Dynamic modeling and response calculation method of shrouded blade considering stick-slip motion

Abstract

The invention discloses a crowned blade dynamic modeling and response calculation method considering stick-slip motion, which comprises the following steps of: 1, establishing a dynamic model of a crowned blade system under two-dimensional rub-impact motion by combining first-order bending vibration of the crowned blade in the x direction and the y direction; 2, combining normal and tangential relative motion between adjacent blade crowns, establishing a viscous-sliding-separation boundary condition, and determining collision force and friction force at different rub-impact stages; and step 3, capturing key points of viscosity-sliding conversion between adjacent blade crowns by adopting a dichotomy method, and establishing a crown blade response calculation method considering the stick-sliding motion according to a differential equation numerical solution to predict vibration responses of the crown blade under different key parameters. According to the method, the dynamic modeling precision of the shrouded blade system and the response prediction precision of the shrouded blade are improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of dynamic modeling and vibration analysis of rotating machinery blades, and in particular is a method for dynamic modeling and response calculation of shrouded blades taking stick-slip motion into consideration. Background Art

[0002] Blades are important components of aircraft engines and are known as the "aorta" of aircraft engines. Their performance directly affects the stability and safety of aircraft engines. Due to the harsh working environment and complex loads of blades, blade failures occur frequently. High-cycle fatigue (HCF) failure caused by cyclic loads is the main cause of blade failure. Engineers often use friction dampers to reduce the damage caused by this factor to blades. The blade crown device is an effective dry friction damper for aircraft engine blades, which can reduce the blade vibration amplitude and vibration stress level, thereby effectively preventing high-cycle fatigue failure of blades. The use of the blade crown device for vibration reduction has the advantages of not being limited by temperature and having a simple structure.

[0003] However, during engine operation, the contact and motion states between adjacent blade shrouds are complex, and different motion states and contact states such as viscosity-slip-separation may occur between adjacent contact surfaces. The continuous change of the motion state of the indirect contact of the blade shrouds causes the contact force to have strong nonlinearity, making it extremely difficult to accurately analyze the friction dynamic characteristics of the shrouded blades.

[0004] Stick-slip motion is a very tricky problem in friction and dynamics, which has a great influence on the dynamic characteristics of the shrouded blade system. Therefore, considering different contact states (contact-separation) and different motion processes (stick-slip) under the combined action of impact and friction to model the shrouded blade system has positive significance for improving the accuracy of dynamic modeling and response prediction. Summary of the Invention

[0005] The present invention aims to provide a shrouded blade dynamics modeling and response calculation method considering stick-slip motion, thereby improving the accuracy of shrouded blade system dynamics modeling and shrouded blade response prediction.

[0006] The present invention is achieved through the following technical solutions:

[0007] A dynamic modeling and response calculation method for a shrouded blade considering stick-slip motion includes the following steps:

[0008] Step 1: Combine the first-order bending vibrations of the shrouded blade in the x-direction and the y-direction to establish a dynamic model of the shrouded blade system under two-dimensional rubbing motion;

[0009] Step 2: Combine the normal and tangential relative motions between adjacent blade crowns to establish the stick-slip-separation boundary conditions and determine the collision and friction forces at different rubbing stages. The specific process is as follows:

[0010] Step 2.1: Assuming that the collision between adjacent leaf crowns is elastic, the normal relative displacement between the left and right leaf crowns and the reference leaf crown is expressed as:

[0011]

[0012] Where: α is the rubbing contact angle; x1, x2, and x3 represent the displacements of the left leaf crown, reference leaf crown, and right leaf crown in the x-direction, respectively; y1, y2, and y3 represent the displacements of the left leaf crown, reference leaf crown, and right leaf crown in the y-direction, respectively.

[0013] The collision forces N1 and N2 between the left and right blade canopies and the reference blade canopy can be expressed as:

[0014]

[0015] Where: k c is the normal collision stiffness; Δ1 and Δ2 are the normal initial gaps between the reference blade canopy and the left and right blade canopies, respectively; N′1 and N′2 are the reaction forces of N1 and N2, respectively.

[0016] Step 2.2: Express the tangential relative displacement between the reference blade crown and the left blade crown and the right blade crown as:

[0017]

[0018] Where: t r1 is the tangential relative displacement between the reference blade crown and the left blade crown; t r2 is the tangential relative displacement between the reference blade crown and the right blade crown.

[0019] The tangential relative velocity between the reference blade crown and the left and right blade crowns is expressed as:

[0020]

[0021] Where: is the tangential relative velocity between the reference blade crown and the left blade crown; is the tangential relative velocity between the reference blade crown and the right blade crown; and represent the velocities of the left blade canopy, reference blade canopy, and right blade canopy in the x direction, respectively; and represent the velocities of the left blade crown, reference blade crown, and right blade crown in the y direction, respectively.

[0022] The friction force is simulated using an exponential dynamic friction model that depends on relative velocity, and the friction coefficient equation is expressed as:

[0023]

[0024] Where: μ s is the maximum static friction coefficient; μ k is the minimum kinetic friction coefficient; λ is the control parameter for the rate of descent of the control curve, v r is the tangential relative velocity between adjacent blade crown contact surfaces.

[0025] Step 2.3: According to the Coulomb friction model, the friction forces f1 and f2 between the reference blade crown and the left and right blade crowns are expressed as:

[0026]

[0027] Where μ represents the friction coefficient; f1 and f2 are the static friction forces between the left and right blade canopies and the reference blade canopy, respectively; f′1 and f′2 are the reaction forces of f1 and f2, respectively.

[0028] Step 2.4: Establish the complex boundary conditions of stick-slip-separation between adjacent blade crowns under different rubbing conditions and derive the analytical expression of static friction. The specific process is as follows:

[0029] Step 2.4.1: Non-dimensionalize the equations corresponding to the kinetic model and set the following parameters, which are expressed as:

[0030]

[0031] τ=ω x t (24)

[0032]

[0033] Where: ω x is the angular frequency of the blade in the x direction; k x and k y are the dynamic bending stiffness coefficients of the shrouded blade in the x and y directions respectively; m is the mass of the shroud; ω y is the angular frequency of the blade in the y direction; ε x is the damping ratio of the blade in the x direction; ε y is the damping ratio of the blade in the y direction; c x and c y are the viscous damping coefficients in the x- and y-directions respectively; Ω is the angular velocity of the blade; η x Ω and ω x The ratio of η y Ω and ω yΔ1 and Δ2 are the normal initial gaps between the reference blade crown and the left and right blade crowns, respectively; K is the ratio of Δ2 to Δ1; x i is the displacement of the leaf crown i in the x direction; X i is the ratio of the displacement of the leaf crown i in the x direction to Δ1; y i is the displacement of leaf crown i in the y direction; Y i is the ratio of the displacement of the leaf crown i in the y direction to Δ1; τ is the dimensionless time; t is the time k c is the normal collision stiffness, γ is k c With k x The ratio of * represents an arbitrary function; (*)′ represents the first-order derivative of dimensionless time τ; (*)″ represents the second-order derivative of dimensionless time τ; Q i is the airflow exciting force acting on the leaf crown i; is the dimensionless airflow exciting force acting on the blade crown i; n r1 is the normal relative displacement between the left leaf canopy and the reference leaf canopy; Nr1 is the normal dimensionless relative displacement between the left leaf canopy and the reference leaf canopy; X1, X2 and X3 are the dimensionless displacements of the left leaf canopy, the reference leaf canopy and the right leaf canopy in the x-direction respectively; Y1, Y2 and Y3 are the dimensionless displacements of the left leaf canopy, the reference leaf canopy and the right leaf canopy in the y-direction respectively; n r2 is the normal relative displacement between the right blade crown and the reference blade crown; N r2 is the dimensionless relative displacement between the right blade canopy and the reference blade canopy; T r1 is the tangential dimensionless relative displacement between the left blade crown and the reference blade crown; T r2 is the dimensionless relative displacement between the right blade crown and the reference blade crown; T r2 is the tangential dimensionless relative displacement between the right blade crown and the reference blade crown; T′ r1 is the tangential dimensionless relative velocity between the left blade crown and the reference blade crown; T′ r2 is the tangential dimensionless relative velocity between the right blade canopy and the reference blade canopy; X′1, X′2 and X′3 represent the dimensionless velocities of the left blade canopy, the reference blade canopy and the right blade canopy in the x-direction, respectively; Y′1, Y′2 and Y′3 represent the dimensionless velocities of the left blade canopy, the reference blade canopy and the right blade canopy in the y-direction, respectively; N j is the collision force between the reference leaf crown and the adjacent leaf crown; is the dimensionless collision force between the reference blade crown and the adjacent blade crown; f j is the static friction between the reference blade crown and the adjacent blade crown; is the dimensionless sliding friction between the reference shroud and the adjacent shroud; and are the dimensionless sliding friction forces between the reference shroud and the left and right shrouds, respectively; is the resultant dimensionless friction force on the reference blade crown; and are the dimensionless collision forces between the left and right blade canopy and the reference blade canopy, is the resultant dimensionless collision force on the reference leaf crown.

[0034] Step 2.4.2: Convert the kinetic equation corresponding to the kinetic model into a dimensionless form, expressed as:

[0035]

[0036] Where: X″1, X″2 and X″3 represent the dimensionless acceleration of the left blade canopy, reference blade canopy and right blade canopy in the x-direction respectively; Y″1, Y″2 and Y″3 represent the dimensionless acceleration of the left blade canopy, reference blade canopy and right blade canopy in the y-direction respectively; and are respectively expressed as the dimensionless airflow exciting forces acting on the left shroud, the reference shroud, and the right shroud, It is expressed as the dimensionless airflow exciting force acting on the blade crown i.

[0037] Step 2.4.3, Derivation and The analytical expression of , the specific process is:

[0038] 1) When N r1 ≤1 and N r2 When ≤K, the reference canopy does not touch the left canopy or the right canopy, then

[0039] 2) When N r1 >1 and N r2 When ≤K, the reference crown contacts the left crown, and the reference crown separates from the right crown.

[0040] 2.1) If T″ r1 ≠0, the reference blade crown slides relative to the left blade crown, and the dimensionless sliding friction force between the reference blade crown and the left blade crown in the sliding state is obtained The expression is expressed as:

[0041]

[0042] Where: sgn() is the sign function.

[0043] 2.2) If T′ r1 = 0, the tangential relative displacement of the reference blade crown relative to the left blade crown remains unchanged, and the dimensionless static friction force is obtained The analytical expression of is expressed as:

[0044]

[0045] if There is a sticking state between the reference canopy and the left canopy. If The sliding friction force is redirected, and the reference blade crown and the left blade crown are still in a sliding state.

[0046] 3) When N r1 ≤1 and N r2 When K >, the reference crown and the left crown are separated, and the reference crown is in contact with the right crown.

[0047] 3.1) If T′ r2 ≠0, the reference blade crown slides relative to the right blade crown, and the dimensionless sliding friction force between the reference blade crown and the right blade crown in the sliding state is obtained The analytical expression of is expressed as:

[0048]

[0049] 3.2) If T′ r2 = 0, the tangential relative displacement of the reference blade crown relative to the right blade crown remains unchanged, and the dimensionless static friction force is obtained The analytical expression of is expressed as:

[0050]

[0051] if There is a sticking state between the reference canopy and the left canopy. If The sliding friction force is redirected, and the reference blade crown and the right blade crown are still in a sliding state.

[0052] 4) If N r1 >1 and N r2 >K, the reference leaf crown contacts the left leaf crown and the right leaf crown, and we get

[0053] 4.1) If T′ r1 ≠0 and T′ r2 ≠0, the reference blade crown slides relative to both the left and right blade crowns, and Calculate according to formula (40) and formula (45).

[0054] 4.2) If T′ r1 = 0 and T′ r2 ≠0, the reference blade crown slides relative to the right blade crown, According to formula (45), the dimensionless static friction force is The analytical expression of is expressed as:

[0055]

[0056] if There is a sticking state between the reference canopy and the left canopy. If The sliding friction force is turned, and the reference blade crown and the left blade crown are in a sliding state.

[0057] 4.3) If T′ r1 ≠0 and T′ r2 = 0, the reference blade crown slides relative to the left blade crown, According to formula (40), the dimensionless static friction force is The analytical expression of is expressed as:

[0058]

[0059] if There is a sticking state between the reference canopy and the right canopy. If The sliding friction force is turned, and the reference blade crown and the right blade crown are in a sliding state.

[0060] 4.4) If T′ r1 = 0 and T′ r2 = 0, dimensionless static friction and The analytical expression of is expressed as:

[0061]

[0062] if and The reference canopy is in a sticky state with both the left and right canopies; if and The sliding friction between the reference blade crown and the left blade crown and the right blade crown is turned at the same time, and the reference blade crown and the left blade crown and the right blade crown are in a sliding state; if and The sliding friction between the reference blade crown and the left blade crown is turned, the reference blade crown and the left blade crown are in a sliding state, and the reference blade crown and the right blade crown are in a sticking state; if and The reference blade crown and the left blade crown will be in a sticky state, the sliding friction between the reference blade crown and the right blade crown will be redirected, and the reference blade crown and the right blade crown will be in a sliding state.

[0063] Step 3. Establish a calculation method for the response of shrouded blades considering stick-slip motion. The specific process is as follows: use the bisection method to capture the key points of stick-slip transition between adjacent blade shrouds, and establish a calculation method for the response of shrouded blades considering stick-slip motion based on the numerical solution of differential equations to predict the vibration response of shrouded blades under different key parameters.

[0064] Furthermore, the dynamic equation of the dynamic model of the shrouded blade system under the two-dimensional rubbing motion in step 1 is expressed as:

[0065]

[0066] Where: m is the mass of the leaf crown; c x and c y are the viscous damping coefficients in the x-direction and y-direction respectively; k x and k y are the dynamic bending stiffness coefficients of the shrouded blade in the x and y directions respectively; β is the angle between the airflow excitation force and the x direction, N1 and N2 are the collision forces between the left and right shrouds and the reference shroud respectively, f1 and f2 are the static friction forces between the left and right shrouds and the reference shroud respectively, N′1 and N′2 are the reaction forces of N1 and N2 respectively, f′1 and f′2 are the reaction forces of f1 and f2 respectively, and α is the rubbing contact angle; x1, x2 and x3 are the displacements of the left, reference and right shrouds in the x direction respectively, y1, y2 and y3 are the displacements of the left, reference and right shrouds in the y direction respectively; Q1, Q2 and Q3 are the airflow excitation forces acting on the three blades, and their frequencies f e is the blade rotation frequency f r l times, so it can be expressed as:

[0067] And n=1, 2, 3… (2)

[0068] Where: Ω is the angular velocity of the blade; l is the number of obstacles in front of the blade.

[0069] Furthermore, the dynamic bending stiffness coefficients k of the shrouded blade in the x-direction and y-direction in step 1 are x and k y It can be expressed as:

[0070] k x =m(2πf dx ) 2 (5)

[0071] k y =m(2πf dy ) 2 (6)

[0072] Where: f dx and f dy are the first-order dynamic bending frequencies in the x and y directions at different rotational speeds of the shrouded blade.

[0073] Furthermore, the f dx and f ay The method is as follows: a three-dimensional model of a shrouded blade is established and imported into the modal analysis module of the finite element software. Constraints are imposed on the blade root of the shrouded blade to obtain the prestress effect of the shrouded blade at different speeds. The first-order bending vibration frequency of the shrouded blade under the prestress effect is obtained through modal analysis. The curve and equation of the dynamic first-order bending vibration frequency with respect to the speed are obtained by fitting the least squares method. The equation is expressed as:

[0074] f dx =a n Ω n +a n-1 Ω n-1 +…+a1Ω+a0 (3)

[0075] f dy =b n Ω n +b n-1 Ω n-1 +…+b1Ω+b0 (4)

[0076] Where Ω is the angular velocity of the blade, and a and b are fitting coefficients.

[0077] Furthermore, the specific process of step 2.2) in step 2.4.3 is as follows: define C1 as the dimensionless displacement difference of the reference blade crown relative to the left blade crown at the end of the last sliding, expressed as:

[0078] T r1 =(X1-X2)cosα+(Y1-Y2)sinα=C1 (41)

[0079] Obviously, T r1 =0, subtract the third row from the first row of formula (39) to obtain formula (42), which is expressed as:

[0080]

[0081] Subtracting the fourth row from the second row of formula (39) yields formula (43), which is expressed as:

[0082]

[0083] First multiply both sides of the equation (42) by cosα, and both sides of the equation (43) by sinα, then add them together, and then T′r1 =0, T″ r1 =0 and T r1 =C1Substitute and we get the dimensionless static friction force The analytical expression of .

[0084] Furthermore, the specific process of step 3.2) in step 2.4.3 is as follows: define C2 as the dimensionless displacement difference between the reference blade crown and the right blade crown at the end of the last sliding, expressed as:

[0085] T r2 =(X2-X3)cosα+(Y2-Y3)sinα=C2 (46)

[0086] Obviously, T r2 =0, subtract the fifth line from the third line of formula (39) to obtain formula (47), which is expressed as:

[0087]

[0088] Subtracting the sixth row from the fourth row of formula (39) yields formula (48), which is expressed as:

[0089]

[0090] First multiply both sides of the equation (47) by cosα, and both sides of the equation (48) by sinα, then add them together, and then T′ r2 =0, T″ r2 =0 and T r2 =C2Substitute and we get the dimensionless static friction force The analytical expression of .

[0091] Furthermore, the specific process of step 4.2) in step 2.4.3 is as follows: r1 =0, so T″ r1 = 0 and formula (41) is automatically satisfied, first multiply both sides of formula (42) by cosα, multiply both sides of formula (43) by sinα, then add the two together, and then T′ r1 =0, T″ r1 =0 and T r1 =C1Substitute and we get the dimensionless static friction force The analytical expression of .

[0092] Furthermore, the specific process of step 4.3) in step 2.4.3 is as follows: r2 =0, so T″ r2= 0, and formula (46) is automatically satisfied. First, multiply both sides of the equation (47) by cosα, and multiply both sides of the equation (48) by sinα, and then T′ r2 =0, T″ r2 =0 and T r2 =C2Substitute and we get the dimensionless static friction force The analytical expression of .

[0093] Furthermore, the specific process of step 4.4) in step 2.4.3 is as follows: r1 =0, so T″ r1 = 0 and equation (41) is automatically satisfied, first multiply both sides of the equation (42) by cosα, multiply both sides of the equation (43) by sinα, then add the two together, and then T′ r1 =0, T″ r1 =0, T r1 =C1 is substituted into formula (52), which is expressed as:

[0094]

[0095] Because T′ r2 =0, so T″ r2 = 0 and equation (42) is automatically satisfied, multiply both sides of the equation (44) by sinα, multiply both sides of the equation (45) by sinα, add the two together, and then T′ r2 =0, T″ r2 =0, T r2 =C2 is substituted into formula (53), which is expressed as:

[0096]

[0097] Combining formulas (52) and (53), we can obtain the dimensionless static friction force and The analytical expression of .

[0098] The present invention has the following beneficial technical effects:

[0099] The present invention takes into account the influence of stick-slip motion on the dynamic characteristics of the shrouded blade system, calculates the contact force with strong nonlinear characteristics in stages, establishes the complex boundary conditions of stick-slip and separation of the shrouded blade system and derives the dynamic equations of the shrouded blade system, thereby improving the accuracy of the dynamic modeling of the shrouded blade system; based on the established dynamic model and equations of the shrouded blade system, the dichotomy method is used to capture the key points of the stick-slip transition between adjacent blade shrouds, and a more accurate calculation method for the vibration response of the shrouded blade is established according to the numerical solution of the differential equation, thereby solving the problem of inaccurate response prediction of the shrouded blade system in the past. BRIEF DESCRIPTION OF THE DRAWINGS

[0100] Figure 1 : Flowchart of the shrouded blade dynamics modeling and response calculation method of the present invention;

[0101] Figure 2 : Local model of the blade system with crown;

[0102] Figure 3 : Dynamic model of blade system with crown;

[0103] Figure 4 : The geometric relationship between collision force, friction force and vibration displacement;

[0104] Figure 5: Relationship between the first-order dynamic bending frequency and the rotational speed, where: Figure 5(a) shows the first-order dynamic bending frequency f in the x-direction dx , 5(b) is the first-order dynamic bending frequency f in the y direction dy ;

[0105] Figure 6 : Four typical contact states between adjacent leaf crowns;

[0106] Figure 7 : Exponential dynamic friction model;

[0107] Figure 8 : MATLAB-based calculation flow chart of the response of the shrouded blade system;

[0108] Figure 9: Vibration response of the reference blade, where: Figure 9(a) is the displacement response, Figure 9(b) is the friction force response, Figure 9(c) is the impact force response, and Figure 9(d) is the state phase response. DETAILED DESCRIPTION

[0109] The present invention will be further described in detail below with reference to specific embodiments, which are intended to explain the present invention rather than to limit it.

[0110] refer to Figure 1 As shown in FIG, a dynamic modeling and response calculation method for a shrouded blade considering stick-slip motion includes the following steps:

[0111] Step 1: Establish a dynamic model of a shrouded blade under two-dimensional friction motion

[0112] In order to consider the influence of the left and right leaf crowns on the middle leaf crown (also called reference leaf crown), a set of Figure 2The three blades shown in the figure are analyzed. During the operation of the engine, the shrouded blades are in a high-speed rotating state. Under the action of the airflow excitation force, the shrouded blades mainly produce bending vibrations in the rotation tangential direction (x direction) and axial direction (y direction). The vibration in the rotation radial direction (z direction) is very small and can be ignored. When only the first-order bending vibration of the blade in the x and y directions needs to be considered, the shrouded blades are simplified as follows Figure 2 The two-dimensional lumped mass model shown is based on Figure 2 , establish as Figure 3 The dynamic model of the blade system with a shroud is shown. Since only the first-order bending vibration of the blade in the x and y directions is considered, the movement of the blade shroud is regarded as a plane translation. During the blade vibration process, the rubbing contact angle α remains unchanged. The geometric relationship between the contact force and the vibration displacement between adjacent contact surfaces is shown as follows: Figure 4 As shown, in the figure: t represents the tangential direction of contact, n represents the normal direction of contact;

[0113] The dynamic equation corresponding to the dynamic model of the shrouded blade is expressed as:

[0114]

[0115] Where: m is the mass of the leaf crown; c x and c y are the viscous damping coefficients in the x-direction and y-direction respectively; k x and k y are the dynamic bending stiffness coefficients of the shrouded blade in the x and y directions respectively; β is the angle between the airflow excitation force and the x direction, N1 and N2 are the collision forces between the left and right shrouds and the reference shroud respectively, f1 and f2 are the static friction forces between the left and right shrouds and the reference shroud respectively, N′1 and N′2 are the reaction forces of N1 and N2 respectively, f′1 and f′2 are the reaction forces of f1 and f2 respectively, and α is the rubbing contact angle; x1, x2 and x3 are the displacements of the left, reference and right shrouds in the x direction respectively, y1, y2 and y3 are the displacements of the left, reference and right shrouds in the y direction respectively; Q1, Q2 and Q3 are the airflow excitation forces acting on the three blades, and their frequencies f e is the blade rotation frequency f r l times, so it can be expressed as:

[0116] And n=1, 2, 3… (2)

[0117] Where: Ω is the angular velocity of the blade; l is the number of obstacles in front of the blade.

[0118] In order to consider the influence of centrifugal stiffening effect, the modal analysis module in the finite element software is used to calculate the first-order dynamic bending frequency of the shrouded blade considering the prestress effect. First, the constructed three-dimensional model of the shrouded blade is imported into the modal analysis module in the finite element software, and corresponding constraints are imposed on the blade root of the shrouded blade. After obtaining the prestress effect of the shrouded blade at different speeds, the first-order dynamic bending frequency of the shrouded blade under the prestress effect is obtained through modal analysis, and the curve and equation of the first-order dynamic bending frequency with respect to the speed as shown in Figures 5(a) and 5(b) are obtained by fitting according to the least squares method, where: the first-order dynamic bending frequency includes the first-order dynamic bending frequency f in the x direction at different speeds of the shrouded blade dx and the first-order dynamic bending frequency f in the y direction dy , f dx and f dy The equations are expressed as:

[0119] f dx =a n Ω n +a n-1 Ω n-1 +…+a1Ω+a0 (3)

[0120] f dy =b n Ω n +b n-1 Ω n-1 +…+b1Ω+b0 (4)

[0121] Where Ω is the angular velocity of the blade, a and b are fitting coefficients, and their specific values ​​can be given according to Figure 5.

[0122] According to f dx and f dy The dynamic bending stiffness coefficient k of the shrouded blade in the x and y directions is obtained by the equation x and k y It can be expressed as:

[0123] k x =m(2πf dx ) 2 (5)

[0124] k y =m(2πf dy ) 2 (6)

[0125] The continuous change of the indirect contact motion state of the shrouded blade causes the collision force and friction force to have strong nonlinearity, making it difficult to obtain the analytical solution of equation (1). Therefore, numerical methods are usually required to calculate the vibration response of the ducted blade. The key to calculating the blade vibration response lies in the accurate simulation of the collision force and friction force at different contact stages. The collision forces N1 and N2 depend on the normal relative motion between the reference blade shroud and the left and right blade shrouds, respectively. The friction forces f1 and f2 are not only related to the normal relative motion, but also depend on the tangential relative motion between the contact surfaces.

[0126] Step 2: Establish the complex boundary conditions of stick-slip and stick-slip motion of the shrouded blade and determine the collision force and friction force at different stages of rubbing.

[0127] During engine operation, the friction motion between adjacent blade crowns is very complex. The normal motion between adjacent blade crowns will cause the adjacent blade crowns to appear in two states: contact and separation. The four typical contact states between adjacent blade crowns are as follows: Figure 6 shown.

[0128] Step 2.1: Assuming that the collision between adjacent leaf crowns is an elastic collision, the collision process includes four stages: contact, compression, recovery, and separation. The normal relative displacements between the reference leaf crown and the left and right leaf crowns are expressed as:

[0129]

[0130] Where: α is the rubbing contact angle; x1, x2, and x3 represent the displacements of the left leaf crown, reference leaf crown, and right leaf crown in the x-direction, respectively; y1, y2, and y3 represent the displacements of the left leaf crown, reference leaf crown, and right leaf crown in the y-direction, respectively.

[0131] The collision forces N1 and N2 between the left and right blade canopies and the reference blade canopy are expressed as:

[0132]

[0133] Where: k c is the normal collision stiffness; Δ1 and Δ2 are the normal initial gaps between the reference blade crown and the left and right blade crowns, respectively.

[0134] Step 2.2: Express the tangential relative displacement between the reference blade crown and the left blade crown and the right blade crown as:

[0135]

[0136] Where: tr1 is the tangential relative displacement between the reference leaf crown and the left leaf crown; tr2 is the tangential relative displacement between the reference leaf crown and the right leaf crown.

[0137] The tangential relative velocity between the reference blade crown and the left and right blade crowns is expressed as:

[0138]

[0139] Where: is the tangential relative velocity between the reference blade crown and the left blade crown; is the tangential relative velocity between the reference blade crown and the right blade crown; and represent the velocities of the left blade canopy, reference blade canopy, and right blade canopy in the x direction, respectively. and represent the velocities of the left blade crown, reference blade crown, and right blade crown in the y direction, respectively.

[0140] In order to consider the influence of tangential relative velocity, the Figure 7 The exponential relative velocity-dependent dynamic friction model shown simulates friction and expresses the friction coefficient equation as:

[0141]

[0142] Where: μ s is the maximum static friction coefficient; μ k is the minimum kinetic friction coefficient; λ is the control parameter for the rate of descent of the control curve, v r is the tangential relative velocity between adjacent blade crown contact surfaces.

[0143] Step 2.3: According to the Coulomb friction model, the friction forces f1 and f2 between the reference blade crown and the left and right blade crowns are expressed as:

[0144]

[0145] Where: μ represents the friction coefficient.

[0146] It can be seen from formulas (13) and (14) that in the viscous state, the friction force between adjacent blade crowns is between the negative maximum static friction and the positive maximum static friction. However, the specific value of the static friction force in the viscous state is still unknown. Therefore, while establishing the stick-slip-separation boundary conditions between adjacent blade crowns in different contact states, it is necessary to derive the analytical expression of the static friction force in different motion processes.

[0147] Step 2.4: Establish the complex boundary conditions of stick-slip-separation between adjacent blade crowns under different rubbing conditions and derive the analytical expression of static friction. The specific process is as follows:

[0148] Step 2.4.1: Non-dimensionalize the equations corresponding to the kinetic model and set the following parameters, which are expressed as:

[0149]

[0150]

[0151] τ=ω x t (24)

[0152]

[0153] Where: ω x is the angular frequency of the blade in the x direction; k x and k y are the dynamic bending stiffness coefficients of the shrouded blade in the x and y directions respectively; m is the mass of the shroud; ω y is the angular frequency of the blade in the y direction; ε x is the damping ratio of the blade in the x direction; ε y is the damping ratio of the blade in the y direction; c x and c y are the viscous damping coefficients in the x- and y-directions respectively; Ω is the angular velocity of the blade; η x Ω and ω x The ratio of η y Ω and ω y Δ1 and Δ2 are the normal initial gaps between the reference blade crown and the left and right blade crowns, respectively; K is the ratio of Δ2 to Δ1; x i is the displacement of the leaf crown i in the x direction; X i is the ratio of the displacement of the leaf crown i in the x direction to Δ1; y i is the displacement of leaf crown i in the y direction; Y i is the ratio of the displacement of the leaf crown i in the y direction to Δ1; τ is the dimensionless time; t is the time k c is the normal collision stiffness, γ is k c With k x The ratio of * represents an arbitrary function; (*)′ represents the first-order derivative of dimensionless time τ; (*)″ represents the second-order derivative of dimensionless time τ; Q i is the airflow exciting force acting on the leaf crown i; is the dimensionless airflow exciting force acting on the blade crown i; n r1 is the normal relative displacement between the left leaf canopy and the reference leaf canopy; Nr1 is the normal dimensionless relative displacement between the left leaf canopy and the reference leaf canopy; X1, X2 and X3 are the dimensionless displacements of the left leaf canopy, the reference leaf canopy and the right leaf canopy in the x-direction respectively; Y1, Y2 and Y3 are the dimensionless displacements of the left leaf canopy, the reference leaf canopy and the right leaf canopy in the y-direction respectively; n r2 is the normal relative displacement between the right blade crown and the reference blade crown; N r2 is the dimensionless relative displacement between the right blade canopy and the reference blade canopy; T r1is the tangential dimensionless relative displacement between the left blade crown and the reference blade crown; T r2 is the tangential dimensionless relative displacement between the right blade crown and the reference blade crown; T′ r1 is the tangential dimensionless relative velocity between the left blade crown and the reference blade crown; T′ r2 is the tangential dimensionless relative velocity between the right blade canopy and the reference blade canopy; X′1, X′2 and X′3 represent the dimensionless velocities of the left blade canopy, the reference blade canopy and the right blade canopy in the x-direction, respectively; Y′1, Y′2 and Y′3 represent the dimensionless velocities of the left blade canopy, the reference blade canopy and the right blade canopy in the y-direction, respectively; N j is the collision force between the reference leaf crown and the adjacent leaf crown; is the dimensionless collision force between the reference blade crown and the adjacent blade crown; f j is the static friction between the reference blade crown and the adjacent blade crown; is the dimensionless sliding friction between the reference shroud and the adjacent shroud; and are the dimensionless sliding friction forces between the reference shroud and the left and right shrouds, respectively; is the resultant dimensionless friction force on the reference blade crown; and are the dimensionless collision forces between the left and right blade canopy and the reference blade canopy, is the resultant dimensionless collision force on the reference leaf crown.

[0154] Step 2.4.2: Convert the kinetic equation (1) corresponding to the kinetic model into a dimensionless form, expressed as:

[0155]

[0156] Where: X″1, X″2 and X″3 represent the dimensionless acceleration of the left blade canopy, reference blade canopy and right blade canopy in the x-direction respectively; Y″1, Y″2 and Y″3 represent the dimensionless acceleration of the left blade canopy, reference blade canopy and right blade canopy in the y-direction respectively; and are expressed as the dimensionless airflow exciting forces acting on the left shroud, reference shroud and right shroud, respectively.

[0157] Step 2.4.3, solution and The specific process is:

[0158] 1) If Figure 6 As shown in (a), when N r1 ≤1 and N r2 When ≤K, the reference canopy does not touch the left canopy or the right canopy, then

[0159] 2) If Figure 6 As shown in (b), when Nr1>1 and Nr2≤K, the reference canopy contacts the left canopy, and the reference canopy separates from the right canopy.

[0160] 2.1) If T′ r1 ≠0, the reference blade crown slides relative to the left blade crown. According to formula (13), the dimensionless sliding friction force between the reference blade crown and the left blade crown in the sliding state is obtained: The expression is expressed as:

[0161]

[0162] Where: sgn() is the sign function.

[0163] 2.2) If T′ r1 = 0, the tangential relative displacement of the reference blade crown relative to the left blade crown remains unchanged, and C1 is defined as the dimensionless displacement difference of the reference blade crown relative to the left blade crown at the end of the last sliding, which is expressed as:

[0164] T r1 =(X1-X2)cosα+(Y1-Y2)sinα=C1 (41)

[0165] Obviously, T r1 =0, subtract the third row from the first row of formula (39) to obtain formula (42), which is expressed as:

[0166]

[0167] Subtracting the fourth row from the second row of formula (39) yields formula (43), which is expressed as:

[0168]

[0169] First multiply both sides of the equation (42) by cosα, and both sides of the equation (43) by sinα, then add them together, and then T′ r1 =0, T″ r1 =0 and T r1 =C1 is substituted into the equation to obtain the dimensionless static friction force, which is expressed as:

[0170]

[0171] if There is a sticking state between the reference canopy and the left canopy. If The sliding friction force is redirected, and the reference blade crown and the left blade crown are still in a sliding state.

[0172] 3) If Figure 6 As shown in (c), when Nr1≤1 and Nr2>K, the reference canopy and the left canopy are separated, and the reference canopy and the right canopy are in contact.

[0173] 3.1) If T′ r2 ≠0, the reference blade crown slides relative to the right blade crown. According to formula (14), the expression of the dimensionless sliding friction force between the reference blade crown and the right blade crown in the sliding state is obtained, which is expressed as:

[0174]

[0175] 3.2) If T′ r2 = 0, the tangential relative displacement of the reference blade crown relative to the right blade crown remains unchanged, and C2 is defined as the dimensionless displacement difference of the reference blade crown relative to the right blade crown at the end of the last sliding, which is expressed as:

[0176] T r2 =(X2-X3)cosα+(Y2-Y3)sinα=C2 (46)

[0177] Obviously, T r2 =0, subtract the fifth line from the third line of formula (39) to obtain formula (47), which is expressed as:

[0178]

[0179] Subtracting the sixth row from the fourth row of formula (39) yields formula (48), which is expressed as:

[0180]

[0181] First multiply both sides of the equation (47) by cosα, and both sides of the equation (48) by sinα, then add them together, and then T′ r2 =0, T″ r2 =0 and T r2 =C2 is substituted into the equation to obtain the dimensionless static friction force, which is expressed as:

[0182]

[0183] if There is a sticking state between the reference canopy and the left canopy. If The sliding friction force is redirected, and the reference blade crown and the right blade crown are still in a sliding state.

[0184] 4) If Figure 6 As shown in (d), if N r1 >1 and N r2 >K, the reference leaf crown contacts the left leaf crown and the right leaf crown, and we get At this point, the stick-slip motion between the reference blade and the left and right blades is very complex. In this case, the establishment of the stick-slip boundary conditions and the derivation of the analytical expressions for the static friction force during different motion processes are as follows:

[0185] 4.1) If T′ r1 ≠0 and T′ r2 ≠0, the reference blade crown slides relative to both the left and right blade crowns, and Calculate according to formula (40) and formula (45).

[0186] 4.2) If T′ r1 = 0 and T′ r2 ≠0, the reference blade crown slides relative to the right blade crown, According to formula (45), since T′ r1 =0, so T″ r1 = 0 and formula (41) is automatically satisfied; first multiply both sides of the equation (42) by cosα, multiply both sides of the equation (43) by sinα, then add the two together, and then T′ r1 =0, T″ r1 =0 and T r1 =C1Substitute and we get the dimensionless static friction force The analytical expression of is expressed as:

[0187]

[0188] if There is a sticking state between the reference canopy and the left canopy. If The sliding friction force is turned, and the reference blade crown and the left blade crown are in a sliding state.

[0189] 4.3) If T′ r1 ≠0 and T′ r2 = 0, the reference blade crown slides relative to the left blade crown, According to formula (40), since T′ r2 =0, so T″ r2 = 0, and formula (46) is automatically satisfied. First, multiply both sides of the equation (47) by cosα, and multiply both sides of the equation (48) by sinα, and then T′ r2 =0, T″ r2 =0 and T r2 =C2Substitute and we get the dimensionless static friction force The analytical expression of is expressed as:

[0190]

[0191] if There is a sticking state between the reference canopy and the right canopy. If The sliding friction force is turned, and the reference blade crown and the right blade crown are in a sliding state.

[0192] 4.4) If T′ r1 = 0 and T′ r2 =0, because T′ r1 =0, so T″ r1 = 0 and equation (41) is automatically satisfied, first multiply both sides of the equation (42) by cosα, multiply both sides of the equation (43) by sinα, then add the two together, and then T′ r1 =0, T″ r1 =0, T r1 =C1 is substituted into formula (52), which is expressed as:

[0193]

[0194] Because T′ r2 =0, so T″ r2 = 0 and equation (46) is automatically satisfied, first multiply both sides of the equation (47) by cosα, multiply both sides of the equation (48) by sinα, then add the two together, and then T′ r2 =0, T″ r2 =0, T r2 =C2 is substituted into formula (53), which is expressed as:

[0195]

[0196] Combining formulas (52) and (53), we can obtain the dimensionless static friction force and The analytical expression of is expressed as:

[0197]

[0198] if and The reference canopy is in a sticky state with both the left and right canopies; if and The sliding friction between the reference blade crown and the left blade crown and the right blade crown is turned at the same time, and the reference blade crown and the left blade crown and the right blade crown are in a sliding state; if and The sliding friction between the reference blade crown and the left blade crown is turned, the reference blade crown and the left blade crown are in a sliding state, and the reference blade crown and the right blade crown are in a sticking state; if and The reference blade crown and the left blade crown will be in a sticky state, the sliding friction between the reference blade crown and the right blade crown will be redirected, and the reference blade crown and the right blade crown will be in a sliding state.

[0199] Step 3: Establish a calculation method for the response of shrouded blades considering stick-slip motion

[0200] First, the material constants and geometric parameters of the crowned blade are given, including: Young's modulus is 2.1×10 11 Pa, Poisson's ratio is 0.3, and density is 7800 kg / m 3 , the blade length is 0.15m, the blade width is 0.025m, and the blade thickness is 0.004m. Considering the different contact states and different motion processes between the blade crowns, the following are given: Figure 8 The flowchart of the MATLAB-based calculation of the response of the shrouded blade system is shown.

[0201] The bisection method is used to capture the key points of stick-slip transition between adjacent blade shrouds. A more accurate calculation method for the response of shrouded blades is established based on the numerical solution of differential equations. This method is used to predict the vibration response of shrouded blades under different key parameters.

[0202] To verify the effectiveness of the dynamic modeling and response calculation method for shrouded blades considering stick-slip motion proposed in this embodiment, a set of key parameters as shown in Table 1 are given to calculate the vibration response of the shrouded blade. Only the first harmonic component of the airflow excitation force is considered in the calculation process.

[0203] Table 1 Key parameters

[0204]

[0205] Figure 9 shows the displacement response of the reference blade, the change curve of the contact force, and the state diagram between adjacent blade crowns under the given key parameters in Table 1. Analysis of Figures 9(a) and 9(d) shows that the reference blade undergoes chaotic motion under the given parameters in Table 1. The points on the Poincare section form a two-dimensional pattern. The motion between the reference blade crown and the blade crown to its left includes positive slip, negative slip, separation, and viscous motion, which is very complex. Figure 9(b) and 9(c) It can be seen that both the collision force and friction force spectra show continuous spectra, which is the reason for the nonlinear phenomenon of the reference blade.

[0206] From the above analysis, it can be seen that the dynamic modeling and response calculation method of the shrouded blade given in this embodiment can accurately predict the dynamic characteristics of the shrouded blade system, accurately obtain the contact and motion state between any adjacent blade shrouds at any instant, and improve the accuracy of the dynamic modeling of the shrouded blade system.

[0207] In order to overcome the problem of inaccurate dynamic modeling and response prediction of existing shrouded blade systems, this embodiment considers the tangential stick-slip motion between adjacent blade shrouds and proposes a dynamic modeling and response calculation method for shrouded blades. Since the shrouded blades are in a high-speed rotation state during engine operation and are affected by the airflow excitation force, under the action of the airflow excitation force, the shrouded blades mainly produce rotational tangential and axial bending vibrations. Therefore, firstly, based on the structural model of the shrouded blades, a dynamic model of the shroud under two-dimensional rubbing motion is given; at the same time, considering the first-order bending vibration of the shroud in the rotational tangential and axial directions, the movement of the shroud in its own plane is a translational motion, and the rubbing contact angle remains unchanged during the rubbing process. The geometric relationship between the rubbing force, friction force and vibration displacement is given, and the dynamic equation of the shrouded blade is derived; then, a complex boundary strip of contact-separation between adjacent blade shrouds in different contact states is established. The calculation formula of the collision force is given, and the exponential dynamic friction model considering the relative velocity is used to simulate the sliding friction force. According to the relationship between the tangential relative displacement, tangential relative velocity and tangential relative acceleration between adjacent blade crowns in the viscous state, the analytical expression of the static friction force in the viscous state is derived in combination with the system dynamics equation. According to the Coulomb friction theorem, the stick-slip boundary conditions between adjacent blade crowns in different motion processes are established; then, the dichotomy method is used to capture the key points of the stick-slip transition between adjacent blade crowns, and a more accurate blade vibration response calculation method is established based on the numerical solution of differential equations. This method considers the influence of stick-slip motion on the dynamic characteristics of the shrouded blade system, which can effectively improve the accuracy of the dynamic modeling of the shrouded blade system. At the same time, the dichotomy method is used to capture the key points of the stick-slip transition of the system, thereby improving the accuracy of the response prediction of the shrouded blade.

Claims

1. A dynamic modeling and response calculation method for shrouded blades considering stick-slip motion, characterized in that: The steps include: Step 1: Combine the first-order bending vibrations of the shrouded blade in the x-direction and the y-direction to establish a dynamic model of the shrouded blade system under two-dimensional rubbing motion; Step 2: Combine the normal and tangential relative motions between adjacent blade crowns to establish the stick-slip-separation boundary conditions and determine the collision and friction forces at different rubbing stages. The specific process is as follows: Step 2.1: Assuming that the collision between adjacent leaf crowns is elastic, the normal relative displacement between the left and right leaf crowns and the reference leaf crown is expressed as: Where: α is the rubbing contact angle; x1, x2, and x3 represent the displacements of the left leaf crown, reference leaf crown, and right leaf crown in the x-direction, respectively; y1, y2, and y3 represent the displacements of the left leaf crown, reference leaf crown, and right leaf crown in the y-direction, respectively. The collision forces N1 and N2 between the left and right blade canopies and the reference blade canopy are expressed as: Where: k c is the normal collision stiffness; Δ1 and Δ2 are the normal initial gaps between the reference blade canopy and the left and right blade canopies, respectively; N′1 and N′2 are the reaction forces of N1 and N2, respectively. Step 2.2: Express the tangential relative displacement between the reference blade crown and the left blade crown and the right blade crown as: Where: t r1 is the tangential relative displacement between the reference blade crown and the left blade crown; t r2 is the tangential relative displacement between the reference blade crown and the right blade crown. The tangential relative velocity between the reference blade crown and the left and right blade crowns is expressed as: Where: is the tangential relative velocity between the reference blade crown and the left blade crown; is the tangential relative velocity between the reference blade crown and the right blade crown; and represent the velocities of the left blade canopy, reference blade canopy, and right blade canopy in the x direction, respectively. and represent the velocities of the left blade crown, reference blade crown, and right blade crown in the y direction, respectively. The friction force is simulated using an exponential dynamic friction model that depends on relative velocity, and the friction coefficient equation is expressed as: Where: μ s is the maximum static friction coefficient; μ k is the minimum kinetic friction coefficient; λ is the control parameter for the rate of descent of the control curve, v r is the tangential relative velocity between adjacent blade crown contact surfaces. Step 2.3: According to the Coulomb friction model, the friction forces f1 and f2 between the reference blade crown and the left and right blade crowns are expressed as: Where μ represents the friction coefficient; f1 and f2 are the static friction forces between the left and right blade canopies and the reference blade canopy, respectively; f′1 and f′2 are the reaction forces of f1 and f2, respectively. Step 2.4: Establish the complex boundary conditions of stick-slip-separation between adjacent blade crowns under different rubbing conditions and derive the analytical expression of static friction. The specific process is as follows: Step 2.4.1: Non-dimensionalize the equations corresponding to the kinetic model and set the following parameters, which are expressed as: τ=ω x t (24) Where: ω x is the angular frequency of the blade in the x direction; k x and k y are the dynamic bending stiffness coefficients of the shrouded blade in the x and y directions respectively; m is the mass of the shroud; ω y is the angular frequency of the blade in the y direction; ε x is the damping ratio of the blade in the x direction; ε y is the damping ratio of the blade in the y direction; c x and c y are the viscous damping coefficients in the x- and y-directions respectively; Ω is the angular velocity of the blade; η x Ω and ω x The ratio of η y Ω and ω y Δ1 and Δ2 are the normal initial gaps between the reference blade crown and the left and right blade crowns, respectively; K is the ratio of Δ2 to Δ1; x i is the displacement of the leaf crown i in the x direction; X i is the ratio of the displacement of the leaf crown i in the x direction to Δ1; y i is the displacement of leaf crown i in the y direction; Y i is the ratio of the displacement of the leaf crown i in the y direction to Δ1; τ is the dimensionless time; t is the time k c is the normal collision stiffness, γ is k c With k x The ratio of * represents an arbitrary function; (*)′ represents the first-order derivative of dimensionless time τ; (*)″ represents the second-order derivative of dimensionless time τ; Q i is the airflow exciting force acting on the leaf crown i; is the dimensionless airflow exciting force acting on the blade crown i; n r1 is the normal relative displacement between the left blade crown and the reference blade crown; N r1 is the normal dimensionless relative displacement between the left blade canopy and the reference blade canopy; X1, X2 and X3 are the dimensionless displacements of the left blade canopy, the reference blade canopy and the right blade canopy in the x direction respectively; Y1, Y2 and Y3 are the dimensionless displacements of the left blade canopy, the reference blade canopy and the right blade canopy in the y direction respectively; n r2 is the normal relative displacement between the right blade crown and the reference blade crown; N r2 is the dimensionless relative displacement between the right blade canopy and the reference blade canopy; T r1 is the tangential dimensionless relative displacement between the left blade crown and the reference blade crown; T r2 is the tangential dimensionless relative displacement between the right blade crown and the reference blade crown; T′ r1 is the tangential dimensionless relative velocity between the left blade crown and the reference blade crown; T′ r2 is the tangential dimensionless relative velocity between the right blade canopy and the reference blade canopy; X′1, X′2 and X′3 represent the dimensionless velocities of the left blade canopy, the reference blade canopy and the right blade canopy in the x-direction, respectively; Y′1, Y′2 and Y′3 represent the dimensionless velocities of the left blade canopy, the reference blade canopy and the right blade canopy in the y-direction, respectively; N j is the collision force between the reference leaf crown and the adjacent leaf crown; is the dimensionless collision force between the reference blade crown and the adjacent blade crown; f j is the friction force between the reference leaf crown and the adjacent leaf crown; is the dimensionless friction force between the reference blade crown and the adjacent blade crown; and are the dimensionless friction forces between the reference blade crown and the left blade crown and the right blade crown, respectively; is the resultant dimensionless friction force on the reference blade crown; and are the dimensionless collision forces between the left and right blade canopy and the reference blade canopy, is the resultant dimensionless collision force acting on the reference leaf crown. Step 2.4.2: Convert the kinetic equation corresponding to the kinetic model into a dimensionless form, expressed as: Where: X″1, X″2 and X″3 represent the dimensionless acceleration of the left blade canopy, reference blade canopy and right blade canopy in the x-direction respectively; Y″1, Y″2 and Y″3 represent the dimensionless acceleration of the left blade canopy, reference blade canopy and right blade canopy in the y-direction respectively; and are expressed as the dimensionless airflow exciting forces acting on the left shroud, reference shroud and right shroud, respectively. Step 2.4.3, Derivation and The analytical expression of , the specific process is: 1) When N r1 ≤1 and N r2 When ≤K, the reference canopy does not touch the left canopy or the right canopy, then 2) When N r1 >1 and N r2 When ≤K, the reference crown contacts the left crown, and the reference crown separates from the right crown. 2.1) If T′ r1 ≠0, the reference blade crown slides relative to the left blade crown, and the dimensionless sliding friction force between the reference blade crown and the left blade crown in the sliding state is obtained The expression is expressed as: Where: sgn() is the sign function. 2.2) If T′ r1 = 0, the tangential relative displacement of the reference blade crown relative to the left blade crown remains unchanged, and the dimensionless static friction force The analytical expression of is expressed as: if There is a sticking state between the reference canopy and the left canopy. If The sliding friction force is redirected, and the reference blade crown and the left blade crown are still in a sliding state. 3) When N r1 ≤1 and N r2 When K >, the reference crown and the left crown are separated, and the reference crown is in contact with the right crown. 3.1) If T′ r2 ≠0, the reference blade crown slides relative to the right blade crown, and the dimensionless sliding friction force between the reference blade crown and the right blade crown in the sliding state is obtained The analytical expression of is expressed as: 3.2) If T′ r2 = 0, the tangential relative displacement of the reference blade crown relative to the right blade crown remains unchanged, and the dimensionless static friction force The analytical expression of is expressed as: if There is a sticking state between the reference canopy and the left canopy. If The sliding friction force is redirected, and the reference blade crown and the right blade crown are still in a sliding state. 4) If N r1 >1 and N r2 >K, the reference leaf crown contacts the left leaf crown and the right leaf crown, and we get 4.1) If T′ r1 ≠0 and T′ r2 ≠0, the reference blade crown slides relative to both the left and right blade crowns, and Calculate according to formula (40) and formula (45); 4.2) If T′ r1 = 0 and T′ r2 ≠0, the reference blade crown slides relative to the right blade crown, According to equation (45), the dimensionless static friction force is The analytical expression of is expressed as: if There is a sticking state between the reference canopy and the left canopy. If Sliding friction force steering, the reference blade crown and the left blade crown are in a sliding state; 4.3) If T′ r1 ≠0 and T′ r2 = 0, the reference blade crown slides relative to the left blade crown, According to formula (40), the dimensionless static friction force is The analytical expression of is expressed as: if There is a sticking state between the reference canopy and the right canopy. If Sliding friction force steering, the reference blade crown and the right blade crown are in a sliding state; 4.4) If T′ r1 = 0 and T′ r2 = 0, dimensionless static friction and The analytical expression of is expressed as: if and The reference canopy is in a sticky state with both the left and right canopies; if and The sliding friction between the reference blade crown and the left blade crown and the right blade crown is turned at the same time, and the reference blade crown and the left blade crown and the right blade crown are in a sliding state; if and The sliding friction between the reference blade crown and the left blade crown is turned, the reference blade crown and the left blade crown are in a sliding state, and the reference blade crown and the right blade crown are in a sticking state; if and The reference blade crown and the left blade crown will be in a sticky state, the sliding friction between the reference blade crown and the right blade crown will be redirected, and the reference blade crown and the right blade crown will be in a sliding state. Step 3. Establish a calculation method for the response of shrouded blades considering stick-slip motion. The specific process is as follows: use the bisection method to capture the key points of stick-slip transition between adjacent blade shrouds, and establish a calculation method for the response of shrouded blades considering stick-slip motion based on the numerical solution of differential equations to predict the vibration response of shrouded blades under different key parameters.

2. The method for dynamic modeling and response calculation of shrouded blades considering stick-slip motion according to claim 1 is characterized in that: The dynamic equation of the dynamic model of the shrouded blade system under the two-dimensional rubbing motion in step 1 is expressed as: Where: m is the mass of the leaf crown; c x and c y are the viscous damping coefficients in the x-direction and y-direction respectively; k x and k y are the dynamic bending stiffness coefficients of the shrouded blade in the x and y directions respectively; β is the angle between the airflow excitation force and the x direction, N1 and N2 are the collision forces between the left and right shrouds and the reference shroud respectively, f1 and f2 are the static friction forces between the left and right shrouds and the reference shroud respectively, N′1 and N′2 are the reaction forces of N1 and N2 respectively, f′1 and f′2 are the reaction forces of f1 and f2 respectively, and α is the rubbing contact angle; x1, x2 and x3 are the displacements of the left, reference and right shrouds in the x direction respectively, y1, y2 and y3 are the displacements of the left, reference and right shrouds in the y direction respectively; Q1, Q2 and Q3 are the airflow excitation forces acting on the three blades, and their frequencies f e is the blade rotation frequency f r l times, so it can be expressed as: Where: Ω is the angular velocity of the blade; l is the number of obstacles in front of the blade.

3. The method for dynamic modeling and response calculation of shrouded blades considering stick-slip motion according to claim 2 is characterized in that: The dynamic bending stiffness coefficient k of the shrouded blade in the x-direction and y-direction in step 1 x and k y It can be expressed as: k x =m(2πf dx ) 2 (5) k y =m(2πf dy ) 2 (6) Where: f dx and f dy are the first-order dynamic bending frequencies in the x and y directions at different rotational speeds of the shrouded blade.

4. The method for dynamic modeling and response calculation of shrouded blades considering stick-slip motion according to claim 3 is characterized in that: The f dx and f dy The method is as follows: a three-dimensional model of a shrouded blade is established and imported into the modal analysis module of the finite element software. Constraints are imposed on the blade root of the shrouded blade to obtain the prestress effect of the shrouded blade at different speeds. The first-order bending vibration frequency of the shrouded blade under the prestress effect is obtained through modal analysis. The curve and equation of the dynamic first-order bending vibration frequency with respect to the speed are obtained by fitting the least squares method. The equation is expressed as: f dx =a n Oh n +a n-1 Oh n-1 +…+a1Ω+a0 (3) f dy =b n Oh n +b n-1 Oh n-1 +…+b1Ω+b0 (4) Where Ω is the angular velocity of the blade, and a and b are fitting coefficients.

5. The method for dynamic modeling and response calculation of shrouded blades considering stick-slip motion according to claim 1 is characterized in that: The specific process of step 2.2) in step 2.4.3 is as follows: define C1 as the dimensionless displacement difference between the reference blade canopy and the left blade canopy at the end of the last sliding, expressed as: T r1 =(X1-X2)cosα+(Y1-Y2)sinα=C1 (41) Obviously, T r1 =0, subtract the third row from the first row of formula (39) to obtain formula (42), which is expressed as: Subtracting the fourth row from the second row of formula (39) yields formula (43), which is expressed as: First multiply both sides of the equation (42) by cosα, and both sides of the equation (43) by sinα, then add them together, and then T′ r1 =0, T″ r1 =0 and Tr1=C1, we get the dimensionless static friction force The analytical expression of .

6. The method for dynamic modeling and response calculation of shrouded blades considering stick-slip motion according to claim 1, characterized in that: The specific process of step 3.2) in step 2.4.3 is as follows: define C2 as the dimensionless displacement difference between the reference blade canopy and the right blade canopy at the end of the last sliding, expressed as: T r2 =(X2-X3)cosα+(Y2-Y3)sinα=C2 (46) Obviously, T r2 =0, subtract the fifth line from the third line of formula (39) to obtain formula (47), which is expressed as: Subtracting the sixth row from the fourth row of formula (39) yields formula (48), which is expressed as: First multiply both sides of the equation (47) by cosα, and both sides of the equation (48) by sinα, then add them together, and then T′ r2 =0, T″ r2 =0 and T r2 =C2Substitute and we get the dimensionless static friction force The analytical expression of .

7. The method for dynamic modeling and response calculation of shrouded blades considering stick-slip motion according to claim 5, characterized in that: The specific process of step 4.2) in step 2.4.3 is as follows: r1 =0, so T″ r1 = 0 and formula (41) is automatically satisfied, first multiply both sides of formula (42) by cosα, multiply both sides of formula (43) by sinα, then add the two together, and then T′ r1 =0, T″ r1 =0 and T r1 Substituting =C1, we can obtain the analytical expression of the dimensionless static friction force f1.

8. The method for dynamic modeling and response calculation of shrouded blades considering stick-slip motion according to claim 6 is characterized in that: The specific process of step 4.3) in step 2.4.3 is as follows: r2 =0, so T″ r2 = 0, and formula (46) is automatically satisfied. First, multiply both sides of the equation (47) by cosα, and multiply both sides of the equation (48) by sinα, and then T′ r2 =0, T″ r2 =0 and T r2 =C2Substitute and we get the dimensionless static friction force The analytical expression of .

9. The method for dynamic modeling and response calculation of shrouded blades considering stick-slip motion according to claim 5, characterized in that: The specific process of step 4.4) in step 2.4.3 is as follows: r1 =0, so T″ r1 = 0 and equation (41) is automatically satisfied, first multiply both sides of the equation (42) by cosα, multiply both sides of the equation (43) by sinα, then add the two together, and then T′ r1 =0, T″ r1 =0, T r1 =C1 is substituted into formula (52), which is expressed as: Because T′ r2 =0, so T″ r2 = 0 and equation (42) is automatically satisfied, multiply both sides of the equation (44) by sinα, multiply both sides of the equation (45) by sinα, add the two together, and then T′ r2 =0, T″ r2 =0, T r2 =C2 is substituted into formula (53), which is expressed as: Combining formulas (52) and (53), we can obtain the dimensionless static friction force and The analytical expression of .

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