To check the quality of manufacturing the surfaces of the teeth of involute cylindrical wheels, in practice, two types of control are very widely used: measuring the size by rollers (balls) and measuring the length of the common normal.
Since to measure the length of the common normal it is often enough to have only a vernier caliper, this method of controlling the thickness of the teeth is practically more accessible and is widely used especially in the single (repair) production of cylindrical gears of a low degree of accuracy. It should be noted that the accuracy is quite high. this method control due to the direct method of measuring the part, as opposed to measuring the size by rollers, which, by their tolerances, introduce an additional error. The length of the common normal refers to the parameters characterizing the lateral clearance norm in gear train.
When the law corresponding to the design of the gear train is complete, run the one that corresponds to the calculation of the wheels and gears that make it up. What we have done so far has been to determine the number of gear teeth, the number of gear leaves, and the speeds of each gear needed to meet the required ratios.
In the example that will keep us busy for the next eleven articles, we will size the values \u200b\u200bof all these components based on what has been obtained so far. First, we need to define a number of parameters required for the calculation.
Consider external gearing. The calculation is regulated by GOST 16532-70. Measurements are performed in the plane of the normal (perpendicular) surface of the teeth. For helical gears (especially at large angles of inclination) it is necessary, after calculations, to make sure that the width of the wheel rim is "sufficient" for the measurement.
To simplify the calculation on the Internet, on the website www.al-vo.ru, a program was found in the form of a table MS Excel, which allows you to quickly find the length of the common normal of cylindrical gears, including helical gears with offset.
This is the path starting from the diameter of the passage. Arc corresponding to the tooth plus space. The layout of an ordinary clock. A typical clock diagram is shown in the following figure. In this example, we will devote ourselves to determining the following values \u200b\u200bfor all the wheels and gears of the motion.
In the previous example, the most suitable ratios were determined to be sufficiently similar to each other so that the overall motion ratio is distributed as evenly as possible over the entire gear train. To draw the previous layout, the diameters of the wheels were chosen proportionally so that they fit into the available space without overlapping or overlapping each other.
This program It is convenient in that it allows you to quickly check the accuracy of field measurements of the existing gear, including the possible presence or absence of displacements, in the field, with a smartphone or tablet.
If you work with KOMPAS-3D, then when calculating in the library "Shafts and mechanical transmission 2D "you will automatically receive the length of the common normal from the calculation results.
A large area of \u200b\u200bthe base plate must be left free to allow the location of protected bases for bridges and hinges. These areas are indicated with a dashed line to show the clearances or available for wheel diameters. The circles depicting wheels are circles with their respective pitch diameters, which will increase to their full diameter when the edges of the teeth are added. It is very important to take this increase into account when determining the relative wheel size.
It can be seen that overlapping teeth can occur when placing additives on the center wheel and on the fourth wheel. In this particular case, it does not matter, since the fourth wheel is below the level of the center wheel. The same does not happen between the fourth wheel and the gear of the third wheel. In the drawing, there is a large space between the passage circle of the fourth wheel and the corresponding gear of the third wheel. One might think that this fourth wheel could be larger, but it is not, because when the addition is added to the teeth of the fourth wheel and expansion to the sheets of the third wheel gear, this space will be minimal.
Having completed the calculation, you must, taking a vernier caliper, measure the length of the common normal of the resulting number of teeth (several times and in different groups) and obtain values \u200b\u200bequal to the calculated one for a well-cut wheel.
In fairness, it should be mentioned that for measuring the length of the common normal, there is special tool - normal meter. The standard gauge is made on the basis of a caliper or micrometer, supplying the latter with special convenient jaws for measuring measurements and a pointer indicator.
It is convenient to draw on a large scale. The values \u200b\u200bof the pitch diameters are taken directly from the drawing, so it will be convenient to zoom in for calibers with a smaller diameter. The procedure for calculating the parameters of each wheel and pinion gear is as follows.
Taking into account the diameter of the previous pitch and taking into account the gear ratios calculated in the previous example, the diameter of the gear driven by the wheel is determined.
- First, the diameter of the drawn wheel is measured on a breadboard.
- This diameter corresponds to the diameter of the wheel.
- The width of the teeth and leaves is calculated.
Spur gear tolerances are regulated by GOST 1643-81. Including, tolerances are assigned to the length of the common normal, depending on the type of mating and the norm of the side clearance.
P.S. In reference books and in GOST, this calculation is written in such a way that "for two days you need to deal with beer," jumping "from table to table." Apparently, in such cases, the authors have always done this to give themselves "the highest importance and significance" ... And ordinary students and engineers need to be "intimidated" with an abundance of transitions from page to page, so that on the fourth or fifth transition to a new table or diagram, they forget that in general do. If, at the end of everything, add something completely scary - like an involute (this is not euros or dollars, but a function like this), then everything will be done. Get one or two for a hundred mechanical engineers who have a little understanding of gears! And if you get into the jungle of displacement of the contours to obtain certain power or qualitative changes, you will find out that in Germany and Japan, they first calculate and optimize the transmission, and then make an instrument for it ... And we still count everything under a standardized instrument - α \u003d 20 ° ...
Dimensional tolerances for cylindrical gears
Within the company, the mechanical service industry plays an important role in the continuity of product flow. After diagnosing the defect, disassembling, cleaning components, replacing damaged parts, assembling, lubricating and making final adjustments to the machine. However, there are often no spare parts to repair a machine, especially when they are out of date.
Because of this, in the service sector, many companies have some machine tools dedicated to the production of mechanical elements to replace machine parts for maintenance... This is a situation that can happen at your company now: the machine was dismantled and one of its gears was noticed to be broken.
The tangent line to the base circle of the gear that intersects z w of its teeth and is the normal to both extreme involutes, they are called common normal.
The distance between the opposite side surfaces of the teeth of a cylindrical gear along the common normal to these surfaces is called common normal length W (fig. 2).
Limit deviations of the external constant chord
Do you think you will be able to pick up the data from this vehicle element from the remaining fragments and perform calculations to create new equipment? If your answer is no, stay tuned for updates in this class. In it we will learn how to calculate the spur cylindrical gears.
Features of calculating the measuring dimensions of the tooth of bevel spur gears
Cylindrical spur gear. A spur gear train is the most common one that exists. To construct it, it is necessary to consider a number of data, namely: the number of teeth of the outer diameter of the module, the primitive diameter, the inner diameter, the height of the tooth height, the height of the foot head, the step.
The length of the common normal does not depend on at what points of the tooth profiles this normal intersects two opposite involutes. The change in the length of the common normal is proportional to the change in the offset of the original contour xm gear cutting tool. It is also important that size control w not associated with any auxiliary base for installing the measuring tool.
The modulus of a gear is a measure that represents the ratio between the primitive diameter of the same gear and its number of teeth. This relationship is represented mathematically as follows. The module and number of teeth determine the tool that will be used to mill the gear.
The module also assists in calculations to find all other dimensions of the mentioned projectile. Because of this, in fact, it is possible to calculate the modulus based on any known measure of the corresponding mechanism. For example, you can calculate the modulus from measuring the outer diameter and the number of teeth of a gear.
The specified properties of the common normal show the advantage of this method of controlling the thickness of the wheel tooth. This dimension can be measured with a vernier caliper, micrometer, special limit bracket.
The length of the common normal for cylindrical wheels with external straight teeth calculated by the following formula [2]
where m - module, mm; a - the angle of the profile of the original contour, according to the GOST 13755-81 standard a \u003d 20 0; z w - the number of teeth in the length of the common normal; x- displacement coefficient; z - the number of teeth of the controlled wheel; inv a - involute angle corresponding to the profile angle a, for spur gears inv a = tg a - a .
Then you measured the outside diameter and got: \u003d 124 mm. Save this data for future reference. The outer diameter is equal to the first diameter plus twice the height of the tooth head, which, in turn, is equal to the modulus. It's easy to check if you notice the following figure.
Because for our problem we already have the value of the outer diameter, we do not need to calculate it. In order to solve the problem of creating the equipment that we present to you, we must calculate the modulus from the measures that we have. Let's then work on this formula so that it helps us do the necessary calculations.
The length of the common normal for cylindrical wheels with external helical teeth calculated by a similar formula
where m n - normal module, mm;
, and the end angle of the profile of the original contour 
... Here
- the pitch angle of the tooth line specified by the drawing of the gear wheel.
Replaced values: 124 \u003d m 124 \u003d m 62 m \u003d 62 m \u003d 2. So the required transfer modulus must be equal to the note as we use the outer diameter formula for this calculation. This can be done using any known module-specific data.
Ufa State Aviation
So far, we have studied formulas for calculating the primitive diameter, modulus, number of teeth, and outer diameter of a cylindrical spur gear. Let's find out all this by doing the following exercises. Exercise 2 Calculate the number of teeth of a gear that has a primitive diameter of 240 mm and a modulus equal to.
For a helical gear, the length of the common normal is measured at the main angle of inclination of the tooth line b to the end of the wheel, and the possibility of measurement is checked with a sufficient width of the gear rim b by condition
b ³ w sin b ,
where sin b \u003d sin cos.
Number of teeth in the length of the common normal z w for cylindrical wheels with straight teeth must satisfy the condition
,
when 
,
,
Here a - the angle of the profile at a point on the circumference of the tops of the teeth; l - the angle of the profile at the boundary point.
With small displacement coefficients ( x 1) to determine z w you can use a simplified formula
with the resulting value rounded to the nearest integer value.
1.3. Dimensional tolerances for cylindrical gears
The formulas considered above for calculating the nominal measuring dimensions of the cylindrical gears guarantee the backlash-free gearing of the wheels in the transmission. In real gear drives, a guaranteed side clearance in order to eliminate jamming of the teeth during operation under load as a result of temperature deformations of the transmission parts, as well as to place a layer of lubricant on the working profiles of the teeth. The side clearance in engagement is also necessary to compensate for errors in the manufacture and installation of the gear. It is mainly determined by the value of the center distance a w transmission and thickness s wheel teeth.
The standard for involute gear cylindrical gears (GOST 1643-81) establishes eight types of lateral clearance tolerances: h, d, c, b, a, z, y, x (designations of tolerances are arranged in ascending order of the tolerance value). The accepted value of the guaranteed side clearance is the basis for the appointment pairing type gear wheels. The same standard provides six types of pairing: H- zero clearance, E - small clearance, C and D - reduced clearance, B - normal clearance, A - increased clearance. View mates H, E and FROM require increased precision in the manufacture of wheel teeth. They are used for reversible transmissions with high requirements for the kinematic accuracy of the transmission, as well as in the presence of torsional vibrations of the transmission shafts. Most often, in medium-sized mechanical engineering, gears with interface types are used IN and FROM... In the absence of special requirements for the gear train, with each type of mating, a certain type of tolerance for the side clearance is used, indicated by a lowercase letter similar to the letter of the type of mating (for example, AND- and, IN - in, FROM - from etc.).
The tolerance field for the measuring size of the gear is always directed to the tooth body, therefore, the maximum deviations of the measuring size (upper and lower) always have negative values \u200b\u200b[1].
