CN102366895B - Vertical sliding plate in horizontal machining center and vertical sliding plate structural optimization method - Google Patents

Vertical sliding plate in horizontal machining center and vertical sliding plate structural optimization method Download PDF

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CN102366895B
CN102366895B CN 201110307644 CN201110307644A CN102366895B CN 102366895 B CN102366895 B CN 102366895B CN 201110307644 CN201110307644 CN 201110307644 CN 201110307644 A CN201110307644 A CN 201110307644A CN 102366895 B CN102366895 B CN 102366895B
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slide plate
value
upright slide
vertical
upright
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CN102366895A (en
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张辉
郭垒
叶佩青
赵彤
刘倚瑞
祝莉平
甘青
凌卫国
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TONTEC TECHNOLOGY INVESTMENT GROUP Co Ltd
Tsinghua University
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TONTEC TECHNOLOGY INVESTMENT GROUP Co Ltd
Tsinghua University
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Abstract

The invention relates to a vertical sliding plate in a horizontal machining center and a vertical sliding plate structural optimization method. The invention is characterized in that the vertical sliding plate comprises two vertical beams and two transverse beams which are integrated into a whole, the two vertical beams are respectively provided with a vertical guide rail, each vertical guide rail is provided with two vertical sliding blocks arranged at intervals, four vertical sliding blocks are connected with a main spindle box; the two vertical beams of the vertical sliding plate are in a ladder shape, and the two vertical beams are hollow. The method comprises the following steps of: 1) setting a rigidity value required to be met by the vertical sliding plate, solving corresponding deflection Omegas and a rotating angle Thetas; 2) taking the deflection Omegas and the rotating angle Thetas as constraint values, traversing all size combinations in numeric areas of Hv, Hp and k to find all the optimization size values meeting rigidity conditions and corresponding vertical sliding plate masses; and 3) selecting and outputting the minimum mass of the vertical sliding plate and corresponding structural size to finish structural optimization of the vertical sliding plate. By applying the vertical sliding plate and the vertical sliding plate structural optimization method provided by the invention, the mass of the vertical sliding plate can be reduced on the basis that rigidity is met, thus the vertical sliding plate and the vertical sliding plate structural optimization method can be widely applied to the design process of a horizontal machining center.

Description

A kind of upright slide plate of horizontal Machining centers and upright skateboard optimization method
Technical field
The present invention relates to a kind of upright slide plate of horizontal Machining centers, particularly the upright slide plate about a kind of horizontal Machining centers reaches upright skateboard optimization method.
Background technology
Digit Control Machine Tool is manufacturing basis, and is significant to development and national economy.In order to enhance competitiveness, require the moving-member of lathe is done light-weight design on the basis of satisfying rigidity.But the machine tool structure of China design at present also is based on the design of experience, to force analysis, and the rigidity check, do not have too many research, therefore, from the angle analysis of rigidity, the setting of machine tool structure is still reasonable inadequately at present.Case raising middle flask version is the trend of horizontal Machining centers, and wherein Zui Da moving-member is upright slide plate.The physical dimension of opposition slide plate is optimized design, alleviates quality on the basis of satisfying rigidity, and very big for machine performance raising influence, still, there be limited evidence currently of has the research of related optimization.
Summary of the invention
At the problems referred to above, the purpose of this invention is to provide and a kind ofly can satisfy on the basis of rigidity, alleviate upright slide plate and the upright skateboard optimization method of the horizontal Machining centers of upright slide plate quality.
For achieving the above object, the present invention takes following technical scheme: a kind of upright slide plate of horizontal Machining centers, it is characterized in that: it comprises two vertical beam and two transverse beams that are wholely set, be respectively arranged with a vertical guide on described two vertical beam, be arranged at intervals with two vertical slide blocks on every described vertical guide respectively, four described vertical slide blocks connect main spindle box; Described two vertical beam of described upright slide plate are trapezoidal, and the inner hollow of described two vertical beam.
The structural optimization method of the upright slide plate of above-mentioned a kind of horizontal Machining centers, it may further comprise the steps: 1) goal-setting module, traversal computing module and evaluation of result module are set; 2) in the goal-setting module, according to operating mode, set the rigidity value that upright slide plate need satisfy, rigidity is cutting force Fz and amount of deflection ω sRatio, the goal-setting module can be obtained amount of deflection ω according to known rigidity value and cutting force Fz s, can obtain corresponding to amount of deflection ω according to geometrical relationship again sThe rotational angle theta of upright slide plate s3) the amount of deflection ω that obtains input step 2 in the traversal computing module) sAnd rotational angle theta sAs binding occurrence; Upright slide plate quality m and H v, H p, k, l is relevant, wherein, H vThe thickness of the upright slide plate transverse beam of expression, H pThe thickness at the upright slide plate vertical beam of expression middle part, l represents the upper base edge lengths of the upright trapezoidal projection of slide plate vertical beam; K represents that the hypotenuse of the upright trapezoidal projection of slide plate vertical beam is in the vertical projected length of upright slide plate bottom; The traversal computing module in H v, H p, k is as optimizing size, and according to matching requirements, sets H v, H p, the span of k; At H v, H p, in the span of k, travel through all size combinations, and calculate the amount of deflection ω of upright slide plate under cutting force Fz effect corresponding to each size combinations iAnd rotational angle theta iIf, ω i≤ ω sAnd θ i≤ θ s, then calculate the quality of the upright slide plate of this moment, and preserve the optimization size value of this moment and the quality of upright slide plate; 4) quality with all size values that satisfy rigidity condition of obtaining in the step 3) and corresponding upright slide plate inputs to the evaluation of result module, the evaluation of result module is drawn out the graph of a relation of quality and the correspondingly-sized value of all upright slide plates, and choose, export the minimum mass of upright slide plate and the physical dimension of correspondence, the structure optimization of upright slide plate finishes.
Described step 2) and in the step 3), the computational methods of asking for upright slide plate amount of deflection ω and rotational angle theta are as follows: the moment of inertia I of upright slide plate vertical beam xFor:
I x = BH 3 - bh 3 12
In the formula, B represents that width, the b of vertical beam lateral cross section periphery represent that width, the H of vertical beam lateral cross section inner hollow represent that height, the h of vertical beam lateral cross section periphery represent the height of vertical beam lateral cross section inner hollow; Upright slide plate is reduced to along two simply supported beams of vertical beam direction is formed in parallel, have two vertical slide blocks that it is applied power on every simply supported beam; The deflection curve differential equation of upright slide plate is:
ω ′ ′ = M ( y ) EI x ( y )
In the formula, M (y) founds slide plate with Y direction, i.e. the moment of flexure of vertical beam direction variation, and E is the elastic modelling quantity of upright slide plate, I x(y) be the moment of inertia that vertical beam changes along Y direction; The expression formula of M (y) is:
M ( y ) = Fz &CenterDot; y / 4 0 &le; y &le; C Fz &CenterDot; C / 4 C < y &le; C + D Fz &CenterDot; ( 2 C + D - y ) / 4 C + D < y &le; 2 C + D
Wherein, D represents two spacings between the vertical slide block, and C represents two vertical slide blocks distance of corresponding and upright slide plate phase proximal end respectively; Substitution I in the deflection curve differential equation of upright slide plate xExpression formula, and along the length direction of upright slide plate vertical beam, y is carried out integration, try to achieve amount of deflection ω and the rotational angle theta of upright slide plate.
In the described step 3), upright all size combinations of slide plate of traversal adopt following method: at H v, H p, in the span of k, set H v, H p, the currency of k is minimum of a value, progressively increases H pValue; If H pBe not increased to maximum, then continue to increase H pValue; If H pValue be increased to maximum, then finished H pOnce traversal, this moment the value of k is made judgement; If the value of k is not maximum, then increase the value of k, give H pValue be minimum of a value, carry out H again pTraversal; If the value of k is maximum, then finished H pWith the once traversal of k, this moment is to H vValue make judgement; If H vValue be not maximum, then increase H vValue, give H pWith the value of k be minimum of a value, carry out H again pTraversal with k; If H vValue be maximum, then the traversal of all sizes finishes.
The present invention is owing to take above technical scheme, it has the following advantages: 1, the present invention is because when the opposition skateboard is optimized, at first determine the rigidity that upright slide plate should satisfy, and obtain amount of deflection and corner value under the cutting force effect when guaranteeing this rigidity, then in the amount of deflection and corner that guarantee upright slide plate, get the minimum of a value of upright slide plate quality, adopt the thought of this kind lightweight optimization, can make upright slide plate satisfy quality minimum under the situation of rigidity, thereby can improve the dynamic property of lathe.2, the present invention's all size combinations of traversal in the determinative optimization parameter of opposition slide plate quality, and calculating is corresponding to amount of deflection and the corner of upright slide plate under the cutting force effect of each size combinations, when guaranteeing amount of deflection and corner, calculate the quality of corresponding upright slide plate, in the quality of all sizes that satisfy rigidity condition and corresponding upright slide plate, therefore the minimum mass of the upright slide plate of output, is optimized the result and is satisfied the minimum possible value of rigidity situation.The present invention can satisfy on the basis of rigidity, alleviates the quality of upright slide plate, therefore, can be widely used in the design process of horizontal Machining centers.
Description of drawings
Fig. 1 is prior art machine tool structure schematic diagram
Fig. 2 is the upright skateboard schematic diagram of the present invention
Fig. 3 is operational module schematic diagram of the present invention
Fig. 4 is operational module specific embodiment schematic diagram of the present invention
Fig. 5 is the upright slide plate vertical beam schematic cross-section of the present invention
Fig. 6 is the upright stressed constraint schematic diagram of slide plate of the present invention
Fig. 7 is the upright slide plate moment of flexure schematic diagram of the present invention
Fig. 8 is the upright skateboard dimensional parameters schematic diagram of the present invention
Fig. 9 is the dimension volume graph of a relation of the neutral slide plate of the specific embodiment of the invention
The specific embodiment
Below in conjunction with drawings and Examples the present invention is described in detail.
As shown in Figure 1 and Figure 2, horizontal machine tool of the prior art comprises a column 1, be arranged at intervals with two cross slide waies 2 on the column 1, be arranged at intervals with two transverse sliders 3 on every cross slide way 2 respectively, four transverse sliders 3 connect upright slide plate 4 of the present invention, and upright slide plate 4 slides along two cross slide waies 2 on the column 1 by four transverse sliders 3.Upright slide plate 4 of the present invention comprises two vertical beam 41 and two transverse beams 42 that are wholely set, article two, be respectively arranged with on the vertical beam 41 one with column 1 on the perpendicular vertical guide 5 of two cross slide waies 2, be arranged at intervals with two vertical slide blocks 6 on every vertical guide 5 respectively, four vertical slide blocks 6 connect main spindle box 7, and main spindle box 7 slides along two vertical guide 5 on the upright slide plate 4 by four vertical slide blocks 6.Two vertical beam 41 of upright slide plate 4 are trapezoidal, and the inner hollow of two vertical beam 41, and the one side of two vertical beam 41 trapezoidal projectioies is arranged between two cross slide waies 2.
The present invention's slide plate 4 that opposes carries out structure optimization based on following thought: as shown in Figure 1 and Figure 2, can be subjected to the power of X, Y, three directions of Z in the machine tooling process.The present invention is that example is elaborated with the power that lathe is subjected to the Z direction.Main spindle box 7 be positioned on the upright slide plate 4 the position not simultaneously, under the identical power effect, the distortion that upright slide plate 4 produces is different.The present invention analyzes the maximum duty of upright slide plate 4 distortion, and namely main spindle box 7 is positioned at the situation when standing slide plate 4 middle parts.Main spindle box 7 is subjected to cutting force Fz in the Z direction, and cutting force Fz is delivered on the upright slide plate 4 by four on two vertical guide 5 vertical slide blocks 6, makes upright slide plate 4 produce distortion.The ratio of the deflection δ of cutting force Fz and upright slide plate 4 is the rigidity of upright slide plate 4.Under identical cutting force Fz effect, amount of deflection and the corner of upright slide plate 4 are more little, and the rigidity of upright slide plate 4 is more big.When upright skateboard is optimized, should determine the rigidity that upright slide plate 4 should satisfy earlier, and obtain the amount of deflection ω under the cutting force Fz effect when guaranteeing this rigidity sAnd rotational angle theta sValue, then in the amount of deflection ω that guarantees newly-designed upright slide plate 4 and rotational angle theta respectively less than ω sAnd θ sThe time, the m minimalization.
As shown in Figure 3, Figure 4, the optimization method of upright slide plate 4 structures of the present invention may further comprise the steps.
1) goal-setting module 8, traversal computing module 8 and evaluation of result module 10 are set.
2) in goal-setting module 8, according to operating mode, setting upright slide plate 4 needs satisfied rigidity value, and rigidity is cutting force Fz and amount of deflection ω sRatio (for convenience of calculation, can set cutting force Fz is 1), goal-setting module 8 can be obtained amount of deflection ω according to known rigidity value and cutting force Fz s, can obtain corresponding to amount of deflection ω according to geometrical relationship again sThe rotational angle theta of upright slide plate 4 sAmount of deflection ω s, rotational angle theta sEmbodied the rigidity requirement of upright slide plate 4, for a kind of design of definite rigidity, amount of deflection ω s, rotational angle theta sBe constant.
The computational methods of upright slide plate 4 amount of deflection ω and rotational angle theta are as follows:
As shown in Figure 5, for founding the wherein lateral cross section A figure of a vertical beam 41 of slide plate 4, the moment of inertia I of upright slide plate 4 vertical beam 41 xFor
I x = BH 3 - bh 3 12
In the formula, B represents that width, the b of vertical beam 41 lateral cross section A peripheries represent width, the Δ of vertical beam 41 lateral cross section A inner hollow bWall thickness, the H of expression vertical beam 41 lateral cross section A widths represent that height, the h of vertical beam 41 lateral cross section A peripheries represent height, the Δ of vertical beam 41 lateral cross section A inner hollow hThe wall thickness of expression vertical beam 41 lateral cross section A short transverses.The moment of inertia I xSize determined the stiffness characteristics of upright slide plate 4.In the parameter of the geometrical property of above-mentioned expression vertical beam 41 lateral cross section A, the value of B determines Δ by the size that the functional parts of installing on the upright slide plate 4 such as lead screw guide rails take up room bAnd Δ hValue determined by casting technique, so B, b, Δ b, Δ hBe determined value; H is variable, and after H determined, the value of h can be definite fully by H.
As shown in Figure 6, upright slide plate 4 can be considered as Y-direction, and namely two simply supported beams of upright slide plate 4 vertical beam 41 directions are formed in parallel, and two simply supported beams are about the center symmetry, has two vertical slide blocks 6 that it is applied power on every simply supported beam.As shown in Figure 7, for founding the bending moment diagram of slide plate 4.The deflection curve differential equation of upright slide plate 4 is:
&omega; &prime; &prime; = M ( y ) EI x ( y )
In the formula, the moment of flexure of M (y) for changing with Y-axis on the upright slide plate 4, E is the elastic modelling quantity of upright slide plate 4, I x(y) be the moment of inertia that vertical beam 41 changes with Y-axis.The expression formula of M (y) is:
M ( y ) = Fz &CenterDot; y / 4 0 &le; y &le; C Fz &CenterDot; C / 4 C < y &le; C + D Fz &CenterDot; ( 2 C + D - y ) / 4 C + D < y &le; 2 C + D
Wherein, D represents two spacings between the vertical slide block 6, and C represents two vertical slide blocks 6 distances of corresponding and upright slide plate 4 phase proximal end respectively.
Substitution I in the deflection curve differential equation of upright slide plate 4 xExpression formula, and along the length direction of upright slide plate 4 vertical beam 41, y is carried out integration, can be in the hope of amount of deflection ω and the rotational angle theta of upright slide plate 4.
3) the amount of deflection ω that obtains input step 2 in traversal computing module 9) sAnd rotational angle theta sAs binding occurrence; As Fig. 2, shown in Figure 8, can determine according to matching requirements owing to found the total length L of slide plate 4, therefore, quality m and the H of upright slide plate 4 v, H p, k, l is relevant, wherein, H vThe thickness of upright slide plate 4 transverse beams 42 of expression, H pThe thickness at upright slide plate 4 vertical beam 41 middle parts of expression, l represents the upper base edge lengths of upright slide plate 4 vertical beam 41 trapezoidal projectioies; K represents the hypotenuse of upright slide plate 4 vertical beam 41 trapezoidal projectioies in the vertical projected length of upright slide plate 4 bottoms, and by l, the vertical beam 41 that k, k will found slide plate 4 is divided into three piecewise intervals, in each piecewise interval, and all linear variations of vertical beam 41 height dimensions; The traversal computing module 9 in H v, H p, k is as optimizing size, and according to matching requirements, sets H v, H p, the span of k, wherein H vThe installing space that depends on the leading screw that drives upright slide plate 4 motions, H pDepend on the assembly relation that founds slide plate 4 and column 1 with k.At H v, H p, in the span of k, travel through all size combinations, and calculate the amount of deflection ω of upright slide plate 4 under cutting force Fz effect corresponding to each size combinations iAnd rotational angle theta iIf, ω i≤ ω sAnd θ i≤ θ s, then calculate the quality of the upright slide plate 4 of this moment, and preserve the optimization size value of this moment and the quality of upright slide plate 4.Amount of deflection ω i, rotational angle theta iEmbodied the rigidity of the neutral slide plate 4 of traversal computational process.
The size combinations that travels through all upright slide plates 4 can adopt following method: at H v, H p, in the span of k, set H v, H p, the currency of k is minimum of a value, progressively increases H pValue; If H pBe not increased to maximum, then continue to increase H pValue; If H pValue be increased to maximum, then finished H pOnce traversal, this moment the value of k is made judgement.If the value of k is not maximum, then increase the value of k, give H pValue be minimum of a value, carry out H again pTraversal; If the value of k is maximum, then finished H pWith the once traversal of k, this moment is to H vValue make judgement.If H vValue be not maximum, then increase H vValue, give H pWith the value of k be minimum of a value, carry out H again pTraversal with k; If H vValue be maximum, then the traversal of all sizes finishes.The general round numbers of step-length of traversal.
4) quality with all sizes that satisfy rigidity condition of obtaining in the step 3) and corresponding upright slide plate 4 inputs to evaluation of result module 10, evaluation of result module 10 is drawn out the graph of a relation of quality and the correspondingly-sized of all upright slide plates 4, and choose the physical dimension of quality minimum of a value correspondence, the quality of upright slide plate 4 minimums of output and corresponding physical dimension, the structure optimization of upright slide plate 4 finishes.
Enumerate a specific embodiment below, as shown in Figure 1, the application of the present invention on the upright slide plate 4 of horizontal Machining centers.As shown in Figure 5, in the sectional dimension of upright slide plate 4 vertical beam 41, H and h are variable, and other parameters are constant, B=275mm wherein, b=240mm, Δ h=20mm, Δ b=17.5mm.As shown in Figure 6, the total length L of upright slide plate 4 vertical beam 41 directions is 1400mm, H vSpan be 100~400mm, the span of k is 100~700mm, so the value of l is 1400-2 * k.As shown in Figure 3, Figure 4,1) rigidity value is input in the goal-setting module 8, obtain corresponding deflection value ω sWith corner value θ s2) ω s, θ s, B, b, Δ h, Δ bValue and optimize Parameter H v, Hp, the span of k is input in the traversal computing module 9, to optimizing Parameter H v, Hp, k travels through, and calculates all physical dimension Hv that satisfy rigidity condition, Hp, k, and the quality of corresponding upright slide plate 4; 3) in evaluation of result module 10, import all and satisfy the Hv of rigidity condition, Hp, the quality of the upright slide plate 4 of the value of k and correspondence, volume (multiply by density and be quality m) and k and the H of the upright slide plate 4 of output vGraph of a relation (as shown in Figure 9).By among Fig. 9 as can be seen, get suitable design parameter, can make the quality minimalization of upright slide plate 4.The minimum k=440mm that this is routine, H v=250mm, Hp=376mm.
The various embodiments described above only are used for explanation the present invention, and wherein the structure of each parts, connected mode etc. all can change to some extent, and every equivalents and improvement of carrying out on the basis of technical solution of the present invention all should do not got rid of outside protection scope of the present invention.

Claims (3)

1. the upright skateboard optimization method of a horizontal Machining centers, it is characterized in that: the upright slide plate of described horizontal Machining centers comprises two vertical beam and two transverse beams that are wholely set, be respectively arranged with a vertical guide on described two vertical beam, be arranged at intervals with two vertical slide blocks on every described vertical guide respectively, four described vertical slide blocks connect main spindle box; Described two vertical beam of described upright slide plate are trapezoidal, and the inner hollow of described two vertical beam;
Described upright skateboard optimization method may further comprise the steps:
1) goal-setting module, traversal computing module and evaluation of result module are set;
2) in the goal-setting module, according to operating mode, set the rigidity value that upright slide plate need satisfy, rigidity is cutting force Fz and amount of deflection ω sRatio, the goal-setting module can be obtained amount of deflection ω according to known rigidity value and cutting force Fz s, can obtain corresponding to amount of deflection ω according to geometrical relationship again sThe rotational angle theta of upright slide plate s
3) the amount of deflection ω that obtains input step 2 in the traversal computing module) sAnd rotational angle theta sAs binding occurrence; Upright slide plate quality m and H v, H p, k, l is relevant, wherein, H vThe thickness of the upright slide plate transverse beam of expression, H pThe thickness at the upright slide plate vertical beam of expression middle part, l represents the upper base edge lengths of the upright trapezoidal projection of slide plate vertical beam; K represents that the hypotenuse of the upright trapezoidal projection of slide plate vertical beam is in the vertical projected length of upright slide plate bottom; The traversal computing module in H v, H p, k is as optimizing size, and according to matching requirements, sets H v, H p, the span of k; At H v, H p, in the span of k, travel through all size combinations, and calculate the amount of deflection ω of upright slide plate under cutting force Fz effect corresponding to each size combinations iAnd rotational angle theta iIf, ω i≤ ω sAnd θ i≤ θ s, then calculate the quality of the upright slide plate of this moment, and preserve the optimization size value of this moment and the quality of upright slide plate;
4) quality with all size values that satisfy rigidity condition of obtaining in the step 3) and corresponding upright slide plate inputs to the evaluation of result module, the evaluation of result module is drawn out the graph of a relation of quality and the correspondingly-sized value of all upright slide plates, and choose, export the minimum mass of upright slide plate and the physical dimension of correspondence, the structure optimization of upright slide plate finishes.
2. the upright skateboard optimization method of a kind of horizontal Machining centers as claimed in claim 1 is characterized in that: described step 2) and in the step 3), the computational methods of asking for upright slide plate amount of deflection ω and rotational angle theta are as follows:
The moment of inertia I of upright slide plate vertical beam xFor
I x = B H 3 - b h 3 12
In the formula, B represents that width, the b of vertical beam lateral cross section periphery represent that width, the H of vertical beam lateral cross section inner hollow represent that height, the h of vertical beam lateral cross section periphery represent the height of vertical beam lateral cross section inner hollow;
Upright slide plate is reduced to along two simply supported beams of vertical beam direction is formed in parallel, have two vertical slide blocks that it is applied power on every simply supported beam; The deflection curve differential equation of upright slide plate is:
&omega; &prime; &prime; = M ( y ) EI x ( y )
In the formula, M (y) founds slide plate with Y direction, i.e. the moment of flexure of vertical beam direction variation, and E is the elastic modelling quantity of upright slide plate, I x(y) be the moment of inertia that vertical beam changes along Y direction; The expression formula of M (y) is:
M ( y ) = Fz &CenterDot; y / 4 0 &le; y &le; C Fz &CenterDot; C / 4 C < y &le; C + D Fz &CenterDot; ( 2 C + D - y ) / 4 C + D < y &le; 2 C + D
Wherein, D represents two spacings between the vertical slide block, and C represents two vertical slide blocks distance of corresponding and upright slide plate phase proximal end respectively;
Substitution I in the deflection curve differential equation of upright slide plate xExpression formula, and along the length direction of upright slide plate vertical beam, y is carried out integration, try to achieve amount of deflection ω and the rotational angle theta of upright slide plate.
3. the upright skateboard optimization method of a kind of horizontal Machining centers as claimed in claim 1 or 2 is characterized in that: in the described step 3), travel through upright all size combinations of slide plate and adopt following method: at H v, H p, in the span of k, set H v, H p, the currency of k is minimum of a value, progressively increases H pValue; If H pBe not increased to maximum, then continue to increase H pValue; If H pValue be increased to maximum, then finished H pOnce traversal, this moment the value of k is made judgement; If the value of k is not maximum, then increase the value of k, give H pValue be minimum of a value, carry out H again pTraversal; If the value of k is maximum, then finished H pWith the once traversal of k, this moment is to H vValue make judgement; If H vValue be not maximum, then increase H vValue, give H pWith the value of k be minimum of a value, carry out H again pTraversal with k; If H vValue be maximum, then the traversal of all sizes finishes.
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