Roller type with cross section as a periodic asymmetric function curve and its design method

Abstract

The application discloses a roll type with a cross section of a periodic asymmetric function curve and a design method thereof, and belongs to the technical field of plate rolling. The cross section roll type curve of the roll type is a periodic asymmetric function curve, and comprises a rear waist section and a front waist section. The curve length of the front waist section is greater than that of the rear waist section. Firstly, a flat roll is taken as a basis to determine an expression of the cross section roll type curve, then a radial size function of the cross section roll type curve is determined, and then a connecting point of the front waist section and the rear waist section and a phase difference are calculated to obtain a parameter equation of the roll type curve with the cross section of the periodic asymmetric curve. The roll type solves the problems of the current metal laminated composite plate rolled by the curve type roll with the cross section of the periodic symmetric curve, such as poor uniformity of the interface bonding strength distribution on the whole composite interface, low average interface bonding strength of the whole composite interface and the like. The design method is simple and easy to implement, is easy to master and implement, is high in precision, and is universal, and can be popularized for the design of other curve type rolls.

Description

Technical Field

[0001] This invention belongs to the field of metal sheet rolling technology, specifically relating to a roll profile with a cross-section of a periodic asymmetric function curve and its design method. Background Technology

[0002] The shape of the rolls (i.e., roll profile) has a significant impact on the forming of sheet metal during rolling. Based on the cross-sectional roll profile curve of sheet metal (including layered metal composites), rolls can be divided into two main categories: flat rolls and curved rolls. Flat rolls have a standard circular cross-sectional roll profile curve. During sheet metal rolling, due to the asymmetrical distribution of residual stress, sheet shape defects such as edge waviness, transverse bending, and lateral bending are prone to occur. Especially when rolling layered metal composites, there are also problems such as severe sheet warping, high residual stress, and low interfacial bonding strength due to the large differences in the deformation capabilities of dissimilar metals. The cross-sectional curve of curved rolls is generally a periodically symmetrical curve, such as corrugated rolls, toothed rolls, and other corrugated rolls. They adopt elliptical, circular arc, and sinusoidal shapes, which can adjust the distribution of residual stress and coordinate metal deformation under non-uniform force. This helps to alleviate problems such as plate shape defects and large residual stress that exist when flat rolls are used to roll metal sheets. Moreover, compared with flat rolls for rolling metal layered composite plates, the use of curved rolls for rolling metal layered composite plates can improve the interfacial bonding strength of metal layered composite plates to a certain extent. However, when using curved rolls with a periodically symmetrical cross-section to roll metal laminated composite plates, there is a drawback: the stress states of the front and rear waist sections are inconsistent. Due to the excessive compressive stress effect, the interfacial bonding strength between the composite interface of the metal laminated composite plate and the area corresponding to the rear waist section of the curved roll is lower than that of the area corresponding to the front waist section of the curved roll. This results in poor uniformity of interfacial bonding strength in different areas of the composite interface of the metal laminated composite plate rolled with curved rolls with a periodically symmetrical cross-section, which seriously affects the further significant improvement of the average interfacial bonding strength of the entire composite interface and makes it difficult to obtain high-performance metal laminated composite plates with excellent interfacial bonding strength throughout the entire composite interface.

[0003] Therefore, it is of great significance to design and develop a roller that can reduce the shape defects of metal layered composite plates, reduce their residual stress, and significantly improve the uniformity of interfacial bonding strength in different regions of the composite interface and the average interfacial bonding strength of the entire composite interface. Summary of the Invention

[0004] The purpose of this invention is to provide a roll profile with a periodic asymmetric function curve in its cross-section and its design method. By adjusting the residual stress distribution and coordinating metal deformation through the periodic asymmetric change of the roll profile curve, problems such as severe plate warping, high residual stress, and low interfacial bonding strength that exist when rolling metal layered composite plates with flat rolls can be avoided. By designing the roll profile curve in a form where the front and rear waist sections are asymmetrically distributed within the period, local uneven deformation can be enhanced, the tensile stress effect at the composite interface can be increased, and the length of the rolling zone can be extended during the rolling of metal layered composite plates. This solves the problems of poor uniformity of interfacial bonding strength distribution and insufficient average interfacial bonding strength of the entire composite interface when using curved rolls with periodic symmetric curves to roll metal layered composite plates. This significantly improves the average interfacial bonding strength of the metal layered composite plate, resulting in a high-performance metal layered composite plate with excellent interfacial bonding strength throughout the composite interface.

[0005] According to a first aspect of the present invention, a roll profile with a cross-section of a periodic asymmetric function curve is provided, characterized in that the cross-sectional roll profile curve of the roll profile is a periodic asymmetric function curve, the cross-sectional roll profile curve includes a front waist section and a rear waist section, the front waist section is a curve running in the same direction as the rotation direction of the roll from the bottom of the roll profile to the top of the roll profile, the rear waist section is a curve running in the same direction as the rotation direction of the roll from the top of the roll profile to the bottom of the next roll profile, and the curve length of the front waist section is greater than the curve length of the rear waist section.

[0006] According to a second aspect of the present invention, a design method for a roller profile with a cross-section that is a periodic asymmetric function curve as described above is provided, comprising the following steps:

[0007] Step 1: Determine the expression of the cross-sectional roll profile curve based on the flat roll, wherein the cross-sectional roll profile curve is a periodic asymmetric curve;

[0008] Step 2: Determine the radial dimension function of the cross-sectional roller curve, and calculate the connection point and phase difference between the front and rear waist sections of the cross-sectional roller curve;

[0009] Step 3: Determine the parametric equation of the cross-sectional roller profile curve based on the radial dimension function of the cross-sectional roller profile curve, the connection point of the front waist section and the rear waist section, and the phase difference.

[0010] Furthermore, step 1 specifically includes:

[0011] Step 11: Take the cross-sectional roller profile curve of the flat roller with radius r0 as the standard circle, take the center point of the cross-section of the flat roller as the origin, and establish a rectangular coordinate system with the horizontal and vertical directions as the x-axis and y-axis respectively. Define r as the radial dimension function of the cross-sectional roller profile curve, and t as the arc of rotation in the counterclockwise direction starting from the positive x-axis.

[0012] Step 12: Project r onto the x-axis and y-axis respectively to obtain the abscissa and ordinate of each point on the cross-sectional roller curve, and determine the expression of the cross-sectional roller curve.

[0013] Further, the expression for the cross-sectional roller profile curve in step 2 is:

[0014]

[0015] Furthermore, step 2 specifically includes:

[0016] Step 21: Select the function type for the cross-sectional roller profile curve;

[0017] Step 22: Set the period of the radial dimension function of the front and rear waist sections of the cross-sectional roller curve;

[0018] Step 23: Determine the radial dimension function r of the cross-sectional roller profile curve;

[0019] Step 24: Determine the connection point t0 between the front waist segment and the rear waist segment, and calculate the phase difference of the front waist segment. Phase difference with the posterior lumbar segment

[0020] Further, in step 21, a periodic function f(t) (|f(t)| << r0) is selected as the function type of the periodic asymmetric curve, and the radial dimension function is r = r0 + f(t), where the position of r = r0 is called the scale line.

[0021] Furthermore, step 22 specifically includes:

[0022] Step 221: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require As a function of the radial dimension of the front waist section, where Let α1 be the phase difference of the radial dimension function of the front waist section, and let α2 be the projected length of the front waist section on the index line. Assume α1 / α2 = ω > 1, and set the period of the radial dimension function of the rear waist section as α0.

[0023] Step 222: Calculate the period of the cross-sectional roller profile curve. Calculate the number of cycles for the cross-sectional roller profile curve. By taking values ​​for ω, ensuring that the number of cycles is a positive integer, we can obtain the magnitude of ω.

[0024] Furthermore, step 23 specifically includes:

[0025] The radial dimension function of the rear waist segment is written as... in The expression for the radial dimension function r of the cross-sectional roll profile is obtained by considering the phase difference of the radial dimension function of the rear waist section:

[0026]

[0027] Furthermore, step 24 specifically includes:

[0028] Step 241: The cross-sectional roller curve is composed of n front waist segments and n rear waist segments, and the cross-sectional roller curve has a total of 2n connection points, where the connection point t0 is:

[0029]

[0030] Step 242: According to The phase difference of the anterior waist segment was calculated using mathematical induction. Phase difference with the posterior lumbar segment

[0031] Furthermore, the parametric equation of the roll profile of the cross-sectional roll profile curve in step 3 is composed of the front waist segment function and the rear waist segment function:

[0032] The function of the anterior waist segment is:

[0033] in,

[0034] The function of the lower back segment is:

[0035] in,

[0036] The beneficial effects of adopting the above technical solution are as follows:

[0037] (1) The cross-section of the present invention is a periodic asymmetric function curve of the roll type. By adjusting the spatial ratio of the front waist section and the rear waist section, the projection length α1 of the rear waist section on the indexing line is less than the projection length α2 of the front waist section on the indexing line, that is, α2 / α1>1. This can enhance local non-uniform deformation, increase the proportion of interface tensile stress effect, extend the length of the rolling zone, promote the cracking of the interface hardening layer and oxide film, and the extrusion of fresh metal, which helps to achieve strong metallurgical bonding of the composite interface of the metal layered composite plate. By adjusting the residual stress distribution and coordinating metal deformation by changing the cross-sectional roll type curve, it helps to solve the problems of plate shape defects, large residual stress, poor uniformity of interface bonding strength in different areas of the composite interface and low average interface bonding strength of the entire composite interface in the field of metal plate (including metal layered composite plate) rolling, and obtain a high-performance metal layered composite plate with excellent interface bonding strength of the entire composite interface.

[0038] (2) The design method of the roller profile with a periodic asymmetric function curve in the present invention controls the cross-sectional roller profile curve by setting and adjusting the radial dimension function of the roller profile. It is simple, easy to master and implement, highly accurate, and universally applicable. It can be extended to the design of other curved roller profiles. Attached Figure Description

[0039] Figure 1 : Schematic diagram of the cross-sectional roller profile (a, standard circle; b, periodically symmetrical curve);

[0040] Figure 2 : The radial dimension function image of a roller profile with a periodically symmetric cross-section;

[0041] Figure 3 : Schematic diagram of the cross-sectional roller profile (a, periodically symmetrical curve; b, periodically asymmetrical curve);

[0042] Figure 4 : The radial dimension function image of a roller profile curve whose cross-section is a periodic asymmetric function;

[0043] Figure 5 A two-dimensional schematic diagram of a roller profile whose cross-section is a periodic asymmetric function;

[0044] Figure 6 : Schematic diagrams of three-dimensional models of rolls with periodic symmetric cross-sections and rolls with periodic asymmetric cross-sections (a, periodic symmetric function; b, periodic asymmetric function). Detailed Implementation

[0045] The present invention will be further described below with reference to specific embodiments.

[0046] The present invention provides a roll profile with a periodic asymmetric function curve in cross-section and its design method. The cross-sectional roll profile curve is a periodic asymmetric function curve, which is composed of a front waist section and a rear waist section. The front waist section is a curve running in the same direction as the rotation of the roll from the bottom to the top of the roll profile, and the rear waist section is a curve running in the same direction as the rotation of the roll from the top of the roll profile to the bottom of the next roll profile. The curve length of the front waist section is greater than the curve length of the rear waist section.

[0047] The design method for rollers with cross-sections that are periodic asymmetric function curves includes the following steps:

[0048] Step 1: Establish a rectangular coordinate system;

[0049] Using the cross-sectional roll profile curve of a flat roll with radius r0 as a standard circle, and taking the center point of the flat roll's cross-section as the origin, a rectangular coordinate system is established with the horizontal and vertical directions as the x-axis and y-axis, respectively. Let r represent the radial dimension function of the cross-sectional roll profile curve, and t represent the radians rotated counterclockwise from the positive x-axis position (see...). Figure 1 ).

[0050] Step 2: Determine the expression for the cross-sectional roll profile curve;

[0051] Projecting r onto the x-axis and y-axis respectively, we obtain the abscissa and ordinate of each point on the cross-sectional roller profile curve. The expression for the cross-sectional roller profile curve is:

[0052]

[0053] Step 3: Select the function type for the periodic asymmetric curve;

[0054] Based on a flat roller with radius r0, a periodic function f(t) (|f(t)| << r0) is chosen as the function type for the periodic asymmetric curve, and the radial dimension function is r = r0 + f(t) (see...). Figure 1 (a) and (b) and Figure 2 ), where the position of r = r0 is called the scale line.

[0055] Step 4: Set the period of the radial dimension function for the front and rear waist sections;

[0056] Will As a function of the radial dimension of the front waist section, where This is the phase difference of the radial dimension function of the front waistline, from which the period T0 of the radial dimension function of the front waistline is calculated. The projected length of the rear waistline on the index line is defined as α1, and the projected length of the front waistline on the index line is defined as α2 (see...). Figure 3 In (a) and (b), let α2 / α1 = ω > 1, and set the period of the radial dimension function of the rear waist segment as follows:

[0057] Step 5: Determine the proportional relationship between α1 and α2;

[0058] Calculate the period of the cross-sectional roll profile curve The number of cycles for obtaining the cross-sectional roller profile curve For the convenience of geometric modeling and roll processing, the value of ω is selected to ensure that the number of cycles is a positive integer, thus obtaining the magnitude of ω.

[0059] Step 6: Obtain the radial dimension function of the cross-sectional roller profile curve;

[0060] Based on step 4, the radial dimension function of the rear waist segment can be written as... (See Figure 2 ),in The phase difference of the radial dimension function of the rear waist section is used to obtain the expression for the radial dimension function of the cross-sectional roll profile curve:

[0061]

[0062] The radial dimension function graph of the cross-sectional roller profile is shown below. Figure 4 .

[0063] Step 7: Determine the connection point t0 between the front and back waist segments;

[0064] As shown in step 4, the cross-sectional roller curve consists of n front waist segments and n rear waist segments, and the cross-sectional roller curve has a total of 2n connection points. The connection point t0 has two cases:

[0065] (1)

[0066] (2)

[0067] Connection point

[0068] Step 8: Calculate the phase difference of the anterior waist segment Phase difference with the lower back

[0069] according to Calculate the phase difference of the anterior midline using mathematical induction. Phase difference with the lower back

[0070] Step 9: Obtain the parametric equation expression for the cross-sectional roll profile curve;

[0071] Based on the above steps, the parametric equations of the roller profile with a periodic asymmetric function curve in cross-section are obtained, consisting of the front waist section function and the rear waist section function:

[0072] The function of the anterior waist segment is

[0073] in,

[0074] The function of the posterior lumbar segment is

[0075] in,

[0076] To verify that the roller curves, whose cross-sections are periodic asymmetric functions, are connected tangentially at the connection points with a smooth transition, the following operations are performed:

[0077] The first derivative k1 of the function in the anterior segment is:

[0078]

[0079] make but

[0080] Similarly, let Calculate the first derivative k2 of the function in the lower right corner. As shown in the following equation:

[0081]

[0082] At the connection point, the slope of the tangent line to the cross-sectional roller profile is zero.

[0083] That is, β = β′ = 0, so k1 = k2 = -cot(t0).

[0084]

[0085]

[0086] Right now The first derivatives of the cross-sectional roller curves are equal and continuous.

[0087] Derivative of the function of the anterior segment x′(t) 2 +y′(t) 2 =α 2 +β 2 ,because Therefore, x′(t) 2 +y′(t) 2 >0; similarly, the function x′(t) of the latter half of the waist segment 2 +y′(t) 2 =α′ 2 +β′ 2 >0.

[0088] The cross-sectional roller curves x(t) and y(t) are continuously differentiable on [0, 2π], and x′ 2 (t)+y′ 2 Since (t)≠0, the cross-sectional roller profile curve is a smooth curve. In summary, the roller profile curves with a periodic asymmetric function in cross-section are connected tangentially at the connection point and have a smooth transition. Figure 5 This is a two-dimensional schematic diagram of a roller profile whose cross-section is a periodic asymmetric function. Figure 6 In the figures (a) and (b), there are three-dimensional model diagrams of a roll with a periodic symmetric function curve and a roll with a periodic asymmetric function curve, respectively.

[0089] Example 1:

[0090] A design method for a roller with a cross-section that is a periodic asymmetric function curve includes the following steps:

[0091] Step 1: Establish a rectangular coordinate system;

[0092] Using the cross-sectional roll profile curve of a flat roller with a radius of 75mm as the standard circle, and the center point of the cross-section of the flat roller as the origin, a rectangular coordinate system is established with the horizontal and vertical directions as the x-axis and y-axis, respectively. r is defined as the radial dimension function of the cross-sectional roll profile curve, and t is defined as the arc of rotation in the counterclockwise direction starting from the positive x-axis position.

[0093] Step 2: Determine the expression for the cross-sectional roll profile curve;

[0094] Projecting r onto the x-axis and y-axis respectively, we obtain the abscissa and ordinate of each point on the cross-sectional roller profile curve. The expression for the cross-sectional roller profile curve is:

[0095]

[0096] Step 3: Select the function type for the periodic asymmetric curve;

[0097] Based on a flat roller with a radius of 75mm, the periodic function f(t) = 0.5cos(100t) is selected as the function type of the periodic asymmetric curve, and the radial dimension function is r = 75 + 0.5cos(100t), where the position of r = 75 is called the index line.

[0098] Step 4: Set the period of the radial dimension function for the front and rear waist sections;

[0099] Will As a function of the radial dimension of the front waist section, where The phase difference of the radial dimension function of the front waistline is used to calculate the period of the radial dimension function of the front waistline. Let α1 be the projected length of the rear waistline on the index line, and α2 be the projected length of the front waistline on the index line. Assume α2 / α1 = ω > 1. The period of the radial dimension function of the rear waistline is set as...

[0100] Step 5: Determine the proportional relationship between α1 and α2;

[0101] Calculate the period of the cross-sectional roll profile curve The number of cycles for obtaining the cross-sectional roller profile curve For ease of geometric modeling and roll processing, the value of ω is chosen to ensure that the number of cycles is a positive integer. The value of ω is obtained by selecting ω = 3 and the number of cycles n = 150.

[0102] Step 6: Obtain the radial dimension function of the cross-sectional roller profile curve;

[0103] Based on step 4, the radial dimension function of the rear waist segment can be written as... in The phase difference of the radial dimension function of the rear waist section is used to obtain the expression for the radial dimension function of the cross-sectional roll profile curve:

[0104]

[0105] Step 7: Determine the connection point t0 between the front and back waist segments;

[0106] As shown in step 4, the cross-sectional roller profile curve consists of 150 front waist segments and 150 rear waist segments, and has a total of 300 connection points. The position of connection point t0 can be divided into two cases:

[0107] (1)

[0108] (2)

[0109]

[0110] Step 8: Calculate the phase difference of the anterior waist segment Phase difference with the lower back

[0111] according to Calculate the phase difference of the anterior midline using mathematical induction. Phase difference with the lower back As shown in the following formula:

[0112]

[0113]

[0114]

[0115] Step 9: Obtain the parametric equation expression for the cross-sectional roll profile curve;

[0116] Based on the above steps, the parametric equations of the roller profile with a periodic asymmetric function curve in cross-section are obtained, consisting of the front waist section function and the rear waist section function:

[0117] The function of the anterior waist segment is

[0118] in

[0119] The function of the posterior lumbar segment is

[0120] in

[0121] To verify that the roller curves, whose cross-sections are periodic asymmetric functions, are connected tangentially at the connection points with a smooth transition, the following operations are performed:

[0122] Calculate the first derivative k1 of the function in the anterior segment:

[0123]

[0124] make but

[0125] Similarly, let Calculate the first derivative k2 of the function in the posterior segment:

[0126]

[0127] At the connection point: That is, β = β′ = 0, so

[0128]

[0129]

[0130] Right now The first derivatives of the cross-sectional roller curves are equal and continuous.

[0131] Derivative of the function of the anterior segment Because |0.5| << 75, Therefore, x′(t) 2 +y′(t) 2 =α 2 +β 2 >0, similarly, for the function x′(t) in the lower half of the body. 2 +y′(t)2 =α′ 2 +β′ 2 >0.

[0132] Therefore, the x(t) and y(t) of the periodic asymmetric cosine function plane curve are continuously differentiable on [0, 2π], and x′ 2 (t)+y′ 2 Since (t)≠0, the planar curve is a smooth curve. In summary, the roller-shaped curve with a periodic asymmetric cosine function as its cross-section is connected tangentially at the connection point and has a smooth transition.

[0133] Example 2:

[0134] A design method for a roller with a cross-section that is a periodic asymmetric function curve includes the following steps:

[0135] Step 1: Establish a rectangular coordinate system;

[0136] Using the cross-sectional roll profile curve of a flat roll with a radius of 50mm as the standard circle, and the center point of the cross-section of the flat roll as the origin, a rectangular coordinate system is established with the horizontal and vertical directions as the x-axis and y-axis, respectively. r is defined as the length of the point on the cross-sectional roll profile curve on the roll cross-section from the origin, and t is defined as the arc of rotation in the counterclockwise direction starting from the positive x-axis position.

[0137] Step 2: Determine the expression for the cross-sectional roll profile curve;

[0138] Projecting r onto the x-axis and y-axis respectively, we obtain the abscissa and ordinate of each point on the cross-sectional roller profile curve. The expression for the cross-sectional roller profile curve is:

[0139]

[0140] Step 3: Select the function type for the periodic asymmetric curve;

[0141] Based on a flat roller with a radius of 50 mm, the periodic function f(t) = 0.2cos(120t) is selected as the function type of the periodic asymmetric curve, and the radial dimension function is r = 50 + 0.2cos(120t), where the position of r = 50 is called the index line.

[0142] Step 4: Set the period of the radial dimension function for the front and rear waist sections;

[0143] Will As a function of the radial dimension of the front waist section, where The phase difference of the radial dimension function of the front waistline is used to calculate the period of the radial dimension function of the front waistline. Let α1 be the projected length of the rear waistline on the index line, and α2 be the projected length of the front waistline on the index line. Assume α2 / α1 = ω > 1. The period of the radial dimension function of the rear waistline is set as...

[0144] Step 5: Determine the proportional relationship between α1 and α2;

[0145] Calculate the period of the cross-sectional roll profile curve The number of cycles for obtaining the cross-sectional roller profile curve For ease of geometric modeling and roll processing, the value of ω is chosen to ensure that the number of cycles is a positive integer. The value of ω is obtained by selecting ω = 5 and the number of cycles n = 200.

[0146] Step 6: Obtain the radial dimension function of the cross-sectional roller profile curve;

[0147] Based on step 4, the radial dimension function of the rear waist segment can be written as... in The phase difference of the radial dimension function of the rear waist section is used to obtain the expression for the radial dimension function of the cross-sectional roll profile curve:

[0148]

[0149] Step 7: Determine the connection point t0 between the front and back waist segments;

[0150] As shown in step 4, the cross-sectional roller profile curve consists of 200 front waist segments and 200 rear waist segments, and has a total of 400 connection points. The position of connection point t0 can be divided into two cases:

[0151] (1)

[0152] (2)

[0153]

[0154] Step 8: Calculate the phase difference of the anterior waist segment Phase difference with the lower back

[0155] according to Calculate the phase difference of the anterior midline using mathematical induction. Phase difference with the lower back As shown in the following formula:

[0156]

[0157]

[0158]

[0159] Step 9: Obtain the parametric equation expression for the cross-sectional roll profile curve;

[0160] Based on the above steps, the parametric equations of the roller profile with a periodic asymmetric function curve in cross-section are obtained, consisting of the front waist section function and the rear waist section function:

[0161] The function of the anterior waist segment is

[0162] in,

[0163] The function of the posterior lumbar segment is

[0164] in

[0165] To verify that the roller curves, whose cross-sections are periodic asymmetric functions, are connected tangentially at the connection points with a smooth transition, the following operations are performed:

[0166] Calculate the first derivative k1 of the function in the anterior segment:

[0167]

[0168] make but

[0169] Similarly, let Calculate the first derivative k2 of the posterior segment. Then we have:

[0170]

[0171] At the connection point: That is, β = β′ = 0, so

[0172]

[0173]

[0174] Right now The first derivatives of the cross-sectional roller curves are equal and continuous.

[0175] Derivative of the function of the anterior segment Because |0.2| << 50, Therefore, x′(t) 2 +y′(t) 2 =α 2 +β 2 >0, similarly, for the function x′(t) in the lower half of the body. 2 +y′(t)2 =α′ 2 +β′ 2 >0.

[0176] Therefore, the x(t) and y(t) of the periodic asymmetric cosine function plane curve are continuously differentiable on [0, 2π], and x′ 2 (t)+y′ 2 Since (t)≠0, the planar curve is a smooth curve. In summary, the roller-shaped curve with a periodic asymmetric cosine function in cross-section is connected tangentially at the connection point and has a smooth transition.

[0177] Example 3:

[0178] A design method for a roller with a cross-section that is a periodic asymmetric function curve includes the following steps:

[0179] Step 1: Establish a rectangular coordinate system;

[0180] Using the cross-sectional roll profile curve of a flat roller with a radius of 60mm as the standard circle, and the center point of the cross-section of the flat roller as the origin, a rectangular coordinate system is established with the horizontal direction and the vertical direction as the x-axis and y-axis, respectively. r is defined as the radial dimension function of the cross-sectional roll profile curve, and t is defined as the arc of rotation in the counterclockwise direction with the positive x-axis as the starting position.

[0181] Step 2: Determine the expression for the cross-sectional roll profile curve;

[0182] Projecting r onto the x-axis and y-axis respectively, we obtain the abscissa and ordinate of each point on the cross-sectional roller profile curve. The expression for the cross-sectional roller profile curve is:

[0183]

[0184] Step 3: Select the function type for the periodic asymmetric curve;

[0185] Based on a flat roller with a radius of 60 mm, the periodic function f(t) = 1.5cos(150t) is selected as the function type of the periodic asymmetric curve, and the radial dimension function is r = 60 + 1.5cos(150t), where the position of r = 60 is called the index line.

[0186] Step 4: Set the period of the radial dimension function for the front and rear waist sections;

[0187] Will As a function of the radial dimension of the front waist section, where The phase difference of the radial dimension function of the front waistline is used to calculate the period of the radial dimension function of the front waistline. Let α1 be the projected length of the rear waistline on the index line, and α2 be the projected length of the front waistline on the index line. Assume α2 / α1 = ω > 1. The period of the radial dimension function of the rear waistline is set as...

[0188] Step 5: Determine the proportional relationship between α1 and α2;

[0189] Calculate the period of the cross-sectional roll profile curve The number of cycles for obtaining the cross-sectional roller profile curve For ease of geometric modeling and roll processing, the value of ω is chosen to ensure that the number of cycles is a positive integer. To obtain the value of ω, we select ω = 2 and the number of cycles n = 200.

[0190] Step 6: Obtain the radial dimension function of the cross-sectional roller profile curve;

[0191] Based on step 4, the radial dimension function of the rear waist segment can be written as... in The phase difference of the radial dimension function of the rear waist section is used to obtain the expression for the radial dimension function of the cross-sectional roll profile curve:

[0192]

[0193] Step 7: Determine the connection point t0 between the front and back waist segments;

[0194] As shown in step 4, the cross-sectional roller profile curve consists of 200 front waist segments and 200 rear waist segments, and has a total of 400 connection points. The position of connection point t0 can be divided into two cases:

[0195] (1)

[0196] (2)

[0197]

[0198] Step 8: Calculate the phase difference of the anterior waist segment Phase difference with the lower back

[0199] according to Calculate the phase difference of the anterior midline using mathematical induction. Phase difference with the lower back As shown in the following formula:

[0200]

[0201]

[0202]

[0203] Step 9: Obtain the parametric equation expression for the cross-sectional roll profile curve;

[0204] Based on the above steps, the parametric equations of the roller profile with a periodic asymmetric function curve in cross-section are obtained, consisting of the front waist section function and the rear waist section function:

[0205] The function of the anterior waist segment is

[0206] in,

[0207] The function of the posterior lumbar segment is

[0208] in

[0209] To verify that the roller curves, whose cross-sections are periodic asymmetric functions, are connected tangentially at the connection points with a smooth transition, the following operations are performed:

[0210] Calculate the first derivative k1 of the function in the anterior segment:

[0211]

[0212] make but Similarly, let Calculate the first derivative k2 of the posterior segment. Then we have:

[0213]

[0214] At the connection point: That is, β = β′ = 0, so

[0215]

[0216] Right now The first derivatives of the cross-sectional roller curves are equal and continuous.

[0217] Derivative of the function of the anterior segment Because |1.5| << 60, Therefore, x′(t) 2 +y′(t) 2 =α 2 +β 2 >0. Similarly, for the function x′(t) in the latter half of the segment... 2 +y′(t) 2 =α′ 2 +β′ 2 >0.

[0218] Therefore, the x(t) and y(t) of the periodic asymmetric cosine function plane curve are continuously differentiable on [0, 2π], and x′ 2 (t)+y′ 2 Since (t)≠0, the planar curve is a smooth curve. In summary, the roller-shaped curve with a periodic asymmetric cosine function in cross-section is connected tangentially at the connection point and has a smooth transition.

[0219] In summary, the cross-sectional curve of the roller in this invention is a periodic asymmetric function curve, consisting of a rear waist section and a front waist section, with the curve length of the front waist section being greater than that of the rear waist section. This invention solves the problems of poor uniformity of interfacial bonding strength distribution and low average interfacial bonding strength across the entire composite interface when using curved rollers with periodically symmetrical cross-sections to roll layered metal composite plates. The design method is simple, easy to master and implement, highly accurate, and universally applicable, and can be extended to the design of other curved rollers.

[0220] The above are merely specific embodiments of the present invention, but the protection of the present invention is not limited thereto. Any equivalent variations or substitutions of the features of the present technical solution that can be conceived by those skilled in the art are covered within the protection scope of the present invention. The protection scope of the present invention should be determined by the scope of the claims.

Claims

1. A roller type with a cross-section of a periodic asymmetric function curve, characterized in that, The cross-sectional curve of the roll is a periodic asymmetric function curve. The cross-sectional curve includes a front waist section and a rear waist section. The front waist section is a curve that runs in the same direction as the roll rotation direction from the bottom of the roll to the top of the roll. The rear waist section is a curve that runs in the same direction as the roll rotation direction from the top of the roll to the bottom of the next roll. The curve length of the front waist section is greater than the curve length of the rear waist section. The roller design method for a cross-section that is a periodic asymmetric function curve includes the following steps: Step 1: Determine the expression of the cross-sectional roll profile curve based on the flat roll, wherein the cross-sectional roll profile curve is a periodic asymmetric curve; Step 2: Determine the radial dimension function of the cross-sectional roller curve, and calculate the connection point and phase difference between the front and rear waist sections of the cross-sectional roller curve; Step 3: Determine the parametric equation of the cross-sectional roll profile curve based on the radial dimension function of the cross-sectional roll profile curve, the connection point of the front waist section and the rear waist section, and the phase difference; In step 3, the parametric equation of the roll profile of the cross-sectional roll profile curve is composed of the front waist segment function and the rear waist segment function: The function of the anterior waist segment is: , in, ; The function of the lower back segment is: , in, ; in, Let the radius of the flat roller be denoted by ; with the center point of the cross-section of the flat roller as the origin, and the horizontal and vertical directions as denoted by . axis, Establish a rectangular coordinate system along the axes. Representative with The positive axis is the arc of rotation from the starting position in a counterclockwise direction; The phase difference of the aforementioned front waist segment. The phase difference of the posterior waist segment; n The number of cycles for the cross-sectional roller profile curve is given. The value is selected such that the number of cycles is a positive integer. Size; Let be the period of the radial dimension function of the front waist section.

2. The design method according to claim 1, characterized in that, Step 1 specifically includes: Step 11: [The following appears to be a separate, unrelated section:] ...with radius... The cross-sectional curve of the flat roller is used as the standard circle, with the center point of the flat roller's cross-section as the origin, and the horizontal and vertical directions as... axis, Establish a rectangular coordinate system with axes and define... The radial dimension function representing the cross-sectional roller profile curve, Representative with The positive axis is the arc of rotation from the starting position in a counterclockwise direction; Step 12: To each axis, By projecting the axis, the abscissa and ordinate of each point on the cross-sectional roller curve are obtained, and the expression of the cross-sectional roller curve is determined.

3. The design method according to claim 2, characterized in that, The expression for the cross-sectional roller profile curve in step 1 is: 。 4. The design method according to claim 2, characterized in that, Step 2 specifically includes: Step 21: Select the function type for the cross-sectional roller profile curve; Step 22: Set the period of the radial dimension function of the front and rear waist sections of the cross-sectional roller curve; Step 23: Determine the radial dimension function of the cross-sectional roll profile curve. ; Step 24: Determine the connection point between the front waist section and the rear waist section. And calculate the phase difference of the front waist segment. Phase difference with the posterior lumbar segment .

5. The design method according to claim 4, characterized in that, In step 21, a periodic function is selected. ( As a function type of the periodic asymmetric curve, the radial dimension function is ,in The position of the graduation line is called the scale line.

6. The design method according to claim 5, characterized in that, Step 22 specifically includes: Step 221: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require As a function of the radial dimension of the front waist section, where The period of the radial dimension function of the front waist section is calculated by using the phase difference of the radial dimension function of the front waist section. The projected length of the front waist segment on the index line is defined as... The projected length of the rear waist section on the indexing line is ,set up The period of the radial dimension function of the rear waist segment is set to ; Step 222: Calculate the period of the cross-sectional roller profile curve. Calculate the number of cycles of the cross-sectional roller curve. ,right The value is selected such that the number of cycles is a positive integer. Size.

7. The design method according to claim 6, characterized in that, Step 23 specifically includes: The radial dimension function of the rear waist segment is written as... ,in The radial dimension function of the cross-sectional roll profile is obtained by using the phase difference of the radial dimension function of the rear waist section. The expression: 。 8. The design method according to claim 7, characterized in that, Step 24 specifically includes: Step 241: The cross-sectional roller profile curve includes The aforementioned front waist section and The rear waist section, the cross-sectional roller curve has a total of 2 A connection point, the connection point for: ； Step 242: According to The phase difference of the anterior waist segment was calculated using mathematical induction. Phase difference with the posterior lumbar segment .

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