The work deals with a problem of estimation of spring sti ness under conditions of uncertainty of its loading and compression measurements. During spring exploitation, the main its parameters sti ness and initial length change since of the metal strain aging. At the spring checking, values of its compression are measured in the necessary range of loading. It is used to describe the spring compression by the Hooke's law. In practice, a sample of measurements is very short and measuring is implemented with errors, which probability data are unknown. So, it is di cult to validate application of standard statistical methods to estimation of the spring sti ness. Under such conditions, an alternative to now existing approaches is application of the Interval Analysis methods that do not use any probabilistic properties of the measuring errors. In the work, the Interval Analysis methods are used for constructing the information set of admissible values of the spring parameters (the sti ness and initial length).
During practical exploitation of steel springs, some changing of its material appear. Under this, check (tests) of the spring is performed with prescribed time period.
Usually, for estimation of the spring parameters (the sti ness and initial length), the standard methods to estimation are used that are based on the mathematical statistics ideology [1], [2], [3].
But in practice, a sample of measurements is very short and measuring the loading force and spring compression are implemented with errors, which probability data are unknown. Moreover, the model of the compression process is approximately described by the linear dependence corresponding to the Hooke's law, and the sample of measurements is rather short. So, under these conditions, it is di cult to strictly validate application of standard methods, and in practice, they are often used rather formally. As an alternative, application of the statistical methods can be completed by using the Interval Analysis ones.
For constructing a set of admissible values of the spring parameters, the Interval Analysis methods use only the model description of the process (the Hooke's law) and the bounds on the measuring errors of the loading force and compression value.
Such a set is used to call the Information Set. It comprises only such parameters of the model that are consistent with its description, accumulated sample of measurements, and the given bounds onto the mentioned measuring errors. On the basis of the determined Information Set, corresponding tube of the admissible dependencies of the spring under compression is formed.
The paper has the following structure. In Section 2, the technical details of the experiment are considered and typical model of the spring compression vs loading is introduced. Section 3 is devoted: to description of the main Interval Analysis procedures used for constructing the Information Set of the spring parameters (the sti ness and initial length) and the problem of investigation is formulated. In Section 4, for illustration of the elaborated estimation algorithms, a model example is investigated on a sample of measurements of a spring compression. Moreover, comparison of this results is made with ones obtained by the standard Least Square Means method (LSQM) [2]. In Section 5, conclusions are given on abilities of the suggested Interval Analysis approach and its applications in addition to known estimation procedures [1], [2], [3]. 2
Details of experiment for investigation of spring parameters The following experiment is performed. 1. The spring is put under the loading press. 2. The loading is performed by a collection of given forces Fn
fFng; n = 1; N ; Fn 2 [F1; FN ]; 0 < F1;
from the working range [F1; FN ] of the spring.
3. For each force value Fn, the spring compression (length) is measured
(
fsng; n = 1; N :
fFn; sng; n = 1; N :
5. Since of the measuring errors both in the loading force and in the spring
compression, data (
Remark 1. We suppose that
{ measurements (
As a rule, actual description of the spring compression (changing its length vs the compression force) is unknown for a researcher (Fig. 1). Here, the true curve is in dashes; true values of the force F and compression s are marked by circles; the rectangles in dots are regions of possible location of authentic measurements.
So, for elaboration of algorithms for estimating the compression parameters,
the linear model corresponding to the Hooke's law is used in the form
S(F; a; s0) = aF + s0;
(
In the linear model (
Often in practice, it is possible to show some useful approximate intervals for
the spring parameters in model (
Interval approach to estimating the spring parameters. Problem formulation Foundations of Interval Analysis and its applications to processing observations under uncertainty conditions were developed on the basis of the fundamental pioneering work of L.V. Kantorovich [4].
Nowadays, e ective theoretical, applied, and numerical methods of the Interval Analysis have been created both abroad [5], [6] and by Russian researchers [7], [8]. Special interval algorithms and software were developed for solving applied problems of estimation of parameters for experimental chemical processes [10], [11], [12], [13], [14].
Remind that the essence of the interval methods is in estimating the process parameters under conditions of a short measurement sample, uncertainty of the measuring errors probability characteristics and under only interval bounding onto the error values.
Introduce the following necessary de nitions (with using the standard on notations in Interval Analysis [9]).
The uncertainty set of the spring loading and compression measurement. Since absence of probability characteristics of measuring errors, uncertainty of each measurement Fn; sn is formalized (Fig. 2) as a rectangle Hn with the left F n and right F n, lower sn and upper sn boundaries.
n = 1; N ; Hn : 7a) F n = [F n; F n];
where F n = Fn 7b) sn = [sn; sn]; where sn = sn emax; F n = Fn + emax; bmax; sn = sn + bmax:
The admissible value of the parameters vector (a; s0) for model (
The Information Set (INFS) is a totality of all admissible values of the
parameter vector for model (
I(a; s0) = f(a; s0) : S(n; a; s0) 2 F n
sn; n = 1; N g:
(
H k _Fn = Fn_ emax Hn
Hn . . .
of the linear type with a, s0. . .
. . .
F 1
Fk
Fn _ Fn = Fn+ emax _ sn = sn + bmax _sn = sn _ bmax
H
N
F, N FN
The input sample (
The tube of admissible dependencies T b(F ) (Fig. 3) is a totality of all
admissible values of the dependence on the loading force F . For model (
F 2 [F1; FN ] : T b(F ) = min(a;s0)2I(a;s0) S(F; a; s0);
T b(F ) = max(a;s0)2I(a;s0) S(F; a; s0):
Problem of estimation is formulated as follows: by means of the Interval
Analysis methods to construct the Information Set (
For the linear model (
Remark 2. As it will be shown below, formal application of the standard
Least Square Means method (LSQM) [2] and corresponding point-wise estimate
of the parameters ((aSQ; s0;SQ) for the linear model) demonstrate to be useful for
(
Hk
Tube of admissible dependences
analysis of the input sample (
Elaborated engineering interval procedures
For simplicity of narration, consider a practical case when for all n
F n < F n+1 and sn > sn+1:
(
Consider a pair of uncertainty sets Hn and Hk, n = 2; N , k = n 1, with their corner points 1, 2, 3, 4 and 5, 6, 7, and 8 (Fig. 4a). For each pair, the bunch of admissible dependences is introduced. The bunch is bounded by the lines LI and LIII with the extremal values of parameters a min; s0;max and amax; s0;min and by the lines LII and LIV with the intermediate values of parameters aII; s0;II and aIV; s0;IV, correspondingly.
Under conditions (
F k); s0;max = sk + aminF k;
F k); s0;min = sk + amaxF k; F k); s0;II = sk + aIIF k;
F k); s0;IV = sk + aIVF k:
(
III
Fk
Hk 2 3 lines with intermediate values of parameters a, s0 Note that for homogeneous sample (Remark 1) aII = aIV.
From Figure 4a it is seen that in the case of mutual location (
For each pair Hk and Hn, the partial information set G(a; s0)k;n of
parameters is determined by the computed apex points (
Hn 6 (amax,s0min)
L
I 7
L (aIIVV,s0,IV)
LII (aII,s0,II) LIII (amin,.s.0.max)
F, N
Having constructed the collection of partial information sets fGk;n(a; s0)g,
n = 2; N , k = n 1, it becomes possible to calculate the desirable information
set I(a; s0) (
I(a; s0) = Tn=2;N; k=n 1 Gk;n(a; s0):
(
_
a
_
a
acnt
a, mm/N
a priori
rectangle
aapr s0,apr
_
s0
s0,cnt
s0
_
(
This procedure is implemented by some appropriate standard program of
intersection of convex polygons. Image of the information set I(a; s0) is presented in
Fig. 5 (shadowed by red). It is determined by the apex points I, II, III, and IV
(a; s0)I; :::; (a; s0)IV:
The information set (
For practical using, the following parameters' values are given to a researcher: { exact description of the information set I(a; s0); { unconditional interval in parameter a : [a; a]; { unconditional interval in parameter s0 : [s0; s0]; { central point acnt = 0:5(a + a); s0;cnt = 0:5(s0 + s0): 4
Model example. Results of estimation of spring parameters For computations, the following input model data are given: { the initial true length of the spring s0 = 120 mm; { the sti ness true value a = 0:1020 mm/N; { number of measurements N = 5; { the true loading forces are fFn g = f196; 392; 588; 784; 980g N; { the true values of compression are fsng = f100; 80; 60; 40; 20g mm; { the bound on the loading force measurement emax = 40 N; { the bound on the compression measuring bmax = 2 mm; { corrupted measured values of the loading forces are fFng = f221:9; 364:2; 611:4; 778:2; 983:5g N; { corrupted measured values of compression are fsng = f98:28; 78:22; 61:34; 41:90; 18:02g mm.
After procession, the resultants information set I(a; s0) is presented in Fig.6. It is a convex polygon with ve apices. It has sizes: a = 0:1189 mm/N, a = 0:0905 mm/N, s0 = 112:76 mm, s0 = 128:317 mm.
The information set central point is acnt = 0:1047 mm/N, s0;cnt = 120:53 mm.
Formal estimation of the input corrupted sample by the Least Squares Means method gave the point aSQ = 0:1009 mm/N, s0;SQ = 119:31 mm; and estimation of the standard deviation is = 3:12 mm.
Central point
LSQM-point
It is seen (Fig. 6) that since of the outliers absence, both the true point and the LSQM one are inside the information set.
Picture of the compression process and results of procession are also shown in Fig. 7. Note that formal application of the standard rule \ 3 " gives approximate very wide \corridor" around the LSQM-line.
In a contrast, application of the described interval approach allows one to build signi cantly narrow tube of admissible dependences. Moreover, it is seen that uncertainty sets of measurements have been enhanced: their inadmissible parts are cut o by the tube.
Underline that due to admissibility of the true and the LSQM points (Fig. 6), corresponding line dependencies are whole inside the tube (Fig. 7). rseaeumtirceann
Mu o L e l b i iss ise 2 9 3 The Interval Analysis methods was applied to estimation of spring parameters in the compression process under conditions of absence of probability data for the measuring errors.
Important case was investigated when errors are both in the loading force measurements and in ones of the spring compression (length).
Investigations are ful lled on the basis of the wide used Hooke's law with the linear dependence of spring compression vs the loading force.
It was shown that under mentioned conditions Interval Analysis approach gives guaranteed estimation of the process parameters and better estimation of the tube of admissible dependencies.
Moreover, simulation results show that using simultaneously, the interval and standard statistical approaches complement each other; and this allows one to perform more detailed analysis and qualitative comparison of the estimation results.
Acknowledgments. The work was supported by Act 211 Government of the Russian Federation, contract no. 02.A03.21.0006.