Method for designing cam curve of paper transfer mechanism of high-speed printing machine

Abstract

The invention discloses a cam curve design method for a paper transfer mechanism of a high-speed printing machine, which belongs to the field of cam design and comprises the following steps of: S1, dividing the movement of a paper transfer gripper into six sections; s2, setting known conditions; s3, designing a fourth section theta3 < = theta < = theta4, a fifth section theta4 < = theta < = theta5 and a sixth section theta5 < = theta < = theta6 by using a standard double-stop cam curve design method; s4, designing a second section of the cam curve by using a polynomial method, wherein theta1 < = theta2 < = theta1; a B spline curve design method is used to design a first section 0 < = theta < = theta 1 and a third section theta < = theta < = theta 2 < = theta < = theta 3 of a paper transfer cam curve, and the angular velocity omega ec of a paper transfer gripper during constant-velocity handover is calculated. According to the cam curve design method, feasible design results are obtained through the polynomial design method and the B spline curve design method, on the basis that the continuity and smoothness of the jump curve can be improved through the B spline curve, the increasing amplitude of the peak acceleration and the increasing amplitude of the peak jump value are effectively controlled, and therefore the optimal design scheme is obtained.

Description

Technical Field

[0001] This invention relates to a cam curve design method, and more particularly to a cam curve design method for a paper feeding mechanism of a high-speed printing press. Background Technology

[0002] The paper feeding mechanism is the most crucial component of the paper feeding section of a printing press. Its structural design, manufacturing precision, and kinematic and dynamic characteristics directly affect many aspects of the printing press's performance, such as printing speed, registration accuracy, and print quality. The downward-swinging paper feeding mechanism is the most mainstream configuration in modern high-speed printing presses, and its actuators are also known as paper grippers. Figure 1 The diagram shows the structure of a high-speed sheet-fed offset printing press with two color units. During printing, the paper stack automatically rises intermittently until the feeder can separate the paper and transport it to the feed table. After the paper is positioned on the feed table by the front and side guides, the oscillating grippers hold the leading edge of the paper and accelerate it to the transfer zone, where it is handed over to the feed cylinder grippers at the same circumferential speed as the transfer cylinder. The paper then passes through the gripping system between the first impression cylinder, the feed cylinder, the second impression cylinder, and the rear feed cylinder, and is transported forward, completing the two-color printing process in the process.

[0003] Figure 2 The cam-linkage mechanism controlled by the conjugate cam shown is a simplified structural diagram of the drive mechanism of a high-speed pendulum paper feeding mechanism. The paper feed teeth are not depicted in the diagram; in actual operation, multiple paper feed teeth are symmetrically fixed on the paper feed shaft 3, meaning they share the same motion law as the paper feed tooth shaft. The motion of the paper feed tooth shaft is controlled by the conjugate cam 1 and its linkage mechanism. The motion design of the paper feed teeth, i.e. Figure 2 The cam curve design of the driving cam shown is the most important part of the paper feeding mechanism design, and the printing process has strict motion requirements for each of its motion stages. Summary of the Invention

[0004] To optimize existing cam curve design methods, this invention provides a cam curve design method for a high-speed printing press paper feeding mechanism, comprising the following steps:

[0005] S1, the movement of the paper-passing teeth is divided into 6 segments, namely: segment 1 0≤θ≤θ1, segment 2 θ1≤θ≤θ2, segment 3 θ2≤θ≤θ3, segment 4 θ3≤θ≤θ4, segment 5 θ4≤θ≤θ5, segment 6 θ5≤θ≤θ6;

[0006] S2, set known conditions: printing speed of the printing press, cam cycle T, and radius R of the paper feed roller. T The length R of the swing arm of the oscillating paper feeder g The maximum angular displacement β of the oscillating paper feedermax And the cornering range of the oscillating paper feeder;

[0007] S3: Design the 4th segment θ3≤θ≤θ4, the 5th segment θ4≤θ≤θ5, and the 6th segment θ5≤θ≤θ6 using the standard double-stop cam curve design method;

[0008] S4: Design the second segment of the cam curve using the polynomial method; design the first segment (0≤θ≤θ1) and the third segment (θ2≤θ≤θ3) of the paper-feeding cam curve using the B-spline curve design method, and calculate the angular velocity ω of the paper-feeding teeth during constant-speed handover. ec .

[0009] Furthermore, in step S4, the polynomial method is used to design the expressions for segments 1 to 3 of the cam curve as follows:

[0010]

[0011] Further, in step S1, θ1 = 60°, θ2 = 74°, θ3 = 139°, θ4 = 169°, θ5 = 320°, and θ6 = 360°.

[0012] Furthermore, in step S4, the boundary conditions for the first segment of the paper-feeding cam curve are designed using the B-spline curve design method as follows:

[0013]

[0014] Furthermore, in step S4, the boundary conditions for the third segment of the paper-feeding cam curve are designed using the B-spline curve design method as follows:

[0015]

[0016] Furthermore, the internal nodes of the B-spline curves in segments 1 and 3 are set as follows:

[0017] T1 = [0, 6, 55, 60]

[0018] T2 = [74, 78, 85, 139].

[0019] In summary, the present invention has the following advantages over the prior art:

[0020] The cam curve design method provided by this invention has achieved feasible design results through both polynomial and B-spline curve design methods. By using B-spline curves, the continuity and smoothness of the jump curve can be improved, and the increase in peak acceleration and peak jump value can be effectively controlled, thereby obtaining the optimal design scheme. Attached Figure Description

[0021] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this invention, illustrate exemplary embodiments of the invention and are used to explain the invention, but do not constitute an undue limitation of the invention. In the drawings:

[0022] Figure 1 This is a schematic diagram of a high-speed sheet-fed offset printing press.

[0023] Figure 2 This is a schematic diagram of a swing-down paper delivery mechanism;

[0024] Figure 3 This is a schematic diagram of the paper-passing teeth movement;

[0025] Figure 4 Design diagram of the first segment of the paper-feeding cam curve using a polynomial method;

[0026] Figure 5 The polynomial design diagram for the second segment of the paper-feeding cam curve;

[0027] Figure 6 The third polynomial design diagram of the paper-feeding cam curve;

[0028] Figure 7 Design diagram of paper-feeding cam curve based on polynomial method;

[0029] Figure 8 The paper delivery cam curve after adding jump constraints;

[0030] Figure 9 Design the first segment of the paper-feeding cam curve using the B-spline method;

[0031] Figure 10 Design diagram of the B-spline method for the third segment of the paper-feeding cam curve;

[0032] Figure 11 Design diagram of the paper-feeding cam curve for the B-spline method;

[0033] Figure 12 Design drawing of B-spline paper delivery cam curve to increase jump constraint.

[0034] The above figures include the following reference numerals:

[0035] 1. Paper stack; 2. Paper stack chain; 3. Gripper shaft; 4. First impression cylinder; 5. Paper feed cylinder; 6. Second impression cylinder; 7. Oscillating paper feeder; 8. Paper stack; 9. Feeder; 10. Drive cylinder; 11. Printing unit; 12. Plate cylinder; 13. Rubber cylinder; 14. Conjugate cam; 15. Active swing arm; 16. Paper feeder shaft; 17. Spring; 18. Wall panel; 19. Driven swing arm. Detailed Implementation

[0036] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other. The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0037] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments of the present invention. As used herein, the singular form may also include the plural form unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0038] Unless otherwise specifically stated, the relative arrangement, numerical expressions, and values ​​of the components and steps set forth in these embodiments do not limit the scope of the invention. It should also be understood that, for ease of description, the dimensions of the various parts shown in the drawings are not drawn to actual scale. Techniques, methods, and devices known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and devices should be considered part of the specification. In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar reference numerals and letters in the following figures denote similar items; therefore, once an item is defined in one figure, it need not be further discussed in subsequent figures.

[0039] See Figures 1 to 12 As shown, this invention provides a method for designing the cam curve of a high-speed printing press paper feeding mechanism, including the following steps:

[0040] S1, the movement of the paper-passing teeth is divided into 6 segments, namely: segment 1 0≤θ≤θ1, segment 2 θ1≤θ≤θ2, segment 3 θ2≤θ≤θ3, segment 4 θ3≤θ≤θ4, segment 5 θ4≤θ≤θ5, segment 6 θ5≤θ≤θ6;

[0041] S2, set known conditions: printing speed of the printing press, cam cycle T, and radius R of the paper feed roller. T The length R of the swing arm of the oscillating paper feeder g The maximum angular displacement β of the oscillating paper feeder max And the cornering range of the oscillating paper feeder;

[0042] S3: Design the 4th segment θ3≤θ≤θ4, the 5th segment θ4≤θ≤θ5, and the 6th segment θ5≤θ≤θ6 using the standard double-stop cam curve design method;

[0043] S4: Design the second segment of the cam curve using the polynomial method; design the first segment (0≤θ≤θ1) and the third segment (θ2≤θ≤θ3) of the paper-feeding cam curve using the B-spline curve design method, and calculate the angular velocity ω of the paper-feeding teeth during constant-speed handover. ec .

[0044] As a preferred option, the polynomial method is used to design the expressions for segments 1 to 3 of the cam curve as follows:

[0045]

[0046] As a preferred option, the boundary conditions for the first segment of the paper-feeding cam curve are designed using the B-spline curve design method as follows:

[0047] θ=0°β g =0,ω g =0,ε g =0

[0048] θ=60°β g =0.41888rad,ω g =27.318 rad / s, ε g =0

[0049] As a preferred option, the boundary conditions for the third segment of the paper-feeding cam curve are designed using the B-spline curve design method as follows:

[0050] θ=74°β g =0.63135 rad, ω g =27.318 rad / s, ε g =0

[0051] θ=139°β g = 1.0472 rad, ω g =0,ε g =0

[0052] Example:

[0053] To ensure print quality and high registration accuracy, each sheet of paper is sequentially and statically positioned on the feed table during printing, guided by the front and side guides. The main task of the pendulum gripper is: to remain stationary in front of the feed table, where the opening and closing gripper mechanism controls its teeth to open and grip the leading edge of the paper; to swing the paper, gradually accelerating it until its speed matches the linear speed of the transfer cylinder; to maintain this speed for a certain period of time, and at the junction, the opening and closing gripper mechanism controls the teeth to open to complete the uniform speed transfer of the paper to the transfer cylinder. During the transfer, the paper is gripped by the teeth of the transfer cylinder and transferred to the impression cylinder to enter the printing unit; the gripper continues to swing to its farthest point and pauses for a certain period of time; the gripper swings back to the feed table to wait for the next sheet of paper; taking a printing speed of 15,000 sheets / hour as an example, its cycle time is 0.24 seconds.

[0054] like Figure 3 As shown, θ is the camshaft rotation angle, and β is the angular displacement of the paper feeder. Based on the above description, the motion of the paper feeder is divided into the following 6 segments:

[0055] In the first segment, 0≤θ≤θ1, the paper feeder bites the leading edge of the paper and accelerates its swing. When it reaches the end of this segment, the linear velocity of its swing is the same as the circumferential linear velocity of the paper feed roller.

[0056] In the second segment, θ1≤θ≤θ2, this segment is a constant speed swing segment. Initially, the paper is still being gripped by the paper feeder. After passing the transfer point C and completing the constant speed transfer, the paper is controlled and driven by the paper feeder roller. The paper feeder without paper continues to swing at a constant speed to the end of this segment.

[0057] In the third segment, θ2≤θ≤θ3, the paper feeder decelerates and swings to the position furthest from the paper feed table. At the end of this segment, its velocity is 0, and its angular displacement reaches its maximum value β. max ;

[0058] In the fourth segment, θ3≤θ≤θ4, the paper feeder remains stationary at the position furthest from the paper feed table.

[0059] In the 5th segment, θ4≤θ≤θ5, the paper feeder swings back to the paper feeder platform. When it reaches the end of this segment, its displacement, velocity, and acceleration are all 0.

[0060] In the 6th segment, θ5≤θ≤θ6, the paper feeder stops still in front of the paper feed table. After the vibration is eliminated, the opening and closing mechanism controls the opening of the feeder and bites the leading edge of the second paper.

[0061] The setting of each angle interval described above is subject to strict limitations and requirements, and must be carefully designed based on the specific printing equipment's process conditions. Analysis of each motion segment reveals that the design of the first three segments is relatively complex. Specifically, segment 2 is a constant-speed motion segment, requiring a specific displacement of the paper feed gripper at the junction. Segments 1 and 3 connect one end of the constant-speed segment and one end of the rest segment, respectively, and cannot be designed using a standard double-stop cam curve; instead, polynomial or spline curve methods can be employed. Segment 5 is the return segment, which can be designed using a standard double-stop cam curve.

[0062] Assume the following known conditions: the printing speed of the printing press is 18,000 sheets / hour, meaning the cam's cycle time is T = 0.2 seconds. The radius of the paper feed roller is R. T =145mm, the length of the paper feed gripper's swing arm is R g =166.75mm, the maximum angular displacement of the paper feed gripper is 60°. Other relevant boundary conditions are shown in Table 1:

[0063] Table 1 Boundary conditions for the paper-passing motion

[0064]

[0065]

[0066] The parameter ω in Table 1 ec This represents the angular velocity of the paper feed gripper during constant-speed handover, and it needs to be calculated based on the process conditions. In this example, the length of the paper feed gripper swing arm is R. g =166.75mm, the radius of the paper feed roller is R T =145mm, and the printing speed is 18,000 sheets / hour, then:

[0067]

[0068] Based on the printing speed, the angular velocity of the camshaft is:

[0069]

[0070] From Table 1, assume the paper feed tooth angle displacement at the junction is β. c =30°, to make it approximately in the middle of the constant speed zone, let the interval length of the third constant speed zone be 14°, and the angular displacement of the paper feeder at the starting point be 24°, then its angular displacement at the end point of this segment is:

[0071] β g =ω ec ×14×0.2 / (2π)+24=36.174° (3)

[0072] Based on the above-mentioned known conditions, this method will design and analyze the cam curve of this high-speed paper delivery mechanism using both polynomial and B-spline methods.

[0073] II. Solving using the polynomial method

[0074] In this section, the first three segments of the cam curve will be designed using a polynomial method. The fifth segment will use a standard 4-5-6-7 degree polynomial to ensure the continuity of its jump curve with the adjacent two resting segments at the start and end points of the motion.

[0075] Based on the boundary conditions given in Table 1, we can convert them into standard polynomial solution boundary conditions using the general form of polynomial curves:

[0076] y = C0 + C1x + C2x 2 +...+C n x n (4)

[0077] In the formula: y represents dimensionless displacement; x represents dimensionless time; C i , i = 0, 1, 2, ..., n — representing the coefficients of each term in the polynomial.

[0078] In equation (4), the polynomial coefficients C i The values ​​of i = 0, 1, 2, ..., n are unknown and need to be determined based on the specific design conditions. The degree of a polynomial is determined by its highest power term; that is, an nth-degree polynomial will have n+1 terms. Since there is also a constant term with coefficient C0, there are a total of n+1 coefficients that need to be determined. Usually, the number of terms in a polynomial is called its order k; that is, the order of an nth-degree polynomial is k = n+1.

[0079] When using polynomials in cam curve design, the polynomial is constructed based on the number of specified boundary conditions in the s,v,a,j curve. The coefficients of the corresponding polynomial are obtained by solving a system of linear equations. For example, if there are k specified boundary conditions, a polynomial of order k and degree k-1 needs to be constructed.

[0080] In this embodiment, the paper feeder displacement is an angular displacement. For ease of description, the cam curve is expressed in a dimensionless form; that is, the parameters in the curve equation are normalized, as shown in equation (5):

[0081]

[0082] In the formula: β—the motion angle of the follower during a certain stroke (push, pause, or return), ° or rad; h—the displacement of the follower during that stroke, mm; x—the dimensionless time of the follower during that stroke; y—the dimensionless displacement of the follower; y'—the dimensionless velocity of the follower; y"—the dimensionless acceleration of the follower; y'"—the dimensionless jerk of the follower; s—the displacement of the follower, mm; θ—the rotation angle of the camshaft, °.

[0083] In this embodiment, it is only necessary to set h = 1 and s = β in equation (5). g Then we have y = β g The remaining formulas and transformation relationships remain unchanged, and the calculation results are shown in Table 2.

[0084] Table 2 sets the boundary conditions for the first three polynomial curves.

[0085]

[0086] Table 2 shows the boundary conditions of the first three polynomials after conversion. The angular displacement of the paper feeder is expressed in radians. For easy verification during plotting, it can be converted to angles.

[0087] Figure 4 , Figure 5 and Figure 6 The design of the polynomials for each segment is shown. Segment 5 uses a standard 4-5-6-7 degree polynomial. Figure 7 This is a diagram showing the complete design results of the paper-feeding cam curve.

[0088] Figure 7 The final design of the high-speed paper-feeding mechanism cam curve is shown. The curves are smooth, with no overshoot, meeting the basic requirements for the paper-feeding gripper motion. Table 3 lists the basic characteristics of this design. Because the jump values ​​at the endpoints were not limited in the design of the first and third polynomial segments, abrupt changes occur at both ends of the second rest segment, resulting in poor smoothness and potential vibration under high-speed operation. This leads to machine noise, affects print quality, and reduces registration accuracy. The solution is to set the jump values ​​at both endpoints of the first and third polynomial segments to 0, based on the original boundary conditions.

[0089] Table 3. Basic characteristic information of the paper delivery cam curve based on the polynomial method.

[0090]

[0091] Figure 8Tables 1 and 3 show the design results and corresponding basic characteristic information after adding the jump boundary constraint. Clearly, adding the boundary condition significantly improves the smoothness and continuity of the jump curve and reduces the peak value of the negative jump. However, a side effect is that the peak acceleration increases by about 25% compared to the original design, increasing the contact force between the cam and the follower rollers and exacerbating wear. Equations (6) and (7) respectively give the polynomial displacement expressions for the first three segments obtained from the two designs. As shown in Equation (7), after adding the jump constraint in the boundary condition, the first and third segments both change from the original 5th-order polynomials to 7th-order polynomials, resulting in an increase in peak acceleration. Therefore, which of the two design results is more desirable depends on the printing speed and further dynamic analysis. If the distribution of the motion intervals of each segment is appropriately adjusted within the allowable range of the process conditions, the peak acceleration can be reduced, but to a limited extent.

[0092]

[0093]

[0094] Table 4. Basic characteristics of the paper delivery cam curve after adding jump constraints.

[0095]

[0096] II. Solving using the B-spline method

[0097] The first and third segments of the paper-feeding cam curve are redesigned using the B-spline curve method, with other conditions remaining unchanged. The design results are then compared and analyzed with those obtained using the polynomial method.

[0098] First, without adding any jump constraints, according to Table 1, the boundary conditions for the first segment of the B-spline curve should be:

[0099]

[0100] As shown in equation (8), there are a total of 6 boundary conditions in this section. If a B-spline curve of order k=6 is used for design, then no internal nodes need to be set. Figure 9 The design results are displayed. The relevant parameters are set as follows:

[0101]

[0102] According to Table 1, the boundary conditions for the third segment of the B-spline curve should be:

[0103]

[0104] As shown in equation (10), this segment also has 6 boundary conditions. If a B-spline curve of order k=6 is used for design, then no internal nodes need to be set. Figure 10The design results are displayed. The relevant parameters are set as follows:

[0105]

[0106] like Figure 11 As shown, using the same boundary conditions, the first and third segments are designed using a 6th-order B-spline curve. The fifth segment still uses a 4th-5th-6th-7th degree polynomial. The resulting paper-feeding cam curve is similar to... Figure 7 The results are exactly the same. In fact, since both segments 1 and 3 have 6 boundary conditions, when designing using a 6th-order B-spline curve, it is not necessary to design internal nodes. Therefore, the result is still two B-spline curves, each composed of a single-segment polynomial. This example also illustrates that, with appropriate settings, B-spline curves can generate arbitrary single-segment polynomials. Or, it can be said that a polynomial is also a special type of B-spline curve.

[0107] So, to increase the smoothness and continuity of the jump curve, what design results would we get by adding control over the jump values ​​at the start and end points in the boundary constraints of the two B-spline curve segments? If we set the jump values ​​at the start and end points of the first and third curve segments to 0, then each segment would have 8 constraints. Still using a B-spline curve of order k=6 for design, two internal nodes would be required for each segment. It is precisely because we can freely choose the position of the internal nodes during design that we have more active control and adjustment capabilities over the resulting cam curve. After several minutes of adjustment using interactive design software, the internal nodes of the first and third B-spline curve segments were set as follows:

[0108]

[0109] Figure 12 Table 5 presents a comparison of the basic characteristic information of the cam curves obtained under the two design methods to achieve the final design results. Clearly, the B-spline curve design method, while improving the smoothness and continuity of the jump curve, significantly controls the increase in peak acceleration. Its peak acceleration is reduced by approximately 15% compared to the polynomial method with jump constraints, and the positive peak jump is also reduced by approximately 23%. The drawback is a slight increase in negative jump.

[0110] In fact, after adding boundary conditions, the spline curve obtained by setting internal nodes is actually an automatic splicing of 3 segments of a 5th degree polynomial. Therefore, after adding boundary conditions, the degree of the constituent polynomial of the B-spline curve does not increase, but the number of segments increases.

[0111] Figure 12The results shown can be considered the optimal design scheme for this high-speed paper-feeding cam curve. Feasible design results were obtained through both polynomial and B-spline curve design methods. By using B-spline curves, the increase in peak acceleration and peak jump value can be effectively controlled while improving the continuity and smoothness of the jump curve, thus obtaining the optimal design scheme.

[0112] Table 5 Comparison of Cam Curve Design Results between Polynomial Method and B-Spline Method

[0113]

[0114] The above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for designing the cam curve of a paper feeding mechanism in a high-speed printing press, characterized in that, Including the following steps: S1, the movement of the paper-passing teeth is divided into 6 segments, namely: segment 1 0≤θ≤θ1, segment 2 θ1≤θ≤θ2, segment 3 θ2≤θ≤θ3, segment 4 θ3≤θ≤θ4, segment 5 θ4≤θ≤θ5, segment 6 θ5≤θ≤θ6; S2, set known conditions: printing speed of the printing press, cam cycle T, and radius R of the paper feed roller. T The length R of the swing arm of the oscillating paper feeder g The maximum angular displacement β of the oscillating paper feeder max And the cornering range of the oscillating paper feeder; S3: Design the 4th segment θ3≤θ≤θ4, the 5th segment θ4≤θ≤θ5, and the 6th segment θ5≤θ≤θ6 using the standard double-stop cam curve design method; S4: Design the second segment θ1≤θ≤θ2 of the cam curve using the polynomial method; design the first segment 0≤θ≤θ1 and the third segment θ2≤θ≤θ3 of the paper-feeding cam curve using the B-spline curve design method, and calculate the angular velocity ω of the paper-feeding teeth during constant-speed handover. ec .

2. The cam curve design method for the paper feeding mechanism of a high-speed printing press according to claim 1, characterized in that, In step S4, the polynomial method is used to design the expressions for segments 1 to 3 of the cam curve as follows:

3. The cam curve design method for the paper feeding mechanism of a high-speed printing press according to claim 1, characterized in that, In step S1, θ1 = 60°, θ2 = 74°, θ3 = 139°, θ4 = 169°, θ5 = 320°, and θ6 = 360°.

4. The cam curve design method for the paper feeding mechanism of a high-speed printing press according to claim 3, characterized in that, In step S4, the boundary conditions for the first segment of the paper-feeding cam curve are designed using the B-spline curve design method as follows:

5. The cam curve design method for the paper feeding mechanism of a high-speed printing press according to claim 3, characterized in that, In step S4, the boundary conditions for the third segment of the paper-feeding cam curve are designed using the B-spline curve design method:

6. The cam curve design method for the paper feeding mechanism of a high-speed printing press according to claim 3, characterized in that, The internal nodes of the B-spline curves in segments 1 and 3 are set as follows: T1＝[0,6,55,60] T2＝[74,78,85,139]。

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