Numerical simulation method for temperature field of single abrasive particle in ultrasonic grinding

Abstract

The application discloses a kind of single abrasive grain ultrasonic grinding temperature field numerical simulation method, comprising: S100. Establish single abrasive grain ultrasonic grinding model, determine the motion trajectory when abrasive grain grinds workpiece;S200. Calculate the heat flow intensity of workpiece when abrasive grain grinds workpiece, according to the motion trajectory of abrasive grain, establish the heat source model of Gaussian distribution;S300. Establish the temperature field algorithm model of workpiece, the specific process includes: S301. The heat conduction process of workpiece is simplified as two-dimensional heat conduction, establishes two-dimensional heat conduction partial differential equation;S302. According to the heat source model of step S200, establish the calculation equation of temperature field generated by arbitrary point heat source to workpiece;S303. Simultaneous solution two-dimensional heat conduction partial differential equation and the calculation equation of temperature field generated by arbitrary point heat source to workpiece. In the application, the temperature field of workpiece in grinding area during single abrasive grain ultrasonic grinding process can be quickly predicted, the problem of temperature increasing effect caused by time-varying movable heat source can be solved simultaneously, the calculation result is very accurate, and the application is suitable for popularization and use.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of ultrasonic grinding, in particular to a single abrasive particle ultrasonic grinding temperature field numerical simulation method. BACKGROUND

[0002] Single abrasive particle ultrasonic grinding is not affected by other abrasive particles in the process of machining workpieces, and is a relatively excellent machining method. In single abrasive particle ultrasonic grinding machining, excessively high grinding temperature often leads to burn and residual tensile stress on the surface of the workpiece, affecting the surface quality and fatigue strength of the workpiece. Therefore, it is crucial to study the influence of grinding temperature on the workpiece.

[0003] In the prior art, the influence of grinding temperature on the temperature field of the workpiece is mainly detected through experiments or simulated and predicted by finite element software. Experimental detection has long time and poor timeliness, and finite element software simulation prediction is only applicable to grinding processes with single abrasive particle movement and simple heat source model, and cannot solve the problem of temperature increase effect caused by time-varying movable heat source, resulting in inaccurate prediction. SUMMARY

[0004] The present application aims to solve at least one of the technical problems existing in the prior art. To this end, the present application provides a single abrasive particle ultrasonic grinding temperature field numerical simulation method, which can quickly predict the temperature field of the workpiece in the grinding area during single abrasive particle ultrasonic grinding, and can also solve the problem of temperature increase effect caused by time-varying movable heat source. The calculation result is very accurate and suitable for popularization and use.

[0005] The single abrasive particle ultrasonic grinding temperature field numerical simulation method according to the first aspect of the present application comprises the following steps:

[0006] S100. Establish a single abrasive particle ultrasonic grinding model to determine the motion trajectory of the abrasive particle when grinding a workpiece;

[0007] S200. Calculate the heat flow intensity flowing into the workpiece when the abrasive particle grinds the workpiece, and establish a Gaussian distribution heat source model according to the motion trajectory of the abrasive particle;

[0008] S300. Establish a temperature field algorithm model of the workpiece, and the specific process comprises:

[0009] S301. Simplify the heat conduction process of the workpiece into two-dimensional heat conduction, and establish a two-dimensional heat conduction partial differential equation;

[0010] S302. According to the heat source model of step S200, establish a calculation equation of the temperature field generated by an arbitrary point heat source on the workpiece;

[0011] S303. Simultaneously solve the two-dimensional heat conduction partial differential equation and the calculation equation of the temperature field generated by an arbitrary point heat source on the workpiece.

[0012] The numerical simulation method for the temperature field of single abrasive grain ultrasonic grinding according to embodiments of the present invention has at least the following beneficial effects:

[0013] The numerical simulation method for the temperature field of single abrasive grain ultrasonic grinding of this invention has several advantages. First, compared to experimental testing, it requires less time and is more timely. Second, compared to existing finite element software, which is only applicable to grinding processes with simple abrasive grain motion and heat source models, the algorithm model of this invention effectively solves the problem of the heating effect caused by time-varying movable heat sources by simultaneously solving the two-dimensional heat conduction partial differential equation and the calculation equation for the temperature field generated by an arbitrary heat source on the workpiece, resulting in higher accuracy. Furthermore, the numerical simulation method of this invention is applicable not only to single abrasive grain ultrasonic grinding but also to ordinary grinding processes. It can help researchers intuitively understand the temperature changes during single abrasive grain ultrasonic grinding and predict thermal damage to the workpiece. It can also further explore the influence of various parameters on grinding heat under different working conditions, promoting the exploration and research of grinding mechanisms and contributing to further research progress in the field of ultrasonic processing, making it suitable for widespread use.

[0014] According to some embodiments of the present invention, in step S200, the heat flux intensity Q flowing into the workpiece is... w The calculation formula is:

[0015] Q w =H u ·η wg ·(Q t -Q ch )

[0016] In the formula, H u η is the ultrasonic grinding coefficient. wg Q is the energy distribution ratio between the heat flux intensity flowing into the workpiece and the heat flux intensity flowing into the abrasive grains. t Q is the total heat flux intensity in the grinding region. ch The intensity of the heat flow carried away by the wear debris;

[0017] The Gaussian heat source model is represented as:

[0018]

[0019] In the formula, x and y represent the coordinates of any point within the grinding area, n represents different time periods during abrasive grinding of the workpiece, and a p This represents the penetration depth of the abrasive grains.

[0020] According to some embodiments of the present invention, the total heat flux intensity Q in the grinding region t Represented as:

[0021] Qt =F gt ·v s ·S -1

[0022] In the formula: F gt The tangential grinding force of the abrasive grains; v s denoted as the linear velocity of the abrasive grain; S is the contact area between the abrasive grain and the workpiece.

[0023] The heat flux intensity Q carried away by the grinding debris ch Represented as:

[0024]

[0025] In the formula: ρ w c is the density of the workpiece material. w T represents the heat capacity of the workpiece material. m v is the melting point temperature of the workpiece material. w For grinding feed rate, φ g The diameter of a single abrasive grain;

[0026] The energy distribution relationship between the heat flux intensity flowing into the workpiece and the heat flux intensity flowing into the abrasive grains is θ. wg Represented as:

[0027]

[0028] In the formula, λ w Let λ be the thermal conductivity of the workpiece. g ρ is the thermal conductivity of the abrasive particles. g c is the density of the abrasive material. g This refers to the heat capacity of the abrasive material.

[0029] According to some embodiments of the present invention, in step S301, the surface temperature distribution of the workpiece is expressed as follows:

[0030] T = T(t,x,y)

[0031] In the formula, x and y represent two-dimensional directions, and t represents time;

[0032] The two-dimensional partial differential equation for heat conduction is:

[0033]

[0034] In the formula, ρ is density, and C p is the specific heat capacity, k is the thermal conductivity, and R is the abrasive grain radius.

[0035] According to some embodiments of the present invention, initial values ​​and boundary conditions are given before solving the two-dimensional heat conduction partial differential equation.

[0036] According to some embodiments of the present invention, the two-dimensional heat conduction partial differential equation is solved by the Ritz-Galerkin numerical method.

[0037] According to some embodiments of the present invention, the Ritz-Galerkin numerical method includes:

[0038] The continuous boundary value region is divided into a grid region consisting of a finite number of elements;

[0039] The continuous variable function on the continuous boundary value region is transformed into a new function through a linear combination of basis functions;

[0040] Transform the differentials in the initial values ​​and boundary conditions into equivalent variational forms.

[0041] According to some embodiments of the present invention, the two-dimensional heat conduction partial differential equation is transformed into a fully discretized Ritz-Galerkin equation in an implicit scheme using the Ritz-Galerkin numerical method:

[0042]

[0043] In the formula, ψ i ψ is a basis function, related to x and y. j Let T be the corresponding trial function, a be the thermal diffusivity, and T be the thermal diffusivity. i The coefficient is related to time t, where T i T is formed using backward difference. i+Δt ;

[0044] The fully discretized Ritz-Galerkin equations were then solved using a computer.

[0045] According to some embodiments of the present invention, the continuous boundary value region is divided into a grid region composed of a finite number of non-overlapping triangles, where each triangle is defined as an element and each vertex of the element is defined as a node. The nodes are numbered in a certain order, and linear elements are used. Then, each node i corresponds to a basis function ψ. i .

[0046] According to some embodiments of the present invention, in step S100, the grinding start point coordinates are defined as (0, Y0), and when time t = t n When = nΔt, the position of the abrasive grain in the xy plane is:

[0047]

[0048] In the formula, n s Where A is the grinding wheel rotation speed, A is the ultrasonic amplitude, and f is the ultrasonic frequency;

[0049] The temperature field generated by an arbitrary heat source on the workpiece is expressed as:

[0050]

[0051] In the formula, v x (n) represents the moving speed of the heat source, i.e., the linear velocity of the abrasive particles;

[0052] K m (p) is a specific function, K m The expression for (p) is:

[0053]

[0054] in,

[0055]

[0056] Other features and advantages of the invention will be set forth in the description which follows, and will be apparent in part from the description, or may be learned by practicing the invention. Attached Figure Description

[0057] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0058] Figure 1 A schematic diagram of a single abrasive grain ultrasonic grinding model;

[0059] Figure 2 This is a schematic diagram of the trajectory simulation during the grinding process of a single abrasive grain;

[0060] Figure 3 A schematic diagram of a Gaussian heat source model during the grinding process of a single abrasive grain;

[0061] Figure 4 This is a schematic diagram of the mesh generation for the continuous boundary value region;

[0062] Figure 5 This is a schematic diagram of a triangular cell within a grid region.

[0063] Figure 6 This is a schematic diagram of one possible location of node i in the grid region;

[0064] Figure 7 This is a schematic diagram of another possible location for node i in the grid region. Detailed Implementation

[0065] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0066] In the description of this invention, it should be understood that the orientation descriptions, such as up, down, front, back, inside, outside, top, bottom, etc., are based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limiting this invention.

[0067] In the description of this invention, "multiple" refers to two or more. The use of "first" and "second" is for distinguishing technical features only and should not be construed as indicating or implying relative importance, or implicitly indicating the number of indicated technical features or their sequential relationship.

[0068] In the description of this invention, unless otherwise explicitly defined, terms such as "set up" and "establish" should be interpreted broadly, and those skilled in the art can reasonably determine the specific meaning of the above terms in this invention in conjunction with the specific content of the technical solution.

[0069] The following is for reference. Figures 1 to 7 A numerical simulation method for the temperature field of ultrasonic grinding of a single abrasive grain according to an embodiment of the present invention is described.

[0070] The numerical simulation method for temperature field during single-abrasive ultrasonic grinding according to an embodiment of the present invention includes the following steps: S100. Establishing a single-abrasive ultrasonic grinding model and determining the motion trajectory of the abrasive during grinding of the workpiece; S200. Calculating the heat flow intensity flowing into the workpiece during grinding, and establishing a Gaussian distributed heat source model based on the motion trajectory of the abrasive; S300. Establishing a temperature field algorithm model for the workpiece, specifically including: S301. Simplifying the heat conduction process of the workpiece into two-dimensional heat conduction and establishing a two-dimensional heat conduction partial differential equation; S302. Based on the heat source model in step S200, establishing a calculation equation for the temperature field generated by an arbitrary heat source on the workpiece; S303. Simultaneously solving the two-dimensional heat conduction partial differential equation and the calculation equation for the temperature field generated by an arbitrary heat source on the workpiece.

[0071] The numerical simulation method for the temperature field of single abrasive grain ultrasonic grinding according to this invention has several advantages. First, compared to experimental testing, it requires less time and is more timely. Second, compared to existing finite element software, which is only applicable to grinding processes with simple abrasive grain motion and heat source models, the algorithm model of this invention effectively solves the problem of the heating effect caused by time-varying movable heat sources by simultaneously solving the two-dimensional heat conduction partial differential equation and the calculation equation for the temperature field generated by an arbitrary heat source on the workpiece, resulting in higher accuracy. Furthermore, the numerical simulation method of this invention is applicable not only to single abrasive grain ultrasonic grinding but also to ordinary grinding processes. It can help researchers intuitively understand the temperature changes during single abrasive grain ultrasonic grinding and predict thermal damage to the workpiece. It can also further explore the influence of various parameters on grinding heat under different working conditions, promoting the exploration and research of grinding mechanisms and contributing to further research progress in the field of ultrasonic processing, making it suitable for widespread use.

[0072] The steps S100 to S300 of the numerical simulation method for the temperature field of ultrasonic grinding of a single abrasive grain according to an embodiment of the present invention will be described in more detail below.

[0073] In some embodiments of the present invention, step S100 includes: establishing a single abrasive grain ultrasonic grinding model and determining the motion trajectory of the abrasive grain when grinding the workpiece.

[0074] Specifically, the abrasive grains grind sinusoidal grooves on the workpiece surface, and their ultrasonic trajectory is mainly controlled by the amplitude and frequency of the ultrasonic vibration. A single abrasive grain ultrasonic grinding model is shown below. Figure 1 As shown, the trajectory of the abrasive grain in space is determined by a fixed coordinate system, and the motion state of the abrasive grain is determined by its linear velocity v. s It is described by three components of velocity along the coordinate axes, where v x The velocity component along the x-axis, v y The velocity component along the y-axis, v z Let f represent the velocity component along the z-axis, A represent the amplitude of the ultrasonic vibration, and f represent the frequency of the ultrasonic vibration.

[0075] The parameter equation for single-grain ultrasonic grinding can be expressed as Formula 1:

[0076]

[0077] In the formula, v w This refers to the grinding feed rate, in mm / min, φ g The diameter of a single abrasive grain is expressed in mm (n). s The grinding wheel speed is given in rpm, A is the ultrasonic amplitude in μm, f is the ultrasonic frequency in kHz, and x is the ultrasonic frequency in kHz. g yg and z gLet t represent the coordinates of the abrasive grain's center point, and t represent the time variable during the abrasive grain's motion. c The time required for the abrasive grains to complete one cycle of processing.

[0078] Among them, the time t required for the abrasive grains to complete one processing cycle. c This can be expressed as Formula 2:

[0079]

[0080] In the formula, a p This indicates the depth of penetration of the abrasive grains.

[0081] In this invention, the individual abrasive grains used are approximately conical in shape, but the shape of the part used for grinding is similar to a circle. Therefore, in the calculation process of this invention, the abrasive grains are defined as regular spheres.

[0082] The processing time t for a grinding arc length c Discretize into n parts, each part being Δt c By calculating the distance between the workpiece position and the abrasive grain center position within the processing area at different times, the processing area Ω of the abrasive grain can be determined. Ω can be expressed as Formula 3:

[0083]

[0084]

[0085] Correspondingly, the z-axis coordinate value of the workpiece within the machining area changes with the penetration depth of the abrasive grains, and its value can be expressed as Equation 4:

[0086]

[0087] Based on the above derivation, the grinding process parameter v is calculated. w =600mm / min, d s =75mm, n s =1500rpm; a p The ultrasonic vibration grinding trajectory of a single abrasive grain at a resolution of 20 μm, A = 10 μm, and f = 25 kHz is as follows: Figure 2 As shown.

[0088] In some embodiments of the present invention, S200 includes: calculating the heat flow intensity flowing into the workpiece when the abrasive grains grind the workpiece, and establishing a Gaussian distributed heat source model based on the movement trajectory of the abrasive grains. The specific process is as follows:

[0089] According to the grinding principle, the total grinding power in the grinding processing area can be expressed as Formula 5:

[0090] Q t =F gt ·vs ·S -1

[0091] In the formula, F gt v is the tangential grinding force of the abrasive grains. s Let S be the linear velocity of the abrasive grain, and S be the contact area between the abrasive grain and the workpiece. Since the linear velocity and depth of penetration of a single abrasive grain into the workpiece are both functions of time during the machining process, the grinding force and the total grinding power also change with time.

[0092] Since axial ultrasonic vibration does not change the radial motion of the abrasive grains, the normal grinding force of a single abrasive grain is considered equal in both conventional and ultrasonic grinding. The normal grinding force of a single abrasive grain during grinding can be expressed as Equation 6:

[0093]

[0094] Where γ1, γ2, and σ0 are constants obtained by solving the linear equation based on experimental data, and n represents different time periods in the grinding process, such as v s (n) represents the linear velocity of the abrasive particles at different time intervals.

[0095] In conventional grinding, according to the binomial theorem of friction, the coefficient of friction can be expressed as Equation 7:

[0096]

[0097] In the formula, α, β, and P0 are coefficients determined by the physical and mechanical properties of the contact interface, obtained from data on material properties.

[0098] Existing research has shown that ultrasonic vibration in ultrasonic grinding can effectively reduce the coefficient of friction between abrasive grains and the workpiece surface. Furthermore, by studying the effect of ultrasonic vibration on the reduction of grinding force, the ratio of the cutting friction coefficient under vibrating motion conditions to that under non-vibrating motion conditions can be expressed as Equation 8:

[0099]

[0100] In the formula, ξ is the ratio of the amplitude of the macroscopic velocity of the object to the amplitude of the vibration velocity, and θ is the angle between the vibration direction and the macroscopic velocity of the object. In axial ultrasonic vibration, the value of θ is taken as π / 2.

[0101] This leads to the derivation of the tangential grinding force F of a single abrasive grain under ordinary grinding. gt_c This can be represented as Formula 9:

[0102]

[0103] By taking θ as π / 2, the tangential grinding force F under ultrasonic grinding is derived. gt_u This can be represented by Formula 10:

[0104]

[0105] Total heat flux Q in the grinding zone t The heat flux intensity Q flowing into the workpiece w The heat flux intensity Q carried away by the grinding debris ch The heat flux intensity Qg flowing into the abrasive particles and the heat flux intensity Q of the air entering the abrasive particles. f The composition can be specifically represented by Formula 11:

[0106] Q t =Q w +Q g +Q ch +Q f

[0107] Abrasive grinding is a type of dry grinding with a short grinding duration, and the heat flux intensity Q carried away by the air is high. f Very little, therefore the intensity of the heat flow Q carried away by the air is very low. f Ignore to simplify calculations.

[0108] For iron-based workpieces, the heat flux intensity Q carried away by the grinding debris is... ch It is approximately equal to the heat required to bring it to a molten state, which can be expressed as formula 12:

[0109]

[0110] In the formula, ρ w c is the material density of the workpiece. w The heat capacity of the workpiece material; T m This is the melting point temperature of the workpiece material.

[0111] In grinding without coolant, the heat flux intensity Q flowing into the workpiece is... w The energy distribution relationship between the heat flux intensity Qg flowing into the abrasive grains can be expressed as Equation 13:

[0112]

[0113] In the formula, λ w Let λ be the thermal conductivity of the workpiece. g ρ is the thermal conductivity of the abrasive particles. g c is the material density of the abrasive grains. g The material heat capacity of the abrasive particles.

[0114] The heat flux intensity Q flowing into the workpiece during single abrasive grain grinding is obtained by calculating the heat distribution ratio. w This can be represented as Formula 14:

[0115] Q w =H u·η wg ·(Q t -Q ch )

[0116] In the formula H u The value represents the ultrasonic grinding coefficient.

[0117] During the machining process of a single abrasive grain, the leading edge of the abrasive grain contacts the workpiece, and the penetration depth at the tip of the abrasive grain is the greatest. Therefore, the Gaussian distribution heat source pattern is more suitable for spherical single abrasive grains. The Gaussian distribution heat source model for spherical abrasive grains can be derived as Equation 15:

[0118]

[0119] In the formula, x and y represent the coordinates of any point within the grinding zone.

[0120] Based on the above derivation, in the grinding process parameter v w =600mm / min, d s =75mm, n s =1500rpm / min, a p =20μm, A=10μm, f=25kHz. The heat source model for a single abrasive grain at the maximum grinding depth position on a cutting trajectory during ultrasonic vibration grinding is calculated as follows: Figure 3 As shown.

[0121] In some embodiments of the present invention, step S300 includes: establishing a temperature field algorithm model for the workpiece, specifically including:

[0122] S301. Simplify the heat conduction process of the workpiece into two-dimensional heat conduction and establish a two-dimensional heat conduction partial differential equation;

[0123] S302. Based on the heat source model in step S200, establish the calculation equation for the temperature field generated by the heat source at any point on the workpiece;

[0124] S303. Solve simultaneously the two-dimensional partial differential equation of heat conduction and the equation for calculating the temperature field generated by an arbitrary heat source on the workpiece.

[0125] In some embodiments of the present invention, step S301 specifically includes the following:

[0126] The heat conduction during ultrasonic grinding of a single abrasive grain can be simplified into a two-dimensional heat conduction problem, represented by the x and y directions. The temperature distribution on the workpiece surface can be expressed as:

[0127] T = T(t,x,y)

[0128] Correspondingly, the two-dimensional partial differential equation for heat conduction can be expressed as Equation 16:

[0129]

[0130] Where k is the thermal conductivity, with units of W / (m*K), C p ρ is the specific heat capacity, expressed in J / (kg*K), and ρ is the density, expressed in kg / m³. 3 .

[0131] To determine the solution to the two-dimensional heat conduction partial differential equation, appropriate initial values ​​and boundary conditions must be given. Assuming the initial temperature of the workpiece is the same as room temperature, set at 300K (approximately 27℃), during single-abrasive ultrasonic grinding, the grinding area differs significantly from the total workpiece area, and the workpiece boundary has sufficient contact with the outside. Therefore, the workpiece boundary can be considered to be at room temperature, i.e., always maintained at 300K. Thus, the initial values ​​and boundary conditions for single-abrasive ultrasonic grinding can be expressed as Equation 17:

[0132]

[0133] The solution to the two-dimensional heat conduction partial differential equation can be obtained using the Ritz-Galerkin numerical method. The specific process is as follows: the continuous boundary value region is divided into a grid region consisting of a finite number of elements; the continuous variable function on the continuous boundary value region is transformed into a new function through a linear combination of basis functions; and the differentials in the initial values ​​and boundary conditions are transformed into equivalent variational forms.

[0134] Through the above process, an approximate solution to the original problem on the element can be obtained. This method not only offers high accuracy and small error but also boasts high computational efficiency and wide applicability, making it highly superior. The specific steps are as follows:

[0135] First, the continuous boundary value region is divided into a grid region:

[0136] D:{(t,x,y)|0≤t≤TIME,0≤x≤X,0≤y≤Y}

[0137] The meshed region is

[0138]

[0139] in For time step, The spatial step size of the x-axis. Let y be the spatial step size of the y-axis. Divide the region D into a finite number of non-overlapping triangles. Each triangle is called a cell, its vertices are called nodes, the other two nodes belonging to the same cell are called adjacent nodes, and two triangles with a common edge are called adjacent cells.

[0140] After determining the mesh partitioning, the nodes are numbered in a certain order, for example, D:{(x,y)|0≤x≤3,0≤y≤0.3}. After selecting an appropriate number of nodes for partitioning, the result is as follows:Figure 4 As shown.

[0141] The coordinates of the i-th node are (x i y i ),like Figure 5 As shown, suppose μ = (μ1, μ2, μ3), use Δ μ Let Δ(i,j,k) represent a triangular unit with vertices numbered μ1, μ2, μ3, and arranged in a counter-clockwise direction. Using linear elements, each node i corresponds to a basis function ψ. i basis function ψ i The support of is all triangular units with i as the vertex.

[0142] Therefore, the basis function ψ can be derived. i In the triangular unit Δ μ The restriction can be expressed as Equation 18:

[0143]

[0144] Among them, a i b i and c i This can be represented as Formula 19:

[0145] a i =x j y k -x k y j ,b i =y j -y k ,c i =x k -x j

[0146] S μ This can be represented as Formula 20:

[0147]

[0148] Assume that all nodes adjacent to node i are l e (e = 1, 2, L, m), then we only need to take j = l sequentially from formula 18 to formula 20. e ,k=l e+1 ,like Figure 6 As shown, when i is an interior point, e = 1, 2, L m, l m+1 =l1, such as Figure 7 As shown, when i is a boundary point, e = 1, 2, ..., m-1, then ψ is obtained. i An expression on a support.

[0149] Secondly, the Ritz-Galerkin format is introduced:

[0150] Use trial function Multiplying Equation 16 by both sides and integrating, we get Equation 21:

[0151]

[0152] By processing Equation 21 using Green's formula and boundary conditions, we can obtain Equation 22:

[0153]

[0154] Introduce a bilinear form and L to Equation 22 2 (D) Inner product, bilinear form, can be expressed as Equation 23:

[0155]

[0156] L 2 (D) The inner product can be expressed as Equation 24:

[0157]

[0158] After the above processing, formula 22 can be simplified to formula 25:

[0159]

[0160] Based on the above expression, the function T can be approximated as a new function T by a linear combination of the basis functions in the Ritz-Galerkin method. h Specifically, this can be expressed as Formula 26:

[0161]

[0162] In the formula, T i T represents the value used to approximate T. h The coefficients in the function are related to time t, ψ i These are basis functions, which are related to x and y.

[0163] Trial function v≈v h =ψ j This yields the semi-discretized Ritz-Galerkin equation of Equation 25, which can be specifically expressed as Equation 27:

[0164]

[0165] In the formula, a is the thermal diffusivity, with units of m. 2 / s.

[0166] The time domain is further discretized independently using the finite difference method, which can be expressed as Equation 28:

[0167]

[0168] Using the backward difference form, that is, when the coefficient T i With T i+Δt When expressed in formal terms, the fully discretized Ritz-Galerkin equation under the implicit scheme can be obtained, which can be specifically expressed as Equation 29:

[0169]

[0170] The fully discretized Ritz-Galerkin equations can be reduced to a large system of sparse linear algebra equations, which can be solved in principle using numerical algebra methods. However, in the process of solving the equations, it is generally necessary to select an appropriate solution method, write a program, and perform computer calculations based on the size of the problem and the capacity and speed of the computer.

[0171] In some embodiments of the present invention, step S302 specifically includes the following:

[0172] When solving the two-dimensional partial differential equation for heat conduction, setting initial values ​​and boundary conditions is equivalent to providing the algorithm with initial values. Through iterative calculation, the temperature at each node in each time layer can be calculated as time progresses, thus simulating the heat conduction process on the workpiece surface. However, relying solely on conventional mathematical calculations with given initial values ​​and boundary conditions is completely inconsistent with the actual heat transfer in single-grain ultrasonic grinding. In single-grain ultrasonic grinding, the abrasive grains continuously transfer heat to the workpiece surface. The amount of heat transferred depends not only on the grinding time but also on the position of the abrasive grains. To accurately calculate the temperature change in each time layer, the following processing is required.

[0173] From step S100, the motion parameter equation for ultrasonic grinding of a single abrasive grain can be obtained. Assuming the starting coordinates of the grinding start point are (0, Y0), when time t = t n When = nΔt, the position of the abrasive grain in the xy plane can be expressed by formula 30:

[0174]

[0175] The heat source of a single abrasive grain in ultrasonic grinding can be considered as a Gaussian-distributed heat source flowing into the workpiece surface from the spherical abrasive head. The temperature field generated on the workpiece by the heat source at any point can be expressed as Equation 31:

[0176]

[0177] In the formula, v x(n) represents the moving speed of the heat source, i.e., the linear velocity of the abrasive particles;

[0178] K m (p) is a specific function, K m (p) can be expressed as formula 32:

[0179]

[0180] in,

[0181]

[0182] In some embodiments of the present invention, step S303 specifically includes the following:

[0183] pass Can be in grid area D h Find a coordinate point (x) that approximates the location of the abrasive grain. * i ,y * j Given that the radius of the abrasive grain is R, based on the following conditions:

[0184]

[0185] Grid region D can be found h All nodes (x) that satisfy the condition i ,y j Then give these nodes a temperature T. i , j In addition, the temperature u generated by the inflow of heat source at this time T (x n ,y n Then, the temperature field calculation formula for the workpiece can be obtained (33).

[0186]

[0187] The aforementioned algorithm model overcomes the limitations of two-dimensional heat conduction partial differential equations in engineering mathematics, which rely on boundary conditions and initial values. It effectively solves the complex calculation problems caused by the constantly changing and moving heat source during single-abrasive ultrasonic grinding. It achieves the simultaneous solution of the temperature field generated by an arbitrary heat source on the workpiece and the numerical calculation of the two-dimensional heat conduction partial differential equation, thus enabling real-time, effective, and accurate calculation of the instantaneous temperature field during single-abrasive ultrasonic grinding. This is particularly useful when the spatial step size of the x and y axes is finely divided, reaching orders of magnitude 10. -6 When the algorithm model predicts, the error will be extremely small. However, because the algorithm logic is highly applicable, it will not increase the computation time. The algorithm model can be effectively applied to industrial production practice, providing theoretical calculation basis and thinking method reference for the optimization of many advanced manufacturing technologies.

[0188] In some embodiments of the present invention, step S400 is the verification of the algorithm model results, specifically including:

[0189] Step S401: Build a single abrasive grain ultrasonic assisted grinding experimental platform and test the grinding temperature field of the workpiece in single abrasive grain ultrasonic grinding and the grinding temperature field of the workpiece in ordinary grinding through experiments.

[0190] Step S402: Simulate and predict the grinding temperature field of the workpiece in ultrasonic grinding of a single abrasive grain and the grinding temperature field of the workpiece in ordinary grinding using finite element software (e.g., COMSOL).

[0191] Step S403: Calculate the grinding temperature field of the workpiece in ultrasonic grinding of a single abrasive grain and the grinding temperature field of the workpiece in ordinary grinding using the algorithm model of the present invention.

[0192] Step S404: Compare the results from steps S401 to S403. The comparison results are as follows:

[0193] In ultrasonic grinding, the measured maximum grinding temperature was 165.33℃, while the algorithm model calculated a maximum grinding temperature of 168.68℃, with a relative error of 2.06%. The finite element method (FEM) simulation predicted a maximum grinding temperature of 156.92℃, with a relative error of 5.09%. In conventional grinding, the measured maximum grinding temperature was 187.48℃, while the algorithm model calculated a maximum grinding temperature of 189.79℃, with a relative error of 1.23%. The FEM simulation predicted a maximum grinding temperature of 177.32℃, with a relative error of 5.42%. Therefore, the algorithm model proposed in this invention can accurately predict the grinding temperature of a single abrasive grain in both ultrasonic and conventional grinding processes, and the prediction accuracy is higher than that of the FEM simulation. This demonstrates the rationality of the single abrasive grain ultrasonic grinding temperature calculation model proposed in this invention.

[0194] In summary, the numerical simulation method for the temperature field of single abrasive ultrasonic grinding of the present invention achieves calculations that closely approximate the actual process. Compared with simulation prediction using finite element software, it is more accurate, requires less time, and has better timeliness. Furthermore, the algorithm model of the present invention is applicable not only to single abrasive ultrasonic grinding but also to ordinary grinding. It can not only help researchers understand the changes in processing temperature during single abrasive ultrasonic grinding and predict thermal damage to the workpiece, but also further explore the influence of various parameters on grinding heat under different working conditions, promote the exploration and research of grinding mechanism laws, and even be further developed and optimized to be applicable to different processing fields, thus playing a greater role.

[0195] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

Claims

1. A numerical simulation method for the temperature field during ultrasonic grinding of a single abrasive grain, characterized in that, Includes the following steps: S100. Establish a single abrasive grain ultrasonic grinding model and determine the motion trajectory of the abrasive grain when grinding the workpiece; S200. Calculate the heat flow intensity into the workpiece when the abrasive grains grind the workpiece, and establish a Gaussian heat source model based on the movement trajectory of the abrasive grains; S300. Establish the temperature field algorithm model for the workpiece, the specific process of which includes: S301. Simplify the heat conduction process of the workpiece into two-dimensional heat conduction and establish a two-dimensional heat conduction partial differential equation; S302. Based on the heat source model described in step S200, establish the calculation equation for the temperature field generated by the heat source at any point on the workpiece; S303. Solve the two-dimensional partial differential equation of heat conduction and the calculation equation of the temperature field generated by the arbitrary heat source on the workpiece simultaneously; In step S200, the heat flux intensity flowing into the workpiece Q w The calculation formula is: In the formula, H u The ultrasonic grinding coefficient is... This represents the energy distribution ratio between the heat flux intensity flowing into the workpiece and the heat flux intensity flowing into the abrasive grains. Q t The total heat flux intensity within the grinding region, Q ch The intensity of the heat flow carried away by the wear debris; The Gaussian heat source model is represented as: In the formula, x , y This represents the coordinates of any point within the grinding area. n Represents different time periods during abrasive grinding of a workpiece. a p This represents the penetration depth of the abrasive grains.

2. The numerical simulation method for the temperature field of single abrasive grain ultrasonic grinding according to claim 1, characterized in that, Total heat flux intensity in the grinding zone Q t Represented as: In the formula, F gt The tangential grinding force of the abrasive grains. v s For abrasive linear velocity, S This represents the contact area between the abrasive grains and the workpiece. The heat flux carried away by the grinding debris Q ch Represented as: In the formula, ρ w For the density of the workpiece material, c w For the heat capacity of the workpiece material, T m The melting point temperature of the workpiece material. For grinding feed rate, The diameter of a single abrasive grain; The energy distribution relationship between the heat flux intensity flowing into the workpiece and the heat flux intensity flowing into the abrasive grains Represented as: In the formula, The thermal conductivity of the workpiece. The thermal conductivity of the abrasive grains is... ρ g For the density of abrasive materials, c g This refers to the heat capacity of the abrasive material.

3. The numerical simulation method for the temperature field of single abrasive grain ultrasonic grinding according to claim 1, characterized in that, In step S301, the surface temperature distribution of the workpiece is expressed as follows: In the formula, x , y Represents two-dimensional coordinates. t Indicates time; The two-dimensional partial differential equation for heat conduction is: In the formula, ρ For density, For specific heat capacity, k Thermal conductivity, R Where is the abrasive grain radius.

4. The numerical simulation method for the temperature field of ultrasonic grinding of a single abrasive grain according to claim 3, characterized in that, Before solving the two-dimensional heat conduction partial differential equation, initial values ​​and boundary conditions are given.

5. The numerical simulation method for the temperature field of single abrasive grain ultrasonic grinding according to claim 4, characterized in that, The two-dimensional partial differential equation of heat conduction was solved using the Ritz-Galerkin numerical method.

6. The numerical simulation method for the temperature field of ultrasonic grinding of a single abrasive grain according to claim 5, characterized in that, The Ritz-Galerkin numerical method includes: The continuous boundary value region is divided into a grid region consisting of a finite number of elements; The continuous variable function on the continuous boundary value region is transformed into a new function through a linear combination of basis functions; Transform the differentials in the initial values ​​and boundary conditions into equivalent variational forms.

7. The numerical simulation method for the temperature field of single abrasive grain ultrasonic grinding according to claim 6, characterized in that, The two-dimensional heat conduction partial differential equation is transformed into a fully discretized Ritz-Galerkin equation in an implicit scheme using the Ritz-Galerkin numerical method: In the formula, As a basis function, and For the corresponding trial function, For thermal diffusivity, The coefficient is the time factor. t related, Formed using backward difference ; The fully discretized Ritz-Galerkin equations were then solved using a computer.

8. The numerical simulation method for the temperature field of ultrasonic grinding of a single abrasive grain according to claim 7, characterized in that, The continuous boundary value region is divided into a grid region consisting of a finite number of non-overlapping triangles. Each triangle is defined as an element, and the vertices of the element are defined as nodes. The nodes are numbered in a certain order, and linear elements are used. Each node... Each corresponds to one of the aforementioned basis functions. .

9. The numerical simulation method for the temperature field of ultrasonic grinding of a single abrasive grain according to claim 1, characterized in that, In step S100, the grinding start point coordinates are defined as (0, ...). ), when time At that time, the abrasive grains are in The position of the plane is: In the formula, The rotational speed of the grinding wheel. This refers to the ultrasonic amplitude. f Ultrasonic frequency; The temperature field generated by an arbitrary heat source on the workpiece is expressed as: In the formula, v x (n) The moving speed of the heat source is the linear velocity of the abrasive particles; K m (p) For a specific function, K m (p) The expression is: in, 。

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