Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

49, 2, pp. 419-438, Warsaw 2011

SENSITIVITY ANALYSIS OF AN IDENTIFICATION

METHOD DEDICATED TO NONLINEAR SYSTEMS

WORKING UNDER OPERATIONAL LOADS

Joanna Iwaniec

AGH University of Science and Technology, Department of Robotics and Mechatronics, Kraków,

Poland; e-mail: jiwaniec@agh.edu.pl

In the paper, the exploitational nonlinear systems identificationmethod
based on algorithms of the restoring force, boundary perturbations and
direct parameter identification methods is presented. The obtained pa-
rameter estimates provide information concerning forces transferred on
the foundation and find application in the model-based diagnostics.
The results of the sensitivity analysis carried out in order to assess the
influence of input parameters uncertainties (accuracy of resonant fre-
quency and amplitude estimates, errors of transfer function estimation
in operational conditions, value of introduced additional mass) on the
accuracy of estimated system parameters are also presented.

Key words: sensitivity analysis, nonlinear system identification, opera-
tional loads

1. Introduction

Each realmechanical structure is nonlinear to some degree. Typical sources of
nonlinearities are of geometrical origin (Kerschen et al., 2006; Nayfeh andPai,
2004) resulting from considerable structure deformations, physical nonlineari-
ties (Kerschen et al., 2006; Schultze et al., 2001) related to nonlinearmaterial
properties, nonlinear damping forces (Al-Bender et al., 2004) deriving from
energy dissipation phenomena (e. g. dry, internal friction), nonlinear bounda-
ry conditions (Babitsky andKrupenin, 2001; Kerschen et al., 2006) related to
appearance of clearances and structural nonlinearities arising fromapplication
of structural elements of discrete nonlinear characteristics, such as springs and
absorbers.
Although the sources of nonlinear system properties can vary, all the non-

linear systems have some common properties. In general, they do not follow



420 J. Iwaniec

the superposition principle and exhibit complex phenomenaunusual for linear
systems, such as jumps, self-excited and chaotic vibrations, changes in natural
frequencies resulting from changes in the excitation amplitudes, co-existence
of many stable equilibrium positions. In view of these properties, classical
identification methods can not be used for the purposes of nonlinear system
identification. It is also impossible to formulate a general identificationmethod
applicable to all nonlinear systems in all instances.

Formany years, linearization methods were the onlymethods used for the
purposes of nonlinear system identification.Themost frequently usedmethods
were the equivalent (Nichols et al., 2004) and stochastic linearization. In the
following years, the concept of nonlinear normalmodeswas introduced (Rand,
1974; Rosenberg, 1962), for weakly nonlinear systems the perturbation theory
wasdeveloped (KevorkianandCole, 1996;Nayfeh, 1981;O’Maley, 1991). Later
publications (Chan et al., 1996; Chen and Cheung, 1996; Qaisi and Kilani,
2000) were dedicated to identification of strongly nonlinear systems. Recently,
the researchers have been taking interest in making use of nonlinear system
properties instead of avoiding or ignoring them (Nichols et al., 2004; Rhoads et
al., 2005).More frequently, themachines andmechanical systems are designed
for work in nonlinear ranges of dynamic characteristics taking advantage of
phenomena characteristic for nonlinear systems.

2. Nonlinear system identification methods

Thefirst research intononlinear system identificationmethodsgoesback to the
seventies of the last century (Ibanez, 1973; Masri and Caughey, 1979). Later
works consider identification of single degree-of-freedom systems with various
types of nonlinearities. Multiple degree-of-freedom identification methods are
relatively new, since they have been elaborated over the last 15 years.

Independently of the applied identification method, the nonlinear system
identification can be considered as a complex process consisting of nonlineari-
ty detection, determination of the nonlinearity location, type and functional
form,model parameters estimation aswell as verification and validation of the
estimated model.

2.1. Classical methods

Classical nonlinear system identification procedures consist of two main
stages. In the first step, linear system parameters are estimated by exciting



Sensitivity analysis of an identification method... 421

the system at an operating point where the system dynamic behaviour is
nominally linear. In the second step, on the basis of nominally linear para-
meters found in the first step, estimation of nonlinear system parameters is
performed. Classical nonlinear system identification methods can be classi-
fied according to the following categories: linearization methods, time domain
methods, frequency domain methods, modal methods, time-frequency analy-
sis methods, methods based on neural networks, wavelet transform methods,
structural model updating. Such a classification is certainly not exhaustive
and the additional categories can be introduced. For instance, it is possible to
make distinction between parametric and nonparametric methods, single and
multiple inputmethods, single andmulti degree-of-freedom methods, etc.
Classical nonlinear system identification procedures require an input me-

asurement or at least estimate, which can be treated as an essential disa-
dvantage. In many mechanical systems, measurement of exciting forces (e.g.
tire-road or wheel-rail contact forces) is difficult or impossible to carry out.
Moreover, behaviour of a large variety of mechanical systems is not linear in
a broad enough frequency range around any operating point.

2.2. Operational nonlinear system identification method

Contrary to the classical nonlinear system identification methods, the al-
gorithm of the operational nonlinear system identification method (Fig.1),
considered in this paper, requires neither input measurement (estimate) nor
linear systembehaviour in a broad frequency range around an operating point
(Haroon et al., 2005; Iwaniec, 2009b). Therefore, it is amethod convenient for
parameter identification of strongly nonlinear systems working under opera-
tional loads the measurement of which is difficult or impossible to carry out.
Themethod canbeused for bothnonlinearity detection and systemparameter
identification.
The algorithm of themethod, presented schematically in Fig.1, consists in

sequential application of the restoring force, boundaryperturbation anddirect
parameter identification techniques (Haroon et al., 2005).
In the first step of the algorithm, the discretemodel of the considered real

system is assumed and dynamic equations ofmotion for individual degrees-of-
freedom are formulated

Mẍ(t)+Cẋ(t)+Kx(t)=Fz(x(t), t)+N(x(t),ẋ(t)) (2.1)

where M, C, K are mass, damping, stiffness matrices, x(t), ẋ(t), ẍ(t) – ti-
me histories of displacements, velocities and accelerations of system masses,
respectively, Fz – reduced force, N – nonlinear restoring force.



422 J. Iwaniec

Fig. 1. Proposed and verified (Iwaniec, 2009b) identificationmethod of nonlinear
systems working under operational loads

For the purposes of research into properties of vehicle suspension systems
or mechanical systems of two ”dominant” degrees of freedom, the model pre-
sented in Fig.2 can be used. That universal model, making it possible to im-
plement the idea of the sky-hooked-damper (Haaron et al., 2005), was applied
by the author for parameter identification of the Skytruck airplane suspen-
sion system (Iwaniec, 2009b; Iwaniec andUhl, 2007), vibratorymachine body
suspension system (Iwaniec, 2009a,b) and rotational machine shaft support
(Iwaniec, 2007, 2009b). Dynamic equations of motion formulated for the par-
ticular model masses are as follows

M1ẍ1+(C1+C2)ẋ1−C2ẋ2+(K1+K2)x1−K2x2+N1+N2 =
=C1ẋb+K1xb

(2.2)
M2ẍ2−C2ẋ1+C2ẋ2−K2x1+(K2+K3)x2 =N1

where

N1 =N1(x1(t),x2(t),ẋ1(t),ẋ2(t))

N2 =N2(x1(t),xb(t),ẋ1(t),ẋb(t))



Sensitivity analysis of an identification method... 423

and M1 – unsprungmass, M2 – sprung mass, K1 – tire stiffness coefficient,
K2 – suspension stiffness coefficient, C1 – tire damping coefficient, C2 – su-
spension damping coefficient, x1 – displacement of mass M1, x2 – displace-
ment of mass M2, xb – tire patch displacement, N1 – nonlinear restoring
force acting on the vehicle suspension, N2 – nonlinear restoring force acting
on the vehicle tire.

Fig. 2. Discrete model of a 2 degree-of-freedom nonlinear system (vehicle
suspension system)

After transformation of the equation of motion formulated for a given
degree-of-freedom into the form (2.3) and substitution of themeasured system
responses (usually systemvibration accelerations), reconstruction of the resto-
ring forces acting on the system of interest is performed. In case of 2 degree-
of-freedom systems, equation of motion for mass M2 (e.g. car body, railway
car body) is transformed to the following form

M2ẍ2 =−C2(ẋ2− ẋ1)−K2(x2−x1)−K3x2+N1 (2.3)

where
N1 =N1(x1,x2,ẋ1,ẋ2) ẍ2 = ẍ2(t)

ẋ2 = ẋ2(t) ẋ1 = ẋ1(t)

x2 =x2(t) x1 =x1(t)

that makes it possible to determine relation between the acceleration of the
sprung mass and difference between velocities of the system masses (relati-
ve velocity) as well as relation between acceleration of the sprung mass and
differencebetween thedisplacements of systemmasses (relative displacement).



424 J. Iwaniec

If the determined restoring forces are nonlinear, in the following step of
the algorithm, they are approximated with the use of parametrical model and
subtracted from the overall force of resistance in the system. Such a procedure
makes it possible to determine the linear component of reactions

ẍ2−fn1−fn2 =
C2
M2
(ẋ1− ẋ2)+

K2
M2
(x1−x2)−

K3
M2
x2 (2.4)

where: fn1,fn2 are functions approximating the identified nonlinear restoring
forces.
On this basis, with the use of the direct parameter estimationmethod, the

parameters of the considered systemare estimated. Since themeasurements of
system responses are carried out in operational conditions, the force exciting
the system remains unknown. Therefore, the number of unknowns is higher
than the number of dynamic equations of motion that can be formulated and
the direct parameter estimation method provides only the ratios of system
parameters.
In order to determine absolute values of the system parameters, the boun-

dary perturbation method consisting in modification of the system dynamic
properties, is applied. In practice, introduction of an additional mass is the
most convenientmethod of structuralmodification.Application of such an ap-
proach makes it possible to formulate an additional equation of motion and,
what follows, to determine values of the demanded parameters.
In case of a 2 degree-of-freedom system, for which the discrete model pre-

sented in Fig.2 was used, application of the direct parameter identification
method results in formulation of the following equations

T21(iω)=
X2(iω)

X1(iω)
⇒K2

(
1− 1
T21(ωk)

)
+K3 =ω

2
kM2 k=1,2, . . . ,Nf

(2.5)

T21(0)=
K2

K2+K3

where T21(iω) is the transfer function between displacements of masses M2
and M1 (determined after elimination of nonlinear restoring forces), X1(iω),
X2(iω) – Fourier transform of signals x1(t) and x2(t), respectively, Nf –
number of system degrees of freedom, T21(0) – transfer function valuated for
ω=0.
In practice, the exact values of the considered systemmasses remain unk-

nown. Therefore, the direct parameter estimation method makes it possible
to formulate 2 equations with 3 unknowns – M2, K2, K3. An additional dy-
namic equation of motion, formulated for the modified system (according to



Sensitivity analysis of an identification method... 425

the algorithm of the boundary perturbation method) by introduction of an
additional mass ∆M2, has the following form

T
′
21(iω)=

X′2(iω)

X′1(iω)
⇒K2

(
1−

1

T ′21(ωp)

)
+K3 =ω

2
pM2 p=1,2, . . . ,N

′
f

(2.6)
where T ′21(iω) is the transfer function between displacements of masses
(M2+∆M2) and M1, N

′
f – number of the system degrees of freedom.

The absolute values of demanded parameters M2,K2,K3 can be determi-
ned by solving a set of equations (2.5) and (2.6).

2.3. Sensitivity analysis

In order to assess the accuracy of the discussed operational nonlinear sys-
tem identification method and the influence of the input variables measu-
rement errors on the accuracy of estimated parameters, it was necessary to
elaborate an analytical model of the system and determine the expected va-
lues of quantities x̂ and ŷ analytically on the basis of theoretical (infinitely
accurate) data. In case of the considered 2 degree-of-freedom system (Fig.2)
excited to vibrations in one direction by basemotion (kinematical excitation),
dynamic equations of motion are as follows

M1ẍ1+C1(ẋ1− ẋ2)+K1(x1−x2)=N1(x1−x2,ẋ1− ẋ2)=0
(2.7)

M2ẍ2+C1(ẋ2− ẋ1)+K1(x2−x1)+C2ẋ2+K2x2 =
=N2(x1−x2,ẋ1− ẋ2)+K2x2+Cẋ2 =N2

Taking into account (Goliński, 1979)
√
K1
M1
=Ω1

√
K2
M2
=Ω2

C1 =2γ1M1Ω1 C2 =2γ2M2Ω2

(2.8)

and
Ω1
Ω2
=Λ

M1
M2
=V

ω

Ω2
=µ2

ω

Ω1
=µ1 =

µ2
Λ

(2.9)

where Ω1, Ω2 are undamped natural frequencies of masses M1 and M2,
γ – dimensionless coefficient of linear (viscotic) damping, µ – frequency ratio
(µ=ω/Ω), relations (2.7) can be written in the following form

ẍ1+2γ1Ω1(ẋ1− ẋ2)+Ω21(x1−x2)=0
(2.10)

ẍ2+2γ1VΩ1(ẋ2− ẋ1)+Ω21V (x2−x1)+2γ2Ω2ẋ2+Ω22x2 =N2



426 J. Iwaniec

Using the Euler substitution

x1 =Ae
iω

x2 =Be
iω (2.11)

velocities and accelerations of the considered systemmasses can be written in
the following form

ẋ1 = iωAe
iω ẋ2 = iωBe

iω

ẍ1 =−ω2Aeiω ẍ2 =−ω2Beiω
(2.12)

Then set of equations (2.10) can be given as

−ω2Aeiω +2iωγ1Ω1(A−B)eiω+Ω21(A−B)eiω =0
(2.13)

−ω2Beiω+2iωγ1VΩ1(B−A)eiω +Ω21V (B−A)eiω +2iωγ2Ω2Beiω+
+Ω22Be

iω =N2

or

−ω2A+2iωγ1Ω1(A−B)+Ω21(A−B)=N1 (2.14)
−ω2B+2iωγ1VΩ1(B−A)+Ω21V (B−A)+2iωγ2Ω2B+Ω22B=N2

Below, set of equations (2.14) is presented in the matrix form
[
−ω2+2iωγ1Ω1+Ω21) −2iωγ1Ω1−Ω21
−2iωγ1VΩ1−Ω21V −ω2+2iωγ1VΩ1+Ω21V +2iωγ2Ω2+Ω22

][
A
B

]
=

(2.15)

=

[
0
N2

]

As a result of division by Ω22

[
−µ22+2iµ2γ1Λ+Λ2 −2iµ2γ1Λ−Λ2
−VΛ(Λ+2iµ2γ1 Λ2V +1−µ22+2iµ2(γ1VΛ+γ2)

]



A

Ω22
B

Ω22



=

[
0
N2

]

(2.16)
Having introduced the following notation

W




A

Ω22
B

Ω22



=

[
0
N2

]

(2.17)

W=

[
−µ22+2iµ2γ1Λ+Λ2 −2iµ2γ1Λ−Λ2

−VΛ(Λ+2iµ2γ1) a+ib

]



Sensitivity analysis of an identification method... 427

where
a=Λ2V +1−µ22 b=2µ2(γ1VΛ+γ2)

the determinant of the matrix W can be described by the relation

|W|=(−µ22+2iµ2γ1Λ+Λ2)(a+ib)−VΛ(Λ+2iµ2γ1)(2iµ2γ1Λ+Λ2) (2.18)

Equation (2.18) can also be written in the following form

|W|=C1+iD1 (2.19)

where

C1 =µ
4
2−µ22[1+Λ2(1+V )+4γ1γ2Λ]+Λ2

D1 =2µ2γ2Λ
2+γ1Λ−µ22[γ1Λ(1+V )+γ2]

Theamplitudes of displacements canbecomputed on thebasis of the following
relations

A=x1
∣∣∣
max
=
N2Λ

2

K1

√
a2+ b2

C21 +D
2
1

(2.20)

B=x2
∣∣∣
max
=
N2
K2

√
Λ4+4γ21Λ

2µ22
C21 +D

2
1

In the further part of the paper, for the purposes of notation simplification,
under the terms x1 and x2 the amplitudes A and B of displacements will be
understood.
In order to verify the correctness of the obtained relations, numerical data

was selected in awaymaking it possible to compare the responses determined
on the basis of formulated relations with the example presented in Goliński
(1979). For that purpose, the ratios of masses M1/M2 = V = 1 and frequ-
encies Ω1/Ω2 =Λ=4 were assumed. In Fig.3 there are presented responses
(displacements) of the systemcharacterised by dimensionless damping factors:
γ1 = 0.01 and γ2 = 0.001, stiffness: K1 = 16000N/m, K2 = 1000N/m and
mass M1 = 20kg. The system has two resonant frequencies, corresponding
to µ2 = 1/

√
2 and µ2 = 5.7. The minimum of displacement of the mass M2

corresponds to µ2 =4.123.
In Fig.4, relation between the amplitudes of two-mass system stationary

vibrations in function of µ2 with damping neglected, taken from Goliński
(1979) (page 178) is shown.
Comparison of the amplitude-frequency characteristics of displacements

x1 and x2 of masses M1 and M2 determined analytically (Fig.3) with the



428 J. Iwaniec

Fig. 3. Displacements x1 and x2 of the consideredmasses presented in functions of
the frequency ratio µ2

Fig. 4. Amplitude-frequency characteristics of displacements of
two-degree-of-freedom system in function of µ2 (with the influence of damping

neglected) (Goliński, 1979)

characteristic presented in Goliński (1979) (see Fig.4) proved correctness of
the considerations presented above.

Perturbation of the considered systemboundary conditions by introducing
a change in the value ofmass M2 (by ∆M2) results in a change in the system
response. In Fig.5, the amplitude-frequency characteristic of displacement x1
of mass M1 and x2 of mass M2 in function of µ2 and the value of mass
(M2+∆M2) is presented.



Sensitivity analysis of an identification method... 429

Fig. 5. Amplitude-frequency characteristic of displacements of mass (a) M1 and
(b) M2 in function of µ2 and the mass value (M2+∆M2), ∆M2 ∈ 〈−10,+10〉,

where M
z
=M2+∆M2kg

The transfer functions T21 and T
′
21, defined by the ratio of displacements

of the systemmasses, can be computed on the basis of the following relations

T21 =
x2
x1
=
K1
K2Λ2

√
Λ4+4γ21Λ

2µ22
a2+ b2

(2.21)

T ′21 =
x′2
x′1
=
K1
K2Λ

′2

√
Λ
′4+4γ21Λ

′2µ
′2
2

a
′2+ b

′2

Taking into account equations (2.7), for the considered method, the relations
describing T21 and T

′
21 are as follows (k=1,2, . . . ,Nf, p=1,2, . . . ,N

′
f)

T21(iω)=
X2(iω)

X1(iω)
⇒K2

(
1−

1

T21(ωk)

)
+K3 =ω

2
kM2

T21(0)=
K2

K2+K3
(2.22)

T ′21(iω)=
X′2(iω)

X′1(iω)
⇒K2

(
1− 1
T ′21(ωp)

)
+K3 =ω

2
p(M2+∆M2)

hence

M2 =∆M2
(T21
T ′21

T ′21−T0
T21−T0

−1
)−1
=∆MW(T ′21,T21) (2.23)

where

T21 =T21(ωk) T
′
21 =T

′
21(ωp)

W(T ′21,T21)=
(T21
T ′21

T ′21−T0
T21−T0

−1
)−1



430 J. Iwaniec

Thevalue ofmass M2 is estimated on the basis of product (2.23) of the va-
lue of additional mass ∆M2 and function W(T

′
21,T21). Function W(T

′
21,T21)

is determined experimentally by measurement of the transfer functions T21
and T ′21.

Fig. 6. Amplitude-frequency characteristic of: (a) transfer function T ′
21
,

(b) T ′
21
−T21, (c) derivative ∂T21/∂M2, (d) T ′21/T21 in the function of mass

(M2+∆M2) and µ2 (values obtained for ∆M2 ∈ 〈−10,+10〉), where
M
z
=M2+∆M2kg

In Fig.6, there is presented the transfer function T ′21 and difference
T ′21 − T21 resulting from a change in the value of mass M2 by ±50%. It
can be easily noticed that in spite of the significant change in themass value,
changes in themagnitudes of functions T21 and T

′
21 are relatively insignificant,

especially at a distance from the perpendicular plane crossing the point deter-
mining the extremumof the function W (in the considered case µ2 ≈ 4.12). A
diagram of the function T ′21 derivative computedwith respect to themass M2
is presented in Fig.7a. Isolines almost parallel to the M2 axis reveal small
sensitivity of the transfer function T ′21 to changes in the M2 values, in con-
trast to much higher sensitivity to changes in µ2, especially for µ2 →∼ 4.12.
That sensitivity is one of the sources of system parameter estimation errors
and results in the necessity of accurate estimation of thethe transfer functions



Sensitivity analysis of an identification method... 431

T21, T
′
21, local minimum of the function W (2.23) and careful selection of the

value of the additional mass ∆M2.

Fig. 7. Sensitivity ∂T ′
21
/∂M2 of the transfer function to changes in mass M2 –

arrows determine the direction of increase in the derivative (a), monotonicity of the
function determining the mass M2 in the neighborhood of frequency corresponding

to the minimum of transfer function and additional mass (b)

Figure 8 depicts the function W plotted in function of µ2 and the addi-
tional mass ∆M2 determined by a relative percentage change d in the mass
value

d=
∆M2
M2
·100% (2.24)

It can be noticed that the function W(T ′21(µ2∆M2),T21(µ2∆M2)) depends
heavily on the frequency and additional mass, especially in the vicinity to
resonant frequencies. The accuracy of mass M2 estimation depends on the
accuracy of estimation of local extremum of the function W

∂W(T ′21(µ2∆M2),T21(µ2∆M2))

∂µ2
=0 (2.25)

Roots of equation (2.25), determining the extremum of the function
W(T ′21(µ2∆M2),T21(µ2∆M2)), in the further part of the paperwill be referred
to as frequency of antiresonance µ2|ant−rez.
The relation determining the value ofmass M2 (2.23) in the neighborhood

of the frequency corresponding to the minimum of transfer function (Fig.8)
is a function increasing or decreasing in four ranges of µ2 and ∆M2 (Fig.7b)
antisimetrically with respect to the plane of frequency of the antiresonance
µ2|ant−rez and nominal mass M2 (∆M2 =0).



432 J. Iwaniec

Fig. 8. Diagram of the function W(T ′
21
(µ2∆M2),T21(µ2∆M2)) with respect to

frequency (determined by µ2) and relative percentage change d in the additional
mass ∆M2

In order to avoid searching for the extremum (minimum or maximum) of
function (2.23), in the regions presented in Fig.7b, it is convenient towrite an
expression describing the mass M2 as follows

M2 =min
∣∣∣∆M2

(T21
T ′21

T ′21−T0
T21−T0

−1
)−1∣∣∣=min |∆M2W(T ′21,T21)| (2.26)

Then theminimum is searched for in thewhole domain of the function (2.23).
An important drawback of the discussedmethod of system parameters es-

timation consists in the requirement of high accuracy of the ”antiresonant”
frequency µ2|ant−rez estimation and the necessity of application of signifi-
cant values of the additional mass ∆M2 that influences the accuracy of the
mass M2 estimation. Taking into account relation (2.23), the relative error of
mass M2 estimation can be calculated as the sum of errors of the function
W(T ′21,T21) estimation and the additional mass ∆M2 determination

δ(M2)= δ(∆M2)+δ(W(T
′
21(µ2,∆M2),T21(µ2,∆M2)) (2.27)

Themass ∆M2 can be determined with an arbitrary technical accuracy and,
therefore, the error of that mass estimation is negligible. Therefore, this er-
ror equals the error of estimation of the function W(T ′21,T21). Themeasured
functions T ′21, T21 (found theoretically by relations (2.5)1 and (2.6)) are expe-
rimentally determined complex functions depending on µ2 and ∆M2. In the
further part of the paper, the influence of the estimation accuracy of the ”an-
tiresonant” frequency µ2|ant−rez and the selection of additionalmass ∆M2 on
the accuracy of mass M2 estimation will be discussed.



Sensitivity analysis of an identification method... 433

In order to determine the influenceof accuracy of the ”antiresonant” frequ-
ency µ2|ant−rez estimation and the selection of the additionalmass ∆M2 value
on the variance of mass M2, relations (2.8) and (2.12) were used.Matrix T is
as follows

T=

[
∂W(T21(µ2,∆M2),T

′
21(µ2,∆M2)

∂µ2

∂W(T21(µ2,∆M2),T
′
21(µ2,∆M2))

∂∆M2

]

(2.28)
Consecutive elements of matrix T, ∂W/∂µ2 and ∂W/∂∆M2 are presented in
Fig.9.

Fig. 9. Elements ∂W/∂∆M2 and ∂W/∂µ2 of the matrix T

In Fig.10, a diagram of the mass M2 variance is presented. Isometric
projection on the plane d,µ2 makes it possible to determine the course of
lines of the same value of variance. The minimum of the mass M2 variance
lies in the vertical plane containing the straight line µ2 =µ2|ant−rez.

Fig. 10. Variance of mass M2 in function of d and µ2, (a) three-dimensional
function shape, (b) isometric projection on the plane d,µ2



434 J. Iwaniec

The variance of mass M2, described by the relation:

var(M2)=
n∑

j=1

(∂M2(T21(µ2,∆M2),T ′21(µ2,∆M2)
∂µ2

)2∣∣∣∣
µ2=µ2|ant−rez

σ2(µ2)+

(2.29)

+
n∑

j=1

(∂M2(T21(µ2,∆M2),T ′21(µ2,∆M2)
∂∆M2

)2∣∣∣∣
∆M2=0

σ2(∆M2)+ . . .

makes it possible to analyse the overall influence of estimation accuracy of
many input quantities on the total relative (percentage) error of themass M2
estimation

δ%(M2)=

√√√√E[(M2−M̂2)2]
M22

=

√
var(M2)

M2
·100% (2.30)

In Fig.11 the characteristic of relative error (2.30) of themass M2 estima-
ted on the basis of change in the system response resulting from introduction
of an additional mass ∆M2(±25%M2) is presented. It was assumed that the
mass ∆M2 is determined with the accuracy of 1%, while the accuracy of the
”antiresonance” frequency µ2|ant−rez estimation equals 0%, 1%, 2%, 3%, 5%
and 7%. In Fig.11, the same relation plotted for the change in themass value
by ±5% is presented.

Fig. 11. Relative error of the mass M2 estimated on the basis of change in the
system response resulting from addition or subtraction of the mass ∆M2 in the

range of d=±25%

On the basis of Fig.11, it can be stated that the relative error of the
mass M2 estimation, for a given accuracy of the ”antiresonant” frequen-
cy δ%(µ2|ant−rez) estimation, depends exponentially on the mass change



Sensitivity analysis of an identification method... 435

d=(∆M2/M2)·100% and approaches infinity for d→ 0. The curve drawn for
δ%(µ2|ant−rez) = 0 determines the method error. Such results are consistent
with the results presented in Fig.10. In order to obtain the required accuracy
of themass M2 estimation, it is necessary to increase the value of ∆M2 for a
constant value of δ%(µ2|ant−rez) or, for the assumed value of ∆M2kg or d [%]
to increase the accuracy of ”antiresonant” frequency δ%(µ2|ant−rez) estima-
tion. Theoretically, the measurements of frequency can be carried out with a
high accuracy, nevertheless, in practice, the estimation of the ”antiresonant”
frequency (µ2|ant−rez) on the basis of the previously determined extremum of
the transfer function T21 and T

′
21 requires selection of the proper excitation,

e.g. of the swept-sine type. When introduction of the further changes to the
mass ∆M2 is not possible, the accuracy of µ2 estimation should be increased,
taking into account longer time of measurements and the necessity of careful
selection of the excitation type.

3. Conclusions and final remarks

The paper concerns the operational identificationmethod thatmakes it possi-
ble to estimate parameters of linear as well as nonlinear mechanical systems
on the basis of system dynamic responsesmeasured during the normal system
work. The algorithm of the method consists in sequential application of the
restoring force, boundary perturbations and direct parameter identification
techniques. The obtained parameter estimates provide information concerning
forces transferred to the foundation that can be used for the purposes of ear-
ly damage detection and, what follows, minimization of negative influence of
vibrations transferred to the foundations and the environment. Information
from the monitoring process can be treated as a basis for making a decision
on further machine exploitation and the input data to the system of auto-
matic control, which would control braking in order to minimize potential
threat.

In the paper, there are also presented the results of the sensitivity analysis
carried out in order to assess the influence of input parameters uncertainties
(accuracy of resonant frequency and amplitude estimates, errors of transfer
function estimation in operational conditions, value of the introduced addi-
tional mass) on the accuracy of estimated system parameters. The presented
analysis makes it possible not only to determine the sensitivity of estimated
system parameters, such as mass M2 or stiffness coefficient K2, to measu-
rement errors of the input variables but also to select the size of introduced



436 J. Iwaniec

structural modification (change in themass M2) minimizing the error of esti-
mated parameters for the given accuracy of measurements of input variables.

Acknowledgements

Scientific research was partially financed by the Ministry of Science and Hi-

gher Education (from 2010 till 2012) within the framework of research project No.

N504493439.

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Analiza wrażliwości metody identyfikacji układów nieliniowych

w warunkach eksploatacyjnych

Streszczenie

Wpracy przedstawionometodę operacyjnej identyfikacji parametrówmodeli nie-
liniowych konstrukcji mechanicznych, realizowaną w oparciu o algorytmymetody sił
resztkowych, zaburzeń brzegowychoraz bezpośredniej identyfikacji parametrów.Uzy-
skane estymaty parametrów dostarczają informacji o siłach przekazywanych na pod-
łoże i znajdują zastosowanie w diagnostyce realizowanej w oparciu o model układu
nieuszkodzonego.
W celu oszacowania wpływu niepewności parametrów wejściowych na dokład-

ność estymowanych parametrów układu, przeprowadzono analizę wrażliwości poszu-
kiwanych parametrówukładu (masy, sztywności, tłumienia) na dokładność estymacji
częstotliwości i amplitud rezonansowych (uwarunkowaną błędami estymacji funkcji
przejścia w warunkach eksploatacyjnych), a także wartość masy wprowadzanej do
układu w celu zmodyfikowania jego własności dynamicznych (zgodnie z algorytmem
metody zaburzeń brzegowych).

Manuscript received April 22, 2010; accepted for print September 15, 2010