Study on Rubbing-Induced Vibration Characteristics Considering the Flexibility of Coated Casings and Blades

Abstract

Rubbing between a blade and its coated casing is one of the main failures in aero-engine systems. This paper aims to study the effects of coated casings on rubbing-induced dynamic responses considering the flexibility of the coated casing and the flexibility of the blade. Firstly, an actual compressor blade is established by the shell element and verified by the experiment and ANSYS 19.2 software. Subsequently, a new dynamic model for the coated casing is proposed based on the laminated shell element, and the proposed dynamic model for the coated casing is verified by comparing the natural characteristics calculated by ANSYS software. Moreover, a comprehensive analysis is conducted to analyze the influences of the casing model, coating parameters, and casing parameters on vibration characteristics. Finally, the results show that the coating can diminish the severity level of rubbing. Notably, the material and thickness of the coating can change the nodal diameter vibrations of the casings (NDVCs) induced by rubbing. This study provides valuable guidance for the optimization and design of blade–casing systems.

1. Introduction

To enhance engine efficiency by changing airtightness, the radial clearance between blades and casings has been gradually reduced [1,2,3]. However, this reduction has aroused the issue of blade–casing rubbing faults, further resulting in economic losses and catastrophic accidents. Technological advancements have led to the application of coatings on the casing to enhance wear resistance and reduce vibration induced by rubbing. Consequently, attention has gradually shifted from blade–casing rubbing to the blade-coated casing, and the latter has become a main concern in aero-engine operation. Rubbing may have adverse effects on blades and casings, as depicted in Figure 1. Consequently, a comprehensive analysis of blade-coated casing rubbing is important to ensure the secure operation of aero-engines.

The dynamic modeling of blades has been a prominent focus in recent decades, with a variety of approaches using the finite element (FE) method, such as the cantilever beam model [6,7,8,9,10,11,12,13,14,15,16], the cantilever plate model [17,18,19,20,21,22], the shell model [23,24,25,26,27,28,29,30,31], and the solid element model [32,33,34,35,36]. It is a challenge to accurately capture the true morphological characteristics of compressor blades when beam element models are employed. Although the solid element model can compensate for this shortcoming, the use of solid elements often results in lower computational efficiency. Hence, the adoption of shell element modeling emerges as a compromise. Based on the shell element, various researchers have made significant contributions to the establishment of dynamic models for rotating blades. For instance, Sinha and Zylka [23] derived the control differential equation for rotating pre-twisted airfoil blades, transforming it into matrix eigenvalue form using the Rayleigh–Ritz method. Sun et al. [24] proposed a modeling method for pre-twisted rotating blades based on shell theory and the Hamilton principle, and they studied the vibration characteristics under different stagger angles and rotational speeds. Kee and Shin [26] developed a modeling approach using composite blades with a mixed element, combining an eight-node solid element with a shell element. Yangui et al. [27] used a three-node triangular shell element with six degrees of freedom to conduct the dynamic modeling of blades. Rafiee et al. [29] used the Shell99 element to establish an FE model of fan blades and focused on the study of the aeroelastic characteristics of blades. Kee and Kim [30] established an FE model for pre-twisted rotating composite blades using a degenerated shell element with nine nodes. They studied the effects of composite material lamination and fiber orientation parameters on the dynamic characteristics of blades. Additionally, Xiong et al. [31] utilized shell elements to establish a dynamic model for actual compressor blades and analyzed crack-induced characteristics. The aforementioned literature underlines the maturity of employing shell elements in blade modeling. Thus, based on our previous study (Ref. [31]), shell elements are utilized to establish compressor blades to investigate the rubbing fault of compressor-blade-coated casings in this paper.

There is a great deal of interest in the modeling of casings for blade–casing systems, with the lumped-mass model and FE model being approaches of interest. To simplify calculations, casings are built by the lumped-mass method. For instance, Sinha [37] simplified the casing as a lumped-mass point and investigated the transient dynamic response considering rubbing. Ma et al. [38] developed an FE model of a rotor–rigid casing and analyzed the vibration response induced by rubbing, and the casing was considered as the lump-mass point. While such simplified models serve certain analytical purposes, the actual casing, which is a thin shell structure, undergoes local deformation due to rubbing. Therefore, the accurate modeling of casings needs to consider casing flexibility. Zeng et al. [39] used Timoshenko beam elements to establish an FE model of casings and studied rubbing-induced dynamic responses. Lesaffre et al. [40] studied the stability of a rotating beam-flexible casing considering rubbing based on the Routh–Hurwitz criterion. Guo et al. [41] introduced an elastic-supported casing model according to the Sanders shell theory and mainly investigated the rubbing-induced vibration response of casings. Zeng et al. [42] also proposed a modeling approach of a rotor-disk-flexible casing considering elastic support using ANSYS/LS-DYNA software. In particular, Batailly and Legrand [43] analyzed the vibration response caused by the unilateral contact between blades and flexible casings based on the contact algorithm. In the context of coated casings, which are prevalent in modern aero-engine designs, more attention is paid to composite casings [44,45,46,47,48,49,50,51]. To enhance calculation accuracy, the FE method is widely employed to establish composite cylindrical shells (i.e., coated casings). For example, Zhang et al. [49,50] used a four-node shell element to conduct the dynamic modeling of a hard-coated cylindrical shell and investigated nonlinear vibration characteristics considering the base excitation. Additionally, Khan et al. [51] also studied the dynamic characteristics of laminated cylindrical shells via the FE method. Despite these advancements, the literature indicates a need for further exploration into utilizing more accurate shell elements in building dynamic models of coated casings. The development of coated casing models using the finite shell element method is still a field worthy of further research to improve accuracy and reliability.

In addition, scholars have also turned their attention to the study of the rubbing dynamic response in blade–casing systems. For example, Ma et al. [52,53] considered the bending deformation of blades and established an improved normal rubbing force model. Zhou et al. [54] investigated rubbing-induced vibration characteristics and found that harmonic frequencies near the natural frequency are more susceptible to the effect of rubbing. Hou et al. [55] analyzed frequency modulation considering rubbing in a rotor system and proposed an indicator to evaluate the severity of rubbing. Wu et al. [56] established a dynamic model with large rotation in a rotor-blade system and found the natural frequency in the spectrum. Shao et al. [57] also investigated the vibration characteristics caused by looseness and rubbing. Li et al. [58] proposed a rubbing model considering the local deformation of the casing during the rubbing process and studied the vibration characteristics of casings. Yang et al. [59,60,61] proposed a novel hysteresis force model to characterize contact force and studied the bending–torsional coupling vibration of the blade–casing system under non-uniform initial clearance conditions. Zeng et al. [62] primarily analyzed the rubbing-induced dynamic responses of blades. However, casing vibration has not received substantial attention in Refs. [63,64,65,66,67,68,69,70]. Although the vibration of an uncoated casing was considered in our previous paper [41], coatings can alter the natural characteristics of a casing and affect the dynamic response. Notably, coating wear has garnered significant attention, including in relation to the wear process [71,72,73], blade–coating interactions [74,75,76], wear models [77,78], and so on. While current research on blade–casing rubbing predominantly analyzes the vibration characteristics of blades, that of casing is often overlooked. Although the rubbing-induced dynamic response of blade–casing systems has gained widespread attention, it is essential to shift focus towards the vibration characteristics of coated casings and the effect of coatings on rubbing, with regard to coating material, coating thickness, etc.

The primary motivation of this paper is to study the rubbing-induced dynamic responses in a blade-coated casing system. The following contributions are listed in this paper on the basis of previous works: (1) A modeling method of the coated casing is developed based on laminated shell elements. Subsequently, a dynamic model of the system with rubbing is established. (2) The influence of the casing model (lumped-mass model, cylindrical shell model, and laminated casing model) and casing parameters (casing support stiffness and casing length) on the dynamic response of the system is systematically analyzed. (3) Notably, this study delves into the analysis of the effects of coatings on dynamic responses considering rubbing, including coating thickness and coating materials. In summary, this article primarily focuses on the response of coated casings and explores the influence of coatings on rubbing-induced dynamic responses, providing valuable insights into the field.

The layout of this article is organized as follows: The establishment of the model is described in Section 2, including blade dynamic modeling, casing dynamic modeling, rubbing force modeling, and system dynamic modeling. Section 3 mainly includes numerical simulation analyses, such as a comparison of casing models and investigations into the influence of coating parameters and casing parameters. Finally, Section 4 displays some conclusions.

2. Dynamic Modeling of the Blade-Coated Casing System with Rubbing

The dynamic model proposed is displayed in Figure 2. Its main components include blade dynamic modeling, casing dynamic modeling, and so on. Throughout the modeling process, several assumptions are considered as follows:

(1)

Similar to Ref. [59], coating wear and temperature effects are not considered when rubbing occurs.

(2)

The origin of the coordinate system is located at the center of mass of the rigid disk.

Figure 2. Schematic of the flexible blade–casing system.

Figure 2. Schematic of the flexible blade–casing system.

2.1. Dynamic Model of the Blade

The compressor blade is shown in Figure 3 and the blade is established by the FE method. To accurately describe the blade shape, a three-dimensional shell element with eight nodes is used to establish the FE model of the blade. This FE model has a total of 400 (20 × 20) elements. Oxyz and O’x’y’z’ are the global and local coordinate systems, respectively. R_d represents the radius of the rigid disk.

On the basis of Ref. [31], the dynamic model of the blade is built. Detailed parameters can be found in Ref. [31]. A spring element is applied at each node of the blade root to simulate the elastic support, as depicted in Figure 3. It is essential to note that the shell element possesses only five degrees of freedom. Consequently, the last torsional stiffness of the spring element is filled with zero N·m/rad. Consequently, the differential equation of motion of the blade with elastic support is written as

$$M\ddot{x}+(G+C)\dot{x}+(K_{\mathrm{e}}+K_{\mathrm{ce}}+K_{\mathrm{co}}-K_{\mathrm{s}})x=\mathbf{0}$$

(1)

where M, C, G, K_e, K_ce, K_co, and K_s represent the mass, damping, Coriolis force, structure stiffness, centrifugal stiffening, elastic support stiffness, and spin softening matrixes of the blade. Detailed modeling processes and descriptions are displayed in Appendix A. In this paper, the stiffness of springs to simulate the elastic support is calculated and the results are found to be k_x = 4.001 × 10⁸ N/m, k_y = 4.61 × 10⁷ N/m, k_z = 1.997 × 10⁸ N/m, k_θx = 1.437 × 10¹¹ N·m/rad, and k_θy = 2.118 × 10¹⁰ N·m/rad during the analysis process in Section 3.

The proposed model of the compressor blade is verified through comparisons of natural frequency, which is determined through the frequency-sweep experiment, as shown in Figure 4. Additionally, ANSYS software is also employed to calculate the natural frequency. Throughout the experiment, the blade is securely fixed in the electromagnetic excitation shake by the fixture and an accelerometer is placed in the middle of the blade to measure the vibration signal. The vibration signal is collected by the DH5956 data acquisition instrument and the mobile workstation is used to process the collected data. By applying loads with different frequencies, the response amplitude is amplified if the resonance phenomenon occurs. The frequency approximately corresponds to the natural frequency of the blade.

The comparative results of the first three natural frequencies (NFs) of the blade are listed in

Table 1. The first three NFs of the blade with fixed support.

Orders	Experiment [31]	ANSYS Model	Proposed Model	Errors between the Experiment and Proposed Model (%)	Errors between ANSYS Model and Proposed Model (%)
fn1b (Hz)	657.5	680.1	679.0	3.27	−0.15
fn2b (Hz)	2008.0	1993.8	1976.1	−1.60	−0.89
fn3b (Hz)	2409.0	2531.4	2500.0	3.77	−1.22

. All frequency errors are less than 5%. This verifies the proposed dynamic model of the blade. Sources of error are analyzed as follows: (1) Material property errors: Physical properties of materials, such as elastic modulus, density, and Poisson’s ratio, may have measurement errors or variability. (2) Geometric dimension errors: The geometric dimensions used in the model (such as thickness, length, width, etc.) may deviate from the actual dimensions. (3) Boundary condition errors: The boundary conditions applied in the simulation (for example, constraints) may differ from the actual conditions. (4) Manufacturing errors: Tolerances in the manufacturing process can cause the actual dimensions of structural components to differ from the design dimensions. (5) Numerical errors: Approximation and rounding errors are induced in numerical calculations.

It is difficult to guarantee the same elastic modulus and density in a simulation and an experiment. Thus, these two typical parameter errors are analyzed, as shown in Figure 5. As the density increases, the first natural frequencies of the blade and error linearly decrease. However, with the increase in elasticity, the first natural frequencies of the blade and error increase linearly.

2.2. Dynamic Model for the Coated Casing

Although the lumped-mass modelling of casings can improve computational efficiency, it fails to consider the local deformation of the casing and potentially leads to errors in calculating the radial clearance at the blade tips. Thus, the laminated shell element is introduced to establish the FE model for flexible coated casings. By defining the distinct material parameters of each casing layer, the stiffness and mass matrices of laminated shell elements are calculated.

Within the laminated shell element, material parameters change in the thickness direction. This leads to different densities, strain matrices, and elastic matrices in different layers. Consequently, when calculating the mass and stiffness matrices of the laminated shell element, integration by parts is employed for precision. To simplify the computation process, a new local coordinate system ξηζ′ is established in each layer of the shell, as depicted in Figure 6. Numbers 1–8 represent 8 nodes within a shell element. The mass matrix and stiffness matrix of each material layer are computed using Gaussian integration in the new local coordinate system ξηζ′, and the corresponding position of the p-th layer is expressed as

$${\zeta }_{p-1}=-1+2\times {\displaystyle \frac{h_{p-1}}{h_{\mathrm{t}}}}, {\zeta }_p=-1+2\times {\displaystyle \frac{h_p}{h_{\mathrm{t}}}},$$

(2)

where ζ_p−1 and ζ_p correspond to −1 and 1 in the local coordinate system ξηζ′, and h_t represents the total thickness of the laminated shell. h_p−1 and h_p represent the upper and lower boundaries of the pth layer.

The relationship between coordinate system ξηζ′ and coordinate system ξηζ is denoted as

$$\zeta =-1+{\displaystyle \frac{\left({\zeta }^{\prime }+1\right)\left({\zeta }_p-{\zeta }_{p-1}\right)}{2}}, \mathrm{d}\zeta ={\displaystyle \frac{{\zeta }_p-{\zeta }_{p-1}}{2}}\mathrm{d}{\zeta }^{\prime }.$$

(3)

The mass matrix of the laminated shell element is written as

$$M^{\mathrm{e}}={\displaystyle {\int }_V\rho }{N}^{\mathrm{T}}N\mathrm{d}V={\displaystyle \sum _{p=1}^q{{\displaystyle {\int }_{-1}^1{\displaystyle {\int }_{-1}^1{\displaystyle {\int }_{-1}^1\rho }}}}_p{N_p}^{\mathrm{T}}{N}_p\left|J\right|\left|{J^{\prime }}_p\right|\mathrm{d}\xi \mathrm{d}\eta \mathrm{d}{\zeta }^{\prime }}$$

(4)

where ζ_p−1 and ζ_p are the material density of the p-th layer and q is the total number of layers. N_p denotes the shape function matrix along the thickness direction ζ′. $\left|{J^{\prime }}_p\right|$ is written as $\left|{J^{\prime }}_p\right|={\displaystyle \frac{{\zeta }_p-{\zeta }_{p-1}}{2}}$.

The stiffness matrix of the laminated shell element can be written as

$$K^{\mathrm{e}}={\displaystyle {\int }_V{B^{\prime }}^{\mathrm{T}}D{B}^{\prime }\mathrm{d}V}={\displaystyle \sum _{p=1}^q{\displaystyle {\int }_{-1}^1{\displaystyle {\int }_{-1}^1{\displaystyle {\int }_{-1}^1{{B^{\prime }}_p}^{\mathrm{T}}}}}{D}_p{B^{\prime }}_p\left|J\right|\left|{J^{\prime }}_p\right|\mathrm{d}\xi \mathrm{d}\eta \mathrm{d}{\zeta }^{\prime }},$$

(5)

where B’_p and D_p denote the strain and elastic matrices in the p-th layer of elements.

By assembling the stiffness and mass matrices of the element, those of the coated casing can be obtained. The equation of motion of the coated casing can be written as

$$M_{\mathrm{c}}{\ddot{u}}_{\mathrm{c}}+C_{\mathrm{c}}{\dot{u}}_{\mathrm{c}}+K_{\mathrm{c}}{u}_{\mathrm{c}}=\mathbf{0},$$

(6)

where M_c, C_c, and K_c denote the mass, damping, and stiffness matrices of the coated casing. The expression for the damping matrix is denoted as

$$C_{\mathrm{c}}={\alpha }_{\mathrm{c}}{M}_{\mathrm{c}}+{\beta }_{\mathrm{c}}{K}_{\mathrm{c}},$$

(7)

where ${\alpha }_{\mathrm{c}}={\displaystyle \frac{4\mathsf{\pi }{f}_{\mathrm{n}1\mathrm{c}}{f}_{\mathrm{n}2\mathrm{c}}(f_{\mathrm{n}1\mathrm{c}}{\xi }_2-f_{\mathrm{n}2\mathrm{c}}{\xi }_1)}{(f_{\mathrm{n}1\mathrm{c}}^2-f_{\mathrm{n}2\mathrm{c}}^2)}}$ and ${\beta }_{\mathrm{c}}={\displaystyle \frac{f_{\mathrm{n}2\mathrm{c}}{\xi }_2-f_{\mathrm{n}1\mathrm{c}}{\xi }_1}{\mathsf{\pi }(f_{\mathrm{n}2\mathrm{c}}^2-f_{\mathrm{n}1\mathrm{c}}^2)}}$. f_n1c (Hz) and f_n2c (Hz) represent the first two NFs of the casings. ξ₁ and ξ₂ represent the damping ratios of the first and second modes, respectively.

The proposed laminated shell model for the coated casing is validated by comparing the NFs with those obtained from the ANSYS software. The coating material is Metco 313 NS, and the material parameters are listed in

Table 2. Parameters of the coated casing.

Parameters	Value	Parameters	Value	Parameters	Value
Thickness of coating ha (mm)	1	Thickness of casing hc (mm)	2	Length of casing Lc (m)	0.1
Density of coating ρa (kg/m3)	2000	Density of casing ρc (kg/m3)	4370	Inner diameter of casing Rc (m)	0.30682
Elastic modulus of coating Ea (GPa)	3.11	Elastic modulus of casing Ec (GPa)	125	Poisson ratio v	0.3

. The first four NFs are presented in

Table 3. The first four NFs of the coated casing with fixed support.

Orders	fn1c (Hz)	fn2c (Hz)	fn3c (Hz)	fn4c (Hz)
Proposed model	1929.1	1930.3	1950.7	1954.9
ANSYS model	1929.8	1930.8	1951.4	1955.4
Errors (%)	−0.036	−0.026	−0.036	−0.026

and the mode shapes are presented in Figure 7. A comparison with the ANSYS results indicates that the errors in the first four NFs are all within 0.2%, and the mode shapes exhibit good consistency. This verifies the proposed dynamic model for the coated casing.

Three-dimensional linear spring elements are used at each node of both ends of the casing to simulate its elastic support. Spring elements are incorporated into the cylindrical coordinate system. This requires transformation to the cartesian coordinate system during stiffness matrix calculations. The stiffness values in each direction in the cylindrical coordinate system are $k_r=5\times {10}^7$ N/m, $k_{\mathrm{t}}=4\times {10}^7$ N/m, $k_{\mathrm{a}}=2\times {10}^8$ N/m, $k_{\mathsf{\theta }\mathrm{r}}=1\times {10}^6$ N·m/rad, and $k_{\mathsf{\theta }\mathrm{t}}=1\times {10}^6$ N·m/rad. The FE model for the coated casing with elastic support is shown in Figure 8. The first six NFs of the casing with elastic support are shown in

Table 4. The first six NFs of the coated casing with elastic support.

Orders	fn1c (Hz)	fn2c (Hz)	fn3c (Hz)	fn4c (Hz)	fn5c (Hz)	fn6c (Hz)
Proposed model	823.54	826.63	827.93	839.19	842.85	858.30
ANSYS model	823.41	826.47	827.83	839.12	842.65	858.29
Errors (%)	0.0157	0.0193	0.0120	0.0080	0.0230	0.0012

, and those of the proposed model show good consistency with the results from ANSYS. Thus, the proposed model for coated casings is verified considering elastic support. The error in the natural frequency of the casing is very small, so the source of this error may be numerical, including approximation and rounding errors.

The proposed casing model consists of 3584 elements and 11,264 nodes (i.e., 67,584 degrees of freedom), which causes a low calculation speed. To enhance computational efficiency, the Craig–Bampton method is used to establish the reduced dynamic model. The reduced dynamic model for the casing is depicted as

$${\widehat{M}}_{\mathrm{c}}{\ddot{q}}_{\mathrm{c}}+{\widehat{C}}_{\mathrm{c}}{\dot{q}}_{\mathrm{c}}+{\widehat{K}}_{\mathrm{c}}{q}_{\mathrm{c}}={\widehat{F}}_{\mathrm{c}}$$

(8)

The reduced casing has 320 degrees of freedom.

Table 5. Comparison of NFs between the reduced model and unreduced model.

Orders	fn1c (Hz)	fn2c (Hz)	fn3c (Hz)	fn4c (Hz)	fn5c (Hz)
Unreduced model	823.54	826.63	827.93	839.19	842.85
Reduced model	824.23	827.32	830.21	840.39	845.44
Errors (%)	0.084	0.084	0.275	0.140	0.307

displays the first five NFs of the reduced dynamic model. The maximum error of the natural frequency is 0.307%. This meets the required accuracy standards and verifies the reduced dynamic model.

2.3. Dynamic Model of the Blade-Coated Casing System

Considering static misalignment e_c, the schematic of a rotating blade-flexible coated casing is shown in Figure 9, considering rubbing. O_c and O represent the centers of the casing and disk in a stationary state, respectively. Mesh refinement is implemented at locations susceptible to rubbing, as depicted in Figure 9a. In order to correspond to the node of the blade, the casing in the rubbing region is divided into 20 elements in the length direction. The non-rubbing regions on both sides of the casing are divided into two elements. Thus, the casing has a total of 24 elements in the length direction in Figure 9a. In Figure 9b, the casing position and blade tip orbit are displayed by the blue and pink lines when the rubbing fault occurs. By analyzing the gap $c_{\mathrm{rub}}^i$ between the blade tip and the casing, whether the rubbing fault occurs is judged and the normal rubbing force is calculated. The calculation process of $c_{\mathrm{rub}}^i$ is shown in Appendix B, and the expression of $c_{\mathrm{rub}}^i$ is shown in Equation (A29). Additionally, Figure 9c mainly shows the local contact between the blade and the casing.

The rubbing model is shown in Figure 10. Figure 10a displays the blade-coated casing system. This rubbing model simultaneously considers the contact deformation between the blades and the casing, the structural deformation of each component, and the energy dissipation phenomenon caused by contact damping. The rubbing stiffness of the hysteresis contact-force model includes the structural stiffnesses k_c and k_b, as well as the local contact stiffness k_h^*. In other words, the rubbing stiffness is equal to the series stiffness of these three stiffness values, as shown in Figure 10b.

The detailed derivation of the rubbing force is shown in Appendix B.

During the operation of an aircraft engine, the blade is subjected to aerodynamic forces, and the overall aerodynamic load vector Q_p is written as

$$Q_{\mathrm{p}}={\displaystyle \sum _{i=1}^n{Q}_i^{\mathrm{p},\mathrm{ext}}},$$

(9)

where $Q_i^{\mathrm{p},\mathrm{ext}}$ is the aerodynamic load of an element. The detailed expression can be found in Ref. [79].

Taking node pair 1 in Figure 9 as an example, the friction force is

$$F_{\mathrm{t}}^1=\mu F_{\mathrm{n}}^1,$$

(10)

where μ represents the frictional coefficient.

The rubbing forces $F_{\mathrm{n}}^i$ and $F_{\mathrm{t}}^i$ ($F_{\mathrm{t}}^i=\mu F_{\mathrm{n}}^i$) are projected in the x and y directions. μ represents the friction coefficient. The resultant forces $F_x^i$ and $F_y^i$ are written as

$$\begin{array}{l}{F}_x^i=F_{\mathrm{n}}^i\cos (\omega t+{\varphi }_{\mathrm{b}})-F_{\mathrm{t}}^i\sin (\omega t+{\varphi }_{\mathrm{b}})\\ F_y^i=F_{\mathrm{n}}^i\sin (\omega t+{\varphi }_{\mathrm{b}})+F_{\mathrm{t}}^i\cos (\omega t+{\varphi }_{\mathrm{b}})\end{array},$$

(11)

where ω represents the rotating speed and ϕ_b represents the blade phase.

The expressions of F_rb and F_rc are

$$\begin{array}{l}{\mathit{F}}_{\mathrm{rb}}=\left[\begin{array}{cccc}\begin{array}{cccc}\cdots & F_x^1& F_y^1& 0\end{array}& 0& 0& \begin{array}{cccc}\cdots & F_x^{21}& F_y^{21}& \begin{array}{ccc}\begin{array}{cc}0& 0\end{array}& 0& \begin{array}{cc}0& \cdots \end{array}\end{array}\end{array}\end{array}\right]\\ {\mathit{F}}_{\mathrm{rc}}=-{\mathit{F}}_{\mathrm{rb}}\end{array}.$$

(12)

Thus, considering the rubbing forces, the dynamic model of the blade-coated casing is written as

$$\left[\begin{array}{cc}{K}_{\mathrm{e}}+K_{\mathrm{ce}}+K_{\mathrm{co}}-K_{\mathrm{s}}& \mathbf{0}\\ \mathbf{0}& {\widehat{K}}_{\mathrm{c}}\end{array}\right]\left[\begin{array}{c}x\\ q_{\mathrm{c}}\end{array}\right]+\left[\begin{array}{cc}G+C& \mathbf{0}\\ \mathbf{0}& {\widehat{C}}_{\mathrm{c}}\end{array}\right]\left[\begin{array}{c}\dot{x}\\ {\dot{q}}_{\mathrm{c}}\end{array}\right]+\left[\begin{array}{cc}M& \mathbf{0}\\ \mathbf{0}& {\widehat{{\rm M}}}_{\mathrm{c}}\end{array}\right]\left[\begin{array}{c}\ddot{x}\\ {\ddot{q}}_{\mathrm{c}}\end{array}\right]=\left[\begin{array}{c}{F}_{\mathrm{c}}\\ \mathbf{0}\end{array}\right]+\left[\begin{array}{c}{Q}_{\mathrm{p}}\\ \mathbf{0}\end{array}\right]+\left[\begin{array}{l}{F}_{\mathrm{rb}}\\ F_{\mathrm{rc}}\end{array}\right]$$

(13)

where F_rb, F_rc, and F_c represent the rubbing force vectors of the blade and casing and the centrifugal force vector of the blade. The rubbing force is nonlinear, so once the system has a rubbing fault, the system will show obvious nonlinear characteristics.

3. Numerical Analysis

The casing is established using a laminated shell element with elastic support, while the blade is built as detailed in Section 2.2. The influence of the parameters related to the coating casing on the dynamic response of the system with rubbing is analyzed in this section.

3.1. Influence of the Casing Model

Guo et al. [41] analyzed the effects of casing models on rubbing, encompassing the lump-mass casing model and the cylindrical shell casing model. Inspired by Ref. [41], this subsection further compares the vibration responses of different casing models based on the hysteresis contact-force model, including the lump-mass casing model, cylindrical shell casing model, and laminated casing model. This comparison is conducted under the same geometric structure parameters and rubbing parameters. The simulation parameters of the blade are detailed in

Table 6. Simulation parameters of the blade.

Parameters	Value	Parameters	Value
Frictional coefficient μ	0.3	Rotating speed ω (r/min)	10,000
Static parallel misalignment ec (mm)	1.6	Minimum clearance gmin (mm)	0.1
Misalignment angle β1 (°)	0.15	Blade tip staggered angle β2 (°)	54.68
Radius of the blade tip Rd (mm)	216.52	Inner diameter of casing Rc (mm)	306.82

, and those of the casing are provided in

Table 7. Simulation parameters of the casing.

Models	Parameters
Lumped-blade mass casing model	mc = 1.69 kg, kc = 1.34 × 106 N/m
Cylindrical-blade shell casing model	Lc=0.1 m, hc = 2 mm, Ec = 125 GPa, ρc = 4370 kg/m3
Laminated-blade casing model	Lc = 0.1 m, hc = 2 mm, ha = 1 mm, Ea = 3.11 Gpa, Ec = 125 GPa, ρa = 2000 kg/m3, ρc = 4370 kg/m3

. The dynamic responses are calculated by the Newmark-β method, and are detailed in Appendix C.

Figure 11 displays the time-domain waveform (TDW) of the rubbing forces and vibration response at the 21st node of the blade tips under three casing models. Notably, the different casing models exhibit a minimal difference in bending displacement and radial displacement. The frequency components remain consistent among these three models. Amplitude amplification is observed at frequency 5f_r because frequency 5f_r is close to the first natural frequency of the blade. Comparatively, the cylindrical shell casing model demonstrates a slight reduction in the maximum normal rubbing force, radial displacement, and bending displacement in contrast to the lumped-mass casing model. This phenomenon can be attributed to the localized deformation of the cylindrical shell casing model under the action of the normal rubbing force. This deformation increases the gap between the blade and the casing in subsequent load steps, ultimately reducing blade penetration and the severity level of rubbing.

The natural characteristics and radial equivalent stiffness of the casing are directly influenced by changing the material parameters. The first NFs of the cylindrical shell casing and laminated shell casing are 906.12 Hz and 823.55 Hz, respectively. The equivalent radial stiffnesses are 1.396 × 10⁶ N/m and 1.336 × 10⁶ N/m, respectively. The equivalent radial stiffness of the laminated shell casing is greater than that of the cylindrical shell casing, but its maximum normal rubbing force, radial displacement, and bending displacement are slightly reduced compared to the cylindrical shell casing model (see Figure 11a), indicating that having a coating on the surface of the casing can reduce the severity level of blade–casing rubbing.

The vibration responses of the blade at the 11th and 1st nodes of the blade tip are shown in Figure 12. When comparing Figure 11 and Figure 12, a notable difference in the radial and bending displacements can be found among the blade tips. Specifically, the radial and bending displacements of the blade tip nodes exhibit a gradual decrease with the change in the position at the blade tip. This is because the thickness of the blade is non-uniform, and the elongation of each node at the blade tip is different under the action of the centrifugal force. Near the 21st node, contact between the blade tip and the casing appears, causing a reduction in the radial and bending displacements. Of course, an obvious conclusion is that changes in the casing model will not cause changes in frequency components, only in amplitude.

The blade passes the coated casing at four positions: points A (0), B (π/2), C (π), and D (3π/2), the deformation diagram of the casing is shown in Figure 13. The definition of the points can be found in Figure 9a. At position A, blade–casing rubbing can obviously be found. When rubbing occurs, the deformation of laminated shell models at point A is greater than that of the cylindrical shell models, and the deformation of the two models at that point is significantly greater than that at the other points. However, for the lumped-mass casing, the overall casing is translated to the right due to rubbing forces. Additionally, obvious NDVCs can be observed in the cylindrical shell model and the laminated shell casing model. However, the induced NDVCs are also different among these two flexible casing models.

The time-domain waveforms and the spectra of radial vibration at points A, B, C, and D are shown in Figure 14, Figure 15 and Figure 16. In these figures, the radial displacement of the rigid casing is much smaller than that of the cylindrical shell casing and laminated shell casing. The difference in radial displacement between the cylindrical shell casing and the laminated shell casing at the rubbing point is not significant, but at the non-rubbing position, the laminated shell casing has a larger vibration displacement compared to the cylindrical shell casing. Additionally, there is a significant difference in the frequency components between the two flexible casing models. The cylindrical shell casing exhibits amplitude amplification at frequency 6f_r, while the laminated shell casing exhibits amplitude amplification at frequency 5f_r, and exhibits dense frequency characteristics at rubbing point A. At non-rubbing points (points B, C, and D), the cylindrical shell casing and the laminated shell casing also exhibit amplitude amplification at 6f_r and 5f_r, respectively. The main reason for the above differences is the difference in natural characteristics between the cylindrical shell casing and the laminated shell casing.

3.2. Influence of Coating Thickness

Changes in coating thickness directly affect the natural characteristics and radial stiffness of a casing. Therefore, the influences of coating thickness are investigated.

The NRF and dynamic response at the 21st node are shown in Figure 17. The equivalent radial stiffnesses of casings with coating thicknesses of $0.5 \mathrm{mm}$, $1 \mathrm{mm}$, and $2 \mathrm{mm}$ are 1.357 × 10⁶ N/m, 1.396 × 10⁶ N/m, and 1.54 × 10⁶ N/m, respectively. Correspondingly, the first NFs are 860.68 Hz, 823.55 Hz, and 768.99 Hz, respectively. Notably, the equivalent radial stiffness increases with the increase in coating thickness, which causes rubbing force to increase. However, in Figure 18, the rubbing force at a 0.5 mm thickness is greater than that at a 1 mm thickness. The main reason is that the radial displacement of the 1 mm coating thickness at the rubbing point is greater than that of the 0.5 mm coating thickness during the initial rubbing time period from 0.3238 s to 0.324 s, resulting in a smaller penetration between the blade and casing, and ultimately a smaller rubbing force. The coating thickness has a slight effect on the response amplitude of the blade.

From Figure 18 and Figure 19, it is evident that the NDVCs induced by rubbing under different coating thicknesses are different. At rubbing point A, the radial displacement and frequency components of the casing under the three coating thicknesses show minimal differences, and an amplitude amplification phenomenon is observed at frequency 5f_r. Conversely, at non-rubbing point B, there is a notable difference in the radial displacement and frequency components of the casing. Specifically, when the coating thicknesses are 1 mm and 2 mm, the amplitude amplification phenomenon occurs at frequency 5f_r. In contrast, when the coating thickness is 0.5 mm, the amplitude amplification phenomenon appears at frequency 6f_r in Figure 19d.

3.3. Influence of Coating Material

A comparative analysis of NRF and vibration response under three coating materials (Metco 308 NS, Metco 601 NS, and Metco 313 NS) is conducted. The corresponding material parameters are detailed in

Table 8. Material parameters, NF, and equivalent radial stiffness of coatings.

Materials	Ingredient	Density ρa (kg/m3)	Elasticity Modulus Ea (GPa)	Natural Frequency f (Hz)	Equivalent Radial stiffness (N/m)
Metco 308NS	Ni-graphite	4900	3.95	841.32	1.376 × 106
Metco 601NS	Al-Si polymer	1500	2.08	731.76	1.412 × 106
Metco 313NS	Al-Si-graphite	2000	3.11	823.55	1.396 × 106

. Variations in coating materials lead to alterations in the NF and equivalent radial stiffness of the casing, and the results are shown in Table 8.

Figure 20, Figure 21 and Figure 22 display the TDW of NRF, the vibration response of the blade, casing deformation, and the vibration response of the casing at points A and B under different coating materials. Observations from these figures reveal that, when the coating material is Metco 308 NS, the NRF is the largest, and the NDVCs induced by rubbing are more pronounced. Conversely, the normal rubbing force, vibration response of the blade, and vibration response of the casing at points A and B are roughly the same when the coating materials are Metco 601 NS and Metco 313 NS. However, it is noteworthy that Metco 601 NS is primarily suitable for low-pressure compressors at low-temperature conditions. Therefore, for the purposes of this simulation and analysis, Metco 313 NS is chosen as the coating material to study in this paper. The vibration response induced by rubbing can serve as a crucial criterion for the selection of coating materials in the design process of blade-coated casing systems.

Figure 21 and Figure 22 reveal variations in radial displacements, frequency components, and induced NDVCs under different coating materials. At rubbing point A, an amplitude amplification phenomenon is consistently observed at frequency 5f_r for all three materials. However, at non-rubbing point B, the amplitude amplification phenomenon occurs at frequency 5f_r for both Metco 308 NS and Metco 313 NS materials, while for the Metco 601 NS material casing, the amplitude amplification phenomenon appears at frequency 6f_r. These differences highlight the sensitivity of the vibration response of casings to the choice of coating materials, emphasizing the importance of this factor in material selection for good performance when rubbing occurs in a blade-coated casing system.

3.4. Influence of the Support Stiffness of the Coated Casing

The elastic support stiffness of casings can influence natural characteristics and the equivalent radial stiffness of the casing. Therefore, this section discusses the effects of the radial, tangential, and axial elastic support stiffnesses of casings on the equivalent radial stiffness of the casing and the vibration response of the system. Simulation parameters are listed in

Table 9. Simulation parameters.

Varying Parameters	Ranges	Constant Parameters
Radial support stiffness kr	[1 × 106, 1 × 109] N/m	kt = 1 × 108 N/m, ka = 1 × 108 N/m, Lc = 0.1 m, Ec = 125 GPa, ρc = 4370 kg/m3, hc = 2 mm, Ea = 3.11 GPa, ρa = 2000 kg/m3, ha = 1 mm
Tangential support stiffness kt	[1 × 106, 1 × 109] N/m	kr =1 × 108 N/m, ka = 1 × 108 N/m, Lc = 0.1 m, Ec = 125 GPa, ρc = 4370 kg/m3, hc = 2 mm, Ea = 3.11 GPa, ρa = 2000 kg/m3, ha = 1 mm,
Axial support stiffness ka	[1 × 106, 1 × 109] N/m	kr =1 × 108 N/m, kt = 1 × 108 N/m, Lc = 0.1 m, Ec = 125 GPa, ρc = 4370 kg/m3, hc = 2 mm, Ea = 3.11 GPa, ρa = 2000 kg/m3, ha = 1 mm,

.

Figure 23, Figure 24 and Figure 25 show the equivalent radial stiffnesses and rubbing forces of the casings under different elastic support stiffnesses. The vibration response of the 21st node of the blade tip and the vibration response at point A of the casing are analyzed. Some conclusions are listed as follows:

(1) Equivalent radial stiffness exhibits an increase with the rise in elastic support stiffness. Specifically, the equivalent radial stiffness and maximum rubbing force are primarily influenced by the radial and tangential elastic support stiffnesses, and the axial elastic support stiffness has a minimal effect on the equivalent radial stiffness and maximum normal rubbing force. The equivalent radial stiffness undergoes significant variations when the radial elastic support stiffness ranges from 1 × 10⁶ N/m to 1 × 10⁸ N/m and the tangential elastic support stiffness ranges from 1 × 10⁸ N/m to 1 × 10⁹ N/m. Overall, the radial elastic support stiffness has the most obvious influence on the equivalent radial stiffness of the coated casing.

(2) As the radial elastic stiffness of the casing increases, there is an obvious increase in the amplitudes of rubbing force and vibration displacement, while the frequency component remains unchanged. The support stiffness in three directions exerts a notable influence on the radial vibration response at point A of the casing. Specifically, with an increase in radial and tangential stiffness, the frequency corresponding to the amplitude amplification phenomenon also increases. This is attributed to the difference in the natural frequency of the casing induced by changes in the support stiffness of the coated casing.

3.5. Influence of the Length of the Coated Casing

The length of the casing directly affects the natural frequency and equivalent radial stiffness of the coated casing. Therefore, this section analyzes the effect of casing length on vibration response. The vibration response of the 21st node of the blade tip and the vibration response at point A of the casing are shown in Figure 26. The first natural frequency, equivalent radial stiffness, and maximum normal rubbing force of the 21st node under different casing lengths are listed in

Table 10. The first NF and equivalent radial stiffness of the casing at different lengths.

Casing Length Lc(m)	0.06	0.08	0.1	0.12	0.14	0.16	0.18	0.2
Equivalent radial stiffness kc (106 N/m)	1.543	1.558	1.396	1.308	1.270	1.248	1.228	1.208
1st natural frequency fn1c (Hz)	1146.20	939.93	860.64	844.81	684.28	633.77	591.70	555.67
Maximum normal rubbing force Fn (N)	33.27	31.36	33.21	31.09	28.22	29.16	28.63	28.87

.

As shown in Figure 26, a change in the length of the casing only slightly affects the amplitude of the bending vibration of the blade and does not affect the frequency component. It can be also observed that the length of the casing has a significant influence on the radial displacement at point A of the casing. As the length of the casing increases, the amplitude amplification gradually decreases from frequency 9f_r to frequency 3f_r, which is caused by the difference in the natural frequency of the casing.

From Table 10, as the length of the casing increases, both the first natural frequency and equivalent radial stiffness of the casing decrease. However, the maximum normal rubbing force does not decrease with the reduction in the radial stiffness of the casing. The primary reason lies in the varying deformation of the casing with different lengths, leading to different penetrations between the blade and casing. Consequently, the rubbing force does not decrease proportionally with the decline in the radial stiffness of the casing. As depicted in Figure 26, changes in the length of the casing only slightly affect the amplitude of the bending vibration of the blade and have no obvious effect on the frequency components. Simultaneously, the length of the casing significantly influences the radial displacement at point A of the casing. With the increase in casing length, the amplitude amplification gradually decreases from frequency 9f_r to frequency 3f_r.

4. Conclusions

To investigate blade-coated casing rubbing, an FE model of a laminated shell casing is established using eight-node shell elements. The accuracy of the model is validated by comparing its natural characteristics with the results obtained from ANSYS software. Additionally, an actual compressor blade with varying sections and a twisted shape is also established using the shell element. Subsequently, based on hysteretic contact force, a dynamic model with rubbing is established to investigate rubbing-induced vibration characteristics. Some key conclusions can be summarized as follows:

(1) The vibration responses of the blade–casing system with rubbing are analyzed among three casing models: the lumped-mass model, the cylindrical shell model, and the laminated shell model. The study reveals that the vibration response amplitude of the laminated-blade shell casing with rubbing is the smallest. This indicates that the coating can reduce the severity level of rubbing. Additionally, the frequency corresponding to the amplitude amplification phenomenon of the cylindrical shell model and the laminated shell model at the rubbing and non-rubbing points are different due to variations in natural characteristics.

(2) Furthermore, the material and thickness of the coating can change the NDVCs induced by rubbing. The frequency corresponding to amplitude amplification is also influenced by the casing support stiffness and casing length, as these parameters change the natural characteristics of the casing.

The primary contribution of this article lies in the establishment of a dynamic model for an actual blade-coated casing, taking into account rubbing faults. The study emphasizes understanding the effects of coating and casing on dynamic responses in the presence of rubbing, with a specific focus on analyzing casing vibration. It is important to note that this article does not address coating wear, which stands out as a potential focus for future work. The study of these dynamics considering wear would provide a more comprehensive understanding of dynamic behavior under operational conditions.