Overview Involute cylindrical gears can be roughly divided into spur gears (including splines) and helical gears (including herringbone gears). For example, they can be divided into external gears and internal gears according to the meshing state. Widely used in various types of mechanical transmission. Gears are always used in pairs. Their parameters are numerous and complex. In machine tool repair, instrument repair and product imitation, accurately mapping the parameters of unknown gears is an important and meticulous task. The purpose of gear mapping is to accurately find the original design parameters. The main parameters of the involute cylindrical gear include the number z of teeth, the normal modulus mn (diameter), the diameter of the index circle d, the pressure angle n, the helix angle, and the variable coefficient. x, tooth height coefficient ha, etc.
Although these parameters have been standardized, the standards used by countries are not consistent (internationally, they can be divided into two major categories: ISO and imperial). If you do not understand the status of the national gear standards, you can not correctly find the original design parameters of the gears. Therefore, the mapping of gears also includes research on the gear standards of various countries. This adds to the difficulty of gear mapping.
Gear mapping has always been regarded as a difficult and troublesome work in the field of measurement. These difficulties are mainly: 1) It is difficult to select a surveying method. Due to the different degree of completeness and degree of wear of the tested gears, the tolerance values ​​of various parameters are different, and the means of surveying and mapping are different, so that there are many alternative surveying methods, but it is difficult to determine an optimal surveying method.
2) Calculation is difficult. Due to the many parameters of the gears, the calculation formulas of some parameters are more complicated, some parameters are difficult to solve with elementary mathematics, and some parameters are simply trial and error.
3) It is difficult to determine the standard. Because the gear standards adopted by different countries are not the same, even if the standards of different countries are different in different years, it is very difficult for gear mapping.
1 Involute cylindrical gear mapping flow chart and surveying program Computer-aided involute cylindrical gear mapping is the application of the advantages of large computer information, fast calculation and accurate judgment to gear mapping in the above three difficulties in gear mapping. Use the computer to store and query the necessary data and charts, perform calculations, and make logical judgments to arrive at the final survey data. The flow chart is shown in Figure 1.
We have compiled a computer-assisted involute cylindrical gear mapping program. The program has the following characteristics: 1) It runs in Chinese prompts and man-machine dialogue mode, and is easy to operate. It can also be used by those who do not understand computer expertise.
2) After analysis and optimization, the main parameters of the spur gear, helical gear, internal tooth and external gear are calculated and verified, so that the program can be used for the mapping of various cylindrical gears at the same time.
3) Store some important data and charts in the computer and automatically query them, such as the basic tooth profile table commonly used in various countries; the standard series of common modulus (diameter) in various countries; base section, modulus (diameter), pressure Angle comparison table; double modulus (two-diameter section) series table, and the like.
4) A solution method for calculating the transcendental equation of the single gear displacement coefficient using the cross-bar M value is proposed.
11 program description 1) For the pair of gears, the pinion is z1 and the large gear is z2.
2) For the mapping of pairs of gears, first measure z1, then measure z2.
3) Unless otherwise stated, in the signed expression, the upper symbol represents the external engagement, and the lower symbol represents the internal engagement.
4) The parameters in the program where the symbol is added in the upper right corner are direct measurement parameters.
5) In order to make the data accurate, the trimming section should be avoided when mapping the trimming gear.
12 program structure program can be roughly divided into three parts: 1) direct measurement parameter input and calculation; 2) indirect measurement parameter determination; 3) check calculation.
13 Direct measurement parameters An unknown measured gear, with the degree of integrity of the gear itself and the existing measurement methods, different direct measurement parameters can be selected to calculate other parameters. When selecting direct measurement parameters, the following principles should generally be mastered. : Easy to measure, the results are also more accurate; easy to use this parameter to calculate other parameters; less tolerance; (less wear.
When selecting the direct measurement parameters, the mapping program should not only consider the above four principles, but also consider the commonality when mapping the spur or helical teeth separately, so that the determination of the selected direct measurement parameters tends to be consistent, so as to facilitate Compilation and normal operation of computer programs.
The direct measurement parameters selected by the mapping program are described below.
1) Number of teeth z This is a simple and important parameter.
2) The length of the common normal line Wk, W (k-1) The length of the common normal is an important parameter for characterizing the tooth thickness. It can calculate the base section Pb and the displacement coefficient x more conveniently, and is easy to measure (compared to special short-toothed gears, With the exception of splines and some internal gears, the common normal of most gears can be measured more accurately.
The common normal cross-counting number k and k-1 are related to the modulus mn pressure angle n and the number of teeth z. Since the measured gear is an unknown gear, the program uses an estimation formula to indicate the number of teeth to be measured k.
INPUTE k= ;01z 05Y/NY: The program runs down; N: Re-enter the appropriate k.
In the gear mapping, since the common normals Wk and W(k-1) are a set of relative numbers, as long as Wk, W(k-1) can be measured, it can be considered that the selection of the number of cross-measuring k is correct, and there is no need to pursue standard value.
Since the wear of the gear has a great influence on the value of the common normal line, after inputting the values ​​of Wk and W(k-1), the program will prompt to add an appropriate amount of thinning, and the value of the thinning amount is between 00503 mm, which will be regarded as wear and tear. Depending on the degree, the surveyor enters the computer based on experience.
3) When the spans M1 and M2 are used to map some special gears (such as dual modulus gears) or short splines and some internal gears, it is possible to detect the common normal length Wk and W(k-1). At this time, it is possible to measure the Wk and W(k-1) instead of measuring the span distance M1, M2.
The M value of the cross-bar distance is easy to measure, and the exact value is stable. However, the equation used to calculate the base section Pb is a transcendental function, which can only be solved by trial and error. It is not only computationally intensive, but also has certain blindness and is difficult to grasp. Therefore, it is not often used. The mapping program appropriately deforms the original equation, and uses the string-cut method to perform iterative operations to find an exact solution.
The transcendental equation of the unknown can be solved by computer, and the solver will not be described again.
The wear of the gear has a great influence on the M value of the span of the rod. Therefore, after inputting the span of the rods M1 and M2, the program will propose a reduction and thinning amount of 0103 mm, and the surveyor inputs the appropriate thinning amount by experience.
4) The diameter of the tip circle diameter da diameter of the tip circle is relatively easy to measure. It can be used to calculate the tooth height coefficient ha and the displacement coefficient x, but the disadvantage is that the da tolerance is large.
5) Tooth circle round helix angle a indexing circle helix angle is very important, but it is not easy to measure, and the tooth tip circle helix angle a is relatively easy to measure. a can be measured by Wangong or directly by coloring. The method is directly measured.
By =arctan(mzdatana) for spur gears, the program makes a=06) The actual center distance a is when the paired gears are mapped, the actual center distance a can determine the displacement form of the gears. For the mapping of individual gears, the procedure will not Request to enter a.
At this point, the surveying program should input up to five sets of direct measurement data. When the data input is completed, the program will transfer to the determined part of the indirect calculation parameter.
14 Indirect calculation parameter mapping program In this part, the measured parameters are directly measured by the measured gear to calculate or query other important parameters, and determine the meshing and displacement forms used. The determined parameters are described separately in the order of the program.
1) Determine the modulus mn (or the diameter of the OP) and the pressure angle n. For the convenience of use, the gear base, modulus (diameter), pressure angle table and spline base, modulus (path) are programmed in the program. Section), pressure angle comparison table, in the previous procedure we have obtained the base section as Pb=Wk-Wk-1, or Pb=2rbz through the computer to query the gear base section, modulus (diameter), pressure angle comparison table (Or spline base, modulus (diameter), pressure angle table), you can get three sets of modulus and pressure angle closest to the Pb value, for the surveyor to choose. The surveyor can also query the national modular (diameter) standard series table according to the country of the measured gear, and use this as a reference to select the best group among the three groups of mn (diameter) and n values. Measuring the modulus of the gear mn (diameter) and pressure angle n.
For helical gears, the end face modulus ms and the end face pressure angle s are calculated here.
2) Determine whether the measured gear is displaced. First, calculate the theoretical common normal line length WkWk=mcosn[(n-05) zinvn] to compare Wk with Wk. When Wk=Wk, it is non-displacement gear, Wk, Wk Displacement gear.
When the measured gear cannot measure Wk, the theoretical M value is compared with the actual M to determine the displacement form: M1=2rbrcoscosx1)dL1R=1 even gear R=cos90z odd gears invx1=)(dL12rb-2z) invi when M= M non-displacement gear, M, M displacement gear.
3) Determine the displacement coefficient x When Wk, Wk or M, M, calculate the displacement coefficient x=Wk-Wk2mnsinn or by the following formula x=(invx1 12-dL12rb inv)z2tann where x1=avccos(2rbRM1-dL1) .
4) Judging the displacement form For the pair of gears, the actual center distance a can be used to judge the displacement form.
First calculate the theoretical center distance a: a = z1 z2) mn2cosa = a height displacement, axial displacement coefficient y = 0; a, a is angular displacement, y = x1 x 2.
For a single gear can not determine its displacement form, the program automatically moves to the next step.
5) Determine the tooth height coefficient ha* value, the tooth height coefficient ha*=1 of the ordinary gear, and the tooth height coefficient ha*=08 of the short tooth series, but some gears have a particularly fat and short tooth height, which may have two kinds of modulus The indexing circle is calculated with a larger modulus m1, and the tooth height is calculated with a smaller number m2. The tooth height coefficient ha* is still 1, so it is possible to have three options for a measured gear, namely normal tooth height, short tooth height and double modulus tooth height. The mapping procedure is calculated using the addendum circle da. Ha*, because the tolerance of da is large, the calculation accuracy of ha* is low and the error is large. The value of ha* can only be judged by the program, ie ha*=)(da-d2mn x)ha*>09 Take ha*=1, which is the normal tooth height; 07
At this point, the determined parameters are z, mn, n, x, ha*. From these parameters, the common parameters of the gear can be basically calculated, and the program is transferred to the verification part.
15 Checking In order to check whether the determined parameters are accurate, the mapping program uses the tip circle da and the common normal line Wk as the checking parameters.
Da=mtz 2mt(ha* xt)(2) Wk=mncosn[(k-05) zinv] 2xnmnsinn(3) The determined parameters are substituted for (2) equation (3).
When da=da, Wk=Wk, the mapping is correct, and the result is printed; when one of the above two items does not match, the mapping is wrong, and the program will turn to re-determine the calculation parameter segment.
2 Supplementary Note 1) The mapping of gears is a relatively complicated task. Due to the different standards of various gears and their different uses, the degree of wear, manufacturing precision and limitations of measuring methods have caused various difficulties in surveying and mapping. It is not easy to make a surveying success. If the measured parameters are completely correct, That is even more difficult. The process of mapping the gear should be a process of repeated measurement, iterative calculation, and repeated verification. Only in this way, it is possible to obtain satisfactory results. Therefore, when the verification does not meet the requirements in the mapping program, it will return, so as to recalculate and determine the measured parameters. .
2) The mapping program in computer-aided involute cylindrical gear mapping is to use the advantages of computer to help the surveyor to map the gear faster, better and more accurately. It can provide surveying and mapping ideas, calculation and judgment for surveyors. , check calculations and various tips; but the original raw data has to be provided by the surveyor, so the accuracy of the surveying program depends mainly on the accuracy of the original data provided by the surveyor. In order to ensure the accuracy of surveying and mapping, the error of various parameters required to be input should be kept within 001mm; according to the previous surveying and mapping experience, the program will set the theoretical error of surveying within 005mm, that is, the equal value of the judgement in the program means the value of both sides of the equal sign. The difference is not greater than 005mm.
We know a medium-precision, medium-modulus, gear with a circle of less than 100mm. The manufacturing tolerance value of most parameters is generally between 001005mm, plus the wear error and measurement error of the gear. The error value is about 005mm, so it is appropriate to set the allowable error to 005mm when mapping most gears.
3) Sometimes we want to get the manufacturing precision of the gears being surveyed. Since most of the surveying gears are old, worn gears, the values ​​of their various parameters have changed, so it is very important to get the gear manufacturing precision in the surveying and mapping. difficult. We can only estimate the original accuracy of the gear based on the gear's use, speed, Machining process, and the deviation of the measured results from the measured values.
4) The use of computer-aided gear mapping is a new topic. It is the first time that each type of cylindrical gear is mapped with a program. There are many loopholes and deficiencies. I hope that the experts will give valuable advice and make The program can be constantly modified and improved.
The author hopes that with the relatively rough work, it will lead to the development of online value measurement in computer applications.
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