Ultrasonic gear honing thermal behavior control method based on force-thermal analysis

Abstract

The invention relates to the technical field of ultrasonic gear honing hot working, in particular to an ultrasonic gear honing thermal behavior control method based on force-thermal analysis, which comprises the following steps: constructing an ultrasonic amplitude transformer three-dimensional resonant frequency characteristic equation with indefinite coupling boundary conditions; a honing wheel single abrasive particle motion track, a wear rate and a material removal rate are studied to obtain a single abrasive particle cutting force, the number of abrasive particles participating in honing in a machining area is obtained, a dynamic grinding force model of an ultrasonic gear honing machining area is established, and on the basis, a high-order function curve and a moving heat source theory are combined to obtain a dynamic grinding force model of the ultrasonic gear honing machining area. A three-dimensional transient temperature field distribution model of an ultrasonic gear honing machining area is constructed, and a thermal control strategy is provided for ultrasonic gear honing; and finally, a thermal-structural coupling dynamic equilibrium equation of ultrasonic gear honing is established, the thermal deformation rule of the amplitude-change pole is disclosed, dynamic compensation of thermal deformation of the amplitude-change pole in the machining process is achieved, and the surface integrity and machining precision of the hardened tooth face gear subjected to ultrasonic gear honing machining are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of ultrasonic honing thermal processing, and particularly to a method for controlling the thermal behavior of ultrasonic honing based on force-thermal analysis. Background Art

[0002] Hard tooth surface gears with low noise and high load-carrying capacity play a decisive role in the transmission of motion and power systems, and are widely used in high-end equipment fields such as high-speed transportation, aerospace, and new energy. Due to their long-term service in harsh environments such as high rotational speeds, heavy / alternating / impact loads, fatigue failure behaviors such as tooth root bending fatigue, tooth surface contact fatigue, corrosion, and wear are likely to occur, and the requirements for the life and reliability of hard tooth surface gears are more stringent. Research shows that the surface integrity and machining accuracy of hard tooth surface gears are the key factors affecting their life and reliability. Therefore, how to improve the surface integrity and machining accuracy of hard tooth surface gears and manufacture high-performance hard tooth surface gears that meet the above service requirements has become an important challenge faced by the innovation of mechanical equipment and the high-end equipment manufacturing industry.

[0003] Ultrasonic honing is commonly used for the finishing of hard tooth surface gears. It has higher processing efficiency, lower equipment cost, and the processing residual stress is compressive stress, which is beneficial to improving the surface integrity of hard tooth surface gears. The tooth surface texture processed is conducive to reducing the transmission noise of hard tooth surface gears and improving the reliability of hard tooth surface gears. Moreover, it can effectively reduce honing wheel clogging, reduce honing force, and improve the tooth surface microtopography. However, during the ultrasonic honing process, the friction between the honing wheel and the hard tooth surface gear will generate a large amount of heat in the processing area, causing the temperature of the surface layer of the hard tooth surface gear to rise rapidly, resulting in thermal damage to the surface layer of the hard tooth surface gear, which will reduce the surface integrity of the hard tooth surface gear, shorten the service life of the hard tooth surface gear, and a large amount of heat diffuses into the horn, causing the horn to generate thermal deformation, changing the relative position between the honing wheel and the processed hard tooth surface gear, exacerbating the tooth profile and pitch errors, reducing the processing accuracy, and generating transmission noise. Summary of the Invention

[0004] To solve the technical defects that during the existing ultrasonic honing process, the friction between the honing wheel and the hard tooth surface gear will generate a large amount of heat in the processing area, resulting in a reduction in the surface integrity of the hard tooth surface gear, causing thermal deformation of the horn, reducing the processing accuracy, and generating transmission noise, the present invention provides a method for controlling the thermal behavior of ultrasonic honing based on force-thermal analysis, exploring a new path for improving the surface integrity and processing accuracy of ultrasonic honing, and having important significance for extending the service life of hard tooth surface gears and reducing transmission noise.

[0005] The present invention provides a method for controlling the thermal behavior of ultrasonic honing based on force-thermal analysis, including the following steps:

[0006] Step 1: Establish a three-dimensional resonance frequency characteristic equation of the ultrasonic horn with indefinite coupling boundary conditions, and accurately solve the resonance frequency of the ultrasonic horn;

[0007] Step 2: Based on the resonance frequency of the ultrasonic horn obtained in Step 1, establish a dynamic grinding force model for ultrasonic honing;

[0008] Step 3: Based on the dynamic grinding force model of the ultrasonic honing processing area established in Step 2, establish a three-dimensional transient temperature field distribution model for ultrasonic honing;

[0009] Step 4: Based on the three-dimensional transient temperature field distribution model of the ultrasonic honing processing area established in Step 3, establish a thermal-structural coupling analysis model for ultrasonic honing, and analyze the thermal deformation law of the horn through the thermal-structural coupling analysis model of ultrasonic honing, so as to realize the dynamic compensation of the thermal deformation of the horn during the processing.

[0010] Preferably, the sub-steps of Step 1 are:

[0011] Step 1.1: Construct a cylindrical coordinate system, and divide the ultrasonic horn composed of the horn and the simplified ring disk of the hard tooth surface gear into integration regions, where the horn is simply referred to as the rod, and the simplified ring disk of the hard tooth surface gear is simply referred to as the disk; based on the penalty function, introduce a penalty factor, and construct the essential boundary condition admissible functions in the direction of each integration region of the rod and the disk in the non-coupled case in the cylindrical coordinate system and , where ;

[0012] When there is an interference fit between the rod and the disk, the essential boundary condition admissible functions in the direction of each integration region in the cylindrical coordinate system and are:

[0013] (1-1),

[0014] When there is a clearance fit between the rod and the disk, the essential boundary condition admissible functions in the direction of each integration region in the cylindrical coordinate system and are:

[0015] (1-2),

[0016] In formulas (1-1) and (1-2), is the The dimensionless coordinate in the direction, which is the dimensionless coordinate of each integral region of the ultrasonic horn in the direction, is the dimensionless coordinate of each integral region of the ultrasonic horn in the , , and take values between ; is time, , , are respectively the penalty factors in the direction, are respectively the boundary condition functions of the horn bar in the direction; , , are all undetermined coefficients; is an orthogonal function, where , where is ; , ω is the angular velocity; are respectively natural numbers greater than 0;

[0017] Step 1.2. The potential energy function and the kinetic energy function of each integral region of the ultrasonic horn are both functionals of , , . According to the matching form of the bar and the disk in the ultrasonic horn, if the bar and the disk are in interference fit, substitute formula (1-1) into and ; if the bar and the disk are in clearance fit, substitute formula (1-2) into and , and then obtain the potential energy function and the kinetic energy function of each integral region with respect to under different matching conditions. Then add the potential energy function and the kinetic energy function of each integral region to obtain the total potential energy function and the total kinetic energy function of the ultrasonic horn as a whole with respect to ;

[0018] Step 1.3. Build an experimental platform, use a washer-type pressure sensor to measure the pre-tightening force of the threaded connection between the bar and the disk, and regard the influence of the pre-tightening force of the threaded connection on the vibration characteristics of the ultrasonic horn as the influence of elastic constraints on the ultrasonic horn. Then the elastic potential energy function generated by the pre-tightening force of the threaded connection is expressed as V e ;

[0019] Step 1.4. Establish the energy equation of the ultrasonic horn It is:

[0020] (1 - 3),

[0021] Step 1.5. Respectively, take the derivative of the energy equation in formula (1 - 3) with respect to to solve for , , to obtain the three - dimensional resonance frequency characteristic equation that does not reflect the coupling relationship of the ultrasonic horn:

[0022] (1 - 4),

[0023] In formula (1 - 4), represents the stiffness matrix generated by the material and geometric parameters of the ultrasonic horn itself, represents the stiffness matrix generated by the pre - tightening force of the threaded connection between the rod and the disk of the ultrasonic horn, is the mass matrix; is , is the zero matrix;

[0024] Step 1.6. Since there is a coupling relationship between the rod and the disk in the ultrasonic horn, assuming that the displacements at the coupling of the rod and the disk are continuous, then the displacements at the coupling of the rod and the disk are equal. Therefore, the matrix [X] in formula (1 - 4) is expressed as:

[0025] (1 - 5),

[0026] In formula (1 - 5), the matrix is the matrix obtained based on the equality of displacements at the coupling of the rod and the disk; is the matrix containing , , obtained based on the displacement continuity condition;

[0027] Step 1.7. Substitute formula (1 - 5) into formula (1 - 4) to obtain the three - dimensional resonance frequency characteristic equation with indefinite coupling boundary conditions:

[0028] (1 - 6),

[0029] In formula (1 - 6), is the inversion of , , ; is the characteristic frequency of the ultrasonic horn. Solve the characteristic frequency of the ultrasonic horn according to formula (1 - 6) , because it is a function of the resonance frequency of the ultrasonic amplitude transformer, and based on this, the resonance frequency of the ultrasonic amplitude transformer is obtained ;

[0030] Preferably, the sub-steps of step 2 are:

[0031] Step 2.1, establish a rectangular coordinate system, conduct a kinematic analysis on a single abrasive grain on the honing wheel during the ultrasonic honing process, and obtain the velocity equation of the single abrasive grain in the direction with respect to time :

[0032] (2-1),

[0033] In formula (2-1), is the feed rate of the hard tooth surface gear, is the linear velocity of the honing wheel, is the rotational speed of the honing wheel, is the amplitude of the ultrasonic amplitude transformer, is the resonance frequency of the ultrasonic amplitude transformer;

[0034] Step 2.2, assume that the volume of a single abrasive grain before ultrasonic honing is , the time taken to complete one honing process is , and the volume of a single abrasive grain becomes after one honing process. Establish a volume change function of a single abrasive grain under the influence of ultrasonic vibration :

[0035] (2-2);

[0036] And in the same way, establish a material removal rate function of a single abrasive grain on the honing wheel for removing the material on the hard tooth surface gear within time as:

[0037] (2-3),

[0038] In formula (2-3), is the volume of the hard tooth surface gear before machining, is the volume of the hard tooth surface gear after one honing process of a single abrasive grain. Then, analyze the cutting force of a single abrasive grain and establish the normal frictional force and of a single abrasive grain with respect to and tangential frictional force as:

[0039] (2 - 4),

[0040] and the normal force caused by the cutting deformation of a single abrasive grain as a function of the tangential force is;

[0041] (2 - 5),

[0042] In formulas (2 - 4) and (2 - 5), is a constant. Assuming that a single abrasive grain is spherical, is the central angle corresponding to the part of a single abrasive grain participating in the processing, is the radius of a single abrasive grain, is the feed rate of the hard tooth surface gear, is the linear velocity of the honing wheel, is the cutting arc length of the honing wheel;

[0043] Step 2.3: Determine the interference length and the number of interferences between the trajectories of two abrasive grains within one vibration period, and obtain the number of abrasive grains participating in honing in the ultrasonic honing tooth - machining area according to the ultrasonic honing tooth - machining parameters and the abrasive grain distribution is:

[0044] (2 - 6);

[0045] In formula (2 - 6), is the time taken to complete one honing process, is the number of abrasive grains per unit area on the honing wheel;

[0046] is the rotational speed of the honing wheel, , is the resonant frequency of the ultrasonic horn, is the feed rate of the hard tooth surface gear; Based on the number of abrasive grains participating in honing in the ultrasonic honing tooth - machining area

[0047] (2 - 7),

[0048] In formula (2 - 7), is the macroscopic normal force in the ultrasonic honing tooth - machining area, is the macroscopic tangential force in the ultrasonic honing tooth - machining area;

[0049] Preferably, the sub - steps of step ③ are:

[0050] Step 3.1: Based on the dynamic grinding force model of the ultrasonic honing machining area established in Step 2, obtain the total heat in the machining area :

[0051] (3-1),

[0052] In formula (3-1), is the honing width, is the cutting arc length of the honing wheel, is the linear velocity of the honing wheel, is the macroscopic tangential force in the ultrasonic honing machining area;

[0053] Meanwhile, obtain the heat entering the surface of the hardened gear , which is:[[]]

[0054] (3-2),

[0055] In formula (3-2), , where , respectively represent the heat transfer coefficients of the cutting fluid and the hardened gear; , , where is the contact radius between a single abrasive grain and the surface of the hardened gear in the ultrasonic honing machining area, is the linear velocity of the honing wheel, is the density of the hardened gear, is the specific heat capacity of the hardened gear, is the thickness of the chip;

[0056] Step 3.2: Assume that a single abrasive grain is spherical. When a single abrasive grain cuts the hardened gear, the single abrasive grain intersects with the plane of the surface of the hardened gear before machining. The cross-sectional shape of the single abrasive grain at the intersection plane is circular, and the radius of this circle is the contact radius between a single abrasive grain and the hardened gear in the ultrasonic honing machining area , establish a function of the cutting depth of a single abrasive grain; assume that there are multiple abrasive grains in the ultrasonic honing machining area, then the distribution of the cutting depth of each abrasive grain conforms to the Rayleigh distribution, and the probability density function of the cutting depth h of each abrasive grain in the ultrasonic honing machining area can be obtained, which is:[[]]

[0057] (3-3),

[0058] In formula (3-3), is the standard deviation, and then the contact radius between the abrasive grain and the hard tooth surface gear at any position in the ultrasonic honing machining area is obtained Regarding 、 and function of, is:[[]]

[0059] (3-4),[[]]

[0060] In formula (3-4), is the radius probability density function of multiple abrasive grains in the machining area, is the area of the ultrasonic honing machining area, is the radius of a single abrasive grain;

[0061] Step 3.3: Based on the moving heat source theory, use the superposition method of the heat source temperature field to calculate the surface temperature of the hard tooth surface gear; the heat generated by the honing process between the honing wheel and the hard tooth surface gear can be simplified as a surface heat source, and the surface heat source is regarded as a combination of countless linear heat sources, and the linear heat source is considered as a combination of multiple point heat sources; assume that the point heat source is located at the coordinate origin of the heat conductor At, the initial surface temperature of the hard tooth surface gear is T = 0 °C, and the heat q w [[]]entering the surface of the hard tooth surface gear is simplified as a point heat source; based on the heat conduction theory, after performing Fourier transform on the three-dimensional heat conduction differential equation, the temperature at the point M at the surface coordinate position in the ultrasonic honing machining area at any time is obtained, which is expressed as:[[]]

[0062] (3-5),[[]] <o:p>< / o:p>

[0063] In formula (3-5), is the density of the hard tooth surface gear, is the specific heat capacity of the hard tooth surface gear, is the thermal diffusivity of the hard tooth surface gear, is the distance from any point in the hard tooth surface gear to the point heat source, , where, is the base circle radius of the hard tooth surface gear; is the rolling angle of the hard tooth surface gear; is the feed speed of the hard tooth surface gear; is any time during the ultrasonic honing machining process;

[0064] Step 3.4: Based on the high-order function curve, on the basis of formula (3-5), combined with the single abrasive grain speed equation established by formula (2-1) and the function shown in formula (3-4)[[]] Establish a three-dimensional transient temperature field distribution model for the ultrasonic honing machining area , That is:

[0065] (3 - 6),

[0066] In formula (3 - 6), is the ultrasonic honing machining duration, is the maximum rolling angle of each point on the surface of the hardened gear, is a function of ;

[0067] Preferably, the sub-steps of step 4 are:

[0068] Step 4.1: Conduct a discretization analysis of the temperature field distribution of the horn based on the three-dimensional transient temperature field distribution model of ultrasonic honing established in step 3. Divide the horn along the axial direction into several units, and obtain the axial force load and the thermal bending moment load received by each unit of the horn. Consider the deformation of each unit of the horn as a linear problem and apply the superposition principle. Then, for each unit of the horn, there is: and the thermal bending moment load it receives . Consider the deformation of each unit of the horn as a linear problem and apply the superposition principle. Then, for each unit of the horn, there is:

[0069] (4 - 1);

[0070] Step 4.2: Establish a rectangular coordinate system, adopt a two-node horn unit, and define the node generalized displacement of the two-node horn unit as: That is:

[0071] (4 - 2),

[0072] In formula (4 - 2), (i = 1, 2) represents the node displacement, (i = 1, 2) represents the rotation vector. In the rectangular coordinate system, assume that the generalized force vector received by the unit node is , the thermal load of the unit node is and the node prestress is . Establish the force balance equation of each unit of the horn:

[0073] (4 - 3),

[0074] In formula (4 - 3), is the stiffness matrix of each unit of the horn, is the generalized displacement of each unit of the horn;

[0075] Step 4.3. The radius vector of any point P on each unit of the horn before deformation is . After deformation occurs in any unit of the horn, the radius vector of any point P on each unit of the horn satisfies the following relationship:

[0076] (4-4),

[0077] In formula (4-4), the initial position vector of point P is expressed as in the rectangular coordinate system; the generalized displacement of each unit of the horn is expressed as in the rectangular coordinate system. From this, the expression of in the rectangular coordinate system, and the linear velocity and acceleration of point P in the rectangular coordinate system are obtained;

[0078] Step 4.4. Each unit of the horn is analyzed according to the Euler-Bernoulli beam. Based on this, the kinetic energy and potential energy of the th unit of the horn are obtained. Assuming the horn is divided into units, the total kinetic energy and potential energy of the entire horn are expressed as:

[0079] (4-5),

[0080] Substitute the total kinetic energy and potential energy of the entire horn into the Lagrange equation to derive its thermodynamics equilibrium equation as:

[0081] (4-6),

[0082] In formula (4-6), is the total structural mass matrix of the horn, is the total damping matrix of the horn, is the total stiffness matrix of the horn determined by , is the load matrix. From this, the total deformation displacement of the horn is obtained, where is the first derivative of , is the second derivative of . Through the total deformation displacement of the horn, to clarify the deformation displacement value of the horn at any time and any position during the ultrasonic honing process. Therefore, the thermal deformation law of the horn can be analyzed through the deformation displacement value to achieve dynamic compensation for the thermal deformation of the horn during the processing.

[0083] The technical solution provided by the present invention has the following technical effects compared with the prior art:

[0084] First, study the three-dimensional coupled boundary conditions at the contact between the horn and the hard tooth surface gear, improve the three-dimensional coupled resonance analysis model of the ultrasonic transducer, and construct the three-dimensional resonance frequency characteristic equation of the ultrasonic transducer with indefinite coupled boundary conditions, laying a foundation for the study of the three-dimensional transient temperature field distribution and thermal-structural coupling in the ultrasonic honing processing area;

[0085] Second, study the movement trajectory, wear rate and material removal rate of a single abrasive grain of the honing wheel to obtain the cutting force of a single abrasive grain. At the same time, considering the trajectory interference situation between different abrasive grains, obtain the number of abrasive grains participating in honing in the processing area, establish a dynamic grinding force model for the ultrasonic honing processing area, and on this basis, considering the characteristic parameters of the abrasive grains and the contact relationship between the abrasive grains and the workpiece, combine the high-order function curve and the moving heat source theory to construct a three-dimensional transient temperature field distribution model for the ultrasonic honing processing area, providing a thermal control strategy for ultrasonic honing;

[0086] Third, based on the finite element method, the three-dimensional transient temperature field distribution and the Lagrange equation, obtain the thermal load and thermal bending moment functions of the horn, establish the thermal-structural coupling dynamic equilibrium equation of ultrasonic honing, reveal the thermal deformation law of the horn, and through the thermal deformation law of the horn, dynamic compensation for the thermal deformation of the horn during the processing can be achieved, providing theoretical support for the ultrasonic honing dynamic compensation technology;

[0087] Fourth, the present invention can improve the surface integrity and processing accuracy of the hard tooth surface gear after ultrasonic honing processing, reduce the transmission noise, and explore a new way for manufacturing hard tooth surface gears with long life and high reliability. Description of the Drawings

[0088] The drawings here are incorporated into the specification and form a part of this specification, showing the embodiments consistent with the present invention and used together with the specification to explain the principles of the present invention.

[0089] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for use in the description of the embodiments or the prior art. Obviously, for those of ordinary skill in the art, other drawings can also be obtained based on these drawings without creative efforts.

[0090] Figure 1It is the overall process block diagram corresponding to a method for controlling the thermal behavior of ultrasonic honing based on force-thermal analysis in an embodiment of the present invention;

[0091] Figure 2 It is the cutting schematic diagram of a single abrasive grain in an embodiment of the present invention;

[0092] Figure 3 It is the schematic diagram of the xyz rectangular coordinate system established in step 4 of the method for controlling the thermal behavior of ultrasonic honing in an embodiment of the present invention. Detailed implementation manners

[0093] In order to be able to more clearly understand the above objects, features and advantages of the present invention, the solution of the present invention will be further described below. It should be noted that, without conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other.

[0094] Many specific details are set forth in the following description in order to fully understand the present invention, but the present invention can also be implemented in other ways different from those described herein; obviously, the embodiments in the specification are only part of the embodiments of the present invention, rather than all of the embodiments.

[0095] The specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0096] In one embodiment, as Figure 1 shown, a method for controlling the thermal behavior of ultrasonic honing based on force-thermal analysis is disclosed, including the following steps:

[0097] Step 1: Establish a three-dimensional resonance frequency characteristic equation of an ultrasonic horn with indefinite coupling boundary conditions, and accurately solve the resonance frequency of the ultrasonic horn. Its sub-steps are:

[0098] Step 1.1: Construct a cylindrical coordinate system, divide the ultrasonic horn composed of the horn and the simplified ring disk of the hard tooth surface gear into integration regions, where the horn is simply referred to as the rod, and the simplified ring disk of the hard tooth surface gear is simply referred to as the disk; based on the penalty function, introduce a penalty factor, and construct the essential boundary condition admissible functions in the , , directions of each integration region of the rod and the disk in the non-coupled case in the cylindrical coordinate system, where ;

[0099] When there is an interference fit between the rod and the disk, the essential boundary condition admissible functions in the , , directions of each integration region are:

[0100] (1 - 1),

[0101] When the rod and the disk are in clearance fit, the essential boundary condition admissible functions in the direction of each integral region are , , as follows:

[0102] (1 - 2),

[0103] In formulas (1 - 1) and (1 - 2), is the dimensionless coordinate of each integral region of the ultrasonic horn in the direction, is the dimensionless coordinate of each integral region of the ultrasonic horn in the direction, is the dimensionless coordinate of each integral region of the ultrasonic horn in the , , and take values between ; is time, , , are respectively the penalty factors in the direction, are respectively the boundary condition functions of the horn bar in the direction; , , are all undetermined coefficients; is an orthogonal function, where , where is ; , ω is the angular velocity; are respectively natural numbers greater than 0;

[0104] Step 1.2. The potential energy function and the kinetic energy function of each integral region of the ultrasonic horn are both , , functionals. According to the fitting form of the rod and the disk in the ultrasonic horn, if the rod and the disk are in interference fit, substitute formula (1 - 1) into and ; if the rod and the disk are in clearance fit, substitute formula (1 - 2) into and , and then obtain the functionals of each integral region with respect to Potential energy function and kinetic energy function , and then add the potential energy function and kinetic energy function of each integration region to obtain the overall total potential energy function of the ultrasonic horn with respect to ; and total potential energy function ;

[0105] Step 1.3: Build an experimental platform, use a washer-type pressure sensor to measure the pre-tightening force of the threaded connection between the rod and the disk, and regard the influence of the threaded connection pre-tightening force on the vibration characteristics of the ultrasonic horn as the influence of elastic constraints on the ultrasonic horn. Then, the elastic potential energy function generated by the threaded connection pre-tightening force is expressed as V e ;

[0106] Step 1.4: Establish the energy equation of the ultrasonic horn as:

[0107] (1-3),

[0108] Step 1.5: Solve the derivatives of the energy equation in formula (1-3) with respect to , , , to obtain the three-dimensional resonance frequency characteristic equation that does not reflect the coupling relationship of the ultrasonic horn:

[0109] (1-4),

[0110] In formula (1-4), represents the stiffness matrix generated by the material and geometric parameters of the ultrasonic horn itself, represents the stiffness matrix generated by the pre-tightening force of the threaded connection between the rod and the disk of the ultrasonic horn, is the mass matrix; is , is the zero matrix;

[0111] Step 1.6: Since there is a coupling relationship between the rod and the disk in the ultrasonic horn, assume that the displacements at the coupling of the rod and the disk are continuous. Then, the displacements at the coupling of the rod and the disk are equal. Therefore, the matrix [X] in formula (1-4) is expressed as:

[0112] (1-5),

[0113] In formula (1-5), the matrix is the matrix obtained based on the equality of the displacements at the coupling of the rod and the disk; is the matrix containing , , Matrix of

[0114] Step 1.7: Substitute formula (1-5) into formula (1-4) to obtain the three-dimensional resonant frequency characteristic equation with indefinite coupling boundary conditions:

[0115] (1-6),

[0116] In formula (1-6), for The inversion of , ; is the characteristic frequency of the ultrasonic horn, and the characteristic frequency of the ultrasonic horn is solved according to formula (1-6) ,because is the resonant frequency of the ultrasonic horn function, on this basis the resonant frequency of the ultrasonic amplifier is obtained ;

[0117] Step 2: Based on the resonant frequency of the ultrasonic horn obtained in step 1, a dynamic grinding force model of ultrasonic gear honing is established, and its sub-steps are:

[0118] Step 2.1, establish In the rectangular coordinate system, the kinematic analysis of a single abrasive grain on the honing wheel during ultrasonic honing is carried out to obtain the time dependence of the single abrasive grain. exist Velocity equation in direction :

[0119] (2-1),

[0120] In formula (2-1), is the feed rate of the hardened gear, is the linear speed of the honing wheel, is the rotation speed of the honing wheel, is the amplitude of the ultrasonic horn, is the resonant frequency of the ultrasonic horn;

[0121] Step 2.2: Assume that the volume of a single abrasive grain before ultrasonic honing is The time required to complete a honing process is , the volume of a single abrasive grain after one honing process becomes , based on the volume change function of a single abrasive particle under the influence of ultrasonic vibration :

[0122] (2-2);

[0123] And in the same way, establish the material removal rate function of a single abrasive grain on the honing wheel for removing the material on the hard tooth surface gear within time. It is:

[0124] (2 - 3),

[0125] In formula (2 - 3), is the volume of the hard tooth surface gear before machining, is the volume of the hard tooth surface gear after a single abrasive grain honing process. Then analyze the cutting force of a single abrasive grain and establish the normal frictional force of a single abrasive grain with respect to and as: and the tangential frictional force is:

[0126] (2 - 4),

[0127] And the functions of the normal force and the tangential force caused by the cutting deformation of a single abrasive grain are;

[0128] (2 - 5),

[0129] In formulas (2 - 4) and (2 - 5), is a constant. Assume that a single abrasive grain is spherical, is the central angle corresponding to the part of a single abrasive grain participating in machining, is the radius of a single abrasive grain, is the feed rate of the hard tooth surface gear, is the linear velocity of the honing wheel, is the cutting arc length of the honing wheel;

[0130] Step 2.3: Determine the interference length and the number of interferences between the trajectories of two abrasive grains within one vibration period, and obtain the number of abrasive grains participating in honing in the ultrasonic honing machining area according to the ultrasonic honing machining parameters and the abrasive grain distribution It is:

[0131] (2 - 6);

[0132] In formula (2 - 6), is the time taken to complete one honing process, is the number of abrasive grains per unit area on the honing wheel;

[0133] is the honing wheel speed, , is the resonant frequency of the ultrasonic horn, is the feed rate of the hard tooth surface gear; based on the number of abrasive grains participating in honing in the ultrasonic honing machining area and formulas (2-4) and (2-5), establish a dynamic grinding force model for the ultrasonic honing machining area:

[0134] (2-7),

[0135] In formula (2-7), is the macroscopic normal force in the ultrasonic honing machining area, is the macroscopic tangential force in the ultrasonic honing machining area;

[0136] Step 3. Based on the dynamic grinding force model of the ultrasonic honing machining area established in Step 2, establish a three-dimensional transient temperature field distribution model for ultrasonic honing, and its sub-steps are:

[0137] Step 3.1. Based on the dynamic grinding force model of the ultrasonic honing machining area established in Step 2, obtain the total heat in the machining area :

[0138] (3-1),

[0139] In formula (3-1), is the honing width, is the cutting arc length of the honing wheel, is the linear velocity of the honing wheel, is the macroscopic tangential force in the ultrasonic honing machining area;

[0140] At the same time, obtain the heat entering the surface of the hard tooth surface gear , is:

[0141] (3-2),

[0142] In formula (3-2), , where , respectively represent the heat transfer coefficients of the cutting fluid and the hard tooth surface gear; , , where is the contact radius between a single abrasive grain in the ultrasonic honing machining area and the surface of the hard tooth surface gear, is the linear velocity of the honing wheel, is the density of the hard tooth surface gear, is the specific heat capacity of the hard tooth surface gear, is the thickness of the chip;

[0143] Step 3.2. Assume that a single abrasive grain is spherical. When a single abrasive grain cuts a hard tooth surface gear, the single abrasive grain intersects with the plane where the surface of the hard tooth surface gear is before machining. The cross-sectional shape of the single abrasive grain at the intersection plane is circular, and the radius of this circle is the contact radius between the single abrasive grain in the ultrasonic honing machining area and the hard tooth surface gear. , establish a function regarding the cutting depth of a single abrasive grain; assume that there are multiple abrasive grains in the ultrasonic honing machining area, then the distribution of the cutting depth of each abrasive grain conforms to the Rayleigh distribution, and the probability density function of the cutting depth h of each abrasive grain in the ultrasonic honing machining area can be obtained , which is:

[0144] (3 - 3),

[0145] In formula (3 - 3), is the standard deviation, and then the contact radius between the abrasive grain and the hard tooth surface gear at any position in the ultrasonic honing machining area is obtained as a function of , and , which is:

[0146] (3 - 4),

[0147] In formula (3 - 4), is the radius probability density function of multiple abrasive grains in the machining area, is the area of the ultrasonic honing machining area, is the radius of a single abrasive grain;

[0148] Step 3.3. Based on the moving heat source theory, use the superposition method of the heat source temperature field to calculate the surface temperature of the hard tooth surface gear; the heat generated by the honing process between the honing wheel and the hard tooth surface gear can be simplified as a surface heat source, and the surface heat source is regarded as a combination of countless linear heat sources, and the linear heat source is considered as a combination of multiple point heat sources; assume that the point heat source is located at the coordinate origin of the heat conductor, the initial surface temperature of the hard tooth surface gear is T = 0 °C, and the heat q w entering the surface of the hard tooth surface gear is simplified as a point heat source; based on the heat conduction theory, after performing the Fourier transform on the three-dimensional heat conduction differential equation, the temperature at the point M of the surface coordinate position in the ultrasonic honing machining area of the hard tooth surface gear at any time is obtained, which is expressed as:

[0149] (3 - 5),

[0150] In formula (3 - 5), is the density of the hard tooth surface gear, is the specific heat capacity of the hard tooth surface gear, is the thermal diffusivity of the hard tooth surface gear, is the distance from any point inside the hard tooth surface gear to the point heat source, , , where is the base circle radius of the hard tooth surface gear; is the rolling angle of the hard tooth surface gear; is the feed speed of the hard tooth surface gear; is any time during the ultrasonic honing process;

[0151] Step 3.4: Based on the high - order function curve, on the basis of formula (3 - 5), combined with the single abrasive grain velocity equation established by formula (2 - 1) and the function shown in formula (3 - 4), establish a three - dimensional transient temperature field distribution model for the ultrasonic honing processing area , which is:

[0152] (3 - 6),

[0153] In formula (3 - 6), is the ultrasonic honing processing duration, is the maximum rolling angle of each point on the surface of the hard tooth surface gear, is a function of ;

[0154] Step 4: Based on the three - dimensional transient temperature field distribution model established in Step 3 for the ultrasonic honing processing area, establish an ultrasonic honing thermal - structure coupling analysis model, and analyze the thermal deformation law of the horn through the ultrasonic honing thermal - structure coupling analysis model to achieve dynamic compensation for the thermal deformation of the horn during the processing; its sub - steps are:

[0155] Step 4.1: Discretize and analyze the temperature field distribution of the horn based on the three - dimensional transient temperature field distribution model established in Step 3 for ultrasonic honing. Divide the horn along the axial direction into several units, and obtain the axial force load and the thermal bending moment load received by each unit of the horn. Consider the deformation of each unit of the horn as a linear problem and apply the superposition principle. Then, for each unit of the horn, there is:

[0156] (4 - 1);

[0157] Step 4.2: Establish a rectangular coordinate system, adopt a two-node horn element, and define the nodal generalized displacements of the two-node horn element as:

[0158] (4-2),

[0159] In formula (4-2), (i = 1, 2) represents the nodal displacement, (i = 1, 2) represents the rotation vector. In the rectangular coordinate system, assume that the generalized force vector acting on the element nodes is , the thermal load on the element nodes is and the nodal prestress is , and establish the force balance equation for each element of the horn:

[0160] (4-3),

[0161] In formula (4-3), is the stiffness matrix of each element of the horn, is the generalized displacement of each element of the horn;

[0162] Step 4.3: The radius vector of any point P on each element of the horn before deformation is , then after any element of the horn deforms, the radius vector of any point P on each element of the horn satisfies the following relationship: <>

[0163] (4-4),

[0164] In formula (4-4), the initial position vector of point P is expressed as in the rectangular coordinate system; the generalized displacement of each element of the horn is expressed as in the rectangular coordinate system. From this, the expression of in the rectangular coordinate system is deduced, as well as the linear velocity and acceleration of point P in the rectangular coordinate system;

[0165] Step 4.4: Analyze each element of the horn according to the Euler-Bernoulli beam, and thus obtain the kinetic energy and potential energy of the th element of the horn. Assume that the horn is divided into elements, then the total kinetic energy of the horn With the potential energy Expressed as:

[0166] (4-5),

[0167] Substitute the overall kinetic energy of the horn and the potential energy into the Lagrange equation, and the derived thermodynamics equilibrium equation is:

[0168] (4-6),

[0169] In formula (4-6), is the overall structural mass matrix of the horn, is the overall damping matrix of the horn, is the overall stiffness matrix of the horn determined by , is the load matrix, from which the overall deformation displacement of the horn is obtained, where is the first derivative of , is the second derivative of . Through the overall deformation displacement of the horn, the deformation displacement value at any time and any position during the ultrasonic gear honing process of the horn can be determined. Therefore, the thermal deformation law of the horn can be analyzed through the deformation displacement value to achieve dynamic compensation for the thermal deformation of the horn during the processing.

[0170] The above is only the specific implementation manner of the present invention, enabling those skilled in the art to understand or implement the present invention. Although the foregoing embodiments have been described in detail, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some or all of the technical features; and these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the foregoing embodiments, and they should all be covered by the protection scope of the claims.

Claims

1. A method for controlling the thermal behavior of ultrasonic honing based on force-thermal analysis, characterized in that, It includes the following steps: Step 1: Establish a three-dimensional resonance frequency characteristic equation of the ultrasonic horn with indefinite coupling boundary conditions, and accurately solve the resonance frequency of the ultrasonic horn; Step 2: Based on the resonance frequency of the ultrasonic horn obtained in Step 1, establish a dynamic grinding force model for ultrasonic honing; Step 3: Based on the dynamic grinding force model of the ultrasonic honing machining area established in Step 2, establish a three-dimensional transient temperature field distribution model for ultrasonic honing; Step 4: Based on the three-dimensional transient temperature field distribution model of the ultrasonic honing machining area established in Step 3, establish a thermal-structural coupling analysis model for ultrasonic honing, and analyze the thermal deformation law of the horn through the thermal-structural coupling analysis model of ultrasonic honing to realize the dynamic compensation of the thermal deformation of the horn during the machining process.

2. The ultrasonic honing thermal behavior control method based on force-thermal analysis according to claim 1, wherein The sub-steps of Step 1 are: Step 1.1, construct a cylindrical coordinate system, and divide the ultrasonic horn composed of the horn and the simplified ring disk of the hard tooth surface gear into integral regions. Herein, the horn is simply referred to as the rod, and the simplified ring disk of the hard tooth surface gear is simply referred to as the disk. Based on the penalty function, introduce a penalty factor, and construct the admissible functions of the essential boundary conditions of each integral region of the rod and the disk in the non - coupled case in the cylindrical coordinate system direction , , , where ; When there is an interference fit between the rod and the disc, the essential boundary condition admissible functions in the direction of each integral region in the cylindrical coordinate system , , are as follows: （1-1）， When the rod and the disk are in clearance fit, the essential boundary condition admissible functions in the direction of each integral region in the cylindrical coordinate system , , are as follows: （1-2）， In Formulas (1-1) and (1-2), is the dimensionless coordinate of each integration region of the ultrasonic horn in the direction, is the dimensionless coordinate of each integration region of the ultrasonic horn in the direction, is the dimensionless coordinate of each integration region of the ultrasonic horn in the , , and take values between ; is time, , , are respectively the penalty factors in the direction, are respectively the boundary condition functions of the horn bar in the direction; , , are all undetermined coefficients; is an orthogonal function, where , where is ; , ω is the angular velocity; are respectively natural numbers greater than 0; Step 1.2, Potential energy functions of each integral region of the ultrasonic horn and kinetic energy functions are both , , functionals. According to the matching form of the rod and disk in the ultrasonic horn, if the rod and disk are in interference fit, substitute formula (1-1) into and ; if the rod and disk are in clearance fit, substitute formula (1-2) into and , and then obtain the potential energy functions of each integral region with respect to and kinetic energy functions under different matching conditions. Then add the potential energy functions and kinetic energy functions of each integral region to obtain the total potential energy function and total kinetic energy function of the whole ultrasonic horn with respect to ; Step 1.3: Build an experimental platform, use a washer-type pressure sensor to measure the pre-tightening force of the threaded connection between the rod and the disc, and regard the influence of the threaded connection pre-tightening force on the vibration characteristics of the ultrasonic horn as the influence of elastic constraint on the ultrasonic horn. Then, the elastic potential energy function generated by the threaded connection pre-tightening force is expressed as V e ; Step 1.

4. Establish the energy equation of the ultrasonic horn It is as follows: （1-3）， Step 1.

5. Respectively, solve the derivative of the energy equation in formulas (1-3) with respect to , , to obtain a three-dimensional resonance frequency characteristic equation that does not reflect the coupling relationship of the ultrasonic horn: （1-4）， In formulas (1-4), represents the stiffness matrix generated by the material and geometric parameters of the ultrasonic horn itself, represents the stiffness matrix generated by the pre-tightening force of the threaded connection between the rod and the disc of the ultrasonic horn, is the mass matrix; is , is the zero matrix; Step 1.6: Since there is a coupling relationship between the rod and the disk in the ultrasonic horn, assuming that the displacements at the coupling of the rod and the disk are continuous, the displacements at the coupling of the rod and the disk are equal. Therefore, the matrix [X] in formula (1-4) is expressed as: （1-5）， In formula (1-5), the matrix is the matrix obtained based on the equal displacements at the coupling of the rod and the disc; is the matrix containing , , obtained based on the displacement continuity condition; Step 1.7: Substitute formula (1-5) into formula (1-4) to obtain a three-dimensional resonance frequency characteristic equation with indefinite coupling boundary conditions: （1-6）， In Formula (1-6), is inverted, , ; is the characteristic frequency of the ultrasonic horn, and the characteristic frequency of the ultrasonic horn is solved according to Formula (1-6) , because is the resonant frequency of the ultrasonic horn is a function of .

3. A method for controlling the thermal behavior of ultrasonic honing based on force-thermal analysis according to claim 1, characterized in that The sub-steps of Step 2 are: Step 2.1: Establish a rectangular coordinate system, conduct kinematic analysis on a single abrasive grain on the honing wheel during the ultrasonic honing process, and obtain the velocity equation of the single abrasive grain with respect to time in the direction : （2-1）， In formula (2-1), is the feed rate of the hard tooth surface gear, is the linear velocity of the honing wheel, is the rotational speed of the honing wheel, is the amplitude of the ultrasonic horn, is the resonance frequency of the ultrasonic horn; Step 2.

2. Assume that the volume of a single abrasive grain before ultrasonic honing is , the time taken to complete one honing process is , and the volume of a single abrasive grain after one honing process becomes . Establish the volume change function of a single abrasive grain under the influence of ultrasonic vibration : （2-2）； And in the same way, establish the material removal rate function of a single abrasive grain on the honing wheel for removing materials on the hardened surface gear within the time as follows: （2-3）， In formula (2-3), is the volume of the hard tooth surface gear before machining, is the volume of the hard tooth surface gear after honing with a single abrasive grain once. Then, analyze the cutting force of a single abrasive grain and establish the normal friction force and of a single abrasive grain and the tangential friction force are as follows: （2-4）， and the normal force caused by the cutting deformation of a single abrasive grain as a function of is given by; （2-5）， In Formulas (2-4) and (2-5), is a constant. Assume that a single abrasive grain is spherical. is the central angle corresponding to the part of a single abrasive grain participating in machining. is the radius of a single abrasive grain. is the feed rate of the hard tooth surface gear. is the linear velocity of the honing wheel. is the cutting arc length of the honing wheel. Step 2.3: Determine the interference length and the number of interference times between the trajectories of two abrasive grains within one vibration period, and calculate the number of abrasive grains participating in honing in the ultrasonic honing machining area according to the ultrasonic honing machining parameters and the abrasive grain distribution It is: （2-6）； In formula (2-6), is the time taken to complete one honing process, is the number of abrasive grains per unit area on the honing wheel; is the rotational speed of the honing wheel, , is the resonance frequency of the ultrasonic horn, is the feed rate of the hard tooth surface gear; Based on the number of abrasive grains participating in honing in the ultrasonic gear honing processing area and formulas (2-4) and (2-5), a dynamic grinding force model for the ultrasonic gear honing processing area is established:​ （2-7）， In Formula (2-7), is the macroscopic normal force in the ultrasonic honing machining area, is the macroscopic tangential force in the ultrasonic honing machining area.

4. A method for controlling the thermal behavior of ultrasonic honing based on force-thermal analysis according to claim 1, characterized in that The sub-steps of Step 3 are: Step 3.1: Based on the dynamic grinding force model of the ultrasonic honing machining area established in Step 2, obtain the total heat in the machining area : （3-1）， In formula (3-1), is the honing width, is the cutting arc length of the honing wheel, is the linear velocity of the honing wheel, is the macroscopic tangential force in the ultrasonic honing gear machining area; Meanwhile, the heat entering the surface of the hard tooth surface gear is obtained , which is （3-2）, In formula (3-2), , where and represent the heat transfer coefficients of the cutting fluid and the hard tooth surface gear, respectively; , , where is the contact radius between a single abrasive grain and the surface of the hard tooth surface gear in the ultrasonic honing machining area, is the linear velocity of the honing wheel, is the density of the hard tooth surface gear, is the specific heat capacity of the hard tooth surface gear, is the thickness of the chip; Step 3.2: Assume that a single abrasive grain is spherical. When a single abrasive grain cuts a hard tooth surface gear, the single abrasive grain intersects with the plane where the surface of the hard tooth surface gear is before machining. The cross-sectional shape of the single abrasive grain at the intersection plane is circular, and the radius of this circle is the contact radius between the single abrasive grain in the ultrasonic honing machining area and the hard tooth surface gear. , establish a function about the cutting depth of a single abrasive grain; assume that there are multiple abrasive grains in the ultrasonic honing machining area, then the distribution of the cutting depth of each abrasive grain conforms to the Rayleigh distribution, and the probability density function of the cutting depth h of each abrasive grain in the ultrasonic honing machining area can be obtained , which is: （3-3）, In formula (3-3), is the standard deviation, and then the contact radius between the abrasive grain and the hard tooth surface gear at any position in the ultrasonic honing machining area is obtained Regarding 、 and function of is: （3-4）, In formula (3-4), is the radius probability density function of multiple abrasive grains in the machining area, is the area of the ultrasonic honing machining area, is the radius of a single abrasive grain; Step 3.3: Based on the moving heat source theory, use the superposition method of the heat source temperature field to calculate the surface temperature of the hard tooth surface gear; the heat generated by the honing process between the honing wheel and the hard tooth surface gear can be simplified as a surface heat source, and the surface heat source is regarded as a combination of countless linear heat sources, while the linear heat source is considered as a combination of multiple point heat sources; assume that the point heat source is located at the coordinate origin of the heat conductor where the initial surface temperature of the hard tooth surface gear is T = 0 °C, and the heat q w entering the surface of the hard tooth surface gear is simplified as a point heat source; based on the heat conduction theory, after performing the Fourier transform on the three-dimensional heat conduction differential equation, the temperature at the point M at the surface coordinate position of the hard tooth surface gear within the ultrasonic honing machining area at any time is expressed as: （3-5）， In formula (3-5), is the density of the hard tooth surface gear; is the specific heat capacity of the hard tooth surface gear; is the thermal diffusivity of the hard tooth surface gear; is the distance from any point inside the hard tooth surface gear to the point heat source; , , where is the base circle radius of the hard tooth surface gear; is the rolling angle of the hard tooth surface gear; is the feed speed of the hard tooth surface gear; is any time during the ultrasonic honing process; Step 3.

4. Based on the high-order function curve, on the basis of formula (3-5), combined with the single abrasive grain velocity equation established by formula (2-1) and the function shown in formula (3-4) , establish a three-dimensional transient temperature field distribution model in the ultrasonic honing machining area , which is as follows: （3-6）， In formula (3-6), is the ultrasonic honing machining time,[[]]END]] is the maximum rolling angle of each point on the surface of the hardened gear,[[]]END]] is a function of .

5. A method for controlling the thermal behavior of ultrasonic honing based on force-thermal analysis according to claim 1, characterized in that The sub-steps of Step 4 are: Step 4.

1. Conduct discretization analysis on the temperature field distribution of the horn based on the three-dimensional transient temperature field distribution model of ultrasonic honing established in Step 3. Divide the horn into several units along the axial direction, and obtain the axial force loads borne by each unit of the horn and the thermal bending moment loads borne by it . Consider the deformation of each unit of the horn as a linear problem. Applying the superposition principle, for each unit of the horn, there is （4-1）； Step 4.2, establish a rectangular coordinate system, adopt a two-node horn element, and define the nodal generalized displacements of the two-node horn element as follows: （4-2）， In Equation (4-2), $(i = 1, 2)$ represents the nodal displacement, $(i = 1, 2)$ represents the rotation vector. In a rectangular coordinate system, let the generalized force vector acting on the element nodes be , the thermal load on the element nodes be and the prestress of the nodes be . Establish the force equilibrium equations for each element of the horn: （4-3）， In Equation (4-3), is the stiffness matrix of each unit of the horn; is the generalized displacement of each unit of the horn; Step 4.

3. The radius vector of any point P on each unit of the horn before deformation is . After deformation occurs in any unit of the horn, the radius vector of any point P on each unit of the horn satisfies the following relationship: （4-4）， In formula (4-4), the initial position vector of point P is represented in the rectangular coordinate system as ; the generalized displacements of each unit of the horn are represented in the rectangular coordinate system as , from which it is deduced that in the rectangular coordinate system, and the linear velocity of point P in the rectangular coordinate system , acceleration ; Step 4.4: Each unit of the horn is analyzed according to the Euler - Bernoulli beam, and based on this, the kinetic energy of the th unit of the horn is obtained. And the potential energy . Suppose the horn is divided into units, then the total kinetic energy and potential energy of the horn are expressed as: （4-5）， Substitute the overall kinetic energy of the horn and potential energy into the Lagrange equation to derive its thermodynamics equilibrium equation as follows: （4-6）， In Formula (4-6), is the overall structural mass matrix of the horn,[ is the overall damping matrix of the horn,[ is the overall stiffness matrix of the horn determined by ,[ is the load matrix, from which the overall deformation displacement of the horn is obtained , where is the first derivative of,[ is the second derivative of. Through the overall deformation displacement of the horn, the deformation displacement values at any time and any position of the horn during the ultrasonic honing process are determined. Therefore, the thermal deformation law of the horn can be analyzed through the deformation displacement values to achieve dynamic compensation for the thermal deformation of the horn during the processing.[

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