Elements of a mathematical model of a measuring device. Model of error in the form of a random elementary function Mathematical model of measurement results of measurement error
4.3.1. Linear model of error change
In general, the error model L O95 (0 can be represented as A ogs (t) = A 0 + F(t), where D 0 is the initial error of the SI; F(t) is a function of time that is random for a set of SI of a given type, caused by physical and chemical processes of gradual wear and aging of elements and blocks. It is practically impossible to obtain an exact expression for the function F(t) based on physical models of aging processes. Therefore, based on data from experimental studies of changes in errors over time, the function F(t) approximated by one or another mathematical relationship.
The simplest model of error change is linear;
\ I (G) = D 0 + vt, (4.1)
where v is the rate of change of error. As studies have shown, this model satisfactorily describes the aging of the SI at the age of one to five years. Its use in other time ranges is impossible due to the obvious contradiction between the failure rates determined by this formula and the experimental values.
Metrological failures occur periodically. The mechanism of their periodicity is illustrated in Fig. 4.2a, where straight line 1 shows the change in the 95% quantile under a linear law.
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.......... A | ||
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Rice. 4.2. Linear (a) and exponential (b, c) laws of change
errors
In the event of a metrological failure, the error D 0 95 (0 exceeds the value D pr = D 0 + D s, where D s is the value of the margin of the normalized error limit necessary to ensure the long-term operability of the measuring instrument. With each such failure, the device is repaired, and its error returns to initial value D 0. After time T = A - t; a failure occurs again (moments t v t 2, t 3, etc.), after which repairs are made again. Consequently, the process of changing the SI error is described by broken line 2 in Fig. 4.2, a, which can be represented by the equation
d 095 (0 = D 0 + p D, (4.2)
where n is the number of failures (or repairs) of the SI. If the number of failures is considered an integer, then this equation describes discrete points on line 1 (Fig. 4.2,a). If we conditionally assume that I can also take fractional values, then formula (4.2) will describe the entire straight line 1 of the change in error D 095 (0 in the absence of failures.
The frequency of metrological failures increases with increasing speed v. It just as strongly depends on the margin of the normalized error value D s in relation to the actual value of the measuring instrument error D 0 at the time of manufacture or completion of repair of the device. The practical possibilities of influencing the rate of change v and the margin of error Dz are completely different. The rate of aging is determined by the existing production technology. The margin of error for the first overhaul interval is determined by the decisions made by the measuring instrument manufacturer, and for all subsequent overhaul intervals - by the level of culture of the user's repair service.
If the metrological service of the enterprise provides, during repairs, an SI error equal to the error D 0 at the time of manufacture, then the frequency of metrological failures will be low. If, during repairs, only the fulfillment of the condition D 0 = (0.9,...,0.95) D pr is ensured, then the error may go beyond the permissible values already in the coming months of operation of the SI and for most of the verification interval it will be operated with error exceeding its accuracy class. Therefore, the main practical means of achieving long-term metrological serviceability of a measuring instrument is to ensure a sufficiently large reserve Dz, normalized in relation to the limit D pr.
The gradual continuous consumption of this reserve ensures the metrologically sound state of the SI for a certain period of time. Leading instrument-making plants provide Dz = (0.4,...,0.5) L pr, which at an average aging rate v = 0.05 L pr per year allows obtaining a repair interval T = D/v = 8. ..,10 years and failure rate ω = 1/71 - 0.1,..., 0.125 year 1.
When the SI error changes in accordance with formula (4.1), all repair intervals T will be equal to each other, and the frequency of metrological failures ω = \/T will be constant throughout the entire service life.
4.3.2. Exponential error model
In reality, for some devices the overhaul intervals are reduced, while for others they are increased. This can be explained by the fact that the SI error increases or decreases exponentially over time. With an accelerating increase in error (Fig. 4.2.6), each subsequent overhaul interval is shorter than the previous one and the frequency of metrological failures a>(0 increases over time. With a slow increase in error (Fig. 4.2, c), each subsequent overhaul interval is longer than the previous one and the frequency of metrological failures failure rate co(0 decreases over time down to zero.
For the considered cases, changes in error over time are described based on an exponential model. It shows the frequency of metrological failures "
w(0 = w 0 e"", (4.3)
where (o 0 is the frequency of metrological failures at the time of manufacture of the measuring instrument (i.e. at t = 0), year "; a is the positive or negative acceleration of the metrological aging process, year 1.
The number of failures n(t) is determined through the failure frequency a>(0 and when it changes exponentially, according to formula (4.3), is calculated as
l(r) = /(o(T)rfr = \^e at dx = -1).
Then the change in time of the SI error, taking into account formula (4.2), has the form
D 0 95 (0 = D 0 + I(0D 3 = d 0 + (4.4)
This dependence is shown by curves 1 in Fig. 4.2,6 and 4.2,c.
Section I. METROLOGY
The practical use of formula (4.4) requires knowledge of four parameters: the initial error value (D 0), the absolute margin of error (D 3), the initial frequency of metrological failures (co 0) at t = 0 and the acceleration (a) of the aging process. The equations for determining these parameters, obtained from equation (4.4), turn out to be transcendental, which significantly complicates their application.
In order to simplify the use of equation (4.4), it is necessary to expand the exponential function into a series and take the first three terms of this expansion. As a result, the dependence of the SI error on time will be presented in the form
A o.95^ = A 0 + A 3°V + Az»o^/2 = \ + vt + af/2, (4.5)
where v is the initial rate of error increase, %; and A is the absolute value of the acceleration of the change in error, %. In the particular case when a = 0, (4.5) turns into a linear equation of the form (4.1).
The service life of a measuring instrument is the calendar time elapsed from the moment of its manufacture until the end of its operation. With a positive acceleration of the aging process (see Fig. 4.2.6), the frequency of failures increases with increasing service life, and after time Gcl it has to be repaired so often that operation becomes economically unprofitable, since it is cheaper to buy a new device. The economic feasibility of repair is determined by the ratio of the average cost of one repair with r to the cost with k of a new measuring instrument, called in the relative depth of repair c = with r / s k. Service life of measuring instruments
Chapter 4. Metrological reliability of measuring instruments
By solving the resulting equation together with the first expression from (4.6), it is possible to calculate the total number of failures (repairs) of the measuring instrument during its service life.
Example 4.1. For electromechanical measuring instruments of a magnetoelectric system of accuracy class 0.5, the repair depth is c = 0.3... 0.4; frequency of metrological failures at the time of manufacture w 0 = 0.11 year 1 , acceleration of the aging process a ~ 0.19 year 1 . Determine the service life of such devices and the total number of failures.
The service life of the device is calculated using formula (4.7):
T sd = ]/ Dz-OD 1-0.19 = 12.63 years.
The equation for calculating the total number of failures is:
p ъ = - [exp(i/ d/a s ■ (0 0)-1],
Substituting numerical data into it, we get
4 = ^19 1? xp (0 "19 /^ 0 "19 "°" 3 "0.1 1)~*]= 0.579(e 5 "8 -l)=5.8.
The calculation data corresponds to experimental data, according to which the average service life of the devices under consideration is 11-12 years, during which they undergo 4-6 repairs.
With a negative acceleration of the aging process of the SI, the overhaul period increases. After a certain number of repairs n L it becomes infinite, metrological failures do not occur and the SI operates until it becomes obsolete. In this case (a< 0) число метрологических отказов
% = = lim n(t) = lim^ (e al -1) = ^ . (->■" (->■" a a.
The SI error tends to a limit equal, according to (4.4),
D 0 95 (oo) = d 0 - -S- A 3 = DO + „_ d. (4.8)
The exponential model of the aging process makes it possible to describe changes in the fashionability of the SI as its age increases from one year to almost infinity. However, this model has a number of disadvantages. For SI with a negative acceleration of the aging process, it predicts at t-> °° the tendency of fashionability to the limit
value (4.8). At the same time, for SI with positive acceleration, the model predicts an unlimited increase in error over time, which contradicts practice.
Some of the disadvantages of the exponential aging model can be eliminated by using the so-called logistic model, as well as polynomial and diffusion Markov models or models based on autoregressive processes of the integrated moving average.
The technology uses a large number of reliability indicators, which are given in the GOST 27.002-89 standard. The main ones are also used in the theory of metrological reliability. Knowledge of metrological reliability indicators allows the consumer to optimally use measuring instruments, plan the capacity of repair areas, the size of the reserve fund of instruments, reasonably assign calibration intervals and carry out measures for maintenance and repair of measuring instruments.
Metrological failures during the operation of measuring instruments are more than 60% in the third year of operation and reach 96% when operating for more than four years.
As indicators of maintainability, the probability and average time of restoration of the SI operability are used. The probability of restoration of the working state is the probability that the time to restore the working state of the SI will not exceed a specified value. It represents the value of the recovery time distribution function at /= where is the specified recovery time. The average recovery time to a working state is the mathematical expectation of the recovery time, determined before its distribution function.
In general, the error model 0.95(t) can be represented as 0.95(t) = 0 + F(t), where D0 is the initial SI error; F(t) is a random function of time for a set of SIs of a given type, caused by physical and chemical processes of gradual wear and aging of elements and blocks. It is practically impossible to obtain an exact expression for the function F(t) based on physical models of aging processes. Therefore, based on data from experimental studies of changes in errors over time, the function F(t) is approximated by one or another mathematical dependence.
The simplest model of error change is linear:
where v is the rate of change of error. As the studies have shown, this model satisfactorily describes the aging of the SI at the age of one to five years. Its use in other time ranges is impossible due to the obvious contradiction between the failure rates determined by this formula and the experimental values.
Metrological failures occur periodically. The mechanism of their periodicity is illustrated in Fig. 1, a, where straight line 1 shows the change in the 95% quantile under a linear law.
![](https://i1.wp.com/studwood.ru/imag_/43/165966/image003.jpg)
Rice. 2.
In the event of a metrological failure, the error D0.95(t) exceeds the value Dpr=D0+nD3, where D3 is the value of the margin of the normalized error limit necessary to ensure the long-term operability of the measuring instrument. With each such failure, the device is repaired and its error returns to the original value D0. After time Tr = ti - ti-1, a failure occurs again (moments tt, t2, t3, etc.), after which repairs are made again. Consequently, the process of changing the SI error is described by broken line 2 in Fig. 1, a, which can be represented by the equation
![](https://i0.wp.com/studwood.ru/imag_/43/165966/image004.jpg)
where n is the number of failures (or repairs) of the SI. If the number of failures is considered to be an integer, then this equation describes discrete points on straight line 1 (Fig. 2, a). If we conditionally assume that n can also take fractional values, then formula (2) will describe the entire straight line 1 of the change in error D0.95(t) in the absence of failures.
The frequency of metrological failures increases with increasing speed v. It just as strongly depends on the margin of the normalized error value D3 in relation to the actual value of the measuring instrument error D0 at the time of manufacture or completion of repair of the device. The practical possibilities for influencing the rate of change v and the margin of error D3 are completely different. The rate of aging is determined by the existing production technology. The margin of error for the first overhaul interval is determined by the decisions made by the measuring instrument manufacturer, and for all subsequent overhaul intervals - by the level of culture of the user's repair service.
If the enterprise's metrological service provides, during repairs, an SI error equal to the D0 error at the time of manufacture, then the frequency of metrological failures will be low. If, during repairs, only the fulfillment of the condition D0 (0.9... 0.95) Dpr is ensured, then the error may go beyond the permissible values in the coming months of operation of the SI and for most of the verification interval it will be operated with an error exceeding its class accuracy. Therefore, the main practical means of achieving long-term metrological serviceability of a measuring instrument is to ensure a sufficiently large reserve D3, normalized in relation to the limit Dpr.
The gradual continuous consumption of this reserve ensures the metrologically sound state of the SI for a certain period of time. Leading instrument-making plants provide D3 = (0.4...0.5) Dpr, which at an average aging rate v = = 0.05AP/year allows us to obtain a repair interval Tp = D3 = 1/T/v = 8... 10 years and failure rate p= 0.1... 0.125 year-1.
When the SI error changes in accordance with formula (1), all repair intervals Тр = 1/Т will be equal to each other, and the metrological failure rate p will be constant throughout the entire service life. However, experimental studies have shown that this is not true in practice.
In general, the error model A 095 (i) can be presented as Do9 5 (?) = Up + F(t), where Do is the initial SI error; F(t)- a random function of time for a set of SI of a given type, caused by physical and chemical processes of gradual wear and aging of elements and blocks. Get the exact expression for a function F(t) Based on physical models of aging processes, it is practically impossible. Therefore, based on data from experimental studies of changes in errors over time, the function F(t) approximated by one or another mathematical relationship.
The simplest model of error change is linear:
Where v- rate of change of error. As the studies have shown, this model satisfactorily describes the aging of SI at the age of one to five years. Its use in other time ranges is impossible due to the obvious contradiction between the failure rates determined by this formula and the experimental values.
Metrological failures occur periodically. The mechanism of their periodicity is illustrated in Fig. 4.2, A, where by a straight line 1 shows the change in the 95% quantile with a linear law.
In the event of a metrological failure, the error D 095 (?) exceeds the value D pr = Up + D 3, where D is the value of the margin of the standardized error limit necessary to ensure the long-term operability of the measuring instrument. With each such failure, the device is repaired, and its error returns to the original value D^ Over time T? = t ( - - t j _ l failure occurs again (moments t u t 2 , t 3 etc.), after which repairs are carried out again. Consequently, the process of changing the SI error is described by broken line 2 in Fig. 4.2, A, which can be represented by the equation
Where P - number of failures (or repairs) of SI. If the number of failures is considered to be an integer, then this equation describes discrete points on a straight line 1
(see Fig. 4.2, A). If we conditionally accept that P can take fractional values, then formula (4.2) will describe the entire straight line 1 changes in error L 095 (() in the absence of failures.
Metrological failure rate increases with speed V. It just as strongly depends on the margin of the normalized error value D 3 in relation to the actual value of the measuring instrument error D 0 at the time of manufacture or completion of repair of the device. Practical possibilities for influencing the rate of change V and the margin of error D are completely different. The rate of aging is determined by the existing production technology. The margin of error for the first overhaul interval is determined by the decisions made by the measuring instrument manufacturer, and for all subsequent overhaul intervals - by the level of culture of the user's repair service.
If the metrological service of the enterprise provides, during repairs, an SI error equal to the error D 0 at the time of manufacture, then the frequency of metrological failures will be low. If, during repairs, only the fulfillment of the condition Up to * (0.9-0.95) D pr is ensured, then the error may go beyond the permissible values in the coming months of operation of the SI and for most of the verification interval it will be operated with an error exceeding its class accuracy. Therefore, the main practical means of achieving long-term metrological serviceability of a measuring instrument is to ensure a sufficiently large reserve D 3, normalized in relation to the limit D pr.
The gradual continuous consumption of this reserve ensures the metrologically sound state of the SI for a certain period of time. Leading instrument-making plants provide D 3 = (0.4-0.5) D pr, which at an average aging rate V= 0.05 D pr/year allows you to obtain a repair interval Г р = A 3 /i= 8-10 years and failure rate co = 1/Gy = 0.1-0.125 year -1.
When the SI error changes in accordance with formula (4.1), all repair intervals T will be equal to each other, and the frequency of metrological failures с = 1 /T will be constant throughout the entire service life.
In general, measurement results and their errors should be considered as functions that vary randomly over time, i.e. random functions, or, as they say in mathematics, random processes. Therefore, the mathematical description of the results and measurement errors (i.e., their mathematical models) should be built on the basis of the theory of random processes. Let us outline the main points of the theory of random functions.
By a random process X(t) is a process (function), the value of which for any fixed value t = tQ is a random variable X(t). A specific type of process (function) obtained as a result of experience is called implementation.
Rice. 4. Type of random functions
Each realization is a non-random function of time. The family of implementations for any fixed value of time t (Fig. 4) is a random variable called cross section random function corresponding to time t. Consequently, a random function combines the characteristic features of a random variable and a deterministic function. With a fixed value of the argument, it turns into a random variable, and as a result of each individual experiment it becomes a deterministic function.
Mathematical expectation random function X(t) is a non-random function which, for each value of the argument t, is equal to the mathematical expectation of the corresponding section:
where p(x, t) is the one-dimensional distribution density of the random variable x in the corresponding section of the random process X(t).
Variance random function X(t) is a non-random function whose value for each moment of time is equal to the dispersion of the corresponding section, i.e. dispersion characterizes the spread of realizations relative to m (t).
Correlation function- non-random function R(t, t") of two arguments t and t", which for each pair of argument values is equal to the covariance of the corresponding sections of the random process:
The correlation function, sometimes called autocorrelation, describes the statistical relationship between the instantaneous values of a random function separated by a given time value t = t"-t. If the arguments are equal, the correlation function is equal to the variance of the random process. It is always non-negative.
Random processes that occur uniformly in time, partial implementations of which oscillate around the average function with a constant amplitude, are called stationary. Quantitatively, the properties of stationary processes are characterized by the following conditions:
The mathematical expectation is constant;
The cross-sectional dispersion is a constant value;
The correlation function does not depend on the value of the arguments, but only on the interval.
An important characteristic of a stationary random process is its spectral density S(w), which describes the frequency composition of the random process for w>O and expresses the average power of the random process per unit frequency band:
The spectral density of a stationary random process is a non-negative function of frequency. The correlation function can be expressed in terms of spectral density
When constructing a mathematical model of measurement error, all information about the measurement being carried out and its elements should be taken into account.
Each of them can be caused by the action of several different sources of errors and, in turn, also consist of a certain number of components.
Probability theory and mathematical statistics are used to describe errors, but first it is necessary to make a number of significant reservations:
The application of methods of mathematical statistics to the processing of measurement results is valid only under the assumption that the individual readings obtained are independent of each other;
Most of the probability theory formulas used in metrology are valid only for continuous distributions, while error distributions due to the inevitable quantization of samples, strictly speaking, are always discrete, i.e. the error can only take a countably many values.
Thus, the conditions of continuity and independence for measurement results and their errors are observed approximately, and sometimes are not observed. In mathematics, the term “continuous random variable” is understood as a significantly narrower concept, limited by a number of conditions, than “random error” in metrology.
In metrology, it is customary to distinguish three groups of characteristics and error parameters. The first group is the measurement errors (standards of errors) specified as required or permissible standards for characteristics of measurements. The second group of characteristics are errors attributed to the totality of measurements performed according to a certain technique. The characteristics of these two groups are used mainly in mass technical measurements and represent probabilistic characteristics of the measurement error. The third group of characteristics—statistical estimates of measurement errors—reflect the proximity of a separate, experimentally obtained measurement result to the true value of the measured quantity. They are used in the case of measurements carried out during scientific research and metrological work.
A set of formulas describing the state, movement and interaction of objects obtained within the framework of selected physical models based on the laws of physics will be called mathematical model of an object or process. The process of creating a mathematical model can be divided into a number of stages:
1) drawing up formulas and equations that describe the state, movement and interaction of objects within the framework of the constructed physical model. The stage includes recording in mathematical terms the formulated properties of objects, processes and connections between them;
2) study of mathematical problems that are approached at the first stage. The main issue here is the solution of the direct problem, i.e. obtaining numerical data and theoretical consequences. At this stage, mathematical apparatus and computing technology (computer) play an important role.
3) finding out whether the results of analysis and calculations or consequences from them are consistent with the results of observations within the accuracy of the latter, i.e. whether the adopted physical and (or) mathematical model satisfies the practice-the main criterion for the truth of our ideas about the world around us.
The deviation of the calculation results from the observation results indicates either the incorrectness of the applied mathematical methods of analysis and calculation, or the incorrectness of the adopted physical model. Determining the sources of errors requires great skill and highly qualified researchers.
Often, when constructing a mathematical model, some of its characteristics or relationships between parameters remain uncertain due to the limited knowledge of our knowledge about the physical properties of the object. For example, it turns out that the number of equations describing the physical properties of an object or process and the connections between objects is less than the number of physical parameters characterizing the object. In these cases, it is necessary to introduce additional relationships characterizing the object of study and its properties, sometimes even trying to guess these properties, so that the problem can be solved and the results correspond to the experimental results within a given error.
Production errors can be considered as random variables described by probabilistic (theoretical) and statistical (experimental) methods. A comprehensive characteristic of error as a random variable is the distribution law with specific values of the corresponding parameters. The description of the distribution of production errors is most consistent with Gauss's law with a probability density calculated by the formula:
Where T and σ – mathematical expectation and standard deviation.
The Gaussian distribution has been repeatedly confirmed by experimental data in the range of values corresponding to the range ±3σ. According to this distribution, the alignment error at a particular point εх in the direction X is perceived as a random variable distributed according to a normal law, with the following characteristics:
(3.16)
Where rx– correlation coefficient between the displacement values of neighboring unit sections in the direction X; C2x– number of combinations of X 2 each, calculated from the expression
From relations (3.15) and (3.16) an analytical notation for the probability density of the distribution of quantities is derived:
Graphs of the dependence of alignment errors on the coordinates of points along one axis, resulting from relation (3.18), are shown in Fig. 3.59.
Rice. 3.59. Diagram of layer alignment errors in direction X
If statistical data is available, the numerical characteristics of the distribution (3.18) can be found for a section of length L with grid spacing h. They are found from the relations:
(3.19)
Where M.L., σ L– respectively, the mathematical expectation and dispersion of the deformation of a section of length L; – number of combinations of L/ h by 2.
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