Calculation Method, Equipment, and Medium for Time-Varying Meshing Stiffness of Rack and Pinion in Rack Railway

Abstract

The present invention relates to the field of rack railway dynamics technology, and specifically to a method, device and medium for calculating the time-varying meshing stiffness of a rack and pinion of a rack railway, wherein the calculation method includes the calculation of the single tooth stiffness of the rack train traction gear, the single tooth stiffness of the rack and the contact stiffness of the rack and pinion, the single tooth stiffness of the rack train traction gear, the single tooth stiffness of the rack and pinion and the contact stiffness of the rack and pinion are connected in series to obtain the meshing stiffness of a single tooth pair, and the meshing stiffness of the gear teeth of each meshing contact point at any time is accumulated in parallel to obtain the comprehensive time-varying meshing stiffness of the rack and pinion of the rack railway. When calculating the rack matrix stiffness, the present invention fully considers the support constraint mode of the rack railway rack structure and the influence of the rack deflection deformation, and uses the deflection superposition method to calculate the rack matrix deformation to obtain the rack matrix stiffness. The calculation method is simple and effective, with high precision and high efficiency.

Description

Technical Field

[0001] The present invention relates to the technical field of mechanical dynamics, and particularly to a calculation method, device, and medium for the time-varying meshing stiffness of a gear-rack in a rack railway. Background Art

[0002] Mountain rack railways mainly overcome the problem of insufficient adhesion of traditional railway wheel-rail systems on large slopes through the meshing transmission between the traction gears of the train bogie and the rack laid at the center line of the track. As the most critical load-bearing structure when a rack train runs on a large slope, the traction gear and rack structures are important systems to ensure the safe and stable operation of the train. Therefore, studying the dynamic characteristics of the gear-rack power system of a rack railway plays an important role.

[0003] The time-varying meshing stiffness is one of the main excitation sources of the gear-rack transmission system. Studying the time-varying meshing stiffness of a gear-rack is of great significance for providing accurate and effective excitation input for the research of its related dynamic characteristics. Currently, the research on time-varying meshing stiffness mainly focuses on gear transmission systems. There are few relevant literatures on the time-varying meshing stiffness of gear-racks, and when analytically calculating the time-varying meshing stiffness of gear-racks, the rack is often directly equated with a gear for calculation. However, there are obvious differences between gear-rack transmission and gear transmission in terms of both structure and function: the matrix cross-section of a gear is circular, and the constraint axis is located at the center of the gear, and gear transmission realizes the transmission function between rotational motions; while the matrix cross-section of a rack is rectangular, and hinge constraints are often set at both ends for fixation, and gear-rack transmission realizes the conversion between rotational and translational motions. The two not only have different structures and functions, but also have different constraint boundary conditions. In addition, in terms of meshing characteristics, the meshing stiffness curve of gear transmission consists of a single-period meshing curve, while the meshing stiffness of a gear-rack supported only by hinges at both ends is non-periodic within the length of a rack. Therefore, although both gear-rack transmission and gear transmission adopt the form of tooth transmission, there are still significant differences. To more accurately reflect the calculation results, in the prior art, finite element analysis modeling is usually used to conduct simulations when studying such dynamic characteristics, but the modeling process is complex, requires a large amount of original data, has a large amount of calculation, and each time a parameter is changed, a new model needs to be built and analyzed again, resulting in a slow response speed.

[0004] Therefore, it is necessary to specifically propose an efficient and accurate analytical calculation method for the time-varying meshing stiffness of a gear-rack in a rack railway. Summary of the Invention

[0005] The purpose of the present invention is to: in view of the problem that the prior art lacks an efficient and accurate calculation method for the time-varying meshing stiffness of a gear-rack, provide a calculation method, device, and medium for the time-varying meshing stiffness of a gear-rack in a rack railway.

[0006] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] A calculation method for determining the time-varying meshing stiffness of a gear and a rack, comprising the following steps:

[0008] Based on the tooth bending stiffness k b of the gear tooth, the tooth shear stiffness k s of the gear tooth, the tooth axial compression stiffness k a of the gear tooth, and the gear matrix stiffness k f , obtain the single-tooth stiffness k c of the gear;

[0009] Based on the tooth bending stiffness k bt of the rack tooth, the tooth shear stiffness k st of the rack tooth, the tooth axial compression stiffness k at of the rack tooth, and the rack matrix stiffness k ft , obtain the single-tooth stiffness k t of the rack; wherein the rack matrix stiffness k ft is obtained by using the following formula:

[0010]

[0011] In the formula, F is the meshing force, β is the pressure angle, x represents the displacement of the action point corresponding to the meshing point on the axis of the rack matrix in the horizontal direction, y represents the deflection of the action point corresponding to the meshing point on the axis of the rack matrix, where when the intersection point of the extended line of the meshing force on the axis of the rack matrix is located outside the rack matrix, the action point on the axis of the rack matrix is regarded as the foot of the perpendicular of the meshing point on the axis of the rack matrix; when the intersection point of the extended line of the meshing force on the axis of the rack matrix is located inside the rack matrix, the action point on the axis of the rack matrix is regarded as this intersection point;

[0012] Calculate the contact stiffness k ht between the gear and the rack;

[0013] Based on the single-tooth stiffness k c of the gear, the single-tooth stiffness k t of the rack, and the contact stiffness k ht between the gear and the rack, obtain the meshing stiffness k i of a single pair of teeth of the gear and the rack, where i is the number of the meshing teeth of the gear and the rack;

[0014] Accumulate the meshing stiffness k i of each meshing contact point at any moment to obtain the time-varying meshing stiffness of the gear and the rack.

[0015] In the present invention, the stiffness of the gear tooth part and the rack tooth part are both decomposed into bending stiffness, shear stiffness, and axial compression stiffness with pairwise perpendicular directions. By using the potential energy principle method and combining with the strain energy formula of material mechanics, the bending stiffness, shear stiffness, and axial compression stiffness of the tooth part on the gear or rack can be obtained respectively; when calculating the stiffness of the gear-rack base, the influence of the support and constraint mode of the gear-rail structure of the gear-rail railway and the deflection deformation of the rack is fully considered, and the deflection superposition method is used to calculate the deformation of the rack base to obtain the stiffness of the rack base. The time-varying meshing stiffness calculation model of the rack and gear established in this way can better fit the actual working conditions of the gear-rack pair, and the calculation accuracy is higher; moreover, the calculation efficiency is greatly improved compared with the finite element method.

[0016] Preferably, the single-tooth stiffness k of the gear c is obtained by the following formula:

[0017] Preferably, the single-tooth stiffness k of the rack t is obtained by the following formula:

[0018] Preferably, when obtaining the stiffness k of the rack base ft , when the intersection point of the extension line of the meshing force and the center line of the rack base is outside the rack base during the meshing of the gear and the rack, the horizontal displacement x and the deflection y of the acting point on the central axis of the rack base at the meshing position are respectively

[0019]

[0020]

[0021] When the intersection point of the extension line of the meshing force and the center line of the rack base is inside the rack base, the horizontal displacement x and the deflection y of the acting point on the central axis of the rack base at the meshing position are respectively

[0022]

[0023]

[0024] In the formula, F a is the component force of the meshing force F in the x direction, F b is the component force of the meshing force F in the y direction, a 1 , b 1 are respectively the x-axis coordinate values of the foot point corresponding to the meshing point on the central axis of the rack base, b 1 is the difference between the full length of the rack and the absolute value of a 1 , a 2 , b 2 are respectively the distances from the intersection point of the extension line of the meshing force and the central axis of the rack base to the hinge ends on both sides of the rack, x B0is the coordinate value of the meshing point in the x-axis direction, E is the elastic modulus, A is the cross-sectional area of the rack base, l is the length of the rack, M is the equivalent torque at the end of the rack, and I is the moment of inertia of the rack base cross-section.

[0025] Preferably, the matrix stiffness correction model is used to calculate the gear matrix stiffness k f , the gear matrix stiffness k f is obtained through the following formula:

[0026]

[0027] In the formula, β is the pressure angle, L is the tooth width, E is the elastic modulus, S F is the critical partial tooth thickness, u f is the tooth height at the meshing line, and L*, M*, P*, Q* are constants related to the base circle and inner hole radius of the gear.

[0028] Preferably, the present invention obtains the gear-rack contact stiffness k through the following formula ht : In the formula, E is the elastic modulus, L is the tooth width (the gear and the rack on the rack railway have the same tooth width), and υ is the Poisson's ratio.

[0029] Preferably, the meshing stiffness k of a single pair of teeth of the gear-rack i is obtained through the following formula:

[0030] An electronic device for data processing, comprising:

[0031] One or more processors;

[0032] A storage device for storing one or more programs, which when executed by the one or more processors cause the one or more processors to implement the above calculation method.

[0033] A computer-readable medium having a computer program stored thereon, and the program implements the above calculation method when executed by a processor.

[0034] In summary, due to the adoption of the above technical solutions, the beneficial effects of the present invention are:

[0035] The time-varying meshing stiffness calculation method for the rack and gear of the present invention fully considers the influence of the rack support method and the rack deflection deformation. The formed time-varying meshing stiffness calculation model of the rack and gear can better fit the actual working conditions of the gear-rack pair, has higher calculation accuracy, and greatly improves the calculation efficiency compared with the finite element method. The present invention can provide accurate and effective excitation input for the related research of the gear-rack power system of the rack railway, so as to provide effective data support. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 It is an analytical calculation flow chart for determining the time-varying meshing stiffness of a gear and rack.

[0037] Figure 2 It is a schematic diagram of the forces acting on the gear.

[0038] Figure 3 It is a schematic diagram of the forces acting on the rack.

[0039] Figure 4 It is a schematic diagram of the finite element modeling model of the gear.

[0040] Figure 5 It is a schematic diagram of the finite element modeling model of the rack.

[0041] Figure 6 It is a comparison chart of the finite element analysis results and the theoretical analysis results obtained by this method (tooth width b = 60 mm, module m = 31.831 mm).

[0042] Figure 7 It is a comparison chart of the finite element analysis results and the theoretical analysis results obtained by this method (tooth width b = 80 mm, module m = 31.831 mm). Specific implementation manners

[0043] The present invention will be described in detail below with reference to the accompanying drawings.

[0044] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0045] Embodiment 1

[0046] An analytical calculation method for determining the time-varying meshing stiffness of a gear and rack provided in this embodiment is based on the principle of potential energy, and comprehensively considers tooth deformation, Hertz contact deformation, and the deformation of the gear and rack matrix to establish an analytical calculation model for the time-varying meshing stiffness of the gear and rack. The comprehensive time-varying meshing stiffness of the gear and rack consists of gear stiffness, rack stiffness, and tooth pair contact stiffness.

[0047] When solving the stiffness of the traction gear of a gear-rail train, the gear teeth are simplified as a cantilever beam on the root circle of the gear; the gear stiffness is decomposed into the bending stiffness k b of the tooth (i.e., the tooth part), the shear stiffness k s and the axial compression stiffness k a , as well as the gear matrix stiffness k f; The tooth stiffness is solved by the potential energy principle method, and the gear matrix stiffness is solved by using the matrix stiffness correction model. The tooth stiffness and matrix stiffness of the gear are connected in series to obtain the gear stiffness. When solving the rack stiffness, the rack is considered as a structure form with both ends hinged and supported. Similarly, the rack stiffness is decomposed into the bending stiffness k bt of the tooth part, the shear stiffness k st and the axial compression stiffness k at , as well as the rack matrix stiffness k ft ; The tooth part stiffness of the rack is solved by the energy method. The deflection superposition method is used to calculate the matrix deformation, so as to obtain the rack matrix stiffness. The tooth stiffness and matrix stiffness are connected in series to obtain the rack stiffness. The single tooth pair meshing stiffness can be obtained by connecting the gear stiffness, rack stiffness and tooth pair contact stiffness in series; The multi-tooth meshing stiffness is obtained by parallel superposition of the single-tooth meshing stiffness. Finally, the comprehensive time-varying meshing stiffness of the gear rack that changes periodically with the number of meshing teeth can be obtained. The main steps are as Figure 1 shown.

[0048] The specific design method is as follows:

[0049] (1) Solve the gear stiffness k c based on the potential energy principle. The force condition of the gear is as Figure 2 shown. The gear stiffness is decomposed into the bending stiffness k b of the tooth part, the shear stiffness k s , the axial compression stiffness k a and the gear matrix stiffness k f . According to the strain energy formula in material mechanics, we can get:

[0050]

[0051] In the formula, U b is the bending potential energy of the tooth part, U s is the shear potential energy of the tooth part, U a is the compression potential energy of the tooth part, U f is the deformation potential energy of the gear matrix, and F is the meshing force.

[0052] Due to energy superposition, the total potential energy stored in the gear is:

[0053]

[0054] Therefore, the derived single-tooth stiffness k c of the gear is:

[0055] Among them: The gear matrix stiffness k f can be calculated by using the existing matrix stiffness correction model, and is specifically obtained through the following formula:

[0056]

[0057] where β is the pressure angle, L is the tooth width, E is the elastic modulus, S F is the critical partial tooth thickness, u f is the tooth height at the line of action, and L*, M*, P*, Q* are constants related to the base circle and inner hole radius of the gear.

[0058] The bending stiffness, shear stiffness, and axial compression stiffness of the gear teeth can be further obtained by combining with the following strain energy formula in mechanics of materials:

[0059]

[0060]

[0061]

[0062] where M 1 , M 2 are the torques generated by the meshing force on any point of the transition curve and involute respectively; I y1 , I y2 are the section moments of inertia of any point on the transition curve and involute respectively; A y1 , A y2 are the cross-sectional areas of any point on the transition curve and involute respectively; α is the shear section coefficient (1.2 for rectangle); E is the material elastic modulus; G is the shear modulus; F a , F b are the components of the meshing force in the x and y directions respectively; y 1 , y 2 are the y-axis coordinates of any point on the transition curve and involute respectively.

[0063] For the convenience of calculation, the displacement is converted into angular displacement for integration, and the converted bending stiffness, shear stiffness, and axial compression stiffness can be obtained by the following formulas respectively:

[0064]

[0065]

[0066]

[0067] (2) Rack stiffness k t Calculation: Similar to the gear stiffness, the rack stiffness can also be decomposed into the tooth bending stiffness k bt , tooth shear stiffness k st , tooth axial compression stiffness k at and rack base stiffness k ft , so the single-tooth stiffness k t of the rack can be similarly deduced and expressed as:

[0068] Specifically, the stiffness k of the rack base ft is calculated by superposing deflections. Considering that the two ends of the rack are hinged, in this embodiment, the displacements of the rack base in the x-direction and y-direction are calculated separately first, and then the displacements are equivalent to the displacements in the direction of the meshing force through the superposition principle, and the stiffness of the rack base is obtained by using Hooke's theorem. Combining Figure 3 with the force diagram of the rack during the meshing of the middle gear and the rack, the meshing force is equivalently transferred to the intersection point of the meshing force direction and the central axis of the rack base along the meshing force direction. As the contact position changes, the intersection point of the meshing force direction and the central axis will also change (such as point B 2 and point B 22 ), and different calculation methods need to be adopted when the intersection point is inside and outside the rack base.

[0069] 1) When the intersection point of the extension line of the meshing force and the center line of the rack base is outside the base, taking the corresponding meshing point B in the figure as an example, the extension line of the meshing force intersects with the central axis of the rack base at point B 2 outside the base, and the perpendicular foot point corresponding to the meshing point on the central axis of the rack base is B 1 :

[0070] The equivalent torque at the left end of the rack is where y B is the y-axis coordinate of the corresponding meshing point B.

[0071] According to the deflection calculation formula, the deflection at point B 1 corresponding to the meshing point B on the central axis of the rack base is where a z is the x-axis coordinate of the meshing contact point B; l is the length of the rack; I is the moment of inertia of the cross-section of the rack base.

[0072] The displacement of point B 1 in the x-direction is where A is the cross-sectional area of the rack base, a z is the x-axis coordinate value of the perpendicular foot point corresponding to the meshing point on the central axis of the rack base, b z is the difference between the full length of the rack and the absolute value of a 1 ; F at a z / (EA) is the compression displacement of section a z , and this displacement is used to equivalently represent the compression displacement of the meshing force in section BB 2 .

[0073] Thus, according to the superposition principle, we get

[0074] 2) When the intersection point is inside the rack base, taking the corresponding meshing point B in the figure 0For example, the extension line of the meshing force intersects the central axis of the rack base at point B within the base: 22 Intersection occurs at:

[0075] According to the deflection calculation formula, the deflection at point B 22 is where a y and b y are respectively the distances from the intersection point B 22 to the left and right hinged ends of the rack.

[0076] The displacement of the intersection point B 22 in the x - direction is where x B0 is the coordinate value of the meshing point B 0 in the x - axis direction, F at (x B - a y ) / (EA) is the compression displacement of section h, and this displacement is used to equivalently represent the compression displacement of the meshing force in section B 0 B 22 section.

[0077] Thus, by the superposition principle, the corresponding stiffness of the rack base is

[0078] Therefore, in summary, according to the displacement superposition principle and Hooke's theorem, the stiffness k ft of the rack base can be expressed as:

[0079]

[0080] where F is the meshing force, β is the pressure angle, x represents the displacement of the acting point on the central axis of the rack base in the horizontal direction, y represents the deflection of the meshing point at the corresponding acting point on the central axis of the rack base. Among them, when the intersection point of the extension line of the meshing force on the central axis of the rack base is outside the rack base, the acting point on the central axis of the rack base is regarded as the foot of the perpendicular of the meshing point on the central axis of the rack base; when the intersection point of the extension line of the meshing force on the central axis of the rack base is inside the rack base, the acting point on the central axis of the rack base is regarded as this intersection point.

[0081] On the other hand, the calculation methods of the bending stiffness k bt , shear stiffness k st and axial compression stiffness k at of the rack teeth refer to the existing calculation methods of gear stiffness, as follows:

[0082]

[0083]

[0084]

[0085] Where M 3 I is the moment generated by the meshing force on any point on the rack contact line; y is the section moment of inertia of any point on the rack contact line; A y is the cross-sectional area of ​​any point on the rack contact line; α is the shear section coefficient (1.2 for rectangle); F at and F bt are the components of meshing force in the x and y directions respectively; 3 is the y-axis coordinate of any point on the rack contact line.

[0086] (3) Calculation of single tooth meshing stiffness of gear rack: According to the relationship between force and strain energy, the energy stored in a pair of teeth during the meshing process of the gear rack is

[0087]

[0088] In the formula, i represents the number of a single meshing tooth pair, k ht is the Hertz contact stiffness of the gear rack. The gear rack contact stiffness is obtained by the Hertz contact stiffness calculation formula:

[0089]

[0090] Where υ is Poisson's ratio and L is the tooth width.

[0091] Therefore, the meshing stiffness of a single pair of teeth on a gear rack can be expressed as

[0092]

[0093] (4) Calculation of gear rack periodic meshing stiffness: The meshing tooth number n is determined by the meshing geometry of the gear rack. Due to the hinge constraints at both ends of the rack, the gear rolls and translates along the rack direction, so the meshing line also translates along the rack direction and is perpendicular to the rack tooth surface; by determining the number of intersections between the meshing line and the rack tooth surface at any time, the number of gear pairs involved in the meshing at that time can be determined, and the composite stiffness of the corresponding meshing contact points can be connected in parallel, that is, superimposed to obtain the composite stiffness of double-tooth or multi-tooth meshing, that is, the comprehensive time-varying meshing stiffness of the gear rack can be expressed as

[0094] For example, for the gear rack meshing, the overlap is 2<ε<3, and the meshing stiffness can be expressed as:

[0095]

[0096] In the formula, k c,i represents the gear meshing stiffness of the i-th pair of teeth, k t,i represents the rack meshing stiffness of the i-th pair of teeth, k ht is the Hertzian contact stiffness of the gear rack.

[0097] Furthermore, taking the gear-rack drive system of a mountain gear-rail train as the research object, the design parameters of its gear-rack drive are shown in Table 1. A three-dimensional model of the gear and rack is constructed using the 3D software Catia, as shown in Figure 4 , Figure 5 . Using the finite element analysis software Abaqus, a finite element model of the gear and rack is established, and the finite element modeling and static analysis of the gear and rack are carried out using the finite element analysis method. The center of the gear is fixed, and all degrees of freedom of the two ends of the rack except rotation are constrained. The C3D8R mesh is used. According to the load magnitude transmitted by the gear-rack of the gear-rail train, a load of 50 N is applied to each node at each meshing position of the gear and rack along the meshing line direction. By extracting the strain energy of the model, the meshing stiffness is calculated using the relationship between stiffness, acting force, and strain energy.

[0098] Table 1 Gear-rack drive parameters

[0099]

[0100] Through the finite element method and the analytical method proposed in this embodiment, the accuracy of the analytical model of the time-varying meshing stiffness of the gear and rack is verified under different conditions, as shown in the result comparison in Figure 6 and Figure 7 under different tooth widths (60 mm, 80 mm). It can be seen from the comparison of the calculation results that the calculation results of the analytical calculation method proposed in this paper are very close to the results of the finite element modeling analysis, with high accuracy. Compared with the finite element method, it is simple, effective, has a small amount of calculation, and greatly improves the calculation efficiency.

[0101] Through the calculation model established by the above analytical calculation method of the time-varying meshing stiffness of the gear and rack, compared with the finite element calculation results, the correctness of the established analytical calculation model can be verified. Based on the theoretical analytical calculation model established in this embodiment, the influence laws of the vertical clearance h z (the clearance between the pitch circle of the gear and the pitch line of the rack during actual use), the transmission pressure angle, and the rack length on the time-varying meshing stiffness of the gear and rack can be further studied, so as to provide effective data support for the parameter design of the gear-rail railway.

[0102] The analytical method of the time-varying meshing stiffness of the gear-rack of the gear-rail railway proposed in this paper is simple, efficient, and highly accurate, and can be effectively used for the excitation input of the gear-tooth meshing system of the gear-rail railway in the later stage.

[0103] Embodiment 2

[0104] An electronic device for data processing, comprising:

[0105] One or more processors;

[0106] A storage device for storing one or more programs, which when executed by the one or more processors cause the one or more processors to implement the calculation method in Embodiment 1.

[0107] Embodiment 3

[0108] A computer-readable medium having stored thereon a computer program, which when executed by a processor implements the parsing calculation method in Embodiment 1.

[0109] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A calculation method for determining the time-varying meshing stiffness of a rack and pinion, characterized in that, it includes the following steps: According to the bending stiffness \(k\) of the gear tooth b , the shear stiffness \(k\) of the gear tooth s , the axial compression stiffness \(k\) of the gear tooth a and the matrix stiffness \(k\) of the gear f , the single-tooth stiffness \(k\) of the gear is obtained c ; the single-tooth stiffness \(k\) of the gear c is obtained by the following formula: According to the tooth bending stiffness k of the rack bt , the tooth shear stiffness k st , the tooth axial compression stiffness k at and the rack base stiffness k ft , the single-tooth stiffness k of the rack is obtained t , the single-tooth stiffness k of the rack t is obtained by the following formula: where the rack base stiffness k ft is obtained using the following formula: In the formula, F is the meshing force, x represents the displacement of the acting point corresponding to the meshing point on the central axis of the rack base in the horizontal direction, y represents the deflection of the acting point corresponding to the meshing point on the central axis of the rack base, and β is the pressure angle; Calculate the contact stiffness k between the gear and the rack ht ; According to the single-tooth stiffness \(k\) of the gear c , the single-tooth stiffness \(k\) of the rack t , and the contact stiffness \(k\) between the gear and the rack ht , the meshing stiffness \(k\) of a single pair of teeth of the gear and rack is obtained i , where \(i\) is the number of the meshing teeth of the gear and rack; Contact stiffness k of gear and rack ht It is obtained by the following formula: In the formula, E is the elastic modulus, L is the tooth width, and υ is the Poisson's ratio; The meshing stiffness k of a single pair of gear teeth i is obtained by the following formula: Accumulate the tooth pair meshing stiffness \(k\) at each meshing contact point at any moment i to obtain the time-varying meshing stiffness of the gear and rack.

2. The calculation method for determining the time-varying meshing stiffness of a rack and pinion according to claim 1, characterized in that, Obtain the stiffness k of the rack base ft When calculating, when the gear meshes with the rack, the intersection point of the extension line of the meshing force and the central axis of the rack base is outside the rack base. The horizontal displacement x and deflection y of the acting point on the central axis of the rack base at the meshing position are respectively when the intersection point of the extension line of the meshing force and the center line of the rack base is within the rack base, the horizontal displacement x and the deflection y of the acting point corresponding to the meshing position on the central axis of the rack base are respectively where F a is the x - component of the meshing force F, F b is the y - component of the meshing force F, a 1 is the x - axis coordinate value of the foot of the perpendicular corresponding to the meshing point on the central axis of the rack base, b 1 is the difference between the full length of the rack and the absolute value of a 1 , a 2 , b 2 are the distances from the intersection point of the extension line of the meshing force and the central axis of the rack base to the hinge ends on both sides of the rack respectively, x B0 is the coordinate value of the meshing point in the x - axis direction, E is the elastic modulus, A is the cross - sectional area of the rack base, l is the length of the rack, M is the equivalent torque at the end of the rack, and I is the moment of inertia of the rack base cross - section.

3. The calculation method for determining the time-varying meshing stiffness of a rack and pinion according to claim 1, characterized in that, Gear matrix stiffness k f Obtained by the following formula: In the formula, β is the pressure angle, L is the tooth width, E is the elastic modulus, S F is the critical partial tooth thickness, u f is the tooth height at the line of action, and L*, M*, P*, and Q* are constants related to the base circle and inner hole radius of the gear.

4. An electronic device for data processing, characterized in that, it includes: one or more processors; a storage device for storing one or more programs, which when executed by the one or more processors cause the one or more processors to implement the calculation method according to any one of claims 1-3.

5. A computer-readable medium having a computer program stored thereon, characterized in that, the program, when executed by a processor, implements the calculation method according to any one of claims 1-3.

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