Methods and means of measuring angles. Angular tolerances. Methods for measuring horizontal angles Methods for measuring angles Tools for measuring angles

The measurement of the horizontal angle is performed by the method of receptions. When measuring several angles that have a common vertex, the method of circular techniques is used.

Work begins with the installation of a theodolite over the center of the sign (for example, a peg) that secures the top of the corner, and sighting targets (milestones, special marks on tripods) at the ends of the sides of the corner.

Installing a theodolite in working position consists of centering the instrument, leveling it and focusing the telescope.

Centering performed with a plumb line. A tripod is set above the peg so that the plane of its head is horizontal and the height corresponds to the height of the observer. Fix the theodolite on a tripod, hang the plumb line on the hook of the anchor screw and, loosening it, move the theodolite along the tripod head until the plumb line tip aligns with the center of the peg. Centering accuracy with a plumb line is 3 - 5 mm.

Using an optical plummet, theodolite (if the theodolite has one), you must first level, and then center. Centering accuracy with optical plummet 1 – 2 mm.

Leveling theodolite is performed in the following order. By turning the alidade, its level is set in the direction of the two lifting screws, and by rotating them in different directions, the level bubble is brought to zero point. Then the alidade is rotated by 90º and the bubble is brought back to zero point with the third lifting screw.

Focusing spotting scope is performed “by the eye” and “by the subject”. By focusing “on the eye”, by rotating the diopter ring of the eyepiece, a clear image of the reticle is achieved. By focusing “on the subject”, by rotating the handle of the rack, a clear image of the observed object is achieved. Focusing should be done in such a way that when the observer's head shakes, the image does not move relative to the strokes of the grid of threads.

Measuring the angle by the method of receptions. The reception consists of two half-receptions. First half perform with the position of the vertical circle to the left of the telescope. Having fixed the limbus and unfastened the alidade, point the telescope at the right sighting target. After the observed sign has entered the field of view of the tube, the fixing screws of the alidade and the telescope are clamped and, acting as the leading screws of the alidade and the tube, point the center of the grid of threads to the image of the sign and take a reading along a horizontal circle. Then, having detached the tube and the alidade, point the tube at the left sighting target and take the second reading. The difference between the first and second readings gives the value of the measured angle. If the first count turned out to be less than the second, then 360º is added to it.

The second half-reception is performed with the position of the vertical circle on the right, for which the pipe is transferred through the zenith. In order for the readings to differ from those taken in the first half-reception, the limb is shifted by several degrees. Then the measurements are performed in the same sequence as in the first half-reception.

If the results of measuring the angle in half steps differ by no more than double the accuracy of the device (that is, 1¢ for the T30 theodolite), the average is calculated, which is taken as the final result.

The concept of measuring by the method of circular techniques several corners that have a common vertex. One of the directions is taken as the initial one. Alternately, in a clockwise direction, with a circle on the left, point the pipe at all sighting targets and take readings. The last guidance is again done on the initial direction. Then, moving the pipe through the zenith, they again observe all directions, but in reverse order- counterclock-wise. From the readings with a circle on the left and a circle on the right, the averages are found and the average value of the initial direction is subtracted from them. Get a list of directions - angles counted from the initial direction.

2.8.1. Basic concepts. For angular dimensions, as well as linear ones, there are rows of normal angles. However, with regard to angles, this concept is used much less frequently, since when developing elements of parts with angular dimensions, the angle value is often obtained either by calculation to ensure certain functions of the developed mechanism design, or is determined by the necessary arrangement of functional units. Therefore, for angular dimensions, it is less often necessary to use the concept of a normal angle.

With regard to angular dimensions, the concept of tolerance is also used, similar to the tolerance for a linear dimension.

Angle tolerance called the difference between the largest and smallest maximum allowable angles. Angle tolerance denoted AT (short for English expression Angle tolerance - angular tolerance).

At normalizing the accuracy of angular dimensions the concept of "deviation" is not applied, but it is provided that the tolerance can be located differently relative to the nominal value of the angle. The tolerance can be located in the plus side of the nominal angle ( +AT ), or negative ( -AT ), or symmetrical about it ( ±AT/2 ). Naturally, in the first case, the lower, and in the second case, the upper deviations are equal to zero, i.e. correspond to cases of deviations both for the main hole and the main shaft when normalizing the accuracy of linear dimensions.

Peculiarity manufacturing And measurements of angular dimensions lies in the fact that the accuracy of the angle largely depends on the length of the sides forming this angle. And in the process of manufacturing parts and when measuring them than less length side of the angle, the more difficult it is to make an exact angle and the more difficult it is to measure it accurately. True, with very long sides of the angles, another nuisance appears in the form of distortion (deviation from a straight line) of the lines forming the Angle. Based on these features of the angular dimensions, when normalizing the requirements for accuracy, the angle tolerance value is set depending on the length of the smaller side that forms the angle, and not on the value of the nominal angle.

2.8.2. Ways to Express Angle Tolerance. Taking into account the fact that the value of the angle to express different ways, when normalizing the requirements for accuracy, the tolerance values ​​are expressed differently ( GOST 2908-81) and the corresponding angle notation is used:

α - nominal angle

AT α - tolerance, expressed in radian measure, and the corresponding exact value in degrees;

AT" α - tolerance, expressed in degrees, but with a rounded value in comparison with the radian expression;

ATh- tolerance, expressed in a linear measure by the length of the segment on the perpendicular to the end of the smaller side of the angle.

The relationship between tolerances in angular and linear units is expressed by the dependence ATh = AT αLi 10 3 , where ATh measured in microns, AT α - in mkrad; Li - length.

2.8.3. Accuracy series for angular dimensions. IN GOST 2908-81 17 series of accuracy are established, called degrees of accuracy (from 1 to 17). The concept of "degree of accuracy" is identical to the concept of "quality", "class of accuracy".

The designation of accuracy is made by indicating symbol angle tolerance and degree of accuracy, for example AT5, AT7.

Tolerance series, i.e. the difference between the tolerances of adjacent degrees, formed using a factor of 1.6, i.e. if it is necessary to obtain angle tolerances for the 18th grade, which is not in the standard, the AT17 tolerances must be multiplied by 1.6, and to obtain the ATO, the ATI tolerances must be divided by 1.6.

There are several ways to measure horizontal angles: method of tricks, method of circular techniques, method of repetitions, method of all combinations. The simplest and most common is the method of receptions. The method of circular receptions It is used when several angles need to be measured at one point. Repetition method it is recommended to use if the accuracy of the theodolite is insufficient and it is required to measure the angle with higher accuracy. Measurement of the horizontal angle by the repetition method can only be performed with a repetition theodolite. Combination method characterized by laboriousness and is used only for high-precision measurements of several angles at one point, when the measurement errors of the angles should be within 1 ".

The measurement of the angle by the method of receptions consists in its measurement by two half-acceptances. Each half-step is to perform the following actions:

• 1) pointing the vertical thread of the net of threads at the right sighting target;
• 2) taking a reference i, along a horizontal circle;
• 3) entry in the reference log i,;
• 4) pointing the vertical thread of the net of threads at the left sighting target;
• 5) taking a count b] in a horizontal circle
• 6) record in the countdown log b(,
• 7) calculation of the value of the horizontal angle = a ( - b ( .

Target targets are

View from above

Rice. 5.11. sighting cylinder

the object or device at which the telescope is aimed. When observing triangulation points, the sighting target is usually low-phase sighting cylinder(Fig. 5.11) geodetic sign. This figure shows the image seen in the field of view of the tube of a straight image theodolite. In this case, the vertical thread of the grid of threads is pointed at an imaginary axis of symmetry of the sighting cylinder. When observing the points of the theodolite traverse, vertically installed milestones or studs from the set of a measuring device for measuring distances are used as sighting targets.

After measuring the angle with the first half-step, the position of the limb is changed. There are two ways to change the position of the limb of the horizontal goniometric circle:

• 1) make 2-3 turns with the leading screw of the limb, the position of the limb may change by 2-3 °;
• 2) with the fixing screw of the alidade fixed, unfasten the fixing screw of the dial, turn the dial to an arbitrary angle (it is recommended by about 90°), fix the fixing screw of the dial.

After performing the described actions, the pipe is transferred through the zenith and the angle is measured by the second half-reception (at a different position of the vertical circle). Calculation of the value of the horizontal angle from the second half-reception is carried out in a similar way:

P2 = n2 - b2.

Thus, the angle will be measured twice. The results of measuring the angle by two half steps, respectively, are equal to p| and p 2 . R as_

the deviation of the angle values ​​from two half-points should not exceed twice the error in measuring the angle with this theodolite, i.e. the condition must be met

Where t- root-mean-square error of angle measurement in one step. For theodolite 2T30, this tolerance is G.

Measurement of angles by two half steps is carried out in order to:

• 1) measurement control;
• 2) improving measurement accuracy: the error of the mean value from several measurements is always less than the error of a single measurement.

The results of measuring horizontal angles are recorded in the appropriate journal (Table 5.1).

Table 5.1

Horizontal Angle Measurement Log

When measuring horizontal angles, it is important to understand the difference between the dial and alidade leads. When any of these screws is turned, the telescope rotates in a horizontal plane, or, as they say, "along the horizon." Although from the side of the observer's actions they seem to be exactly the same, the difference between them is fundamental. If the limb is fixed and the telescope is aimed at different points only with the help of alidade screws, then the readings will differ, since the limb remains motionless. If you act in the opposite way, i.e. fix the alidade, and when aiming the telescope at different points, use only the screws of the dial, the reading to any points will be the same, since the dial and the alidade with the telescope located on it will rotate together with the dial as a whole. It follows that if, when measuring the horizontal angle, the pipe was pointed to the right point and a reading was taken, and when pointing to the left point, the pointing or fixing screw of the limb was randomly turned, then it makes no sense to perform further actions, since the zero diameter of the horizontal circle will change its position . And in this case, it is necessary to start the execution of the half-reception again. The confusion between limbus screws and alidade screws is the most common mistake new theodolite students make.

If the accuracy of measuring angles in one go using the existing theodolite is slightly lower than required, then two options are possible:

• use a theodolite of higher accuracy;
• measure the angle not in one step, but P tricks. Then, as the final value of the angle, the average of P receptions, root mean square error M measurement of the angle in this case will be equal to

Where T- root-mean-square error of angle measurement in one step.

It should be noted that the error of multiple angle measurements decreases proportionally square root from the measurements. For example, to reduce the angle measurement error by 3 times, it is necessary to measure the angle in nine steps. Therefore, repeated measurement of the angle in order to improve the accuracy of measurements is justified only when the required accuracy differs slightly from the accuracy of the instrument used.

The state standard GOST 10529-86 distinguishes three groups of theodolites: high-precision, precise and technical.

High-precision theodolites provide measurement of angles with an error of no more than 1 "; types T1, T05.

Accurate theodolites provide measurement of angles with an error of 2" to 7"; types T2, T5.

Technical theodolites provide measurement of angles with an error from 10 "to 30"; types T15, T30.

An additional letter in the theodolite cipher indicates its modification or constructive solution: A - astronomical, M - mine surveying, K - with a compensator in a vertical circle, P - direct image pipe (terrestrial).

The state standard for theodolites provides, in addition, the unification of individual components and parts of theodolites; the second modification has the number 2 in the first position of the cipher - 2T2, 2T5, etc., the third modification has the number 3 - 3T2, 3T5KP, etc.

Before measuring the angle, it is necessary to bring the theodolite into working position, that is, perform three operations: centering, leveling and installing the telescope.

Centering the theodolite is setting the axis of rotation of the alidade above the apex of the angle being measured; the operation is performed using a plumb line suspended on the hook of the anchor screw, or using an optical plummet.

Leveling the theodolite is setting the axis of rotation of the alidade to vertical position; the operation is performed with the help of lifting screws and a level while alidade of a horizontal circle.

Tube placement is the placement of a tube over the eye and over the subject; the operation is performed with the help of a movable eyepiece ring (setting according to the eye - focusing the grid of threads) and the screw for focusing the tube on the object (pos.15 in Fig.4.4).

Angle measurements are carried out strictly according to the method corresponding to the measurement method; there are several ways to measure horizontal angles: this is a method separate corner(method of techniques), the method of circular techniques, the method in all combinations, etc.

Separate corner method. Measurement of a single angle consists of the following:

aiming the pipe at a point that fixes the direction of the first side of the corner (Fig. 4.16), with a circle to the left (CL), taking a reading L1;

turning the alidade clockwise and pointing the pipe at a point that fixes the direction of the second side of the corner; taking a reading L2,

calculation of the angle at CL (Fig. 4.16):

permutation of the dial by 1o - 2o for theodolites with one-sided reading and by 90o - for theodolites with two-sided reading,

transferring the pipe through the zenith and pointing it to a point that fixes the direction of the first side of the corner, with a right circle (KP); taking a reading R1,

turning the alidade clockwise and pointing the pipe at a point that fixes the direction of the second side of the corner; taking a reading R2,

angle calculation at CP:

when the condition |vl - vp|< 1.5 * t, где t - точность теодолита, вычисление среднего значения угла:

vsr = 0.5 * (vl + vp).

Measurement of the angle at one position of the circle (KL or KP) is one half-step; a full cycle of measuring the angle at two positions of the circle is one step.

Recording readings along the limb and calculating the angle are made in the journals of the established form.

Roundabout method. If more than two directions are observed from one point, then the method of circular techniques is often used. To measure angles in this way, you must perform the following operations (Fig. 4.17):

at CL, set a reading close to zero on the limb and point the pipe at the first point; take a reading on the limb.

rotating the alidade clockwise, point the pipe successively to the second, third, etc. points and then back to the first point; each time to take readings on the limb.

move the pipe through the zenith and, at the CP, direct it to the first point; take a reading on the limb.

rotating the alidade counterclockwise, point the pipe sequentially at (n-1), ..., third, second points and again at the first point; each time to take readings on the limb.

Then, for each direction, the averages are calculated from the readings at CL and CS, and after that, the values ​​of the angles relative to the first (initial) direction.

The method of circular receptions makes it possible to weaken the influence of errors acting proportionally to time, since the average readings for all directions refer to one physical moment in time.

Influence of theodolite eccentricity on readings along the limb. Let in Fig. 4.18 the axis of rotation of the alidade intersect horizontal plane at point B", and point B is the projection of the vertex of the measured angle onto the same plane. The distance between points B and B" will be denoted by l, the distance between points B and A - S.

If the theodolite were at point B, then when the pipe was pointed at point A, the reading on the limb would be equal to b. Let's move the theodolite to point B", while maintaining the orientation of the limb; in this case, the reading along the limb when the pipe is pointed at point A will change and become equal to b"; the difference between these readings is called theodolite centering error and is denoted by the letter c.

From the triangle BB "A we have:

or by the smallness of the angle c

The value l is called the linear centering element, and the angle Q is called corner element centering; Angle Q is constructed by projecting the theodolite's axis of rotation and is measured from the linear element in a clockwise direction to the direction of the observed point A.

The correct reading on the limb will be:

b = b" + c. (4.19)

Influence of target target reduction on readings along the limb.

If the projection of the sighting target A "on the horizontal plane does not coincide with the projection of the center of the observed point A, then an error occurs in the reduction of the sighting target (Fig. 4.19). The segment AA" is called the linear element of the reduction and is denoted by l1; the angle Q1 is called the corner element of the reduction; it is built during the projection of the sighting target and is measured from the linear element in a clockwise direction to the direction to the theodolite installation point. We denote the correct reading along the limb - b, the actual - b ", the error in the direction BA is r. From the triangle BAA" we can write:

or by the smallness of the angle r

The correct reading on the limbus will be

b = b" + r. (4.21)

The corrections c and r reach the greatest value at I = I1 = 90o (270o), when.

In this case

In the practice of measuring angles, two methods are used to take into account the eccentricity of the theodolite and the sighting target.

The first way is that the centering is performed with such an accuracy that it allows not to take into account the eccentricity error. For example, when working with technical theodolites, the allowable influence of errors in the centering of the theodolite and sighting target can be taken c = r = 10 "; with an average distance between points S = 150 m, it turns out that l = l1 = 0.9 cm, that is, theodolite or sighting it is enough to set the target above the center of the point with an error of about 1 cm.To center with such an accuracy, you can use an ordinary plumb line.Centering a theodolite or sighting target with an accuracy of 1-2 mm can only be done using an optical plummet.The second method is to directly measure the elements l and And, l1 and I1, calculating the corrections c and r using formulas (4.18) and (4.20) and correcting the measurement results with these corrections using formulas (4.19) and (4.21).The method for measuring the elements of theodolite centering and sighting target is described in.

The corners of products are measured by three main methods: comparison method with strict controls corner measures, squares, cone gauges and templates; absolute goniometric method, based on the use of instruments with a goniometric scale; indirect trigonometric method, which consists in determining the linear dimensions associated with the measured angle by a trigonometric function.

Universal means of measuring angles include vernier, optical and indicator goniometers, as well as other devices. The angles of inclination of the surfaces of products are measured by levels and optical squares.

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Link Nominal size, mm Size tolerance, mm Type of link Аδ

The results of the calculation of the constituent links
Link Nominal diameter, mm Tolerance, mm Lower deviation, mm Upper deviation, mm

Educational materials
References 1. Krylova G.D. Fundamentals of standardization, certification, metrology: Textbook for universities. – M.: Audit-UNITI.1998. 2. Lifits I.M. Fundamentals of standardization, metrolo