Dynamic power modeling method for face gear worm grinding wheel gear grinding

Abstract

The invention discloses a dynamic power modeling method for gear grinding of a face gear worm grinding wheel, which comprises the following steps of: 1, acquiring surface topography parameters of a grinding worm, including abrasive particle density, abrasive particle projection height distribution and worm geometric parameters; step 2, establishing a motion trail model of the abrasive particles in the dynamic contact area based on the cooperative rotation motion of the grinding worm and the face gear; step 3, kinematics analysis based on an abrasive particle track is obtained, and the face gear continuous generating grinding process is controlled by periodic phenomena of a meshing period and a feeding period; 4, a single abrasive particle cutting model, a friction force calculation model, a ploughing force calculation model and a cutting force calculation model are constructed respectively, and the grinding force in the face gear continuous generating grinding process is obtained; and 5, the grinding power is determined to be the product of the torque and the rotating speed, the grinding force vector of each abrasive particle is converted from O2t to Owt, and a grinding power prediction model is constructed based on the coordinates of the abrasive particles in the Owt.

Description

Technical Field

[0001] The invention belongs to the technical field of gear grinding, and specifically relates to a dynamic power modeling method for face gear worm grinding wheel gear grinding. Background Art

[0002] The continuous generative grinding process for face gear worm wheels differs fundamentally from traditional static grinding processes (such as flat, cylindrical, and form grinding) due to its inherent, position-dependent dynamic contact characteristics. This complexity presents four fundamental modeling challenges for power variation analysis: First, the grinding wheel surface is composed of randomly distributed abrasive particles with uncertain cutting edges, which is significantly different from deterministic tool geometry (such as milling, turning, and hobbing), introducing significant uncertainty into material removal modeling. Second, the conjugate meshing between the worm wheel and the face gear involves hyperboloid matching. The curvature combination of the contact zone and the tooth profile geometry continuously changes along the feed direction, resulting in a dynamic evolution of the contact state, making traditional static contact models difficult to apply. Third, the synchronous rotational motion of the grinding wheel and workpiece produces a complex, high-order spatiotemporal relative motion trajectory. This requires simultaneous consideration of the multi-scale coupling effects of macroscopic tool motion and microscopic abrasive particle kinematics, making it difficult to accurately describe using analytical mathematical forms. Fourthly, the simulated meshing motion of the grinding wheel causes the number of teeth involved in grinding and the cutting depth to change dynamically with position and time, triggering transient material removal rate fluctuations and dynamic load redistribution, thus invalidating the traditional quasi-static modeling framework based on the steady-state cutting assumption.

[0003] In summary, these factors jointly determine that the force-energy characteristics of FGCGG are highly dynamic and complex, resulting in the current lack of a power prediction method for face gear worm grinding wheel processing. This further highlights the urgent need to develop a dedicated modeling method that can accurately capture its dynamic changes in power, which is of great significance for in-depth understanding of the processing mechanism, prediction of processing effects and optimization of process parameters. Summary of the Invention

[0004] In view of this, the purpose of the present invention is to provide a dynamic power modeling method for face gear worm grinding wheel grinding, which effectively reveals the periodic power mechanism in the continuous generating grinding of end gears and provides a theoretical basis for process optimization.

[0005] In order to achieve the above object, the present invention provides the following technical solutions:

[0006] A dynamic power modeling method for face gear worm grinding wheel gear grinding comprises the following steps:

[0007] Step 1: Reconstruct the surface topography of the grinding worm:

[0008] Obtaining surface topography parameters of the grinding worm, including abrasive density, abrasive protrusion height distribution, and worm geometric parameters; reconstructing the digital surface topography of the grinding worm through optical measurement and three-dimensional clustering algorithm.

[0009] Step 2: Kinematic analysis of wear particle trajectory

[0010] Based on the coordinated rotational motion of the grinding worm and the face gear, a motion trajectory model of the abrasive particles in the dynamic contact area is established. The motion trajectory model maps the abrasive particle position to the face gear motion coordinate system through coordinate transformation, and calculates the projected relationship between the local cutting speed and the feed speed.

[0011] Step 3: Analysis of the mechanism of periodic change of contact area

[0012] Based on the kinematic analysis of the abrasive particle trajectory, the continuous generation grinding process of the face gear is controlled by the periodic phenomena of the meshing cycle and the feed cycle; during the meshing process between the grinding worm and the face gear, the rotation angle of the grinding worm The change of causes the tooth surface alignment to change periodically, and the contact area The single meshing cycle shows a U-shaped periodic change that first decreases and then increases; the radial feed motion passes through the center distance l w The gradual adjustment of the contact area results in an n-shaped periodic change along the feed direction, which first decreases and then increases.

[0013] Step 4: Construct grinding force model

[0014] The single abrasive cutting model, friction force calculation model, plowing force calculation model and cutting force calculation model were constructed respectively to obtain the grinding force during the continuous generation grinding process of face gears.

[0015] Step 5: Build a grinding power prediction model

[0016] The grinding power is determined as the product of torque and speed, and the grinding force vector of each abrasive particle is converted from the face gear motion coordinate system O to 2t Convert to the grinding worm motion coordinate system O wt , based on the motion coordinate system O of the grinding worm wt The coordinates in are used to construct the grinding power prediction model.

[0017] Furthermore, in step 1, the method steps for obtaining the surface morphology parameters of the grinding worm are as follows:

[0018] 11) Construct a three-dimensional surface model of the grinding worm;

[0019] 12) Use digital microscope to measure surface morphology;

[0020] 13) Use three-dimensional clustering algorithm based on image recognition to identify and extract wear particles;

[0021] 14) Analysis of abrasive particle protrusion height distribution;

[0022] 15) Surface topography is reconstructed using key statistical parameters including abrasive particle density, mean value, and standard deviation of protrusion height.

[0023] Furthermore, in step 11), the method for constructing the surface three-dimensional model of the grinding worm is:

[0024] Construct the fixed coordinate system O of the virtual production wheel s0 , the motion coordinate system O of the virtual forming wheel st , Fixed coordinate system O of grinding worm w0 and the motion coordinate system O of the grinding worm wt , then for the virtual forming wheel with involute profile, its surface is in the moving coordinate system O st The position vector in is:

[0025]

[0026] Where: r bs is the base circle radius of the virtual shape-generating wheel; θ k represents the involute angle; θ0 represents half the width of the corner space on the base circle; u s Represents the position parameter on the z-axis along the width direction of the gear;

[0027] Then in the motion coordinate system O wt In the virtual wheel surface r wt The position vector is expressed as:

[0028]

[0029] in: Indicates the coordinate system O st To the motion coordinate system O wt The transformation matrix of is the rotation angle of the grinding worm.

[0030] Furthermore, in step 15), the protrusion height distribution of the abrasive particles obeys a Gaussian distribution, and its mean and standard deviation are calibrated by actual measurement data.

[0031] Furthermore, in step 2, the kinematic analysis method of the wear particle trajectory is as follows:

[0032] 21) Establish a motion trajectory model of abrasive particles in the dynamic contact area

[0033] 211) Establish the relationship between the face gear rotation angle and the grinding worm rotation angle:

[0034]

[0035] in: is the grinding worm rotation angle; is the face gear rotation angle; N2 is the number of face gear teeth;

[0036] 212) The abrasive grains are placed in the grinding worm motion coordinate system O wt The position vector P in wt Transform to the face gear motion coordinate system O 2t :

[0037]

[0038] Where: P 2t is the motion coordinate system O of the abrasive particle in the face gear 2t The position vector in ; is the face gear motion coordinate system O 2t To the grinding worm motion coordinate system O wt The transformation matrix of

[0039] 22) Calculate the projection relationship between local cutting speed and feed speed

[0040] 221) Calculate the face gear motion coordinate system O 2t Nominal feed rate in:

[0041]

[0042] in: is the face gear motion coordinate system O 2t Nominal feed rate in ; is the grinding worm motion coordinate system O wt Nominal feed rate in M 2tw0 From the fixed worm coordinate system O w0 To the face gear motion coordinate system O 2t The transformation matrix of

[0043] Get the local feed speed v wi and nominal local cutting depth a pi_local :

[0044]

[0045] a pi_local =a p cosβ pi

[0046] in: is the face gear motion coordinate system O 2t The local feed rate in ; α piRepresents the projection angle between the motion plane and the feed speed direction; β pi Indicates the projection angle between the moving surface and the nominal cutting depth direction; a p Indicates the nominal cutting depth;

[0047] 222) The angular velocity of the grinding worm is converted from the grinding worm fixed coordinate system O to w0 Transform to the face gear motion coordinate system O 2t , and the local cutting speed is obtained:

[0048] ω 2t =M 2tw0 ·ω w0

[0049] Where: 2t is the angular velocity of the face gear in the motion coordinate system; ω w0 is the angular velocity of the grinding worm; M 2tw0 The fixed coordinate system O of the grinding worm w0 Transform to the face gear motion coordinate system O 2t The transformation matrix of

[0050] Get the local cutting speed:

[0051] v si =|ω 2t ×P 2ti |

[0052] Where: v si is the local cutting speed; P 2ti Indicates the abrasive particle i in the face gear moving coordinate system O 2t The position vector in .

[0053] Furthermore, the face gear motion coordinate system O 2t To the grinding worm motion coordinate system O wt The transformation matrix for:

[0054]

[0055] From the fixed worm coordinate system O w0 To the face gear motion coordinate system O 2t The transformation matrix M 2tw0 for:

[0056]

[0057] in: is the grinding worm rotation angle; is the rotation angle of the grinding worm; l w is the center distance; w is the helix angle of the grinding worm; E wsIndicates the center distance between the virtual forming wheel and the worm shaft.

[0058] Furthermore, in step 4, the single abrasive cutting model includes:

[0059] Maximum cutting depth of a single abrasive grain:

[0060]

[0061] Where: h mi is the maximum cutting depth of a single abrasive particle; C is the abrasive particle density; r represents the coefficient of the relationship between the height and width of the abrasive particle; v wi Indicates the local feed speed; v si Indicates the local cutting speed; a pi Indicates the local cutting depth; d ei represents the local wheel diameter;

[0062] Actual local cutting depth a pi_real :

[0063] a pi_real =H i ―H max +a pi_local

[0064] Among them: H i Indicates the protruding height of a single abrasive particle; H max is the assumed maximum protruding height of the abrasive particles; a pi_real Indicates the actual local cutting depth obtained based on wear particle trajectory analysis;

[0065] Instantaneous cutting depth h i :

[0066]

[0067] Where: E * is the equivalent elastic modulus derived from E1 and E2, where E1 and E2 are the elastic moduli of the abrasive and the workpiece, respectively; i is the average contact pressure; d i is the abrasive particle diameter;

[0068] Plowing occurs when the average contact pressure exceeds half the Brinell hardness of the workpiece material; cutting occurs when the average contact pressure exceeds the Brinell hardness; expressed as:

[0069]

[0070] in: It is the critical cutting depth at which plowing of the material occurs; h is the critical cutting depth when the material is cut; BrinellThe Brinell hardness of the workpiece material corresponds to the cutting depth.

[0071] Furthermore, in step 4, the friction force calculation model includes:

[0072] Normal friction force F ni rubbing :

[0073]

[0074] in: is the average pressure; A 0i is the contact area;

[0075] Tangential friction force F ti rubbing :

[0076] F ti rubbing =μ i ·F ni rubbing

[0077] Where: μ i is the friction coefficient.

[0078] Furthermore, in step 4, the plowing force calculation model includes:

[0079] Normal plowing force F of a single abrasive particle ni :

[0080] F ni =H Brinell

[0081] Among them: H Brinell is the Brinell hardness of the workpiece material;

[0082] Tangential plowing force F of a single abrasive particle ti ploweing :

[0083] F ti plowing =μ pi Fniplowing

[0084] Where: μ pi is the plowing friction coefficient.

[0085] Furthermore, in step 4, the cutting force calculation model includes the resultant normal cutting force and the resultant tangential cutting force, and is solved by integrating the improved Ernst-Merchant model:

[0086]

[0087] in: is the tangential cutting force of a single abrasive grain; is the normal cutting force of a single abrasive particle; α cri is the abrasive contact angle; r i is the abrasive radius; t0 is the critical thickness in the cutting stage; τ s is the shear strength of the workpiece material; α i is the rake angle; β i is the friction angle.

[0088] Furthermore, in step 5, the grinding power prediction model is:

[0089]

[0090] Where: P grinding is the grinding power; F i is the grinding force vector of a single abrasive particle; and is the coordinate system O of the i-th abrasive particle in the grinding worm motion wt The coordinates in ; and are the axis components of the grinding force vector of a single abrasive grain in the dynamic coordinate system of the worm grinding wheel; n worm is the worm grinding wheel speed; N is the total number of effective abrasive particles in the contact area.

[0091] The beneficial effects of the present invention are:

[0092] Compared to the empirical power prediction methods commonly used in gear grinding research, the proposed dynamic power modeling approach for face gear worm grinding establishes a multi-physics interaction approach that dynamically tracks the transient contact zone geometry and the microscopic interaction mechanisms of discrete abrasive particles, enabling a multiscale characterization of time-dependent material removal phenomena. This approach fundamentally advances FGCGG (face gear continuous generating grinding) modeling by simultaneously addressing both macroscopic process energetics and microscopic abrasive-workpiece interactions through kinematic analysis. The proposed approach integrates three fundamental parameter domains: tool parameters that define the grinding worm geometry and surface topography; machining parameters that control chip formation kinematics; and workpiece parameters that determine the material mechanical properties that regulate microcontact mechanics. By discretely calculating the instantaneous power contribution of individual abrasive particles and then performing spatiotemporal aggregation, the proposed method reconstructs the total grinding power by solving the geometric series and transient interaction dynamics along the feed motion. By explicitly incorporating the time-varying mesh position and its cascading effect on the contact geometry evolution, a high-fidelity estimation of the power waveform variation is achieved, effectively linking macroscopic process observables with microscopic tribological phenomena while overcoming the temporal resolution limitations inherent in traditional steady-state methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0093] In order to make the purpose, technical solutions and beneficial effects of the present invention more clear, the present invention provides the following drawings for illustration:

[0094] Figure 1 The figure is a flow chart of the dynamic power modeling method for face gear worm grinding wheel gear grinding according to the present invention.

[0095] Figure 2 Schematic diagram of the FGCGG coordinate system.

[0096] Figure 3 Schematic diagram of the reconstruction process of the grinding worm surface morphology; (a) CAD model of the grinding worm in FGCGG; (b) surface morphology measurement using a digital microscope; (c)-(e) wear particle identification and extraction using a 3D clustering algorithm based on image recognition.

[0097] Figure 4 Schematic diagram of the kinematic analysis based on abrasive particle trajectory; (a) A single abrasive particle trajectory in the face gear coordinate system during one complete rotation cycle of the grinding worm in FGCGG; (b) Abrasive particle trajectory and motion plane, which is inconsistent with the travel plane defined by the feed direction and cutting depth; (c) Contact line in the contact area, where the intersection between the abrasive particle contact line and the forming wheel contact line represents the ideal processing point in FGCGG.

[0098] Figure 5 Schematic diagram of the contact area evolution mechanism in FGCGG; (a) Geometric evolution of tool-workpiece engagement at different grinding worm rotation angles A complete meshing cycle corresponds to the worm rotation angle -p to p, representing the entire meshing cycle; (b) Cyclic contact area variation within one worm rotation cycle, with the grinding area showing a "U-shaped" variation pattern - first decreasing and then increasing within each meshing cycle; (c) Continuous variation of the contact area over multiple meshing cycles, where each cycle follows the same "U-shaped" variation pattern; (d) Macroscopic progression of the contact area along the feed direction, showing an "n-shaped" evolution; the contact area changes accordingly when the worm gear moves from the outer radial position to the inner radial position of the face gear, which is determined by the parameter l w From R max Change to R min Definition; (e) Schematic diagram of the contact area shape at the external, intermediate, and internal feed positions. The contact area varies periodically with meshing position (resulting in a "U-shaped" pattern), while the feed motion in the radial direction introduces additional variation (resulting in an "n-shaped" pattern); these two mechanisms together explain the micro- and macro-periodic variations in grinding power observed in FGCGG.

[0099] Figure 6Schematic diagram of the abrasive-workpiece interaction mechanism in FGCGG; (a) Chip formation process; (b) Friction phase interaction, where the material undergoes elastic deformation without significant displacement; (c) Plowing phase interaction, where the material undergoes plastic displacement but is not removed; (d) Cutting phase interaction, where the material is sheared and forms chips. The cutting plane of each abrasive particle varies depending on its position relative to the face gear workpiece, resulting in a misalignment between the nominal cutting plane defined by the feed motion and the actual cutting plane. This misalignment leads to deviations between the nominal and actual machining parameters.

[0100] Figure 7 (a) is the interaction between the abrasive particle protrusion height and the workpiece surface, where the maximum particle protrusion height corresponds to the nominal depth of the cut surface; Figure 7 (b) Cutting phase classification framework, in which the abrasive contact state is divided into friction, plowing, and cutting phases based on the undeformed chip thickness. During the grinding process, the instantaneous depth of cut varies along the cutting path, dividing material removal into friction, plowing, and cutting phases. Due to the different contact conditions of the abrasive particles, these phases exhibit different mechanical responses and energy characteristics.

[0101] Figure 8 Figure 3 Micro-contact models for different contact phases; (a) Contact model for the friction phase, where the surface material undergoes plastic deformation without material removal; (b) Contact model for the plowing phase, which is characterized by a combination of elastic and plastic deformation; part of the material is displaced, forming ridges along the contact trace; (c) Contact model for the cutting phase, where chip formation occurs simultaneously with friction and plowing characteristics, and material is effectively removed. DETAILED DESCRIPTION

[0102] The present invention will be further described below with reference to the accompanying drawings and specific embodiments so that those skilled in the art can better understand the present invention and implement it. However, the embodiments are not intended to limit the present invention.

[0103] Specifically, such as Figure 1 As shown, the dynamic power modeling method for face gear worm grinding wheel gear grinding in this embodiment includes the following steps.

[0104] Step 1: Reconstruct the surface topography of the grinding worm:

[0105] The surface topography parameters of the grinding worm are obtained, including abrasive density, abrasive protrusion height distribution and worm geometric parameters; the surface topography parameters are used to reconstruct the digital surface topography of the grinding worm through optical measurement and three-dimensional clustering algorithm.

[0106] In this embodiment, the method steps for obtaining the surface topography parameters of the grinding worm are as follows:

[0107] 11) Construct a three-dimensional surface model of the grinding worm.

[0108] A crowned ground worm is used to simulate the rotational motion of a virtual profile gear meshing with a face gear. At every rotation angle, the surface of the ground worm continuously meshes with the interior of a virtual profile wheel driven by the face gear. As the worm wheel rotates, the virtual profile wheel also rotates simultaneously at a predefined speed ratio. Consequently, the surface of the ground worm can be viewed as the envelope of a family of contact lines generated by a series of profile wheel profiles.

[0109] Specifically, the method for constructing the surface three-dimensional model of the grinding worm is as follows:

[0110] Construct the fixed coordinate system O of the virtual production wheel s0 , the motion coordinate system O of the virtual forming wheel st , Fixed coordinate system O of grinding worm w0 and the motion coordinate system O of the grinding worm wt , then for the virtual forming wheel with involute profile, its surface is in the moving coordinate system O st The position vector in is:

[0111]

[0112] Where: r bs is the base circle radius of the virtual shape-generating wheel; θ k represents the involute angle; θ0 represents half the width of the corner space on the base circle; u s Represents the position parameter on the z-axis along the width of the gear.

[0113] Then in the motion coordinate system O wt In the virtual wheel surface r wt The position vector is expressed as:

[0114]

[0115] in: Indicates the coordinate system O st To the motion coordinate system O wt The transformation matrix of is the grinding worm rotation angle.

[0116] like Figure 2 As shown, considering the kinematic relationship between the stationary coordinate system and the moving coordinate system, the grinding worm rotation angle is expressed as The rotation angle of the generating wheel is expressed as The center distance between the grinding worm and the forming wheel gear is expressed as E ws and the helix angle of the grinding worm is expressed as λ w From O st to O wtThe transformation matrix can be expressed as follows:

[0117]

[0118] Since the surface of the worm wheel is the envelope of the surface of the generating wheel, the surface S of the worm wheel w The following meshing equations must be satisfied:

[0119]

[0120] Where: θ k is the worm grinding wheel forming angle parameter, r wt is the point vector on the worm grinding wheel surface, u s is the gear shaping cutter tooth width parameter, is the worm grinding wheel angle.

[0121] Therefore, the worm gear surface can be and θ k Discretization is performed and a two-parameter envelope process is formed.

[0122] 12) Use digital microscope to measure surface morphology;

[0123] 13) Use three-dimensional clustering algorithm based on image recognition to identify and extract wear particles;

[0124] 14) Analysis of abrasive particle protrusion height distribution;

[0125] 15) Surface topography is reconstructed using key statistical parameters including abrasive particle density, mean value, and standard deviation of protrusion height.

[0126] Through this method, the distribution and size of abrasive particles can be accurately characterized, providing a data basis for contact analysis. Specifically, due to the randomness of the grinding process, the surface morphology of the grinding worm cannot be accurately described by pure geometric expressions. Extensive research has been conducted to develop an accurate grinding wheel modeling method. However, existing studies have shown that the protrusion height distribution of abrasive particles generally follows a Gaussian distribution. In order to digitally reconstruct the surface of the grinding worm, the spatial distribution and protrusion height of the abrasive particles are extracted from actual surface measurements. This embodiment uses a Keyence VHX-1000C digital microscope to ensure high-precision surface acquisition. Multiple locations on the surface of the grinding worm are scanned to collect three-dimensional morphology data. A 3D clustering algorithm is then used to identify abrasive particles, which enables the quantification of spatial particle density and individual particle protrusion height. These measurements provide an analytical basis for surface morphology modeling. Key parameters such as particle density, average protrusion height and protrusion height standard deviation are derived to construct a digital grinding worm model suitable for further contact analysis. The modeling process is as follows Figure 3 shown.

[0127] Step 2: Kinematic analysis of wear particle trajectory

[0128] Based on the coordinated rotational motion of the grinding worm and the face gear, a motion trajectory model of the abrasive particles in the dynamic contact area is established. The motion trajectory model maps the abrasive particle position to the face gear motion coordinate system through coordinate transformation, and calculates the projection relationship between the local cutting speed and the feed speed.

[0129] Specifically, in this embodiment, the steps of the kinematic analysis of the wear particle trajectory are as follows:

[0130] 21) Establish a motion trajectory model of abrasive particles in the dynamic contact area

[0131] 211) Establish the relationship between the face gear rotation angle and the grinding worm rotation angle. Specifically, the grinding worm and face gear rotate simultaneously. To analyze the relative motion of the abrasive particles with respect to the face gear, the grinding trajectory of the abrasive particles must be examined. Based on the manufacturing principle of FGCGG, the rotation angle relationship between the grinding worm and the face gear is as follows:

[0132]

[0133] in: is the grinding worm rotation angle; is the face gear rotation angle; N2 is the face gear teeth number. Specifically, in this embodiment, the number of grinding worm gear heads is 1.

[0134] 212) The abrasive grains are placed in the grinding worm motion coordinate system O wt The position vector P in wt Transform to the face gear motion coordinate system O 2t :

[0135]

[0136] Where: P 2t is the motion coordinate system O of the abrasive particle in the face gear 2t The position vector in ; is the face gear motion coordinate system O 2t To the grinding worm motion coordinate system O wt The transformation matrix; M 2t20 is the transfer matrix from the static coordinate system to the dynamic coordinate system of the face gear; M 20w0 is the transfer matrix from the static coordinate system of the worm grinding wheel to the static coordinate system of the face gear; M w0wt is the transfer matrix from the dynamic coordinate system to the static coordinate system of the worm grinding wheel; x wt 、y wt and z wt is the motion coordinate system O of the abrasive in the grinding worm wt The coordinates in .

[0137] That is, in this embodiment, the face gear motion coordinate system O2t To the grinding worm motion coordinate system O wt The transformation matrix is:

[0138]

[0139] in: is the grinding worm rotation angle; is the rotation angle of the grinding worm; l w is the center distance; w is the helix angle of the grinding worm; E ws Indicates the center distance between the virtual forming wheel and the worm grinding wheel axis.

[0140] According to the above formula, the cutting abrasive particles are in the face gear motion coordinate system O 2t The position in the worm is determined by the worm rotation angle and the center distance l w Determine the center distance l w is reduced due to the feed motion. Considering that the rotational speed is significantly higher than the feed speed, the trajectory of the abrasive particles forms a complex spatial pattern with cyclic speed characteristics, e.g. Figure 4 This unique kinematic feature enables the FGCGG to perform a continuous grinding process, effectively grinding each individual tooth using a grinding worm of limited width.

[0141] The motion plane near the engagement point is defined by the abrasive particle trajectory. Computational analysis investigates the correlation between nominal machining parameters and trajectory-based parameters for different abrasive particle positions. The trajectory-based parameters are derived by projecting vectors onto the particle motion plane.

[0142] 22) Calculate the projection relationship between local cutting speed and feed speed

[0143] 221) According to the relative motion and position between the coordinate systems, calculate the face gear motion coordinate system O 2t Nominal feed rate in:

[0144]

[0145] in: is the face gear motion coordinate system O 2t Nominal feed rate in ; is the grinding worm motion coordinate system O wt Nominal feed rate in; v f Indicates the nominal feed rate included in the machining parameters; M 2tw0 From the fixed worm coordinate system O w0 To the face gear motion coordinate system O 2t The transformation matrix of , and:

[0146]

[0147] in: is the grinding worm rotation angle; is the rotation angle of the grinding worm; l w is the center distance; w is the helix angle of the grinding worm; E ws Indicates the center distance between the virtual forming wheel and the worm shaft.

[0148] Get the local feed speed v wi and nominal local cutting depth a pi_local :

[0149]

[0150] a pi_local =a p cosβ pi

[0151] in: is the face gear motion coordinate system O 2t The local feed rate in ; α pi Represents the projection angle between the motion plane and the feed speed direction; β pi Indicates the projection angle between the moving surface and the nominal cutting depth direction; a p Indicates the nominal cutting depth.

[0152] 222) The angular velocity of the grinding worm is converted from the grinding worm fixed coordinate system O to w0 Transform to the face gear motion coordinate system O 2t , and the local cutting speed is obtained:

[0153]

[0154] Where: 2t is the angular velocity of the face gear in the motion coordinate system; ω w0 is the angular velocity of the grinding worm; M 2tw0 The fixed coordinate system O of the grinding worm w0 Transform to the face gear motion coordinate system O 2t The transformation matrix of w Indicates the helix angle of the grinding worm; E ws Indicates the center distance between the virtual gear shaping machine and the grinding worm wheel.

[0155] Get the local cutting speed:

[0156] v si =|ω 2t ×P 2ti |

[0157] Where: vsi is the local cutting speed; P 2ti Represents the abrasive particle i in the face gear moving coordinate system O 2t The position vector in .

[0158] In this way, the nominal machining parameters are converted into trajectory-based parameters for individual abrasive grains, enabling a more accurate calculation of the undeformed chip thickness.

[0159] Step 3: Analysis of the mechanism of periodic change of contact area

[0160] Based on the kinematic analysis of the abrasive particle trajectory, the continuous generation grinding process of the face gear is controlled by the periodic phenomena of the meshing cycle and the feed cycle; during the meshing process between the grinding worm and the face gear, the rotation angle of the grinding worm The change of causes the tooth surface alignment to change periodically, and the contact area The single meshing cycle shows a U-shaped periodic change that first decreases and then increases; the radial feed motion passes through the center distance l w The progressive adjustment results in an n-shaped periodic change of the contact area along the feed direction, which first decreases and then increases.

[0161] Specifically, the FGCGG process exhibits inherent dynamic instability, which is governed by two superimposed periodic phenomena: the meshing periodicity caused by the continuous worm gear meshing and the progression caused by the feed through the gear profile. Figure 5 As shown in Figures 5(a) and 5(b), the meshing cycles show that the contact area evolves along a characteristic “U-shaped” trajectory driven by the rotational phase-dependent tooth alignment changes between the grinding worm and the workpiece. At the same time, high-speed worm rotation causes cyclic contact area modulation between adjacent tooth meshes (see Figure 5 (c)), resulting in high-frequency interaction dynamics.

[0162] In addition, trajectory analysis showed that the irregular boundary of the contact zone between the grinding worm and the face gear caused the O wt The abrasive contact line L2 at different positions changes, such as Figure 4 (c) These changes affect the number of actively contacting abrasive particles, the penetration depth, and the shape of the wear chips. This kinematic characteristic leads to highly dynamic contact conditions, which in turn causes changes in the grinding forces and grinding power.

[0163] Considering the contact law that changes along the feed direction, the feed cycle is determined by the feed speed v f Controlled center distance modulation l w Introduced a macroscopic "n-shaped" contact area variation (see Figure 5(d) and 5(e)). This dual-scale interaction mechanism directly regulates two key process parameters: (1) effective abrasive density and (2) instantaneous depth of cut distribution. The superposition of high-frequency meshing oscillations and low-frequency feed progression produces a unique modulation of grinding conditions, fundamentally distinguishing FGCGG from traditional gear grinding methods such as profile grinding. This coupled periodic mechanism establishes the characteristic process fingerprint of FGCGG, where transient contact area dynamics are the main driver of the power variation characteristics.

[0164] Step 4: Construct grinding force model

[0165] A single abrasive cutting model, friction force calculation model, plowing force calculation model and cutting force calculation model were constructed respectively to obtain the grinding force during the continuous development grinding process of face gears.

[0166] During gear grinding, the contact area and the number of meshing teeth change with the rotational motion (see Figure 6 (a)), resulting in unstable grinding area and variations in the number of effective abrasive grains. These dynamics lead to variations in grinding force and power, making traditional force and power prediction models designed for stable grinding processes unsuitable for FGCGG.

[0167] However, at the micro-contact scale, the material removal mechanism in FGCGG is similar to other grinding processes. The chip formation process can be divided into three distinct stages: friction, plowing, and cutting. Each stage has its own unique mechanical and geometric interactions, such as Figure 6 (b), 6(c) and 6(d).

[0168] (1) Single abrasive cutting model

[0169] In order to identify the cutting phase of a single abrasive grain, it is necessary to calculate the cutting depth based on the kinematics of the abrasive grain and its interaction with the workpiece surface. To achieve this goal, it is first necessary to determine a calculation model for the maximum cutting depth, such as Figure 6 (a). The maximum cutting depth of a single abrasive grain (i.e., the maximum undeformed chip thickness) can be expressed as follows:

[0170]

[0171] Where: h mi is the maximum cutting depth of a single abrasive particle; C is the abrasive particle density; r represents the coefficient of the relationship between the height and width of the abrasive particle, which can be determined by measurement; v wi Indicates the local feed speed; v si Indicates the local cutting speed; a pi Indicates the local cutting depth; d ei Indicates the equivalent grinding wheel diameter of the abrasive grain position.

[0172] The proposed FGCGG model is fundamentally different from the traditional model. Where the original framework assumes uniform abrasive kinematics, FGCGG requires that the machining parameters be explicitly projected onto a single abrasive motion plane to account for the inherent differences between the nominal settings and the actual cutting geometry. Thus, based on the abrasive trajectory, the nominal cutting depth a p Projected onto a pi_real ,like Figure 7 (b) This vector space correction enables the precise determination of the local material removal dynamics.

[0173] Considering the matching process during machining, it is reasonable to assume that the maximum protrusion height H of the abrasive grain is max Corresponds to the nominal surface of the workpiece (see Figure 7 (a)). Under this assumption, the protrusion height is less than H max ―a pi_local The particles are considered to be inactive and do not contribute to the grinding process. Therefore, the actual local cutting depth a pi_real It can be expressed as:

[0174] a pi_real =H i ―H max +a pi_local

[0175] Among them: H i Indicates the protruding height of a single abrasive particle; H max is the assumed maximum protruding height of the abrasive particles; a pi_real Indicates the actual local cutting depth obtained based on wear particle trajectory analysis;

[0176] Specifically, Figure 8 (a) Represents the interaction between the abrasive grain protrusion height and the workpiece surface, where the maximum grain protrusion height corresponds to the nominal depth of the cut face. Figure 8 (b) Cutting phase classification framework, in which the abrasive contact state is divided into friction, plowing, and cutting phases based on the undeformed chip thickness. During the grinding process, the instantaneous depth of cut varies along the cutting path, dividing material removal into friction, plowing, and cutting phases. Due to the different contact conditions of the abrasive particles, these phases exhibit different mechanical responses and energy characteristics.

[0177] Using the local contact parameters of a single abrasive particle, the local maximum cutting depth h can be calculated mi .like Figure 8 As shown in (a), the instantaneous cutting depth h i Depends on the rotation angle θ iGiven the undeformed chip thickness model, the change in cutting depth is related to the rotation angle. Through the geometric penetration calculation, the cutting angle and cutting angle of each abrasive particle can be determined, and the instantaneous cutting depth h can be calculated by identifying the corresponding position along the contact trajectory. i .

[0178] Specifically, in order to classify the active abrasive particles into the friction, plowing and cutting stages, it is necessary to define the critical boundary of the cutting depth. It is reasonable to approximate the shape of the abrasive particles as a sphere, and the protrusion height corresponds to the diameter d of the sphere. i Therefore, the contact model of ball-plane interaction can be applied. Based on the Hertz contact model, the instantaneous cutting depth h i It can be expressed as:

[0179]

[0180] Where: E * is the equivalent elastic modulus derived from E1 and E2, where E1 and E2 are the elastic moduli of the abrasive and the workpiece, respectively; i is the average contact pressure; d i is the abrasive particle diameter.

[0181] Plowing occurs when the average contact pressure exceeds half the Brinell hardness of the workpiece material; cutting occurs when the average contact pressure exceeds the Brinell hardness. These conditions can be expressed as:

[0182]

[0183] in: It is the critical cutting depth at which plowing of the material occurs; h is the critical cutting depth when the material is cut; Brinell The Brinell hardness of the workpiece material corresponds to the cutting depth.

[0184] By considering the instantaneous cutting depth and critical cutting depth of a single abrasive particle, the cutting stage of the effective abrasive particle can be determined, e.g. Figure 6 Once the contact phase of each individual abrasive particle is determined, the corresponding grinding force can be calculated, laying the foundation for grinding power prediction.

[0185] (2) Friction calculation model

[0186] For particles in frictional contact, the forces generated are mainly frictional and are related to a combination of elastic, elastoplastic and plastic deformations. In order to calculate the friction forces, it is necessary to analyze the geometry of the contact interface. Figure 7 As shown in (a), in the friction stage, the diameter is d i The spherical abrasive particles will produce a depth of h i The indentation produces a contact angle αi and contact width L i ; According to contact mechanics, the contact angle α i for:

[0187]

[0188] Get the contact area A 0i :

[0189]

[0190] Based on the Hertz contact model, the average pressure for:

[0191]

[0192] Where: E * is the equivalent elastic modulus derived from E1 and E2, where E1 and E2 are the elastic moduli of the abrasive and workpiece, respectively;

[0193] Normal friction force F ni rubbing :

[0194]

[0195] in: is the average pressure; A 0i is the contact area;

[0196] Tangential friction force F ti rubbing :

[0197] F ti rubbing =μ i ·F ni rubbing

[0198] Where: μ i is the friction coefficient.

[0199] (3) Plowing force calculation model

[0200] During the plowing stage, the surface material of the workpiece is displaced from its original position to both sides of the abrasive track, forming a "pile" along the path (see Figure 8 (b)). According to existing research, the plowing contact stage can be modeled as an indentation test scenario, where the normal force is equal to the Brinell hardness of the workpiece material. Therefore, the normal plowing force F of a single abrasive particle is ni :

[0201] F ni =H Brinell

[0202] Among them: HBrinell is the Brinell hardness of the workpiece material;

[0203] Tangential plowing force F of a single abrasive particle ti plowing :

[0204] F ti plowing =μ pi Fniplowing

[0205] Where: μ pi is the plowing friction coefficient.

[0206] The plowing friction coefficient can be estimated as:

[0207]

[0208] Where: L i is the width of the abrasive contact area; d i is the abrasive particle diameter.

[0209] (4) Cutting force calculation model

[0210] During the cutting stage, the surface material is removed to form grinding chips. In order to simulate the mechanical interaction of individual abrasive particles during the cutting stage, this embodiment uses the improved Ernst Merchant model to describe the material removal process. Figure 8 As shown in (c), the chip forming force is calculated by integrating the force components corresponding to the cutting depth increment. That is, in this embodiment, the cutting force calculation model includes the resultant normal cutting force and the resultant tangential cutting force, and is solved using the improved Ernst-Merchant model integration to obtain:

[0211]

[0212] in: is the tangential cutting force of a single abrasive grain; is the normal cutting force of a single abrasive particle; α cri is the abrasive contact angle; r i is the abrasive radius; t0 is the critical thickness in the cutting stage; τ s is the shear strength of the workpiece material; α i is the rake angle; β i is the friction angle.

[0213] Step 5: Build a grinding power prediction model

[0214] In order to predict the grinding power of the spindle, the torque at each grinding position must be calculated. The grinding power is determined as the product of torque and rotational speed. Since the directions of the grinding force components (normal and tangential) are intrinsically related to the orientation of the individual abrasive grains, a transformation is required. This transformation decomposes the force components into planes perpendicular to the axis of rotation, since the components in other directions are constrained by the machine tool structure. The grinding forces are calculated in the face gear moving coordinate system O. 2t Calculated in, and the main axis is in O wt Therefore, it is necessary to transform the grinding force vector of each abrasive grain from the face gear motion coordinate system O 2t Convert to the grinding worm motion coordinate system O wt , based on the motion coordinate system O of the grinding worm wt The grinding power prediction model is constructed based on the coordinates in . Specifically, the grinding power prediction model is:

[0215]

[0216] Where: P grinding is the grinding power; F i is the grinding force vector of a single abrasive particle; and is the coordinate system O of the i-th abrasive particle in the grinding worm motion wt The coordinates in ; and are the axis components of the grinding force vector of a single abrasive grain in the dynamic coordinate system of the worm grinding wheel; n worm is the worm grinding wheel speed; N is the total number of effective abrasive particles in the contact area.

[0217] Face gear continuous generation grinding (FGCGG) exhibits distinct periodic power variation patterns due to the simultaneous rotational motion of the grinding worm and face gear. These variations include a macroscopic "n-shaped" pattern along the radial feed direction and a "U-shaped" pattern during the meshing motion. To reveal the mechanisms underlying these patterns, this paper proposes a dynamic power modeling approach for face gear worm grinding wheel grinding, accounting for the contact geometry and position-dependent properties of the abrasive particle kinematics. First, a geometric kinematic model of FGCGG is proposed to provide more accurate chip parameter estimation by considering the time-varying superimposed motion of the grinding worm and face gear. Then, the grinding power is calculated by accumulating abrasive particles at different grinding stages over the dynamic contact area. Finally, the proposed model is experimentally validated in terms of the amplitude and periodicity of the power variations. Geometric analysis of the abrasive particle motion quantitatively reveals a maximum 6% difference between the nominal and actual machining parameters, highlighting the necessity of parameter modification. Furthermore, further analysis based on penetration calculations revealed that variations in contact area—including across the meshing position and along the radial feed position—are the primary source of the periodic power variations, with the random nature of the grinding wheel topography also contributing to minor fluctuations. This example provides clear insights into the mechanism behind periodic power variations in FGCGG and offers a theoretical basis for process optimization.

[0218] The above embodiments are merely preferred embodiments for the purpose of fully illustrating the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are within the scope of protection of the present invention. The scope of protection of the present invention shall be subject to the claims.

Claims

1. A dynamic power modeling method for face gear worm grinding wheel grinding, characterized by: The steps include: Step 1: Reconstruct the surface topography of the grinding worm: Obtaining surface topography parameters of the grinding worm, including abrasive density, abrasive protrusion height distribution, and worm geometric parameters; reconstructing the digital surface topography of the grinding worm through optical measurement and three-dimensional clustering algorithm. Step 2: Kinematic analysis of wear particle trajectory Based on the coordinated rotational motion of the grinding worm and the face gear, a motion trajectory model of the abrasive particles in the dynamic contact area is established. The motion trajectory model maps the abrasive particle position to the face gear motion coordinate system through coordinate transformation, and calculates the projected relationship between the local cutting speed and the feed speed. Step 3: Analysis of the mechanism of periodic change of contact area Based on the kinematic analysis of the abrasive particle trajectory, the continuous generation grinding process of the face gear is controlled by the periodic phenomena of the meshing cycle and the feed cycle; during the meshing process between the grinding worm and the face gear, the rotation angle of the grinding worm The change of causes the tooth surface alignment to change periodically, and the contact area The single meshing cycle shows a U-shaped periodic change that first decreases and then increases; the radial feed motion passes through the center distance l w The progressive adjustment results in an n-shaped periodic change of the contact area along the feed direction, which first decreases and then increases; Step 4: Construct grinding force model The single abrasive cutting model, friction force calculation model, plowing force calculation model and cutting force calculation model were constructed respectively to obtain the grinding force during the continuous generation grinding process of face gears. Step 5: Build a grinding power prediction model The grinding power is determined as the product of torque and speed, and the grinding force vector of each abrasive particle is converted from the face gear motion coordinate system O to 2t Convert to the grinding worm motion coordinate system O wt , based on the motion coordinate system O of the grinding worm wt The coordinates in are used to construct the grinding power prediction model.

2. The dynamic power modeling method for face gear worm grinding wheel gear grinding according to claim 1, characterized in that: In step 1, the method steps for obtaining the surface topography parameters of the grinding worm are as follows: 11) Construct a three-dimensional surface model of the grinding worm; 12) Use digital microscope to measure surface morphology; 13) Use three-dimensional clustering algorithm based on image recognition to identify and extract wear particles; 14) Analysis of abrasive particle protrusion height distribution; 15) Surface topography is reconstructed using key statistical parameters including abrasive particle density, mean value, and standard deviation of protrusion height.

3. The dynamic power modeling method for face gear worm grinding wheel gear grinding according to claim 2, characterized in that: In the step 11), the method for constructing the three-dimensional surface model of the grinding worm is as follows: Construct the fixed coordinate system O of the virtual production wheel s0 , the motion coordinate system O of the virtual forming wheel st , Fixed coordinate system O of grinding worm w0 and the motion coordinate system O of the grinding worm wt , then for the virtual forming wheel with involute profile, its surface is in the moving coordinate system O st The position vector in is: Where: r bs is the base circle radius of the virtual shape-generating wheel; θ k represents the involute angle; θ0 represents half the width of the corner space on the base circle; u s Represents the position parameter on the z-axis along the width direction of the gear; Then in the motion coordinate system O wt In the virtual wheel surface r wt The position vector is expressed as: in: Indicates the coordinate system O st To the motion coordinate system O wt The transformation matrix of is the grinding worm rotation angle.

4. The dynamic power modeling method for face gear worm grinding wheel gear grinding according to claim 1, characterized in that: In step 2, the kinematic analysis method of the wear particle trajectory is as follows: 21) Establish a motion trajectory model of abrasive particles in the dynamic contact area 211) Establish the relationship between the face gear rotation angle and the grinding worm rotation angle: in: is the grinding worm rotation angle; is the face gear rotation angle; N2 is the number of face gear teeth; 212) The abrasive grains are placed in the grinding worm motion coordinate system O wt The position vector P in wt Transform to the face gear motion coordinate system O 2t : Where: P 2t is the motion coordinate system O of the abrasive particle in the face gear 2t The position vector in ; is the face gear motion coordinate system O 2t To the grinding worm motion coordinate system O wt The transformation matrix of 22) Calculate the projection relationship between local cutting speed and feed speed 221) Calculate the face gear motion coordinate system O 2t Nominal feed rate in: in: is the face gear motion coordinate system O 2t Nominal feed rate in ; is the grinding worm motion coordinate system O wt Nominal feed rate in M 2tw0 From the fixed worm coordinate system O w0 To the face gear motion coordinate system O 2t The transformation matrix of Get the local feed speed v wi and nominal local cutting depth a pi_local : to pi_local =a p ·cosβ pi in: is the face gear motion coordinate system O 2t The local feed rate in ; α pi Represents the projection angle between the motion plane and the feed speed direction; β pi Indicates the projection angle between the moving surface and the nominal cutting depth direction; a p Indicates the nominal cutting depth; 222) The angular velocity of the grinding worm is converted from the grinding worm fixed coordinate system O to w0 Transform to the face gear motion coordinate system O 2t , and the local cutting speed is obtained: oh 2t =M 2tw0 ·oh w0 Where: 2t is the angular velocity of the face gear in the motion coordinate system; ω w0 is the angular velocity of the grinding worm; M 2tw0 The fixed coordinate system O of the grinding worm w0 Transform to the face gear motion coordinate system O 2t The transformation matrix of Get the local cutting speed: v si =|ω 2t ×P 2ti | Where: v si is the local cutting speed; P 2ti Indicates the abrasive particle i in the face gear moving coordinate system O 2t The position vector in .

5. The dynamic power modeling method for face gear worm grinding wheel gear grinding according to claim 4, characterized in that: Face gear motion coordinate system O 2t To the grinding worm motion coordinate system O wt The transformation matrix for: From the fixed worm coordinate system O w0 To the face gear motion coordinate system O 2t The transformation matrix M 2tw0 for: in: is the grinding worm rotation angle; is the rotation angle of the grinding worm; l w is the center distance; w is the helix angle of the grinding worm; E ws Indicates the center distance between the virtual forming wheel and the worm shaft.

6. The dynamic power modeling method for face gear worm grinding wheel gear grinding according to claim 1, characterized in that: In step 4, the single abrasive cutting model includes: Maximum cutting depth of a single abrasive grain: Where: h mi is the maximum cutting depth of a single abrasive particle; C is the abrasive particle density; r represents the coefficient of the relationship between the height and width of the abrasive particle; v wi Indicates the local feed speed; v si Indicates the local cutting speed; a pi Indicates the local cutting depth; d ei represents the local wheel diameter; Actual local cutting depth a pi_real : a pi_real =H i ―H max +a pi_local Among them: H i Indicates the protruding height of a single abrasive particle; H max is the assumed maximum protruding height of the abrasive particles; a pi_real represents the actual local cutting depth obtained based on wear particle trajectory analysis; Instantaneous cutting depth h i : Where: E * is the equivalent elastic modulus derived from E1 and E2, where E1 and E2 are the elastic moduli of the abrasive and the workpiece, respectively; i is the average contact pressure; d i is the abrasive particle diameter; Plowing occurs when the average contact pressure exceeds half the Brinell hardness of the workpiece material; cutting occurs when the average contact pressure exceeds the Brinell hardness; expressed as: in: It is the critical cutting depth at which plowing of the material occurs; h is the critical cutting depth when the material is cut; Brinell The Brinell hardness of the workpiece material corresponds to the cutting depth.

7. The dynamic power modeling method for face gear worm grinding wheel gear grinding according to claim 1, characterized in that: In step 4, the friction force calculation model includes: Normal friction force F ni rubbing : in: is the average pressure; A 0i is the contact area; Tangential friction force F ti rubbing : F ti rubbing =μ i ·F ni rubbing Where: μ i is the friction coefficient.

8. The dynamic power modeling method for face gear worm grinding wheel gear grinding according to claim 1, characterized in that: In step 4, the plowing force calculation model includes: Normal plowing force F of a single abrasive particle ni : F ni =H Brinell Among them: H Brinell is the Brinell hardness of the workpiece material; Tangential plowing force F of a single abrasive particle ti plowing : F ti plowing =μ pi F ni plowing Where: μ pi is the plowing friction coefficient.

9. The dynamic power modeling method for face gear worm grinding wheel gear grinding according to claim 1, characterized in that: In step 4, the cutting force calculation model includes the normal cutting force resultant and the tangential cutting force resultant, and is solved by integrating the improved Ernst-Merchant model: in: is the tangential cutting force of a single abrasive grain; is the normal cutting force of a single abrasive particle; α cri is the abrasive contact angle; r i is the abrasive radius; t0 is the critical thickness in the cutting stage; τ s is the shear strength of the workpiece material; α i is the rake angle; β i is the friction angle.

10. The dynamic power modeling method for face gear worm grinding wheel gear grinding according to claim 1, characterized in that: In step 5, the grinding power prediction model is: Where: P grinding is the grinding power; F i is the grinding force vector of a single abrasive particle; and is the coordinate system O of the i-th abrasive particle in the grinding worm motion wt The coordinates in ; and are the axis components of the grinding force vector of a single abrasive grain in the dynamic coordinate system of the worm grinding wheel; n worm is the worm grinding wheel speed; N is the total number of effective abrasive particles in the contact area.

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