Basic calculation formulas of the “maximum-minimum” method. Collection of examples and tasks in metrology Processing of observation results and estimation of measurement errors

If with a change in the basic unit n times the derivative of the units changes n P times, then they say that this derivative unit has a dimension p relative to the basic unit.

The simplest example: the dimension of area or volume for those systems of units where the main unit is length. The dimension of area is two, the dimension of volume is three, because...

In more complex cases, if a unit of a certain quantity A has dimensions p, q and r relative to units of length, mass and time, then the dimension formula is written as:

where the symbols L, M and T are generalized designations for units of length, mass and force without specifically indicating the size of the units. This means that if each of the basic units is increased by 10 times, then the derived unit is increased by 10 pqr times.

It may turn out that the size of the derived unit is independent of any of the base units. In this case, the derived unit is said to be dimensionless or have zero dimension. For any choice of base units dimension formula is a monomial made up of symbols of base units, and these powers can be positive, negative, integer or fractional.

When forming dimension formulas, use the following theorems:

Theorem 1. If the numerical value of quantity C is equal to the product of the numerical values ​​of quantities A and B, then dimension C is equal to the product of dimensions A and B, i.e.

(2.2)

Theorem 2. If the numerical value of a quantity C is equal to the ratio of the numerical values ​​of A and B, then the dimension C is equal to the ratio of the dimensions A and B, i.e.

Theorem 3. If the numerical value of the quantity C is equal to the power n of the numerical value of the quantity A, then the dimension C is equal to the power n of the dimension A, i.e.

(2.4)

The proofs of these theorems are very simple, which can be illustrated by the proof of the first of them.

Let the numerical value C be equal to the product of the numerical values ​​A and B. When measured by units c 1, a 1 and b 1 we have

(2.5)

where C 1 = C/c 1; A 1 = A/a 1 ; in, = in/b 1.

Accordingly, when measuring the same quantities with units c 2, a 2 and b 2

(2.6)

where C 2 = C/c 2; A 2 = A/a 2 ; B 2 = B/b 2 .

From a comparison of C, A and B, expressed in different units, we obtain:

(2.7)

If now

(2.8)

(2.9)

(2.10)

Q.E.D.

Similarly, it is not difficult to prove the other two theorems. It is important to note that the dimension does not depend on the presence or absence of constant dimensionless factors or dimensionless quantities in the construction of the derived unit. This means, for example, that the dimension of the area of ​​a square

(2.11)

and area of ​​a circle

(2.12)

will be the same, since the coefficient does not depend on the size of the main units.

To conclude our consideration of the concepts of dimension, let us consider what changes in the formulas of dimension will occur with different choices of basic units. Obviously, in this case, the dimensional formulas will contain completely different expressions, since the connection of derived units, for example in mechanics, will change significantly when the basic unit of mass is replaced by the basic unit of force. For example, denoting the dimension of the basic unit of the MKGSS-force system with the symbol F, we obtain the dimension of mass:

(2.13)

The dimension of energy in the MKGSS system will be

(2.14)

From this expression, the attractiveness of the MKGSS system for mechanical calculations immediately becomes clear, since energy so simply depends on the basic units - force and length.

In conclusion of the section devoted to an overview of various systems of units, we mention that the dimension of derived units does not depend on the definition of the size of the derived unit. For example, if you express the areas of flat figures in square meters, when the unit of area is the area of ​​a square with a side equal to a unit of length, and then express the same area in “round” meters, i.e., define a unit of area as the area of ​​a circle with a diameter equal to one length, then the dimension of the area with such a redefinition will not change and will be equal to .

As mentioned above, the SI system includes seven basic, i.e., arbitrarily chosen, units of physical quantities. These units and their designations are given in table. 2.1.

Table 2.1.

Basic units of the international SI system

The basic units of the SI system were given appropriate definitions. Let us consider in more detail each of these units with explanations of the so-called implementation, i.e., the basic principles of their independent reproduction in international standards.

1.1. Definition of metrology.

1.2. Definition of measurement.

1.3. Types of measuring instruments.

1.4. Types and methods of measurements.

1.5. Accuracy of measurements.

1.6. Presentation of measurement results.

1.7. Rounding rules.

1.8. Unity of measurements.

1.9. Conclusion on the section.

2. Assessment of measurement errors based on the given metrological characteristics of measuring instruments.

2.1. Standardized metrological characteristics of measuring instruments.

2.1.1. Appointment of N.M.H.

2.1.2. Nomenclature of N.M.H., currently accepted.

2.1.2.1. N.M.H. necessary to determine the measurement result.

2.1.2.2. N.M.H., necessary to determine the measurement error.

2.1.3. The development trend of N.M.H. complexes

2.2. Estimates of errors in direct measurements with single observations.

2.2.1. Components of measurement error.

2.2.2. Summation of measurement error components.

2.2.3. Examples of estimating the error of direct measurements.

2.3. Estimation of errors of indirect measurements.

2.3.1. Components of errors in indirect measurements.

2.3.2. Summation of errors.

2.3.3. Examples of estimating errors of direct measurements.

2.4. Estimation of errors of indirect measurements.

2.4.1. Components of errors in indirect measurements.

2.4.2. Summation of direct measurement errors

2.4.3. Examples of estimating the error of indirect measurements.

3. Ways to reduce measurement errors.

3.1. Ways to reduce the influence of random errors.

3.1.1. Multiple observations with direct measurements.

3.1.2. Multiple observations with indirect measurements.

3.1.3. Smoothing of experimental dependencies using the least squares method for joint measurements.

3.2. Ways to reduce the influence of systematic errors.

4. Standardization.

Fundamentals of metrology and standardization.

Tyurin N.I. Introduction to metrology. - M.: Standards Publishing House, 1976.

1. Basic concepts of metrology.

Metrology cf.: biology, geology, meteorology.

Logos is a word, a relation (logometer).

"Logia" is the science of...

Subway metrology? metro - underground (French) - literally: capital (1863 - London; 1868 - New York; 1900 - Paris; 1935 - Moscow)

Metro policy- metropolis, main city.

Head waiter - head waiter, main, first - ratio, measure of primacy.

The meter is a measure of length, but: metrology is much older than the meter; meter was “born” in 1790, meter - from the Greek - measure.

Metrology - the study of measures (ancient dictionary).

“Russian metrology or a table comparing Russian measures, weights and coins with French ones.”

Linear and linear measures:

1 vershok=4.445 cm;

1 arshin=16 vershoks=28 inches - pipes

1 fathom = 3 arshins;

1 verst=500 fathoms

Capacity measures:

1 barrel=40 buckets;

1 bucket = 10 mugs (damask glasses);

1 mug=10 glasses=2 bottles=20 scales=1.229 l

Weights:

1 pood = 40 pounds = 16.380 kg;

1 pound=32 lots;

1 lot=3 spools;

1 spool=96 shares=4.266 g.

"Small spool but precious".

1 pound of medical weight = 12 ounces = 96 drams = 288 = 5760 grains = 84 spools.

Meticulous:not a grain.

Coins:

1 imperial=10 rubles (gold);

Silver: ruble, fifty dollars, quarter, two-kopeck piece, ten-kopeck piece, nickel.

Copper: three-kopek coin, penny (2 kopecks), 1 kopeck = 2 money = 4 half rubles.

The rich man fell in love with the poor woman,

A scientist fell in love with a stupid woman,

I fell in love with ruddy - pale,

Gold - copper half...

M. Tsvetaeva.

We are talking about concepts such as measures of length, measures of capacity, measures of weight...

Accordingly, there is a concept of length; capacity, or in modern language - volume; weight, or, as we now know, better to say mass, temperature, etc.

How to combine all these concepts?

Now we say that all these are physical quantities.

How to determine what a physical quantity is? How are definitions given in such an exact science as, for example, mathematics? For example, in geometry. What is an isosceles triangle? It is necessary to find a higher one in the hierarchical ladder of concepts; what concept stands above the concept of physical quantity? The superior concept is the property of an object.

Length, color, smell, taste, mass - these are different properties of an object, but not all of them are physical quantities. Length and mass are physical quantities, but color and smell are not. Why? What is the difference between these properties?

Length and mass are what we know how to measure. You can measure the length of the table and find out that it is so many meters. But you can't measure the smell, because... Units of measurement have not yet been established for it. However, smells can be compared: this flower smells stronger than this one, i.e. the concept applies to smell more less.

Comparing the properties of objects by type more or less is a more primitive procedure compared to measuring something. But this is also a way of knowing. There is an alternative representation when all parameters and relationships of objects and phenomena are designated as three classes of physical quantities.

The first class of physical quantities includes :

quantities, based on the number of sizes of which, are harder, softer, colder, etc. Hardness (the ability to resist penetration), temperature as the degree of heating of the body, the strength of the earthquake.

Second view: relations of order and equivalence not only between the sizes of quantities, but also between the differences in pairs of their sizes. Time, potential, energy, temperature associated with the thermometer scale.

Third type: additive physical quantities.

Additive physical quantities are quantities on the set of sizes of which not only the relations of order and equivalence, but also the operations of addition and subtraction are defined.

The operation is considered certain, if its result is also the size of the same physical quantity and there is a method for its technical implementation. For example: length, mass, thermodynamic temperature, current strength, emf, electrical resistance.

How does a child perceive the world? At first, of course, he doesn’t know how to measure anything. At the first stage, he develops the concepts of more and less. Then comes the stage that is closer to measurement - this is the counting of objects, events, etc. There is already something in common with measurement. What? That the result of counting and measuring is a number. Not relations like more - less, but a number. How do these numbers differ, i.e. number as a result of counting and number as a result of measurement?

The measurement result is a named number, for example 215m. The number 2.15 itself expresses how many units of length are contained in a given length of a table or other object. And the result of counting 38 pieces is something. Counting is counting, and measurement is measurement.

This is how the process of development of a child’s knowledge of the world proceeds, the same or approximately this is how the development of primitive man proceeded, i.e. at the first stage of comparing things by type more - less, then - counting.

Then comes the next stage, when you want to express in the form of a number something that cannot be counted by piece - the volume of liquid, the area of ​​a piece of land, etc., i.e. something continuous rather than discrete.

So, various physical quantities are measured, and a physical quantity is a property of an object, which is qualitatively common to many objects, and quantitatively individual for each given object.

Are there many physical quantities? With the development of human society, their list is constantly increasing. At first there were only length, area, volume, spatial quantities and time, then mechanical quantities were added - mass, force, pressure, etc., thermal quantities - temperature, etc. In the last century, electrical and magnetic quantities were added - current strength, voltage, resistance, etc. Currently there are more than 100 physical quantities. For brevity, in what follows, the word “physical” can be omitted and simply said size..

Concept magnitude contains qualitative sign, i.e. what is this quantity, for example length, and quantitative sign, for example, the length became 2.15m. But the same length of the same table can be expressed in other units, for example, in inches, and you get a different number. However, it is clear that the quantitative content of the concept “length of a given table” remains unchanged.

In this regard, the concept is introduced size quantities and concept meaning quantities. The size does not depend on the units in which the value is expressed, i.e. He invariant in relation to the choice of unit.

1.6.2 Processing observation results and estimating measurement errors

The error of the measurement result is assessed during the development of the MVI. Sources of errors are the OM model, measurement method, SI, operator, influencing factors of measurement conditions, algorithm for processing observation results. As a rule, the error of the measurement result is estimated using the confidence probability R= 0,95.

When choosing the P value, it is necessary to take into account the degree of importance (responsibility) of the measurement result. For example, if a measurement error could result in loss of life or severe environmental consequences, the P value should be increased.

1. Measurements with single observations. In this case, the result of a measurement is taken to be the result of a single observation x (with the introduction of a correction, if any), using previously obtained (for example, during the development of MVI) data on the sources that make up the error.

Confidence limits of the NSP measurement result Θ( R) is calculated using the formula

Where k(P) is the coefficient determined by the accepted R and number m 1 components of the NSP: Θ( R) - boundaries found by non-statistical methods j th component of the NSP (the boundaries of the interval within which this component is located, determined in the absence of information about the probability of its location in this interval). At P - 0.90 and P = 0.95 k(P) is equal to 0.95 and 1.1, respectively, for any number of terms m 1. At P=0.99 values k(P) the following (Table 3.3): Table 3.3

If the components of the NSP are distributed uniformly and are specified by confidence limits 0(P), then the confidence limit of the NSP of the measurement result is calculated using the formula

The standard deviation (RMS) of a measurement result with a single observation is calculated in one of the following ways:

2. Measurements with multiple observations. In this case, it is recommended to start processing the results by checking for the absence of errors (gross errors). A miss is the result of x n an individual observation included in a series of n observations, which, for given measurement conditions, differs sharply from the other results of this series. If the operator discovers such a result during the measurement and reliably finds its cause, he has the right to discard it and carry out (if necessary) additional observation to replace the discarded one.

When processing existing observation results, individual results cannot be arbitrarily discarded, as this can lead to a fictitious increase in the accuracy of the measurement result. Therefore, the following procedure is used. Calculate the arithmetic mean x of the observation results x i using the formula

Then the estimate of the standard deviation of the observation result is calculated as

expected miss x n from x:

Based on the number of all observations n(including x n) and the value accepted for measurement R(usually 0.95) according to or any reference book, but probability theories find z( P, n)— normalized sample deviation of the normal distribution. If Vn< zS(x), then observation x n is not a miss; if V n > z S(x), then x n is a miss to be excluded. After eliminating x n, repeat the determination procedure X And S(x) for the remaining series of observation results and checking for a miss of the largest of the remaining series of deviations from the new value (calculated based on n - 1).

The arithmetic mean x is taken as the measurement result [see. formula (3.9)] of the observation results xh The error x contains random and systematic components. The random component, characterized by the standard deviation of the measurement result, is estimated using the formula

It is easy to check whether the observation results x i belong to the normal distribution for n ≥ 20 by applying the 3σ rule: if the deviation from X does not exceed 3σ, then the random variable is normally distributed. Confidence limits of random error of measurement result with confidence probability R find by formula

Confidence limits Θ( R) The NSP of a measurement result with multiple observations is determined in exactly the same way as in a measurement with a single observation - using formulas (3.3) or (3.4).

Summation of the systematic and random components of the error of the measurement result when calculating Δ( R) is recommended to be carried out using criteria and formulas (3.6-3.8), in which S(x) is replaced by S(X) = S(X)/√n;

3. . The value of the measured quantity A is found from the results of measurements of the arguments alf ait at associated with the desired quantity by the equation

The type of function ƒ is determined when establishing the OP model.

The desired value A is related to the measured arguments by the equation

Where b i are constant coefficients

It is assumed that there is no correlation between measurement errors a i. Measurement result A calculated by the formula

Where and i— measurement result and i with the amendments introduced. Estimation of the standard deviation of the measurement result S(A) calculated using the formula

Where S(a i)- assessment of the standard deviation of the measurement result a i.

Confidence limits ∈( R) random error A with a normal distribution of errors a i

Where t(P, neff)— Student’s coefficient corresponding to the confidence probability R(usually 0.95, in exceptional cases 0.99) and the effective number of observations n eff calculated by the formula

Where n i-number of observations during measurement a i.

Confidence limits Θ( R) NSP of the result of such a measurement, the sum Θ( R) and ∈( R) to obtain the final value Δ( R) is recommended to be calculated using criteria and formulas (3.3), (3.4), (3.6) - (3.8), in which m i ,Θ i, And S(x) are replaced accordingly by m, b i Θ i, And s(A)
Indirect measurements with nonlinear dependence. For uncorrelated measurement errors a i the linearization method is used by expanding the function ƒ(a 1 ,…,a m) into a Taylor series, that is

where Δ a i = a i - a— deviation of an individual observation result a i from a i ; R- remainder term.

The linearization method is acceptable if the increment of the function ƒ can be replaced by its total differential. Remaining member neglected if

Where S(a)— estimation of the standard deviation of random errors in the measurement result a i. In this case, deviations Δ a i(must be taken from possible error values ​​and such that they maximize R.
Measurement result A calculated using the formula  = ƒ(â …â m).

Estimation of the standard deviation of the random component of the error in the result of such an indirect measurement s(Â) calculated by the formula

a ∈( P) - according to formula (3.13). Meaning n eff NSP boundary Θ( P) and error Δ( P) the result of indirect measurement with a linear dependence is calculated in the same way as with a linear dependence, but with the replacement of coefficients b i by δƒ/δa i

Casting method(for indirect measurements with nonlinear dependence) is used for unknown distributions of measurement errors and i and with correlation between errors and i to obtain the result of an indirect measurement and determine its error. In this case, it is assumed that there is a series n observation results and ij. measured arguments a i. Combinations and ij received in j experiment, substitute into formula (3.12) and calculate a series of values A j measured quantity A. The measurement result  is calculated using the formula

Estimation of the standard deviation s(Â)— the random component of the error  — is calculated using the formula

a ∈ ( R) - according to formula (3.11). Boundaries of the NSP Θ( R) and error Δ( R) measurement result  is determined by the methods described above for a nonlinear relationship.

The “maximum-minimum” method is based on the assumption that when assembling a mechanism, it is possible to combine increasing links made to the largest maximum dimensions with decreasing links made to the smallest maximum dimensions, or vice versa.

This calculation method ensures complete interchangeability during the assembly and operation of products. However, the tolerances of the component dimensions calculated using this method, especially for dimensional chains containing many links, can turn out to be unreasonably small in technical and economic terms, therefore this method is used to design dimensional chains with a small number of component links of low accuracy.

First task

The nominal size of the closing link can be determined by the formula (see example of the first problem).

If we take the total number of chain links n, then the number of components will be n – 1. Let's accept: m– number of increasing links, R – number of decreasing ones, then

n – 1 = m + p.

In general, the formula for calculating the nominal size of the closing link will be as follows:

(8.1)

For example (see section 8.1)

A0 = A 2 – A1 = 64 – 28 = 36 mm.

Based on equality (8.1), we obtain:

; (8.2)

. (8.3)

Subtract term by term from equality (8.2) equality (8.3), we obtain:

.

Since the sum of increasing and decreasing links is all the constituent links of the chain, the resulting equality can be simplified:

. (8.4)

Thus, the tolerance of the closing link is equal to the sum of the tolerances of all the constituent links in the chain.

To derive formulas for calculating the maximum deviations of the closing link, subtract term by term from equality (8.2) equality (8.1) and from equality (8.3) equality (8.1), we obtain:

; (8.5)

. (8.6)

Thus, the upper deviation of the closing dimension is equal to the difference between the sums of the upper deviations of the increasing and lower deviations of the decreasing dimensions; the lower deviation of the closing dimension is equal to the difference between the sums of the lower deviations of the increasing and upper deviations of the decreasing dimensions.

For the example of the first problem (see section 8.1) we get:

= 0.04 + 0.08 = 0.12 mm;

Thus,

Let us determine the tolerance of the closing link through the obtained maximum deviations:

This value coincides with the previously found tolerance value, which confirms the correctness of the problem solution.

Second task

When solving the second problem, the tolerances of the component dimensions are determined by the given tolerance of the closing dimension TA0 in one of the following ways: equal tolerances or tolerances of the same quality.

1. When deciding equal tolerance method – approximately equal tolerances are assigned to the component dimensions, guided by the average tolerance.

So, we assume that

then the sum of the tolerances of all component sizes is equal to the product of the number of component links and the average tolerance, i.e.:

.

Let's substitute this expression into equality (8.4): , from here

. (8.7)

By found value Tcp AI establish tolerances for component sizes, taking into account the size and responsibility of each size.

In this case, the following conditions must be met: the accepted tolerances must correspond to standard tolerances, the sum of the tolerances of the component dimensions must be equal to the tolerance of the trailing dimension, i.e. equality (8.4) must be satisfied. If equality (8.4) cannot be ensured with standard tolerances, then a non-standard tolerance is established for one component size, determining its value using the formula

. (8.8)

The equal tolerance method is simple and gives good results if the nominal sizes of the constituent links of the dimensional chain are in the same interval.

Let's solve the example of the second problem (see Section 8.1) using the equal tolerance method (8.7):

mm.

A1 = 215; TA1 = 0.04;

A2 = 60; TA2 = 0.04;

A3 = 155; TA3 = 0.04.

In this example, equality (8.4) is observed, and there is no need to adjust the tolerance of one of the component dimensions.

Let us write down equality (8.5) for this example:

0,12 = 0,06 – (-0,03 – 0,03).

(The numerical values ​​of the maximum deviations of the component dimensions are chosen conditionally.)

TA1 = 0.04, which means Ei(A1) = +0.02;

Ei(A2) = -0.03; TA2 = 0.04, which means Es(A2) = +0.01;

Ei(A3) = -0.03; TA3 = 0.04, which means Es(A3) = +0.01.

Let's check that equality (8.6) is satisfied:

0 = 0,02 – (0,01 +0,01);

Thus, we get the answer:

; ; .

2. A more universal and simplified selection of tolerances for any variety of sizes of component links is way tolerances of one qualification .

With this method, the dimensions of all component links (except for the corrective Aj) assign tolerances from one quality level, taking into account the nominal dimensions of the links.

To derive the formula, the initial dependence is equality (8.4):

.

However, the tolerance of any size can be calculated using the formula

Where A– the number of tolerance units, constant within one qualification (Table 8.1); - the tolerance unit depends on the nominal size of the component link (Table 8.2).

Table 8.1

Number of tolerance units

Meaning of tolerance units