Mechanical analysis method for flexible sleeve tooth connection considering three-dimensional friction force of contact surface

Abstract

The invention discloses a mechanical analysis method for flexible sleeve tooth connection considering three-dimensional friction force of a contact surface, which comprises the following steps of: firstly, dividing the flexible sleeve tooth connection into a plurality of shaft section infinitesimal elements along the axis direction, and simulating the contact behavior of a meshing tooth pair of each shaft section infinitesimal element by adopting a three-dimensional contact unit; wherein parameters of each contact unit are calculated according to geometric parameters and material parameters of the teeth. Summing the output results of all the contact units to obtain the relation between displacement and force or torque, and finally drawing a hysteresis loop and solving the rigidity and damping. According to the method, the non-linear rigidity can be calculated according to the size parameters of flexible sleeve tooth connection, a basis is provided for relatively accurately estimating the resonance rotating speed of the rotor system, and the method has important theoretical value and engineering significance for the safety design of the engine rotor system.

Description

Technical Field

[0001] The invention relates to the technical field of rotor dynamics, and in particular to a mechanical analysis method for a flexible sleeve tooth connection taking into account three-dimensional friction force of a contact surface. Background Art

[0002] With the vigorous development of my country's aviation industry and the pursuit of independent control of the entire process technology, the independent design and manufacturing of aircraft engines has become a challenge that must be faced. As a sophisticated and complex industrial product, aircraft engines are known as the "flower of industry", "the jewel in the crown of industry" and "the heart of the aircraft". Aircraft engines need to ensure normal operation under harsh conditions of high temperature, high pressure and high speed, which is undoubtedly a huge challenge for the design and manufacture of engines. Due to the limitations of processing, installation, disassembly and other processes, the rotor system of the engine is usually not processed as a whole. The two rotors are connected as a whole through a connecting structure. Among them, the flexible sleeve tooth connection is widely used in the low-pressure rotor of aircraft engines due to its compact structure, strong load-bearing capacity, high reliability, and ability to accommodate the misalignment of the two rotors.

[0003] As a high-speed rotating component, aircraft engines are subject to various loads during use. These loads will be transmitted in the rotor system of the aircraft engine. For the flexible sleeve-tooth connection structure, the loads are transmitted by many contact surfaces. Due to the structural characteristics of the flexible sleeve-tooth connection with multiple contact surfaces, its mechanical properties are essentially nonlinear. When the load is large, the state of the contact surface of the sleeve-tooth connection will change significantly, which will stimulate the nonlinear mechanical properties of the flexible sleeve-tooth connection, and the stiffness of the rotor system will be reduced at the sleeve-tooth connection, thereby reducing the resonant speed of the rotor system, which may invade the operating speed range of the aircraft engine. In addition, under some working conditions, the friction force on the contact surface of the sleeve-tooth connection will be converted into negative damping of the rotor system, which will cause self-excited vibration of the rotor system, which may cause serious destructive damage.

[0004] In the traditional stiffness / damping estimation method, the stiffness / damping of the flexible sleeve gear connection is regarded as a constant, but these two parameters of the flexible sleeve gear connection actually have nonlinear characteristics, which makes the estimation of the resonant speed and vibration response of the aircraft engine inaccurate, resulting in the need for additional redundant design during design. Therefore, it is very important to establish a calculation method that takes into account the friction of the contact surface to accurately estimate the nonlinear stiffness / damping characteristics of the flexible sleeve gear connection. Summary of the invention

[0005] The technical problem to be solved by the present invention is to provide a mechanical analysis method for a flexible sleeve tooth connection taking into account the three-dimensional friction force of the contact surface in view of the defects involved in the background technology.

[0006] The present invention adopts the following technical solutions to solve the above technical problems:

[0007] The mechanical analysis method of the flexible sleeve tooth connection considering the three-dimensional friction force of the contact surface includes the following steps:

[0008] Step 1), dividing the flexible sleeve tooth connection into a number of shaft segment microelements, obtaining the stiffness of the inner shaft tooth and the outer shaft tooth, connecting the stiffness of the inner shaft tooth and the outer shaft tooth in series, and obtaining the meshing stiffness of the teeth of the flexible sleeve tooth connection;

[0009] Step 1.1), divide the flexible sleeve tooth connection with a tooth width of L evenly into several shaft segment microelements along its axis, and the width of each shaft segment microelement is recorded as L d ;

[0010] Step 1.2), for any shaft segment microelement, according to potential energy and beam theory, combined with the geometric dimensions of the involute and transition circle of the flexible sleeve connecting the inner shaft, the bending stiffness of the inner shaft tooth is obtained: Shear stiffness Axial compression stiffness

[0011]

[0012] Among them, L d is the tooth width of the flexible sleeve gear shaft segment microelement, E is the elastic modulus of the flexible sleeve gear connection inner shaft, υ is the Poisson's ratio of the flexible sleeve gear connection inner shaft, d is the radial distance between the meshing point of the flexible sleeve gear connection inner shaft and the base circle of the flexible sleeve gear connection inner shaft, r b is the base circle radius of the inner shaft of the flexible sleeve gear connection, r f is the root circle radius of the flexible sleeve gear connecting the inner shaft, is the shear modulus of the inner shaft of the flexible sleeve gear connection;

[0013] The midline of the involute tooth of the flexible sleeve gear connection internal shaft is called the midline of the flexible sleeve gear connection internal shaft tooth, A x1 =(2h x1 )L is the area of ​​the cross section on the transition circle of the flexible sleeve gear connection inner shaft with a radial distance of x1 from the base circle, h is the moment of inertia of the section on the transition curve with a radial distance x1 from the base circle, x1 The distance from the intersection of the section at a radial distance x1 from the base circle of the flexible sleeve gear connection inner shaft on the transition circle of the flexible sleeve gear connection inner shaft to the tooth centerline of the flexible sleeve gear connection inner shaft;

[0014] γ Ais one of the variables in the integral formula, which means: on the involute of the flexible sleeve gear connection inner shaft, a moving point between the base circle of the flexible sleeve gear connection inner shaft and the meshing point of the flexible sleeve gear connection inner shaft is temporarily recorded as moving point A, and a tangent line of the base circle of the flexible sleeve gear connection inner shaft is drawn through moving point A, and the tangent point is temporarily recorded as point B, γ A is the angle between the line connecting the center of the inner shaft of the flexible sleeve gear connection and point B and the tooth centerline of the inner shaft of the flexible sleeve gear connection; and γ1 is the γ when the moving point A is located at the fitting point of the inner shaft of the flexible sleeve gear connection. A γ2 is the value of γ when the moving point A is located at the intersection of the involute of the flexible sleeve gear connection inner shaft and the base circle of the flexible sleeve gear connection inner shaft. A The value of

[0015] Rigidity k of the inner shaft tooth int for:

[0016]

[0017] Step 1.3), for any shaft segment microelement, according to potential energy and beam theory, combined with the involute of the flexible sleeve connecting the external shaft, the bending stiffness of the external shaft tooth is obtained: Shear stiffness and axial compressive stiffness

[0018]

[0019] Wherein, under the action of torque, the flexible sleeve gear connection inner shaft teeth and the contacting flexible sleeve gear connection outer shaft teeth are called a pair of meshing teeth, and the flexible sleeve gear connection outer shaft teeth in the pair of fitting teeth have two involute tooth profiles, and the tooth center line of the flexible sleeve gear connection outer shaft is the center line of the two involute tooth profiles;

[0020] γ C The meaning is: a moving point on the involute of the flexible sleeve gear connecting the outer shaft is temporarily recorded as moving point C. A tangent line of the base circle of the flexible sleeve gear connecting the inner shaft is drawn through moving point C. The tangent point is temporarily recorded as point D. C is the angle between the line connecting the center of the inner shaft of the flexible sleeve gear connection and point D and the center line of the tooth of the flexible sleeve gear connection external shaft; γ3 is the angle between the moving point C and the root circle of the tooth of the flexible sleeve gear connection external shaft when the moving point C is located at the intersection of the involute of the tooth of the flexible sleeve gear connection external shaft and the root circle of the tooth of the flexible sleeve gear connection external shaft. C γ4 is the value of when the moving point C is located at the meshing point of the flexible sleeve gear connecting the external shaft. C γ5 is the value when the moving point C is located at the intersection of the extended line of the involute of the flexible sleeve gear connecting the external shaft teeth and the base circle of the flexible sleeve gear connecting the internal shaft. C The value of

[0021] Rigidity k of the outer shaft tooth ext for:

[0022]

[0023] Step 1.4), calculate the meshing stiffness of a single tooth pair for:

[0024]

[0025] Step 2), construct the equilibrium equation according to the deformation of the flexible sleeve tooth connection:

[0026] Step 2.1), calculate the intrusion amount at any meshing point according to the iteration variable;

[0027] Step 2.1.1), establish a spatial rectangular coordinate system, the origin of which is on the axis of the flexible sleeve-tooth connection when it is not subjected to any load, the positive direction of the transverse shear force transmitted from the turbine to the flexible sleeve-tooth connection is recorded as the positive direction of the y-axis, the positive direction of the x-axis is the direction from the fan / compressor to the turbine, the origin is recorded as point o, and the end face of the flexible sleeve-tooth connection close to the fan is located on the yoz plane; if the load is a bending moment, the positive direction of the z-axis is the same as the positive direction of the bending moment;

[0028] When the flexible sleeve tooth connection is deformed by shear load or bending moment load, there is an angle θ between the axis of the inner shaft of the flexible sleeve tooth connection and the axis of the outer shaft of the flexible sleeve tooth connection in the xoy plane. z The coordinate of the intersection point between the two axes is x a ; When the flexible sleeve tooth connection is subjected to torque, the inner shaft of the flexible sleeve tooth connection has a torsion angle φ around the x-axis relative to the outer shaft of the flexible sleeve tooth connection;

[0029] Step 2.1.2), then construct the iteration variable X = [x a ,θ z ,φ] T For a certain shaft segment microelement of a flexible sleeve gear connection, the coordinates of the meshing point of the i-th tooth are (x i ,y i ), at this meshing point, the relative displacement of the inner shaft relative to the outer shaft and the vertical relative displacement δ i for:

[0030]

[0031] Step 2.1.3), for a certain shaft segment microelement of the flexible sleeve gear connection, a local coordinate system nkt is constructed at the meshing point of the i-th tooth of the shaft segment microelement; the origin of the coordinate system coincides with the fitting point, the direction from the origin of the local coordinate system to the turbine is the positive direction of the k-axis, the normal direction pointing to the external shaft of the flexible sleeve gear connection at the meshing point is the positive direction of the n-axis, and the positive direction of the t-axis is determined by the right-hand rule;

[0032] Vertical relative displacement δ iThe resulting intrusion Δ along the n-axis of the local coordinate system b n i and the tangential displacement Δ along the positive t-axis of the local coordinate system t bi for:

[0033]

[0034] Among them, α i is the angle between the normal line at the meshing point of the flexible sleeve tooth of the i-th tooth connected to the inner shaft and the positive direction of the y-axis; at this meshing point, the intrusion amount caused by the torsion angle φ Tangential displacement along the t axis for:

[0035]

[0036] Among them, p i is the pressure angle at the meshing point, and the meshing radius is the angle between the center line of the tooth of the flexible sleeve connecting the inner shaft and the positive direction of the y-axis, r m is the pitch radius; therefore, the total intrusion of the i-th tooth is i n and the tangential displacement Δ along the t axis i t for:

[0037]

[0038] In summary, under the combined action of lateral load and torque, the relative displacement vector of each tooth at the meshing point is for: n, t, and k are unit vectors in the positive directions of the n-axis, t-axis, and k-axis in the local coordinate system nkt, respectively;

[0039] Step 2.2), calculating the reaction force and reaction moment of the flexible sleeve tooth connection;

[0040] Step 2.2.1), the tangential stiffness at the meshing point of the i-th tooth of the flexible sleeve connecting the inner shaft in the shaft segment microelement is κ is the proportional coefficient; the contact point of the i-th tooth of the flexible sleeve connecting the inner shaft in the shaft segment microelement is recorded as a, and the contact point on the external meshing tooth in the same meshing pair is recorded as b; the displacement of point a in space is the relative displacement of the meshing point Point b represents the sliding displacement of the slider, which is recorded as It will be updated according to the force state; at this time, the force dF along the normal direction of the meshing point received by the i-th tooth of the flexible sleeve gear connection inner shaft in the shaft segment microelement is i n for:

[0041]

[0042] in, is the Heaviside function, which has the following form:

[0043]

[0044] Step 2.2.2), at the mth time step, define a circle with point a(m) as the center and a radius r0 as:

[0045]

[0046] The coordinates of points a and b coincide at the initial zero moment when the flexible sleeve is not loaded. Let the vector from a to b be recorded as Then at any time vector r ab for:

[0047]

[0048] Step 2.2.3), according to the basic idea of ​​the time advancement method, solve the tangential friction force at the meshing point of the i-th tooth of the flexible sleeve connecting the inner shaft in the shaft segment microelement:

[0049] Step 2.2.3.1), obtain the sliding displacement corresponding to the slider b (m.1) in the previous iterative calculation result Record it as If it is the first iteration calculation, the sliding displacement of the slider is (0,0);

[0050] Step 2.2.3.2), according to the tangential relative displacement of contact point a in step m Sliding displacement of the slider in step m.1 Calculate the assumed friction as well as The vector r ab-pre for:

[0051]

[0052] Where dx represents the tooth width of the shaft segment microelement, and its value is the same as L in step 1). d Equal, used for integral calculation in subsequent steps;

[0053] Step 2.2.3.3), if the vector r ab-pre The magnitude is less than the vector r ab The actual friction force in the mth step is equal to the friction force under assumption, and its direction is the same as r ab-pre The same; otherwise, it means that the friction force at the meshing point is equal to the critical friction force, and the direction is the same as r ab-preIn the same direction, friction The calculation formula is:

[0054]

[0055] in, is the Heaviside function, which is the same as in step 2.2.1);

[0056] Step 2.2.3.4), update the sliding displacement of the slider. If no sliding occurs, set If slippage occurs, the slippage of the slider in step m is solved as:

[0057]

[0058] So far, the tangential friction force on the microplane at the meshing point of the i-th tooth of the flexible sleeve connecting the inner shaft in the shaft segment microelement is obtained: Decompose it into the t and k directions in the local coordinate system, and the components in the corresponding directions and for:

[0059]

[0060] Step 2.2.4), the force on the i-th tooth of the flexible sleeve connecting the inner shaft in the shaft segment microelement is obtained. Since a shaft segment microelement contains a total of z teeth, the force on the microelement is as follows:

[0061]

[0062] where y i and z i are the y-axis and z-axis coordinates of the meshing position of the i-th tooth of the flexible sleeve gear connected to the inner shaft in a shaft segment microelement, and further integration along the axial direction is used to obtain the resultant force and torque of the entire flexible sleeve gear structure, as shown below:

[0063]

[0064] Step 3), iteratively solving the equations by Newton's iteration method;

[0065] Step 3.1), construct the equilibrium equation:

[0066] Given the torque T, select the lateral shear amplitude or bending moment amplitude It is called "load amplitude". The load sequence to be solved is constructed with the load amplitude, and the number of cycles to be solved is determined. c , each cycle is T h data points, a total of T c ·T hstep. Generally speaking, the number of cycles and the data points in one cycle need to be sufficient until the calculation results converge. The shear load to be solved in step j is or moment load for:

[0067]

[0068] The shear load amplitude is When, for a flexible sleeve-tooth connection structure with an axial length of L, the equation F(X)=0 for solving the corresponding linear displacement of the jth step is:

[0069]

[0070] The magnitude of the moment load is When , the equation F(X)=0 used to solve the corresponding angular displacement of step j is:

[0071]

[0072] Step 3.2), solving the equilibrium equation by Newton iteration method;

[0073] In step 2.1.2), the independent variable X=[x a ,θ z ,φ] T , use Newton iteration method to solve, the iteration format is as follows:

[0074]

[0075] When solving iteratively, first give the initial value of the variable in, is a very small non-zero real number, φ0 is the torsion angle when only torque exists, substitute the initial value for iterative solution; when X k+1 With X k When the difference between the two norms of is less than the convergence limit set by the user, the Newton iteration method solution of the jth step is considered to be completed; when the load of the jth step is solved through k-step iterations, the obtained iteration variable is recorded as For moment loads, the angular displacement at step j is solved as For shear load, solve for the linear displacement σ at step j j for:

[0076]

[0077] Continue to iterate until the number of iterations set in step 3.1) is completed, and finally a total of T c ·T h Linear or angular displacement of a

[0078] Step 4), calculating the equivalent stiffness and damping according to the hysteresis loop;

[0079] Step 4.1), draw the hysteresis loop:

[0080] According to the shear load obtained in step 3.2) (or moment load ) and the corresponding linear displacement σ j (or angular displacement is ), one by one as data points, with displacement as the horizontal axis, represented by the symbol θ, and load as the vertical axis, the hysteresis loop is drawn; in the hysteresis loop, the minimum displacement -θ s and the maximum displacement θ s The hysteresis loop is divided into two lines, and the line with a larger load value is recorded as M t1 , the other line is marked as M b1 , M t1 The direction of the minimum displacement is -θ s To the maximum displacement θ s , M b1 The direction of the maximum displacement θ s To minimum displacement -θ s ;

[0081] Step 4.2), calculate the equivalent stiffness and damping;

[0082] According to the hysteresis loop obtained in step 4.1), the equivalent stiffness k is given eq The calculation formula is as follows:

[0083]

[0084] The loss factor η is calculated as follows:

[0085]

[0086] The damping coefficient c eq The calculation formula is as follows:

[0087]

[0088] Where ω is the rotor speed.

[0089] Compared with the prior art, the present invention adopts the above technical solution and has the following technical effects:

[0090] Based on the existing methods, the present invention takes into account the three-dimensional full contact surface friction and develops a mechanical analysis method for a flexible sleeve tooth connection that takes into account the three-dimensional friction of the contact surface. The method can calculate the nonlinear stiffness according to the dimensional parameters of the flexible sleeve tooth connection, providing a basis for relatively accurately estimating the resonant speed of the rotor system, and has important theoretical value and engineering significance for the safety design of the engine rotor system. BRIEF DESCRIPTION OF THE DRAWINGS

[0091] Figure 1 This is the overall idea of ​​the present invention;

[0092] Figure 2 Schematic diagram of the structural parameters used to solve the meshing stiffness of the internal shaft teeth;

[0093] Figure 3 Schematic diagram of the structural parameters used to solve the meshing stiffness of the outer shaft teeth;

[0094] Figure 4 It is a schematic diagram of the first two elements in the iteration variable in the flexible sleeve tooth connection;

[0095] Figure 5 Schematic diagram of structural parameters of the i-th internal shaft tooth of the flexible sleeve tooth connection;

[0096] Figure 6 It is a schematic diagram of the three-dimensional full-contact friction model;

[0097] Figure 7 A schematic diagram showing the comparison between the blocking state and the sliding state using the three-dimensional full-contact friction model;

[0098] Figure 8 Schematic diagram of the hysteresis loop used to solve the equivalent stiffness and damping coefficient. DETAILED DESCRIPTION

[0099] The technical solution of the present invention is further described in detail below in conjunction with the accompanying drawings:

[0100] The present invention can be implemented in many different forms and should not be considered to be limited to the embodiments described herein. On the contrary, these embodiments are provided to make this disclosure thorough and complete, and will fully express the scope of the present invention to those skilled in the art. In the accompanying drawings, components are enlarged for clarity.

[0101] The overall idea of ​​the present invention is as follows Figure 1As shown. For a certain target external load, set the iteration variable and use the Newton iteration method for iterative solution. The specific process is roughly as follows: divide the flexible sleeve tooth connection into several shaft segment micro-elements along the axial direction, and use a three-dimensional contact unit to simulate the contact behavior of the meshing tooth pair of each shaft segment micro-element, where the parameters of each contact unit are calculated based on the geometric parameters and material parameters of the tooth. Then sum the output results of all contact units to obtain the relationship between displacement and force or torque, and then draw a hysteresis loop to obtain stiffness and damping. The specific steps are as follows:

[0102] Step 1), dividing the sleeve gear connection into a number of shaft segment microelements, obtaining the stiffness of the inner shaft gear and the outer shaft gear, connecting the stiffness of the inner shaft gear and the outer shaft gear in series, and obtaining the meshing stiffness of the teeth of the flexible sleeve gear connection;

[0103] Step 1.1), divide the flexible sleeve tooth connection with a tooth width of L evenly into several shaft segment microelements along its axis, and the width of each shaft segment microelement is recorded as L d ;

[0104] Step 1.2), for any shaft segment microelement, according to potential energy and beam theory, combined with the geometric dimensions of the involute and the preset transition circle, the bending stiffness of the inner shaft tooth is obtained: Shear stiffness Axial compression stiffness

[0105]

[0106] Among them, L d is the tooth width of the flexible sleeve gear shaft segment microelement, E is the elastic modulus of the flexible sleeve gear connection inner shaft, υ is the Poisson's ratio of the flexible sleeve gear connection inner shaft, d is the radial distance between the meshing point of the flexible sleeve gear connection inner shaft and the base circle of the flexible sleeve gear connection inner shaft, r b is the base circle radius of the inner shaft of the flexible sleeve gear connection, r f is the root circle radius of the flexible sleeve gear connecting the inner shaft, A is the shear modulus of the inner shaft of the flexible sleeve gear connection. x1 =(2h x1 )L is the area of ​​the cross section on the transition curve with a radial distance of x1 from the base circle, is the moment of inertia of the section on the transition curve with a radial distance x1 from the base circle; r b is the base circle radius, r f is the tooth root radius, γ A ,γ1,γ2 see Figure 2 ; Then the stiffness k of the inner shaft tooth int for:

[0107]

[0108] Step 1.3), for any shaft segment microelement, calculate the internal shaft tooth stiffness: Based on potential energy and beam theory, the bending stiffness is obtained by combining the geometric dimensions of the involute and the transition circle: Shear stiffness and axial compressive stiffness The calculation formula is:

[0109]

[0110] Among them, γ C The definitions of γ4, γ5, and γ3 are given in Figure 3 .

[0111] Then the stiffness k of the outer shaft tooth is ext for:

[0112]

[0113] Step 1.3) Calculate the meshing stiffness k of a single tooth pair g n for:

[0114]

[0115] Step 2) Construct the equilibrium equation based on the deformation of the flexible sleeve tooth connection:

[0116] Step 2.1) Calculate the intrusion at any meshing point based on the iteration variable

[0117] The format of the iteration variable is X=[x a ,θ z ,φ] T , where x a is the distance from the intersection of the inner shaft axis and the outer shaft axis to the fixed end face after the flexible sleeve tooth connection is deformed, θ z is the angle between the inner and outer axes, see Figure 4 , φ is the torsion angle of the inner shaft relative to the outer shaft. Figure 4 In the coordinate system in, for the iteration variable X = [x a ,θ z ,φ] T , the coordinates of the i-th tooth are (x i ,y i ) relative axial displacement of the inner flexible sleeve gear and vertical relative displacement δ i for:

[0118]

[0119] The vertical relative displacement δ i The resulting normal intrusion along the n-axis and the tangential displacement along the t-axis are calculated as follows:

[0120]

[0121] Among them, α i is the angle between the normal line of the meshing point of the i-th tooth and the positive direction of the y-axis, such as Figure 5 At this meshing point, the normal intrusion caused by the torsion angle φ Tangential displacement along the t axis The calculation is shown below:

[0122]

[0123] Among them, p i is the pressure angle, the meshing radius is the angle between the tooth centerline and the positive direction of the y-axis, r m is the pitch radius. Therefore, the total normal intrusion and the tangential displacement along the t axis for:

[0124]

[0125] In summary, under the combined action of lateral load and torque, the relative displacement vector of each tooth at the meshing point is for: The directions of the unit vectors k, t, and n are as follows: Figure 6 shown.

[0126] Step 2.2) Calculate the reaction force and reaction moment of the flexible sleeve tooth connection

[0127] The tangential stiffness of each tooth at the meshing point is κ is the proportionality coefficient. The three-dimensional full contact friction unit is as follows Figure 6 As shown in the figure, the contact point on the inner meshing tooth is marked as a, and the contact point on the outer meshing tooth is marked as b. The displacement of point a in space is the relative displacement of the meshing point. Point b represents the sliding displacement of the slider, which is recorded as It will be updated according to the force state. At this time, the force along the normal direction of the meshing point is for:

[0128]

[0129] in, is the Heaviside function, which has the following form:

[0130]

[0131] The friction force caused by tangential relative sliding is described in detail below: Figure 7As shown, at the mth time step, a circle with point a(m) as the center is defined, and the radius r0 satisfies the following formula:

[0132]

[0133] The coordinates of points a and b coincide at the initial zero moment when the flexible sleeve is not loaded. Let the vector from a to b be recorded as Then at any time vector r ab for:

[0134]

[0135] According to the basic idea of ​​the time advancement method, the basic process of solving the tangential friction force at the meshing point of the i-th pair of teeth is summarized as follows:

[0136] ① Obtain the sliding displacement corresponding to the slider b (m.1) in the previous iterative calculation result Record it as If it is the first iteration calculation of the program, the sliding displacement of the slider is (0,0).

[0137] ② Assuming that the mth step in the calculation is a "blocking state", according to the tangential relative displacement of the contact point a in the mth step Sliding displacement of the slider in step m.1 Calculate the friction force under the assumption as well as The vector r ab-pre for:

[0138]

[0139] ③ Compare the assumed friction force with the critical friction force at this time. If it is less than the critical friction force, the meshing point in the mth step is in a "blocking state". At this time, the actual friction force is equal to the assumed friction force, and the direction is the same as r. ab-pre The same; otherwise, it means that the meshing point is in a "slip state", at which time the friction force is equal to the critical friction force, and the direction is the same as r ab-pre In the same direction, the friction force dF i p The calculation formula is:

[0140]

[0141] ④ Update the sliding displacement of the slider. If no sliding occurs, If slippage occurs, the slippage of the slider in step m is solved as:

[0142]

[0143] So far, the tangential friction force on the microplane at the meshing point of the i-th pair of teeth is obtained: Decompose it into the t and k directions in the local coordinate system, and the components in the corresponding directions and for:

[0144]

[0145] The force on the i-th tooth in a shaft segment microelement dx, the shaft segment microelement contains a total of z teeth, then the force on the microelement is as follows:

[0146]

[0147] where y i and z i are the coordinates of the y-axis and z-axis of the meshing position of the i-th tooth, respectively. Further integration along the axial direction is performed to obtain the resultant force and torque of the entire flexible sleeve gear structure, as shown in the following formula:

[0148]

[0149] Step 3) Newton iteration method to iteratively solve the equations

[0150] Step 3.1) Construct the balanced equation

[0151] Given the torque T, select the lateral shear amplitude or bending moment amplitude It is called "load amplitude". Use the load amplitude to construct the load sequence to be solved and determine the number of cycles to be solved T c , each cycle is T h data points, a total of T c ·T h Step. The shear load to be solved in step j or moment load for:

[0152]

[0153]

[0154] For a flexible sleeve-tooth connection structure with an axial length of L, the equation group of linear displacement to be solved in the jth step with F(X)=0 is:

[0155]

[0156] The F(X) of the equation system of angular displacement to be solved in step j is:

[0157]

[0158] Step 3.2) Newton iteration method solution

[0159] Independent variable X = [x a ,θ z ,φ] T , use Newton iteration method to solve, the iteration format is as follows:

[0160]

[0161] When solving iteratively, first give the initial value of the variable in, is a very small non-zero real number, φ0 is the torsion angle when only torque exists, and the initial value is substituted for the iterative solution. When the load of the jth step is solved through k steps of iteration, the obtained iteration variable is recorded as For moment loads, the angular displacement at step j is solved as For shear load, solve for the linear displacement σ at step j j for:

[0162]

[0163] Continue to iterate and solve until the required number of steps in step 3.1) is completed, and a total of T c ·T h Linear or angular displacement of a.

[0164] Step 4) Calculate the equivalent stiffness and damping based on the hysteresis loop

[0165] Step 4.1) Draw the hysteresis loop

[0166] According to the shear load obtained in 3.2) (or moment load ) and the corresponding linear displacement σ j (or angular displacement is ), and draw them one by one to get the hysteresis loop, which is roughly in the form of Figure 8 As shown, M is a whole symbol, representing shear force or bending moment, M t1 is a dotted line, M b1 is a dotted line, M1 is M t1 and M b1 The average value in the first quadrant, θ is the linear displacement or angular displacement; M and θ with s subscripts indicate the corresponding maximum value.

[0167] Step 4.2) Calculate equivalent stiffness and damping

[0168] According to the hysteresis loop obtained in 4.1), the equivalent stiffness k is given eq The calculation formula is as follows:

[0169]

[0170] The loss factor η is calculated as follows:

[0171]

[0172] The damping coefficient c eq The calculation formula is as follows:

[0173]

[0174] Where ω is the rotor speed.

[0175] It will be understood by those skilled in the art that, unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as those generally understood by those skilled in the art in the art to which the present invention belongs. It should also be understood that terms such as those defined in common dictionaries should be understood to have meanings consistent with the meanings in the context of the prior art, and will not be interpreted with idealized or overly formal meanings unless defined as herein.

[0176] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A mechanical analytical method for a flexible sleeve-tooth connection considering three-dimensional friction of the contact surface, characterized in that: The following steps are involved: Step 1), dividing the flexible sleeve tooth connection into a number of shaft segment microelements, obtaining the stiffness of the inner shaft tooth and the outer shaft tooth, connecting the stiffness of the inner shaft tooth and the outer shaft tooth in series, and obtaining the meshing stiffness of the teeth of the flexible sleeve tooth connection; Step 1.1), divide the flexible sleeve tooth connection with a tooth width of L evenly into several shaft segment microelements along its axis, and the width of each shaft segment microelement is recorded as L d ; Step 1.2), for any shaft segment microelement, according to potential energy and beam theory, combined with the geometric dimensions of the involute and transition circle of the flexible sleeve connecting the inner shaft, the bending stiffness of the inner shaft tooth is obtained: Shear stiffness Axial compression stiffness Among them, L d is the tooth width of the flexible sleeve gear shaft segment microelement, E is the elastic modulus of the flexible sleeve gear connection inner shaft, υ is the Poisson's ratio of the flexible sleeve gear connection inner shaft, d is the radial distance between the meshing point of the flexible sleeve gear connection inner shaft and the base circle of the flexible sleeve gear connection inner shaft, r b is the base circle radius of the inner shaft of the flexible sleeve gear connection, r f is the root circle radius of the flexible sleeve gear connecting the inner shaft, is the shear modulus of the inner shaft of the flexible sleeve gear connection; The midline of the involute tooth of the flexible sleeve gear connection internal shaft is called the midline of the flexible sleeve gear connection internal shaft tooth, A x1 =(2h x1 )L is the area of ​​the cross section on the transition circle of the flexible sleeve gear connection inner shaft with a radial distance of x1 from the base circle, h is the moment of inertia of the section on the transition curve with a radial distance x1 from the base circle, x1 The distance from the intersection of the section at a radial distance x1 from the base circle of the flexible sleeve gear connection inner shaft on the transition circle of the flexible sleeve gear connection inner shaft to the tooth centerline of the flexible sleeve gear connection inner shaft; γ A is one of the variables in the integral formula, which means: on the involute of the flexible sleeve gear connection inner shaft, a moving point between the base circle of the flexible sleeve gear connection inner shaft and the meshing point of the flexible sleeve gear connection inner shaft is temporarily recorded as moving point A, and a tangent line of the base circle of the flexible sleeve gear connection inner shaft is drawn through moving point A, and the tangent point is temporarily recorded as point B, γ A is the angle between the line connecting the center of the inner shaft of the flexible sleeve gear connection and point B and the tooth centerline of the inner shaft of the flexible sleeve gear connection; and γ1 is the γ when the moving point A is located at the fitting point of the inner shaft of the flexible sleeve gear connection. A γ2 is the value of γ when the moving point A is located at the intersection of the involute of the flexible sleeve gear connection inner shaft and the base circle of the flexible sleeve gear connection inner shaft. A The value of Rigidity k of the inner shaft tooth int for: Step 1.3), for any shaft segment microelement, according to potential energy and beam theory, combined with the involute of the flexible sleeve connecting the external shaft, the bending stiffness of the external shaft tooth is obtained: Shear stiffness and axial compressive stiffness Wherein, under the action of torque, the flexible sleeve gear connection inner shaft teeth and the contacting flexible sleeve gear connection outer shaft teeth are called a pair of meshing teeth, and the flexible sleeve gear connection outer shaft teeth in the pair of fitting teeth have two involute tooth profiles, and the tooth center line of the flexible sleeve gear connection outer shaft is the center line of the two involute tooth profiles; γ C The meaning is: a moving point on the involute of the flexible sleeve gear connecting the outer shaft is temporarily recorded as moving point C. A tangent line of the base circle of the flexible sleeve gear connecting the inner shaft is drawn through moving point C. The tangent point is temporarily recorded as point D. C is the angle between the line connecting the center of the inner shaft of the flexible sleeve gear connection and point D and the center line of the tooth of the flexible sleeve gear connection external shaft; γ3 is the angle between the moving point C and the root circle of the tooth of the flexible sleeve gear connection external shaft when the moving point C is located at the intersection of the involute of the tooth of the flexible sleeve gear connection external shaft and the root circle of the tooth of the flexible sleeve gear connection external shaft. C γ4 is the value of when the moving point C is located at the meshing point of the flexible sleeve gear connecting the external shaft. C γ5 is the value when the moving point C is located at the intersection of the extended line of the involute of the flexible sleeve gear connecting the external shaft teeth and the base circle of the flexible sleeve gear connecting the internal shaft. C The value of Rigidity k of the outer shaft tooth ext for: Step 1.4), calculate the meshing stiffness of a single tooth pair for: Step 2), construct the equilibrium equation according to the deformation of the flexible sleeve tooth connection: Step 2.1), calculate the intrusion amount at any meshing point according to the iteration variable; Step 2.1.1), establish a spatial rectangular coordinate system, the origin of which is on the axis of the flexible sleeve-tooth connection when it is not subjected to any load, the positive direction of the transverse shear force transmitted from the turbine to the flexible sleeve-tooth connection is recorded as the positive direction of the y-axis, the positive direction of the x-axis is the direction from the fan / compressor to the turbine, the origin is recorded as point o, and the end face of the flexible sleeve-tooth connection close to the fan is located on the yoz plane; if the load is a bending moment, the positive direction of the z-axis is the same as the positive direction of the bending moment; When the flexible sleeve tooth connection is deformed by shear load or bending moment load, there is an angle θ between the axis of the inner shaft of the flexible sleeve tooth connection and the axis of the outer shaft of the flexible sleeve tooth connection in the xoy plane. z The coordinate of the intersection point between the two axes is x a ; When the flexible sleeve tooth connection is subjected to torque, the inner shaft of the flexible sleeve tooth connection has a torsion angle φ around the x-axis relative to the outer shaft of the flexible sleeve tooth connection; Step 2.1.2), then construct the iteration variable X = [x a ,θ z ,φ] T For a certain shaft segment microelement of a flexible sleeve gear connection, the coordinates of the meshing point of the i-th tooth are (x i ,y i ), at this meshing point, the relative displacement of the inner shaft relative to the outer shaft and the vertical relative displacement δ i for: Step 2.1.3), for a certain shaft segment microelement of the flexible sleeve gear connection, a local coordinate system nkt is constructed at the meshing point of the i-th tooth of the shaft segment microelement; the origin of the coordinate system coincides with the fitting point, the direction from the origin of the local coordinate system to the turbine is the positive direction of the k-axis, the normal direction pointing to the external shaft of the flexible sleeve gear connection at the meshing point is the positive direction of the n-axis, and the positive direction of the t-axis is determined by the right-hand rule; Vertical relative displacement δ i The resulting intrusion along the n-axis of the local coordinate system and the tangential displacement along the positive t-axis of the local coordinate system for: Among them, α i is the angle between the normal line at the meshing point of the flexible sleeve tooth of the i-th tooth connected to the inner shaft and the positive direction of the y-axis; at this meshing point, the intrusion amount caused by the torsion angle φ Tangential displacement along the t axis for: Among them, p i is the pressure angle at the meshing point, and the meshing radius is the angle between the center line of the tooth of the flexible sleeve connecting the inner shaft and the positive direction of the y-axis, r m is the pitch radius; therefore, the total intrusion of the i-th tooth is and the tangential displacement along the t axis for: In summary, under the combined action of lateral load and torque, the relative displacement vector of each tooth at the meshing point is for: n, t, and k are unit vectors in the positive directions of the n-axis, t-axis, and k-axis in the local coordinate system nkt, respectively; Step 2.2), calculating the reaction force and reaction moment of the flexible sleeve tooth connection; Step 2.2.1), the tangential stiffness at the meshing point of the i-th tooth of the flexible sleeve connecting the inner shaft in the shaft segment microelement is κ is the proportional coefficient; the contact point of the i-th tooth of the flexible sleeve connecting the inner shaft in the shaft segment microelement is recorded as a, and the contact point on the external tooth in the same meshing pair is recorded as b; the displacement of point a in space is the relative displacement of the meshing point Point b represents the sliding displacement of the slider, which is recorded as It will be updated according to the force state; at this time, the force along the normal direction of the meshing point on the i-th tooth of the flexible sleeve gear connection inner shaft in the shaft segment microelement is for: in, is the Heaviside function, which has the following form: Step 2.2.2), at the mth time step, define a circle with point a(m) as the center and a radius r0 as: The coordinates of points a and b coincide at the initial zero moment when the flexible sleeve is not loaded. Let the vector from a to b be recorded as Then at any time vector r ab for: Step 2.2.3), according to the basic idea of ​​the time advancement method, solve the tangential friction force at the meshing point of the i-th tooth of the flexible sleeve connecting the inner shaft in the shaft segment microelement: Step 2.2.3.1), obtain the sliding displacement corresponding to the slider b (m.1) in the previous iterative calculation result Record it as If it is the first iteration calculation, the sliding displacement of the slider is (0,0); Step 2.2.3.2), according to the tangential relative displacement of contact point a in step m Sliding displacement of the slider in step m.1 Calculate the assumed friction as well as The vector r ab-pre for: Where dx represents the tooth width of the shaft segment microelement, and its value is the same as L in step 1). d Equal, used for integral calculation in subsequent steps; Step 2.2.3.3), if the vector r ab-pre The magnitude is less than the vector r ab The actual friction force in the mth step is equal to the friction force under assumption, and its direction is the same as r ab-pre The same; otherwise, it means that the friction force at the meshing point is equal to the critical friction force, and the direction is the same as r ab-pre In the same direction, friction The calculation formula is: in, is the Heaviside function, which is the same as in step 2.2.1); Step 2.2.3.4), update the sliding displacement of the slider. If no sliding occurs, set If slippage occurs, the slippage of the slider in step m is solved as: So far, the tangential friction force on the microplane at the meshing point of the i-th tooth of the flexible sleeve connecting the inner shaft in the shaft segment microelement is obtained: Decompose it into the t and k directions in the local coordinate system, and the components in the corresponding directions and for: Step 2.2.4), the force on the i-th tooth of the flexible sleeve connecting the inner shaft in the shaft segment microelement is obtained. Since a shaft segment microelement contains a total of z teeth, the force on the microelement is as follows: where y i and z i are the y-axis and z-axis coordinates of the meshing position of the i-th tooth of the flexible sleeve gear connected to the inner shaft in a shaft segment microelement, and further integration along the axial direction is used to obtain the resultant force and torque of the entire flexible sleeve gear structure, as shown below: Step 3), iteratively solving the equations by Newton iteration method; Step 3.1), construct the equilibrium equation: Given the torque T, select the lateral shear amplitude or bending moment amplitude It is called "load amplitude". The load sequence to be solved is constructed with the load amplitude, and the number of cycles to be solved is determined. c , each cycle is T h data points, a total of T c ·T h step. Generally speaking, the number of cycles and the data points in one cycle need to be sufficient until the calculation results converge. The shear load to be solved in step j is or moment load for: The shear load amplitude is When, for a flexible sleeve tooth connection structure with an axial length of L, the equation F(X)=0 for solving the corresponding linear displacement of the jth step is: The magnitude of the moment load is When , the equation F(X)=0 used to solve the corresponding angular displacement of step j is: Step 3.2), solving the equilibrium equation by Newton iteration method; In step 2.1.2), the independent variable X=[x a ,θ z ,φ] T , use Newton iteration method to solve, the iteration format is as follows: When solving iteratively, first give the initial value of the variable in, is a very small non-zero real number, φ0 is the torsion angle when only torque exists, substitute the initial value for iterative solution; when X k+1 With X k When the difference between the two norms of is less than the convergence limit set by the user, the Newton iteration method solution of the jth step is considered to be completed; when the load of the jth step is solved through k-step iterations, the obtained iteration variable is recorded as For moment loads, the angular displacement at step j is solved as For shear load, solve for the linear displacement σ at step j j for: Continue to iterate until the number of iterations set in step 3.1) is completed, and finally a total of T c ·T h Linear or angular displacement of a Step 4), calculating the equivalent stiffness and damping according to the hysteresis loop; Step 4.1), draw the hysteresis loop: According to the shear load obtained in step 3.2) (or moment load ) and the corresponding linear displacement σ j (or angular displacement is ), one by one as data points, with displacement as the horizontal axis, represented by the symbol θ, and load as the vertical axis, the hysteresis loop is drawn; in the hysteresis loop, the minimum displacement -θ s and the maximum displacement θ s The hysteresis loop is divided into two lines, and the line with a larger load value is recorded as M t1 , and the other line is denoted as M b1 , M t1 The direction of the minimum displacement is -θ s To the maximum displacement θ s , M b1 The direction of the maximum displacement θ s To minimum displacement -θ s ; Step 4.2), calculate the equivalent stiffness and damping; According to the hysteresis loop obtained in step 4.1), the equivalent stiffness k is given eq The calculation formula is as follows: The loss factor η is calculated as follows: The damping coefficient c eq The calculation formula is as follows: Where ω is the rotor speed.

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