ARTICLE IN PRESS
Mechanical Systems and Signal Processing 23 (2009) 1548–1553
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing
journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Adaptive diagnosis of the bilinear mechanical systems
L. Gelman , S. Gorpinich, C. Thompson
Cranfield University, Cranfield, UK
a r t i c l e i n f o abstract
Article history: A generic adaptive approach is proposed for diagnosis of the bilinear mechanical
Received 25 July 2007 systems. The approach adapts the free oscillation method for bilinearity diagnosis of
Received in revised form mechanical systems. The expediency of the adaptation is proved for a recognition
8 January 2009
feature, the decrement of the free oscillations. The developed adaptation consists of
Accepted 19 January 2009
variation of the adaptive likelihood ratio of the decrement with variation of the
Available online 29 January 2009
resonance frequency of the bilinear system. It is shown that in the cases of the
Keywords: frequency-independent and the frequency-dependent internal damping, the adaptation
Bilinear mechanical systems is expedient. To investigate effectiveness of the adaptation in these cases, a numerical
Diagnosis
simulation was carried out. The simulation results show that use of the adaptation
Adaptation
increases the total probability of the correct diagnosis of system bilinearity.
Decrement of free oscillations
Likelihood ratio & 2009 Elsevier Ltd. All rights reserved.
Resonance frequency
1. Introduction
Bilinear mechanical systems abound in many settings and applications. Some examples of bilinear mechanical systems
discussed in literature include:
oscillating mechanical systems with clearances and motion limiting stops[1];
offshore structures: free-hanging risers, tension leg platforms and suspended loads [2–4], articulated loading towers,
constrained by a connection to a massive tanker or vessels moored against fenders[5];
damaged (e.g., cracked, pitted, etc.) mechanical systems [6–10]; and
un-damaged gearboxes [11].
Among the most widely used approaches for vibro-acoustical diagnosis of bilinearity of mechanical systems is the free
oscillation method [7,8,12]. The method consists of the impact excitation of the free mechanical oscillations of the bilinear
system and evaluation of a diagnosis feature from the vibro-acoustical signals radiated by these oscillations. As a diagnosis
feature, the decrement of the free oscillations has been widely used [8,9].
Normally, diagnosis of the bilinear mechanical systems is carried out under presence of variable nuisance parameters
[13,14] of the systems. Changes of nuisance parameters lead to the deterioration of recognition effectiveness [13,14]. Typical
examples of the variable nuisance parameters for mechanical systems are the performance parameters: e.g., the shaft
speed, a load, etc.
The problem is to preserve diagnosis effectiveness of the bilinear mechanical systems in presence of the variable
nuisance parameters. It is important to solve this problem for various bilinear mechanical systems: systems with
Corresponding author.
E-mail address: L.Gelman@Cranfield.ac.uk (L. Gelman).
0888-3270/$ - see front matter & 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ymssp.2009.01.007
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L. Gelman et al. / Mechanical Systems and Signal Processing 23 (2009) 1548–1553 1549
clearances and motion limiting stops, offshore structures, damaged and un-damaged machinery. This problem has not been
investigated in literature.
To preserve the recognition effectiveness, a new generic adaptive approach for system diagnosis is proposed here based
on an adaptive likelihood ratio. In general, the proposed adaptive likelihood ratio depends on the diagnosis feature vector
and also on the vector of measurable variable nuisance parameters. This is an improvement over most published
applications concerning recognition and diagnostics of various systems; normally the classical likelihood ratio [15]
averaged over ranges of variable nuisance parameters is used for system recognition and diagnostics.
The purposes of this paper are to:
propose a generic adaptive approach for system diagnosis;
apply an adaptive approach for the free oscillation method of diagnosis of system bilinearity with the decrement as a
recognition feature; and
compare the proposed and traditional approaches for diagnosis of the bilinear mechanical systems.
2. The bilinear system and decrement of the free oscillations
Let us consider a generic bilinear mechanical system, a single degree of freedom oscillator, in which the stiffness is
bilinear
8
< X€ þ 2hS X_ þ o2 X ¼ 0; XX0;
S
(1)
: X€ þ 2hC X_ þ o2C X ¼ 0; Xo0;
where X is the displacement, m p is ffiffiffiffiffiffiffiffiffi
the mass, hS ¼p oS, ffi hC ¼ zCop
zSffiffiffiffiffiffiffiffiffi C, ffiffiffiffiffiffiffiffiffiffiffi
zS and zC arep
the damping
ffiffiffiffiffiffiffiffiffiffiffi
ffi ratios at the positive and
negative displacements, zS ¼ c=2 kS m; zC ¼ c=2 kC m; oS ¼ kS =m; oC ¼ kC =m, c is the damping, kS and kC are
the stiffness at the positive and negative displacements.
At the positive displacement, the stiffness decreases with the quantity Dk ¼ kCkS [1–11].
Using Eq. (1), the resonance frequencies at the positive and negative displacements can be written as follows,
respectively:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
oSd ¼ oS 1 z2S ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2)
oCd ¼ oC 1 z2C :
Using Eqs. (1) and (2), the resonance frequency of the bilinear system after transformations can be written as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o2Cd o2Sd c2
o0 ¼ 4 2
þ . (3)
ðoCd þ oSd Þ 4m2
Eq. (3) is generic. In the important case of the low-damped system (i.e., the damping ratios are less than 0.1) substituting
Eq. (2) into Eq. (3) yields
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o2C o2S c2
o0 ¼ 4 2
þ . (4)
ðoC þ oS Þ 4m2
Finally, equation for resonance frequency can be written as follows (see Appendix A) [16]:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
2 1k
o0 ¼ oCd pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,
n
(5)
1þ 1k
where k* is the stiffness ratio, k* ¼ Dk/kC.
The logarithmic decrement of the free oscillations of the bilinear system under the impact initial conditions [x(0) ¼ 0,
ẋ(0) ¼ n0] is obtained from Eq. (1) after transformations as
2pz o
d¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi C Cd ½1 h þ ð1 k Þ1=2 ,
n n
(6)
2 n 1=2
o0 1 zC ½1 þ ð1 k Þ
where h* is the normalized damping ratio [7]
n hC hS
h ¼ . (7)
hC
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3. Adaptation of system diagnosis
Procedures of the proposed generic adaptive approach are to
find a vector of measurable variable nuisance parameters for the selected recognition feature vector;
measure these parameters; and
use an adaptation of the likelihood ratio (i.e., the adaptive likelihood ratio): to vary the likelihood ratio of diagnosis
feature vector with variation of a vector of measurable variable nuisance parameters in order to decrease the influence
of these parameters and thus, preserve the diagnosis effectiveness.
The adaptation of the free oscillation method using the logarithmic decrement as a recognition feature could be
achieved with the use of measurable variable nuisance parameters, on which the decrement depends. As shown
by Eq. (6), the decrement depends on the damping ratio zC, the resonance frequency oCd at the negative displacement of
the bilinear system, the resonance frequency o0 of the bilinear system, the stiffness ratio k* and the normalized damping
ratio h*.
The stiffness ratio k* is a basic diagnosis parameter [1–11]; the considered diagnosis is based on the difference in the
stiffness ratios between the linear and the bilinear systems. According to Eq. (7), the normalized damping ratio h* depends
on the basic recognition parameter k*. The damping ratio zC at the negative displacements (i.e., the linear system), is a
nuisance parameter.
One can see from Eq. (5) that the resonance frequency o0 of the bilinear system depends on a basic diagnosis parameter,
the stiffness ratio k* and the resonance frequency at the negative displacement oCd (i.e., the resonance frequency of the
linear system). Therefore, the resonance frequency o0 can be used as a recognition feature [6], when the resonance
frequency at the negative displacement oCd is known. Otherwise, the resonance frequencies o0 and oCd also become
nuisance parameters.
An important case is considered here: the resonance frequency at the negative displacement oCd is the unknown
measurable nuisance parameter. Thus, the resonance frequency of the bilinear system o0 is also the unknown measurable
nuisance parameter. Normally, the basic parameters of the bilinear system: mass m, damping c and stiffness k are variable
random parameters due to the manufacturing tolerances of bilinear systems. Therefore, the resonance frequencies of the
linear and bilinear systems are also variable random parameters.
For these cases, the adaptation depending on the measurable variable random resonance frequency of the bilinear
system is proposed and investigated.
The adaptive method consists of estimating the decrement
_
d, the resonance frequency o0 and the adaptive likelihood
ratio La which depends on the decrement estimate d and the estimate o ^ 0 of the resonance frequency
^ o Wðdjo
^ 0 ; S1 Þ
La ðd; ^ 0Þ ¼ _ , (8)
Wðdjo
^ 0 ; S0 Þ
_ _
where Wðdjo ^ 0 ; Sj Þ is the one-dimensional conditional probability density function (pdf) of the decrement estimate d which
depends on the status (class) Sj of the system and the estimate o ^ 0 of the resonance frequency, j ¼ 0, 1; class S0 corresponds
to the linear system with resonance frequency oCd, damping ratio zC and the zero stiffness ratio; class S1 corresponds to the
bilinear system with the bilinear stiffness and non-zero stiffness ratio.
In contrast, the classical non-adaptive method consists
_
of estimating the decrement d and the classical likelihood ratio
Lc which depends only on the decrement estimate d:
^ ¼ WðdjS1 Þ
Lc ðdÞ _ , (9)
WðdjS0 Þ
_ _
where Wð ^ djSj Þ is the one-dimensional conditional pdf of the decrement estimate d which depends on class Sj of the system.
_
The pdf Wð^ djSj Þ of the classical likelihood ratio is averaged over a range of the variable resonance frequency for class Sj. In
_
contrast, the pdf Wðdjo ^ 0 ; Sj Þ of the adaptive likelihood ratio is not averaged over a range of the variable resonance
frequency.
Then the standard decision-making procedure [15] should be used: the likelihood ratio La or Lc is compared with one or
several thresholds depending on the selected effectiveness criterion. For example, if the maximum likelihood criterion [15]
is used, then a threshold for the likelihood ratios is unity.
It should be highlighted that the proposed adaptation is expedient only if the resonance frequency of the bilinear system is
variable.
To prove the expediency of the adaptation, dependencies between the decrement d and the stiffness ratio k* were
studied for various values of the resonance frequency o0 and two different proportional internal dampings [7].
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L. Gelman et al. / Mechanical Systems and Signal Processing 23 (2009) 1548–1553 1551
4. The decrement of the free oscillations for different internal damping
4.1. The frequency-independent internal damping
For the frequency-independent internal damping, the normalized damping ratio h* can be obtained from Eq. (7) after
transformations as
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n n
h ¼1 1k (10)
Substituting Eq. (10) into Eq. (6) yields, after transformations
" #
n
2phC 2ð1 k Þ
d¼ . (11)
o0 1 þ ð1 kn Þ1=2
The dependencies of the decrement (11) on the stiffness ratio are shown in Fig. 1 for various values of the resonance
frequency. As follows from Eq. (11) and Fig. 1, the decrement depends on the stiffness ratio k* and the resonance frequency
o0. The adaptation is expedient since, for constant values of the nuisance parameter o0, the decrement depends on the
stiffness ratio k*.
4.2. The frequency-dependent internal damping
For the frequency-dependent internal damping, the normalized damping ratio, h* can be obtained from Eq. (7) after
transformations as
n n
h ¼k (12)
Substituting Eq. (12) into Eq. (6) yields, after transformations
" #
2phC 1 k þ ð1 k Þ1=2
n
d¼ . (13)
o0 n
1 þ ð1 k Þ1=2
The dependencies of the decrement (13) on the stiffness ratio k* are shown in Fig. 2 for various values of the resonance
frequency o0. As follows from Eq. (13) and Fig. 2, the decrement depends on the stiffness ratio k* and the resonance
frequency o0. As in the previous case, the adaptation is expedient since, for constant values of the nuisance parameter o0,
the decrement depends on the stiffness ratio k*.
5. Numerical simulation
To estimate the effectiveness of the adaptation, a numerical simulation was carried out for cases 4.1–4.2 and classes S0
and S1. The variable random resonance frequency of the linear system oCd is assumed to be uniformly distributed in the
range (1000–1200) Hz, i.e., 20% variation. The random stiffness ratio of the bilinear system is uniformly distributed in the
range (0–0.4). The considered variation values correspond to real cases occurring in the bilinear systems listed in the
chapter 1.
It is assumed, without loss of generality, that the decrement and the resonance frequency were measured without errors
for both classes. The damping parameter is hC ¼ 0.05 rad/s.
Fig. 1. Dependencies of the decrement of the free oscillation on the stiffness ratio k* and the resonance frequency o0 for the frequency-independent
internal damping (hC ¼ 0.05 rad/s): (a) o0 ¼ 2p100 rad/s; (b) o0 ¼ 2p1000 rad/s; and (c) o0 ¼ 2p10,000 rad/s.
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1552 L. Gelman et al. / Mechanical Systems and Signal Processing 23 (2009) 1548–1553
Fig. 2. Dependencies of the decrement of the free oscillation on the stiffness ratio k* and the resonance frequency o0 for the frequency-dependent
internal damping (hC ¼ 0.05 rad/s): (a) o0 ¼ 2p100 rad/s; (b) o0 ¼ 2p1000 rad/s; and (c) o0 ¼ 2p10,000 rad/s.
Table 1
Estimates of the total probabilities of bilinearity diagnosis.
Damping model Non-adaptive approach Adaptive approach
The frequency-dependent internal damping 0.8 1.0
The frequency-independent internal damping 0.81 1.0
5.1. Non-adaptive approach
Estimates of the conditional pdfs of the decrement averaged over the resonance frequency ranges for classes S0 and S1 for
different damping models were obtained using the Monte-Carlo procedure and Eqs. (5), (11) and (13). To estimate these
pdfs, 950 runs were simulated for each combination of the class and the damping model. Then, the classical likelihood
ratios (9) were estimated on the basis of the averaged pdfs of the decrement.
Estimates P of the total probabilities of the correct diagnosis for the non-adaptive and the adaptive approaches are
calculated as follows: P ¼ mlPl+mbPb, where Pl and Pb are estimates of the probability of the correct diagnosis of the linear
and the bilinear systems, respectively; ml, and mb are a priori probabilities of the linear and the bilinear systems,
respectively. Taking into account the equal number of the simulated runs for the linear and the bilinear systems, a priori
probabilities are ml ¼ mb ¼ 0.5.
These estimates are obtained by the simulation, using the classical likelihood ratios and the maximum likelihood
criterion [15] and presented in Table 1.
5.2. Adaptive approach
According to the assumption that the decrement and the resonance frequency were measured without errors for both
classes and due to the monotonous character of the dependencies presented in Figs. 1 and 2, the decrement values of the
class S0 do not overlap with the decrement values of the class S1. Therefore, when using the adaptive likelihood ratio (8) and
measuring the decrement and the resonance frequency without errors, the estimates of the total probability of the correct
diagnosis are equal to unity.
Taking into account these estimates, one can see from Table 1 that when using the adaptive approach, the estimates of
the total probability of the correct diagnosis increase by (19–20%). This increase highlights the efficiency of the proposed
adaptive approach for bilinearity diagnosis with the logarithmic decrement as a diagnosis feature.
6. Conclusions
1. A generic adaptive approach was proposed for system diagnosis. The main idea of the approach is a variation of the
adaptive likelihood ratio with variation of the vector of measurable variable nuisance parameters. The approach
decreases the influence of these parameters on diagnostic features and thus preserves diagnosis effectiveness in the
presence of variable nuisance parameters.
2. The proposed approach was applied for the free oscillation method for bilinearity diagnosis of the bilinear mechanical
systems. Expediency of the adaptation was proved for the following diagnosis feature: the decrement of the free
oscillations of the bilinear system. The resonance frequency of the bilinear system was selected as the measurable
ARTICLE IN PRESS
L. Gelman et al. / Mechanical Systems and Signal Processing 23 (2009) 1548–1553 1553
variable random nuisance parameter. The new generic analytical equation for the resonance frequency of the bilinear
system was obtained.
3. The adaptation consists of a variation of the adaptive likelihood ratio of the decrement of the free oscillations with
variation of the variable random resonance frequency of the bilinear system. It was shown that in the cases of the
frequency-independent and the frequency-dependent internal dampings, the proposed adaptation is expedient.
Generally, the proposed adaptation is expedient if a diagnostic feature depends on a variable nuisance parameter.
4. To investigate the adaptation effectiveness in the above-mentioned cases, a numerical simulation was carried out. The
simulation results have shown that when the adaptive approach was used, the estimates of the total probability of the
correct recognition increased by (19–20)%. This increase in probabilities is obtained for the considered numerical
example.
Generally, increase depends mainly on the dependency between a diagnostic feature and a nuisance parameter and an
error in estimation of a nuisance parameter.
5. The paper results indicate that use of the proposed adaptation for the decrement of the free oscillations improves the
effectiveness of bilinearity diagnosis for bilinear mechanical systems in presence of the variable nuisance parameter, the
resonance frequency of the system.
Appendix A
Eq. (4) after transformations can be written as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o2 o2 ½1 þ ððoC þ oS Þ2 z2S =o2C Þ
o0 ¼ 4 C S (A1)
ðoC þ oS Þ2
Eq. (A1) after the additional transformation can be presented as follows:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h pffiffiffiffiffiffiffiffiffiffiffiffi i
u
u o2 o2 1 þ ð1 þ kS =kC Þ2 z2S
t C S
o0 ¼ 4 (A2)
ðoC þ oS Þ2
Taking into account that kSpkC [1–11], Eq. (2) and expression for the stiffness ratio, Eq. (A2) for low-damped system
finally can be presented as follows:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
2 1k
o0 ¼ oCd pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
(A3)
1þ 1k
References
[1] S. Narsiavas, On the dynamics of oscillators with bilinear damping and stiffness, International Journal of Non-linear Mechanics 25 (5) (1990)
535–554.
[2] C. Hsu, The response of a parametrically excited hanging string in fluid, ASME Journal of Sound and Vibration 39 (1975) 305–316.
[3] M. Patel, D. Brown, J. Witz, Operability analysis for a mono-hull crane barge, in: Proceedings of the RINA Spring Meeting, 1986.
[4] J.M.T. Thompson, H.B. Stewart, Nonlinear Dynamics and Chaos, Willey, New York, 1986.
[5] E. Jefferys, M. Patel, Dynamic analysis models of the tension leg platform, ASME Journal of Energy Resources Technology 104 (1982) 317–323.
[6] M. Imregun, K. Saniturk, Natural frequency and damping changes produced by fatigue cracks, in: Proceedings of the Fifteenth International Seminar
on Modal Analysis, Leuven, Belgium, 1990, pp. 791–805.
[7] L.M. Gelman, S.V. Gorpinich, Non-linear vibro-acoustical free oscillation method for crack detection and evaluation, Mechanical Systems and Signal
Processing 14 (3) (2000) 343–351.
[8] F.P. Banas, L.M. Gelman, S.V. Gorpinich, Nonlinear vibro-acoustic free oscillation method for crack detection and evaluation, in: Proceedings of Third
International Conference ‘‘Acoustical and Vibratory Surveillance Methods and Diagnostic Techniques,’’ CETIM-Senlis, France, 1998, pp. 521–529.
[9] A. Dimarogonas, Vibration of cracked structures: a state-of-the-art review, Engineering Fracture Mechanics 55 (5) (1996) 831–855.
[10] L. Gelman, P. Anuzis, M. Sanderson, C. Thompson, Intelligent Recognition: Components of The Short Time Fourier Transform vs. Conventional
Approaches, Chapter 22 in the Book: Computer Aided Intelligent Recognition Techniques and Applications, Wiley, New York, 2005.
[11] L. M Gelman, Piece-wise model and estimates of damping and natural frequency for a spur gear, Mechanical Systems and Signal Processing 21 (2007)
1192–1196.
[12] S.G. Braun, Mechanical Signature Analysis, Academic Press, New York, 1986.
[13] V.A. Kovalevsky, Image Pattern Recognition, Springer, Berlin, 1980.
[14] M. DeGroot, Optimal Statistical Decisionsm, McGraw-Hill, New York, 1970.
[15] C. Chen, L. Pau, P. Wang, Handbook on Pattern Recognition and Computer Vision, World Scientific Publishing Company, Inc., Singapore, 1993.
[16] L. Gelman, I. Petrunin, M. Sanderson, C. Thompson, S. Yakasai, Novel fatigue and damping diagnostics, in: Proceedings of the Eighteenth International
Congress and Exhibition on Condition Monitoring and Diagnostic Engineering Management (COMADEM 2005 International), Cranfield, 2005, UK.
