A method for designing non-uniform thermally adaptable clearances for Roots rotors and related mechanical devices.

Abstract

This invention discloses a design method and related mechanical devices for non-uniform thermally adaptive clearance of Roots rotors. The design method, based on rotor meshing characteristics, constructs a clearance-free theoretical rotor profile and discretizes it to obtain a set of discrete points. A thermo-structural coupling analysis is performed using the temperature field as a load to solve the rotor's thermal deformation law and extract the thermal coordinate parameters of each discrete point. Combined with a preset target operating clearance, the cold-state normal compensation amount corresponding to each discrete point is calculated. Based on the cold-state normal compensation amount, the theoretical discrete points are reconstructed by normal offset, generating an actual rotor profile with non-uniform clearance characteristics. This invention achieves non-uniform clearance pre-compensation from the design source by combining the rotor's thermal deformation distribution law, and performs differentiated correction on the initial cold-state clearance, effectively avoiding friction and jamming faults caused by thermal deformation. It significantly improves the operating stability and overall service life of the Roots rotor under extreme conditions such as high pressure differential and high speed, adapting to the needs of multiple operating scenarios.

Description

Technical Field

[0001] This invention relates to the field of fluid machinery technology, specifically to a method for designing a non-uniform thermally adaptable clearance for a Roots rotor and related mechanical devices. Background Technology

[0002] Roots compressors and vacuum pumps play a crucial role in semiconductor manufacturing, photovoltaics, petrochemicals, nuclear power, and large-scale scientific facilities due to their core advantages of compact structure, high pumping speed, lack of internal compression, and adaptability to harsh process environments. The structure of a Roots compressor or vacuum pump mainly consists of two figure-eight shaped, vertically arranged rotor assemblies, along with a matching housing, end caps, drive motor, sealing components, and cooling components. The motor drives the rotor assemblies to rotate synchronously in opposite directions. Through the periodic changes in volume between the rotors and between the rotor and the housing, gas is drawn in, transported, and discharged. To balance performance and safety, existing equipment, during cold assembly, has precision clearances reserved between rotors and between the rotor and the housing.

[0003] However, under extreme conditions such as high pressure differential and high speed, the enormous heat of compression generated by the isochoric compression of gas can significantly raise the rotor temperature. Due to the rotor's complex geometry and mass distribution, its internal temperature field is extremely uneven, leading to significantly different non-uniform thermal expansion in various parts. This thermally induced deformation directly changes the original mating clearance preset in the cold state. This thermally induced clearance reduction is the main cause of rotor jamming, decreased equipment reliability, and even failure.

[0004] In existing technologies, there are various typical cooling solutions to suppress the thermal expansion of the rotor or housing. For example, for cooling housing components, passive or active cooling of the housing is achieved by installing a water jacket inside the housing or using forced air cooling, indirectly reducing the rotor temperature. For cooling the medium in the pump chamber or compression chamber, low-temperature gas is introduced into the chamber through counter-current cooling technology to lower the temperature of the medium inside the chamber and reduce the generation of compression heat. For cooling bearing components, oil is sprayed onto the bearing to achieve lubrication and remove the frictional heat generated during bearing operation, preventing the bearing heat from being conducted to the rotor. In addition, some research has attempted to directly cool the rotor itself to more precisely control the rotor temperature. For example, Chinese invention patent CN118881557B discloses "A rotor temperature control system and control method for a Roots vacuum pump". This technical solution embeds a wireless temperature measuring chip inside the rotor to monitor the rotor temperature distribution in real time, and then adjusts the flow rate of the circulating coolant entering the hollow rotor based on the monitored temperature data to achieve active control of the rotor temperature, thereby suppressing the thermal expansion of the rotor. However, the above solutions all focus on "thermal suppression" during equipment operation, that is, reducing the temperature of the rotor or related components through external cooling intervention, thereby indirectly maintaining the preset fit clearance. These methods often require additional complex systems, such as circulating cooling systems, wireless temperature measurement and automatic control systems, and oil-lubricated cooling systems. This not only significantly increases the manufacturing cost, assembly difficulty, and maintenance cost of the equipment, but also increases the potential for failure and reduces the overall operational stability of the equipment. Furthermore, under extreme conditions such as sudden temperature changes and extremely uneven rotor temperature distribution, the response speed of external cooling intervention is slow, and the cooling effect cannot quickly cover all parts of the rotor, failing to suppress the non-uniform thermal expansion of the rotor in a timely manner. This makes it difficult to meet the high reliability and high stability requirements of high-end applications for Roots compressors and vacuum pumps. Summary of the Invention

[0005] The purpose of this invention is to address the problems in the prior art by providing a non-uniform thermal adaptation clearance design method and related mechanical device for Roots rotors. This method compensates for the non-uniform clearance of the rotor from the design stage by predicting the temperature field and thermal deformation distribution of the rotor under extreme operating conditions and correcting the initial cold clearance accordingly. This ensures that the actual clearance of the rotor tends to be uniform and meets the optimal operating requirements after reaching thermal equilibrium, thereby improving the reliability and lifespan of the equipment under extreme operating conditions.

[0006] To achieve the above objectives, the present invention provides the following technical solution: Firstly, a method for designing non-uniform thermally adaptable clearances for Roots rotors is provided, including: Based on the Roots rotor meshing principle, a pair of fully meshed, gapless theoretical rotor profiles are generated, and the generated theoretical rotor profiles are discretized to obtain the discrete point set of the theoretical rotor profiles; The temperature field is used as a load applied to the Roots rotor. A thermo-structural coupling analysis is performed to calculate the thermal deformation of the rotor under free expansion or constrained conditions, and the coordinates of each discrete point in the theoretical rotor profile discrete point set under thermal conditions are extracted. Based on the coordinates of each discrete point in the theoretical rotor profile discrete point set under hot conditions, and combined with the preset target gap, the cold state normal compensation amount of each discrete point in the theoretical rotor profile discrete point set is determined. Based on the cold-state normal compensation, each discrete point in the theoretical rotor profile discrete point set is moved along the normal direction to generate the actual rotor profile discrete point set under cold-state non-uniform gap, so as to form the actual Roots rotor profile.

[0007] As a preferred embodiment, the step of generating a pair of fully meshed, gapless theoretical rotor profiles based on the Roots rotor meshing principle involves determining the basic geometric parameters of the Roots rotor according to design requirements. The basic geometric parameters of the Roots rotor include the number of rotor teeth, rotor blade profile, center distance, and rotor length. The rotor blade profile includes any one or more of the following: circular arc profile, cycloidal profile, and involute profile.

[0008] As a preferred embodiment, the step of discretizing the generated theoretical rotor profile to obtain a discrete point set of the theoretical rotor profile involves discretizing the theoretical rotor profile according to the principle of equal arc length or equal angle. The expression for the discrete point set of the theoretical rotor profile is as follows:

[0009] Discrete point set It can characterize the original profile shape and meet the machining accuracy requirements of CNC machine tools.

[0010] As a preferred embodiment, the steps of using the temperature field as a load applied to the Roots rotor, performing thermo-structural coupling analysis, calculating the thermal deformation of the rotor under free expansion or constrained conditions, and extracting the coordinates of each discrete point in the theoretical rotor profile discrete point set under thermal conditions include: The input parameters are obtained and preprocessed in the finite element analysis software; the input parameters include rotor material, temperature, load, and geometric parameters. The temperature field is applied as a load to the rotor structure, and a thermo-structural coupling analysis is performed to calculate the thermal deformation of the rotor under free expansion or constrained conditions, and to solve for the temperature field distribution and equivalent displacement results inside the rotor. Solve for the discrete points of the hot rotor profile and extract each discrete point. P i exist xdirection and y thermal displacement vector in direction To obtain the new coordinates under thermal conditions .

[0011] As a preferred embodiment, the step of determining the cold-state normal compensation amount of each discrete point in the theoretical rotor profile discrete point set based on the coordinates of each discrete point in the hot state and in combination with the preset target gap includes: Calculate the discrete points in the discrete point set of the theoretical rotor profile. P i tangent vector; Based on each discrete point P i Given the tangent vector, calculate the left and right normal vectors perpendicular to the direction of the tangent vector at the discrete point; The direction of the theoretical rotor profile towards the discrete points is determined by judging whether the left and right normal vectors point through the internal region enclosed by the discrete points of the theoretical rotor profile. P i outer unit normal vector N i ; Discrete point displacement considering thermal expansion compensation The rotor clearance under cold conditions is designed, with a preset target clearance of [value missing]. d target The cold-state normal compensation amount for each discrete point in the theoretical rotor profile discrete point set is determined as follows:

[0012] For each original discrete point P i Along its unit outward normal vector N i Move and calculate the offset distance d i ,when d i When the value is positive, the discrete point shifts outward along the normal direction; when... d i When the value is negative, the discrete point shifts inward along the normal direction; The discrete points of the non-uniform gap are calculated using the following formula: .

[0013] Furthermore, the discrete points in the set of discrete points for the calculated theoretical rotor profile are... P i The steps for finding the tangent vector include: Will P 1. P 2. P n-1, P n Define the boundary points as points, and define the remaining points as interior points; For interior points, the tangent vector at each discrete point is calculated using the five-point difference method as follows:

[0014] For boundary points, the tangent vector of the discrete point is calculated using the three-point difference method according to the following formula: .

[0015] Furthermore, the statement based on each discrete point P i The steps for calculating the left and right normal vectors perpendicular to the direction of the tangent vector at discrete points include: Calculate the left normal vector using the following formula:

[0016] Calculate the right normal vector using the following formula:

[0017] In the formula, the subscript x , y Representing the discrete points respectively P i Consistent definition x and y Directional components; Normalize according to the following formula:

[0018] In the formula, For small-scale prevention of zeroing.

[0019] As a preferred embodiment, the non-uniform thermally adaptive gap design method for Roots rotors further includes a reliability assessment of the discrete point set of the actual rotor profile of the generated non-uniform gap. The reliability assessment includes: The hot clearance distribution of the Roots rotor, formed by the discrete point set of the actual rotor profile with non-uniform clearance, is calculated under the working temperature field to evaluate the uniformity of the tooth clearance. Check for negative clearances between rotors and between rotors and housing under hot conditions; If the uniformity of the inter-tooth clearance is not met or there is a negative clearance, adjust the preset target clearance. d target or discrete point displacement at the design operating temperature The cold normal compensation amount of each discrete point in the theoretical rotor profile discrete point set is re-determined, and based on the cold normal compensation amount, each discrete point in the theoretical rotor profile discrete point set is moved along the normal until the non-uniform gap actual rotor profile discrete point set that meets the reliability requirements is obtained.

[0020] Secondly, a Roots rotor is provided, which is designed and manufactured using the non-uniform thermal adaptation gap design method for Roots rotors as described in the first aspect.

[0021] Thirdly, a Roots positive displacement fluid machine is provided, comprising a Roots rotor as described in the second aspect; the Roots positive displacement fluid machine includes a Roots vacuum pump and a Roots compressor.

[0022] Compared with the prior art, the present invention has at least the following beneficial effects: This invention presents a non-uniform thermal adaptation clearance design method for Roots rotors. Through thermal deformation calculations, it performs non-uniform compensation for rotor clearance, optimizing the hot-state clearance and precisely quantifying the non-uniform thermal deformation of the rotor under extreme operating conditions. Based on this, it transforms the "thermal suppression" approach into a "thermal adaptation" design. The core of this method lies in its approach: instead of passively cooling or controlling after thermal deformation occurs, it performs a "pre-deformation" non-uniform correction on the rotor profile while it is still in the cold-processed state. Specifically, it generates a non-uniform cold-state clearance based on the principle of more compensation for areas with large thermal deformation and less or even negative compensation for areas with small thermal deformation. When the rotor reaches thermal equilibrium under operating conditions, its non-uniform thermal expansion is precisely offset by the pre-set non-uniform clearance, thus achieving a highly uniform and optimized actual working clearance between rotors and between the rotor and the housing. This design fundamentally solves the problems of clearance mismatch and local jamming caused by uneven thermal expansion, significantly improving the operational reliability and service life of Roots compressors or vacuum pumps under extreme conditions such as high pressure differentials and high speeds without relying on complex active cooling control systems. Meanwhile, the method of the present invention has wide applicability and can be flexibly applied to the design of various rotor profiles such as circular arc, cycloidal, and involute. Its parameterized design process and evaluation iteration mechanism are conducive to the formation of standardized product design specifications and serial development, which can significantly shorten the new product development cycle and improve design iteration efficiency. Attached Figure Description

[0023] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention. For those skilled in the art, other related drawings can be obtained from these drawings without creative effort.

[0024] Figure 1This is a flowchart of the non-uniform thermal adaptation gap design method for Roots rotor according to an embodiment of the present invention; Figure 2 This is a schematic diagram of a typical arc-shaped two-lobe rotor profile and its equidistant gap profile according to an embodiment of the present invention; Figure 3 This is a graph showing the variation of the circumferential thermal deformation displacement of the Roots rotor with the characteristic length under thermal load according to an embodiment of the present invention. Figure 4 This is a graph showing the variation of the inter-tooth clearance of Roots rotors with equidistant and non-uniform clearances as a function of characteristic length in embodiments of the present invention. Figure 5 This is a graph showing the variation of the inter-tooth clearance of a non-uniform gap Roots rotor with characteristic length under thermal load according to an embodiment of the present invention. Detailed Implementation

[0025] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, those skilled in the art can obtain other embodiments without creative effort.

[0026] Please see Figure 1 The non-uniform thermal adaptation clearance design method for Roots rotors according to embodiments of the present invention mainly includes four steps: generation of the theoretical profile of a fully meshing Roots rotor, calculation of the thermal deformation of a fully meshing Roots rotor, generation of the actual Roots rotor profile with non-uniform clearance, and reliability assessment of the inter-tooth clearance of the Roots rotor. Specifically, the method of this embodiment is as follows: Based on the Roots rotor meshing principle, a pair of fully meshed, gapless theoretical rotor profiles are generated, and the generated theoretical rotor profiles are discretized to obtain the discrete point set of the theoretical rotor profiles; The temperature field is used as a load applied to the Roots rotor. A thermo-structural coupling analysis is performed to calculate the thermal deformation of the rotor under free expansion or constrained conditions, and the coordinates of each discrete point in the theoretical rotor profile discrete point set under thermal conditions are extracted. Based on the coordinates of each discrete point in the theoretical rotor profile discrete point set under hot conditions, and combined with the preset target gap, the cold state normal compensation amount of each discrete point in the theoretical rotor profile discrete point set is determined. Based on the cold-state normal compensation, each discrete point in the theoretical rotor profile discrete point set is moved along the normal direction to generate the actual rotor profile discrete point set under cold-state non-uniform gap, so as to form the actual Roots rotor profile.

[0027] In one possible implementation, when generating the theoretical profile of the fully meshing Roots rotor, the basic geometric parameters of the Roots rotor are determined according to the design requirements. These basic geometric parameters include the number of rotor teeth, rotor blade profile, center distance, and rotor length. Typical rotor blade profiles include any one or more of the following: circular arc profile, cycloidal profile, and involute profile.

[0028] In one possible implementation, the theoretical rotor profile is discretized according to the principle of equal arc length or equal angle. The expression for the discrete point set of the theoretical rotor profile is as follows:

[0029] Discrete point set It can accurately represent the original profile shape and meet the machining accuracy requirements of CNC machine tools.

[0030] In one possible implementation, the steps of treating the temperature field as a load applied to the Roots rotor, performing thermo-structural coupling analysis, calculating the thermal deformation of the rotor under free expansion or constrained conditions, and extracting the coordinates of each discrete point in the theoretical rotor profile discrete point set under thermal conditions include: The input parameters are obtained and preprocessed in the finite element analysis software; these input parameters include rotor material, temperature, load, and geometric parameters. The temperature field is applied as a load to the rotor structure, and a thermo-structural coupling analysis is performed to calculate the thermal deformation of the rotor under free expansion or constrained conditions, and to solve for the temperature field distribution and equivalent displacement results inside the rotor. Solve for the discrete points of the hot rotor profile and extract each discrete point. P i exist x direction and y thermal displacement vector in direction To obtain the new coordinates under thermal conditions .

[0031] In one possible implementation, the step of determining the cold-state normal compensation amount of each discrete point in the theoretical rotor profile discrete point set based on the coordinates of each discrete point in the hot state, combined with a preset target gap, includes: Calculate the discrete points in the discrete point set of the theoretical rotor profile. P i tangent vector; Based on each discrete point P i Given the tangent vector, calculate the left and right normal vectors perpendicular to the direction of the tangent vector at the discrete point; The direction of the theoretical rotor profile towards the discrete points is determined by judging whether the left and right normal vectors point through the internal region enclosed by the discrete points of the theoretical rotor profile.P i outer unit normal vector N i ; Discrete point displacement considering thermal expansion compensation The rotor clearance under cold conditions is designed, with a preset target clearance of [value missing]. d target The cold-state normal compensation amount for each discrete point in the theoretical rotor profile discrete point set is determined as follows:

[0032] For each original discrete point P i Along its unit outward normal vector N i Move and calculate the offset distance d i ,when d i When the value is positive, the discrete point shifts outward along the normal direction; when... d i When the value is negative, the discrete point shifts inward along the normal direction; The discrete points of the non-uniform gap are calculated using the following formula: .

[0033] Furthermore, the discrete points in the theoretical rotor profile discrete point set are calculated. P i The steps for finding the tangent vector include: Will P 1. P 2. P n-1 , P n Define the boundary points as points, and define the remaining points as interior points; For interior points, the tangent vector at each discrete point is calculated using the five-point difference method as follows:

[0034] For boundary points, the tangent vector of the discrete point is calculated using the three-point difference method according to the following formula: .

[0035] Furthermore, based on each discrete point P i The steps for calculating the left and right normal vectors perpendicular to the direction of the tangent vector at discrete points include: Calculate the left normal vector using the following formula:

[0036] Calculate the right normal vector using the following formula:

[0037] In the formula, the subscript x , y Representing the discrete points respectively P i Consistent definition x and y Directional components; Normalize according to the following formula:

[0038] In the formula, For small-scale prevention of zeroing.

[0039] In one possible implementation, the Roots rotor non-uniform thermal adaptation gap design method of this embodiment further includes a reliability assessment of the generated non-uniform gap actual rotor profile discrete point set; specifically, the reliability assessment includes: The hot clearance distribution of the Roots rotor, formed by the discrete point set of the actual rotor profile with non-uniform clearance, is calculated under the working temperature field to evaluate the uniformity of the tooth clearance. Check for negative clearances (i.e. interference) between rotors and between rotors and the housing under hot conditions to ensure safe rotor operation. If the uniformity of the inter-tooth clearance is not met or there is a negative clearance, adjust the preset target clearance. d target or discrete point displacement at the design operating temperature The cold normal compensation amount of each discrete point in the theoretical rotor profile discrete point set is re-determined, and based on the cold normal compensation amount, each discrete point in the theoretical rotor profile discrete point set is moved along the normal until the non-uniform gap actual rotor profile discrete point set that meets the reliability requirements is obtained.

[0040] Compared with the prior art, the present invention performs non-uniform compensation for rotor clearance through thermal deformation calculation, thereby optimizing the thermal clearance uniformly. This is beneficial to improving the operational reliability of Roots compressors or vacuum pumps under extreme operating conditions.

[0041] Meanwhile, the thermally adaptable non-uniform gap design method proposed in this invention is applicable to various rotor profile designs. By segmenting the operating conditions of different markets, the rotor profile type and gap design can be flexibly adjusted, which will help broaden the application and promotion of Roots vacuum pumps in different market segments. The non-uniform gap design method of this invention achieves gap design through simulation calculations for specific rotor types, which will facilitate the design and development of product series, shorten product design cycles, and improve product iteration efficiency.

[0042] The invention will be further illustrated below with an example of a non-uniform clearance design for a Roots rotor.

[0043] Step 1: Theoretical profile generation of the fully meshing Roots rotor.

[0044] The circular arc profile is the most common profile type for Roots vacuum pumps. The tooth tip radius of a two-lobe circular arc Roots vacuum pump rotor is designed as follows: R m =71mm, rotor pitch circle radius is R =45mm, the theoretical profile equation is as follows:

[0045]

[0046]

[0047]

[0048] In the formula, b The distance from the center of the peak circle to the center of the rotor; r n The radius of the rotor peak circle; d For equidistant design gaps, when d = 0 represents the theoretical profile of full engagement; Design the rotation angle for the rotor profile; x 1, y 1] is the coordinate equation of the top circular line; [ x 2, y [2] is the coordinate equation of the conjugate curve of the circular arc.

[0049] Solving for the results d Discrete points of the profile under different equidistant design gaps in [0mm, +0.45mm] are as follows: Figure 2 As shown.

[0050] Step 2: Calculation of thermal deformation of the fully meshing Roots rotor.

[0051] Define Δ T Given different temperature rise thermal boundaries ∈ [10℃, 150℃], with the rotor material set as cast iron and the ambient temperature set as 20℃, the circumferential thermal deformation of the rotor is obtained as follows: Figure 3 As shown.

[0052] Step 3: Generating the actual Roots rotor profile with non-uniform gap.

[0053] The rotor temperature is set to 100℃ under extreme operating conditions. The maximum rotor thermal deformation is calculated to be +0.05455mm. A certain operating margin is allowed, and the target clearance is defined. d target=0.1mm, along the circumferential discrete points of the rotor ( Figure 3 Non-uniform gap design (as shown in the direction) Figure 4 As shown.

[0054] Step 4: Reliability assessment of the tooth clearance of the Roots rotor.

[0055] Redefining Δ T Given different temperature rise thermal boundaries ∈ [10℃, 150℃], with the rotor material set as cast iron and the ambient temperature set as 20℃, the variation of the non-uniform gap rotor tooth clearance is obtained as follows: Figure 5 As shown, at rotor temperatures below 100°C under extreme operating conditions, the tooth clearance can be guaranteed to be greater than the target clearance. d target =0.1mm, which ensures the reliability of the Roots rotor operation.

[0056] To address the problem of uneven clearance distribution and jamming caused by rotor thermal expansion in existing Roots equipment under extreme conditions, this invention provides non-uniform compensation for rotor clearance from the design stage. This ensures that the rotor achieves a uniform and optimized fit clearance at operating temperatures, fundamentally preventing thermal jamming and improving the operational reliability of the equipment under extreme conditions such as high pressure differentials and high speeds.

[0057] Another embodiment of the present invention also proposes a Roots rotor, which is designed and manufactured using the non-uniform thermal adaptation gap design method for Roots rotors described in the embodiments of the present invention.

[0058] Another embodiment of the present invention provides a Roots positive displacement fluid machine, including the aforementioned Roots rotor; wherein the Roots positive displacement fluid machine includes a Roots vacuum pump and a Roots compressor.

[0059] Finally, it should be noted that the above embodiments are merely specific implementations of the present invention, used to illustrate the technical solutions of the present invention, and not to limit it. The scope of protection of the present invention is not limited thereto. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that any person skilled in the art can still modify or easily conceive of changes to the technical solutions described in the foregoing embodiments within the technical scope disclosed in the present invention, or make equivalent substitutions for some of the technical details; and these modifications, changes, or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention, and should all be covered within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method of designing a Roots rotor non-uniform thermal adaptation gap, characterized in that, include: Based on the Roots rotor meshing principle, a pair of fully meshed, gapless theoretical rotor profiles are generated, and the generated theoretical rotor profiles are discretized to obtain the discrete point set of the theoretical rotor profiles; The temperature field is used as a load applied to the Roots rotor. A thermo-structural coupling analysis is performed to calculate the thermal deformation of the rotor under free expansion or constrained conditions, and the coordinates of each discrete point in the theoretical rotor profile discrete point set under thermal conditions are extracted. Based on the coordinates of each discrete point in the theoretical rotor profile discrete point set under hot conditions, and combined with the preset target gap, the cold state normal compensation amount of each discrete point in the theoretical rotor profile discrete point set is determined. Based on the cold-state normal compensation, each discrete point in the theoretical rotor profile discrete point set is moved along the normal direction to generate the actual rotor profile discrete point set under cold-state non-uniform gap, so as to form the actual Roots rotor profile.

2. The method of Roots rotor non-uniform thermal fit clearance design of claim 1, wherein, The step of generating a pair of fully meshed, gapless theoretical rotor profiles based on the Roots rotor meshing principle involves determining the basic geometric parameters of the Roots rotor according to design requirements. The basic geometric parameters of the Roots rotor include the number of rotor teeth, rotor blade profile, center distance, and rotor length. The rotor blade profile includes any one or more of the following: circular arc profile, cycloidal profile, and involute profile.

3. The non-uniform thermal adaptation gap design method for Roots rotors according to claim 1, characterized in that, The step of discretizing the generated theoretical rotor profile to obtain a discrete point set of the theoretical rotor profile involves discretizing the theoretical rotor profile according to the principle of equal arc length or equal angle. The expression for the discrete point set of the theoretical rotor profile is as follows: Discrete point set It can characterize the original profile shape and meet the machining accuracy requirements of CNC machine tools.

4. The non-uniform thermal adaptation gap design method for Roots rotor according to claim 1, characterized in that, The steps of using the temperature field as a load applied to the Roots rotor, performing thermo-structural coupling analysis, calculating the thermal deformation of the rotor under free expansion or constrained conditions, and extracting the coordinates of each discrete point in the theoretical rotor profile discrete point set under thermal conditions include: The input parameters are obtained and preprocessed in the finite element analysis software; the input parameters include rotor material, temperature, load, and geometric parameters. The temperature field is applied as a load to the rotor structure, and a thermo-structural coupling analysis is performed to calculate the thermal deformation of the rotor under free expansion or constrained conditions, and to solve for the temperature field distribution and equivalent displacement results inside the rotor. Solve for the discrete points of the hot rotor profile and extract each discrete point. P i exist x direction and y thermal displacement vector in direction To obtain the new coordinates under thermal conditions .

5. The non-uniform thermal adaptation gap design method for Roots rotors according to claim 1, characterized in that, The step of determining the cold-state normal compensation amount of each discrete point in the theoretical rotor profile discrete point set based on the coordinates of each discrete point in the hot state and in combination with the preset target gap includes: Calculate the discrete points in the discrete point set of the theoretical rotor profile. P i tangent vector; Based on each discrete point P i Given the tangent vector, calculate the left and right normal vectors perpendicular to the direction of the tangent vector at the discrete point; The direction of the theoretical rotor profile towards the discrete points is determined by judging whether the left and right normal vectors point through the internal region enclosed by the discrete points of the theoretical rotor profile. P i outer unit normal vector N i ; Discrete point displacement considering thermal expansion compensation The rotor clearance under cold conditions is designed, with a preset target clearance of [value missing]. d target The cold-state normal compensation amount for each discrete point in the theoretical rotor profile discrete point set is determined as follows: For each original discrete point P i Along its unit outward normal vector N i Move and calculate the offset distance d i ,when d i When the value is positive, the discrete point shifts outward along the normal direction; when... d i When the value is negative, the discrete point shifts inward along the normal direction; The discrete points of the non-uniform gap are calculated using the following formula: 。 6. The non-uniform thermal adaptation gap design method for Roots rotor according to claim 5, characterized in that, The discrete points in the set of discrete points of the calculated theoretical rotor profile P i The steps for finding the tangent vector include: Will P 1. P 2. P n-1 , P n Define the boundary points as points, and define the remaining points as interior points; For interior points, the tangent vector at each discrete point is calculated using the five-point difference method as follows: For boundary points, the tangent vector of the discrete point is calculated using the three-point difference method according to the following formula: 。 7. The non-uniform thermal adaptation gap design method for Roots rotor according to claim 6, characterized in that, The basis of each discrete point P i The steps for calculating the left and right normal vectors perpendicular to the direction of the tangent vector at discrete points include: Calculate the left normal vector using the following formula: Calculate the right normal vector using the following formula: In the formula, the subscript x , y Representing the discrete points respectively P i Consistent definition x and y Directional components; Normalize according to the following formula: In the formula, For small-scale prevention of zeroing.

8. The method for designing non-uniform thermal adaptation gaps for Roots rotors according to claim 1, characterized in that, It also includes a reliability assessment of the generated discrete point set of the actual rotor profile with non-uniform gaps; The reliability assessment includes: The hot clearance distribution of the Roots rotor, formed by the discrete point set of the actual rotor profile with non-uniform clearance, is calculated under the working temperature field to evaluate the uniformity of the tooth clearance. Check for negative clearances between rotors and between rotors and housing under hot conditions; If the uniformity of the inter-tooth clearance is not met or there is a negative clearance, adjust the preset target clearance. d target or discrete point displacement at the design operating temperature The cold normal compensation amount of each discrete point in the theoretical rotor profile discrete point set is re-determined, and based on the cold normal compensation amount, each discrete point in the theoretical rotor profile discrete point set is moved along the normal until the non-uniform gap actual rotor profile discrete point set that meets the reliability requirements is obtained.

9. A Roots rotor, characterized in that, It is designed and manufactured using the non-uniform thermal adaptation gap design method of the Roots rotor as described in any one of claims 1 to 8.

10. A Roots-type positive displacement fluid machine, characterized in that, Includes the Roots rotor as described in claim 9; the Roots positive displacement fluid machinery includes a Roots vacuum pump and a Roots compressor.

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