Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

2, 40, 2002

APPLICATION OF THE PROJECTIVE INPUT RESIDUAL

METHOD TO DIAGNOSIS OF FAILURE MECHANICAL

SYSTEMS

Bartosz Powałka

Krzysztof Marchelek

Institute of Manufacturing Engineering, Technical University of Szczecin

e-mail: bartps@safona.tuniv.szczecin.pl

The paper is concernedwith themethod of failure detection on the basis
of incompletemeasurements, whichmeans that the number ofmeasured
signals is lower than the number of degrees of freedom of mechanical
systems. The method does not require the reduction of the number of
degrees of freedom of the mathematical model and therefore the iden-
tified parameters can be physically interpreted. The proposed indicator
used to detect the damaged element is based on the comparison of par-
tial derivatives of the input residuals of the damaged and undamaged
models. The method will be illustrated with an example of FWD-32J
milling machine.

Key words: milling machine, damage diagnosis

1. Introduction

Machines during their exploitation are exposed towear of the co-operating
elements. The wear, in the case of responsible constructions such as bridges
etc. May lead tragic consequences if not detected sufficiently early. Therefore,
the development ofmethods focused on the detection of damages (or changes)
in mechanical systems is justified.

A number of dynamics-based methods have been developed in the past
for detection of changes due to damage (Doebling et al., 1996). To provide
efficient inspections of complexmechanical systemsmodel-dependentmethods
are used. These methods in combination with real measurements, are gene-
rally more accurate in locating damage than measurement alone (Natke et



402 B.Powałka, K.Marchelek

al., 1993). The methods which use only dynamic responses can detect the
existence of damage but do not allow one to indicate its location. At the
forefront of the hybrid methods (Li et al., 1995), i.e. combinations of model
and measurement based treatments are the approaches which do not require
tomeasure all co-ordinates of the structure (Kidder, 1973; Baruch, 1998). This
feature is of paramount importancewhen investigating the location of damage
in large, multidimensional structures such as machine tools since, in general,
the number of sensors available is a constrained number. Hence, one has to
derive information on damage on the basis of incomplete measurements.

The paper presents an application of the Projective Input Residual Me-
thod (PIRM) (Oeljeklaus, 1998), which handles the problem of measurement
incompleteness, to the fault diagnosis of a machine tool. The application of
the projection onto the subspace (Kurnik, 1997) is focused on the reduction
of the problem dimensionality to formulate it uniquely. The PIRM can also
be effectively used in the procedure of experimental updating of structural
parameters that describe the computationalmodel of an object. Then, the pa-
rameters chargedwith errors resulted from inaccurate estimation conductedon
the basis of literature data can be treated similarly to ”damaged” parameters.

2. Mathematical model

As previously mentioned, the methods of damage detection are based on
the examination of the agreement between the information delivered by the
mathematical model and that obtained from the measurements of the real
object.Thisproblemconsists in thedetection of thosemodel parameterswhich
undergo changes as the result of the damage. Many real mechanical systems
can be represented by amodel built bymeans of the finite element method or
rigid finite element method. The sample model of the milling machine FWD-
32J developed by making use of the rigid finite element method is shown in
Fig.1. Mathematical model can be presented in the form

MÜ +HU̇+KU =P (2.1)

where
M,H,K – inertia, damping and stiffness matrices, respectively

U,U̇,Ü – vectors of generalized displacements, velocities and ac-
celerations, respectively

P – vector of the generalized force.



Application of the projective input residual method... 403

Since the determination of all 3n(n+1)/2 (n — number of degrees of
freedom) elements of M,H,Kmatrices for a system of n DOF is a time con-
suming task, the modelling employed the well known topology elaborated by
Natke et al. (1995). Application of the topology is especially convenient when
used along with the Rigid Finite Element (RFE) method (Kruszewski et al.,
1999), which is practiced inmodelling ofmachine tools. Thismethod assumes
that the structure consists of discrete spring-less masses (bodies) connected
by mass-less spring-damping elements (SDE). Machine tools are intrinsical-
ly amenable to lumped parameter representations, comprising rigid massive
structural members such as beds, columns etc., connected by light compliant
interfaces at guideways, slides etc. The advantage of the RFE is the preserva-
tion of the geometrical similarity of the physical model and the real object,
which significantly simplifies the localization of the damaged elements of the
examined object (Marchelek, 1991).

Fig. 1. Model of the FWD-32 J milling machine in the RFEmethod

The initial computational model described by equation (2.1) is built of a
sum of n×nmatrices

M=
µ∑

i=1

M
∗
i H=

β∑

j=1

H
∗
j K=

β∑

k=1

K
∗
k (2.2)

where µ is the number of rigid finite elements, β – the number of spring-



404 B.Powałka, K.Marchelek

damping elements

M
∗
i = diag[0,0, ...,0, Mi︸︷︷︸

block No. i

,0, ...,0]

Mi = diag[m1i,m2i,m3i,m4i,m5i,m6i] (2.3)

H
∗
j =






0 · · · 0 0 0 · · · 0 0 0 · · · 0
...
...
...
...
...
...
...

...
...
...
...

0 · · · 0 Hjpp 0 · · · 0 −Hjpr 0 · · · 0
...
...
...
...
...
...
...

...
...
...
...

0 · · · 0 Hjrp 0 · · · 0 Hjrr 0 · · · 0
...
...
...
...
...
...
...

...
...
...
...

0 · · · 0 0 0 · · · 0 0 0 · · · 0






horizontal band
No. p

horizontal band
No. r

vertical vertical
band band
No. p No. r

The matrix M∗j is located between the rigid elements p and r. For deta-
ils concerning the structure of the matrices Hjpp, Hjpr, Hjrp and Hjrr (see
Kruszewski et al., 1986). Thematrix K∗k has the same form as thematrix H

∗
j.

Once the model of the machine tool is constructed, its validity is often
checked by comparing its dynamic characteristics with those obtained froman
experiment. If the correlation between these two is poor, then themodelmust
be corrected, which is knownalso as themodel updating.The systemmatrices
are corrected in the computational model independently by the dimensionless
correction factors

M(a)=
µ∑

i=1

aiM
∗
i H(a)=

β∑

j=1

ajH
∗
j K(a)=

β∑

k=1

akK
∗
k (2.4)

Hence, the dynamic stiffness matrix in the frequency domain has the form

Sω(a) = −ω
2
M(a)+ jωH(a)+K(a)=

(2.5)

=
µ∑

i=1

ai(−ω
2
M
∗
i)+

β∑

j=1

ai(jωH
∗
j)+

β∑

k=1

akK
∗
k



Application of the projective input residual method... 405

3. Projective input residual method

If all output components aremeasured, the classical input residualmethod
can be applied as the minimization of the L2 norm of P

M −Sω(a)U
M, whe-

re PM and UM are the measured vectors of excitation and displacements,
respectively.
If not even component of the displacement vector UM measured, then

Sω(a)U
M =Sω(a)C

⊤
CU
M +Sω(a)C

⊤
CU
M (3.1)

where the matrix C ∈ {0,1}m×n, and m is the number of the measured
displacements. Thematrix C consists of m rows with 1 corresponding to the
measured coordinate, C is the complementarymatrix to C.The samplematrix
C for a n = 5 degree of freedom system and m = 3 measured co-ordinates
can have the form

C=





1 0 0 0 0
0 1 0 0 0
0 0 0 0 1




 (3.2)

and its complementary matrix

C=

[
0 0 1 0 0
0 0 0 1 0

]

(3.3)

Hence, the matrix Sω(a)C
⊤
CU
M corresponds to the measured CUM di-

splacements, while Sω(a)C
⊤
CU
M includes the unmeasured CUM displace-

ments. Thus, it is known that

Sω(a)U
M ∈ εa

where εa is the set of calculated inputs, determined as

εa =
{
Sω(a)C

⊤
CU
M +P0 : P0 ∈R

(
Sω(a)C

⊤
)}

(3.4)

where R
(
Sω(a)C

⊤
)
is the image of Sω(a)C

⊤
. The aim functional of the pro-

jective input residual method is defined as follows

Jc(a)= min
Pγ∈εa

‖PM −Pγ‖22 =dist(P
M,εa)

2 (3.5)

where Pγ is the calculated input vector (force) that is equal to the distance
between PM and the vector space εa · [CFω(a)]

−1 (see Fig.2), where Fω(a)=



406 B.Powałka, K.Marchelek

Fig. 2. Illustration of the projective input residual method (Oeljeklaus, 1998)

S
−1
ω (a), is not the inverse of [CFω(a)], which does not exists, but it is a map
of the R([CFω(a)]) (an image of the [CFω(a)]) onto the domain of [CFω(a)].
Jc(a) can be calculated as

Jc(a)=
∥∥∥P(a)(PM −Sω(a)C

⊤
CU
M)
∥∥∥
2

2
(3.6)

where P(a) is the projector onto N⊥a (see Fig.2), which is the orthogonal
complement of Na =R(Sω(a)C

⊤) – kernel of CFω(a).
Thus

P(a)= [CFω(a)]
+[CFω(a)] (3.7)

where [CFω(a)]
+ is theMoore-Penrose pseudo-inverse matrix.

Therefore, we obtain

Jc(a)=
∥∥∥[CFω(a)]

+[CFω(a)]P
M − [CFω(a)]

+
CU
M)
∥∥∥
2

2
(3.8)

The residuum v(a)=P(a)(PM−Sω(a)C
⊤
CU
M) is a non-linear function due

to the non-linearity of P(a). Thus, the minimization of Jc(a) consists in the
following non-linear optimization find

â=argmin
a
vH(a)v(a) (3.9)



Application of the projective input residual method... 407

Moreover, the partial derivatives are determined

vυ =
∂v(a)

∂aυ
υ=1, ...,np (3.10)

which are especially important when using the methods such as sequential
linear programming or the Levenberg-Marquardt method.

4. Damage indicator

The aim of the damage indication is not an accurate system identification
but the damage location that is equivalent to the determination of parameters
that have changed with respect to the original model. Since the gradients
vν(a) are very sensitive to the structural changes, therefore they can be used
for the damage detection and can be applied in the form of the indicator

τν =
‖vν(1)‖

‖vνD(1)‖
­ 0 (4.1)

where vD(a) are the residuals of the measured U
M
D of the probably damaged

structure for the excitation vector PM, which was also used to excite the
undamaged structure (UM).

Since ‖vνD(1)‖ is always different from 0, the proposed indicator is de-
termined uniquely. Because of measurement errors, ‖vν(1)‖ is a real number
greater than zero, and its division by ‖vνD(1)‖which is calculated for the U

M
D

corresponding to the damaged system, gives information on which parame-
ter is changed and, in consequence, localizes the damage. If the damage does
not exist, one can expect that ‖vν(1)‖ ≈ ‖vνD(1)‖ and τ ≈ 1. Otherwise
0<τν ≪ 1, since ‖v

ν(1)‖≪‖vνD(1)‖.

5. Measurement data preparation and damage simulation

Themilling machine was excited using an electrohydraulic exciter located
between bodiesNo. 1 and7,whichwere the head and the table of themachine,
respectively. The hydraulic exciter produced a force impulse (very short time
of action) of 370daN amplitude. Its location is given in Fig.3. The exciter
was located along the direction corresponding to the direction of the resultant



408 B.Powałka, K.Marchelek

Fig. 3. Scheme of the experimental stand (a) location of the sensors and excitation
force P , (b) direction of the excitation force P , (c) transformation of the measured

displacements q to the displacement vector u

cutting force (Fig.3b). This made that the results of the analysis reflect real
machining conditions.Thedisplacements of bodiesNo. 1 and7weremeasured.
For this purpose 16 sensors were used (8 for each body). This enabled more
accurate determination of the displacements of themeasured bodies. Themi-
nimum number of sensors required was 6 (Fig.3c). The relative locations of
the sensors thatmeasure thedisplacements of thehead are shown inFigure 3a.
After FFT (frequency range 1-300Hz) of the displacements of the head and
the table and their transformations into the generalized co-ordinate system
the results were recorded. Hence, 6 complex characteristics for the head and
the table in the frequency range 1-300Hz were registered (see Fig.4). It me-
ans that 12 of 42 co-ordinates were measured. The detailed description of the
experimental investigations and signal processing can be found in (Gutowski
and Berczyński, 1995).

The force impulse can be approximated with the Dirac function because
of a very short time of duration. The Fourier transform of such a signal is 1.
That unit force (in the frequency domain) acted simultaneously on the head
and table but in the opposite directions. Since the model was built in the
generalized co-ordinate system there was a necessity (as for displacements) to



Application of the projective input residual method... 409

Fig. 4. AF characteristics of (a) head (b) table of the milling machine



410 B.Powałka, K.Marchelek

transform the excitation vector to the gravity centers of the excited bodies.
The transform resulted in the excitation vector (n×1) that contained 12 non-
zero elements: 1-6 excitation components of the head (bodyNo. 1) and 37-42
excitation components of the table (body No. 7).

To minimize the errors of measurements the displacements for N = 4
resonance frequencies were assumed for further analysis. It was so, because
the coherence function for those resonance frequencies was close to 1, which
meant that the influence of noise and other error carrying factors was small. It
resulted in N complex displacement vectors with 12 non-zero elements. Since
there were nomeasurements taken from the damagedmachine tool there was
a need to simulate them. The simulation considered the changes appearing
during the exploitation of the machine.

The relative locations of the elements that constitute a slideway connection
are associated with friction forces. Therefore themachine tool is accompanied
with progressing wear of co-operating elements. Thus, a lower stiffness of the
contact joint is reflected in the measurements by higher values of the displa-
cements of the adjacent bodies.

The displacements of theworn elementswere simulated bymultiplying the
displacements thatweremeasured for the elements of the unwornmachine. To
simulate the wear of the slideway connection between bodiesNo. 6 and 7, ele-
mentsNo. 37-42 (table) of the displacement vector weremultiplied by 2,which
was the representative for damage at that location of themillingmachine.The
wear of theheadwas simulated similarly, and componentsNo. 1-6were subjec-
ted to the amplification. In both simulations themodified displacement vector
was denoted UMD .

6. Examination results

It appears that some groups of parameters are responsible for the larger
values of the table displacements (Fig.5a) and their changes, in comparison
with the undamagedmachine, which was detected by the proposed indicator.
Thefirst group consists of stiffness elementsNo. 8, 9, 10, 11 and the correspon-
ding dampingparameters 38, 39, 40, 41 that describe the contact properties of
bodies No. 6 and 7 (slideway connection). It could be expected as their lower
values resulted in higher values of the table displacements.Higher values of the
displacements of the table can also be caused by the damage in the connection
of bodiesNo. 5 and6.Theparameters characterising this connection (No. 12 –
stiffness and No. 42 damping) are strongly coupled with the parameters from



Application of the projective input residual method... 411

Fig. 5. Values of the damage indicator for (a) table; (b) head of the milling machine

the first mentioned group by the displacements of body No. 6 (base of the
slideway connection 6-7). The indicator also detected the damage to slideway
connection of bodies 4 and 5 (stiffness parameters No. 12, 13, 14, 16, 18, 19,
20 and corresponding damping parametersNo. 42, 43, 44, 46, 48, 49, 50). This
can be explained by the high value of the stiffness of the bolted connection
between bodies 5 and 6 (about 100 times greater than the stiffness of the con-
nection of body 4 and 5). It causes that bodies 5 and 6 behave almost like
one rigid element, which automatically couples the parameters of the slideway
connection of bodies 4 and 5 with the displacements of the table.

The indicator detected damage (for the increased displacements of the he-
ad) in the following elements (Fig.5b): 1-8, 21-38, 51-60. Stiffness parameter 1
and damping parameter 31 characterise the physical properties of the connec-
tion of the head (body No. 1) with the horizontal beam (body No. 2). These
parameters influence directly the displacements of the head. Motion of the
head depends also on the stiffness parameters (parameters No. 2-8) and the
corresponding damping parameters (parameters No. 32-38). It is obvious that
higher values of the amplitudes of the head along the x1 axis might be caused
by the lower stiffness of the slideway connection between body 3 and 4.

References

1. BaruchM., 1998,Damage detection based on reducedmeasurements,Mecha-
nical Systems and Signal Processing, 12, 23-46



412 B.Powałka, K.Marchelek

2. Doebling S.W, FarrarC.R., PrimeM.B., ShevitzD.W., 1996,Damage
identification and healthmonitoring of structural andmechanical systems from
changes in their vibration characteristics: a LiteratureReview,ReportNo. LA-
13070-MS, Los Alamos National Laboratory

3. Gutowski P., Berczyński S., 1995, Parameters identification of dynamic
models of machine tool supporting systems, Proceedings of the Ninth World
Congress on the Theory of Machines and Mechanisms, Politecnico di Milano,
Italy, 2, 1349-1354

4. Kidder R.L., 1973, Reduction of structural frequency equations, American
Institute of Aeronautics and Astronautics, 32, 892

5. Kruszewski J.I. et al., 1999,Rigid Finite Element Method in Dynamics of
Construction, (in Polish)Warsaw,WNT

6. Kurnik W., 1997, Divergent and Oscillational Biffurcation, (in Polish) War-
saw,WNT

7. Li C., Smith C., Smith S.W., 1995, Hybrid approach for damaged detection
in flexible structures, Journal of Guidance, Control andDynamics, 18, 419-425

8. MarchelekK., 1991,Machine Tool Dynamics, (inPolish)Edition 2.Warsaw,
WNT

9. Natke H.G., CollmanD., ZimmermanH., 1995, Beitrag zur korrektur des
rechenmodells eines elastomechanischen systems anhand von versuchsergebnis-
sen,VDI-Berichte, 221, 23-32

10. Natke H.G., Tomlinson G.R., Yao J.T.P., 1993, Safety Evaluation Based
on Identification Approaches Related to Time-Variant and Nonlinear Structu-

res, Wiesbaden, Vieweg

11. Oeljeklaus M., 1998, Ein Beitrag zur Systemidentifikation: Das Projektive
Einganggrossenverfahren und das Regularisierte Ausganggrossenverfahren im
Frequenzbereich fur unvollstandigeMessungen. Habilitation, Universitat Han-
nover,.Reports of theCurt-Risch-Institute of theUniversity ofHannover:CRI-
F- 3/99

Zastosowania rzutowanego residuum sygnału wejściowego do diagnostyki

uszkodzeń układów mechanicznych

Streszczenie

Wpracy przedstawiono zastosowaniemetody rzutowanego residuum sygnałuwej-
ściowego do wykrywania uszkodzeń na podstawie niekompletnych pomiarów obiektu
(liczba pomierzonych sygnałów jest mniejsza niż liczba stopni swobody modelu ba-
danego układu mechanicznego) oraz do wskazania parametrów opisujących model



Application of the projective input residual method... 413

obliczeniowy, wymagającychmodyfikacji przy praktycznej realizacji procedur identy-
fikacji parametrycznej. Metoda nie wymaga redukcji liczby stopni swobody modelu
matematycznego, a tym samym identyfikowane parametry zachowują interpretację
fizyczną. Zaproponowanywskaźnik służący do lokalizacji uszkodzonego elementu po-
równuje pochodne cząstkowe residuów wielkości wejściowych dla modelu bez uszko-
dzeń z modelem reprezentującym obiekt z uszkodzeniami. Rozważania teoretyczne
zilustrowane są przykładem obliczeniowym dla frezarki FWD-32J.

Manuscript received May 15, 2000; accepted for print August 14, 2001