Calculation method of dynamic mesh stiffness of involute cylindrical gears incorporating inter-tooth wear mechanism

Abstract

The present invention discloses a method for calculating the dynamic meshing stiffness of involute cylindrical gears that incorporates the inter-tooth wear mechanism, and belongs to the field of stability analysis of involute cylindrical gear transmission systems. First, the dynamic inter-tooth wear amount is obtained by the Archard wear formula, and the gear meshing stiffness in a healthy state is obtained by the variable-section cantilever beam deformation theory. The real-time dynamic meshing stiffness containing tooth wear faults that is associated with the inter-tooth wear amount and the gear meshing stiffness is obtained according to the meshing relationship of the gear pair, and the gear wear state is observed in real time through the stiffness values ​​of the two wear cycles before and after. The present invention realizes the calculation of the real-time dynamic meshing stiffness that takes into account the meshing time and inter-tooth wear while considering the wear fault state of the involute cylindrical gear. It helps to clarify the changes in the internal coupling contact mechanism caused by the wear of the tooth surface of the involute cylindrical gear, and provides a new idea for further analysis of the nonlinear vibration characteristics between the tooth surfaces.

Description

Technical Field

[0001] The present invention broadly relates to the fields of vibration excitation, gear wear failure, and system stability design and analysis of high-speed cylindrical gear transmission systems, and particularly relates to a method for calculating the dynamic meshing stiffness of high-speed cylindrical gears taking into account inter-tooth wear. Background Art

[0002] As one of the most commonly used gear transmission forms, cylindrical gears are constantly developing towards high speed and heavy load. With the gradual improvement of safety, reliability and comfort requirements for products and equipment, gear tooth wear, as the main early failure form of gears, is the main source of vibration and noise in high-speed cylindrical gear transmissions. The negative effects it brings in certain important situations have become an urgent problem to be solved. For example:

[0003] (1) In the transmission system of wind turbine gearboxes, cylindrical planetary gears have the advantages of compact structure, high transmission efficiency, smooth movement and strong impact resistance. However, due to the large transmission ratio and high torque of the system itself, wind turbine gearboxes frequently fail, reducing the efficiency of wind power generation. (2) In the transmission system of new energy vehicles, due to the rapid rise of new energy vehicles such as pure electric vehicles and hybrid vehicles, there are higher requirements for vehicle driving quietness, which has led to significant changes in the application structure of gears in electric vehicles. Among them, wheel-side reducers are a typical example. Early prevention of gear failure can enable gear transmission to effectively control noise under high speed and high load.

[0004] The excitation components in cylindrical gear transmission systems are complex and varied. Traditional simulations treat them as rigid couplings connected by springs. Domestic and international researchers have conducted extensive research on gear transmission systems, from dynamic modeling to vibration characteristic analysis. It is worth noting that research on time-varying mesh stiffness, one of the dynamic excitations during gear meshing, is generally considered to be healthy teeth. However, since gears inevitably experience early failure forms such as tooth wear under harsh operating conditions such as high speed and heavy load, it is unreasonable to ignore the impact of tooth wear on time-varying mesh stiffness. Under this dynamic excitation, the system exhibits different vibration characteristics. Given that the probability of nonlinear meshing, such as tooth wear, increases significantly at high speeds, research on time-varying mesh stiffness models that account for time-varying mesh stiffness and tooth surface wear is essential.

[0005] Therefore, it is necessary to propose a calculation method for the meshing stiffness that takes into account the wear contact state of the cylindrical gear tooth surface and the meshing time, so as to more accurately and efficiently analyze and calculate the influence of the real-time dynamic meshing stiffness of high-speed cylindrical gears that integrates the tooth surface wear mechanism on the nonlinear vibration response characteristics of the cylindrical gear. Summary of the Invention

[0006] The present invention aims to provide a stiffness calculation method that considers the wear state of cylindrical gear tooth surfaces and time-varying mesh stiffness. This method allows for more accurate and efficient analysis and calculation of the real-time dynamic mesh stiffness of high-speed cylindrical gears that incorporates the tooth wear mechanism, thereby enabling the determination of gear health. The Archard wear formula is used to calculate the dynamic wear of the meshing teeth, and the variable-section cantilever beam theory is used to calculate the normal mesh stiffness of cylindrical gear teeth under static load. Based on this, a dynamic mesh stiffness function for cylindrical gears that is related to both time and tooth wear is established.

[0007] The technical solution of the present invention is: a method for calculating the dynamic meshing stiffness of involute cylindrical gears that integrates the tooth wear mechanism. First, the Archard wear formula is used to obtain the dynamic wear between meshing teeth. The meshing stiffness model of the cylindrical gear transmission system vibration model is obtained based on the deformation theory of the variable-section cantilever beam. Combined with the motion angle relationship of the meshing gear pair, the inter-tooth meshing stiffness including tooth wear is obtained. The real-time dynamic meshing stiffness of the teeth associated with the meshing time and tooth surface wear is obtained through Fourier series fitting. Finally, the health status of the gear is judged by the return value of the real-time meshing stiffness.

[0008] Furthermore, the specific process of using Archard's wear formula to obtain the dynamic wear between gear teeth is as follows:

[0009] First, the dynamic wear h between the teeth of the general gear meshing is calculated by the Archard wear formula P Specifically: Consider the dynamic load p of the gear meshing P , meshing point slip speed s P , calculate the accumulated wear of the gear teeth after a certain period of load operation according to the following formula, and determine the rotation period for the normally meshing gear teeth to maintain healthy meshing:

[0010] h P,n (α1)=h P,n-1 (α1)+kp P,n-1 (α1)s P (α1)

[0011] Among them, n is the number of wear times, k is the wear coefficient, and α1 is the meshing angle at the meshing point of the driving wheel.

[0012] Furthermore, based on the deformation theory of variable-section cantilever beams, the meshing stiffness model of the cylindrical gear transmission system with tooth wear is obtained: Based on material mechanics and elastic mechanics, the energy method is used to obtain the bending potential energy U stored in the axial compression deformation, bending deformation and shear deformation of the gear teeth. b , shear potential energy U s , compression potential energy U a ;

[0013]

[0014]

[0015]

[0016] Where d is the horizontal distance from the meshing point to the base circle; h is the distance between the meshing point and the symmetry line of the gear teeth; dx is the width of the microsection at a distance x from the meshing point; E is the elastic modulus; G is the shear modulus; I x and A x are the moment of inertia and cross-sectional area of ​​the cross section at a distance x from the meshing point; F a is the lateral force between teeth; F b is the radial force between teeth;

[0017]

[0018] A x =2h x L

[0019]

[0020]

[0021] Among them, R b is the base circle radius, h x is half the length of the microsection at the meshing point x; α is the pressure angle at the meshing point x; ν is the Poisson's ratio; L is the tooth width; α2 is the half-tooth arc of the tooth base circle.

[0022] Furthermore, combined with the rotation angle relationship of the meshing gear pair, the specific process of obtaining the inter-tooth meshing stiffness including tooth wear is as follows:

[0023] The single tooth meshing stiffness k s It is understood as the parallel connection of multiple meshing stiffnesses, the double-tooth meshing stiffness k d It is understood as the parallel connection of the single tooth meshing stiffness of the meshing gear teeth, and the comprehensive meshing time-varying meshing stiffness k(t) of the alternating meshing of single and double teeth of the gear is obtained, which is calculated as follows;

[0024]

[0025]

[0026]

[0027]

[0028]

[0029]

[0030]

[0031] k(t)=k(α1 / ω)=k d (α1 / ω)+k s (α1 / ω)

[0032] Where ω is the angular velocity of the driving wheel; k b,i 、k s,i 、k a,i 、k h,i 、k f,i (i=1,2) are the bending stiffness, shear stiffness, axial compression stiffness, Hertz contact stiffness and matrix deformation stiffness between the front and rear pairs of meshing teeth respectively; δ f represents the tooth displacement caused by the elastic deformation of the matrix; θ f Indicates the central angle of the half tooth on the tooth root circle; u f Indicates the height from the meshing point to the top of the tooth root circle; S f Indicates the arc length corresponding to a gear tooth on the tooth root circle; h r The ratio r of the tooth root radius to the shaft hole radius f / r int ; L * 、M * 、P * and Q * is with θ f and h r The relevant constants conform to the following polynomials and Table 1;

[0033] u f =R b [(α1+α2)sinα1+cosα1]-r f

[0034] S f =2θ f r f

[0035] X(h r ,θ f )=A / θ 2 f +B / h 2 r +C / θ f +D / θ f +Eh r +F

[0036] Table 1

[0037]

[0038] Where X represents L * 、M* 、P * and Q * , constants A, B, C, D, E and F are given in Table 1.

[0039] Furthermore, the real-time dynamic mesh stiffness of the gear teeth, which is associated with the meshing time and tooth surface wear, is obtained through Fourier series fitting. Finally, the health status of the gear is judged by the return value of the real-time mesh stiffness. Specifically, the continuous real-time dynamic mesh stiffness taking into account the meshing time and inter-tooth wear in adjacent meshing cycles is used to make a ratio, and the health status of the gear is observed in real time to find the critical point of complete wear failure of the gear teeth. The expression is as follows:

[0040]

[0041] Among them, k c n (t), k c n-1 (t) is the time-varying meshing stiffness of the gear teeth after two wear cycles; λ is its ratio.

[0042] Furthermore, through sixth-order Fourier series fitting, the short-period tooth meshing stiffness taking into account the tooth surface wear with the pressure angle at the meshing point as the independent variable is converted into the real-time dynamic meshing stiffness of the gear teeth taking into account the time and tooth surface wear for several periods with the meshing time as the independent variable.

[0043] By making full use of the existing relevant foundations on the calculation of static mesh stiffness of cylindrical gears and combining them with the dynamic wear characteristics of the tooth surface, a reasonable and efficient calculation program for the dynamic mesh stiffness of high-speed cylindrical gears that integrates the gear tooth wear mechanism is compiled.

[0044] Based on the time-varying meshing stiffness of the normally paired tooth surfaces, the present invention fully considers the contact state of the gear teeth when gear tooth wear occurs, and takes into account the changes in the real-time meshing stiffness of the meshing gear teeth and the health status of the gears when the gear tooth surfaces are worn in actual transmission.

[0045] The present invention aims at the actual wear state of the tooth surface of the cylindrical gear during high-speed transmission, analyzes the meshing stiffness under the wear state, and obtains the actual meshing stiffness of the cylindrical gear.

[0046] The present invention observes the health status of the gear in real time by taking into account the time-varying comprehensive stiffness of the time and wear status.

[0047] The present invention is based on the dynamic meshing contact process of the tooth surface, comprehensively considers the normal meshing stiffness of the tooth surface and the meshing stiffness of the tooth surface under wear, and establishes a three-dimensional coupling mechanism between the meshing stiffness of the tooth surface and the meshing time and gear tooth wear.

[0048] Beneficial effects of the present invention:

[0049] The present invention realizes the real-time dynamic meshing stiffness calculation taking into account the tooth wear state of cylindrical gears at high speeds, while taking into account the meshing time and tooth surface wear. It helps to clarify the internal coupling contact mechanism of cylindrical gear tooth surface vibration and provides new ideas for further analysis of nonlinear vibration characteristics between tooth surfaces. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 This is the healthy state of the cylindrical gear teeth after multiple wears;

[0051] Figure 2 This is a schematic diagram of the load-bearing contact analysis of the tooth surface of a cylindrical gear;

[0052] Figure 3 is the mesh stiffness of healthy gear mesh;

[0053] Figure 4 is the dynamic time-varying mesh stiffness considering meshing time and tooth wear; (a) is wear 0 times; (b) is wear 1×10 10 times; (c) is the wear 1×10 12 Second-rate;

[0054] Figure 5 The calculation process for observing the actual dynamic mesh stiffness and health status of cylindrical gears;

[0055] Figure 6 Comparison of mesh stiffness between healthy and worn cylindrical gears. DETAILED DESCRIPTION

[0056] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but the protection scope of the present invention is not limited thereto.

[0057] 1. First, analyze the meshing contact relationship of the active and passive tooth surfaces in the high-speed cylindrical gear transmission system. When the gear teeth wear out and fail to mesh, the meshing spacing between the active and passive gears will change. After multiple wears, the tooth surface wear is unevenly distributed along the tooth profile ( Figure 1 ).

[0058] 2. When the gear tooth surface wears, the spacing of the tooth surface profiles involved in the meshing changes. It is necessary to analyze the tooth surface load contact of the actual meshing state of the high-speed cylindrical tooth surface and the tooth surface wear failure state respectively ( Figure 2 ), and the bearing transmission deformation of each tooth surface is obtained according to Archard wear formula (1).

[0059] h P,n (α1)=h P,n-1 (α1)+kp P,n-1 (α1)s P (α1) (1)

[0060] Among them, n is the number of wear times, k is the wear coefficient, p P is the contact pressure of gear meshing; s P is the relative sliding speed at the gear meshing point; α1 is the meshing angle between the driving gear teeth.

[0061] 3. According to the dynamic contact state between teeth, the bending potential energy U stored by the axial compression deformation, bending deformation and shear deformation generated by the wear characteristics of the gear teeth is obtained through the variable cross-section cantilever beam theory of material mechanics as shown in Equations (2) to (4): b , shear potential energy U s , compression potential energy U a .

[0062]

[0063]

[0064]

[0065] Where d is the horizontal distance from the meshing point to the base circle; h is the distance between the meshing point and the symmetry line of the gear teeth; dx is the width of the microsection at a distance x from the meshing point; E is the elastic modulus; G is the shear modulus; I x and A x are the moment of inertia and cross-sectional area of ​​the cross section at a distance x from the meshing point; F a is the lateral force between teeth; F b is the radial force between teeth. The specific expressions of the parameters in equations (2) to (4) are shown in equations (5) to (8).

[0066]

[0067] A x =2h x L (6)

[0068]

[0069]

[0070] Among them, R b is the base circle radius, h x is half the length of the microsection at the meshing point x; ν is the Poisson's ratio; L is the tooth width; α2 is the half-tooth arc of the tooth base circle.

[0071] 4. The single tooth meshing stiffness k s It can be understood as the parallel connection of multiple meshing stiffnesses as shown in formula (9), and the double-tooth meshing stiffness k dIt can be understood as the parallel connection of the single tooth meshing stiffness of the meshing gears as shown in formula (10), and the comprehensive time-varying meshing stiffness k(t) associated with the meshing time of the alternating single and double teeth meshing of the gears as shown in formula (11) is obtained. The system exhibits strong nonlinear characteristics.

[0072]

[0073]

[0074]

[0075]

[0076]

[0077] k(t)=k(α1 / ω)=ks(α1 / ω)+ks(α1 / ω) (14)

[0078] Where ω is the angular velocity of the driving gear, k h is the Hertzian contact stiffness, k b is the bending stiffness, k s is the shear stiffness, k a is the axial compression stiffness, k f The matrix deformation stiffness, the conversion relationship between stiffness and potential energy is shown in Equation (15), the expressions of bending stiffness, shear stiffness and axial compression stiffness are shown in Equations (11) to (13), and the expressions of Hertz stiffness and matrix deformation height are shown in Equations (16) to (17).

[0079]

[0080]

[0081]

[0082] Among them, δ f represents the tooth displacement caused by the elastic deformation of the matrix; θ f Indicates the central angle of the half tooth on the tooth root circle; u f Indicates the height from the meshing point to the top of the tooth root circle; S f Indicates the arc length corresponding to a gear tooth on the tooth root circle; h r γ is the ratio of the tooth root radius to the shaft hole radius f / γ int ;L * 、M * 、P * and Q * is with θ f and h rThe relevant constants conform to the following polynomials (18) to (19) and Table 1.

[0083] u f =R b [(α1+α2)sinα1+cosα1]-r f (18)

[0084] S f =2θ f r f (19)

[0085] X(h r ,θ f )=A / θ 2 f +B / h 2 r +C / θ f +D / θ f +Eh r +F (20)

[0086] Table 1

[0087]

[0088] 5. The real-time meshing stiffness ratio of the two wear cycles is used to obtain the health status of the gear meshing in real time, as shown in formula (18), to determine whether the gear teeth have completely worn out.

[0089]

[0090] Among them, k c n (t), k c n-1 (t) is the time-varying meshing stiffness of the gear teeth after two wear cycles; λ is its ratio.

[0091] In summary, the calculation method of dynamic meshing stiffness of high-speed cylindrical gears integrated with gear tooth wear mechanism proposed in this invention can be expressed as follows: Figure 5 First, the dynamic wear between teeth is calculated by the Archard wear formula, and the gear meshing stiffness in a healthy state is obtained by the deformation theory of a variable-section cantilever beam. According to the meshing relationship of the gear pair, the real-time dynamic meshing stiffness containing tooth wear faults, which is associated with the inter-tooth wear and the gear meshing stiffness, is obtained. The gear wear state is observed in real time through the stiffness values ​​before and after two wear cycles.

[0092] Example

[0093] Taking the parameters of a single-stage cylindrical gear shown in Table 2 as an example, the dynamic mesh stiffness of the cylindrical gear is calculated considering the tooth wear characteristics. Table 2 shows the parameters and motion state quantities of a single-stage cylindrical gear pair.

[0094] Table 2

[0095]

[0096] Figure 3 is the meshing stiffness of the healthy teeth in the cylindrical gear transmission under normal meshing conditions, Figure 4 This is a dynamic, time-varying mesh stiffness that accounts for meshing time and tooth wear. It can be seen that mesh stiffness k(t) decreases with repeated wear cycles. After the wear reaches trillions of times, the mesh stiffness plummets, leading to complete gear failure. Ignoring real-time monitoring of tooth wear will result in significant input errors in gear nonlinear vibration analysis. Figure 6 The comparative relationship between the meshing stiffness of healthy gears and that of gears under different wear times is given.

[0097] Through the above analysis, the authors correlated the tooth surface meshing stiffness with the meshing time and tooth surface wear at the same time, which helps to clarify the internal coupling contact mechanism of cylindrical gear tooth surface vibration, quickly and accurately obtain more realistic dynamic excitation factors, and provide new ideas for further analysis of the nonlinear vibration characteristics between tooth surfaces.

[0098] The series of detailed descriptions listed above are only specific descriptions of feasible implementation methods of the present invention. They are not intended to limit the scope of protection of the present invention. Any equivalent implementation methods or changes that do not deviate from the technical spirit of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for calculating the dynamic meshing stiffness of involute cylindrical gears that incorporates the tooth wear mechanism is characterized by: First, the dynamic wear between meshing teeth is calculated using the Archard wear formula. The meshing stiffness model of the cylindrical gear transmission system vibration model is derived based on the deformation theory of a variable-section cantilever beam. Combined with the kinematic angle relationship of the meshing gear pair, the inter-tooth meshing stiffness including tooth wear is derived. Fourier series fitting is used to obtain the real-time dynamic meshing stiffness of the gears, which is associated with the meshing time and tooth surface wear. Finally, the health status of the gear is determined by the return value of the real-time meshing stiffness. Combined with the rotation angle relationship of the meshing gear pair, the specific process of obtaining the inter-tooth meshing stiffness including tooth wear is as follows: The single tooth meshing stiffness k s It is understood as the parallel connection of multiple meshing stiffnesses, the double-tooth meshing stiffness k d It is understood as the parallel connection of the single tooth meshing stiffness of the meshing gear teeth, and the comprehensive meshing time-varying meshing stiffness k(t) of the alternating meshing of single and double teeth of the gear is obtained, which is calculated as follows; k(t)=k(α1 / ω)=k d (α1 / ω)+k s (α1 / ω) Where ω is the angular velocity of the driving wheel; k b,i 、k s,i 、k a,i 、k h,i and k f,i are the bending stiffness, shear stiffness, axial compression stiffness, Hertz contact stiffness and matrix deformation stiffness between the front and rear pairs of meshing teeth, respectively, where i = 1, 2; δ f represents the tooth displacement caused by the elastic deformation of the matrix; θ f Indicates the central angle of the half tooth on the tooth root circle; u f Indicates the height from the meshing point to the top of the tooth root circle; S f Indicates the arc length corresponding to a gear tooth on the tooth root circle; h r The ratio r of the tooth root radius to the shaft hole radius f / r int ;L * 、M * 、P * and Q * is with θ f and h r The relevant constants conform to the following polynomials and Table 1; u f =R b [(α1+α2)sinα1+cosα1]-r f S f =2θ f r f X(h r ,i f )=A / θ 2 f +B / h 2 r +C / θ f +D / θ f +Eh r +F Table 1 Where X represents L * 、M * 、P * and Q * , constants A, B, C, D, E and F are given in Table 1.

2. The method for calculating the dynamic meshing stiffness of involute cylindrical gears incorporating the tooth wear mechanism according to claim 1 is characterized in that: The specific process of using Archard wear formula to obtain the dynamic wear between gear teeth is as follows: First, the dynamic wear h between the teeth of the general gear meshing is calculated by the Archard wear formula P Specifically: Consider the dynamic load p of the gear meshing P , meshing point slip speed s P , calculate the cumulative wear of the gear teeth after a certain period of load operation according to the following formula, and determine the rotation cycle for which the normally meshing gear teeth maintain healthy meshing: h P,n (α1)=h P,n-1 (α1)+kp P,n-1 (α1)s P (α1) Among them, n is the number of wear times, k is the wear coefficient, and α1 is the meshing angle at the meshing point of the driving wheel.

3. The method for calculating the dynamic meshing stiffness of involute cylindrical gears incorporating the tooth wear mechanism according to claim 1 is characterized in that: According to the deformation theory of variable cross-section cantilever beam, the meshing stiffness model of cylindrical gear transmission system including gear tooth wear is obtained: According to material mechanics and elastic mechanics, the energy method is used to obtain the bending potential energy U stored by the axial compression deformation, bending deformation and shear deformation of the gear teeth. b , shear potential energy U s , compression potential energy U a ; Where d is the horizontal distance from the meshing point to the base circle; h is the distance between the meshing point and the symmetry line of the gear teeth; dx is the width of the microsection at a distance x from the meshing point; E is the elastic modulus; G is the shear modulus; I x and A x are the moment of inertia and cross-sectional area of ​​the cross section at a distance x from the meshing point; F a is the lateral force between teeth; F b is the radial force between teeth; <h2 style=";text-align:left;direction:ltr">A<h2 style=";text-align:left;direction:ltr"> x <h2 style=";text-align:left;direction:ltr"> =2h<h2 style=";text-align:left;direction:ltr"> x <h2 style=";text-align:left;direction:ltr"> L Among them, R b is the base circle radius, h x is half the length of the microsection at the meshing point x; α is the pressure angle at the meshing point x; ν is the Poisson's ratio; L is the tooth width; α2 is the half-tooth arc of the tooth base circle.

4. The method for calculating the dynamic meshing stiffness of involute cylindrical gears incorporating the tooth wear mechanism according to claim 1 is characterized in that: The real-time dynamic mesh stiffness of the gear teeth, which is associated with the meshing time and tooth surface wear, is obtained through Fourier series fitting. Finally, the health status of the gear is judged by the return value of the real-time mesh stiffness. Specifically, the ratio of the continuous real-time dynamic mesh stiffness taking into account the meshing time and inter-tooth wear in adjacent meshing cycles is used to observe the health status of the gear in real time and find the critical point of complete wear failure of the gear teeth. The expression is as follows: Among them, k c n (t), k c n-1 (t) is the time-varying meshing stiffness of the gear teeth after two wear cycles; λ is its ratio.

5. The method for calculating the dynamic meshing stiffness of involute cylindrical gears incorporating the tooth wear mechanism according to claim 4 is characterized in that: Through the sixth Fourier series fitting, the short-period tooth meshing stiffness taking into account the tooth surface wear with the pressure angle at the meshing point as the independent variable is converted into the real-time dynamic meshing stiffness of the gear teeth taking into account the time and tooth surface wear for several periods with the meshing time as the independent variable.

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