Long-period gear mesh stiffness calculation method considering superposition of tooth surface manufacturing errors

Abstract

This disclosure relates to a method for calculating the meshing stiffness of long-cycle gears considering the superposition of manufacturing errors on the tooth surface. This method transforms tooth surface measurement error data into the contact points of the gear meshing surfaces, considers the error coupling and superposition between the driving and driven gear contact points under long meshing periods, and performs load-bearing contact analysis at the meshing contact points. This yields a gear pair meshing stiffness that more closely reflects actual conditions, more realistically reproducing the contact situation during gear meshing and improving the calculation accuracy of gear meshing stiffness.

Description

Technical Field

[0001] This disclosure relates to the field of gear transmission technology, and in particular to a method for calculating the meshing stiffness of long-cycle gears considering the superposition of tooth surface manufacturing errors. Background Technology

[0002] Gear meshing stiffness and manufacturing errors are two important internal excitation factors that cause vibration in gear systems. Meshing stiffness excitation arises from the periodic change in stiffness and is the most obvious characteristic distinguishing gear system vibration from that of most mechanical systems. Gear meshing stiffness refers to the load on the line of action required to produce a 1-micron deflection on a 1-millimeter tooth width for one or more pairs of precisely meshing teeth to mesh simultaneously. The key to calculating gear meshing stiffness is accurately calculating the tooth surface contact, which is directly related to tooth surface errors. Existing methods for calculating gear meshing stiffness that consider tooth surface errors mostly obtain the error limit value for the corresponding accuracy grade according to ISO standards, and then assume the error distribution of the tooth surface. This method provides a relatively ideal consideration of errors but does not account for the superimposed coupling effect of meshing teeth on tooth surface errors. Summary of the Invention

[0003] To avoid the shortcomings of the prior art, this application provides a method for calculating the meshing stiffness of long-period gears that considers the superposition of tooth surface manufacturing errors, in order to solve the problem that the prior art does not consider the superposition and coupling effect of meshing teeth on tooth surface errors.

[0004] According to an embodiment of this disclosure, a method for calculating the meshing stiffness of long-period gears considering the superposition of tooth surface manufacturing errors is provided. The method includes:

[0005] Gear measuring equipment is used to measure the tooth surface error of each tooth of the driving gear and the driven gear, so as to obtain the tooth surface measurement point error data of all teeth of the driving gear and the driven gear;

[0006] For a pair of gear teeth, the gear meshing surfaces, contact lines, and contact points are generated and arranged at equal intervals for the driving gear and the driven gear, respectively. The contact points of the driving gear and the driven gear are numbered to obtain the correspondence between the contact points of the driving gear meshing surfaces and the contact points of the driven gear meshing surfaces.

[0007] For a pair of gear teeth, based on the positional relationship between the tooth surface measurement coordinates and the meshing surfaces of the driving gear and the driven gear, respectively, the tooth surface measurement point error data are interpolated to the contact points of the driving gear meshing surface and the driven gear meshing surface, respectively, to construct the initial contact point error matrix of the driving gear and the initial contact point error matrix of the driven gear.

[0008] During the long meshing cycle, for all gear teeth, in different meshing stages, the action surfaces of all driving gear meshing surfaces and driven gear meshing surfaces are matched with the corresponding contact lines and contact points to obtain the initial contact point error matrices of all driving gears and driven gears; wherein, the simultaneously meshing gear pairs include reference gear pairs and related gear pairs, and the initial error matrix of each gear tooth is the reference gear pair error matrix of that gear tooth;

[0009] The contact point error of the reference tooth pair is replaced by the contact point error of the relevant tooth pair of the driving gear and the driven gear, respectively, to correct the initial contact point error matrix of all driving gears and the initial contact point error matrix of the driven gear, as well as the superposition of the corresponding contact point errors of the driving gear and the driven gear.

[0010] Based on the target contact point error matrix of the driving wheel and the target contact point error matrix of the driven wheel, as well as the superposition of the corresponding contact point errors of the driving wheel and the driven wheel, the bending-shear compliance matrix of the gear tooth surface mesh is calculated using the finite element algorithm. The coefficients in the bending-shear compliance matrix of the tooth surface mesh are interpolated to the contact points of the meshing surfaces of the driving wheel and the driven wheel. The deformation of each contact point is calculated using the analytical formula of line contact. Then, the contact equation considering the superposition error of the tooth surface is solved. The meshing stiffness of the gear pair is solved based on the static equilibrium of the contact points.

[0011] Furthermore, the process of measuring tooth surface error includes:

[0012] Input the number of teeth, module, pressure angle, helix angle, displacement coefficient, measurement start and end points of the involute and the helix in advance; the tooth profile measurement range is from the diameter of the tooth profile control circle to 0.95 times the diameter of the tooth tip forming circle, and the tooth direction measurement range is the smaller of the two values: the length of the tooth surface area between the two ends minus 5% of the tooth width or one module length at each end of the axis, and the number of measurement points along the tooth profile direction and the tooth width direction is at least 16.

[0013] Furthermore, taking the actual meshing line length of the gear end face as the width and the tooth width as the length, the meshing action surface of the gear pair is formed. Contact lines and contact points are arranged at equal intervals within the action surface. Among them, the contact line from the engagement point to the disengagement point within the action surface, including both the engagement point and the disengagement point, represents the complete meshing process of the gear teeth. The contact lines on the driving gear and the driven gear are numbered sequentially according to the meshing sequence. The contact points on each contact line are numbered sequentially from the end face where the engagement point is located to the end face where the disengagement point is located. Contact points with the same number on the driving gear and the driven gear are corresponding contact points.

[0014] Furthermore, based on the coordinate relationship between the contact point of the meshing action surface and the error measurement point, the random point interpolation function griddata is used to interpolate the tooth surface contact point error; where the random point interpolation function is vq=griddata(x,y,v,xq,yq), where x and y are the coordinates of the tooth surface error measurement point, v is the measurement error corresponding to the measurement point, xq and yq are the coordinates of the contact point of the meshing action surface, and vq is the error on the action surface contact point obtained based on the tooth surface measurement error interpolation;

[0015] The initial contact error matrices of the driving wheel and the driven wheel are as follows: The interpolation errors are written into m according to the tooth numbering order and the contact point numbering order. p or m g A matrix with n rows and n columns, m p and m g These represent the number of teeth on the driving gear and the number of teeth on the driven gear, respectively, where n is the number of contact points on a single tooth surface. The initial contact error matrix of the driving gear and the initial contact error matrix of the driven gear are denoted as efp and efg, respectively.

[0016] The tooth numbering sequence is defined as follows: arbitrarily select one tooth on the driving gear and one on the driven gear as the initial reference tooth, and number the teeth on the driving gear and the driven gear sequentially according to the meshing direction.

[0017] Furthermore, the meshing phase includes a multi-tooth meshing phase and a few-tooth meshing phase. The action surfaces of simultaneously meshing tooth pairs are matched according to the multi-tooth meshing phase and the few-tooth meshing phase. Among them, the time it takes for the gear teeth to travel one base pitch length on the meshing line from the start of engagement is one base pitch cycle. Within one base pitch cycle, the multi-tooth meshing phase is the process from the reference tooth pair entering engagement to a related tooth pair exiting engagement, and the few-tooth meshing phase is the remaining part within the base pitch cycle. The correspondence of the action surface area, contact line, and contact point is achieved by numbering the simultaneously meshing tooth pairs, contact lines, and contact points, and by controlling a variable within the numbering.

[0018] Furthermore, methods for correcting the initial contact error matrices of all driving gears and driven gears include:

[0019] Find the number of contact points n1 and n2 within the first and second base period of the action surface. Then, perform a matrix row shift transformation on the data in columns (n1+1) to (n1+n2) and columns (n1+n2+1) to the last column of the initial contact error matrices efp and efg. The row numbers after the transformation of columns (n1+1) to (n1+n2) in the initial contact error matrices efp and efg are i-1, and the row numbers after the transformation of columns (n1+n2+1) to the last column are i-2, where i is the matrix row number before the transformation. If i-1 or i-2 is less than or equal to 0, for the initial contact error matrix efp of the driving wheel, i-1 = i-1 + m p i-2 = i-2 + m p For the initial contact error matrix efg of the driven wheel, i-1=i-1+m g i-2 = i-2 + m g m p and m g These represent the row numbers of the initial contact error matrices efp and efg for the driving and driven wheels, respectively.

[0020] Furthermore, the formula for calculating the bending-shear compliance matrix of the tooth surface mesh nodes is as follows:

[0021] [δ b ]=[δ total ]-[δ local ]

[0022] In the formula, [δ b [δ] represents the gear bending-shear compliance matrix. total ] represents the overall deformation compliance matrix of the gear, [δ local [ ] represents the local deformation compliance matrix of the gear;

[0023] The analytical formula for calculating the deformation at each contact point in line contact is:

[0024]

[0025]

[0026] In the formula, λ ci For the contact deformation of the i-th segmented contact line, F i For the load on the segmented contact wire, l i E represents the length of the segmented contact wire. * For the equivalent elastic modulus, R1 is the normal radius of curvature of the driving wheel at the contact point, R2 is the normal radius of curvature of the driven wheel at the contact point, E1 is the elastic modulus of the driving wheel, E2 is the elastic modulus of the driven wheel, v1 is the Poisson's ratio of the driving wheel, and v2 is the Poisson's ratio of the driven wheel.

[0027] The contact equation considering the superposition error of the tooth surface is:

[0028] -[δ b ]{F}-{λ c}+LSTE+{c}={ε}

[0029] In the formula, [δ b Let {F} be the local deformation matrix at the contact point, and {λ} be the load vector at each contact point. c {c} represents the contact deformation vector of each contact point, LSTE represents the static transmission error of the gear pair along the meshing line direction, {c} represents the remaining clearance vector of each contact point, and {ε} represents the initial contact clearance of the contact point, including the superposition error of tooth surface contact points, the modification amount, and the meshing misalignment amount.

[0030] The static equilibrium equation at the contact point is:

[0031]

[0032] In the formula, k m For the overall meshing stiffness of the gear pair, k i Let F be the meshing stiffness at the i-th contact point. i Let ε be the normal load at the i-th contact point. i Let be the initial contact gap at the i-th contact point.

[0033] The technical solutions provided by the embodiments of this disclosure may include the following beneficial effects:

[0034] In the embodiments of this disclosure, the long-cycle gear meshing stiffness calculation method that considers the superposition of tooth surface manufacturing errors is used to convert the tooth surface measurement error data to the contact point of the gear meshing surface. Considering the error coupling superposition between the contact points of the driving gear and the driven gear under long meshing period, the load-bearing contact analysis is performed on the meshing contact point, thereby obtaining a gear pair meshing stiffness that is more in line with the actual situation, more realistically restoring the contact situation of gear meshing under actual conditions, and improving the calculation accuracy of gear meshing stiffness. Attached Figure Description

[0035] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this disclosure and, together with the description, serve to explain the principles of this disclosure. It is obvious that the drawings described below are merely some embodiments of this disclosure, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.

[0036] Figure 1 A step diagram illustrating the method for calculating the long-cycle gear meshing stiffness considering the superposition of tooth surface manufacturing errors in an exemplary embodiment of this disclosure;

[0037] Figure 2 This diagram illustrates a gear meshing process in an application scenario of the long-cycle gear meshing stiffness calculation method that considers the superposition of tooth surface manufacturing errors in an exemplary embodiment of the present disclosure.

[0038] Figure 3 This diagram shows a unfolded view of the meshing surface of a gear pair in an application scenario according to an exemplary embodiment of this disclosure.

[0039] Figure 4 This is a schematic diagram illustrating the distribution of the drive wheel error measurement points and contact points in an application scenario as shown in an exemplary embodiment of this disclosure;

[0040] Figure 5 This is a schematic diagram illustrating the distribution of error measurement points and contact points of the driven wheel in an application scenario according to an exemplary embodiment of this disclosure;

[0041] Figure 6 This is a schematic diagram illustrating the interpolation error distribution at the contact points of the action surface in an application scenario of an exemplary embodiment of this disclosure;

[0042] Figure 7 This is a schematic diagram illustrating the matching of the action surface area during the multi-tooth meshing stage in an application scenario of an exemplary embodiment of this disclosure;

[0043] Figure 8 This is a schematic diagram illustrating the matching of the action surface area during the few-tooth meshing stage in an application scenario of the exemplary embodiments of this disclosure;

[0044] Figure 9 This is a schematic diagram illustrating the tooth number and number of rotations in an application scenario of an exemplary embodiment of this disclosure;

[0045] Figure 10 This is a graph illustrating the meshing stiffness history of a gear pair in an application scenario as described in the exemplary embodiments of this disclosure. Detailed Implementation

[0046] Exemplary embodiments will now be described more fully with reference to the accompanying drawings. However, these exemplary embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, they are provided so that this disclosure will be more comprehensive and complete, and will fully convey the concept of the exemplary embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments.

[0047] Furthermore, the accompanying drawings are merely illustrative diagrams of embodiments of this disclosure and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and therefore repeated descriptions of them will be omitted. Some block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities.

[0048] This example implementation provides a method for calculating the meshing stiffness of long-period gears considering the superposition of tooth surface manufacturing errors. (Reference) Figure 1 As shown, the method for calculating the meshing stiffness of long-cycle gears that takes into account the superposition of tooth surface manufacturing errors may include steps S101 to S106.

[0049] Step S101: Use a gear measuring device to measure the tooth surface error of each tooth of the driving gear and the driven gear, so as to obtain the tooth surface measurement point error data of all teeth of the driving gear and the driven gear;

[0050] Step S102: For a pair of gear teeth, generate the gear meshing surface, arrange the contact lines and contact points equidistantly for the driving gear and the driven gear, and number the contact points of the driving gear and the driven gear respectively to obtain the correspondence between the contact points of the driving gear meshing surface and the contact points of the driven gear meshing surface.

[0051] Step S103: For a pair of gear teeth, based on the positional relationship between the tooth surface measurement coordinates and the meshing surfaces of the driving gear and the driven gear, respectively, the tooth surface measurement point error data are interpolated to the contact points of the driving gear meshing surface and the driven gear meshing surface, respectively, to construct the initial contact point error matrix of the driving gear and the initial contact point error matrix of the driven gear.

[0052] Step S104: During the long meshing period (the time it takes for the initially paired teeth to pair up again), for all teeth, in different meshing stages, the action surfaces of the meshing surfaces of all driving and driven gears are matched with the corresponding contact lines and contact points to obtain the initial contact point error matrix of all driving gears and the initial contact point error matrix of driven gears; wherein, the simultaneously meshing tooth pairs include reference tooth pairs and related tooth pairs, and the initial error matrix of each tooth is the reference tooth pair error matrix of that tooth;

[0053] Step S105: Replace the contact point error of the reference tooth pair with the contact point error of the relevant tooth pair of the driving gear and the driven gear respectively, so as to correct the initial contact point error matrix of all driving gears and the initial contact point error matrix of the driven gear, as well as the superposition of the corresponding contact point errors of the driving gear and the driven gear.

[0054] Step S106: Based on the target contact point error matrix of the driving wheel and the target contact point error matrix of the driven wheel, as well as the superposition of the corresponding contact point errors of the driving wheel and the driven wheel, the bending-shear compliance matrix of the gear tooth surface mesh is calculated using the finite element algorithm. The coefficients in the bending-shear compliance matrix of the tooth surface mesh are interpolated to the contact points of the meshing surfaces of the driving wheel and the driven wheel. The deformation of each contact point is calculated using the analytical formula of line contact. Then, the contact equation considering the superposition error of the tooth surface is solved. The meshing stiffness of the gear pair is solved based on the static equilibrium of the contact points.

[0055] By using the above-mentioned method for calculating the long-cycle gear meshing stiffness that considers the superposition of tooth surface manufacturing errors, the tooth surface measurement error data is transformed into the contact points of the gear meshing surface. Considering the error coupling and superposition between the contact points of the driving gear and the driven gear under long meshing cycles, a load-bearing contact analysis is performed on the meshing contact points. This yields a gear pair meshing stiffness that is more consistent with actual conditions, more realistically reproducing the contact situation during gear meshing under actual conditions, and improving the calculation accuracy of gear meshing stiffness.

[0056] Below, we will refer to Figures 1 to 10 The steps of the long-period gear meshing stiffness calculation method considering the superposition of tooth surface manufacturing errors in this example embodiment will be explained in more detail.

[0057] In step S101, the tooth surface deviation is measured using a gear measuring center. Parameters such as the number of teeth, module, pressure angle, helix angle (left-hand or right-hand), displacement coefficient, starting and ending points of the involute measurement, and starting and ending points of the helix measurement need to be input in advance. The corresponding parameters are shown in Table 1. Due to the large amount of error measurement data, only partial error measurement data for the tooth profile and tooth direction of one gear are given here, as shown in Tables 2 and 3.

[0058] Table 1. Gear pair parameters for detailed implementation instructions

[0059]

[0060]

[0061] Table 2 shows the measurement deviation of the driven gear tooth profile when the tooth width is 16mm.

[0062]

[0063] Table 3 shows the measurement deviation of the driven gear tooth direction when the tooth profile measurement circle diameter is 232.3479 mm.

[0064]

[0065] In addition, the tooth profile measurement range is from the diameter of the tooth profile control circle to 0.95 times the diameter of the tooth tip forming circle, and the tooth direction measurement range is the smaller of the following two values: the length of the tooth surface area between the two ends along the axis minus 5% of the tooth width or one module. Furthermore, the number of measurement points along both the tooth profile direction and the tooth width direction should not be less than 16.

[0066] In step S102, the gear meshing process is illustrated in the diagram below. Figure 2 As shown in the figure, N1N2 are the end face meshing lines, B2 is the starting point of the end face meshing of the tooth under examination, and B1 is the exit point of the end face meshing of the tooth under examination. Taking the length of the actual meshing line segment B1B2 on the gear end face as the height and the tooth width as the length, the meshing surface of the gear pair is developed. The unfolded diagram of the gear pair meshing surface is shown below. Figure 3 As shown. In this embodiment, the process of the gear teeth engaging and disengaging is divided into 40 engagement moments. The longest contact line is divided into 18 contact points at equal intervals, and the remaining contact lines are divided into different numbers of contact points according to their respective lengths. The meshing surface is divided into a total of 476 contact points. Furthermore, the driving and driven gears respectively start from tooth root B... p1 and tooth tip B g1 The engagement begins, and the engagement point is numbered as the first contact point of the working surface. The remaining contact points are numbered according to the contact line sequence. For contact points on the same contact line, they are numbered sequentially from the end face closest to the engagement point to the end face furthest from the engagement point. This method is used to complete the numbering of the contact points of the driving and driven wheels.

[0067] In step S103, after obtaining the error data of the tooth surface measurement points, the coordinate systems of the measurement points and the contact points of the action surface are first unified. The unified distribution of measurement points and contact points is as follows: Figure 4 and Figure 5 As shown, these correspond to the driving and driven gears, respectively. Based on the positional relationship between the contact points of the meshing surfaces and the tooth surface error measurement points in the figure, the error at the contact points of the meshing surfaces is interpolated using the Matlab function vq = griddata(x,y,v,xq,yq). In this function, x and y are the coordinates of the tooth surface error measurement points, v is the measurement error corresponding to the measurement point, xq and yq are the coordinates of the contact points of the meshing surfaces, and vq is the error at the contact points of the meshing surfaces obtained based on the tooth surface measurement error interpolation. The interpolation results are shown below. Figure 6As shown, the three-dimensional surface in the figure is a distribution diagram of tooth surface measurement errors, and the * point represents the interpolation error at the contact point. Following this method, the error interpolation of the contact points of all meshing surfaces of the driving and driven wheels can be completed sequentially. After completing the interpolation of the contact point errors of all tooth surfaces, the contact point error matrices efp and efg of the driving and driven wheels are listed. In this embodiment, efp and efg are both matrices of 1653 rows and 476 columns, where 1653 represents the number of teeth traversed by the driving wheel as it rotates through the number of teeth of the driven wheel, and 476 represents the number of contact points on a single tooth meshing surface. The initial contact point error matrix of the driving wheel and the initial contact point error matrix of the driven wheel are: the interpolation errors are written into m according to the tooth numbering order and the contact point numbering order. p (or m) g A matrix with n rows and n columns, m p and m g These represent the number of teeth on the driving gear and the number of teeth on the driven gear, respectively. n is the number of contact points on a single tooth surface. The initial contact error matrix of the driving gear and the initial contact error matrix of the driven gear are denoted as efp and efg, respectively. The tooth numbering order is defined as follows: arbitrarily select one tooth on the driving gear and the driven gear as the initial reference tooth, and number the teeth of the driving gear and the driven gear in sequence according to the meshing direction.

[0068] In step S104, the meshing stage includes a multi-tooth meshing stage and a few-tooth meshing stage. The action surfaces of simultaneously meshing tooth pairs are matched according to the multi-tooth meshing stage and the few-tooth meshing stage. Specifically, a tooth pair's process of going through one multi-tooth and one few-tooth meshing stage from the start of engagement constitutes one base pitch cycle, meaning the tooth pair traverses one base pitch p on the meshing line. bt The length of the tooth pair is called the reference tooth pair. Within the base pitch cycle of the reference tooth pair, the tooth pairs that mesh simultaneously with it are called related tooth pairs. Within a base pitch cycle, the multi-tooth meshing stage is the process of the reference tooth pair entering engagement and a related tooth pair exiting engagement, while the few-tooth meshing stage is the remaining part of the base pitch cycle. The correspondence of the action surface area, contact line, and contact point is achieved by numbering the simultaneously meshing tooth pairs, contact lines, and contact points, and controlling a variable within the numbering. In the embodiments of this application, the meshing action surface situations of the multi-tooth and few-tooth stages are as follows: Figure 7 and Figure 8 As shown. First, the active surface region is matched. For example... Figure 7 As shown, in the multi-tooth stage, there are three pairs of teeth meshing simultaneously. The tooth pairs from the engagement to the disengagement direction are numbered sequentially as reference tooth pairs. Related tooth pairs Related tooth pairs Where i and i' are the tooth numbers of the driving and driven gears, respectively, and j and j' are the number of rotations of the driving and driven gears, respectively. During the stage with fewer teeth, there are two pairs of teeth meshing simultaneously. The tooth pairs from the engagement to the disengagement direction are sequentially numbered as reference tooth pairs. Related tooth pairs The corresponding active surface area is as follows Figure 8 As shown. The diagram illustrating the tooth number and number of revolutions is as follows. Figure 9 As shown. After the matching of the action surface area is completed, the contact lines are mapped in the multi-tooth and few-tooth stages respectively. The contact lines of the reference tooth pair in the multi-tooth meshing stage are numbered as follows. The contact lines on related tooth pair 1 and related tooth pair 2 are respectively numbered as follows: Where, N lm N represents the number of contact lines to divide the multi-tooth meshing stage. mesh The number of meshing positions within a base pitch cycle is defined by the superscripts p and g, which refer to the driving and driven gears, respectively. The contact lines of the reference tooth pairs during the low-tooth meshing phase are numbered as follows: The contact line number of the relevant tooth pair is Where, N ls The number of contact lines is determined for the few-tooth meshing stage. For both the many-tooth and few-tooth meshing stages, the correspondence of contact lines between different tooth pairs only requires ensuring that the variables q and r in the contact line numbering are consistent.

[0069] In step S105, the specific method for correcting the tooth surface error matrices efp and efg of the driving and driven wheels is as follows: First, find the number of contact points n1 and n2 in the first and second base pitch cycles within the action surface. Then, perform a matrix row translation transformation on the data in columns (n1+1) to (n1+n2) and columns (n1+n2+1) to the last column of the initial contact error matrices efp and efg. The row number after the transformation of columns (n1+1) to (n1+n2) in the initial contact error matrices efp and efg is i-1, and the row number after the transformation of columns (n1+n2+1) to the last column is i-2, where i is the matrix row number before the transformation. If i-1 or i-2 is less than or equal to 0, for the initial contact error matrix efp of the driving wheel, i-1 = i-1 + m p i-2 = i-2 + m p For the initial contact error matrix efg of the driven wheel, i-1=i-1+m g i-2 = i-2 + m g m p and m g These represent the row numbers of the initial contact error matrices efp and efg for the driving and driven wheels, respectively.

[0070] In step S106, after obtaining the superposition error of the tooth surface contact points at each meshing moment of the gear pair through steps S101 to S105, the bending-shear deformation of the gear is first calculated using the finite element method. A global finite element model of the gear and a local finite element model of a single tooth are established in Ansys. A tooth surface is selected for consideration, and a unit normal load is applied to the tooth surface of the global finite element model to obtain the overall deformation compliance coefficient δ of the tooth surface.total Then, a reverse unit normal load is applied to the local finite element model to obtain the local deformation compliance coefficient δ of the gear teeth. local The bending-shear compliance coefficient matrix of the tooth surface mesh nodes is obtained by the following formula.

[0071] [δ b ]=[δ total ]-[δ local ]

[0072] In the formula, [δ b [δ] represents the gear bending-shear compliance matrix. total ] represents the overall deformation compliance matrix of the gear, [δ local ] represents the local deformation compliance matrix of the gear.

[0073] Then, the mesh nodes of the finite element model of the tooth surface and the contact points of the meshing surface are unified into a single coordinate system. i and j are the contact points of the meshing surface, referred to as tooth surface contact points, and i' and j' are the mesh nodes of the finite element model of the tooth surface, referred to as tooth surface mesh nodes. The flexibility of the tooth surface mesh nodes... The coordinate relationship between the tooth surface contact point i and all mesh nodes i' is used to interpolate and obtain the compliance coefficient of the tooth surface contact point i with respect to the tooth surface mesh node j'. Then, based on the coordinate relationship between the tooth surface contact point j and all mesh nodes j', Interpolation yields the compliance coefficient of tooth surface contact point i with respect to tooth surface contact point j, thus obtaining δ. bij .

[0074] Secondly, the deformation of each meshing contact point is calculated using the following analytical formula for line contact.

[0075]

[0076]

[0077] In the formula, λ ci For the contact deformation of the i-th segmented contact line, F i For the load on the segmented contact wire, l i E represents the length of the segmented contact wire. * R1 and R2 are the equivalent elastic moduli, and R1 and R2 are the normal radii of curvature of the two gears at the contact point. E1, E2, v1, and v2 are the elastic moduli and Poisson's ratio of gear 1 and gear 2, respectively.

[0078] Finally, the contact equation considering the superposition error of the tooth surface is solved, based on the load F at each tooth surface contact point. i Based on the transmission error LSTE at each meshing moment, the static equilibrium equation at the contact point is listed, and the meshing stiffness of the gear pair is obtained by solving it.

[0079] The nonlinear contact equation considering the superposition error of the tooth surface is as follows:

[0080] -[δ b ]{F}-{λ c}+LSTE+{c}={ε}

[0081] In the formula, [δ b Let {F} be the local deformation matrix at the contact point, and {λ} be the load vector at each contact point. c {} represents the contact deformation vector of each contact point, LSTE represents the static transmission error of the gear pair along the meshing line, {c} represents the remaining clearance vector of each contact point, and {ε} represents the initial contact clearance of the contact point, including the superposition error of tooth surface contact points, the amount of modification, and the amount of meshing misalignment.

[0082] The static equilibrium equation at the contact point is as follows:

[0083]

[0084] In the formula, k m For the overall meshing stiffness of the gear pair, k i F i ε i These represent the meshing stiffness, normal load, and initial contact gap at the i-th contact point, respectively.

[0085] According to the method proposed in this application, the tooth surface deviation of the gear pair in Table 1 is measured. The normal meshing force per unit tooth width of the gear pair is taken as 427.6 N. Each base pitch period is divided into 16 meshing moments. Since the number of teeth of the small and large gears is coprime, the long meshing period is the number of revolutions (57 revolutions) of the small gear (driving gear) through the large gear (driven gear). Taking the long meshing period as the solution time, the long-period gear pair meshing stiffness considering the superposition of tooth surface manufacturing errors is obtained as follows: Figure 10 As shown in the figure, the horizontal axis represents the number of the meshing teeth, and the vertical axis represents the meshing stiffness, with the unit being N / (um*mm).

[0086] By using the above-mentioned method for calculating the long-cycle gear meshing stiffness that considers the superposition of tooth surface manufacturing errors, the tooth surface measurement error data is transformed into the contact points of the gear meshing surface. Considering the error coupling and superposition between the contact points of the driving gear and the driven gear under long meshing cycles, a load-bearing contact analysis is performed on the meshing contact points. This yields a gear pair meshing stiffness that is more consistent with actual conditions, more realistically reproducing the contact situation during gear meshing under actual conditions, and improving the calculation accuracy of gear meshing stiffness.

[0087] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this disclosure. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. In addition, those skilled in the art can combine and integrate the different embodiments or examples described in this specification.

[0088] Other embodiments of this disclosure will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of this disclosure that follow the general principles of this disclosure and include common knowledge or customary techniques in the art not disclosed herein. The specification and examples are to be considered exemplary only, and the true scope and spirit of this disclosure are indicated by the appended claims.

Claims

1. A method for calculating the meshing stiffness of long-period gears considering the superposition of manufacturing errors on the tooth surface, characterized in that, The method includes: Gear measuring equipment is used to measure the tooth surface error of each tooth of the driving gear and the driven gear, so as to obtain the tooth surface measurement point error data of all teeth of the driving gear and the driven gear; For a pair of gear teeth, the gear meshing surfaces, contact lines, and contact points are generated and arranged at equal intervals for the driving gear and the driven gear, respectively. The contact points of the driving gear and the driven gear are numbered to obtain the correspondence between the contact points of the driving gear meshing surfaces and the contact points of the driven gear meshing surfaces. For a pair of gear teeth, based on the positional relationship between the tooth surface measurement coordinates and the meshing surfaces of the driving gear and the driven gear, respectively, the tooth surface measurement point error data are interpolated to the contact points of the driving gear meshing surface and the driven gear meshing surface, respectively, to construct the initial contact point error matrix of the driving gear and the initial contact point error matrix of the driven gear. During the long meshing cycle, for all gear teeth, in different meshing stages, the action surfaces of all driving gear meshing surfaces and driven gear meshing surfaces are matched with the corresponding contact lines and contact points to obtain the initial contact point error matrices of all driving gears and driven gears; wherein, the simultaneously meshing gear pairs include reference gear pairs and related gear pairs, and the initial error matrix of each gear tooth is the reference gear pair error matrix of that gear tooth; The contact point error of the reference tooth pair is replaced by the contact point error of the relevant tooth pair of the driving gear and the driven gear, respectively, to correct the initial contact point error matrix of all driving gears and the initial contact point error matrix of the driven gear, as well as the superposition of the corresponding contact point errors of the driving gear and the driven gear. Based on the target contact point error matrix of the driving wheel and the target contact point error matrix of the driven wheel, as well as the superposition of the corresponding contact point errors of the driving wheel and the driven wheel, the bending-shear compliance matrix of the gear tooth surface mesh is calculated using the finite element algorithm. The coefficients in the bending-shear compliance matrix of the tooth surface mesh are interpolated to the contact points of the meshing surfaces of the driving wheel and the driven wheel. The deformation of each contact point is calculated using the analytical formula of line contact. Then, the contact equation considering the superposition error of the tooth surface is solved. The meshing stiffness of the gear pair is solved based on the static equilibrium of the contact points.

2. The method for calculating the meshing stiffness of long-period gears considering the superposition of tooth surface manufacturing errors according to claim 1, characterized in that, The process of measuring tooth surface error includes: Input the number of teeth, module, pressure angle, helix angle, displacement coefficient, measurement start and end points of the involute and the helix in advance; the tooth profile measurement range is from the diameter of the tooth profile control circle to 0.95 times the diameter of the tooth tip forming circle, and the tooth direction measurement range is the smaller of the two values: the length of the tooth surface area between the two ends minus 5% of the tooth width or one module length at each end of the axis, and the number of measurement points along the tooth profile direction and the tooth width direction is at least 16.

3. The method for calculating the meshing stiffness of long-period gears considering the superposition of tooth surface manufacturing errors according to claim 2, characterized in that, The actual meshing line length of the gear end face is taken as the width, and the tooth width is taken as the length to form the meshing action surface of the gear pair. Contact lines and contact points are arranged at equal intervals within the action surface. Among them, the contact line from the engagement point to the disengagement point within the action surface, including both the engagement point and the disengagement point, represents the complete meshing process of the gear teeth. The contact lines on the driving gear and the driven gear are numbered sequentially according to the meshing sequence. The contact points on each contact line are numbered sequentially from the end face where the engagement point is located to the end face where the disengagement point is located. Contact points with the same number on the driving gear and the driven gear are corresponding contact points.

4. The method for calculating the meshing stiffness of long-period gears considering the superposition of tooth surface manufacturing errors according to claim 3, characterized in that, Based on the coordinate relationship between the contact point of the meshing action surface and the error measurement point, the random point interpolation function griddata is used to interpolate the tooth surface contact point error. The random point interpolation function is vq = griddata(x,y,v,xq,yq), where x and y are the coordinates of the tooth surface error measurement point, v is the measurement error corresponding to the measurement point, xq and yq are the coordinates of the contact point of the meshing action surface, and vq is the error on the action surface contact point obtained based on the tooth surface measurement error interpolation. The initial contact point error matrix of the driving wheel and the initial contact point error matrix of the driven wheel are: The interpolation error is written in the matrix of m p or m g rows and n columns, m p and m g are the number of teeth of the driving gear and the number of teeth of the driven gear respectively, n is the number of contact points on a single tooth surface, and the initial contact point error matrix of the driving wheel and the initial contact point error matrix of the driven wheel are denoted as efp and efg respectively. The tooth numbering sequence is defined as follows: arbitrarily select one tooth on the driving gear and one on the driven gear as the initial reference tooth, and number the teeth on the driving gear and the driven gear sequentially according to the meshing direction.

5. The method for calculating the meshing stiffness of long-period gears considering the superposition of tooth surface manufacturing errors according to claim 4, characterized in that, The meshing phase includes a multi-tooth meshing phase and a few-tooth meshing phase. The action surfaces of simultaneously meshing tooth pairs are matched according to the multi-tooth meshing phase and the few-tooth meshing phase. The time it takes for the teeth to travel one base pitch length on the meshing line from the start of engagement is called a base pitch cycle. Within a base pitch cycle, the multi-tooth meshing phase is the process from the reference tooth pair entering engagement to a related tooth pair exiting engagement, and the few-tooth meshing phase is the remaining part within the base pitch cycle. The correspondence of the action surface area, contact line, and contact point is achieved by numbering the simultaneously meshing tooth pairs, contact lines, and contact points, and controlling a variable within the numbering.

6. The method for calculating the meshing stiffness of long-period gears considering the superposition of tooth surface manufacturing errors according to claim 5, characterized in that, Methods for correcting the initial contact error matrices of all driving gears and driven gears include: Find the number of contact points n1 and n2 within the first and second base period of the action surface. Then, perform a matrix row shift transformation on the data in columns (n1+1) to (n1+n2) and columns (n1+n2+1) to the last column of the initial contact error matrices efp and efg. The row numbers after the transformation of columns (n1+1) to (n1+n2) in the initial contact error matrices efp and efg are i-1, and the row numbers after the transformation of columns (n1+n2+1) to the last column are i-2, where i is the matrix row number before the transformation. If i-1 or i-2 is less than or equal to 0, for the initial contact error matrix efp of the driving wheel, i-1 = i-1 + m p i-2 = i-2 + m p For the initial contact error matrix efg of the driven wheel, i-1=i-1+m g i-2 = i-2 + m g m p and m g These represent the row numbers of the initial contact error matrices efp and efg for the driving and driven wheels, respectively.

7. The method for calculating the meshing stiffness of long-period gears considering the superposition of tooth surface manufacturing errors according to claim 6, characterized in that, The formula for calculating the bending-shear compliance matrix of the tooth surface mesh nodes is as follows: [d b ]=[d total ]-[d local ] In the formula, [δ b [δ] represents the gear bending-shear compliance matrix. total ] represents the overall deformation compliance matrix of the gear, [δ local [ ] represents the local deformation compliance matrix of the gear; The analytical formula for calculating the deformation at each contact point in line contact is: In the formula, λ ci For the contact deformation of the i-th segmented contact line, F i For the load on the segmented contact wire, l i E represents the length of the segmented contact wire. * For the equivalent elastic modulus, R1 is the normal radius of curvature of the driving wheel at the contact point, R2 is the normal radius of curvature of the driven wheel at the contact point, E1 is the elastic modulus of the driving wheel, E2 is the elastic modulus of the driven wheel, v1 is the Poisson's ratio of the driving wheel, and v2 is the Poisson's ratio of the driven wheel. The contact equation considering the superposition error of the tooth surface is: -[d b ]{F}-{λ c }+LSTE+{c}={e} In the formula, [δ b Let {F} be the local deformation matrix at the contact point, and {λ} be the load vector at each contact point. c {c} represents the contact deformation vector of each contact point, LSTE represents the static transmission error of the gear pair along the meshing line direction, {c} represents the remaining clearance vector of each contact point, and {ε} represents the initial contact clearance of the contact point, including the superposition error of tooth surface contact points, the modification amount, and the meshing misalignment amount. The static equilibrium equation at the contact point is: In the formula, k m For the overall meshing stiffness of the gear pair, k i Let F be the meshing stiffness at the i-th contact point. i Let ε be the normal load at the i-th contact point. i Let be the initial contact gap at the i-th contact point.

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