The subscripts s and c indicate the internal and the boundary DOFs of sector, respectively. K_mp is the coupling matrix representing the electromechanical coupling effect. C_p denotes the capacitance of the piezo patch. Then, the governing equation of the blisk coupled with piezo patches can be expressed as

\begin{array}{l} [\begin{array}{l} M_{ma} & 0 \\ 0 & 0 \end{array}] [\begin{array}{c} \overset{. .}{u} \\ \overset{. .}{V} \end{array}] + [\begin{array}{l} C_{ma} & 0 \\ 0 & 0 \end{array}] [\begin{array}{c} \dot{u} \\ \dot{V} \end{array}] + \\ [\begin{array}{l} K_{ma} & K_{mpa} \\ {K_{mpa}}^{T} & K_{ea} \end{array}] [\begin{array}{c} u \\ V \end{array}] = [\begin{array}{c} f \\ Q \end{array}] \end{array}

(4)

2.2. Mistuning identification methodology

In the experiments of bladed disks, the deviation between the natural frequency of a single blade and the nominal value is often used to quantify the mistuning. However, the mistuning caused by the assembly is not considered and this method cannot be applied to the blisk due to the integrality. Besides, the intrusive experimental system such as actuators, vibrometers, and attachments introduce additional mistuning. For the blisk with piezoelectric excitation components, the mistuning arises mainly from the differences between material parameters, boundary conditions during assembly, electrical parameters and bonding position of piezo patches, mechanical properties of adhesives, amplification factors for channels of a high voltage amplifier, etc. The sources of mistuning are similar for the magnetic excitation system mainly introduced by the attachment of magnets to the blades. Furthermore, the identification results are difficult to be verified under working condition.

To overcome these challenges, a mistuning identification and model updating method with traveling wave excitation verification is developed. Besides, the identification of the damping ratio is also studied for the prediction of response. The flow chart of the methodology is shown in Figure 1. First of all, the nominal blisk with piezo patches is designed and manufactured. The deviations between the blades are unavoidable turning the blisk into mistuned state. Then, the detuning strategy is used to distinguish the blade-dominated mode related to the blade under measurement. Specifically, magnetic cylinders are attached on other blades as additional masses. In this way, the numerous closely spaced natural frequencies are separated and easier for the measurement. The piezo patch on the blade under measurement is activated and the excitation frequency sweeps through the nominal natural frequency. The displacement response near the tip of the blade is measured. Hence, the actual natural frequency can be obtained from the response curve retrieved by data reconstruction based on the analytical expression of amplitude of single degree of freedom (SDOF) system under harmonic excitation. The damping ratio can also be obtained with the half-power bandwidth method. The mistuning pattern of elastic modulus widely used as mistuning parameter (Lim et al., 2007; Deng et al., 2020; Liang et al., 2023) can be identified from the Kirchhoff-plate theory. This identification process will be iterated until all blades are measured. Subsequently, the model of the blisk can be updated by the modification of material parameters. Finally, the accuracy of identification results is evaluated by TWE test. The magnetic masses attached on blades are removed. All the piezo patches are activated and phase-shifted mutually to simulate the engine order excitation. The same TWE test is conducted on the updated model. The differences of natural frequencies and response amplitudes between the numerical simulation and experiment can be acquired and thus the errors can be calculated.

3. Experimental studies

In this section, a dummy blisk is designed for the experimental studies and simulated numerically, whereupon the dynamic properties of the model are calculated. Then, the experimental system consisting of a piezoelectric TWE system, a measurement system, and a data acquisition (DAQ) system is set up. Finally, the experimental scheme is introduced. The isolated natural frequencies and the damping ratio of blades are identified from the blade-by-blade measurement with the detuning strategy.

3.1. Design of the dummy blisk

The finite element (FE) model of the dummy blisk is shown in Figure 2(a). There are 12 straight blades with 15° stagger angle. The flanges are provided on both sides of the model. The model is meshed in ICEM using structural meshing for good mesh quality. The blisk is modeled by SOLID 45 element and the piezo patches are modeled by SOLID 5 element. The material of the blisk is aluminum alloy. The material properties are Young’s modulus 50 GPa, density 2700 kg/m³, and Poisson ratio 0.3. The piezo patch with geometric parameters 0.04 × 0.04 × 0.0005 m is bonded near the bottom of blade as shown in Figure 2(b). The density of piezoelectric material is 7500 kg/m³. The other material parameters are given in Supplemental material. Specifically, the strain at the bottom of blade is always great leading to the high coupling strength between the mechanical and electric fields (Fan et al., 2021; Shi et al., 2021). Hence, more electrical energy could be transformed into mechanical energy. The element size is 0.01 m. The coupling between the piezo patch and the blade is achieved by sharing nodes on the interface.

The result of the modal analysis for the FE model with a flange fixed is shown in Figure 3. It can be found that the first three orders of mode groups are blade dominant among which the modes with ND = 0 are displayed in Figures 4(a), (b) and 5(a). Specifically, the first two are bending modes of blades and the third is torsional mode of blades. The fourth order of mode group contains modes that the disk and the blades dominates, and the veering region. The mode with ND = 0 in this group is displayed in Figure 5(b) which is bending mode of disc. The curves for the first three mode groups are flat lines. The distribution of natural frequencies is quite dense which can also confirm the blade dominant modes. For the fourth mode group, the curve veers after ND = 0 and then becomes flat. This veering phenomenon is the characteristic of the coupled system and the modes in the veering region are more sensitive to the mistuning (Perkins and Mote, 1986; Afolabi and Alabi 1992). Hence, we should pay primary attention on these modes. The third order of bending vibration mode with nodal diameter ND = 1 marked in Figure 3 is located in the veering region, thereby selected as the test mode. The nominal natural frequency, that is, natural frequency of this mode, is 646 Hz and the mode shape is shown in Figure 6. The coupling phenomenon can be observed that the blades and the disc are all involved in the mode. Notice that the disk-blade coupling is weak for this mode due to the relatively high stiffness of the disk.

Although the disk and the blades are not integrated for facilitating manufacture, they can still be treated as a blisk if the blades are not removed in the process of mistuning identification. The mistuning due to the assembly of blades disturbs the original mistuning pattern for blisks. Besides, the intrusive experiment system also introduces additional mistuning. Multiple sources of mistuning can be synthesized in the mistuning identification method because it is the total mistuning pattern that is identified. Moreover, the total mistuning pattern can be regarded as the original mistuning of a blisk since the identification of a certain mistuning pattern is our concern in this paper, not the difference between mistuning patterns.

3.2. Experimental system

The experimental system consists of a piezoelectric TWE system, a measurement system, and a DAQ system shown in Figure 7. The piezo patches sprayed with a coat of insulation paint are bonded on the bottom of blades with epoxy adhesive. They are connected to the high voltage amplifier. The terminal blocks are used to connect wires. The amplifier receives signals from the digital I/O via the junction box. The laser displacement vibrometer is mounted on the swivel frame achieving the circular field displacement measurement. The displacement response at the tip of each blade is measured. The measurement signal is collected by the DAQ system. The strain gauges and the strain gauge conditioner are used also for verification.

The TWE system utilizes piezoelectric actuators as excitation sources. The implementation of the TWE system is as follows: A multi-channel signals are generated with the digital I/O board and transmitted through the junction box. The multi-channel signals are phase-shifted according to the appropriate engine order of the excitation. The signals are amplified to the required amplitude by the high voltage amplifier and finally applied to the piezo patches to realize the TWE. Specifically, the multi-channel signals are alternating voltage signals as

U_{j} = U \sin (ω t + (j - 1) θ), j = 1,2, \dots, n

(5)

θ = 2 π \frac{e}{n}

(6)

where U_j is the voltage applied to the jth blade, U is the voltage amplitude, ω is the voltage frequency, e is the order of excitation, n is the number of blades.

3.3. Experimental scheme

The blade-by-blade measurement is used with the detuning strategy. The measure point is marked in Figure 8(a) as a reference. The detuning masses of 3.18 g as shown in Figure 8(b) are attached to all blades except the blade under test. The mass of a blade is 40.13 g, and thus the relative mass of the detuning masses is 8%. The masses are magnetic cylinders and a pair of them sticks firmly on blades by magnetic force. They can be removed without destruction and residue. Notice that the response at the tip for each blade in the test mode is the maximum. The masses are attached at the tips to increase the kinetic energy efficiently while the strain energy is almost not affected. Therefore, natural frequencies can be changed as much as possible. Only the piezo patch corresponding to the blade under test is activated. Then, the response data of blades with frequencies near the third order of bending vibration mode can be acquired. The FRFs of isolated blades based on the analytical expression of amplitude under harmonic excitation can be retrieved by data reconstruction, given as follow:

A = \frac{1}{\sqrt{{(1 - λ^{2})}^{2} + {(2 ξ λ)}^{2}}} \frac{P}{k}

(7)

where λ = ω/ω_n, ω is the excitation frequency, ω_n is the natural frequency, P is the amplitude of excitation force, k is the stiffness, and ξ is the structural damping ratio. Therefore, the isolated natural frequency and the amplitude can be acquired from the peak frequency and the peak value of the retrieved FRFs. Subsequently, the mistuning pattern of blades can be identified according to the deviation of the natural frequencies defined as follow:

{}^{f}{δ_{m, i}} = \frac{f_{m, i}}{f_{m}^{Tuned}} - 1

(8)

where ^fδ_m,i denotes the mistuning coefficient of the mth natural frequency for the ith blade.

f_{m}^{Tuned}

is the nominal mth natural frequency of the tuned blisk.

Then, the model can be updated from the isolated natural frequency. The modification of the elastic modulus for each blade is widely used. The relation between the elastic modulus and the natural frequency for a single blade can be derived from the Kirchhoff-plate differential equation, namely

f \propto \sqrt{E}

(9)

Then, the mistuning pattern of elastic modulus can be expressed with the mistuning coefficient of natural frequency as follow:

{}^{E}{δ_{m, i}} = 2^{f} δ_{m, i} + {({}^{f}{δ_{m, i}})}^{2}

(10)

where the mistuning coefficient of the elastic modulus for the ith blade is defined as

{}^{E}{δ_{m, i}} = \frac{E_{m, i}}{E_{m}^{Tuned}} - 1

(11)

Furthermore, the structural damping ratio can be estimated from the retrieved FRFs with the half-power bandwidth method. The half-power points ω₁ and ω₂ (ω₁ < ω₂) are determined when the amplitude is $1 / \sqrt{2}$ of the resonant peak. The structural dampling ratio can be estimated by

ξ = \frac{ω_{2} - ω_{1}}{2 ω_{n}}

(12)

Then, the Rayleigh damping can be acquired as follow:

β = \frac{2 ξ}{ω_{l} + ω_{u}}

(13)

where ω_l and ω_u are the lower frequency and upper frequency of the frequency band concerned. The mass proportional damping coefficient is ignored due to the high frequency where the stiffness proportional damping coefficient β is dominant.

The elastic modulus and Rayleigh damping ratio of blades can be updated. Then, the TWE test is conducted numerically and experimentally to measure the response. By the comparison of results, the accuracy of the updated model can be evaluated. All the results are presented and discussed in the next section.

4. Results and discussions

In this section, the results of the mistuning identification are listed and the material properties of blades are updated in the FE model accordingly. Then, the vibration responses of the updated model are calculated in blade-by-blade measurement and under TWE to verify the outcomes of the mistuning identification and the model updating.

4.1. Results of the mistuning identification

The isolated natural frequencies and the amplitudes are listed in Table 1 from the retrieved FRFs of blades shown in Supplemental material. It can be found that the natural frequency varies between 631.92 Hz and 660.29 Hz which is around the nominal natural frequency, that is, 646 Hz. Next, the mistuning coefficients of natural frequencies for blades are calculated by equation (8) and plotted in Figure 9. The mean value of mistuning coefficients is −0.08% and the maximum absolute value is 2.2% of the 5th blade. Then, Young’s moduli of blades are calculated by equation (10) and equation (11) listed in Table 2. Notice that the nominal Young’s modulus is 49.6 GPa. Moreover, the structural damping ratio and the Rayleigh damping ratio are calculated by equation (12) and equation (13) listed in Table 3.

Table 1.

The isolated natural frequencies f and the amplitudes A acquired from the retrieved FRFs of blades.

Blade number	1	2	3	4	5	6
Natural frequency/Hz	633.42	658.71	653.26	635.21	660.29	655.24
Amplitude/mm	0.1470	0.1235	0.1144	0.1268	0.1134	0.1117
Blade number	7	8	9	10	11	12
Natural frequency/Hz	641.26	645.97	640.24	651.86	637.96	631.92
Amplitude/mm	0.1104	0.1062	0.1564	0.0984	0.1377	0.1404

Table 2.

Identification results of Young’s moduli E of blades. The nominal value is 49.6 GPa.

Blade number	1	2	3	4	5	6	7	8	9	10	11	12
Young’s modulus/GPa	47.8	51.7	50.8	48.0	51.9	51.1	49.0	49.7	48.8	50.6	48.4	47.5

Table 3.

Identification results of the structural damping ratio ξ and the Rayleigh damping ratio β of blades.

Blade number	1	2	3	4	5	6
Structural damping ratio	0.0053	0.0044	0.0060	0.0061	0.0049	0.0066
Rayleigh damping ratio/× 10⁻⁶	1.3246	1.0652	1.4694	1.5162	1.1827	1.6111
Blade number	7	8	9	10	11	12
Structural damping ratio	0.0072	0.0052	0.0054	0.0059	0.0048	0.0066
Rayleigh damping ratio/× 10⁻⁶	1.7736	1.3599	1.3490	1.4387	1.1929	1.6741

4.2. Results of the model updated from blade-by-blade measurement

The model is updated in the material properties identified from the experiment, that is, Tables 2 and 3. To verify the updated model, the simulation of the blade-by-blade measurement with the detuning strategy is carried out. The model is detuned by attaching detuning mass on the tip of all blades except for the blade under test. The detuning mass is achieved by MASS21 element with mass parameter 3.18 g which is consistent with the mass of the magnet in the actual experiment. The natural frequency and response amplitude of each blade under harmonic excitation can be obtained as shown in Table 4. The deviations of the natural frequency Δf and the amplitude of response ΔA between the blisk and the updated simulation model are clearly illustrated in Figure 10. It can be found that the natural frequencies of the updated model are in excellent agreement with the natural frequencies measured experimentally, especially for the 3rd blade and 10th blade with |Δf| = .01%. |Δf| of all blades are less than 0.2%. Nonetheless, the deviations of the response amplitude |ΔA| are larger than Δf mainly due to the difference in the amplitude of the excitation force.

Table 4.

Natural frequencies and response amplitudes calculated from the simulation of the blade-by-blade measurement with the detuning strategy for the updated model. The detuning mass is achieved by MASS21 element with mass parameter 3.18 g which is consistent with the mass of the magnet in the experiment.

Blade number	1	2	3	4	5	6
Natural frequency/Hz	634.5	658.3	653.2	636.2	659.8	655.1
Amplitude/mm	0.1204	0.1426	0.1045	0.1048	0.1281	0.0949
Blade number	7	8	9	10	11	12
Natural frequency/Hz	641.9	646.3	640.9	651.9	638.8	633.1
Amplitude/mm	0.0886	0.1144	0.1166	0.1070	0.1325	0.0956

4.3. Results of the updated model under traveling wave excitation

The experiment of the blisk under the piezoelectric TWE with circumferential wavenumber 1 and voltage amplitude 75 V is conducted. Accordingly, the simulation of the updated model under the same piezoelectric excitation is also carried out. The FRFs of each blade are shown in Supplemental material. The natural frequencies and the response amplitudes are acquired from the FRFs and shown in Figure 11. The blisk, the updated model, and the original model are considered. It can be seen from Figure 11(a) that the natural frequencies of the updated model are still in excellent agreement with the measured natural frequencies. This remarkable result validates the identified mistuning pattern. The average deviation is 0.11% and the maximum deviation is 0.25% from the 1st blade. In Figure 11(b), the response amplitude of the updated model is more consistent with the response measured under TWE experimentally than the original model. Specifically, the response deviation of the updated model is decreased by 89.4% on average compared with the original model. Besides, the average deviation of the updated model is 7.4%. Hence, the outcomes of the mistuning identification and the model updating for the mistuned blisk are verified by the improved prediction of the vibration response under TWE.

Note that in Figure 11(b), the deviations of some blades are relatively low like the 4th, 7th, and 11th blades. However, they can be relatively high for the 5th, 8th, and 10th blades. The distribution pattern of the updated model, that is, the red line, and the blisk, that is, the blue line, are not the same. Two reasons are mainly responsible for the difference. First, there are deviations between the output voltages of the amplifier channels. Secondly, the modal electromechanical coupling factors of piezo patches are not identical. The inconsistency results in the distortion of the traveling wave. The response amplitudes are thus affected but the peak frequencies remain unchanged.

5. Conclusion

In this paper, a mistuning identification and model updating method with traveling wave excitation verification is developed for the first time, which can verify more comprehensively the effectiveness of the mistuning identification method from the view of forced response. The mistuning identification is based on data reconstruction and half-power bandwidth. The outline of this method can be summarized as follows: First, the detuning strategy is used to distinguish the blade-dominated mode related to the blade under measurement. Magnetic cylinders are attached on other blades as additional masses. The piezo patch on the blade under measurement is activated and the excitation frequency sweeps through the nominal natural frequency. The displacement response near the tip of the blade is measured. Then, the response curve is retrieved by data reconstruction based on the analytical expression of amplitude of SDOF system under harmonic excitation. The actual natural frequency can be obtained by peak extraction and the damping ratio can be obtained with the half-power bandwidth method both from the response curve. The mistuning pattern of elastic modulus can be identified from the Kirchhoff-plate theory. This identification process will be iterated until all blades are measured. Subsequently, the model of the blisk can be updated by the modification of material parameters. Finally, the accuracy of identification results is evaluated by TWE test. The magnetic masses attached on blades are removed. All the piezo patches are activated and phase-shifted mutually to simulate the engine order excitation. The same TWE test is conducted on the updated model. The differences of natural frequencies and response amplitudes between the numerical simulation and experiment can be acquired and thus the errors can be calculated.

The experimental studies are conducted to validate the method. A blisk with piezoelectric patches is designed and the dynamic properties are calculated. The TWE system is set up. The blade-by-blade measurement and TWE test are conducted. The mistuning patterns of elastic modulus and damping ratio are identified. Regarding the results from the blade-by-blade measurement, the natural frequencies of the updated model are in excellent agreement with the natural frequencies measured experimentally. Specifically, the deviations of natural frequencies for all blades are less than 0.2%. This is also confirmed in the results from the TWE test where the average deviation of natural frequencies is 0.11% and the maximum deviation is 0.25%. The response deviation of the updated model is decreased by 89.4% on average compared with the original model. Besides, the average response deviation of the updated model from the experiment is 7.4%.

With the identified and fully verified mistuning pattern, the FE model can be updated and the vibration response of mistuned blisks can be predicted more accurately, so that the dynamic reliability and HCF life of blisks are able to be evaluated. Moreover, when a tuning or a specific mistuning pattern is expected to be achieved on a blisk by some structural modifications, the identified mistuning pattern of the blisk provides a basis for modification of blades. In addition, the whole mistuning identification and verification system can support the experimental studies of the retuning or the intentional detuning.

Supplemental Material

Supplemental Material - Mistuning identification and model updating of a blisk with piezoelectric excitation components

Supplemental Material for Mistuning identification and model updating of a blisk with piezoelectric excitation components by Anlue Li, Yu Fan, Hui Wang, Yaguang Wu and Lin Li in Journal of Vibration and Control.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by Major Projects of Aero-Engines and Gas Turbines (J2019-IV-0023-0091), Aeronautical Science Foundation of China (2019ZB051002 and 20220015051002), China Postdoctoral Science Foundation (2021M700326), and Advanced Jet Propulsion Creativity Center (HKCX2020-02-013 and HKCX2020-02-016).

ORCID iDs

Yu Fan

Yaguang Wu

Supplemental Material

Supplemental material for this article is available online.

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