A method and system for designing a curved surface of a log-busbar bearing roller-driven cylinder

Abstract

The application provides a logarithmic generatrix bearing roller driving roller curved surface design method and system, and the method comprises the following steps: establishing a contact locus curve and constraint conditions in a world coordinate system; converting a normal vector at a contact point and a principal normal vector of the contact locus curve to a roller coordinate system, and combining the constraint conditions to obtain constraint conditions in the roller coordinate system; based on a cylindrical contact locus curve, a cylindrical locus normal vector expressed by a Z-axis coordinate value in the roller coordinate system is obtained; the cylindrical contact locus curve is a cylindrical coordinate expression of the contact locus curve in the roller coordinate system; based on the cylindrical locus normal vector, the Z-axis coordinate value in the roller coordinate system expressed by time, and the constraint conditions in the roller coordinate system, a relationship between a roller curved surface radius and the Z-axis coordinate value in the roller coordinate system is obtained, and then a design equation of the driving roller curved surface is obtained. The application solves the instability problem in the rotation of the logarithmic generatrix roller caused by the existing standard cylindrical roller driving.

Description

Technical Field

[0001] This invention relates to the field of precision mechanical design, specifically to a method and system for designing the curved surface of a logarithmic generatrix bearing roller drive drum. Background Technology

[0002] Logarithmic generatrix bearing rollers offer optimal stress distribution, making them widely used in demanding applications such as high-speed and heavy-load applications. In rolling bearing manufacturing, the profile accuracy of the generatrix is ​​a key factor determining bearing performance, lifespan, and reliability. Currently, industrial applications commonly employ line-scan camera vision inspection systems for non-contact measurement of roller generatrixes. This technology continuously scans the rotating roller surface with a camera and reconstructs its three-dimensional shape. A core prerequisite for achieving high-precision inspection is that the roller must maintain extremely stable rotation around its ideal axis throughout the scanning process. Any axial misalignment or radial runout will cause blurring or distortion of the acquired image, resulting in unacceptable measurement errors.

[0003] In existing technologies, the most common driving scheme utilizes two parallel standard cylindrical rollers to support and frictionally drive the roller. However, this scheme has inherent drawbacks for rollers with complex generatrices (such as logarithmic curves). The fundamental problem lies in geometric mismatch: the straight generatrice of the cylindrical roller cannot form stable contact with the curved generatrice of the roller along its entire length. In actual driving, the contact between the two is dynamically changing point contact or extremely unstable short-line contact. This unstable contact causes fluctuations in frictional torque, inducing small but critical periodic or random changes in the roller's rotation axis, such as jumping and swerving, leading to unstable roller rotation. Moreover, at the moment of poor contact, local slippage is prone to occur, disrupting the strict synchronization between the roller's rotation angle and the camera's line scanning frequency. These factors combined prevent the line scan camera from acquiring a clear image of the roller surface, ultimately resulting in a significant decrease in the measurement accuracy and repeatability of the generatrice profile. Summary of the Invention

[0004] This invention addresses the shortcomings of existing technologies by providing a surface design method and system for a logarithmic busbar bearing roller drive drum.

[0005] To solve the above-mentioned technical problems, the present invention provides the following technical solution:

[0006] A method for designing the surface of a logarithmic generatrix bearing roller-driven drum includes the following steps:

[0007] Establish the contact trajectory curve and constraints in the world coordinate system; wherein, the contact trajectory curve is the path line formed by the point of tangency of the contact line between the roller and the drive roller on the roller surface as time moves on the roller surface, and the constraint is that the normal vector at the contact point between the contact trajectory curve and the drive roller surface is parallel to the principal normal vector of the contact trajectory curve;

[0008] Transform the normal vector at the contact point and the principal normal vector of the contact trajectory curve to the drum coordinate system, and combine them with the aforementioned constraint conditions to obtain the constraint conditions in the drum coordinate system; wherein, the Z-axis of the drum coordinate system is parallel to the Z-axis of the world coordinate system.

[0009] Based on the cylindrical contact trajectory curve, the cylindrical trajectory normal vector is obtained using the Z-axis coordinate value of the drum coordinate system; the cylindrical contact trajectory curve is the cylindrical coordinate representation of the contact trajectory curve in the drum coordinate system.

[0010] Based on the cylindrical trajectory normal vector and the Z-axis coordinate value of the roller coordinate system expressed in time, combined with the constraint conditions under the roller coordinate system, the relationship between the roller surface radius and the Z-axis coordinate value of the roller coordinate system is obtained, and then the design equation of the driving roller surface is obtained; wherein, the Z-axis coordinate value of the roller coordinate system expressed in time is obtained based on the constraint conditions under the roller coordinate system.

[0011] As one possible implementation, the contact trajectory curve in the world coordinate system is represented as follows:

[0012]

[0013] in, This represents the contact trajectory curve in the world coordinate system. Indicates time, The generatrix equation of the logarithmic generatrix bearing roller is represented here. Indicates time The Z-axis coordinate value of the logarithmic generatrix bearing roller in the world coordinate system. Indicates time The circumferential angle of the rollers in a time-logarithmic busbar bearing. This represents the transpose of a matrix.

[0014] As one possible implementation, the step of transforming the normal vector at the contact point and the principal normal vector of the contact trajectory curve to the roller coordinate system, and combining this with the constraint conditions to obtain the constraint conditions in the roller coordinate system, includes the following steps:

[0015] Based on the rotation matrix between the world coordinate system and the drum coordinate system, the normal vector at the contact point and the principal normal vector of the contact trajectory curve are transformed to the drum coordinate system, respectively, to obtain the normal vector at the contact point and the principal normal vector of the contact trajectory curve in the drum coordinate system.

[0016] Combining the aforementioned constraints, the normal vector at the contact point in the roller coordinate system, and the principal normal vector of the contact trajectory curve in the roller coordinate system, the constraint in the roller coordinate system is that the normal vector at the contact point in the roller coordinate system is parallel to the principal normal vector of the contact trajectory curve in the roller coordinate system.

[0017] As one possible implementation, the cylindrical contact trajectory curve is represented as follows:

[0018]

[0019] in, Represents any point on the contact trajectory curve of the cylinder. Represents the roller in the roller coordinate system axis coordinate values, Represents the coordinate system of a point on the cylindrical contact trajectory curve. Distance between axes This represents the circumferential angle of the roller in the roller coordinate system.

[0020] As one possible implementation, the normal vector of the cylindrical trajectory is represented as follows:

[0021]

[0022] in, This represents the normal vector of the cylindrical trajectory, expressed in terms of the Z-axis coordinates of the drum coordinate system. Represents the roller in the roller coordinate system axis coordinate values, Represents the coordinate system of a point on the cylindrical contact trajectory curve. Distance between axes .

[0023] As one possible implementation, the relationship between the radius of the roller surface and the Z-axis coordinate value of the roller coordinate system, based on the normal vector of the cylindrical trajectory and the time-expressed roller coordinate value, combined with the constraints under the roller coordinate system, is obtained, thereby deriving the design equation of the driving roller surface, including the following steps:

[0024] The cross product of the cylindrical trajectory normal vector and the principal normal vector of the contact trajectory curve in the roller coordinate system is performed to obtain the cross product result; wherein, the cross product result is equal to the zero vector;

[0025] Based on the correspondence between the cross product result and the zero vector, the relationship between the slope of the roller radius and the corresponding principal normal vector component is obtained;

[0026] Based on the constraints in the roller coordinate system, the Z-axis coordinate value of the roller coordinate system expressed in terms of time is obtained;

[0027] By combining the Z-axis coordinates of the roller coordinate system expressed in terms of time, the relationship between the slope of the roller radius and the corresponding principal normal vector component is solved to obtain the unique relationship between the roller radius and the Z-axis coordinates of the roller coordinate system, and then the design equation of the driving roller surface is obtained.

[0028] As one possible implementation, the relationship between the slope of the roller radius and the corresponding principal normal vector component is expressed as follows:

[0029]

[0030] in, Indicates the drum radius In position The slope at that point and They represent time respectively The component values ​​of the principal normal vector along the Z-axis and X-axis in the drum coordinate system. Indicates time.

[0031] A surface design system for a logarithmic busbar bearing roller driven drum, used to implement the method described in any of the above-mentioned methods, the system comprising a trajectory constraint construction module, a constraint transformation module, a cylinder transformation module, and a solution module;

[0032] The trajectory constraint construction module is used to establish the contact trajectory curve and constraint conditions in the world coordinate system; wherein, the contact trajectory curve is the path line formed by the point of tangency of the contact line between the roller and the drive roller on the roller surface as time moves on the roller surface, and the constraint condition is that the normal vector at the contact point between the contact trajectory curve and the drive roller surface is parallel to the principal normal vector of the contact trajectory curve.

[0033] The constraint transformation module is used to transform the normal vector at the contact point and the principal normal vector of the contact trajectory curve to the drum coordinate system, and combined with the constraint conditions, obtain the constraint conditions in the drum coordinate system; wherein, the Z-axis of the drum coordinate system is parallel to the Z-axis of the world coordinate system.

[0034] The cylinder conversion module is used to obtain the cylinder trajectory normal vector represented by the Z-axis coordinate value of the roller coordinate system based on the cylinder contact trajectory curve; the cylinder contact trajectory curve is the cylindrical coordinate representation of the contact trajectory curve in the roller coordinate system.

[0035] The solution module is used to obtain the relationship between the radius of the roller surface and the Z-axis coordinate value of the roller coordinate system based on the normal vector of the cylindrical trajectory and the Z-axis coordinate value of the roller coordinate system in terms of time, combined with the constraint conditions under the roller coordinate system, and then obtain the design equation of the driving roller surface; wherein, the Z-axis coordinate value of the roller coordinate system in terms of time is obtained based on the constraint conditions under the roller coordinate system.

[0036] A computer-readable storage medium storing a computer program that, when executed by a processor, implements the method described in any one of the preceding methods.

[0037] An apparatus includes a memory, a processor, and a computer program stored in the memory and running on the processor, wherein the processor, when executing the computer program, implements the method described in any one of the preceding methods.

[0038] This invention, employing the above technical solutions, achieves significant technical effects: It provides a surface design method and system for a logarithmic generatrix bearing roller drive drum. First, based on a kinematic model, a contact trajectory curve under ideal pure rolling conditions in the world coordinate system is constructed. Differential geometric analysis is used to extract higher-order geometric constraints on the contact trajectory curve, requiring that the normal vector at the contact point between the contact trajectory curve and the drive drum surface be parallel to the principal normal vector of the contact trajectory curve. Then, through kinematic transformation, the contact trajectory curve and its variables are converted to the drum coordinate system, yielding the constraints in the drum coordinate system. Finally, by synthesizing the above constraints, a differential equation determining the drum generatrix is ​​established and solved, resulting in an accurate surface equation capable of achieving higher-order conformal contact. This invention fully considers the external surface geometric characteristics of the logarithmic generatrix bearing rollers, designing the outer surface shape of the drive drum through contact stress and mathematical geometry, making the roller rotation more stable during relative motion. It solves the instability problem in the rotation of logarithmic generatrix rollers caused by existing standard cylindrical roller drives, thus meeting the requirements of precise detection of roller cylindrical surface images by line scan cameras. Attached Figure Description

[0039] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0040] Figure 1 This is a flowchart illustrating an embodiment of the method of the present invention;

[0041] Figure 2This is a schematic projection of the curve corresponding to the roller surface equation in an embodiment of the present invention;

[0042] Figure 3 This is an overall schematic diagram of an embodiment of the system of the present invention. Detailed Implementation

[0043] The present invention will be further described in detail below with reference to embodiments. These embodiments are illustrative of the invention and the invention is not limited thereto. Unless otherwise specified, the features in the following embodiments can be combined with each other.

[0044] Example 1:

[0045] In one embodiment, a method for designing the curved surface of a logarithmic generatrix bearing roller drive drum is provided, such as... Figure 1 As shown, it includes the following steps:

[0046] S100: Establish the contact trajectory curve and constraints in the world coordinate system; wherein, the contact trajectory curve is the path line formed by the tangent point moving over time on the surface of the logarithmic generatrix bearing roller, the tangent point is the point of tangency between the contact line between the logarithmic generatrix bearing roller and the drive roller on the surface of the logarithmic generatrix bearing roller, the constraint is that the normal vector at the contact point is parallel to the principal normal vector of the contact trajectory curve, and the contact point is the contact point between the contact trajectory curve and the surface of the drive roller;

[0047] S200: Transform the normal vector at the contact point and the principal normal vector of the contact trajectory curve to the drum coordinate system, and combine the constraint conditions to obtain the constraint conditions in the drum coordinate system; wherein, the Z-axis of the drum coordinate system is parallel to the Z-axis of the world coordinate system, and the X-axis and Y-axis of the drum coordinate system rotate with the drum around the Z-axis of the drum coordinate system.

[0048] S300: Based on the cylindrical contact trajectory curve, obtain the cylindrical trajectory normal vector represented by the Z-axis coordinate value of the drum coordinate system; the cylindrical contact trajectory curve is the cylindrical coordinate representation of the contact trajectory curve in the drum coordinate system;

[0049] S400: Based on the cylindrical trajectory normal vector and the time-expressed Z-axis coordinate value of the roller coordinate system, combined with the constraint conditions under the roller coordinate system, the relationship between the roller surface radius and the X-axis and Y-axis values ​​of the roller coordinate system is obtained, and then the design equation of the driving roller surface is obtained; wherein, the time-expressed Z-axis coordinate value of the roller coordinate system is obtained based on the constraint conditions under the roller coordinate system.

[0050] In another embodiment, in S100, the contact trajectory curve and constraints in the world coordinate system are established, specifically as follows:

[0051] (1) First, it is necessary to define two coordinate systems: the world coordinate system and the drum coordinate system:

[0052] Using the conventional world coordinate system as the fixed system, ,origin The roller is fixed in space and does not move with any other component. The Z-axis coincides with the ideal axis of rotation of the roller, which is the reference axis for the entire system. The X-axis and Y-axis lie in a plane perpendicular to the Z-axis and are determined according to the right-hand rule. Typically, the X-axis can be set to point from the roller axis to the drum axis. The motion of the roller is absolute and non-relative when observed in the world coordinate system.

[0053] Using the roller coordinate system as the moving frame, The drive roller is a rotating body formed by a generatrix rotating around a rotation axis. Although the shape of this generatrix is ​​unknown, it can be written as: Roller radius = The purpose of this invention is to find this relationship, thereby completing the surface design of the drive roller. The origin of the roller coordinate system. Fixed on the rotation axis of the drum, its initial position in the world coordinate system can be set to... ,in This indicates the theoretical center distance between the roller and the drum. The axis of rotation of the shaft coincides with that of the roller and is parallel to the Z-axis of the fixed system. shaft and Initially, the axis is parallel to the X and Y axes of the fixed system. As the drum rotates, the drum coordinate system revolves along with the drum. Rotation of the axis. The drum coordinate system can be understood as an "observer on the drum" used to describe the geometry of the drum itself.

[0054] (2) Define the contact trajectory curve in the world coordinate system

[0055] The surface of the logarithmic generatrix roller is a surface of revolution, formed by rotating a generatrix about its axis. The equation of the roller generatrix is ​​a function of the roller radius r with respect to the axial coordinate z, expressed as: ,in It is a deterministic logarithmic function, in the form of: .

[0056] In a fixed system In the middle, any point on the roller surface The position can be set to two parameters. If it means: ,in, Point axial coordinates, . Point Circumferential angle around the Z-axis, For fixed ,when From 0 to This formula describes the cross-sectional circle of the roller at height z. For a fixed... ,when When the curve changes, the formula describes a generatrix of the roller surface.

[0057] It is important to note that the purpose of this invention is to design an ideal drive roller, thus ensuring that the transmission between the roller and the drive roller is ideal pure rolling without any slippage. Under ideal pure rolling conditions, at any given time t, there is line contact between the roller and the roller. This contact line, observed in a fixed frame, is a curve that varies in space with time t. The point of tangency of this instantaneous contact line on the roller surface moves across the roller surface with time t, forming a path. This path is defined as the ideal contact trajectory and denoted as the space curve. From the perspective of the world coordinate system / fixed coordinate system, the spatial position of the contact point between the roller and the driving roller moves over time, eventually forming an ideal contact trajectory. However, from the perspective of the roller moving coordinate system / moving coordinate system, if the drive is ideal (pure rolling, no slippage), then the relative position of the contact point between the roller and the roller on the roller surface is fixed. The set of all contact points constitutes a stationary spatial curve (ideal contact trajectory), which is the roller generatrix that this invention aims to determine. The roller generatrix connects the roller surface and the roller surface.

[0058] Space curves It must also be located on the roller surface Therefore, points on the ideal contact trajectory can be represented by the roller surface parameters. The relationship between the ideal contact trajectory and the roller surface is expressed as a function that varies with time t: ,in, and The unknown time function describes the motion of the contact point on the roller surface, but its determination requires complex kinematic constraints, which is the problem to be solved in subsequent steps.

[0059] (3) Determine the constraints in the world coordinate system

[0060] For space curves Find the first derivative with respect to the parameter t (time). To obtain the instantaneous rate of change at that point, the first derivative of the position vector (velocity vector) is expressed as follows:

[0061]

[0062] in, The derivative of the busbar function is represented. and These represent the rates of change of angular and axial position, respectively.

[0063] The direction of the tangent vector is the direction of the velocity vector. Normalizing it to a unit vector gives us the unit tangent vector. Represented as: ,in, It represents the direction of instantaneous motion of the contact point in a fixed coordinate system; it is the first-order geometric property of the curve.

[0064] To describe the curvature and direction of the curve, we calculate the second derivative, the second derivative of the position vector (acceleration vector). Represented as: ,in, yes The derivatives of each component with respect to time t are then taken again, resulting in a more complex form, which can be systematically expressed as follows: , , , as well as Linear combination of terms.

[0065] Ideal contact trajectory curve curvature The curvature measures the degree to which it deviates from a straight line. Represented as: curvature The larger the value, the more curved the trajectory is at that point.

[0066] Principal Normal Vector If the direction points to the center of the curve, then the principal normal vector is... They are represented as follows: Principal normal vector This represents the second-order geometric properties, defining the direction of the curve's "bending axis" at that point, which is crucial for the design of roller surfaces.

[0067] To achieve slip-free rolling, the roller surface and the drum surface must be tangent at the contact point, meaning they share the same tangent plane (first-order contact condition). This tangent plane is determined by the tangent vector. The contact is defined by a certain direction vector, which is implicit in the definition of "contact". To achieve a higher level of pure rolling (suppressing micro-slip), higher-order contact constraints are required: the roller surface should not only be tangent to the roller surface at the contact point, but its normal curvature along the contact trajectory direction should also match that of the roller surface. Mathematically, this condition is equivalent to requiring the roller surface to be tangent to the roller surface at the contact point. Unit normal vector at the location It must be related to the principal normal vector of the trajectory. Parallel, that is This condition ensures that the bending morphology of the two surfaces is highly consistent within the local region of their contact point, thus greatly promoting uniform stress distribution and the realization of a pure rolling state, which is the core of achieving ultra-stable rotation. Establishing higher-order contact constraints is a crucial step in transforming the geometric properties of the curve into surface design constraints.

[0068] binormal vector Represented as: Binormal vector tangent vector Principal Normal Vector Together they form a right-handed coordinate system, with the binormal vector The plane spanned by the instantaneous direction of motion and the direction of bending.

[0069] In another embodiment, in S200, the normal vector at the contact point and the principal normal vector of the contact trajectory curve are transformed to the roller coordinate system. Combined with the aforementioned constraint conditions, the constraint conditions in the roller coordinate system are obtained, specifically:

[0070] Assuming the drive roller has an angular velocity Around its axis ( The axis rotates at a constant speed, and its rotation angle is within time t. Represented as: ,in, To represent the initial phase angle, for simplicity without loss of generality, we can set it as follows: .

[0071] A certain point from a fixed system Transformation to the moving frame It requires translation first, followed by rotation. This transformation can be performed using a... homogeneous pose transformation matrix This can also be simplified to a rotation matrix. and a translation vector The combination of kinetic systems. The origin In the world coordinate system The coordinates in are ,in The translation vector represents the theoretical center distance between the roller and the drum. Represented as: Dynamic system Relative to fixed system Around The axis rotated Angle, its rotation matrix Represented as: .

[0072] Fixed system One point Its dynamic system coordinates in Represented as: ,in, The transpose (i.e., inverse) of a rotation matrix is ​​the core formula for coordinate transformation.

[0073] Contact trajectory in a fixed system Mapping onto the roller (coordinate system) for observation, for each time t, the contact point at that time... (exist Transformation from the system in the moving system to the system in the moving system In the middle, we get points ,point Represented as: ,in, This represents the position of the contact point as seen by an observer "located on the roller" at time t, i.e., the representation of the contact point in the roller coordinate system.

[0074] To achieve stable drive, the position of the contact point relative to the roller surface should not change randomly over time. That is, in the roller coordinate system... See, the contact point should always follow the same fixed curve on the roller surface. Movement, i.e. This means that although t is changing, This set of points, in the moving frame This forms a static spatial curve that does not change with time. This is the "contact line." This is a fixed spatial curve. It is precisely the generatrix of the driving roller's rotating surface that this invention seeks to solve that will curve... Around the roller As the axis rotates, the surface scanned by the camera becomes the curved surface of the roller. The desired curved surface of the roller is a fixed spatial curve. Around A surface of revolution formed by rotating an axis. It is a dynamic system A curve in the diagram.

[0075] The coordinate transformation of a vector (such as a normal vector) differs from that of a point; it only involves rotation, not translation. Therefore, in a moving frame of reference... The normal vector of the roller surface at the contact point observed in the image Represented as: At the same time, the principal normal vector in the fixed frame... Transformation to the moving frame In the moving frame, the principal normal vector is... Represented as: .

[0076] Geometric constraints Multiply both sides by left , obtain in the dynamic system Equivalence constraints in: . The roller surface is on a fixed curve The normal vector at a certain point on the roller is only related to the static geometry of the roller. related. It is the transformed principal normal vector. This equation represents the static geometry of the roller ( ) and dynamic geometry of motion ( They got in touch. Because It is fixed, and It changes over time; this condition is actually a very strong constraint used to uniquely determine the curve. The shape.

[0077] In another embodiment, in S300, based on the cylindrical contact trajectory curve, the cylindrical trajectory normal vector represented by the Z-axis coordinate value of the roller coordinate system is obtained, including the following steps:

[0078] (1) Represent the contact trajectory curve in the roller coordinate system using cylindrical coordinates to obtain the cylindrical contact trajectory curve.

[0079] because The curve is the generatrix of the rotating surface of the driving roller, in the moving system Cylindrical coordinates can be used in this context. To indicate, among which, Indicates to Distance between axes Indicates the circumferential angle. express Axis coordinates. For the busbar In other words, For example, a constant (e.g., ).

[0080] Generatrix of the roller surface Defined in the moving coordinate system of the drum In the middle. Because the roller is a surface of revolution, its generatrix lies on a fixed plane (such as...). A planar curve within a plane. Within this plane, the generatrix... The position of any point on the axis can be determined by its axial coordinates. Uniquely determined. Point relative to the roller axis ( vertical distance of axis This depends on the axial position of that point. That is, there exists a functional relationship. It should be noted that, although analysis reveals... It can be used This indicates that the functional relationship between them is unknown and requires further solution. (Curve) It can be parameterized as follows: In rectangular coordinates: , The final solution is the roller radius function. .

[0081] (2) Based on the cylindrical contact trajectory curve, the cylindrical trajectory normal vector represented by the Z-axis coordinate value of the roller coordinate system is obtained.

[0082] To establish the constraint equations, it is first necessary to obtain the curvature of the roller surface on the generatrix. any point above The normal vector at that point. The generatrix Around Rotating the axis yields the parametric equations of the roller surface expressed in cylindrical coordinates: .

[0083] The normal vector of a surface at a point can be obtained by the cross product of its two tangent vectors. Therefore, for... Find them separately and The partial derivative of, for The partial derivative (circumferential tangent vector) is expressed as: ,right The partial derivative (axial tangent vector) is expressed as: ,in, .

[0084] Non-unit normal vector Represented as: On the busbar superior, Substituting into the above formula, we get the busbar. Normal vector on: Unit normal vector Represented as: , Using the drum coordinate system axis coordinate values ​​( The formula represents the unit normal vector of the cylindrical trajectory, and it expresses how the normal vector of the roller surface depends on the shape of its generatrix. and slope .

[0085] In another embodiment, in S400, based on the cylinder trajectory normal vector, the Z-axis coordinate value of the roller coordinate system expressed in time, and the constraints under the roller coordinate system, the relationship between the roller surface radius and the Z-axis coordinate value of the roller coordinate system is obtained, thereby obtaining the design equation of the driving roller surface, including the following steps:

[0086] (1) Perform cross product processing on the normal vector of the cylindrical trajectory and the principal normal vector of the contact trajectory curve in the roller coordinate system to obtain the cross product result; wherein the cross product result is equal to the zero vector.

[0087] Because the busbar is positioned in the moving system flat ,and of The component is 0. This means that the constraint requires the transformed principal normal vector to be 0. of The component must also be zero (or negligible compared to other components, indicating that it lies in the same plane), which is a strong condition.

[0088] Using the fact that parallel vectors have a zero cross product, then... ,set up ,and Then the cross product equation can be expressed as:

[0089] .

[0090] (2) Based on the correspondence between the cross product result and the zero vector, the relationship between the slope of the roller radius and the corresponding principal normal vector component is obtained.

[0091] According to the cross product equation: This shows Must be located flat. This is related to It matches. This indicates that the roller busbar is in position. slope This is entirely determined by the ratio of the components of the principal normal vector corresponding to the point of contact at that moment, from which we obtain:

[0092]

[0093] The right side of the above equation is a function of time t, and the left side is... axis coordinates To obtain a pure differential equation for a function, kinematic relations must be used. use and express.

[0094] (3) Based on the constraints under the roller coordinate system, the Z-axis coordinate value of the roller coordinate system expressed in terms of time is obtained.

[0095] Constraints To elaborate, the constraint equation takes a specific form: parallel vectors mean that their components are proportional, let the proportionality coefficient be... Then we have: , It is time The function, and yes The function, then at t and An implicit correspondence can be established between them. .

[0096] (4) Combining the Z-axis coordinate values ​​of the roller coordinate system expressed in terms of time, solve the relationship between the slope of the roller radius and the ratio of the components of the corresponding principal normal vector, determine the unique relationship between the roller radius and the Z-axis coordinate values ​​of the roller coordinate system, and then obtain the unique design equation of the driving roller surface.

[0097] In this invention, all variables are linked together through the "contact" event and coordinate transformation. As previously stated, the generatrix equation of the roller is... Under the assumption of pure rolling motion, the ratio of the angular velocity of the roller to that of the drum is constant, and in the drum coordinate system, the generatrix of the driving drum is represented as... .

[0098] Principal Normal Vector Derived from the ideal contact trajectory on the roller Any point on the roller surface The location can be used and Therefore, It can be used , , These quantities represent... and the principal normal vector in the moving frame... Represented as And rotation matrix Depends solely on the roller angle ,and .

[0099] because , ,but Includes , and .and ,but and , and The three components are strongly constrained, resulting in three equations. Adding these three equations to the kinematic constraints of pure rolling (fixed angular velocity ratio), we obtain: expressed in terms of time t. , , and It can also be used and This indicates that, due to the rotation angle of the roller... and If there is a fixed ratio, then It can also be used and This indicates that... Therefore, It can be used and It is said that, due to It was eliminated, therefore it is processed by the rotation matrix. After coordinate transformation, It can also be used and It means, that is and It can also be used and express, It can be represented as and The function, assuming this relation has been derived. This yields a first-order ordinary differential equation:

[0100]

[0101] The above ordinary differential equations are the governing equations that determine the shape of the roller generatrix. According to the function... The specific form can be solved using analytical methods (such as the separation of variables method) or numerical methods (such as the Runge-Kutta method). An initial condition is required for the solution, for example, in the middle of the roller. place, That is, the nominal distance minus the center radius of the roller, thus determining a unique solution.

[0102] After determining the unique relationship between the drum radius and the Z-axis coordinate value of the drum coordinate system, since Indicates to The distance along the axis, then This represents the radius function of the roller. ,and Point Axis coordinates, then in the drum coordinate system In this context, the equation for the roller surface can be expressed as: This is the final output of the invention: a clear mathematical model of a roller surface that can guide CNC machining. For example... Figure 2 The diagram shown is a projection schematic of the curve corresponding to the roller surface equation in an embodiment of the present invention. In this embodiment, there are two driving rollers, and the roller surface equations of the two driving rollers are the same.

[0103] Example 2:

[0104] A surface design system for a logarithmic busbar bearing roller-driven drum, used to implement any of the above embodiments, such as... Figure 3 As shown, the system includes a trajectory constraint construction module 100, a constraint transformation module 200, a cylinder transformation module 300, and a solution module 400;

[0105] The trajectory constraint construction module 100 is used to establish the contact trajectory curve and constraint conditions in the world coordinate system; wherein, the contact trajectory curve is the path line formed by the point of tangency of the contact line between the roller and the driving roller on the roller surface as time moves on the roller surface, and the constraint condition is that the normal vector at the contact point between the contact trajectory curve and the driving roller surface is parallel to the principal normal vector of the contact trajectory curve.

[0106] The constraint transformation module 200 is used to transform the normal vector at the contact point and the principal normal vector of the contact trajectory curve to the roller coordinate system, and combined with the constraint conditions, obtain the constraint conditions in the roller coordinate system; wherein, the Z-axis of the roller coordinate system is parallel to the Z-axis of the world coordinate system.

[0107] The cylindrical transformation module 300 is used to obtain the cylindrical trajectory normal vector represented by the Z-axis coordinate value of the roller coordinate system based on the cylindrical contact trajectory curve; the cylindrical contact trajectory curve is the cylindrical coordinate representation of the contact trajectory curve in the roller coordinate system.

[0108] The solution module 400 is used to obtain the relationship between the radius of the roller surface and the Z-axis coordinate value of the roller coordinate system based on the normal vector of the cylindrical trajectory and the time-expressed Z-axis coordinate value of the roller coordinate system, combined with the constraint conditions under the roller coordinate system, and then obtain the design equation of the driving roller surface; wherein, the time-expressed Z-axis coordinate value of the roller coordinate system is obtained based on the constraint conditions under the roller coordinate system.

[0109] Various changes and modifications made without departing from the spirit and scope of this invention, and all equivalent technical solutions, also fall within the scope of this invention.

[0110] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. The same or similar parts between the various embodiments can be referred to each other.

[0111] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, apparatus, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0112] This invention is described with reference to flowchart illustrations and / or block diagrams of the method, terminal device (system), and computer program product according to the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing terminal device to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing terminal device, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0113] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing terminal device to operate in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0114] These computer program instructions can also be loaded onto a computer or other programmable data processing terminal equipment, causing a series of operational steps to be performed on the computer or other programmable terminal equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable terminal equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0115] It should be noted that:

[0116] The phrase "an embodiment" or "an embodiment" used in this specification means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the invention. Therefore, the phrase "an embodiment" or "an embodiment" appearing in various places throughout the specification does not necessarily refer to the same embodiment.

[0117] Furthermore, it should be noted that the shapes and names of the parts and components described in the specific embodiments described in this specification may differ. All equivalent or simple variations made to the structure, features, and principles described in this patent concept are included within the protection scope of this patent. Those skilled in the art to which this invention pertains may make various modifications or additions to the described specific embodiments or use similar methods to replace them, as long as they do not depart from the structure of this invention or exceed the scope defined in these claims, they should all fall within the protection scope of this invention.

Claims

1. A method for designing the curved surface of a logarithmic generatrix bearing roller drive drum, characterized in that, Includes the following steps: Establish the contact trajectory curve and constraints in the world coordinate system; wherein, the contact trajectory curve is the path line formed by the point of tangency of the contact line between the roller and the drive roller on the roller surface as time moves on the roller surface, and the constraint is that the normal vector at the contact point between the contact trajectory curve and the drive roller surface is parallel to the principal normal vector of the contact trajectory curve; The normal vector at the contact point and the principal normal vector of the contact trajectory curve are transformed to the drum coordinate system. Combined with the aforementioned constraint conditions, the constraint conditions in the drum coordinate system are obtained. The Z-axis of the drum coordinate system is parallel to the Z-axis of the world coordinate system. The constraint condition in the drum coordinate system is that the normal vector at the contact point in the drum coordinate system is parallel to the principal normal vector of the contact trajectory curve in the drum coordinate system. Based on the cylindrical contact trajectory curve, the cylindrical trajectory normal vector is obtained using the Z-axis coordinate value of the drum coordinate system; the cylindrical contact trajectory curve is the cylindrical coordinate representation of the contact trajectory curve in the drum coordinate system. Based on the cylindrical trajectory normal vector and the Z-axis coordinate value of the roller coordinate system expressed in time, combined with the constraints under the roller coordinate system, the relationship between the roller surface radius and the Z-axis coordinate value of the roller coordinate system is obtained, and then the design equation of the driving roller surface is obtained; including the following steps: The cross product of the cylindrical trajectory normal vector and the principal normal vector of the contact trajectory curve in the roller coordinate system is performed to obtain the cross product result; wherein, the cross product result is equal to the zero vector; Based on the correspondence between the cross product result and the zero vector, the relationship between the slope of the roller radius and the corresponding principal normal vector component is obtained; Based on the constraints in the roller coordinate system, the Z-axis coordinate value of the roller coordinate system expressed in terms of time is obtained; By combining the Z-axis coordinates of the roller coordinate system expressed in terms of time, the relationship between the slope of the roller radius and the corresponding principal normal vector component is solved to obtain the unique relationship between the roller radius and the Z-axis coordinates of the roller coordinate system, and then the design equation of the driving roller surface is obtained.

2. The method according to claim 1, characterized in that, The contact trajectory curve in the world coordinate system is represented as follows: in, This represents the contact trajectory curve in the world coordinate system. Indicates time, The generatrix equation of the logarithmic generatrix bearing roller is represented here. Indicates time The Z-axis coordinate value of the logarithmic generatrix bearing roller in the world coordinate system. Indicates time The circumferential angle of the rollers in a time-logarithmic busbar bearing. This represents the transpose of a matrix.

3. The method according to claim 1, characterized in that, The process of transforming the normal vector at the contact point and the principal normal vector of the contact trajectory curve to the roller coordinate system, and then combining this with the constraint conditions to obtain the constraint conditions in the roller coordinate system, includes the following steps: Based on the rotation matrix between the world coordinate system and the drum coordinate system, the normal vector at the contact point and the principal normal vector of the contact trajectory curve are transformed to the drum coordinate system, respectively, to obtain the normal vector at the contact point and the principal normal vector of the contact trajectory curve in the drum coordinate system. By combining the aforementioned constraints, the normal vector at the contact point in the drum coordinate system, and the principal normal vector of the contact trajectory curve in the drum coordinate system, the constraints in the drum coordinate system are obtained.

4. The method according to claim 1, characterized in that, The cylindrical contact trajectory curve is represented as follows: in, Represents any point on the contact trajectory curve of the cylinder. Represents the roller in the roller coordinate system axis coordinate values, Represents the coordinate system of a point on the cylindrical contact trajectory curve. Distance between axes This represents the circumferential angle of the roller in the roller coordinate system.

5. The method according to claim 1, characterized in that, The normal vector of the cylindrical trajectory is represented as follows: in, This represents the normal vector of the cylindrical trajectory, expressed in terms of the Z-axis coordinates of the drum coordinate system. Represents the roller in the roller coordinate system axis coordinate values, Represents the coordinate system of a point on the cylindrical contact trajectory curve. Distance between axes .

6. The method according to claim 1, characterized in that, The relationship between the slope of the roller radius and the corresponding principal normal vector component is expressed as follows: in, Indicates the drum radius In position The slope at that point and They represent time respectively The component values ​​of the principal normal vector along the Z-axis and X-axis in the drum coordinate system. Indicates time.

7. A surface design system for a logarithmic generatrix bearing roller-driven drum, used to implement the method as described in any one of claims 1 to 6, characterized in that, The system includes a trajectory constraint construction module, a constraint transformation module, a cylinder transformation module, and a solution module; The trajectory constraint construction module is used to establish the contact trajectory curve and constraint conditions in the world coordinate system; wherein, the contact trajectory curve is the path line formed by the point of tangency of the contact line between the roller and the drive roller on the roller surface as time moves on the roller surface, and the constraint condition is that the normal vector at the contact point between the contact trajectory curve and the drive roller surface is parallel to the principal normal vector of the contact trajectory curve. The constraint transformation module is used to transform the normal vector at the contact point and the principal normal vector of the contact trajectory curve to the drum coordinate system, and combined with the constraint conditions, obtain the constraint conditions in the drum coordinate system; wherein, the Z-axis of the drum coordinate system is parallel to the Z-axis of the world coordinate system. The cylinder conversion module is used to obtain the cylinder trajectory normal vector represented by the Z-axis coordinate value of the roller coordinate system based on the cylinder contact trajectory curve; the cylinder contact trajectory curve is the cylindrical coordinate representation of the contact trajectory curve in the roller coordinate system. The solution module is used to obtain the relationship between the radius of the roller surface and the Z-axis coordinate value of the roller coordinate system based on the normal vector of the cylindrical trajectory and the Z-axis coordinate value of the roller coordinate system in terms of time, combined with the constraint conditions under the roller coordinate system, and then obtain the design equation of the driving roller surface; wherein, the Z-axis coordinate value of the roller coordinate system in terms of time is obtained based on the constraint conditions under the roller coordinate system.

8. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by a processor, it implements the method as described in any one of claims 1 to 6.

9. An apparatus comprising a memory, a processor, and a computer program stored in the memory and running on the processor, characterized in that, When the processor executes the computer program, it implements the method as described in any one of claims 1 to 6.

Images