=Paper=
{{Paper
|id=Vol-1814/paper-08
|storemode=property
|title=Estimation of Spring Stiffness Under Conditions of Uncertainty. Interval Approach
|pdfUrl=https://ceur-ws.org/Vol-1814/paper-08.pdf
|volume=Vol-1814
|authors=Sergey I. Kumkov
}}
==Estimation of Spring Stiffness Under Conditions of Uncertainty. Interval Approach==
https://ceur-ws.org/Vol-1814/paper-08.pdf
Estimation of Spring Stiffness
under Conditions of Uncertainty.
Interval Approach
Sergey I. Kumkov1,2
1
Krasovskii Institute of Mathematics and Mechanics, Ural Branch RAS
and 2 Ural Federal University, Yekaterinburg, Russia.
kumkov@imm.uran.ru
Abstract. The work deals with a problem of estimation of spring stiff-
ness under conditions of uncertainty of its loading and compression mea-
surements. During spring exploitation, the main its parameters stiffness
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 difficult to validate application of standard statistical methods to es-
timation of the spring stiffness. 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 in-
formation set of admissible values of the spring parameters (the stiffness
and initial length).
Keywords: Spring, parameters, stiffness, initial length, estimation, in-
terval analysis methods
1 Introduction
During practical exploitation of steel springs, some changing of its material ap-
pear. Under this, check (tests) of the spring is performed with prescribed time
period.
Usually, for estimation of the spring parameters (the stiffness 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 pro-
bability 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 difficult to strictly validate application of standard methods, and in prac-
tice, they are often used rather formally. As an alternative, application of the
statistical methods can be completed by using the Interval Analysis ones.
64
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 parame-
ters 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 stiffness 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
{Fn }, n = 1, N , Fn ∈ [F1 , FN ], 0 < F1 , (1)
from the working range [F1 , FN ] of the spring.
3. For each force value Fn , the spring compression (length) is measured
{sn }, n = 1, N . (2)
4. As a result of the test, the following sample is obtained:
{Fn , sn }, n = 1, N . (3)
5. Since of the measuring errors both in the loading force and in the spring
compression, data (3) are corrupted as follows:
n = 1, N ,
(4)
Fn = Fn∗ + en , |en | ≤ emax , sn = s∗n + bn , |bn | ≤ bmax ,
where Fn is the loading measurement; Fn∗ is the unknown true value of loading;
en is the additive measuring error of the loading force; emax is the bound (on
65
modulus) on the loading measuring error; sn is the compression measurement;
s∗n is the unknown true value of the compression; bn is the additive measuring
error; bmax is the bound (on modulus) on the compression measuring error. It is
assumed that in the neighbor measurements errors are independent.
Remark 1. We suppose that
– measurements (4) are homogeneous, i.e., the bounding values emax and bmax
in (4) are the same in all measuring;
– sample (3) is authentic i.e., it does not contain outliers.
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.
s, mm
s*0
s1 Unknown true dependence
Region of possible
sk location of authentic
measurement
Measurement
sn of loading force F
and compression s
Unknown
true value Fn*,sn*
sN under measuring
... ... ...
F, N
F1 Fk Fn FN
Fig. 1. True compression dependence and results of its measuring.
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 , (5)
where S(...) is the current value of the spring compression (length), mm; F is
the loading force, N; a < 0 is the stiffness, mm/N; s0 is the spring initial length,
mm.
In the linear model (4), its parameters s0 and a are assumed to be constant
during the time interval of the spring exploitation.
66
Often in practice, it is possible to show some useful approximate intervals for
the spring parameters in model (5)
aapr = [aapr , aapr ], s0,apr = [s0,apr , s0,apr ]. (6)
(Here and further in the text, we keep at the standard notations accepted
for the interval variables [9]).
3 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, effective theoretical, applied, and numerical methods of the Inter-
val Analysis have been created both abroad [5], [6] and by Russian researchers
[7], [8]. Special interval algorithms and software were developed for solving ap-
plied 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 definitions (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 H n with the left
F n and right F n , lower sn and upper sn boundaries.
n = 1, N , H n :
7a) F n = [F n , F n ],
where F n = Fn − emax , F n = Fn + emax ; (7)
7b) sn = [sn , sn ],
where sn = sn − bmax , sn = sn + bmax .
The admissible value of the parameters vector (a, s0 ) for model (4),(5)
is a pair
(a, s0 ) : S(n, a, s0 ) ∈ F n × sn for all n = 1, N , (8)
and corresponding dependence S(n, a, s0 ) is also called admissible.
The Information Set (INFS) is a totality of all admissible values of the
parameter vector for model (4),(5) satisfying the following system of interval
inequalities:
I(a, s0 ) = {(a, s0 ) : S(n, a, s0 ) ∈ F n × sn , n = 1, N }. (9)
67
s0 s, mm _
H1 _Fn = Fn _ emax Fn = Fn+ emax
s1
_
sn = sn + bmax
Hn
Hk _sn = sn _ bmax
sk
Uncertainty Hn
sn set of measurement
HN
Admissible dependence
sN of the linear type with a, s0
... ... ...
F, N
F1 Fk Fn FN
Fig. 2. Measurements, their uncertainty sets, and admissible dependence.
The input sample (3) is called consistent in the interval sense if by (9) there
exists at least one admissible value of the parameter vector and corresponding
admissible dependence.
The tube of admissible dependencies T b(F ) (Fig. 3) is a totality of all ad-
missible values of the dependence on the loading force F . For model (5) and
the information set I(s0 , a), the tube boundaries are calculated with taking into
account the loading force F interval
F ∈ [F1 , FN ] :
T b(F ) = min(a,s0 )∈I(a,s0 ) S(F, a, s0 ), (10)
T b(F ) = max(a,s0 )∈I(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 (9) of the spring coefficients
values consistent with the given data (3) and to build the tube (10) of the
admissible values of the spring compression.
For the linear model (5), fast procedures for constructing the Information Set
(9) with exact description of its boundaries have been elaborated and applied to
solving many practical problems [10], [11], [12], [13], [14]. Due to direct using the
linearity of model (4), these procedures are more fast and give exact description
of the Information Set in comparison with even very powerful procedures of the
SIVIA-type [5].
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
68
s, mm
s1
H1
Upper boundary of the tube
sk
Hn
Hk
sn
Lower boundary of the tube HN
sN
Tube of admissible dependences
... ... ...
F, N
F1 Fk Fn FN
Fig. 3. Tube of admissible dependencies.
analysis of the input sample (3) and for qualitative comparison with the results
on the basis of Interval Analysis.
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 . (11)
For other mutual location of uncertainty sets, computation formulas are derived
in the similar way.
Consider a pair of uncertainty sets H n and H k , 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 (10), parameters of these lines are computed as follows:
LI : a min = (sn − sk )/(F n − F k ), s0,max = sk + amin F k ;
LIII : amax = (sn − sk )/(F n − F k ), s0,min = sk + amax F k ;
(12)
LII : aII = (sn − sk )/(F n − F k ), s0,II = sk + aII F k ;
LIV : aIV = (sn − sk )/(F n − F k ), s0,IV = sk + aIV F k .
69
Note that for homogeneous sample (Remark 1) aII = aIV .
From Figure 4a it is seen that in the case of mutual location (11) of the pair
H n and H k , the bunch of admissible dependencies is completely determined
by the diagonals “2–4” and “6–8”. This allows to use techniques elaborated for
one-dimensional uncertainty sets of measurements [10], [13], [14].
For each pair H k and H n , the partial information set G(a, s0 )k,n of
parameters is determined by the computed apex points (11). Note (Fig. 4b) that
it is a convex four-apex polygon with linear boundaries.
s, mm а)
Hk
1 2 lines with the extremal
values of parameters s0 ,a
sk
3 Hn
4 6 (amax,s0min)
5 LI
sn LIV
(aIV,s0,IV)
lines with intermediate 7
values of parameters a, s0 8 LII (aII,s0,II)
...
LIII (amin,.s. 0max
.
)
...
F, N
Fk Fn
b) s0 , mm
III Gk,n(a, s0)
(amin,s0max)
IV ~ LIV
II ~ LII
(amax,s0min) I
a, mm/N
Fig. 4. Constructing the partial information set; a) a pair of uncertainty sets; b) partial
information set of possible values of compression parameters.
Having constructed the collection of partial information sets {Gk,n (a, s0 )},
n = 2, N , k = n − 1, it becomes possible to calculate the desirable information
set I(a, s0 ) (9) of admissible values of the compression parameters for the whole
sample (3) of measurements
T
I(a, s0 ) = n=2,N , k=n−1 Gk,n (a, s0 ). (13)
70
s0 , mm
U
I(a, s0,emax,bmax) = _ Gk,n
n = 2, N
k = n _1
_
s0
...
Gk,n(a, s0 )
s0,cnt
...
a priori s_0
rectangle
aapr s0,apr
_
_a acnt a a, mm/N
Fig. 5. Output information set I(a, s0 ) of admissible values of compression parameters
a, s0 ; a priori data rectangle (in dots).
This procedure is implemented by some appropriate standard program of inter-
section 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 . (14)
The information set (12) can be compared with the a priori data rectangle
aapr × s0,apr (6), and there is possibility of its enhancing (Fig. 5).
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];
(15)
– 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 s∗0 = 120 mm;
– the stiffness true value a∗ = −0.1020 mm/N;
– number of measurements N = 5;
– the true loading forces are {Fn∗ } = {196, 392, 588, 784, 980} N;
– the true values of compression are {s∗n } = {100, 80, 60, 40, 20} 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 {Fn } = {221.9, 364.2,
71
611.4, 778.2, 983.5} N;
– corrupted measured values of compression are {sn } = {98.28, 78.22, 61.34,
41.90, 18.02} mm.
After procession, the resultants information set I(a, s0 ) is presented in Fig.6.
It is a convex polygon with five 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 estima-
tion of the standard deviation is σ = 3.12 mm.
s0 , mm _
s0
125
Central point
LSQM-point
120
True point
115
_s 0
_
a_ a
a,
mm/N
_ _ _ _
0.12 0.11 0.10 0.09
Fig. 6. Model example; resultant information set.
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 “±3σ” gives appro-
ximate very wide “corridor” around the LSQM-line.
In a contrast, application of the described interval approach allows one to
build significantly narrow tube of admissible dependences. Moreover, it is seen
that uncertainty sets of measurements have been enhanced: their inadmissible
parts are cut off 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).
72
F, N
Measurements and
980
True dependence
uncertainty sets
and true values
Upper boundary
Tube of admissible
of tube
784
dependencies
Lower boundary
of tube
588
+3s
392
_3s
LSQM-line
s, mm
196
100
80
20
40
60
Fig. 7. Model example; results of processing.
5 Conclusions
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 fulfilled on the basis of the wide used Hooke’s law with the
linear dependence of spring compression vs the loading force.
73
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 re-
sults.
Acknowledgments. The work was supported by Act 211 Government of the
Russian Federation, contract no. 02.A03.21.0006.
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