CN107045581B - Method for reducing process difficulty of theoretical gradient pitch screw rod - Google Patents

Method for reducing process difficulty of theoretical gradient pitch screw rod Download PDF

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CN107045581B
CN107045581B CN201710297745.2A CN201710297745A CN107045581B CN 107045581 B CN107045581 B CN 107045581B CN 201710297745 A CN201710297745 A CN 201710297745A CN 107045581 B CN107045581 B CN 107045581B
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reconstruction
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screw rod
array
deviation
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张雷
李杨
孔佳元
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Dongying Zhengnuo Technology Service Co ltd
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Zhejiang Sci Tech University ZSTU
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Abstract

The invention relates to a design method of screw drive, aiming at replacing the original theoretical gradual change screw rod with high equivalent of the screw rod processed by the method so as to effectively ensure the drive precision and greatly reduce the process difficulty and the processing cost. The technical scheme is as follows: a method for reducing the process difficulty of a theoretical gradient pitch screw rod comprises the following steps: step 1, designing a space parameter equation of a spiral line of a theoretical spiral rod; step 2, collecting enough 2n sampling points at equal intervals; step 3, continuously decomposing and reconstructing according to the subdivision reverse engineering rule to obtain a plurality of sets of vertex data; step 4, calculating corresponding total approximate deviation EnThen judging, step 5, calculating the ratio η of the corresponding maximum value in the matrix Q to the corresponding sumQ(ii) a Step 6, taking a sufficient number of data points to perform secondary reverse decomposition reconstruction; 7, respectively calculating the total approximation deviation E of the array B againB(ii) a Obtaining an optimal reconstruction scheme point after judgment; and 8, writing a processing program by using the obtained data.

Description

Method for reducing process difficulty of theoretical gradient pitch screw rod
Technical Field
The invention relates to the field of numerical control machining and various mechanism designs through spiral transmission, in particular to a structural design aiming at realizing special target track motion through a variable-lead screw, and provides a method capable of greatly reducing the process difficulty of a theoretical gradually-changed-pitch screw rod.
Background
The screw drive mechanism is widely used in aerospace ships, textile equipment, resource exploitation equipment and medical appliances nowadays. The screw drive mechanism used in the early days was a constant pitch screw rod, limited by the manufacturing process level. The variable pitch screw drive can theoretically realize various required output motion characteristics by designing the pitch distribution of the screw rod. China is in an industrial transformation period, and with the development of automation equipment towards high speed and precision, higher requirements are put forward for a spiral transmission mechanism. There are few reports on the design of variable pitch screws and the research on the processing method in the prior published countries, and only a few institutional researchers in germany, japan and the usa have conducted the research work. The design and the processing of the variable-pitch screw mechanism which is particularly applied to a high-speed shaft or needs to realize a special motion rule still have a plurality of problems.
The screw rod with the gradually changed pitch, which is designed according to the theoretical target motion law, usually needs to be continuously adjusted when a numerically controlled lathe turns threads with the gradually changed pitch; on the one hand, the method is difficult to realize, and on the other hand, the process difficulty and the processing cost are increased, so that the method is unnecessary. At present, a common method is to equally divide the variable pitch screw rod to be processed into n sections, and take the average value of each section as the pitch of the section for processing. The screw rod actually processed by the simplified method can be applied to equipment with low speed or low precision requirement, but the precision of the screw rod can not meet the requirement when the screw rod is applied to precision instruments with high precision requirement or high-speed and ultrahigh-speed equipment.
Disclosure of Invention
In order to overcome the defects of the background art, the invention provides a method for reducing the process difficulty of a theoretical gradient pitch screw rod.
The technical scheme provided by the invention is as follows:
a method for reducing the process difficulty of a theoretical gradient pitch screw rod sequentially comprises the following steps:
step 1, designing and obtaining a space parameter equation of a spiral line of a theoretical spiral rod through a target motion rule to be achieved, wherein the equation is as follows:
Figure BDA0001283475080000021
in the formula: r is spiroThe radius of the rotating rod is equal to the radius of the rotating rod,
Figure BDA0001283475080000022
is the rotating angle of the screw rod,
Figure BDA0001283475080000023
for axial displacement with respect to
Figure BDA0001283475080000024
A function of (a);
the axis of the screw rod is coaxial with the Z axis in the coordinate system;
the function relationship of the screw pitch of the screw rod with respect to the rotation angle is
Figure BDA0001283475080000025
According to a variable pitch stroke calculation formula
Figure BDA0001283475080000026
It can be known that
Figure BDA0001283475080000027
I.e. the corresponding screw rod angle
Figure BDA0001283475080000028
The corresponding pitch size.
Step 2, collecting enough 2n sampling points at equal intervals for the screw rod axial displacement curve, and recording the sampling points as a vertex set
Figure BDA0001283475080000029
Wherein n must satisfy the condition that n is 2iOr 3 or 2i-1I is an integer; wherein
Figure BDA00012834750800000210
Has a coordinate vector of
Figure BDA00012834750800000211
Step 3, applying cubic B-spline subdivision reverse engineering principle to collect dense vertex points of the (k + 1) th layer
Figure BDA00012834750800000212
Continuously decomposing and reconstructing to the next layer of sparse vertex set according to the subdivision reverse engineering rule
Figure BDA00012834750800000213
Obtaining a plurality of sets of vertex data; the recursion rule is: c. Ck＝Pk+1ck+1；
Wherein P isk+1The method is a decomposition reconstruction matrix from the k +1 th vertex set to the k layer vertex set, and is specifically expressed as follows:
Figure BDA00012834750800000214
further, the line segment formed by the reconstructed k-th layer adjacent vertex can be easily obtained
Figure BDA0001283475080000031
The coordinate equation of (a):
Figure BDA0001283475080000032
in the step 4, the step of mixing,
1) to stop the reconstruction at the right precision position for the decomposition reconstruction program, the total approximation deviation E at n vertices is definedn：
Figure BDA0001283475080000033
Wherein:
Figure BDA0001283475080000034
for the original curve
Figure BDA0001283475080000035
The function of (a) is calculated,
Figure BDA0001283475080000036
is ckThe line segment function equations of the jth 1 and jth vertices in the array,n is the number of vertices after reconstruction.
2) Setting allowable total approximation deviation E according to precision requirement, and according to recursion rule ck＝Pk+1ck+1Continuously reconstructing a sparse vertex set of the next layer, and calculating corresponding total approximate deviation En(ii) a Judging the reconstructed total approximation deviation EnIf the total approximation deviation is larger than the allowable total approximation deviation E, continuing to perform next reverse decomposition reconstruction if the total approximation deviation is smaller than the allowable total approximation deviation E; when the approximate total deviation is larger than the allowable total approximate deviation E, respectively recording the vertex data before and after reconstruction in the array A and the array B which are temporarily stored for later use; the piece-wise approximation deviation data at this time is temporarily stored in the matrix Q.
Step 5, setting a relative deviation judgment coefficient η, and calculating a ratio η of the corresponding maximum value in the matrix Q to the corresponding sumQIf ηQIf the reconstruction scheme is less than η, the array A is directly output, the procedure is ended, the vertex data in the array A is the optimal reconstruction scheme point, otherwise, the procedure is started in the step 5.
Step 6, finding out two end points of the corresponding maximum approximation deviation segment in the array B, and equally taking a sufficient number of 3.2 of the corresponding areas of the original curves corresponding to the end pointsi-1And (4) carrying out secondary reverse decomposition and reconstruction on the data points, and directly carrying out reverse reconstruction to the simplest peak, namely 3 peaks. And inserting the 3 vertexes after secondary reconstruction and replacing the 2 vertexes at the original positions in the original array B.
7, respectively calculating the total approximation deviation E of the array B againB(ii) a If EBIf the number of the vertex points is less than E, outputting an array B, and ending the program, wherein the vertex data in the array B is the optimal reconstruction scheme point; otherwise, the array A is output, the program is ended, and the vertex data in the array A is the optimal reconstruction scheme point.
And 8, using the vertex data in the array A to compile a machining macro program of the numerical control machine.
The only thing to be noted is that, because the number of vertexes in the finally expected vertex set is small, in the first round of reverse decomposition and reconstruction, the initial data points respectively take an odd number and an even number to bring great difference to the result, and need to separately run and compare.
The invention has the beneficial effects that: the broken line obtained by the substitution method can well reproduce the geometric characteristics of the original curve, and the movement rule of the screw rod which is obtained according to the reconstructed broken line and can be theoretically required can be accurately realized; through the transitional splicing of a plurality of sections of spiral sections with equal screw pitches, the aim of highly equivalent substitution of a theoretical spiral rod is achieved, and the process difficulty and the processing cost are greatly reduced on the premise of effectively ensuring the transmission precision.
Drawings
FIG. 1 is a schematic flow chart of the present invention.
Fig. 2 is an approximation effect diagram of the screw axial displacement curve in the embodiment of the invention.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
Taking a high-speed rapier loom of a certain type for realizing weft insertion working conditions by adopting variable-lead spiral transmission as an example, according to the actual weft insertion working condition requirements, the corresponding theoretical variable-lead screw rod axial displacement zz of the rapier loom is obtained about the screw rod rotating angle
Figure BDA0001283475080000041
The equation of (a) is, in particular,
Figure BDA0001283475080000042
when in use
Figure BDA0001283475080000051
The original curve belongs to a straight-line segment, and only the original curve needs to be subjected to inverse decomposition and reconstruction without further inverse decomposition and reconstruction
Figure BDA0001283475080000052
And performing reverse decomposition reconstruction on the segments. On one hand, the node characteristic points are reserved for more approximating the original curve, and meanwhile, the approximation deviation E of the original curve is calculated by utilizing the piecewise function of the original curve for conveniencenTo, for
Figure BDA0001283475080000053
And
Figure BDA0001283475080000054
and respectively carrying out reverse decomposition reconstruction on the segments.
1. First to each other
Figure BDA0001283475080000055
Inverse decomposition reconstruction of segment original curve
According to the above algorithm, it is noted that the original sampling points are respectively odd and even numbers to be decomposed and reconstructed.
a) When the initial vertex number n is even, 512 (i.e. 2.2) is equally spaced in the original curve8) The allowable approximate deviation E is set to 5, and the relative deviation judgment coefficient η is set to 30%, then it can be seen from table 1 that the first round of reconstruction is stopped when the number of vertices is calculated to 8 (the evaluation parameters of the vertices in the period are as shown).
TABLE 1 decomposition/reconstruction data list of 512 original sample points
Figure BDA0001283475080000056
Due to ηQSince 94.8% is larger than the relative deviation determination coefficient, the 5 th operation is performed according to the program. And finally, a vertex set with the number of vertices being 9 is obtained, and the new total approximation deviation is 1.54744, so that the set allowable deviation is met. Under the set precision requirement, the vertex of the segment when the initial sampling point is even number is finally selected as:
Figure BDA0001283475080000061
b) when the initial vertex number n is odd, 768 (namely 3.2) is equidistantly taken in the original curve8) The data points are initial data points, and are also continuously subdivided in the same way, and the precision conditions are the same as the above; then, as shown in table 2, the first round of decomposition reconstruction is stopped until the number of vertices is 6 (the evaluation parameters of the number of vertices in the period are shown in table 2).
TABLE 2 summary of decomposed and reconstructed data when original sample number is 768
Figure BDA0001283475080000062
Due to ηQAnd (6) when the deviation is equal to 96.3%, the deviation is also larger than the relative deviation judgment coefficient, and the 5 th step is carried out according to the program. And obtaining a vertex set with the vertex number of 7, wherein the new total approximation deviation is 1.9072 and is within the set allowable deviation range. Under the set precision requirement, the vertex of the segment when the initial sampling point is odd is finally selected as:
Figure BDA0001283475080000063
c) comparing two groups of optimal schemes when the initial point is respectively odd number and even number, and under the condition that allowable approximation errors can be met, the number of vertexes contained in the optimal scheme when the initial sampling point is the odd number is less, so that the section of the screw rod after equivalent substitution is smaller, and the processing and manufacturing of the variable-lead screw rod are more facilitated, therefore, the optimal reverse decomposition reconstruction vertexes under the condition that the allowable approximation deviation E is 5 and the relative deviation judgment coefficient η is 30% are as follows:
Figure BDA0001283475080000071
2. to pair
Figure BDA0001283475080000072
Inverse decomposition reconstruction of segment original curve
Considering that the original curve span area of the section is smaller, the processing of the section of the screw rod is convenientIn manufacturing, the number of the top points after final reconstruction is controlled to be 2 to 4, and the allowable approximation deviation is reduced to 5/7 in an equal proportion. According to ck＝Pk+1ck+1Recursion rules to the number of vertices 2, 4 and 3; the corresponding total approximation deviations are 1.1629,0.06365, and 0.0276. Under the condition of satisfying the allowable approximation, a scheme with the number of the top points being 3 is finally selected, and the corresponding top point set of the segment is as follows:
Figure BDA0001283475080000073
3. pitch distribution of reconstructed helix
According to the finally selected vertex control set, obtaining the screw axial displacement zz related to the screw corner after the reverse decomposition reconstruction
Figure BDA0001283475080000074
The function of (d) is as follows:
Figure BDA0001283475080000075
the substitution effect of the reconstructed broken line segment and the theory screw rod axial displacement curve is shown in fig. 2.
As can be seen from fig. 2: the original curve can well reproduce the geometric characteristics of the original curve based on the broken line obtained by the substitution method, and the movement rule of the screw rod which is obtained according to the reconstructed broken line and can realize theoretical requirements at high precision is also ensured. According to the formula (5), information such as the corresponding pitch of each segmented screw rod can be conveniently obtained, and the information is shown in table 3. And finally, compiling a machining macro program for the numerical control machine tool according to the information shown in the table 3.
TABLE 3 sectional screw information
Figure BDA0001283475080000081

Claims (2)

1. A method for reducing the process difficulty of a theoretical gradient pitch screw rod sequentially comprises the following steps:
step 1, designing and obtaining a space parameter equation of a spiral line of a theoretical spiral rod through a target motion rule to be achieved:
Figure FDA0002374051320000011
in the formula: r is the radius of the screw rod,
Figure FDA0002374051320000019
is the rotating angle of the screw rod,
Figure FDA00023740513200000110
for axial displacement with respect to
Figure FDA00023740513200000111
A function of (a);
the axis of the screw rod is coaxial with the Z axis in the coordinate system;
step 2, collecting 2n sampling points at equal intervals on the axial displacement curve of the screw rod, and recording the sampling points as a vertex set
Figure FDA0002374051320000012
Wherein n must satisfy the condition that n is 2iOr 3 or 2i-1I is an integer; wherein
Figure FDA0002374051320000013
Has a coordinate vector of
Figure FDA0002374051320000014
Step 3, dense vertex set of the (k + 1) th layer
Figure FDA0002374051320000015
Continuously decomposing and reconstructing to the next layer of sparse vertex set according to the subdivision reverse engineering rule
Figure FDA0002374051320000016
Obtaining a plurality of sets of vertex data; delivery deviceThe push rule is: c. Ck＝Pk+1ck+1；
Wherein P isk+1Decomposing and reconstructing a matrix from the k +1 layer vertex set to the k layer vertex set;
in the step 4, the step of mixing,
1) to stop the reconstruction at a suitable precision position, a total approximation deviation E at n vertices is definedn
Figure FDA0002374051320000017
Wherein:
Figure FDA00023740513200000112
for axial displacement with respect to
Figure FDA00023740513200000113
The function of (a) is calculated,
Figure FDA0002374051320000018
is ckThe line segment function equations of the jth vertex and the jth vertex in the array are obtained, and n is the number of the reconstructed vertices;
2) setting allowable total approximation deviation E according to precision requirement
According to recurrence rule ck＝Pk+1ck+1Continuously reconstructing a next layer of sparse vertex sets, and calculating the segmented approximation deviation data and the total approximation deviation E of each corresponding vertex set after reconstructionn(ii) a Judging the reconstructed total approximation deviation EnIf the total approximation deviation is larger than the allowable total approximation deviation E, continuing to perform next reverse decomposition reconstruction if the total approximation deviation is smaller than the allowable total approximation deviation E; when the total approximation deviation is larger than the allowable total approximation deviation E, respectively recording the vertex data before and after reconstruction in the array A and the array B which are temporarily stored for later use; the piecewise approximation deviation data at this time is temporarily stored in the matrix Q;
step 5, setting a relative deviation judgment coefficient η, and calculating a ratio η of the corresponding maximum value in the matrix Q to the corresponding sumQIf ηQIf < η, the array A is directly output, and the program is endedThe vertex data in the group A is the optimal reconstruction scheme point; otherwise, entering the 6 th step of the program;
step 6, two end points corresponding to the maximum approximation deviation segment in the array B are found, and enough 3.2 is equally spaced in the corresponding areas of the original curves corresponding to the two end pointsi-1Performing secondary reverse decomposition reconstruction on the data points, and directly performing reverse reconstruction to the simplest 3 vertexes; inserting the 3 vertexes after secondary reconstruction and replacing 2 vertexes of the original position in the original array B;
7, recalculating the total approximation deviation E of the array BBIf E isBIf the number of the vertex points is less than E, outputting an array B, and ending the program, wherein the vertex data in the array B is the optimal reconstruction scheme point; otherwise, outputting the array A, and ending the program, wherein the vertex data in the array A is the optimal reconstruction scheme point;
and 8, using the vertex data in the optimal reconstruction scheme point to write a machining macro program of the numerical control machine.
2. The method for reducing the process difficulty of a theoretical progressive pitch screw rod according to claim 1, wherein: the decomposed reconstruction matrix in the step 3 is expressed as follows:
Figure FDA0002374051320000021
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