Investigation on the vibration localization of mistuned bladed disk with frictional contact

Abstract

The mistuning phenomenon is commonly seen in the bladed assemblies of turbomachinery, leading to an unreasonably localized energy distribution that may increase the risk of high cycle fatigue and potentially cause blade failure. The friction coupling interaction is often introduced to attenuate the effect of excessive amplification on the structural vibration. To investigate the vibration localization of damped and mistuned bladed-disk system, a typical mistuning characterization is applied to an improved four-degree-of-freedom lumped parameter model to explicit the vibratory properties with coupled frictional contact. The statistical findings of natural frequency and modal localization factor are obtained by employing the Monte Carlo simulation to explore the influence of mistuned variance and coupled stiffness on the modal localization of free and damped bladed-disk system, respectively. The forced response sensitivity featured in the mistuned variance is quantitatively analyzed from the perspective of vibration localization and energy distribution by the nonlinear solution method. The influence of excitation order and normal load are further, respectively, discussed on the response localization level of the mistuned system. The results can provide fundamental knowledge of the impact of the mistuning pattern on vibration localization for the early design of the aeroengine.

Keywords

Mistuned bladed disk frictional contact modal localization nonlinear solution method response localization

1. Introduction

As a critical work component of turbomachinery, the bladed disk is a common example of a cyclically symmetric structure, where a periodic sector copies itself around the axis of rotation to form the entire circle. It is ideally assumed that all the blades are identical and each sector has the same vibratory property in such a bladed-disk assembly, namely, a tuned system. The vibration analysis can benefit from the single sector-level calculation to realize the rapid dynamic solution rather than the expensive consumption of the entire modeling. However, due to the material tolerances, manufacturing deviations, or operational damage in practical engineering, the mistuning phenomenon of the bladed disk may inevitably occur, which breaks the original periodic symmetry of the ideal tuned system (Castanier and Pierre, 2006; Tan et al., 2019). These mistuning features can hinder the transfer of vibration energy through the sectors, resulting in a much larger response level at certain limited region than the tuned design (Judge et al., 2001). This unreasonable localized energy distribution may severely amplify the vibration level, enhance the risk of high cycle fatigue (HCF), and even lead to premature fracture or failure of the blades. Therefore, it is of vital significance to estimate and further alleviate the negative influence of mistuning features on the vibration of a bladed-disk system.

Different analysis methods are developed to characterize the mistuning pattern. The deterministic approach is suggested to accurately determine the probabilistic dynamic properties of mistuned bladed disks. While the application of this analytic method in mistuning is limited due to the complicated derivation and large computational efforts to obtain each solution (Scarselli and Lecce, 2004), the statistical method is further developed on the assumption that the randomly distributed mistuning pattern can be well described as the statistical variables. Monte Carlo simulation (MCS) is the most frequently employed method to obtain random mistuning parameters subject to the statistical distribution for the repetitive analysis.

Despite the rapid development of large-scale computing technology, it is still difficult to perform a large amount of MCS cases using high-fidelity FE models or FE-based numerical reduction alternatives with numerous degrees of freedom (DOF). With suitable design parameters, the lumped parameter model (LPM) might adequately capture the vibrational features of families of modes, such as frequency veering, modal localization, and vibration amplification, which is more eligible for exploring vibrational mechanisms of the mistuned bladed disk (Pourkiaee and Zucca, 2019). For instance, Anthony et al. (2023) established a cyclic symmetric mass-spring model to figure out the Coriolis effect on the aeroelastic stability of a single-piece bladed disk considering the mistuning pattern. Song et al. (2023) proposed the modular modeling method based on a lumped parametric model to improve the solution efficiency of vibration characteristics for the mistuned blisk. The LPM-based mathematical simulator was used to provide the simulated data of a mistuned bladed-disk system for experimental validations of blade tip timing by Bornassi et al. (2020, 2022). The lumped parameters can be identified using the FE model, and the computation requires much less resource and time than the consumption of FE analysis.

In the ideal tuned system, it is typical to introduce the dissipative effects of dry friction at contact interfaces to passively mitigate structural vibrations (Popp et al., 2003; Rizvi et al., 2014; Zucca et al., 2013). However, the localized amplification is prone to occur in the actual mistuned system, which has an adverse impact on vibration attenuation due to friction coupling (Avramov and Awrejcewicz, 2010; Ma et al., 2017). Regarding this, many scholars have focused on the effects of contact conditions on the nonlinear vibration features of the mistuned coupling system. Wang et al. (2023) carried out the LPM-based parametrical study to investigate the attenuation performance of dynamic vibration absorbers in a mistuned blisk. Zhang et al. (2021) investigated the influence of mistuning parameters on the nonlinear dynamics of the tenon and mortise of aeroengine compressor blades based on the lumped parameter model. Liu et al. (2019) established a lumped parameter model considering friction damping to figure out the influence of damper design parameters on the vibratory response of the bladed-disk system. Zhao et al. (2019) discussed the effect of three mistuned parameters on the coupling vibrational characteristics of a bladed-disk assembly with the contact–separation coupling model, including structural stiffness of blades, tangential stiffness of the contact surface, and initial shroud gap. Joannin et al. (2016) developed a lumped parameter model considering dry friction forces at blade roots to qualitatively evaluate the impact of the mistuning magnitude on the nonlinear dynamics of the cyclic structure, obtaining the varying modal properties with the vibration amplitude.

The literature review indicates that most research has mainly described the contact features among the blade root and disk in the mistuned system, characterized by the parametric model of a single DOF (Avalos and Mignolet, 2010; Yu et al., 2010), two DOF (Anthony et al., 2023; Bornassi et al., 2020, 2022; Song et al., 2023; Sun et al., 2019; Yao et al., 2011; Zhang et al., 2021), or four DOF (Joannin et al., 2016; Liu et al., 2019). However, the damping effect at the blade root joints can be ignored at the rotating condition, which is considered a fully linear elasticity in the presence of friction contacts like shrouds and dampers (Pourkiaee and Zucca, 2019). In the present work, individual blades are modeled as an improved four-DOF lumped parameter system coupled together with spring elements characterizing the contact shroud based on the fundamental parametric model by Joannin et al. (2016), including 3 DOF accounting for the blade so as to simulate the dynamic characteristics of each representative part and 1 DOF accounting for the disk. The MCS is first employed to obtain the statistical results of modal frequency and localization with different mistuned variances and coupled stiffness of free and damped system, respectively. The forced response analysis is carried out using the nonlinear solution method to explore the influence of interested factors on the response localization of the mistuned system, inclusive of engine order (EO) and the normal load N₀. The phenomenological understanding of the role of excitation modes and contact states on the mistuning sensitivity is further obtained to provide analytical support for the preliminary design of the aeroengine.

2. Modeling and methodology

2.1. Lumped parameter modeling

The paper establishes an improved single-sector four-DOF dimensionless LPM as a phenomenological representation to investigate the mistuning effects on the vibration characteristics of bladed-disk assembly instead of the widely used one-DOF or two-DOF parametric model by previous scholars. As illustrated in Figure 1, a total of 10 sectors are reserved, and each symmetric sector (including tip, middle, root, and disk) is simplified to four DOF. The nonlinear frictional contact interface between shrouds is introduced at the adjacent blade tip DOF and modeled by the Oden friction model. The linear excitation force f_e is applied at the middle DOF, and two adjacent sectors are coupled at the disk DOF. The ground stiffness and damping are introduced into the boundary conditions of the system. The basic normalized parameter values of LPM are listed in Table 1 based on the numerical values in Joannin et al. (2016) and Liu et al. (2019). In the initial modal analysis, only the coupling stiffness k_c between contact interfaces is considered since the damping in the turbine blade is generally slight (Chatterjee, 2016). And k_c = 0 when it refers to the free blade system. While in the forced response analysis, the coupling stiffness and damping are both involved to investigate the influence of different factors on the response characteristics of the mistuned system.

Table 1.

The basic normalized parameter values of LPM.

Normalized parameter	Tip	Middle	Root	Disk	Ground
Mass	m_b1 = 0.5	m_b2 = 0.5	m_b3 = 1.5	m_b4 = 12	—
Damping	c₁ = 0.2	c₂ = 0.2	c₃ = 0.2	c₄ = 0.2	c_d = 0.2
Stiffness (*10⁴)	k_b1 = 0.1	k_b2 = 0.2	k_b3 = 6	k_b4 = 2	k_d = 0.5

To better understand the contact behavior of a friction damping structure in the blade disk system, several types of dry friction theoretical models describing the relationship between friction and relative motion of the contact surface have been developed in the last few decades, which can be usually expressed as the hysteresis loop. According to the elastic Coulomb friction model (Oden and Pires, 1983) as shown in Figure 2(a), the contact behavior can be simplified as contact DOF under constant normal load N₀, and the nonlinear friction can be expressed as

f_{n} = {\begin{cases} k_{x} (x (t) - x (τ)) & \dot{x} (τ) = 0 \\ sgn (\dot{x} (τ)) μ N_{0} & \dot{x} (t) = \dot{x} (τ) \\ 0 & N_{0} = 0 \end{cases} \begin{array}{l} stick \\ slip \\ lift off \end{array},

(1)

where f_n is the tangential friction force on the contact surface, k_x is the tangential contact stiffness, μ is the friction coefficient, N₀ is the normal load, x(t) is the relative displacement of contact DOF, and x(τ) is the amount of tangential slip between the contact surface. Three conditions between the contact surface can be obtained during the relative motion, including stick, slip, and lift off, which can be described by the implicit variable x(τ). When the harmonic periodic excitation is applied to the elastic friction model, the response can also be considered a single harmonic wave. Hence, the hysteresis loop of the friction force can be established by adopting the stick–slip theory, as depicted in Figure 2(b).

2.2. Governing formulation

The general time-domain governing equation of a tuned bladed-disk system subjected to linear excitations and nonlinear friction forces can be expressed as follows:

M_{t} \ddot{x} + C_{t} \dot{x} + K_{t} x = f_{e} + f_{n},

(2)

where M _t, C _t and K _t are the tuned mass, damping, and stiffness matrices of the system; x ,

\dot{x}

, and

\ddot{x}

are the displacement, velocity, and acceleration; f _e denotes the linear excitation force vector independent of the vibration amplitudes; and f _n denotes the nonlinear force vector correlated to the displacement and velocity of the contact nodes on the friction interfaces.

Generally, the periodic excitation, caused by the periodic air flow disturbance between the stator and rotor, belongs to a harmonic excitation force in the specific order (engine order excitation) (Castanier and Pierre, 2006). Based on this assumption, it is considered that each component of the low-frequency excitation force vector f _e has the following form:

f_{e}^{j} = F_{a}^{j} \sin (ω_{e} t + \frac{j - 1}{N_{b}} 2 π EO),

(3)

where

f_{e}^{j}

and

F_{a}^{j}

are the time-variant excitation force and its amplitude acting on the jth blade, ω_e is the angular frequency of excitation, t is the time, N_b is the blade number, and EO is the engine order.

Based on the lumped parameter modeling above, without considering the mistuning features, the tuned mass matrix M _t and tuned stiffness matrix K _t can be described as the form in equation (4):

M_{t} = {[\begin{array}{l} M_{i} & 0 & \dots & 0 \\ 0 & M_{i} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & M_{i} \end{array}]}_{N_{b} \times N_{b}}, K_{t} = {[\begin{array}{l} K_{i} & K_{j} & 0 & \dots & K_{j} \\ K_{j} & K_{i} & K_{j} & \dots & 0 \\ 0 & K_{j} & K_{i} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ K_{j} & 0 & 0 & \dots & K_{i} \end{array}]}_{N_{b} \times N_{b}},

(4)

where the submatrix in equation (4) can be composed of

M_{i} = [\begin{array}{l} m_{b 1} & 0 & 0 & 0 \\ 0 & m_{b 2} & 0 & 0 \\ 0 & 0 & m_{b 3} & 0 \\ 0 & 0 & 0 & m_{b 4} \end{array}],

(5)

K_{i} = [\begin{array}{l} k_{b 1} + 2 k_{c} & - k_{b 1} & 0 & 0 \\ - k_{b 1} & k_{b 1} + k_{b 2} & - k_{b 2} & 0 \\ 0 & - k_{b 2} & k_{b 2} + k_{b 3} & - k_{b 3} \\ 0 & 0 & - k_{b 3} & k_{b 3} + 2 k_{b 4} + k_{d} \end{array}], K_{j} = [\begin{array}{l} - k_{c} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - k_{b 4} \end{array}] .

(6)

When considering the impact of mistuning on the system, the main approach is to modify the stiffness of each blade by introducing the small mistuned disturbance ξ, and the mistuned stiffness can be calculated as

Δ K_{m i s} = [\begin{array}{l} ξ_{1} \\ ξ_{2} \\ ⋱ \\ ξ_{N_{b}} \end{array}] \otimes [\begin{array}{l} k_{b 1} & - k_{b 1} \\ - k_{b 1} & k_{b 1} + k_{b 2} & - k_{b 2} \\ - k_{b 2} & k_{b 2} + k_{b 3} & - k_{b 3} \\ - k_{b 3} & k_{b 3} \end{array}],

(7)

where

ξ_{1}

ξ_{2}

, …,

ξ_{N_{b}}

are the ratio of mistuned stiffness to blade-tuned stiffness of the original system, which is called mistuned disturbance in the paper, and

\otimes

denotes the Kronecker product. Then, the total stiffness considering mistuned features can be obtained by adding the mistuned stiffness to the tune one as:

K_{t o t} = K_{t} + Δ K_{m i s} .

(8)

To better reveal the characteristic attributes of the lumped parameter modeling adopted in the paper, the stiffness and mass of each component is normalized with respect to the total stiffness K_b and the total mass M_b of the blade, respectively, as shown in equation (9).

M_{b} = m_{b 1} + m_{b 2} + m_{b 3}, K_{b} = \frac{1}{k_{b 1}^{- 1} + k_{b 2}^{- 1} + k_{b 3}^{- 1}} .

(9)

2.3. Nonlinear solution method

Different from the linear system, the solution of the energy-dependent nonlinear term involved in equation (2) needs much more computational effort. To achieve an efficient solution to the problem, x , f _e, and f _n in the time domain in equation (2) can be converted into the frequency domain by complex harmonic components through complex Fourier transform according to the multi-harmonic balance method (MHBM) (Zhu et al., 2022), as shown in the following:

x = \sum_{k = 0}^{2 N_{h} + 1} X_{k} e^{i k ω t}, f_{e} = \sum_{k = 0}^{2 N_{h} + 1} F_{e k} e^{i k ω t}, f_{n} = \sum_{k = 0}^{2 N_{h} + 1} F_{n k} e^{i k ω t},

(10)

where N_h is the total number of harmonics, and X _k, F _ek and F _nk are the complex harmonic components of the displacement, excitation, and nonlinear force, respectively. And the dynamic governing equation of the bladed-disk system in the frequency domain can be obtained by substituting equation (10) into equation (2):

S X - F_{e} - F_{n} = 0,

(11)

where S is a dynamic stiffness matrix of dimension (2N_h+1) × (2N_h+1) with block diagonal, which can be demonstrated as

S = diag ([K + i k ω C - k^{2} ω^{2} M]), k = 0, \dots, 2 N_{h} + 1 .

(12)

Then, the displacement vector X in the frequency domain can be obtained by solving equation (11), which can be transformed into the time domain x by adopting equation (10), and the nonlinear force vector f _n in the time domain can be obtained by the Coulomb friction model. And each order harmonic component of friction F _n can be further obtained through fast Fourier transform (FFT) by adopting the alternating frequency–time (AFT) method. To avoid the non-convergence, a continuation parameter ds along the arc-length is utilized to trace the trajectory of the response curve in the given frequency spectrum, namely, the arc-length continuation technique (Groll and Ewins, 2001). By regarding the excitation frequency as a variable, the solution can be obtained by constantly modifying the displacement harmonic vector X . Integrating the excitation frequency ω_e and displacement harmonic vector X , a new displacement vector Q is defined to the system vibration equations:

Q = {ω_{e}, X^{T}}^{T} .

(13)

Suppose that R is the residual matrix with respect to the approximate solution X _k at the current iteration step k:

R = S X_{k} - F_{e} - F_{n} .

(14)

where S is the dynamic stiffness matrix of dimension (2N_h+1) × (2N_h+1) with block diagonal. In the arc-length continuation technique, the corresponding residual matrix can be established with the new displacement vector Q :

R^{*} (Q) = {R_{ω}, R^{T}}^{T},

(15)

where R can still be obtained by solving equation (14) and the residual of frequency R_ω is defined by the formula below:

R_{ω (k + 1)} = {‖ Q_{k + 1} - Q_{k}^{0} ‖}^{2} - s_{k + 1}^{2},

(16)

where

Q_{k}^{0}

represents the convergent result of the kth iteration step,

s_{k + 1}^{2} = Q_{k + 1}^{T} Q_{k + 1}

. Based on the Newton–Raphson method, the following iteration process is conducted to obtain

Q_{k + 1}

Q_{k + 1}^{m + 1} = Q_{k + 1}^{m} - \frac{\partial R^{*} (Q)}{\partial Q_{k + 1}^{m}} R^{*} (Q),

(17)

where the superscript m denotes the mth cycle in the iteration step. The (k+1)th iteration step is considered to be convergent when the residual satisfies || R ^*( Q )||<ε in the finite cycles; otherwise, the arc length will be modified to change the load step in the iteration.

The paper employs the MHBM-based continuation technique to obtain the amplitude–frequency response in the frequency domain at the blade tip of each sector and simultaneously adopts the AFT method to compute the nonlinear dynamics and friction in the time domain.

3. Numerical results and discussion

3.1. Mistuning characterization

In the paper, a mistuned variance σ is defined to characterize the mistuning level of all sectors relative to the tuned system. Given the quantitative uncertainty of the manufacture processing and in-service wear on each sector, a group of random mistuned disturbance ξ, which represents the ratio of the elastic modulus of each sector to the tuned system, is subjected to the normal distribution with the mean value of 0 and standard deviation of σ by reference to She et al. (2021), that is, $ξ \sim N (0, σ)$ . Hence, the mistuned elastic modulus E_m can be obtained by equation (18) relative to the tuned one E_t.

E_{m} = E_{t} (1 + ξ) .

(18)

In order to quantify the degree of vibration localization in each sector due to intentional mistuning, the localization factor η is introduced as follows:

η = \frac{(N_{b} - 1)}{(\sum_{i = 1}^{N_{b}} W_{i} / W_{\max} - 1)},

(19)

where N_b is the sector number, W_max is the maximum absolute modal energy of the system, and W_i is the modal energy of each sector. In terms of the modal localization, the modal energy W_i can be obtained by calculating the elastic potential energy of each sector. When it characterizes the response localization, the modal energy W_i is described by integrating the sum of elastic potential energy and kinetic energy corresponding to the modal displacement of the sector in one period:

W_{i} = \frac{2 π \sum_{k = 1}^{N_{t}} (P_{k} + D_{k})}{N_{t}} i = 1, L, N_{b},

(20)

where N_t denotes the number of sampling points in the time domain; and P_k and D_k denote, respectively, the potential energy and kinetic energy corresponding to each sampling point of the sector. The localization factor η can reflect the energy discrepancy between the blade with the largest modal amplitude and others to some extent to measure the degree of localization.

3.2. Modal localization of mistuned system

3.2.1. The influence of mistuned variance

The free blade system with k_c = 0 is first established to discuss the effect of the mistuned variance on modal properties. In Figure 3, natural frequency and modal localization factor under different mistuned variances from σ = 0.01 to σ = 0.05 are obtained using the MCS to explore the effects of mistuning degree on the modal characteristics with respect to the first 40th modes (Castanier and Pierre, 2006). The curve with the point denotes the mean value of each mode, and the band denotes the standard deviation of the MCS results in the figure. The normalized natural frequency clearly increases in Figure 3(a) for modes belonging to the disk-dominant mode family, that is, 1st–3rd and 16th–20th modes, where the strain energy is concentrated in the disk. The increased mode number in these disk-dominant modes translates to greater structural stiffness and, as a result, a higher natural frequency, resulting in a scattered frequency distribution. The mistuning pattern of the blades has little influence on the disk; therefore, the modal localization factor fluctuates at a low level in these disk-dominant modes, as shown in Figure 3(b).

On the contrary, for the modes belonging to a blade-dominated mode family, that is, 4th–15th, 21st–30th, and 31st–40th modes in Figure 3(a), the strain energy tends to be stuck in the isolated blades; therefore, the increased number of modes has little influence on the modal properties of the weak coupling blade-to-disk system, and the average frequency locus appears nearly flat. The fluctuation of the modal localization factor remains substantial in Figure 3(b), demonstrating that the mistuning characteristics have a considerable impact on the modal properties of the blade-dominated family of the assembly. In particular, for the 21st–30th and 31st–40th blade-dominant modes, the mean value of the modal localization factor decreases and subsequently increases as the mode number increases. This indicates that the most severe mode localization is likely to occur at the lowest and highest modes in the blade-dominated mode family, which should be paid attention to in engineering practice.

3.2.2. The influence of coupled stiffness

Further exploration is conducted on the modal localization of the mistuned bladed-disk system with frictional contact. The influence of frictional contact on the mistuned system is investigated by varying the normalized coupled stiffness k_c from 0 (free blade) to 0.4 under σ = 0.01 in Figure 4. Compared to the results of free blades in Figure 3, the natural frequency of damped blades has undergone a notable change throughout the full circle under the mistuning pattern. The introduced stiffness at the damping shroud distracts the originally dense modal distribution. For the bladed-dominant mode family in the free blade system, including 4th–15th, 21st–30th, and 31st–40th modes, the natural frequency is gathered with high modal density and the modal aggregation level. With the increasing coupled stiffness, the natural frequency becomes dispersed, and the mode localization factor gradually decreases. The shrinking standard deviation also reflects the gradual narrowing of modal differences and moderate distribution of modal energy. In general, the proper enhancement of coupling interaction can better smooth the modal aggregation degree and alleviate the modal localization level to avoid excessive vibratory amplification due to the mistuning pattern. It reminds the researcher that frictional contact can be an effective precaution for the inevitable mistuning pattern at the design phase of the bladed-disk system in aeroengine industry.

3.3. Response localization of mistuned system

Although the modal analysis can evaluate the sensitivity of inherent characteristics to the mistuning pattern, the response localization of mistuned systems can be complicated considering the complex excitation forms and contact conditions at the friction model, which cannot be simply predicted by the modal characteristic analysis (She and Li, 2022). Therefore, it is necessary to obtain the frequency–amplitude response considering different excitation orders and normal loads, to study the response localization of the damped and mistuned system under actual operating conditions.

A set of ξ is adopted in the paper, and the same scale is maintained in different cases of mistuned variances σ to avoid the impact of different distributions on the results. The basic LPM parameters are listed in Table 1. The normalized parameters of the friction model are given as follows: normal load N₀ = 0.01, coupled stiffness k_c = 0.2 and the friction coefficient μ = 0.3 at the contact, and the amplitude of excitation force f_e = 0.1 at each sector (Liu et al., 2019).

Figure 5 shows the frequency–amplitude response of the blade tip under different mistuned variances σ at the first excitation order (named 1EO) when normalized normal load N₀ = 0.01. It can be observed that there exist two resonance peaks in the system response within the given frequency range, with different sensitivity to the mistuned variance σ. The increasing variance makes a slight difference to the first modal resonance while significantly affecting the amplitude and frequency of the second one.

Computed by equation (19), the characteristic of response localization is quantified and shown in Figure 6. Generally, the vibration localization level increases with the increase of mistuned variance. Specifically, the response localization factor η remains at a low level at the first resonance near the normalized excitation frequency f = 0.128, while at the second resonance near f = 0.231, it lies in a relatively high state with several local peaks in the frequency range studied. Compared with the tuned counterpart, the response localization factor increases up to 30% at the second resonance point.

From the perspective of energy, the characteristics of vibratory mode are identified by four DOF in Figure 7. At the first resonance frequency range f = 0.1∼0.15, nearly 97% of the vibration response energy concentrates at the disk DOF (DOF4), while at the second resonance f = 0.2∼0.25, the vibration energy of the blade DOF (DOF1, 2, and 3) is predominant. With high energy occupied by the disk DOF, corresponding to the disk-dominant family, the mistuning pattern has little effect on the resonant amplitude and frequency, resulting in a low level of response localization. For the blade-dominant response, the increasing mistuned variance will significantly affect the resonant vibration performance, and the response localization factor will rise to a high level. It reminds the designers that it is necessary to quantitatively evaluate the correlation of the natural vibration characteristics with mistuning sensitivity of the system.

3.3.1. The influence of excitation order

The discussion above is mainly based on the excitation force at 1EO. However, the different arrangements between the static and rotor blades may cause a varying spatial distribution of the excitation at the blades (Sinha and Chen, 1989), which will introduce high-order components and reshape the response localization property. Figure 8 details the forced response change of the mistuned system considering different excitation orders. A remarkable distinction can be interpreted from the response results. In the case of σ = 0.01 in Figure 8(a), the resonance frequency does not deviate significantly from the natural frequency of the pristine system. It can be considered that the periodic excitation force at a specific engine order will mainly excite the blade vibration of the corresponding nodal diameter under a slight mistuning pattern; for instance, the vibration of the 1st nodal diameter mode will be triggered by the excitation force at 1EO. Table 2 compares the frequency discrepancy of the tuned and mistuned system in Figure 8. It can be seen that the resonant frequency due to the intentional mistuning is consistent with the natural frequency of the tuned counterpart under small variance, with only 1.24% as the maximum relative deviation. However, when it comes to σ = 0.1 shown in Figure 8(b), the mistuning pattern has a great impact on the vibration characteristics, and the obvious sensitivity is presented as multiple local peaks at the second resonance in terms of the excitation force at 1EO. Meanwhile, for the vibration incited by 3EO-5EO, the occurrence of multi-peak response behaves evidently at the first resonance. This dissonance between the excitation and response indicates that the excitation force at certain EO will activate not only the corresponding vibration mode but also the others in the system, leading to more complicated response forms. The sudden increase of the energy proportion at the disk DOF can be responsible for this cross correlation, as shown in Figure 9(a), indicating that the vibration mode can also be stimulated by the non-corresponding excitation force at 2EO-5EO. Accordingly, similar phenomena can be verified by obtaining the response localization factor under σ = 0.1 in Figure 9(b). On account of mono-excitation to the multi-response mechanism, several local response localization factors can be observed in the frequency range studied, showing that the mistuned blade may also have a high status of vibration energy localization at a remote resonance with the low-level response, which should be paid much attention to in the initial design of engineering.

Table 2.

Comparison of the resonance frequency of tuned and mistuned system (σ = 0.01).

EO	1	2	3	4	5
Tuned, 1st resonance frequency	0.1281	0.2102	0.2201	0.2216	0.2219
Mistuned, 1st resonance frequency	0.1279	0.2102	0.2196	0.2209	0.2213
Error/%	0.16	0.00	0.23	0.32	0.27
Tuned, 1st resonance frequency	0.2318	0.2653	0.3478	0.4057	0.4258
Mistuned, 1st resonance frequency	0.2314	0.2672	0.3521	0.41	0.4284
Error/%	0.17	0.72	1.24	1.06	0.61

3.3.2. The influence of normal load

Assuming that the normal load of each sector is equal under the small mistuned deviation, the frequency-domain response induced by excitation at 1EO relative to normal loads can be obtained in Figure 10(a). With the increasing normal load at the contact element, the maximum amplitude shows a trend of first decreasing and then increasing, reaching the optimal response level when N₀ = 0.30, and the resonance frequency deviates gradually. To better investigate the contact transition due to the normal load, the corresponding friction force at the second resonance with respect to the relative displacement at the contact interface is further obtained in Figure 10(b). The normalized resonance frequency is 0.231, and the corresponding amplitude is 0.683 under a negligible normal load (N₀ = 0); namely, the contact interface is in a separate condition. As the N₀ increases, the maximum dry friction also increases, leading to a stronger damping effect and a weaker vibration response. The contact interface lies in the transition between the stick and slip condition, where the proportion of the slip condition gradually shrinks and that of the stick condition increases in Figure 10(b) (N₀ = 0.02∼0.30). As the N₀ reaches 0.30, the stick and slip account for half of the contact condition, respectively, with the maximum area of the hysteresis loop and maximum dissipation of vibration energy, yielding an optimal response level of the mistuned system. When the N₀ further increases to 1.0, the relative frictional motion becomes difficult under the stick-dominant condition, restraining the friction damping performance, and the resonance amplitude rises again but remains less than that of the separate condition.

Considering that the blade-dominant response is more sensitive to the mistuning pattern, the response localization characteristics near the second resonance are investigated under the variation of normal loads. It can be concluded from Figure 11 that the change in normal load has a substantial impact on the localization level at the resonance frequency, where the factor has an obvious fluctuation. While little sensitivity is reflected during the remote frequency range under large normal loads, for example, in the case of N₀ > 0.30 and f = 0.26∼0.35 filled by shadows in Figure 11, the response localization level remains unchanged due to the stick-dominant condition at the frictional interface. It is noteworthy that a local extreme appears at f = 0.267 and f = 0.292, validating the conclusion from Figure 11 that it cannot afford to overlook the possible high-level vibration localization at frequency away from the resonance as well.

To further elucidate the change of the response localization factor in the mistuned system, the variance is coupled with the normal load to obtain the climax of factor η within the resonance frequency range of f = 0.22∼0.27, as shown in Figure 12. With the increase of N₀, the response localization factor first increases, then decreases to the constant value. In the meantime, the density of isolines reflects that the change rate of the maximum factor with the variance has the same tendency as well and finally remains unchanged because of the stick condition of the damping structure.

4. Conclusion

In this paper, an improved four-degree-of-freedom lumped parameter model coupled with frictional contact is developed to figure out the nonlinear vibration characteristics of the mistuned bladed-disk structure. A typical mistuning characterization is employed to investigate the effects of mistuned variance and coupled stiffness on the modal localization by Monte Carlo simulation. The influence of complex excitation forms and contact conditions are further illustrated on the response localization level of the mistuned system from the perspective of vibration localization and energy distribution, respectively. The main conclusions are as follows:

(1) The modal localization of the blade-dominated mode family is significantly affected by the mistuning pattern. The level of modal localization can be reduced by incorporating interblade-coupled stiffness at the frictional contact to avoid excessive vibration amplification due to mistuning effect.

(2) The excitation force at a specific EO not only activates the corresponding vibration mode but also triggers additional modes, leading to a cross-correlation with more complicated response characteristics. Additionally, it should be carefully considered during the turbine design that a high level of vibration energy localization can occur at a remote resonance, even with a low response level.

(3) An optimal vibratory response level of the mistuned system exists when considering the influence of a normal load at the contact interface, which can also be observed through the transition between slip-to-stick conditions. However, the stick-dominant condition at the contact interface limits the frictional damping performance at the remote frequency range under large normal loads.

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: The authors gratefully acknowledge the support by the National Science and Technology Major Project (J2019-IV-0022–0090) and the Fundamental Research Funds for the Central Universities (xtr072021002).

ORCID iDs

Zhufeng Liu

Di Zhang

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