Risk Analysis Based on Acceleration Measurements
Florin Popentiu Vladicescu, City University London, DEEIE, Northampton Square, London EC1V,
OHB, e-mail: Fl.Popentiu@city.ac.uk
Radu Mihaela, Aerofina Enterprise, Bucharest, Romania
Erez Friedman, Elbit Defence Systems, Haifa, Israel
Abstract
This paper discusses two methods to perform a comparative analysis of the cinematic parameters, which functionally
characterize the ballistic systems, as well as the results of their application within a safety-testing context. The first method
permits the evaluation of the risks and the formulation of the acceptance criteria by using the correlation coefficient
matrixes. The stress-strength analysis version, integrated within the second method, offers a probabilistic support for the
risks assertion and acceptance criteria definition.
1. Introduction
Part of the risks associated with the functioning of
a ballistic system are connected with the variation
of the cinematic parameters that define its
evolution on the trajectory. The upgrading of such
a system, usually classified as safety critical item,
imposes a thorough investigation of the hazardous
entities and situations, introduced by the new
subsystems, either in normal functioning or in a
circumstance of critical failure. In order to
determine the functional differences between the
reference system “old” and the hybrid prototype
“old-new”, tests, relevant measurements and
comparative analysis are made. Based on the
knowledge experimentally captured, the package of
preliminary assertions, regarding the components
of the risk vectors (<probability of occurrence,
severity of effects/ consequences> in conformity
with the definitions in [1], [2], or < probability of
occurrence, severity of effects/ consequences,
probability of detection> in conformity with the
definition in [3]) is refined, the main elements
being retained to support the decision for
certification and procedure to the series production.
The methods to be described further on deals with
the assessment of the effects/ consequences of the
relative risks, implied by the constructive and
functional modifications, brought about to a
ballistic system by the modernization of some of its
subsystems,
within
the
conditions
of
acknowledgment and acceptance of the absolute
risks that are associated with the reference system
“old’. The analysis are focused both to determine
the level of alteration of the trajectory on which the
ballistic system moves, alteration that may have as
a consequence the modification of the utilization
envelopes, and to quantify the effects on the
operator of the job station, which can be
unacceptable from the point of view of human
resistance. By using the accelerations measured
during the tests with the reference system “old”,
the hybrid system “old-new” in normal
functioning, and the hybrid system “old-new” in
critical failure, the cinematic parameters
(acceleration gradients, velocities, displacements)
are calculated; then are assessed the differences
between the 2 systems, by regression in the first
method and by a probabilistic technique in the
second method. The comparative analysis is
distributed both to the ballistic department and to
the human engineering department. Their
conclusions are used to decide the acceptance/
rejection and the modification (if necessary) of the
1
risk vectors, on the effects/ consequences
component. In Part I the 2 methods are described
and in Part II some results of an analysis already
performed are given.
Part I
2. Risk assessment methods
Notation
m
n
p
T

=T, N, 0-1
D
t=dT, dN, d1N,
d1dd1+D-1
Xi(), iN, 1im
Yj(), jN, 1jn
Zk(), kN, 1kp
Xai(t), Yaj(t), Zak(t)
Xgi(t), Ygj(t), Zgk(t)
Xvi(t), Yvj(t), Zvk(t)
Xsi(t), Ysj(t), Zsk(t)
R squared
Number of tests with the reference system “old”
Number of tests with the system “old-new” in normal functioning
Number of tests with the system “old-new” having the subsystems “new”
in critical failure
Sampling rate
Total number of acquired samples/ record
Moments in which the values of acceleration have been acquired
Length of the analysis interval
Moments in the analysis interval; the interval was tailored from the whole
record; if the lower bound of the interval coincides with the initiation of the
movement then d1 is to be considered “moment 0”
Acceleration signals acquired during the tests with the system “old”
Acceleration signals acquired during the tests with the system “old-new” in
normal functioning
Acceleration signals acquired during the tests with the system “old-new”
having the subsystems “new” in critical failure
Acceleration signals to which the numerical filtration and the scale
transformation have been applied; to these signals have also been chosen
the initiation moment and the length of the analysis interval
Acceleration gradient signals obtained by the numerical derivation of the
acceleration signals
Velocity signals obtained by the numerical integration of the acceleration
signals
Displacement signals obtained by the numerical integration of the velocity
signals
The square of the Pearson product moment correlation coefficient through
data points in X, Y sets:
n
1
N1 n .
n
1
XY
i= 0
Rsquared
n
1
X.
i= 0
n
Y
N2
i= 0
1
n
X
i= 0
2
1
2
n.
X
i= 0
n
1
n
Y
i= 0
2
1
2
. n.
Y
i= 0
N1
N2
Raij, Rgij, Rvij, Rsij
Raik, Rgik, Rvik, Rsik
Correlation coefficients between the signals from the tests with the system
“old” (independant variables) and the signals from the tests with the system
“old-new” in normal functioning (dependant variables), computed by using
the Rsquared formula
Correlation coefficients between the signals from the tests with the system
“old” (independant variables) and the signals from the tests with the system
“old-new” having the subsystems “new” in critical failure (dependant
variables), computed by using the Rsquared formula
2
Ca, Cg, Cv, Cs
AavgX, AavgY, AavgZ
GavgX, GavgY, GavgZ
VmaxX, VmaxY, VmaxZ
SDX, SDY, SDZ
<aX, aX>, <aY, aY>,
<aZ, aZ>
<gX, gX>, <gY, gY>,
<gZ, gZ>
<vX, vX>, <vY, vY>,
<vZ, vZ>
<sX, sX>, <sY, sY>,
<sZ, sZ>
P(1-2>0)
Extreme correlation coefficients by which the acceptance/ rejection
decision is made
Average values of acceleration over the interval selected for analysis
Average values of the acceleration gradient over the interval selected for
analysis
Maximum values of velocity over the interval selected for analysis
Total displacements in the analysed time interval
Normal distribution parameters (mean and standard deviation) for AavgX,
AavgY, AavgZ variables
Normal distribution parameters (mean and standard deviation) for GavgX,
GavgY, GavgZ variables
Normal distribution parameters (mean and standard deviation) for VmaxX,
VmaxY, VmaxZ variables
Normal distribution parameters (mean and standard deviation) for SDX,
SDY, SDZ variables
If 1, 2 are normally distributed variables, having <1, 1>, and <2,
2> respectively, then 1-2 is a normally distributed variable, having
=1-2 si =(12+22), and the probability is:
P=(1/(2*)(-/,)e-y^2/2dy
PA1(AavgX-AavgY>0),
PA2(AavgX-AavgZ>0)
PG1(GavgX-GavgY>0),
PG2(GavgX-GavgZ>0)
PV1(VmaxX-VmaxY>0),
PV2(VmaxX-VmaxZ>0)
PS1(SDX-SDY>0),
PS2(SDX-SDZ>0)
PAmin, PAmax, PGmin,
PGmax, PVmin, PVmax,
PSmin, PSmax
g, g/s, m/s, mm
Probabilities that the differences between the “average values of
acceleration” variables are positive (computed by using P(1-2>0
formula))
Probabilities that the differences between the “average values of
acceleration gradient” variables are positive (computed by using P(12>0))
Probabilities that the differences between the “maximum values of
velocity” variables are positive (computed by using P(1-2>0))
Probabilities that the differences between the “total displacement” variables
are positive (computed by using P(1-2>0))
Extreme probabilities by which the acceptance/ rejection decision is made
Measurement units for acceleration, acceleration gradient, velocity,
displacement; 1g 9.81 m/s2
2.1 Method I
“old-new”. In [4] are discussed some issues about
the numerical filtering of acceleration signals.
Apply the scale transformation on the 2 axis and
establish the start-up moments and the analysis
interval, obtaining the significant acceleration
fragments (time): Xai(t), Yaj(t), Zak(t);
Step 3: Compute the acceleration gradients(time):
Xgi(t), Ygj(t), Zgk(t), velocities(time) Xvi(t), Yvj(t),
Zvk(t) and displacements(time) Xdi(t), Ydj(t),
Zdk(t);
Step 4: Built up the correlation coefficient
matrixes:
Step 1: Perform the tests, acquiring for each of
them, the acceleration signal on the main direction
of movement (or on any other direction of interest),
then choose the records for which plausible
functioning and normal running of the measuring
instrumentation are get; be Xi(), iN, 1im,
Yj(), jN, 1jn, Zk(), kN, 1kp, the sets of
such values;
Step 2: Approximate the recorded signals by
means of numerical filters of high selectivity,
chosen in such a way so as not to erase the traces
(frequency bands) that characterise the functional
differences between the 2 system classes: “old” and
3
R squared
Xai(t)
Yaj(t)
Raij
R squared
Xgi(t)
Ygj(t)
Rgij
R squared
Xvi(t)
R squared Yak(t)
R squared Ygk(t)
Xai(t)
Raik
Xgi(t)
Rgik
Step 5: Compare the correlation coefficients with
the Ca, Cg, Cv, Cs extreme values, which are given
in the test plan, or are established after having
performed the tests and the regression analysis:
MIN(Raij,
Raik)Ca,
MIN(Rgij,
Rgik)Cg,
MIN(Rvij, Rvik)Cv, MIN(Rsij, Rsik)Cs;
Step 6: Analyse, by direct graphic comparison or
by tracing the regression line, the signals between
which there are some unsatisfactory correlation’s,
Yvj(t)
Rvij
R squared
Xsi(t)
Ysj(t)
Rsij
R squared Yvk(t)
R squared Ysk(t)
Xvi(t)
Rvik
Xsi(t)
Rsik
then determine the causes. In the second part, there
is a presentation of the decision-making orientation
for some of them.
Step 7: Decide on the acceptance/ rejection and on
the modification of the risk vectors, on the effects/
consequence component.
.
2.2 Method II
Step 1: Identical with the step 1, method I;
Step 2: Identical with the step 2, method I;
Step 3: Identical with the step 3, method I;
Step 4: Choose the variables that may characterise
statistically the systems functioning within the
interval selected for analysis, for example: the
average values of accelerations, the average values
of acceleration gradient, the maximum values of
velocity, the total displacements: AavgX, AavgY,
AavgZ, GavgX, GavgY, GavgZ , VmaxX, VmaxY, VmaxZ, SDX,
SDY, SDZ;
Step 5: Compute <aX, aX>, <aY, aY>, <aZ,
aZ>, <gX, gX>, <gY, gY>, <gZ, gZ>, <vX,
vX>, <vY, vY>, <vZ, vZ>, <sX, sX>, <sY,
sY>, <sZ, sZ>;
Step 6: Compute PA1(AavgX-AavgY>0), PA2(AavgXAavgZ>0), PG1(GavgX-GavgY>0), PG2(GavgX-GavgZ>0),
PV1(VmaxX-VmaxY>0), PV2(VmaxX-VmaxZ>0), PS1(SDXSDY>0), PS2(SDX-SDZ>0);
Step 7: Compare the obtained probabilities with
the PAmin, PAmax, PGmin, PGmax, PVmin, PVmax, PSmin,
PSmax values, established in the test plan or chosen
after having performed the tests and the statistical
computations:
PAmin< PA1(AavgX-AavgY>0)<PAmax, PAmin< PA2(AavgXAavgZ>0)<PAmax
PGmin<
PG1(GavgX-GavgY>0)<PGmax,
PGmin<
PG2(GavgX-GavgZ>0)<PGmax
PVmin<
PV1(VmaxX-VmaxY>0)<PVmax,
PVmin<
PA2(VmaxX-VmaxZ>0)<PVmax
PSmin< PS1(SDX-SDY>0)<PSmax, PSmin< PS2(SDXSDZ>0)<PSmax
Step 8: If there are any non- conformities,
determine the causes. The second part details the
decision-making orientation for some of them.
Step 9: Decide on the acceptance/ rejection and on
the modification of the risk vectors, on the effects/
consequence axis.
Part II
3. Results
The 2 methods have been recently used to analyse
the experimental results obtained during a safetytesting program. The reference “old” is a ballistic
system powered by a cartridge with solid
propellant. The “old-new” hybrid prototype has the
mass and the centre of gravity slightly modified
against the reference. Moreover, at a certain
moment during the functioning, a quick-disconnect
pin-socket connector is released. In these
conditions, we can anticipate that, for the upgraded
system, the level of accelerations is lesser over the
whole period of functioning and, near to the
moment of disconnection, there is a braking force,
both for a normal functioning and for a critical
failure of the connector.
These effects are visible on the acceleration
diagrams, if we can assure the repeatability of the
tests, from the point of view of the propulsion
system functioning, and if mechanical interlocking
or significant deterioration do not appear.
4
After tests performance, 7 records in the
configuration m=1, n=5, p=1 have been validated
for analysis. In Figures 1-4 are plotted (within the
same system axis) the fragments: acceleration [g]/
time [ms], acceleration gradient [g/s]/ time [ms],
velocity [m/s]/ time [ms], displacement [mm]/ time
[ms], resulted from the application of steps 1-3,
common for both methods.
Figure 1. Acceleration [g]/ Time [ms]
Figure 2. Acceleration gradient [g/s]/ Time [ms]
Figure 3. Velocity [m/s]/ Time [ms]
Figure 4. Displacement [mm]/ Time [ms]
5
3.1 Method I
Step 4: Built up the correlation coefficient
matrixes:
R squared
Xa1
Ya1
0.68
Ya2
0.99
Yg2
0.80
Ya3
0.97
Yg3
0.51
Ya4
0.95
Yg4
0.40
Ya5
0.94
Yg5
0.41
R squared
Xa1
Za1
0.56
R squared
Xg1
Zg1
0.00
R squared
Xg1
Yg1
0.03
R squared
Xv1
Yv1
0.9843
Yv2
0.9997
Yv3
0.9968
Yv4
0.9982
Yv5
0.9850
R squared
Xv1
Zv1
0.9820
R squared
Xs1
Ys1
0.9975
Ys2
0.9999
Ys3
0.9997
Ys4
0.9999
Ys5
0.9993
R squared
Xs1
Zs1
0.9993
One of the causes for the bad correlation is the
delay (by 6 ms in the first case and 16 ms in the
second case); by overlapping the moments when
the signals reach the maximum value we can
improve the correlation coefficients: 0.80 (Xa1 and
Ya1) is 0.94 (Xa1 and Za1).
Another cause for the bad correlation of Xa1 and
Ya1 signals is the temporary mechanical
interlocking (visible on Ya1 diagrams), having no
connection with the “new” subsystems functioning,
which are integrated on the upgraded prototype.
In this situation we can choose one of the
solutions: (a) Ya1 is given up (automatically Yg1,
Yv1, Ys1 are given up) since there are satisfactory
results for the other 4 tests of the same type; (b) the
Ca criterion is given up (or is modified to include
the value 0.80).
Step 5: Compare the obtained correlation
coefficients with the values specified in the test
program: Ca=0.9, Cg=0.40, Cv=0.98, Cs=0.99.
Step 6: Analyse the diagrams between which there
are some unsatisfactory correlation. We notice that
these are Xa1 and Ya1 (correlation coefficient
0.68), Xa1 and Za1 (correlation coefficient 0.56);
the bad correlation of the above signals determines
the worse correlation of the acceleration gradient
signals: Xg1 and Yg1 (correlation coefficient 0.03),
Xg1 and Zg1 (correlation coefficient 0.00). We can
easily notice that, in spite of these, the velocity
signals and the total displacement signals are
satisfactorily correlated. In Figures 5-6 graphical
comparisons between Xa1 and Ya1 (by tracing the
regression line, and by overlapping) are made. Xa1
and Za1.are plotted in figures 7-8 by the same
techniques.
6
Figure 5. Acceleration [g]/ Acceleration [g]
Figure 7. Acceleration [g]/ Acceleration [g]
Figure 6. Acceleration [g]/ Time [ms]
Figure 8. Acceleration [g]/ Time [ms]
so the risk vectors, on effects/ consequences
component, does not change.
Step 7: Decide on acceptance if one of the
solutions (a) or (b) has been adopted at step 6. The
hybrid system does not introduce additional risks,
3.2 Method II
this method hard to apply. However, we can go to
step6.
Step 4, 5: In Table 1 the chosen variables and the
values <mean, standard deviation> are presented.
The lack of information on X, Z variables makes
(some of the probabilities will be 0 or 1).
7
Table 1.
AavgX [g]
AavgY [g]
AavgZ [g]
GavgX [g/s]
GavgY [g/s]
GavgZ [g/s]
VmaxX [m/s]
VmaxY [m/s]
VmaxZ [m/s]
SDX [mm]
SDY [mm]
SDZ [mm]
Test 1
9.63
8.29
7.75
-16.26
-18.39
-15.81
19.91
17.18
16.49
2537
2026
1892
Test 2
8.85
-20.84
18.70
2371
-
Test 3
8.34
-14.70
17.97
2320
-
Test 4
8.65
-27.59
18.79
2365
-
Test 5
7.36
-54.76
18.07
2323
-

9.63
8.30
7.75
-16.26
-27.26
-15.81
19.91
18.14
16.49
2537
2281
1892

0
0.51
0
0
14.38
0
0
0.58
0
0
129.20
0
Table 2.
PA1(AavgX-AavgY>0)
PA2(AavgX-AavgZ>0)
PG1(GavgX-GavgY>0)
PG2(GavgX-GavgZ>0)
PV1(VmaxX-VmaxY>0)
PV2(VmaxX-VmaxZ>0)
PS1(SDX-SDY>0)
PS2(SDX-SDZ>0)
0.9955
1
0.7764
0
0.9989
1
0.9761
1
Step 8: The probabilities that the differences
between the X, Z variables be positive do not
frame within the imposed limits. Because of the
lack of information on X, Z dispersions, we can
choose one of the solutions: (a) allocate standard
deviation values for X, Z variables in a reasonable
way, then recalculate the probabilities; (b) adopt
some determinist acceptance criteria, the simplest
being the framing of |AavgX-AavgZ|, |GavgX-GavgZ|,
|VmaxX-VmaxZ|, |SDX-SDZ| between some limits.
Step 6: The computed probabilities are in
conformity with Table 2.The data in Table 2
indicate that the “average values for acceleration”,
“the maximum values of velocity” and the “total
displacements” are greater for the reference system
“old”. The results obtained for the “average values
of acceleration gradient” are contradictory.
Step 9: Both in case (a) and case (b) we can easily
reach the acceptance decision and conclude that
the hybrid system “old-new” does not introduce
additional risks, so the risk vectors, on effects/
consequences component, does not change.
Step 7: Compare the computed probabilities with
the values: PAmin= PGmin = PVmin = PSmin =0.5 si
PAmax= PGmax = PVmax = PSmax=0.999.
4. Conclusions
Both methods permit the investigation of the
differences between the signals acquired during the
tests with the reference system “old” and the
signals acquired during the tests with the upgraded
system “old-new”, the last being in normal
functioning/ critical failure.
As the result of their application in the mentioned
context, the conclusions are similar. Method II is
mostly indicated if there is an apriori information
on the distribution parameters of the variables, or if
a great number of tests are performed.
8
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