Blade-coating casing normal rub-impact force calculation method considering heat-wear effect

Abstract

The invention belongs to the technical field of rub-impact force calculation, and discloses a blade-coating casing normal rub-impact force calculation method considering a heat-wear effect. Based on a one-dimensional unsteady state heat conduction theory, analytic expressions of temperature rise and linear thermal expansion of the blade and the casing are respectively established, and instantaneous temperature change in each contact process is considered. Secondly, physical modeling of various abrasion mechanisms (such as cutting type abrasion, adhesive abrasion and microcrack abrasion) is introduced, and the influence of coating gap changes on the contact force is effectively captured; in addition, the composite stiffness characteristics of the casing coating and the supporting structure are further integrated, so that the rub-impact force calculation can cover the response characteristics under different structural stiffness conditions. Through finite element comparison and experimental verification, the calculation method disclosed by the invention shows relatively high precision in the aspects of predicting thermal deformation, contact force and wear trend, and provides powerful support for subsequent rub-impact fault diagnosis and structural design.

Description

Technical Field

[0001] This invention relates to the field of friction force calculation technology, and in particular to a method for calculating the normal friction force of a blade-coated casing considering the heat-wear effect. Background Technology

[0002] To improve aero-thermal efficiency, aero-engines are designed to minimize the radial clearance between the blades and the casing. Ground-based test benches are used to establish this clearance, leading to rubbing being a common phenomenon during the development and factory testing of high-performance turbofan engines. Furthermore, gyroscopic torque effects during engine maneuvers can also induce rubbing, causing blade damage and coating wear. To minimize tip clearance, improve fuel efficiency, and prevent blade wear, abrasive sealing coatings with good abrasiveness and erosion resistance are typically used. Sealing coatings are widely applied in compressor sections under varying temperatures and loads, located between the casing and blades, sealing the airflow passage and protecting the blade tips. Due to their complex operating environment, sealing coatings not only need to be soft to prevent wear but also possess strong erosion and corrosion resistance, maintaining performance under high temperatures and high-speed airflow. The application of sealing coatings effectively reduces tip wear and radial clearance, significantly improving engine efficiency and stall resistance. Therefore, blade-coated casing rubbing not only causes nonlinear contact forces, but also leads to significant wear and thermal effects.

[0003] Blade-coated casing rubbing is a common failure in aero-engines, potentially leading to thermal and wear effects. Therefore, the effects caused by rubbing have attracted scholarly attention, including wear, nonlinear vibration, and rubbing forces. The main cause of rubbing is the need to reduce the clearance between rotating and stationary components to improve the thrust-to-weight ratio and reduce fuel consumption in aero-engines, which increases the likelihood of rotor-stator contact. Furthermore, with increasing turbine inlet temperature and engine operating speed, rubbing between the rotor and stator is easily triggered under alternating loads from temperature, centrifugal, and aerodynamic fields, potentially leading to casing wear, blade cracks, blade tip loss, and even engine combustion failures. Therefore, establishing a refined rubbing force model is of great significance.

[0004] Establishing accurate rubbing models has always been a research focus. Rubbing models include contact models, penalty function models, Lagrange multiplier models, augmented Lagrange models, piecewise linear force models, Kelvin-Voigt models, piecewise nonlinear force models, Lankarani-Nikravesh models, and pulse force models. For example, "Padovan J, Choy F K. Nonlinear dynamics of rotor / blade / casing rub interactions. 1987, 109(4): 527-53" simplifies the blade as a cantilever beam with a uniform cross section, considers the first-order bending mode of the blade, and derives a rubbing characterization model (normal rubbing force model) for blade-casing contact. Building upon the Padovan model, “Jiang J, Ahrens J, Ulbrich H, et al. A contact model of a rotating rubbing blade. Proceeding of the 5th IFToMM, Darmstadt, 1998: 478-489” further refined the blade-casing (BC) characterization equations by introducing centrifugal effects and incorporating rotational speed as a parameter into the expression. Furthermore, “Ma H, Tai X, Han Q, et al. A revised model for rubbing between rotating blade and elastic casing. Journal of Sound and Vibration, 2015, 337: 301-320” not only considered variable cross-section blades and rotational effects but also incorporated the equivalent stiffness of the casing, making the force model applicable to different casing conditions."Kang Y, Cao S, Gao T, et al. Development and validation of a rotating blade-casing rubbing model by considering the blade deformation and abradable coating. Journal of Sound and Vibration, 2023, 563: 117853" modified the model to account for the casing's flexibility. "Tang T, Wang Y, Wang S, et al. Investigation on dynamic rubbing characteristics of a bladed rotor system with multi-mode rubbing fault. Journal of Sound and Vibration, 2025, 596: 118790" developed a blade-coated casing rubbing model that considers the effects of coating and blade deformation. In addition, "Legrand M, Pierre C, Peseux B. Structural modal interaction of a four degree-of-freedom bladed disk and casing model. 2010, 5(4): 13-41" considered shaft deformation and blade coupling effects to establish a rubbing force model and analyze coupled vibrations. "Legrand M, Batailly A, Magnain B, et al. Full three-dimensional investigation of structural contact interactions in turbomachines. Journal of Sound and Vibration, 2012, 331(11): 2578-2601" simulated the blade-casing (BC) contact process using a developed contact algorithm.The paper "Cao D, Yang Y, Chen H, et al. A novel contact force model for the impact analysis of structures with coating and its experimental verification. Mechanical Systems and Signal Processing, 2016, 70: 1056-1072" also established a contact model for the blade-casing (BC) system and analyzed coating wear. Although the contact model may have higher accuracy, its complexity leads to lower computational efficiency. The paper "Yang Y, Cao D, Xu Y. Rubbing analysis of anonlinear rotor system with surface coatings. International Journal of Non-Linear Mechanics, 2016, 84: 105-115" proposed a novel blade-coated casing rubbing model and experimentally verified the proposed model. "Wu Z, Zhao L, Yan H, et al. Multi-blade rubbing characteristics of the shaft-disk-blade-casing system with large rotation. Applied Mathematics and Mechanics, 2024, 45(1): 111-136" considers large rotation to establish a rubbing model in a shaft-disk-blade system and explores the vibration characteristics caused by rubbing. Among these rubbing force models, many researchers have adopted the normal rubbing force model due to its adaptability. However, most current normal contact force models do not consider thermal effects.

[0005] The main drawbacks of existing collision and friction models can be summarized in the following two points:

[0006] Ignoring the coupling effect of thermal effects and wear: Most existing normal friction force models (such as piecewise linear models, Kelvin-Voigt models, etc.) do not consider the transient temperature rise (up to 600℃) caused by frictional heat generation and the resulting thermal deformation (such as blade tip expansion), and generally simplify wear as a temperature-independent process. Current models are mostly based on experiments or finite element analysis to handle thermal effects, lacking analytical expressions; wear models (such as the Archard formula) do not incorporate the dynamic influence of temperature on material properties (such as hardness, yield strength), leading to prediction errors in contact force and clearance changes.

[0007] Insufficient modeling of coating characteristics: Existing models do not adequately characterize the equivalent stiffness effect of the coating on the casing. Traditional models assume the casing is rigid or only consider the stiffness of the matrix, neglecting the elastoplastic behavior of the coating (such as compression and cutting) and the dynamic weakening effect of its wear on the system stiffness. This causes the contact force prediction to deviate from the measured results under high penetration (e.g., linear models cannot reproduce the power-law growth trend).

[0008] To address the aforementioned issues, this invention proposes a method for calculating the normal contact force between the blade and the coating casing, considering the thermal-wear effect, significantly improving the accuracy and completeness of the physical coupling modeling. First, based on one-dimensional unsteady-state heat conduction theory, analytical expressions for the temperature rise and linear thermal expansion of both the blade and casing are established, considering the instantaneous temperature changes during each contact process. Second, physical modeling of various wear mechanisms (such as cutting wear, adhesive wear, and microcrack wear) is introduced to effectively capture the influence of coating gap changes on the contact force. Furthermore, the model integrates the composite stiffness characteristics of the casing coating and the supporting structure, enabling the contact force calculation to cover the response characteristics under different structural stiffness conditions. Through finite element comparison and experimental verification, the calculation method of this invention demonstrates high accuracy in predicting thermal deformation, contact force, and wear trends, providing strong support for subsequent contact fault diagnosis and structural design. Summary of the Invention

[0009] To address the aforementioned problems in existing technologies, this invention provides a method for calculating the normal friction force of a blade-coated casing considering the thermal-wear effect. The method includes: calculating the intrusion amount of the blade-coated casing under the combined effects of temperature rise due to frictional heat, blade deformation, and gap changes caused by coating wear; considering the influence of the support stiffness and structural stiffness of the flexible coated casing on the normal friction force, proposing a calculation model for the normal friction force of the blade-coated casing under different intrusion amounts and verifying it based on experimental testing; furthermore, substituting the influence of the thermal-wear effect on the gap into the normal friction force calculation model, ultimately constructing a method for calculating the normal friction force of a blade-coated casing based on the thermal-wear effect.

[0010] The technical solution of this invention is as follows: A method for calculating the normal friction force of a blade-coated casing considering thermal-wear effects, constructing a blade-coated casing normal friction force model that considers thermal effects, wear effects, and coating stiffness effects; based on one-dimensional unsteady-state heat conduction theory and the principle of linear thermal expansion, analytical models of the temperature field and thermal deformation of the blade and casing during the instantaneous friction process are established respectively, and the thermal deformation Δα of the blade is calculated. b Thermal deformation Δα of the casing cThis allows us to obtain the change in the blade tip-coating gap Δh2 and the dynamic influence of frictional heat on the rubbing force. By comprehensively considering multiple wear mechanisms, a dynamic model of the coating gap change is established to obtain the feedback of wear effect on the rubbing process. The elastic characteristics of the coating, casing substrate and support system are integrated into an equivalent stiffness parameter, and the influence of thermal effect and wear effect on the gap is considered to obtain the rubbing force under the influence of composite stiffness.

[0011] The specific process for establishing the analytical model of temperature field and thermal deformation of the blade and casing during instantaneous friction is as follows:

[0012] Treating the heat conduction of the blade as a one-dimensional unsteady-state heat conduction problem, according to the law of energy conservation and Fourier's law, the heat diffusion equation near the friction interface on the coating side is:

[0013]

[0014] In the formula, a c Let a be the thermal diffusivity of the coating. c =K c / ρc、K c ρ is thermal conductivity, measured in W / (m·K); ρ is density, measured in kg / m³. 3 c is the specific heat capacity, with units of J / (kg·K); for the casing, the boundary conditions of equation (1) are:

[0015]

[0016] In the formula, q c T represents the heat flux density flowing into the coating from the friction interface on the coating side. c0 The initial temperature of the coating during a single friction process;

[0017] The solution to equation (1) is the temperature distribution function at distance x and time t in the coating, and its expression is:

[0018]

[0019] In the formula, t cp It is the duration of a single rubbing process, that is, the contact time between the blade and the coating during a single rubbing; It is a Gaussian complementary error function;

[0020] Similarly, the spatiotemporal distribution function of blade temperature can be obtained as follows:

[0021]

[0022] In the formula, a b K is the thermal diffusivity of the blade. b Let q be the thermal conductivity of the blade. bT represents the heat flux density flowing into the blade from the friction interface on the coating side. b0 Let t be the initial temperature of the blade during one friction process. bp The duration of a single friction process for the blade;

[0023] During the rubbing process, all friction is converted into heat, and the frictional heat source is Q. f for,

[0024] Q f =F f ·(R d +L b )·ω d (5)

[0025] In the formula, F f For friction, R d Let L be the radius of the disk. b ω is the length of the blade. d The rotational speed of the disk;

[0026] During the friction process, the blade and the coating share the total frictional heat source. The higher the thermal conductivity, the greater the heat flux density shared. Therefore, the heat flux density q of the blade... b and the heat flux density q of the casing c The expression is,

[0027]

[0028] In the formula, L s a is the length of the coating scratch. s A represents the width of the scrape, i.e., the width of the blade. b This represents the cross-sectional area of ​​the blade.

[0029] Considering the bending of the blade due to the normal force of friction, the radius of rotation of the blade tip is .

[0030] R b =R d +L b (7)

[0031] For a standard circular casing, its parametric equation is:

[0032] X = R c cosθ+d,Y=R c sinθ (8)

[0033] For an elliptical casing, the parametric equations for the casing are as follows:

[0034] X = acosθ + d, Y = bsinθ (9)

[0035] When the blade contacts the circular casing, the contact phase angle θ is calculated.c Its expression is,

[0036] θ=arccos[(R d +L b -δ) / R c (10)

[0037] When the blade contacts the elliptical casing, the contact phase angle θ is calculated. c Its expression is,

[0038] θ c =arccos[(R d +L b -δ) / a] (11)

[0039] Therefore, the scratch distance L of the coating s for,

[0040] L s =2(R) d +L b sinθ c (12)

[0041] Finally, the temperature distribution function of the blade with respect to distance x and time t is:

[0042]

[0043] For the blades, the contact time is the heating time t. bp The expression is,

[0044] t bp =L s / [(R d +L b )·ω d (14)

[0045] During one rotation cycle, the blade experiences only one rubbing failure, lasting for t seconds. bp The remaining time is spent undergoing convective heat transfer and cooling with the air; therefore, the convective heat transfer time of the blades is the cooling time t. bd for,

[0046] t bd =2π / ω d -t bp (15)

[0047] Based on the concept of linear thermal expansion, the elongation of a rectangular blade is calculated; linear thermal expansion describes the change in length of an object as temperature changes, and its formula is:

[0048]

[0049] In the formula, α b It is the linear thermal expansion coefficient of the blade, ΔT b It is the change in leaf temperature;

[0050] During the contact process, the heat has not yet dissipated to the entire blade; the temperature rise is mainly concentrated at the contact point, and the blade's temperature rise ΔT b and leaf elongation Δα b for:

[0051]

[0052] Using the integral formula of the error function:

[0053]

[0054] In the formula, erf(·) is the Gaussian error function;

[0055] Therefore, the leaf elongation Δα b The final expression is:

[0056]

[0057] Before the next blade makes contact, the heat generated by the contact between the previous blade and the next blade has already been lost through convective heat transfer, so there is no heat accumulation. Therefore, the temperature rise of the casing ΔT is... c and the deformation of the casing △α c The calculation expression is:

[0058]

[0059] △α c =η c ·α c ·△T c ·h c (twenty two)

[0060] In the formula, η c α c and h c These are the deformation correction factor, the material thermal expansion coefficient, and the casing thickness, all based on finite element analysis.

[0061] The formula for calculating the change in the blade tip-coating gap, Δh2, is as follows:

[0062] △h2=△α b +△α c (twenty three).

[0063] The dynamic model for the change in coating gap is constructed as follows:

[0064] The force F generated by cutting-type wear ma and wear amount Δh ma;

[0065] During the rubbing process, the coating mainly undergoes elastic deformation, resulting in adhesive wear and abrasive wear. For adhesive wear, a force F is generated. adh and wear amount Δh adh ;

[0066] For abrasive wear, a force F is generated. ab and wear amount Δh ab ;

[0067] For wear caused by microfractures, a force F is generated. mr and wear amount Δh mr ;

[0068] When the blades and the casing come into contact, in addition to coating wear, there is also the effect of coating plastic deformation on the gap. Plastic deformation is described by a plastic constitutive model.

[0069] Total strain ∈ decomposes into elastic strain ∈ e and plastic strain ∈ p :

[0070] ∈=∈ e +∈ p (twenty four)

[0071] For the elastic component, stress and strain have a linear relationship:

[0072] σ=E∈ e (25)

[0073] In the formula, E is the elastic modulus;

[0074] For the plastic part, a plastic constitutive model is used to describe it. The evolution of plastic strain is controlled by the hardening variable α, and the plastic stress satisfies the following relationship:

[0075] f(σ,α)=σ-(σ Y +Kα) (26)

[0076] In the formula, σ Y K is the yield strength, K is the plastic modulus, and α is the hardening variable;

[0077] Assume that during the plastic flow stage, the increment of plastic strain Δ∈ p The increment Δα of the hardening variable satisfies the following relationship:

[0078] △α=△∈ p (27)

[0079] The stress update formula is:

[0080] f = f trial -γ(E+K) (28)

[0081] In the formula, γ is the consistency parameter, and its expression is:

[0082] γ=f trial / (E+K) (29)

[0083] Stress σ, Plastic strain increment Δ∈ p The increment Δα of the hardening variable is updated as follows:

[0084] σ=σ trial -Eγ,△∈ p =γ, △α=γ (30)

[0085] The clearance variation of the coated casing is determined by both wear and plastic deformation, and its clearance variation expression is as follows:

[0086] △h1=△h adh +△h ab +p mr △h mr +p ma △h ma +∈ p F = F fr +F ab +p mr F mr +p ma F ma (31);

[0087] Among them, p ma p mr This represents the ratio of cutting to microfracture.

[0088] Force F ma and wear amount Δh ma The calculation expression is,

[0089]

[0090] In the formula, τ y Indicates shear yield strength, φ is shear angle, I e For the allowable elastic deformation, I is the intrusion amount, σ y E represents the yield strength. a e is the elastic modulus a This refers to the coating thickness.

[0091] Force F adh and wear amount Δh adh The calculation expression is,

[0092] F adh =μF n ,△h adh =kadh e b σ n / H co (33)

[0093] In the formula, μ is the coefficient of friction, and F n It is the normal contact force, k adh It is the Arcard wear coefficient for adhesive wear, e b It is the blade thickness, σ n It is the normal contact pressure, H co Indicates the coating hardness.

[0094] Force F ab and wear amount Δh ab The calculation expression is,

[0095] F ab =η pa F pp ,△h ab =k ab e b σ n / H co (34)

[0096] In the formula, η pa F is the number of particles. pp For the force exerted by a particle, k ab It is the Arcard wear coefficient for abrasive wear.

[0097] Force F mr and wear amount Δh mr The calculation expression is,

[0098] F mr =k mr γ,△h mr =l pa (35)

[0099] In the formula, k mr It is a factor that depends on the particle type, γ is the cohesive energy of the wearable material, l pa It refers to particle size.

[0100] The friction force is calculated as follows:

[0101] Based on the geometry of the torsion-type variable cross-section blade, the cross-sectional area and moment of inertia of the blade are written as:

[0102]

[0103] Among them, b t h t b represents the width and thickness of the leaf at the leaf tip.r h r The width and thickness of the leaf blade at the leaf base; A r and I r These are the cross-sectional area and moment of inertia at the leaf root, respectively.

[0104] Due to the quasi-static assumption, during the contact time Δt, the blade satisfies the following function relationship:

[0105] U e +U c =W (38)

[0106] Where W is the normal force F at the blade tip. n and tangential force F t The work done, U e U is the bending deformation energy of the blade. c U is the centrifugal potential energy under the action of centrifugal force on the blade. e and U c The expressions are as follows:

[0107]

[0108] In the formula, E is the elastic modulus of the blade, L is the blade length, v is the bending displacement function of the blade, ρ is the blade density, and R... d Let Ω be the radius of the blade disk, and Ω be the angular velocity of the blade.

[0109] In addition to potential energy, the contact force when the blade and the coated casing come into contact also does work W on the blade, including the normal contact force F. n and tangential friction force F t F t =μF n And μ is the coefficient of friction when the blade and the coated casing are in contact; therefore, the total work done by the normal force and the tangential force is:

[0110]

[0111] Among them, v t u is the bending displacement at the blade tip. t u is the radial displacement at the blade tip caused by bending deformation. t The expression is:

[0112]

[0113] Assuming the coated housing is a linear elastic body during blade-coated housing contact, the normal contact force on the coated housing is:

[0114] F n =k a u a =k ce uc =k s u cn (43)

[0115] Where, k c d represents the equivalent stiffness of the casing. c This refers to the radial displacement of the casing when the blade and the coated casing are in contact.

[0116] Due to the quasi-static assumption, the blade and casing maintain an instantaneous equilibrium state, which can be represented by the intrusion amount δ and the tip radial displacement u. t Translational displacement u of the casing along the normal direction of contact friction cn and elastic displacement u sn , where u sn Due to the elastic deformation of the casing u c and the elastic deformation of the coating u a Composition, written as the following expression:

[0117] δ=u t +u cn +u sn =u t +u cn +u c +u a (44)

[0118] The formula for the blade deflection curve is:

[0119]

[0120] Substituting equation (46) into equations (39) and (40), and combining it with equation (42), the bending displacement v at the blade tip is finally obtained through the mechanical energy conservation equation. t for:

[0121]

[0122] In the formula, Furthermore, the expressions for A1 and A2 are:

[0123]

[0124] Based on equation (46), combining equations (41) and (42), we obtain u t Then u t Substituting the expression into equation (44):

[0125]

[0126] In the formula, k a For the stiffness of the coating, k ce For the stiffness of the casing base, k s Provides elastic support stiffness for the casing;

[0127] After omitting the higher-order terms in equation (47), we get:

[0128]

[0129] Finally, we get F. n The parsing expression:

[0130]

[0131] In the formula,

[0132] After considering wear and thermal effects, equation (50) becomes:

[0133]

[0134] In the formula, Δh1 is the change in clearance caused by wear, and Δh2 is the change in clearance caused by thermal effects.

[0135] The beneficial effects of this invention are as follows: Compared with existing friction force models, the friction force calculation method proposed in this invention has significant advantages. For the first time, it systematically couples and models thermal effects, wear effects, and coating structural stiffness, breaking through the limitations of traditional models that only consider contact force or some physical factors. This method can not only accurately describe the thermal expansion deformation of the blades and casing caused by frictional heat, but also dynamically reflect the influence of coating wear on contact gap and friction force. Simultaneously, it introduces the regulating effect of multi-source stiffness (coating, substrate, support) on contact response, significantly improving the adaptability and prediction accuracy of the friction force calculation method to actual complex working conditions. The model has a clear physical basis in its theoretical derivation and demonstrates good accuracy and consistency in experimental verification, possessing strong engineering application value. Attached Figure Description

[0136] Figure 1 The following are schematic diagrams of the impact: (a) shows the impact between the blade and the casing, and (b) shows the area of ​​the coating being scratched.

[0137] Figure 2 For plastic deformation analysis: (a) is the cross-sectional analysis of the blade-coated casing, and (b) is the elastoplastic constitutive model;

[0138] Figure 3 This represents the ratio between cutting and micro-fracture; the solid line represents the cutting ratio, and the dashed line represents the micro-fracture ratio.

[0139] Figure 4 A schematic diagram of the friction between a single blade and an elastically coated casing;

[0140] Figure 5The friction force of the casing with a 3mm coating at different speeds and intrusion amounts: (a) 900 r / min, (b) 1100 r / min, (c) 1200 r / min. Detailed Implementation

[0141] Thermal analysis includes three modes of heat transfer: conduction, convection, and radiation. Conduction refers to the transfer of internal energy down a temperature gradient between two objects in complete contact or between different parts of an object. Convection is the process of heat transfer between a solid surface and a fluid in contact with it, and includes natural convection and forced convection. Radiation is the process of heat exchange where electromagnetic energy emitted by one object is absorbed by another object and converted into heat energy; the amount of heat radiated by a high-temperature object per unit time increases with its temperature. Both conduction and convection require a heat transfer medium, while radiation does not, and its radiation efficiency is highest in a vacuum.

[0142] In this calculation method, only heat conduction and heat convection are considered while heat radiation is ignored. During the contact process, the heat conduction process of frictional heat is mainly considered, which causes the blade temperature to rise; during the non-contact process, the heat convection process between the blade and the air is mainly considered, which causes the blade temperature to drop.

[0143] Treating the heat conduction of the blade as a one-dimensional unsteady-state heat conduction problem, according to the law of energy conservation and Fourier's law, the heat diffusion equation near the friction interface on the coating side is:

[0144]

[0145] In the formula, a c (a c =K c / ρc) is the thermal diffusivity of the coating, K c (W / (m·K)) is thermal conductivity, ρ(kg / m 3 ) is density, and c (J / (kg·K)) is specific heat capacity. For the casing, the boundary condition of equation (1) is:

[0146]

[0147] In the formula, q c T represents the heat flux density flowing into the coating from the friction interface. c0 The initial temperature of the coating during a single friction process.

[0148] The solution to equation (1) is the temperature distribution function at distance x and time t in the coating, and its expression is:

[0149]

[0150] In the formula, t cp It is the duration of a single rubbing process, that is, the contact time between the blade and the coating during a single rubbing. The Gaussian complementary error function is calculated using MATLAB's erfc function.

[0151] Similarly, the spatiotemporal distribution function of blade temperature can be obtained as follows:

[0152]

[0153] In the formula, a b K is the thermal diffusivity of the blade. b q is the thermal conductivity of the blade. b T represents the heat flux density flowing into the blade from the friction interface on the coating side. b0 Let t be the initial temperature of the blade during one friction process. bp This represents the duration of a single friction cycle for the blade.

[0154] During the rubbing process, all friction is converted into heat, and the frictional heat source is Q. f for,

[0155] Q f =F f ·(R d +L b )·ω d (5)

[0156] In the formula, F f For friction, R d Let L be the radius of the disk. b ω is the length of the blade. d This represents the rotational speed of the disk.

[0157] During the friction process, the blade and the coating share the total frictional heat source. The higher the thermal conductivity, the greater the heat flux density shared. Therefore, the heat flux density q of the blade... b and the heat flux density q of the casing c The expression is,

[0158]

[0159] In the formula, L s a is the length of the coating scratch. s The width of the scrape, i.e., the width of the blade, such as... Figure 1 As shown. A b This represents the cross-sectional area of ​​the blade.

[0160] Considering the bending of the blade due to the normal force of friction, the radius of rotation of the blade tip is .

[0161] R b =Rd +L b (7)

[0162] For a standard circular casing, its parametric equation is:

[0163] X = R c cosθ+d,Y=R c sinθ (8)

[0164] For an elliptical casing, the parametric equations for the casing are as follows:

[0165] X = acosθ + d, Y = bsinθ (9)

[0166] When the blade contacts the circular casing, the contact phase angle θ can be calculated. c Its expression is,

[0167] θ=arccos[(R d +L b -δ) / R c (10)

[0168] When the blade contacts the elliptical casing, the contact phase angle θ can be calculated. c Its expression is,

[0169] θ c =arccos[(R d +L b -δ) / a] (11)

[0170] Therefore, the scratch distance L of the coating s for,

[0171] L s =2(R) d +L b sinθ c (12)

[0172] Finally, the temperature distribution function of the blade with respect to distance x and time t is:

[0173]

[0174] For the blade, the contact time (i.e., the heating time) t bp The expression is,

[0175] t bp =L s / [(R d +L b )·ω d (14)

[0176] During one rotation cycle, the blade experiences only one rubbing failure, lasting for t seconds. bp The blades spend the remaining time undergoing convective heat transfer and cooling with the air. Therefore, the convective heat transfer time (i.e., cooling time) of the blades is t. bd for,

[0177] t bd =2π / ω d -t bp (15)

[0178] Based on the concept of linear thermal expansion, the elongation of a rectangular blade is calculated. Linear thermal expansion describes the change in length of an object as temperature changes, and its formula is:

[0179]

[0180] In the formula, α b It is the linear thermal expansion coefficient of the blade, ΔT b It is the change in leaf temperature.

[0181] During the contact process, the heat has not yet dissipated to the entire blade; the temperature rise is concentrated near the contact point (x=0). The temperature rise of the blade is ΔT. b and leaf elongation Δα b It can be approximated as:

[0182]

[0183] Using the integral formula of the error function:

[0184]

[0185] In the formula, erf(·) is the Gaussian error function.

[0186] Therefore, the leaf elongation Δα b The final expression is:

[0187]

[0188] Since the blades may contact multiple blades, if the heat generated by the contact between the previous blade and the casing has already been lost through convective heat transfer before the next blade contacts it, there is no heat accumulation. Therefore, each blade-casing contact can concentrate the temperature rise near the contact point (x=0). In this case, the normal elongation of the casing can be approximately calculated using the blade elongation formula. Because casing deformation becomes difficult to calculate analytically when heat accumulation occurs, only calculations are performed for cases without heat accumulation. The casing temperature rise ΔT c and the deformation of the casing △α c The calculation expression is:

[0189]

[0190] The formula for calculating the change in the blade tip-coating gap, Δh2, is as follows:

[0191] △h2=△α b +△α c (twenty three)

[0192] Besides thermal effects, wear effects also exist during the blade-coated casing contact process. Coating wear leads to an increase in the clearance between the blade and the casing, thereby reducing the contact force. Coatings are widely used in the compressor and turbine stages of modern aircraft engines to reduce the operating clearance between the blade tip and the casing, thus improving energy efficiency. Coating materials are diverse, including ceramics, aluminum-silicon alloys, and nickel-graphite composites. Contact between the blade tip and the coating can lead to excessive blade vibration and even damage to the blade's structural integrity due to fatigue effects. Therefore, a deep understanding and accurate modeling of this interaction is crucial for predicting potential hazardous events. Coating wear involves various complex physical phenomena, such as microfracture, material transfer, ploughing wear, elastic deformation, plastic deformation, and cutting.

[0193] For cutting-type wear, the applied force F ma and wear amount Δh ma The calculation expression is,

[0194]

[0195] In the formula, τ y Indicates shear yield strength, φ is shear angle, I e Let I represent the allowable elastic deformation and the intrusion amount. σ y E represents the yield strength. a e is the elastic modulus a This refers to the coating thickness.

[0196] Under low intrusion conditions, the coating primarily undergoes elastic deformation, resulting in adhesive wear and abrasive wear. For adhesive wear, the applied force F... adh and wear amount Δh adh The calculation expression is,

[0197] F adh =μF n ,△h adh =k adh e b σ n / H co (25)

[0198] In the formula, μ is the coefficient of friction, and F n It is the normal contact force, k adh It is the Arcard wear coefficient for adhesive wear, eb It is the blade thickness, σ n It is the normal contact pressure, H co Indicates the coating hardness.

[0199] For abrasive wear, the force F ab and wear amount Δh ab The calculation expression is,

[0200] F ab =η pa F pp ,△h ab =k ab e b σ n / H co (26)

[0201] In the formula, η pa F is the number of particles. pp For the force exerted by a particle, k ab It is the Arcard wear coefficient for abrasive wear.

[0202] For wear caused by microfractures, the applied force F mr and wear amount Δh mr The calculation expression is as follows

[0203] F mr =k mr γ,△h mr =l pa (27)

[0204] In the formula, k mr It is a factor that depends on the particle type, γ is the cohesive energy of the wearable material, l pa It refers to particle size.

[0205] When the blades and casing come into contact, in addition to coating wear, plastic deformation of the coating also occurs. This plastic deformation is described by a plastic constitutive model. Therefore, plastic strain is the primary measure of plastic deformation. When considering elastoplastic deformation, the coating is often simplified as a rod, with each rod independent of the others, such as... Figure 2 As shown.

[0206] Total strain ∈ decomposes into elastic strain ∈ e and plastic strain ∈ p :

[0207] ∈=∈ e +∈ p (28)

[0208] For the elastic component, stress and strain have a linear relationship:

[0209] σ=E∈e (29)

[0210] In the formula, E is the elastic modulus.

[0211] For the plastic part, a plastic constitutive model is used to describe it. The evolution of plastic strain is controlled by the hardening variable α, and the plastic stress satisfies the following relationship:

[0212] f(σ,α)=σ-(σ Y +Kα) (30)

[0213] In the formula, σ Y Let K be the yield strength, K be the plastic modulus, and α be the hardening variable.

[0214] Assume that during the plastic flow stage, the increment of plastic strain Δ∈p and the increment of hardening variable Δα satisfy the following relationship:

[0215] △α=△∈ p (31)

[0216] The test stress is a temporary, hypothetical stress value designed to determine whether the material has undergone plastic flow by checking if it satisfies the yield criterion. If the yield criterion is not met, it is corrected using a regression mapping algorithm to ensure that the stress state of the material correctly reflects its yielding behavior. If the test stress σ trial If the yield condition is met, the current state is directly adopted. If not, the stress is projected onto the yield surface using a regression mapping algorithm. The stress update formula is:

[0217] f = f trial -γ(E+K) (32)

[0218] In the formula, γ is the consistency parameter, and its expression is:

[0219] γ=f trial / (E+K) (33)

[0220] Stress σ, Plastic strain increment Δ∈ p The hardening variable Δα is updated as follows:

[0221] σ=σ trial -Eγ,△∈ p =γ, △α=γ (34)

[0222] Ultimately, the contact force F c It is calculated by balancing with internal forces:

[0223]

[0224] Plastic deformation causes changes in the material's profile, and this profile update is achieved through accumulated plastic strain. Contact force Fc This will induce plastic deformation, which in turn will affect the change in wear profile (morphology). The method of this invention accumulates plastic strain ∈ p This is used to update the wear profile. At each time step, the calculated plastic strain causes a change in the coating surface profile, thus forming the actual wear morphology. This process is simulated through the interaction of plastic strain and contact forces. In short, contact forces affect the coating morphology through plastic deformation, while accumulated plastic strain ∈ p Together with contact force, it determines the wear profile of the coating surface.

[0225] During the contact process between the blade and the coated casing, cutting and microfracture occur first under high frictional forces, followed by adhesive wear, abrasive wear, and plastic deformation. Assuming that each wear mechanism is independent, the wear amount of each mechanism can be calculated independently. Therefore, we first calculate the contact force and wear amount caused by cutting and microfracture, update the blade-coated casing clearance, calculate the total contact force between the blade and the coated casing, further calculate adhesive wear, abrasive wear, and plastic deformation, and finally output the frictional force and the changing clearance value. It is worth noting that the ratio p between cutting and microfracture... ma and p mr like Figure 3 As shown.

[0226] Therefore, the clearance change of the coated casing is determined by both wear and plastic deformation, and its clearance change expression and force expression are as follows:

[0227] △h1=△h adh +△h ab +p mr △h mr +p ma △h ma +∈ p F = F fr +F ab +p mr F mr +p ma F ma (36)

[0228] Blade-casing collision can be viewed as a contact between a flexible body and a dense elastic body, or between two flexible bodies. The collision force depends not only on the bending deformation of the blade but also on the elastic deformation of the casing and its coating. As an abrasive material, the casing coating may undergo localized compression or abrasion deformation during the collision, thus affecting the transmission of the overall contact force. Therefore, when analyzing blade-casing collisions, in addition to considering the bending deformation of the blade, it is also necessary to consider the elastic and plastic deformation of the casing coating during the contact collision process, as well as the influence of the coating's wear characteristics on the collision force. In particular, the abrasiveness of the coating prevents blade-casing substrate rubbing, thus preventing serious rubbing accidents.

[0229] Based on the geometry of the torsion-type variable cross-section blade, the cross-sectional area and moment of inertia of the blade are written as:

[0230]

[0231] Among them, b t h t b represents the width and thickness of the leaf at the leaf tip. r h r The width and thickness of the leaf blade at the leaf base; A r and I r These are the cross-sectional area and moment of inertia at the leaf root, respectively.

[0232] Due to the quasi-static assumption, the blades should satisfy the function-energy relation during the contact time Δt:

[0233] U e +U c =W (39)

[0234] Where W is the normal force F at the blade tip. n and tangential force F t The work done, U e U is the bending deformation energy of the blade. c U is the centrifugal potential energy under the action of centrifugal force on the blade. e and U c The expressions are as follows:

[0235]

[0236] In the formula, E is the elastic modulus of the blade, L is the blade length, v is the bending displacement function of the blade, ρ is the blade density, and R... d Let Ω be the radius of the blade disk, and Ω be the angular velocity of the blade.

[0237] In addition to potential energy, the contact force when the blade and the coated casing come into contact also does work W on the blade, including the normal contact force F. n and tangential friction force F t (Ft =μF n And μ is the coefficient of friction when the blade and the coated casing are in contact. Therefore, the total work done by the normal and tangential forces is:

[0238]

[0239] Among them, v t u is the bending displacement at the blade tip. t u is the radial displacement at the blade tip caused by bending deformation. t The expression is:

[0240]

[0241] Assuming the coated casing is a linear elastic body during blade-coated casing contact, the normal contact force on the coated casing is:

[0242] F n =k a u a =k ce u c =k s u cn (44)

[0243] Where, k c d represents the equivalent stiffness of the casing. c This refers to the radial displacement of the casing when the blade and the coated casing are in contact.

[0244] Due to the quasi-static assumption, the blade and casing maintain an instantaneous equilibrium state, which can be represented by the intrusion amount δ and the tip radial displacement u. t Translational displacement u of the casing along the normal direction of contact friction cn and elastic displacement u sn , where u sn Due to the elastic deformation of the casing u c and the elastic deformation of the coating u a Composition, such as Figure 4 As shown, it can be written as the following expression:

[0245] δ=u t +u cn +u sn =u t +u cn +u c +u a (45)

[0246] The formula for the blade deflection curve is:

[0247]

[0248] Substituting equation (46) into equations (40) and (41), and combining it with equation (43), the bending displacement v at the blade tip is finally obtained through the mechanical energy conservation equation. t for:

[0249]

[0250] In the formula, Furthermore, the expressions for A1 and A2 are:

[0251]

[0252] Based on equation (47), by combining equations (42) and (43), we can obtain u t Then u t Substituting the expression into equation (45):

[0253]

[0254] In the formula, k a For the stiffness of the coating, k ce For the stiffness of the casing base, k s This provides elastic support stiffness for the casing.

[0255] After omitting the higher-order terms in equation (48), we get:

[0256]

[0257] Finally, we get F. n The parsing expression:

[0258]

[0259] In the formula,

[0260] After considering wear and thermal effects, equation (51) is rewritten as follows;

[0261]

[0262] In the formula, Δh1 is the change in clearance caused by wear, and Δh2 is the change in clearance caused by thermal effects.

[0263] To measure the blade-coated casing normal impact force under different penetration levels, a dedicated test bench was constructed, including a drive system, control system, rotor system, stator system, and measurement and acquisition system. An electric motor drives the rotor system, with its speed regulated by frequency converter control, reaching a maximum speed of 3000 r / min, which will not exceed the first critical speed. The rotor system consists of a rotating shaft, four blades, and a disk. The shaft is supported at both ends by two bearing seats, which are bolted to the base. Bolt holes are drilled at the four corners of the casing test piece. Casing test pieces of different thicknesses are bolted to the casing mounting frame for quick replacement of the outer casing test piece. Gaskets are used to isolate the casing and mounting frame to complete the impact force test. The feed system mainly consists of a screw mechanism, with the feed speed controlled by a servo motor, achieving an accuracy of 1 micrometer. It quantitatively controls the feed amount of the coated casing, completing the impact force test under different penetration levels. The blades are fixed to the disk by pressing tenons and grooves onto it. The contact force was measured using a Kistler 9367C sensor, primarily used to acquire the normal contact force and tangential friction force between the blade and the coated casing. This force sensor, manufactured by Kistler, is a piezoelectric dynamic force sensor capable of converting force signals into charge signals. The sensitivity in the x-direction is 7.676 pC / N, in the y-direction it is 7.647 pC / N, and in the z-direction it is 3.874 pC / N. The measurement accuracies in the three directions are 0.1%, 1.0%, and 0.1%, respectively, with a range of 60 kN. In addition, an eddy current displacement sensor was used to measure the shaft vibration displacement, and three-dimensional accelerometers were used to measure the acceleration response of the bearing housing and casing, respectively. The acceleration response was acquired by an LMS data acquisition system, while the contact force and displacement signals were acquired using a DH5956 acquisition system. The experimental results and the results calculated using this method at different speeds are shown below. Figure 5 As shown, since the friction force changes linearly, the slope of the straight line intuitively reflects the accuracy of the model. When the rotational speed is 1100 r / min, the error between the slope of the data-fitted curve and the slope of the theoretical model is the largest, and the maximum error is 7.9%, which meets the engineering requirements.

Claims

1. A method for calculating the normal friction force of a blade-coated casing considering thermal-wear effects, characterized in that, Construct a blade-coated casing normal friction force model that considers the effects of thermal effects, wear effects, and coating stiffness; Based on the one-dimensional unsteady-state heat conduction theory and the principle of linear thermal expansion, analytical models of the temperature field and thermal deformation of the blade and casing during the instantaneous friction process are established respectively, and the thermal deformation Δα of the blade is calculated. b Thermal deformation Δα of the casing c This allows us to obtain the change in the blade tip-coating gap Δh2 and the dynamic influence of frictional heat on the rubbing force. By comprehensively considering multiple wear mechanisms, a dynamic model of the coating gap change is established to obtain the feedback of wear effect on the rubbing process. The elastic characteristics of the coating, casing substrate and support system are integrated into an equivalent stiffness parameter, and the influence of thermal effect and wear effect on the gap is considered to obtain the rubbing force under the influence of composite stiffness.

2. The method for calculating the normal friction force of the blade-coated casing considering the thermal-wear effect according to claim 1, characterized in that, The specific process for establishing the analytical model of temperature field and thermal deformation of the blade and casing during instantaneous friction is as follows: Treating the heat conduction of the blade as a one-dimensional unsteady-state heat conduction problem, according to the law of energy conservation and Fourier's law, the heat diffusion equation near the friction interface on the coating side is: In the formula, a c Let a be the thermal diffusivity of the coating. c =K c / ρc、K c ρ is thermal conductivity, measured in W / (m·K); ρ is density, measured in kg / m³. 3 c is the specific heat capacity, with units of J / (kg·K); for the casing, the boundary conditions of equation (1) are: In the formula, q c T represents the heat flux density flowing into the coating from the friction interface on the coating side. c0 The initial temperature of the coating during a single friction process; The solution to equation (1) is the temperature distribution function at distance x and time t in the coating, and its expression is: In the formula, t cp It is the duration of a single rubbing process, that is, the contact time between the blade and the coating during a single rubbing; It is a Gaussian complementary error function; Similarly, the spatiotemporal distribution function of blade temperature can be obtained as follows: In the formula, a b K is the thermal diffusivity of the blade. b q is the thermal conductivity of the blade. b T represents the heat flux density flowing into the blade from the friction interface on the coating side. b0 Let t be the initial temperature of the blade during one friction process. bp The duration of a single friction process for the blade; During the rubbing process, all friction is converted into heat, and the frictional heat source is Q. f for, Q f =F f ·(R d +L b )·ω d (5) In the formula, F f For friction, R d Let L be the radius of the disk. b ω is the length of the blade. d The rotational speed of the disk; During the friction process, the blade and the coating share the total frictional heat source. The higher the thermal conductivity, the greater the heat flux density shared. Therefore, the heat flux density q of the blade... b and the heat flux density q of the casing c The expression is, In the formula, L s a is the length of the coating scratch. s A represents the width of the scrape, i.e., the width of the blade. b This represents the cross-sectional area of ​​the blade. Considering the bending of the blade due to the normal force of friction, the radius of rotation of the blade tip is . R b =R d +L b (7) For a standard circular casing, its parametric equation is: X=R c cosθ+d,Y=R c sinθ (8) For an elliptical casing, the parametric equations for the casing are as follows: X = acosθ + d, Y = bsinθ (9) When the blade contacts the circular casing, the contact phase angle θ is calculated. c Its expression is, θ=arccos[(R d +L b -d) / R c ] (10) When the blade contacts the elliptical casing, the contact phase angle θ is calculated. c Its expression is, i c =arccos[(R d +L b -d) / a] (11) Therefore, the scratch distance L of the coating s for, L s =2(R d +L b )sinθ c (12) Finally, the temperature distribution function of the blade with respect to distance x and time t is: For the blades, the contact time is the heating time t. bp The expression is, t bp =L s / [(R d +L b )·ω d ] (14) During one rotation cycle, the blade experiences only one rubbing failure, lasting for t seconds. bp The remaining time is spent undergoing convective heat transfer and cooling with the air; therefore, the convective heat transfer time of the blades is the cooling time t. bd for, t bd =2π / ω d -t bp (15) Based on the concept of linear thermal expansion, the elongation of a rectangular blade is calculated; linear thermal expansion describes the change in length of an object as temperature changes, and its formula is: In the formula, α b It is the linear thermal expansion coefficient of the blade, ΔT b It is the change in leaf temperature; During the contact process, the heat has not yet dissipated to the entire blade; the temperature rise is mainly concentrated at the contact point, and the blade's temperature rise ΔT b and leaf elongation Δα b for: Using the integral formula of the error function: In the formula, erf(·) is the Gaussian error function; Therefore, the leaf elongation Δα b The final expression is: Before the next blade makes contact, the heat generated by the contact between the previous blade and the next blade has already been lost through convective heat transfer, so there is no heat accumulation. Therefore, the temperature rise of the casing ΔT is... c and the deformation of the casing △α c The calculation expression is: △α c =the c ·a c ·△T c ·h c (22) In the formula, η c α c and h c These are the deformation correction factor, the material thermal expansion coefficient, and the casing thickness, all based on finite element analysis.

3. The method for calculating the normal friction force of the blade-coated casing considering the thermal-wear effect according to claim 2, characterized in that, The formula for calculating the change in the blade tip-coating gap, Δh2, is as follows: △h2＝△α b +△α c (23)。 4. The method for calculating the normal friction force of the blade-coated casing considering the thermal-wear effect according to claim 1, characterized in that, The dynamic model for the change in coating gap is constructed as follows: The force F generated by cutting-type wear ma and wear amount Δh ma ; During the rubbing process, the coating mainly undergoes elastic deformation, resulting in adhesive wear and abrasive wear. For adhesive wear, a force F is generated. adh and wear amount Δh adh ; For abrasive wear, a force F is generated. ab and wear amount Δh ab ; For wear caused by microfractures, a force F is generated. mr and wear amount Δh mr ; When the blades and the casing come into contact, in addition to coating wear, there is also the effect of coating plastic deformation on the gap. Plastic deformation is described by a plastic constitutive model. Total strain ∈ decomposes into elastic strain ∈ e and plastic strain ∈ p : ∈＝∈ e +∈ p (24) For the elastic component, stress and strain have a linear relationship: σ=E∈ e (25) In the formula, E is the elastic modulus; For the plastic part, a plastic constitutive model is used to describe it. The evolution of plastic strain is controlled by the hardening variable α, and the plastic stress satisfies the following relationship: f(σ,α)=σ-(σ Y +Ka) (26) In the formula, σ Y K is the yield strength, K is the plastic modulus, and α is the hardening variable; Assume that during the plastic flow stage, the increment of plastic strain Δ∈ p The increment Δα of the hardening variable satisfies the following relationship: △α＝△∈ p (27) The stress update formula is: f=f trial -γ(E+K) (28) In the formula, γ is the consistency parameter, and its expression is: γ=f trial / (E+K) (29) Stress σ, Plastic strain increment Δ∈ p The hardening variable increment Δα is updated as follows: s = s trial -Eγ,△∈ p =γ,△α=γ (30) The clearance variation of the coated casing is determined by both wear and plastic deformation, and its clearance variation expression is, Δh1=Δh adh +△h ab +p mr △h mr +p ma △h ma +∈ p F = F fr +F ab +p mr F mr +p ma F ma (31); Among them, p ma p mr This represents the ratio of cutting to microfracture.

5. The method for calculating the normal friction force of a blade-coated casing considering thermal-wear effects according to claim 4, characterized in that, Force F ma and wear amount Δh ma The calculation expression is, In the formula, τ y Indicates shear yield strength, φ is shear angle, I e For the allowable elastic deformation, I is the intrusion amount, σ y E represents the yield strength. a e is the elastic modulus a This refers to the coating thickness.

6. The method for calculating the normal friction force of a blade-coated casing considering thermal-wear effects according to claim 4, characterized in that, Force F adh and wear amount Δh adh The calculation expression is, F adh =μF n ,△h adh =k adh e b s n / H co (33) In the formula, μ is the coefficient of friction, and F n It is the normal contact force, k adh It is the Arcard wear coefficient for adhesive wear, e b It is the blade thickness, σ n It is the normal contact pressure, H co Indicates the coating hardness.

7. The method for calculating the normal friction force of a blade-coated casing considering thermal-wear effects according to claim 4, characterized in that, Force F ab and wear amount Δh ab The calculation expression is, F ab =the pa F pp ,△h ab =k ab e b s n / H co (34) In the formula, η pa F is the number of particles. pp For the force exerted by a particle, k ab It is the Arcard wear coefficient for abrasive wear.

8. The method for calculating the normal friction force of a blade-coated casing considering thermal-wear effects according to claim 4, characterized in that, Force F mr and wear amount Δh mr The calculation expression is, F mr =k mr γ,△h mr =l pa (35) In the formula, k mr It is a factor that depends on the particle type, γ is the cohesive energy of the wearable material, l pa It refers to particle size.

9. The method for calculating the normal friction force of a blade-coated casing considering thermal-wear effects according to claim 1, characterized in that, The friction force is calculated as follows: Based on the geometry of the torsion-type variable cross-section blade, the cross-sectional area and moment of inertia of the blade are written as: Among them, b t h t b represents the width and thickness of the leaf at the leaf tip. r h r The width and thickness of the leaf blade at the leaf base; A r and I r These are the cross-sectional area and moment of inertia at the leaf root, respectively. Due to the quasi-static assumption, during the contact time Δt, the blade satisfies the following function relationship: U e +U c =W (38) Where W is the normal force F at the blade tip. n and tangential force F t The work done, U e U is the bending deformation energy of the blade. c U is the centrifugal potential energy under the action of centrifugal force on the blade. e and U c The expressions are as follows: In the formula, E is the elastic modulus of the blade, L is the blade length, v is the bending displacement function of the blade, ρ is the blade density, and R... d Let Ω be the radius of the blade disk, and Ω be the angular velocity of the blade. In addition to potential energy, the contact force when the blade and the coated casing come into contact also does work W on the blade, including the normal contact force F. n and tangential friction force F t F t =μF n And μ is the coefficient of friction when the blade and the coated casing are in contact; therefore, the total work done by the normal force and the tangential force is: Among them, v t u is the bending displacement at the blade tip. t u is the radial displacement at the blade tip caused by bending deformation. t The expression is: Assuming the coated housing is a linear elastic body during blade-coated housing contact, the normal contact force on the coated housing is: F n =k a in a =k ce in c =k s in cn (43) Where, k c d represents the equivalent stiffness of the casing. c This refers to the radial displacement of the casing when the blade and the coated casing are in contact. Due to the quasi-static assumption, the blade and casing maintain an instantaneous equilibrium state, which can be represented by the intrusion amount δ and the tip radial displacement u. t Translational displacement u of the casing along the normal direction of contact friction cn and elastic displacement u sn , where u sn Due to the elastic deformation of the casing u c and the elastic deformation of the coating u a Composition, written as the following expression: δ=u t +in cn +in sn =in t +in cn +in c +in a (44) The formula for the blade deflection curve is: Substituting equation (46) into equations (39) and (40), and combining it with equation (42), the bending displacement v at the blade tip is finally obtained through the mechanical energy conservation equation. t for: In the formula, Furthermore, the expressions for A1 and A2 are: Based on equation (46), combining equations (41) and (42), we obtain u t Then u t Substituting the expression into equation (44): In the formula, k a For the stiffness of the coating, k ce For the stiffness of the casing base, k s Provides elastic support stiffness for the casing; After omitting the higher-order terms in equation (47), we get: Finally, we get F. n The parsing expression: Where, After considering wear and thermal effects, equation (50) becomes: In the formula, Δh1 is the change in clearance caused by wear, and Δh2 is the change in clearance caused by thermal effects.

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