44
3.5 Formulation of the Relationship between the Observed Data and
the Hidden State
In order to specify the relationship between the observed data
i
y
and the hidden state
i
x , we propose to use the method of floating scale parameters. This can be achieved by
assuming that
i
y
will follow a two-parameter family of distributions, such as gamma or Weibull, denoted by a probability density function pdf.,
, ;
i
y f
, where is the
scale and is the shape parameter respectively. Then, by letting
2 ,
; 1
, ;
|
2 2
1 1
X if
y f
X if
y f
x y
p
i i
i i
3-17
the relationship between
i
y
and
i
x can be established. The details of this establishment of the relationship will be shown when presenting the development of the model using
simulated and real data.
3.6 Modelling Development
We started the development of the model by observing the data in Wang 2002, which shows a typical stochastic nature of a bearing life distribution with vibration
information. See Figure 3-4 below.
Figure 3-4: Vibration data for six bearings
45 Figure 3-4
shows the data of the overall vibration level obtained from six rolling element bearings, which is conducted in a laboratory fatigue experiment several years
ago. Generally, the overall vibration level is a measure of the total vibration amplitude over a wide range of frequencies. The velocity amplitude in this case is expressed in
term of root mean square rms value, which tells us the energy of the vibration. It is noted, this is the only condition monitoring data that are available to us, since other
associated information about the experiment such as size and type of the bearings,
running speed and loading have been lost. Figure 3-4 shows that for every bearing, the
overall vibration level was fairly flat until a time where it started to increase rapidly. This illustrates the idea of the two-stage failure process stated by Christer and Waller
1984 as discussed in Chapter 2.
Since this is an early developmental stage of model building, we are reluctant to use this dataset to develop our model. Hence, a simulation approach was purposely used to
approximate the pattern of the data and test the feasibility of our model. By using simulation, we know the true parameters and the model used, so it helps us to check
whether we can successfully recover the model parameters from simulated data. Also, using these simulated
i
y ~
, we tested our model to see whether it was able to predict the underlying state accurately. Generally, by simulation it is easy to validate a model under
development, by comparing the results produced by the model, since we know all the
1
l ‘s and
2
l
‘s. Figure 3-5 below shows the set-up algorithm for the simulation.
46
Figure 3-5: Set-up algorithm to generate simulated pattern data
We now demonstrate the modelling development through a numerical example based on our simulated data. For this example, to calculate
|
i i
x P
we assume:
1 1
l f
=
1 1
1 l
e
3-18
2 2
l f
=
2 2
2 l
e
3-19
1 1
1
1 1
1 1
l
e l
L P
l F
3-20
2 2
1
2 2
2 2
l
e l
L P
l F
3-21
47 with
025 .
1
and 05
.
2
purposely selected in order to repeat the pattern in
Figure 3-4
. As the relationship between
i
y
and
i
x can be treated through a conditional distribution
|
i i
x y
p
, a Weibull distribution is chosen for the time being. The distribution of
i
y
given
i
x is given below:
2 1
ˆ ˆ
|
ˆ 1
2 2
1 1
1
2 1
i i
y i
y i
i i
x x
if if
e y
e y
x y
p
i i
where
2
ˆ
i i
t t
i
dl l
f dl
l f
l t
b
1 1
1 1
1 1
1 1
1
1
3-22
Equation 3-22 requires explanation, particularly when
2
i
X
. It is known that during the normal working period of
1
l , the measured condition information data may display no particular trend and fluctuate around a constant mean; then by letting the scale
parameter be
1
while the shape parameter is fixed, we have
1
1 1
|
i i
X Y
E
. During the second stage, the readings may start to increase and display a trend, in which
case we assume
2 |
i i
X Y
E
2
1
1
1 1
l t
b
i
; that is, the expected value
of the observed data when
2
i
X
is approximately a linear function of the time since the start of
2
l , with the intercept approximately equal to the expected value of the observed data in stage 1. It is straightforward to show that
2
1
1 1
1
l t
b
i
3-23
Since we don‘t know exactly when
1
L finishes, we use an expected value for
2
conditional on
i
t L
1
, which is given by equation 3-22. Figure 3-6 demonstrates the
working principle of the above modelling idea.
48
Figure 3-6: Relationship between
i
y
and
i
x
The above is just one of many possible ideas to model the relationship between
i
y
and
i
x motivated by the vibration-based monitoring cases and the real-life data we have. It is possible to have an exponential function instead of a linear relationship between
i
y
and
i
x , but let us use the linear case first.
3.7 Numerical Examples
Kategori