Thanks for the code.
Just a remark on the notations, you should write :
F(T) = Int_{0}^{T} PHI(t) . f(T-t) . dt
i.e. not F(t) since t is mute.
However, you should pay attention to the delay notion associated with
convolution and the relationships between discrete convolution and
continuous convolution. I am not sure that the output of conv matches
with a given discretization of the integral above. Maybe rectangle
method, but I am not sure at all. Anyway, you should have F(0)=0 which
does not seem to be the case in your graph.
S.
Le 03/02/2023 à 04:53, Heinz Nabielek a écrit :
Dear SciLab Friends:
I have an object consisting of many (~10,000) small components that can fail in
a statistical way during long-term operation at extreme conditions.
My primary failure model is described by PHI(t) going monotonically from zero
to one at times from t=0 to T. In the computer, this is realized in n timesteps.
Mechanism 2: There is a secondary failure mechanism that starts only when a
component has previously failed by mechanism 1 and occurs after some delay. The
delay function is f(t) and final expression for the evolution of the mechanism
2 failure is given by:
F(t) = Int_{0}^{T} PHI(t) . f(T-t) . dt
a classical convolution Faltungsintegral.
In Scilab, I write
F= conv(PHI, f, "same")/ n;
and the resulting function F looks reasonable for most of the time. Under some
conditions, however, I can have
F>PHI
at early stages, but this is physically impossible. Because it comes later, F
must always be below PHI.
What do I do wrong?
Heinz
_________________________
PS: massively simplified code below......
k=1E-7; // corrosion rate(s-1) in mechanism #1
n=100; // number of timesteps
t=(1:n)'; // time in days
ts= 3600*t; // time in seconds
PHI= 1 - exp(-((k*ts)^2)); // failure fraction in mechanism #1
plot("nl", t, PHI, 'r--');
f= 1 - exp(-ts/3E5); // "pure" failure fraction in mechanism #2
F=conv(PHI,f,"same")/length(t); //"convoluted" failure fraction mechanism #2
plot(t,F,'b--');
legend('failure fraction #1','subsequent failure fraction #2',4);
xlabel('time (hours)');
ylabel('failure fraction');
title('Component failure evolution #1 is followed by #2');
--
Stéphane Mottelet
Ingénieur de recherche
EA 4297 Transformations Intégrées de la Matière Renouvelable
Département Génie des Procédés Industriels
Sorbonne Universités - Université de Technologie de Compiègne
CS 60319, 60203 Compiègne cedex
Tel : +33(0)344234688
http://www.utc.fr/~mottelet
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