Cylindrical spur gear inversion reconstruction design method considering tooth surface thermal deformation

Abstract

The invention discloses a spur gear inversion reconstruction design method considering tooth surface thermal deformation, and belongs to the field of gear transmission design. The method comprises the following steps: firstly, establishing a full tooth profile model containing an involute tooth profile and a tooth root transition curve according to a gear generating principle; then calculating temperature field distribution during steady-state work of the gear based on a heat conduction differential equation without an internal heat source; solving the thermal deformation of the base circle through a thermoelastic mechanics theory, and further obtaining the thermal deformation of the tooth surface distributed along the meshing line; and finally, the deviation minimization of the hot-state tooth profile and the target tooth profile is taken as a target, and the pre-deformation tooth profile required at the normal temperature is solved through inversion of an optimization algorithm. Through the design thought of active pre-compensation, the gear can automatically recover the ideal tooth shape when working in the thermal state, the problem that the gear transmission precision is reduced under the high-temperature working condition is effectively solved, the transmission performance of the gear under the high-speed heavy-load working condition is remarkably improved, and the service life of the gear under the high-speed heavy-load working condition is remarkably prolonged.

Description

Technical Field

[0001] This invention belongs to the field of gear transmission design technology, specifically relating to a method for inverting and reconstructing cylindrical spur gears that takes into account thermal deformation of the tooth surface. It is particularly suitable for the tooth profile design of cylindrical spur gears that need to maintain high transmission accuracy under operating temperature rise. Background Technology

[0002] Gear transmission systems are widely used in critical equipment such as automotive gearboxes, industrial reducers, wind turbine generators, and aerospace vehicles due to their advantages of accurate transmission ratios, high efficiency, and compact structure. However, during gear meshing, the relative sliding and rolling between the tooth surfaces generates frictional heat, causing the gear tooth temperature to rise. Due to the thermal expansion and contraction characteristics of materials, this temperature rise induces microscopic and macroscopic thermal deformation of the gear teeth, thereby altering the precisely designed tooth profile profile. This alteration disrupts the smoothness of the transmission, exacerbates vibration and noise, and affects the service life of the gears.

[0003] Under harsh operating conditions such as high speed and heavy load, the instantaneous temperature rise of gear pairs is more significant. Furthermore, due to the discontinuity of the meshing process and the existence of thermal inertia, a complex non-uniform temperature field is formed in the gear body and even the entire transmission system. The asymmetric thermal stress induced by this non-uniform thermal field distribution can cause unpredictable and complex thermoelastic deformation of the gears, thereby introducing transmission errors and directly damaging meshing accuracy, load distribution uniformity, and system reliability.

[0004] Numerical analysis and practical application show that under high-temperature conditions, the thermal deformation of gear tooth surfaces is significant, reaching tens of micrometers. This level of deformation exceeds the effective compensation range of traditional tooth profile modification techniques (such as profile modification and tooth direction modification). Traditional modification techniques are mainly based on static or quasi-static design concepts, providing minor compensation for manufacturing errors, installation errors, and elastic deformation under static loads. Their compensation capabilities are limited, and they cannot accurately predict and offset the nonlinear, large-scale thermal deformation effects caused by complex temperature fields.

[0005] Currently, while some studies have addressed the problem of thermal deformation, their methods are mostly limited to the "forward modeling" approach. This involves first predicting the potential thermal deformation under specific operating conditions through complex thermo-mechanical coupling simulations, and then performing corresponding compensation and shaping. This method is cumbersome and has limited accuracy in predicting and compensating for complex nonlinear thermal deformations. Summary of the Invention

[0006] The purpose of this invention is to solve the problems that existing gear design methods do not fully consider thermal deformation during operation and its impact on meshing performance, and that the "forward compensation" approach is difficult to accurately and efficiently offset large-scale nonlinear thermal deformation. Therefore, this invention proposes a cylindrical spur gear design method that can accurately invert the thermal deformation process and actively reconstruct the pre-deformed tooth profile at room temperature. This allows the gear to recover its tooth profile exactly to the ideal theoretical tooth profile after thermal expansion at a specific operating temperature, thereby ensuring transmission accuracy and smoothness.

[0007] To achieve the above objectives, the technical solution provided by this invention is:

[0008] A method for inverting and reconstructing cylindrical spur gears that considers thermal deformation of the tooth surface is provided, including the following steps:

[0009] Step 1, Gear full tooth profile modeling: Based on the gear generating principle, the involute tooth profile of the external meshing gear is generated, and the root transition curve equation of the external meshing gear is established by using homogeneous coordinate transformation and tooth profile normal method, so as to realize the geometric modeling of the gear full tooth profile.

[0010] Step 2, steady-state temperature field analysis: Divide the meshing position and contact line of the established gear tooth profile, and calculate the temperature field inside the base circle of the gear under steady-state conditions based on the thermal conductivity differential equation in cylindrical coordinates without internal heat source.

[0011] Step 3, Calculation of thermal deformation of base circle: Based on the established steady-state temperature field inside the gear base circle, and according to thermoelasticity and plane stress theory, the base cylinder is regarded as a plane stress state. The thermal deformation equation inside the base cylinder is listed and solved according to the deformation compatibility conditions to obtain the thermal deformation amount of the base circle.

[0012] Step 4, Quantification of thermal deformation along the meshing line: Based on the obtained thermal deformation amount of the base circle, derive the polar coordinate parameter equation of the actual involute tooth profile after thermal deformation, and solve for the thermal deformation amount distributed along the meshing line.

[0013] Step 5, pre-deformed tooth profile inversion solution: With the goal of minimizing the deviation between the tooth profile of the gear after thermal expansion at the working temperature and the ideal theoretical tooth profile, the inversion objective function is defined, and the required pre-deformed tooth profile at room temperature is solved by optimization algorithm.

[0014] Furthermore, in step 1, the tooth root transition curve of the gear is generated by machining with a double-rounded rack-type tool, and the full tooth profile modeling is achieved in the following way:

[0015] Step 1.1: Based on the geometric parameters of the double-rounded corner rack cutter, construct the Cartesian coordinate parametric equation of the tooth root transition curve with the cutter tooth profile angle as the parameter through homogeneous coordinate transformation and tooth profile normal method.

[0016] Step 1.2: Based on the gear generating principle, generate the involute tooth profile equation with the generating angle as the parameter;

[0017] Step 1.3: Calculate the connection point between the tooth root transition curve and the involute tooth profile, and combine the involute tooth profile, tooth root transition curve, tooth tip arc and tooth root arc to form a complete tooth profile model.

[0018] Furthermore, in step 1.3, when synthesizing the full tooth profile, a geometric interference check is performed through a parametric program to ensure that the tooth profile is free of distortion.

[0019] Furthermore, in step 2, the process of calculating the temperature field within the base circle under steady-state conditions includes:

[0020] Treating the base cylinder of the spur gear as an axisymmetric body, and assuming that the temperature change along the axial direction is zero and only depends on the radial coordinate, the heat conduction differential equation in the cylindrical coordinate system is simplified to an ordinary differential equation.

[0021] Substituting the first type of boundary conditions, the radial temperature distribution function within the base circle is obtained by solving.

[0022] Furthermore, in step 4, the amount of thermal deformation distributed along the meshing line is determined in the following way:

[0023] Step 4.1: Based on the obtained thermal deformation amount of the base circle, determine the radius of the new base circle after thermal deformation, and obtain the theoretical involute based on the new base circle;

[0024] Step 4.2: Taking into account the thermal deformation of tooth thickness and tooth height, the actual tooth profile polar coordinate parametric equation after thermal deformation is obtained;

[0025] Step 4.3: Quantitatively characterize the thermal deformation of the tooth surface by calculating the normal distance between the actual tooth profile and the cold-state theoretical involute tooth profile.

[0026] Furthermore, in step 5, the process of defining the inversion objective function and optimizing the solution includes:

[0027] Step 5.1, define the inversion objective function as the sum of squared deviations of multiple sampling points on the tooth profile, where the deviation of a single point is jointly limited by the thermal expansion scaling factor, the pre-deformed tooth profile coordinates, and the ideal tooth profile coordinates;

[0028] Step 5.2: Using the coordinates of the pre-deformed tooth profile as design variables, a numerical optimization algorithm is used for iterative optimization to minimize the objective function. The final converged tooth profile coordinates are the ideal inverted tooth profile.

[0029] Furthermore, in step 5.2, the numerical optimization algorithm is gradient descent, genetic algorithm, or sequential quadratic programming.

[0030] The advantages of this invention are:

[0031] 1. The cylindrical spur gear inversion reconstruction design method considering tooth surface thermal deformation provided by this invention fundamentally changes the design paradigm of gear thermal deformation compensation, upgrading from the traditional "passive forward compensation" to "active inversion reconstruction." By accurately establishing a full tooth profile model of the gear and calculating its thermal deformation law under working conditions, the pre-deformed tooth profile at room temperature is solved in reverse, so that the compensation amount for thermal deformation is pre-included in the gear during the manufacturing stage. When the gear operates under specific working conditions and reaches the expected working temperature, its tooth profile can automatically and accurately recover to the ideal theoretical tooth profile due to the thermal expansion effect of the material. This method solves the transmission error problem caused by thermal deformation from the source of geometric design, significantly improving the transmission accuracy and meshing smoothness of the gear under hot working conditions.

[0032] 2. Traditional static profile modification techniques are difficult to effectively deal with nonlinear large-scale deformation caused by complex temperature fields. However, this method uses systematic thermo-mechanical coupling analysis to accurately quantify the thermal deformation distribution along the meshing line and perform inverse design based on this. This can effectively offset the tooth profile distortion caused by significant temperature rise under harsh conditions such as high speed and heavy load, ensure the uniformity of load distribution, reduce vibration and noise, and extend the service life of gears.

[0033] 3. The design process constructed in this invention has a clear logic and well-defined steps, decomposing the complex multiphysics coupling problem into a series of interconnected and efficiently calculable steps. This process combines a rigorous theoretical foundation with good engineering applicability, greatly improving the efficiency and reliability of gear design considering thermal deformation factors, and providing an effective new solution to the long-standing problem of maintaining thermal accuracy in the transmission field of high-end equipment. Attached Figure Description

[0034] The above and / or other features and advantages of the present invention will become more readily understood from the following description with reference to the accompanying drawings, in which:

[0035] Figure 1 This is a flowchart of the inversion and reconstruction design method for cylindrical spur gears that takes into account the thermal deformation of the tooth surface, according to the present invention.

[0036] Figure 2 This is a schematic diagram showing the normal distance between the actual tooth profile and the theoretical tooth profile after thermal deformation of the gear in this invention.

[0037] Figure 3 This is a schematic diagram comparing the ideal inverted tooth profile obtained by optimization solution in this invention with the theoretical tooth profile. Detailed Implementation

[0038] The present invention will now be described in detail with reference to the accompanying drawings and exemplary embodiments thereof. It should be noted that the following detailed description of the present invention is for illustrative purposes only and is not intended to limit the scope of the invention.

[0039] Reference Figure 1 The cylindrical spur gear inversion reconstruction design method considering tooth surface thermal deformation, as an exemplary embodiment of the present invention, achieves active compensation for gear thermal deformation through accurate modeling, thermal deformation analysis, and inversion solution. The implementation method of each step is described in detail below.

[0040] Step S1: Precise modeling of the entire gear tooth profile

[0041] Accurate full-profile modeling is based on the gear generating principle. The gear machining method and the type of cutting tool used directly affect the geometry of the tooth root transition curve. When using rack-type cutting tools for gear machining, the tooth root transition curve is a complex yet smooth geometric curve due to the double fillet structure designed at the top of the cutting tool teeth. This curve can be divided into three characteristic segments: the first and second segments are equidistant curves extending the involute, which have the same curvature characteristics mathematically; the third segment is a standard tooth root arc, used to achieve a smooth transition at the tooth root and avoid stress concentration.

[0042] For double-rounded toothed rack cutters, the tooth geometry is defined by a series of key geometric parameters, which together determine the shape of the final machined tooth root transition curve.

[0043] Based on the geometric parameters of the double-rounded corner rack cutter, the Cartesian coordinate parametric equation of the tooth root transition curve is constructed using homogeneous coordinate transformation and the tooth profile normal method, as follows:

[0044] (1)

[0045] In the formula, α ′ ∈ [ α , 9 0 ° ] For the tooth profile angle of the cutting tool, The base circle pressure angle, For the angle of the cutting tool, The radius of the tool fillet. Let the radius be any point on the cutting tool. When the cutting tool tooth profile angle is equal to the rack tooth profile angle, the tangent point connecting the transition curve and the involute tooth profile is obtained. The relationships between other parameters are as follows:

[0046] (2)

[0047] (3)

[0048] In the formula, This is the actual distance from the center of the fillet at the tip of the tool to the centerline. This is the theoretical distance from the center of the fillet at the tip of the tool to the centerline. For displacement coefficients, The distance from the center of the fillet at the top of the tool to the center line of the tool tooth groove. This is the tool module.

[0049] After generating the tooth root transition curve, the theoretical involute is generated based on the base circle according to the involute equation. The coordinates of any point on the involute are... From the angle of the exhibition Decide:

[0050] { x i n v = r b [ s i n ( c − d ) − c c o s ( c − d ) ] y i n v = r b [ c o s ( c − d ) + c s i n ( c − d ) ] (4)

[0051] In the formula, As the independent variable, α k ∈ [ α M , a r c c o s ( r b / r a ) ] It is the pressure angle of the pitch circle. It is the pressure angle at the junction of the tooth root transition curve and the involute. It is the base circle radius of the gear. It is the radius of the gear tooth tip circle. It is the angular offset. ,and , It is the pitch circle tooth thickness. It is the pitch circle radius. It is the pressure angle of the pitch circle.

[0052] Next, the entire tooth profile is synthesized. The generated involute tooth profile, root transition curve, addendum arc, and root arc are combined. During this process, the connection point between the root transition curve and the involute tooth profile is calculated. Its coordinates The relationship is as follows:

[0053] (5)

[0054] In the formula, The angle at which the tool turns at the intersection of the transition curve and the involute.

[0055] according to radius at point Calculate its pressure angle This parameter is crucial for defining the effective segment and transition curve of the involute. Finally, the calculated involute tooth profile (whose parameter range is from...) Pressure angle of the point Pressure angle to tooth tip circle The tooth root transition curve, tooth tip arc, and tooth root arc are combined to construct a complete, continuous, and smooth gear tooth profile model. Throughout the entire tooth profile synthesis process, geometric interference verification can be continuously performed through a parametric program to ensure that the tooth profile is free of distortion and that all segments are smoothly connected.

[0056] This step, through precise modeling, fully reproduces the actual geometry of the gear, particularly accurately describing the geometric characteristics of the stress concentration region at the tooth root. This provides reliable geometric input for subsequent accurate calculations of the temperature field and thermal deformation, avoiding computational errors caused by model simplification.

[0057] Step S2: Steady-state temperature field analysis

[0058] Discretize the gear model established in step S1 and divide the meshing positions and contact lines. Consider the base cylinder of the spur gear as an axisymmetric body; the heat conduction differential equation in cylindrical coordinates without an internal heat source is:

[0059] (6)

[0060] In the formula, Thermal conductivity, For angle, Here are the radial coordinates within the gear base cylinder. Let the direction be along the length of the cylinder. Assuming the gear teeth are in ideal meshing condition and there is no eccentric load along the tooth surface, the axial temperature change can be assumed to be zero; and within the steady-state temperature field, the temperature is uniformly distributed on the inner and outer walls of the base cylinder, meaning the axial temperature change is considered zero and only related to the radial coordinate. Based on this, the heat conduction differential equation in cylindrical coordinates without an internal heat source can be simplified to:

[0061] (7)

[0062] Substituting the first type of boundary condition, i.e. when hour, ; hour, The radial temperature distribution function within the base circle is obtained by solving for:

[0063] t ( r ) = t ( r 2 ) + [ t ( r b ) − t ( r 2 ) ] l n r − l n r 2 l n r b − l n r 2 (8)

[0064] In the formula, Where is the gear shaft radius. The base circle radius of the gear. To stabilize the gear shaft temperature after operation, this temperature field is the direct input for subsequent thermal deformation calculations.

[0065] This step, based on an axisymmetric simplified model and the steady-state heat conduction assumption, transforms the complex three-dimensional unsteady heat transfer problem into a one-dimensional steady-state problem that can be analytically solved. This method significantly improves computational efficiency while maintaining the accuracy of engineering calculations, making the inversion design process practical for engineering applications.

[0066] Step S3: Calculation of thermal deformation of the base circle

[0067] Based on the steady-state temperature field obtained in step S2, and according to thermoelasticity and plane stress theory, the base cylinder is considered to be in a plane stress state. Since the temperature field is axisymmetrically distributed, the radial displacement... Only depends on Furthermore, the tangential displacement is zero. Therefore, based on the deformation compatibility conditions, the equation for the thermal deformation inside the base cylinder is as follows:

[0068] (9)

[0069] In the formula, For radial normal strain, For circumferential (or tangential) normal strain, and For shear strain, For elastic modulus, Poisson's ratio, For temperature difference, It is radial normal stress. It is the circumferential normal stress.

[0070] Combining equations (8) and (9), the thermal deformation of the base circle after the spur gear has stabilized is obtained:

[0071] x b = r b l t ( r 2 ) + l r b 3 [ t ( r b ) − t ( r 2 ) ] r b 2 − r 2 2 − l r b [ t ( r b ) − t ( r 2 ) ] 2 ( l n r b − l n r 2 ) (10)

[0072] This step transforms the temperature field into a structural deformation field through thermo-mechanical coupling analysis. The plane stress assumption conforms to the stress characteristics of a thin-plate gear structure, and the deformation compatibility condition ensures the continuity of the displacement field, making the calculation results both consistent with physical laws and meet engineering accuracy requirements.

[0073] Step S4: Quantification of thermal deformation of meshing lines

[0074] First, based on the base circle thermal deformation amount obtained in step S3 Determine the radius of the new base circle after thermal deformation. According to the principle of involute formation, a change in the base circle radius will lead to a change in the involute tooth profile. The theoretical involute polar coordinate parametric equation for the new base circle is:

[0075] (11)

[0076] In the formula, the subscript 1 represents the new base circle. Let the radius be the new base circle. For Let be the pressure angle at any point on the involute of the base circle radius. Let be the involute function of the new base circle, that is, the angle of development at any point on its involute tooth profile.

[0077] Next, taking into account the thermal deformation of tooth thickness and tooth height, the actual polar coordinate parametric equation of the tooth profile after thermal deformation is obtained as follows:

[0078] r k 2 = r k + D r k = r b + x b c o s α k + D T l r b ( 1 − c o s α k ) c o s α k i k 2 = i k − D i k = i n v α k − D T l c o s α k ( s r k r − 2 r k ( i n v α k − i n v α ) ) 2 [ r b + x b c o s α k + D T l r b ( 1 − c o s α k ) ] (12)

[0079] In the formula, the subscript 2 represents the tooth profile. For Let be the pressure angle at any point on the involute of the base circle radius. The pressure angle of the pitch circle before temperature change. For the pitch circle tooth thickness, Let be the radius of any point on the tooth profile before temperature change. Let be the development angle at any point on the tooth profile after thermal deformation. Any point on the tooth profile before temperature change The original display angle, For point Radial deformation at the location For point Total angular displacement caused by thermal deformation.

[0080] Any point on the working side tooth profile Corresponding to heat deformation The radius of the point is . The angle with the y-axis is ,in and They are represented as follows:

[0081] (13)

[0082] In the formula, For any point on the working side of the involute radius and Angle between axes for Tooth thickness, The pitch circle radius of the gear. for Point pressure angle, This refers to the change in the included angle caused by thermal deformation. This refers to the circumferential thermal deformation. Any point on the tooth profile before thermal deformation The radial coordinate (i.e., the distance from the point to the center of the gear).

[0083] Finally, to quantitatively describe the degree of thermal deformation, the normal distance between the actual gear tooth profile and the theoretical involute tooth profile in the cold state is taken. The amount of thermal deformation on the tooth surface is expressed by the following formula:

[0084] D f t ≈ ( i k 2 − i k 1 ) r k 1 c o s α k = − D T l r b ( r b + x b ) × [ s r − 2 ( i n v α k − i n v α ) ] 2 [ r b + x b c o s α k + D T l r b ( 1 − c o s α k ) ] (14)

[0085] The total thermal deformation distribution along the meshing line is as follows Figure 2 As shown, the quantification results provide a clear compensation target for the inversion design.

[0086] This step, through rigorous geometric derivation, precisely transfers the thermal expansion of the base circle to the deformation of the tooth surface profile, and quantifies the degree of deformation using the explicit indicator of normal distance. This quantification method intuitively reflects the impact of thermal deformation on meshing performance, providing accurate input for inversion design.

[0087] Step S5: Solving the inversion of the pre-deformed tooth profile

[0088] The objective is to minimize the deviation between the tooth profile of the gear after thermal expansion at operating temperature and the ideal theoretical tooth profile. The first... The deviation of the point is ,in This is the thermal expansion scaling factor, which is set to 0.994036 in this embodiment. The coefficient of thermal expansion of the gear is taken as 1.2e-0.5 in this embodiment. For the first The pre-deformed tooth profile coordinates of the point, For the first The ideal tooth profile coordinates of the points. The inversion objective function is defined as the sum of squared deviations of multiple sampling points on the tooth profile, particularly all sampling points, and its expression is:

[0089] (15)

[0090] In the formula, This represents the number of sampling points on the tooth profile.

[0091] The coordinates of the pre-deformed tooth profile are used as design variables, and iterative optimization is performed using a numerical optimization algorithm. Optimization algorithms can include gradient descent, genetic algorithms, or sequential quadratic programming. By continuously adjusting the geometric coordinates of the pre-deformed tooth profile, the objective function is optimized. Minimize. When the optimization algorithm converges, When the minimum value is reached, the obtained tooth profile coordinates are the desired ideal inverted tooth profile (pre-deformed tooth profile).

[0092] This step transforms the complex compensation design into a clear mathematical optimization problem by establishing an "objective-inversion" optimization model. Utilizing the powerful search capabilities of numerical optimization algorithms, the optimal pre-compensation tooth profile is automatically solved.

[0093] A comparison between the ideal inverted tooth profile (pre-deformed tooth profile) obtained by the optimized solution and the theoretical tooth profile (pre-deformed and expanded tooth profile). Figure 3 As shown, a cylindrical spur gear with 27 teeth, a module of 3mm, and a pressure angle of 20° is used as an example for design verification. The "original tooth profile" in the figure is the design reference tooth profile. Based on this original tooth profile, the "pre-deformed tooth profile," i.e., the inverted tooth profile, is designed by inversion using the method of this invention. It can be seen that the inverted tooth profile is no longer a standard tooth profile at room temperature, but rather a special geometric contour that includes thermal deformation compensation. Then, thermal expansion simulation is performed on the inverted tooth profile for verification. The tooth profile formed after thermal expansion due to a working temperature rise (500℃) is shown as the "pre-deformed expanded" tooth profile in the figure. The verification results show that the thermally expanded tooth profile highly matches the original reference tooth profile. This closed loop fully verifies the correctness and effectiveness of the inversion design method of this invention: the pre-deformed tooth profile designed by this method can accurately recover to the expected ideal tooth profile under working hot conditions. The scatter coordinate data of the inverted tooth profile is output and can be directly used for CNC machining.

[0094] Therefore, this invention, through the aforementioned complete inversion and reconstruction design process, achieves a fundamental shift from passive compensation to active design. Compared with traditional gear design methods, this invention can achieve high-precision compensation for thermal deformation on the order of tens of micrometers, effectively solving the problem of decreased transmission accuracy caused by gear temperature rise under harsh operating conditions such as high speed and heavy load. By pre-embedding thermal deformation compensation in the room temperature tooth profile, it ensures that the gear can automatically recover to the ideal tooth profile at operating temperature, significantly improving transmission smoothness, load distribution uniformity, and service life.

[0095] Finally, it should be noted that the features mentioned and / or shown in the above description of exemplary embodiments of the present invention can be combined in the same or similar manner with one or more other embodiments, combined with or substituted for corresponding features in other embodiments. These combined or substituted technical solutions should also be considered to be included within the scope of protection of the present invention.

Claims

1. A method for inverting and reconstructing cylindrical spur gears considering thermal deformation of the tooth surface, characterized in that, Includes the following steps: Step 1, Gear full tooth profile modeling: Based on the gear generating principle, the involute tooth profile of the external meshing gear is generated, and the root transition curve equation of the external meshing gear is established by using homogeneous coordinate transformation and tooth profile normal method, so as to realize the geometric modeling of the gear full tooth profile. Step 2, steady-state temperature field analysis: Divide the meshing position and contact line of the established gear tooth profile, and calculate the temperature field inside the base circle of the gear under steady-state conditions based on the thermal conductivity differential equation in cylindrical coordinates without internal heat source. Step 3, Calculation of thermal deformation of base circle: Based on the established steady-state temperature field inside the gear base circle, and according to thermoelasticity and plane stress theory, the base cylinder is regarded as a plane stress state. The thermal deformation equation inside the base cylinder is listed and solved according to the deformation compatibility conditions to obtain the thermal deformation amount of the base circle. Step 4, Quantification of thermal deformation along the meshing line: Based on the obtained thermal deformation amount of the base circle, derive the polar coordinate parameter equation of the actual involute tooth profile after thermal deformation, and solve for the thermal deformation amount distributed along the meshing line. Step 5, pre-deformed tooth profile inversion solution: With the goal of minimizing the deviation between the tooth profile of the gear after thermal expansion at the working temperature and the ideal theoretical tooth profile, the inversion objective function is defined, and the required pre-deformed tooth profile at room temperature is solved by optimization algorithm.

2. The cylindrical spur gear inversion and reconstruction design method according to claim 1, characterized in that, In step 1, the tooth root transition curve of the gear is generated by machining with a double-rounded rack cutter, and the full tooth profile modeling is achieved in the following way: Step 1.1: Based on the geometric parameters of the double-rounded corner rack cutter, construct the Cartesian coordinate parametric equation of the tooth root transition curve with the cutter tooth profile angle as the parameter through homogeneous coordinate transformation and tooth profile normal method. Step 1.2: Based on the gear generating principle, generate the involute tooth profile equation with the generating angle as the parameter; Step 1.3: Calculate the connection point between the tooth root transition curve and the involute tooth profile, and combine the involute tooth profile, tooth root transition curve, tooth tip arc and tooth root arc to form a complete tooth profile model.

3. The cylindrical spur gear inversion and reconstruction design method according to claim 2, characterized in that, In step 1.3, when synthesizing the full tooth profile, a geometric interference check is also performed through a parametric program to ensure that the tooth profile is free of distortion.

4. The cylindrical spur gear inversion and reconstruction design method according to claim 1, characterized in that, Step 2, the process of calculating the temperature field inside the base circle under steady-state conditions includes: Treating the base cylinder of the spur gear as an axisymmetric body, and assuming that the temperature change along the axial direction is zero and only depends on the radial coordinate, the heat conduction differential equation in the cylindrical coordinate system is simplified to an ordinary differential equation. Substituting the first type of boundary conditions, the radial temperature distribution function within the base circle is obtained by solving.

5. The cylindrical spur gear inversion and reconstruction design method according to claim 1, characterized in that, In step 4, the amount of thermal deformation distributed along the meshing line is determined in the following way: Step 4.1: Based on the obtained thermal deformation amount of the base circle, determine the radius of the new base circle after thermal deformation, and obtain the theoretical involute based on the new base circle; Step 4.2: Taking into account the thermal deformation of tooth thickness and tooth height, the actual tooth profile polar coordinate parametric equation after thermal deformation is obtained; Step 4.3: Quantitatively characterize the thermal deformation of the tooth surface by calculating the normal distance between the actual tooth profile and the cold-state theoretical involute tooth profile.

6. The cylindrical spur gear inversion and reconstruction design method according to claim 1, characterized in that, Step 5, defining the inversion objective function and the optimization solution process, includes: Step 5.1, define the inversion objective function as the sum of squared deviations of multiple sampling points on the tooth profile, where the deviation of a single point is jointly limited by the thermal expansion scaling factor, the pre-deformed tooth profile coordinates, and the ideal tooth profile coordinates; Step 5.2: Using the coordinates of the pre-deformed tooth profile as design variables, a numerical optimization algorithm is used for iterative optimization to minimize the objective function. The final converged tooth profile coordinates are the ideal inverted tooth profile.

7. The cylindrical spur gear inversion and reconstruction design method according to claim 6, characterized in that, In step 5.2, the numerical optimization algorithm is gradient descent, genetic algorithm, or sequential quadratic programming.

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